Observing our Changing Earth: Proceedings of the 2007 IAG General Assembly, Perugia, Italy, July 2 - 13, 2007 (International Association of Geodesy Symposia)

International Association of Geodesy Symposia M ich ael G . Sideris, Series Editor

International Association of Geodesy Symposia M ich ael G . Sideris, Series Editor Symposium 101: Global and Regional Geodynamics Symposium 102: Global Positioning System: An Overview Symposium 103: Gravity, Gradiometry, and Gravimetry Symposium 104: Sea SurfaceTopography and the Geoid Symposium 105: Earth Rotation and Coordinate Reference Frames Symposium 106: Determination of the Geoid: Present and Future Symposium 107: Kinematic Systems in Geodesy, Surveying, and Remote Sensing Symposium 108: Application of Geodesy to Engineering Symposium 109: Permanent Satellite Tracking Networks for Geodesy and Geodynamics Symposium 110: From Mars to Greenland: Charting Gravity with Space and Airborne Instruments Symposium 111: Recent Geodetic and Gravimetric Research in Latin America Symposium 112: Geodesy and Physics of the Earth: Geodetic Contributions to Geodynamics Symposium 113: Gravity and Geoid Symposium 114: Geodetic Theory Today Symposium 115: GPS Trends in Precise Terrestrial, Airborne, and Spaceborne Applications Symposium 116: Global Gravity Field and Its Temporal Variations Symposium 117: Gravity, Geoid and Marine Geodesy Symposium 118: Advances in Positioning and Reference Frames Symposium 119: Geodesy on the Move Symposium 120: Towards an Integrated Global Geodetic Observation System (IGGOS) Symposium 121: Geodesy Beyond 2000: The Challenges of the First Decade Symposium 122: IV Hotine-Marussi Symposium on Mathematical Geodesy Symposium 123: Gravity, Geoid and Geodynamics 2000 Symposium 124: Vertical Reference Systems Symposium 125: Vistas for Geodesy in the New Millennium Symposium 126: Satellite Altimetry for Geodesy, Geophysics and Oceanography Symposium 127: V Hotine Marussi Symposium on Mathematical Geodesy Symposium 128: A Window on the Future of Geodesy Symposium 129: Gravity, Geoid and Space Missions Symposium 130: Dynamic Planet - Monitoring and Understanding … Symposium 131: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles Symposium 132: VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy Symposium 133: Observing our Changing Earth

Observing our Changing Earth Proceedings of the 2007 IAG General Assembly, Perugia, Italy, July 2 - 13, 2007

Edited by Michael G. Sideris

Editor Prof. Dr. Michael G. Sideris University of Calgary Fac. Engineering Dept. Geomatics Engineering 2500 University Drive NW., Calgary AB T2N 1N4 Canada [email protected]

ISBN: 978-3-540-85425-8

e-ISBN: 978-3-540-85426-5

International Association of Geodesy Symposia ISSN: 0939-9585 Library of Congress Control Number: 2008933707 c Springer-Verlag Berlin Heidelberg 2009  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover Design: Integra Software Services Pvt. Ltd. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

At the XXIV General Assembly of the International Union of Geodesy and Geophysics (IUGG), held July 2–13, 2007 in Perugia, Italy, the International Association of Geodesy (IAG) also had its quadrennial General Assembly. The IAG coorganized and contributed to several Union Symposia, as well as to Joint Symposia with other Associations. It also organized five Symposia of its own, one dedicated to each of its four Commissions and a fifth one dedicated to the Global Geodetic Observing System (GGOS). This volume contains the proceedings of these five Symposia, which are listed below: Symposium GS001: Reference Frames Convener: H. Drewes Co-convener: A. Dermanis Symposium GS002: Gravity Field Convener: C. Jekeli Co-conveners: U. Marti, S. Okubo, N. Sneeuw, I. Tziavos, G. Vergos, M. Vermeer, P. Visser Symposium GS003: Earth Rotation and Geodynamics Convener: V. Dehant Co-convener: Chengli Huang Symposium GS004: Positioning and Applications Convener: C. Rizos Co-convener: S. Verhagen Symposium GS005: The Global Geodetic Observing System (GGOS) Conveners: M. Rothacher Co-conveners: R. Neilan, H.-P. Plag The Symposia were organized based on the structure of the IAG (i.e., one per Commission) and covered the there pillars of geodesy, namely geometry, Earth rotation, and gravity field, plus their applications. The inclusion of the Symposium on GGOS – which is no longer a project but a major component of the IAG – integrated all geodetic areas and highlighted the importance of multidisciplinarity in, and for, geodetic research. It is fair to say that this was probably the first time that the breadth of the presented papers demonstrated without a doubt how vital geodesy has become in contributing to the solution of current pressing needs of society, such as monitoring v

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our environment and understanding global change. For example, the number and quality of the papers on GRACE and its applications illustrated clearly geodesy’s valuable contribution to the monitoring of temporal phenomena. It is evident that the era of neglecting time variations has past and that four-dimensional geodesy is now at the forefront. For this reason, we named these proceedings “Observing our Changing Earth”, which also aligns well with the general theme chosen for the IUGG General Assembly (“Earth: Our Changing Planet”). The 2007 Assembly attracted over 400 geodesists from 49 countries, including 120 students. There were 225 oral and 474 poster presentations made at the IAG Symposia. Approximately 20% of those were submitted as full papers for review and inclusion in these Proceedings. The 99 papers that were accepted are contained in this volume, each allocated to one of the five Symposia sections. There are several colleagues who contributed to the success of the IAG General Assembly and should be acknowledged here. We are grateful to Christian Tscherning, the long-serving past Secretary General of the IAG, who coordinated together with the IUGG and IAG Executive Committees and the Local Organizing Committee the venue selection, as well as the scheduling and organization of the Symposia which IAG led or partially contributed to. The Symposia conveners, Herman Drewes, Chris Jekeli, Veronique Dehant, Chris Rizos and Markus Rothacher, as well as their coconveners from the IAG Commissions, Services, GGOS and the Inter-commission Committee on Theory listed on the previous page, are gratefully acknowledged for the selection and organization of the exciting and stimulating scientific content of the Symposia. These quality Proceedings would not have been possible without the tireless work of the Presidents of the four IAG Commission and the Chair of GGOS who acted as conveners and editors for their individual Symposia. Herman Drewes, Chris Jekeli, Veronique Dehant, Chris Rizos and Markus Rothacher guided the reviews of the submitted papers, selected and communicated with the reviewers, and finally accepted the papers that comprise this volume. I am personally indebted to them, as I could have not put this volume together without their invaluable assistance. The contributing authors are also gratefully acknowledged. Finally, I would also like to thank Dr. Elena Rangelova, postdoctoral fellow at the University of Calgary, for her help with the preparation of the camera-ready copy of this volume. Last, but definitely not least, I want to sincerely thank all participating scientists and graduate students, and in particular those who made oral and poster presentations, who came to beautiful Perugia and made our General Assembly an unqualified success. Calgary, May 20, 2008

Michael G. Sideris

Contents

Symposium GS001 Reference Frames Conveners: A. Dermanis, H. Drewes Reference Systems, Reference Frames, and the Geodetic Datum – Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Hermann Drewes Investigations into a Dynamic Geocentric Datum . . . . . . . . . . . . . . . . . . . . . . . . . 11 Jian Wang, Jinling Wang and Craig Roberts On the Strength of SLR Observations to Realize the Scale and Origin of the Terrestrial Reference System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 D. Angermann and H. M¨uller Impact of the Network Effect on the Origin and Scale: Case Study of Satellite Laser Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 X. Collilieux and Z. Altamimi Satellite Laser Ranging Biases and Terrestrial Reference Frame Scale Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 D. Coulot, P. Berio, P. Bonnefond, P. Exertier, D. F´eraudy, O. Laurain and F. Deleflie Status of the European Reference Frame (EUREF) . . . . . . . . . . . . . . . . . . . . . . . 47 Jo˜ao Agria Torres, Zuheir Altamimi, Claude Boucher, Elmar Brockmann, Carine Bruyninx, Alessandro Caporali, Werner Gurtner, Heinz Habrich, Helmut Hornik, Johannes Ihde, Ambrus Kenyeres, Jaakko M¨akinen, Hans v. d. Marel, Hermann Seeger, Jaroslav Simek, Guenter Stangl and Georg Weber Reprocessing of a Regional GPS Network in Europe . . . . . . . . . . . . . . . . . . . . . . 57 C. V¨olksen Latest Enhancements in the Brazilian Active Control Network . . . . . . . . . . . . 65 Luiz P.S. Fortes, Sonia M.A. Costa, Mario A. Abreu, Alberto L. Silva, Newton J.M. J´unior, Jo˜ao G. Monico, Marcelo C. Santos and Pierre T´etreault vii

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Searching for the Optimal Relationships Between SIRGAS2000, South American Datum of 1969 and C´orrego Alegre in Brazil . . . . . . . . . . . . . . . . . . . 71 Leonardo C. Oliveria, Marcelo C. Santos, Felipe G. Nievinski, Rodrigo F. Leandro, Sonia M.A. Costa, Marcos F. Santos, Jo˜ao Magna, Mauricio Galo, Paulo O. Camargo, Jo˜ao G. Monico, Carlos U. Silva, and Tule B. Maia The Permanent Tide in Height Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Jaako M¨akinen and Johannes Ihde Combined Adjustment of National Astro-Geodetic Network and National GPS2000 Geodetic Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Cheng Pengfei, Cheng Yingyan and Mi Jinzhong Symposium GS002 Gravity Field Conveners: C. Jekeli, U. Marti, S. Okubo, N. Sneeuw, I. Tziavos, G. Vergos, M. Vermeer, P. Visser Improved Resolution of a GRACE Gravity Field Model by Regional Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 A. Eicker, T. Mayer-Guerr and K.H. Ilk The Role of the Atmosphere for Satellite Gravity Field Missions . . . . . . . . . . .105 Th. Gruber, Th. Peters and L. Zenner Assessment of GPS-only Observables for Gravity Field Recovery from GRACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 A. J¨aggi, G. Beutler, L. Prange, R. Dach and L. Mervart Detecting the Baltic Sea Level Surface with GPS-Measurements and Comparing it with the Local Geoid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 Aive Liibusk and Harli J¨urgenson Strengthening the Vertical Reference in the Southern Baltic Sea by Airborne Gravimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135 Henriette Skourup, Rene Forsberg, Sofie Louise Sandberg Sørensen, Christian Jermin Andersen, Uwe Sch¨afer, Gunter Liebsch, Johannes Ihde and Uwe Schirmer Downward Continuation of Airborne Gravimetry and Gradiometry Data Using Space Localizing Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 F. Mueller, T. Mayer-G¨urr and A.A. Makhloof The Earth’s Gravity Field Components of the Differences Between Gravity Disturbances and Gravity Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 R. Tenzer, A. Ellmann, P. Nov´ak and P. Vajda Research on the Calibration of Onboard Accelerometer by Dynamic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161 Zou Xiancai, Li Jiancheng, Jiang Weiping, Xu Xinyu and Chu Yonghai

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Current Status of Gravity Measurements in the Republic of Croatia with the Fundamental Gravity Network Finalization Project . . . . . . . . . . . . . . . . . . .169 I. Grgi´c, M. Luˇci´c, M. Liker, T. Baˇsi´c, B. Bariˇsi´c and M. Repani´c The Development of the European Gravimetric Geoid Model EGG07 . . . . . .177 H. Denker, J.-P. Barriot, R. Barzaghi, D. Fairhead, R. Forsberg, J. Ihde, A. Kenyeres, U. Marti, M. Sarrailh and I.N. Tziavos An Attempt for an Amazon Geoid Model Using Helmert Gravity Anomaly .187 D. Blitzkow, A.C.O.C. de Matos, I.O. Campos, A. Ellmann, P. Van´ıcˇ ek and M.C. Santos Combination of Gravimetry, Altimetry and GOCE Data for Geoid Determination in the Mediterranean: Evaluation by Simulation . . . . . . . . . . .195 R. Barzaghi, A. Maggi, N. Tselfes, D. Tsoulis, I.N. Tziavos and G.S. Vergos An Attempt Towards an Optimum Combination of Gravity Field Wavelengths in Geoid Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203 Hussein A. Abd-Elmotaal and Nobert K¨uhtreiber BVP, Global Models and Residual Terrain Correction . . . . . . . . . . . . . . . . . . . .211 M. Elhabiby, D. Sampietro, F. Sans`o and M.G. Sideris Domain Transformation, Boundary Problems and Optimization Concepts in the Combination of Terrestrial and Satellite Gravity Field Data . . . . . . . . .219 P. Holota and O. Nesvadba Numerical Solution of the Fixed Altimetry-Gravimetry BVP Using the Direct BEM Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .229 ˇ R. Cunderl´ ık and K. Mikula On the Combination of Gravimetric Quasi-Geoids and GPS-Levelling Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237 R. Klees and I. Prutkin An Alternative Approach for the Determination of Orthometric Heights Using a Circular-Arc Approximation for the Plumbline . . . . . . . . . . . . . . . . . .245 G. Manoussakis, D. Delikaraoglou and G. Ferentinos On Evaluation of the Mean Gravity Gradient Within the Topography . . . . .253 R. Tenzer and A. Ellmann Comparison of Techniques for the Computation of a Height Reference Surface from Gravity and GPS-Levelling Data . . . . . . . . . . . . . . . . . . . . . . . . . .263 R. Tenzer, R. Klees, I. Prutkin, T. Wittwer, B. Alberts, U. Schirmer, J. Ihde, G. Liebsch and U. Sch¨afer The Inversion of Poisson’s Integral in the Wavelet Domain . . . . . . . . . . . . . . .275 M. Elhabiby and M.G. Sideris

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On the Principal Difficulties and Ways to Their Solution in the Theory of Gravitational Condensation of Infinitely Distributed Dust Substance . . . . . .283 A. M. Krot Representation of Regional Gravity Fields by Radial Base Functions . . . . . .293 M. Antoni, W. Keller and M. Weigelt Precise Gravity Time Series and Instrumental Properties from Combination of Superconducting and Absolute Gravity Measurements . . . .301 H. Wziontek, R. Falk, H. Wilmes and P. Wolf On a Feasibility of Modeling Temporal Gravity Field Variations from Orbits of Non-dedicated Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307 P. Ditmar, A. Bezdek, X. Liu and Q. Zhao EOF Analysis on the Variations of Continental Water Storage from GRACE in China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .315 W. Hanjiang, Z. Guangbin, C. Pengfei and C. Xiaotao Influence of Hydrology-Related Temporal Aliasing on the Quality of Monthly Models Derived from GRACE Satellite Gravimetric Data . . . . . . . .323 J. Encarnac¸a˜ o, R. Klees, E. Zapreeva, P. Ditmar and J. Kusche Image Super-Resolution via Filtered Scales Integral Reconstruction Applied to GOCE Geoid Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329 Ciro Caramiello, Guido Vigione, Alessandra Tassa, Alessandra Buongiorno, Eric Monjoux and Rune Floberghagen An Error Model for the GOCE Space-Wise Solution by Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .337 F. Migliaccio, M. Reguzzoni, F. Sans`o and N. Tselfes Accuracy Analysis of External Reference Data for GOCE Evaluation in Space and Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .345 K.I. Wolf and J. M¨uller Gravity Field Determination at the AIUB – The Celestial Mechanics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353 L. Prange, A. J¨aggi, G. Beutler, R. Dach and L. Mervart Robust Trend Estimation from GOCE SGG Satellite Track Cross-Over Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363 F. Jarecki and J. M¨uller Reference Frame Consistency in CHAMP and GRACE Earth Gravity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .371 Christopher Kotsakis The Fast Analysis of the GOCE Gravity Field . . . . . . . . . . . . . . . . . . . . . . . . . . .379 Xu Xinyu, Li Jiancheng, Jiang Weiping, Zou Xiancai and Chu Yonghai

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GRACE Application to the Receding Lake Victoria Water Level and Australian Drought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .387 J.L. Awange, M.A. Sharifi, W. Keller and M. Kuhn Updated OCTAS Geoid in the Northern North Atlantic – OCTAS07 . . . . . . .397 O.C.D. Omang, A. Hunegnaw, D. Solheim, D.I. Lysaker, K. Ghazavi and H. Nahavandchi New Combined Geoid Solution HGTUB2007 for Hungary . . . . . . . . . . . . . . . .405 Gy. T´oth Regional Astrogeodetic Validation of GPS/Levelling Data and Quasigeoid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .413 Christian Voigt, Heiner Denker and Christian Hirt Insights into the Mexican Gravimetric Geoid (GGM05) . . . . . . . . . . . . . . . . . . .421 D. Avalos, M.C. Santos, P. Van´ıcek and A. Hern´andez An Improved Geoid in North Eastern Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .427 P. Sterzai, F. Coren, N. Creati, I. Marson and M. Maso Towards a New Global Digital Elevation Model . . . . . . . . . . . . . . . . . . . . . . . . . .431 P.A.M. Berry, R.G. Smith, J.A. Freeman and J. Benveniste Symposium GS003 Earth Rotation and Geodynamics Conveners: V. Dehant, C.L. Huang Contribution of Non-Tidal Oceanic Mass Variations to Polar Motion Determined from Space Geodesy and Ocean Data . . . . . . . . . . . . . . . . . . . . . . . .439 F. G¨ottl and F. Seitz Simulation of Historic and Future Atmospheric Angular Momentum Effects on Length-of-day Variations with GCMs . . . . . . . . . . . . . . . . . . . . . . . . .447 Timo Winkelnkemper, Florian Seitz, Seung-Ki Min and Andreas Hense Contributions of Tidal Poisson Terms in the Theory of the Nutation of a Nonrigid Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455 V. Dehant, M. Folgueira, N. Rambaux and S.B. Lambert Consistency of Earth Rotation, Gravity, and Shape Measurements . . . . . . . .463 Richard S. Gross, David A. Lavall´ee, Geoffrey Blewitt and Peter J. Clarke Current Estimation of the Earth’s Mechanical and Geometrical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .473 A.N. Marchenko The Earth’s Global Density Distribution and Gravitational Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483 A.N. Marchenko

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Analysis of Geophysical Variations of the C20 Coefficient of the Geopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .493 L.I. Fern´andez Analysis of Mass Variations in Northern Glacial Rebound Areas from GRACE Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .501 Holger Steffen, J¨urgen M¨uller and Heiner Denker Report of GGP Activities to Commission 3, Completing 10 Years for the Worldwide Network of Superconducting Gravimeters . . . . . . . . . . . . . . . . . . . .511 David Crossley and Jacques Hinderer European Tidal Gravity Observations: Comparison with Earth Tide Models and Estimation of the Free Core Nutation (FCN) Parameters . . . . . . .523 B. Ducarme, S. Rosat, L. Vandercoilden, Xu Jian-Qiao and Sun Heping Physical Modelling to Remove Hydrological Effects at Local and Regional Scale: Application to the 100-m Hydrostatic Inclinometer in Sainte-Croix-aux-Mines (France) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533 L. Longuevergne, L. Oudin, N. Florsch, F. Boudin and J.P. Boy Monitoring Crustal Movement in the Qinghai-Tibetan Plateau Using GPS Measurements from 1993 to 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .541 Weiping Jiang, Xiaohui Zhou, Caijun Xu and Jingnan Liu Impact of Local GNSS Permanent Networks in the Study of Geodynamics in Central Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .549 G. Fastellini, F. Radicioni and A. Stoppini New Results Based on Reprocessing of 13 years Continuous GPS Observations of the Fennoscandia GIA Process from BIFROST . . . . . . . . . . .557 M. Lidberg, J.M. Johansson, H.-G. Scherneck, G.A. Milne and J.L. Davis Analysis of GPS Data from An Antarctic Ice Stream . . . . . . . . . . . . . . . . . . . . .569 R. Dach, G. Beutler and G.H. Gudmundsson Dynamic Analysis of Crustal Movements Along the Dead-Sea Rift . . . . . . . . .581 L. Shahar and G. Even-Tzur Symposium GS004 Positioning and Applications Conveners: C. Rizos, S. Verhagen GPS-Based Monitoring of Surface Displacements in the Mud Volcano Area, Sidoarjo, East Java . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595 H.Z. Abidin, M.A. Kusuma, H. Andreas, M. Gamal and P. Sumintadireja Differential and Precise Point Positioning in the South American Region with lonosphere Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605 S.M. Alves Costa, Alberto Luis da Silva, Newton Jos´e de Moura Jr, Mauricio Alfredo Gende and Claudio Antonio Brunini

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Current State of Precise Point Positioning and Future Prospects and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .615 S. Bisnath and Y. Gao Nonstationary Tropospheric Processes in Geodetic Precipitable Water Vapor Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .625 O.J. Botai, W.L. Combrinck and C.J. deW Rautenbanch Combination of Multiple Repeat Orbits of ENVISAT for Mining Deformation Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .631 H.C. Chang, L. Ge, A.H. Ng, C. Rizos, H. Wang and M. Omura An Investigation into Robust Estimation Applied to Correlated GPS Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .639 R.C. Erenoglu and S. Hekimoglu Geometry-Based TCAR Models and Performance Analysis . . . . . . . . . . . . . . .645 Yanming Feng and Chris Rizos Real-Time Kinematic OTF Positioning Using a Single GPS Receiver . . . . . . .655 Y. Gao and M. Wang Concept of a Multi-Scale Monitoring and Evaluation System for Landslide Disaster Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .669 M. Haberler-Weber, A. Eichhorn, H. Kahmen and B. Theilen-Willige Ionospheric Modelling in the North of Algeria . . . . . . . . . . . . . . . . . . . . . . . . . . .679 H. Dekkiche, S. Kahlouche, C.B. Kadri and R. Mir Kinematic Precise Point Positioning During Marginal Satellite Availability . .691 N.S. Kjørsvik, O. Øvstedal and J.G.O. Gjevestad Establishing a GNSS Receiver Antenna Calibration Field in the Framework of PROBRAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .701 C.P. Krueger, J. Freiberger, B. Heck, M. Mayer, A. Kn¨opfler and B. Sch¨afer Trimble’s RTK and DGPS Solutions in Comparison with Precise Point Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .709 Herbert Landau, Xiaoming Chen, S¨oren Klose, Rodrigo Leandro and Ulrich Vollath The Precision and Accuracy of Shanghai VRS Network . . . . . . . . . . . . . . . . . . .719 Lizhi Lou and Yi Chen Improving the Stochastic Model of GNSS Observations by Means of SNR-based Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .725 X. Luo, M. Mayer and B. Heck

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Evaluating the Brazilian Vertical Datum Through Improved Coastal Satellite Altimetry Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .735 Roberts Teixeira Luz, Wolfgang Bosch, Silvio Rog´eris Correia de Freitas, Bernhard Heck and Regiane Dalazoana Land Subsidence Monitoring in Australia and China using Satellite Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .743 Alex Hay-Man Ng, Hsing-Chung Chang, Kui Zhang, Linlin Ge and Chris Rizos Analysis on Temporal-Spatial Variations of Australian TEC . . . . . . . . . . . . . . .751 Gary Ouyang, Jian Wang, Jinling Wang and David Cole Use of Global and Regional Ionosphere Maps for Single-Frequency Precise Point Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .759 A.Q. Le, C.C.J.M. Tiberius, H. van der Marel and N. Jakowski Quality Control for Building Industry by Means of a New Optical 3D Measurement and Analysis System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .771 A. Reiterer, M. Lehmann, J. Fabiankowitsch and H. Kahmen New Solutions to Classical Geodetic Problems on the Ellipsoid . . . . . . . . . . . . .781 Lars E. Sj¨oberg GNSS Carrier Phase Ambiguity Resolution: Challenges and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .785 P.J.G. Teunissen and S. Verhagen Static Stress Change from the 8 November, 1997 Ms 7.9 Manyi, Tibet Earthquake as Inferred from InSAR Observation . . . . . . . . . . . . . . . . . . . . . . . .793 Wen Yangmao and Xu Caijun Application of Computer Algebra System to Geodesy . . . . . . . . . . . . . . . . . . . . .803 P. Zaletnyik, B. Pal´ancz, J.L. Awange and E.W. Grafarend Crustal Velocity Field Modelling with Neural Network and Polynomials . . . .809 Khosro Moghtased-Azar and Piroska Zaletnyik Next-Generation Algorithms for Navigation, Geodesy and Earth Sciences Under Modernized Global Navigation Satellite Systems (GNSS) . . . . . . . . . . .817 Marcelo C. Santos, Richard B. Langley, Rodrigo F. Leandro, Spiros Pagiatakis, Sunil Bisnath, Rock Santerre, Marc Cocard, Ahmed El-Rabbany, Ren´e Landry, Herb Dragert, Pierre H´eroux and Paul Collins Linear Combinations for Differential Radar Interferometry . . . . . . . . . . . . . . .825 L. Ge, H. Wang, H.C. Chang and C. Rizos

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Symposium GS005 The Global Geodetic Observing System (GGOS) Conveners: H.-P. Plag, M. Rothacher, R. Neilan Recent Progress in the VLBI2010 Development . . . . . . . . . . . . . . . . . . . . . . . . . .833 D. Behrend, J. B¨ohm, P. Charlot, T. Clark, B. Corey, J. Gipson, R. Haas, Y. Koyama, D. MacMillan, Z. Malkin, A. Niell, T. Nilsson, B. Petrachenko, A. Rogers, G. Tuccari and J. Wresnik The Contribution of GGP Superconducting Gravimeters to GGOS . . . . . . . .841 David Crossley and Jacques Hinderer Combined Analysis of Earth Orientation Parameters and Gravity Field Coefficients for Mutual Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .853 A. Heiker, H. Kutterer and J. M¨uller Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .861

Symposium GS001

Reference Frames Conveners: A. Dermanis, H. Drewes

Reference Systems, Reference Frames, and the Geodetic Datum – Basic Considerations Hermann Drewes

Abstract Unambiguous reference systems are a fundamental requirement for accurate and reliable geodetic products. The definition of the reference system, the realization by a reference frame, and the allocation of the geodetic datum have to be strictly coherent. In traditional geodetic reference systems used in triangulation and trilateration networks, the datum was given through independent (astronomic) observations in selected fundamental stations, which fixed the origin and the orientation of the coordinate system. The datum of modern geocentric reference systems must also be determined by independent methods, namely by gravity field parameters and physical models. If it is derived from the reference frame itself, i.e., by coordinate transformations between selected reference stations, the definition of the system will be changed: It does no longer refer to the geo-centre but to the centre of the reference network. Therefore it is indispensable to strictly pay attention that the datum is not affected by the measurements of the frame, and that the realization of the frame does not change the definition of the system.

unique global basis. Monitoring physical processes of global change and geodynamics by precise point positioning requires highly accurate and permanent reference frames relating all the data to a clear and long-term stable datum. We shall start with some nomenclature Reference systems define constants, conventions, models, and parameters, which serve as the necessary basis for the mathematical representation of geometric and physical quantities. An example is a three-dimensional Cartesian system with the origin in the geo-centre, equatorial orientation, metric scale and rotating with the Earth. Reference frames realize the reference system physically, i.e., by a solid materialization of points, and mathematically, i.e., by the determination of parameters (e.g., geometric coordinates). Examples are terrestrial monuments or spatial matters, whose coordinates are computed from the measurements following the definition of the reference system. The geodetic datum fixes unequivocally the relation between a reference frame and a reference system by allocating a set of “given” parameters, e.g., the coorKeywords Reference system · Reference frame · dinates of the origin of the system (X0 , Y0 , Z0 ), the directions of coordinate axes X, Y, Z, and the scale as Geodetic datum · Origin · Orientation · Scale · ITRF a unit of length (e.g., metre). The present paper discusses the principles of the realization of a reference system by a reference frame fo1 Introduction and Motivation cussing on the fixation of the datum parameters. This is the fundamental issue for its establishment. It is Geodetic reference systems are needed to refer the demonstrated by the terrestrial reference frame, but geodetic observations and estimated parameters to a it holds in a similar way for the celestial reference frame, too. Hermann Drewes Deutsches Geod¨atisches Forschungsinstitut, Alfons-Goppel-Str. 11, D-80539 M¨unchen, Germany

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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2 Fundamentals Reference systems, the geodetic datum, and reference frames form an order of hierarchy:

r

r

r

The definition of a reference system must be completely unaffected by the realization of the reference frame and the geodetic datum, i.e., the realization of the system by the frame and the allocation of the datum must not change the definition. The realization of the datum has to be done by methods independent of the measurements of the reference frame, i.e., measurement errors or physical changes altering the observations of the frame must not affect the datum. The mathematical realization of the reference frame has to be done by algorithms that keep the datum parameters fixed and follow strictly the principles defined by the reference system.

seven, datum parameters: three to locate the origin of the coordinate system, three for the orientation of the coordinate axes, and one for the scale, i.e. the unit length. The parameters must be given (Latin datum = given) and cannot be measured. The International Terrestrial Reference System (ITRS) is defined geocentric, equatorial, and metric (McCarthy, 2003). This means that the coordinates given to the origin must be zero in order to refer to the geo-centre: X0 =Y0 =Z0 =0. The orientation of the Z-axis is close to the (varying) Earth’s rotation axis, the X- and Y-axes are in the equatorial plane. The scale is metric according to the metre-convention (http://www.bipm.org/fr/convention/). The realization of the datum parameters has to be done by conventions or observations independent of the measurements of the reference frame, just as in the traditional geodetic reference systems. For the geocentric origin, this can be done by gravimetric parameters, because the geo-centre is defined as the Earth’s centre of mass (M= total mass of the Earth):

In traditional two-dimensional (horizontal) geodetic reference systems (control networks) one has to fix  four datum parameters: two for the origin, one for the X0 = X dm/M orientation and one for the scale. Fixing the origin  means to set the coordinates of the origin of the coor(1) Y dm/M Y0 = dinate system (ϕ0 , λ0 ) to constant numbers. They are  normally given to a fundamental point, e.g., the top of Z dm/M. Z0 = a tower. The orientation of the coordinate axes is given by the azimuths between selected network stations. Both the coordinates of the origin and the azimuths If we compare these equations with the spherical are derived from astronomic observations, which are harmonic coefficients of the Earth’s gravity field, we independent of the angle or distance measurements find nearly identity (a = semi-major Earth axis): in the network. The scale is realized by the prototype  metre. C11 = X dm/a M These datum parameters are fixed for ever; any ob servation errors detected by later repetitions, move(2) Y dm/a M S11 = ments of the stations of the reference frame, adding or    withdrawing of reference stations, do not change the Z dm/a M. C10 = coordinates of the origin, the orientation, or the scale. In this way, any movements of the network with respect to the origin and orientation can be detected even over This means, if we use in satellite positioning a gravlong time intervals. ity field model with C11 = S11 = C10 = 0 for the determination of the satellite orbits, we introduce automatically a coordinate system with the origin X0 = 3 Modern Reference Systems Y0 = Z0 = 0, i.e., in the geo-centre. There is no and Frames degree of freedom for these three datum parameters. All satellite ephemerides are unequivocally and at any In modern, three-dimensional reference systems with time estimated in the geocentric reference system. The Cartesian coordinates we have to fix seven, and only geocentric ephemerides are transferred to the station

Reference Systems, Reference Frames, and the Geodetic Datum

coordinates of the terrestrial reference frame by distance measurements between satellites and ground stations, e.g., by Satellite Laser Ranging (SLR). They are also geocentric unless additional constraints are introduced, e.g., by fixing some terrestrial coordinates. Such constraints are not allowed in order to conserve the datum. This holds true for all observation epochs, the origin is always in the geo-centre, there cannot be a motion of the origin with respect to the geo-centre or vice-versa. The orientation of the coordinate axes of the reference frame could, theoretically, also be defined by the Earth gravity field, namely by the second degree spherical harmonics C21 and S21 , which are functions of the direction of the Earth’s principal axis of inertia. With C21 = S21 = 0, the Z-axis of the coordinate system coincides with the Earth’s axis of the maximum moment of inertia. This definition of the Z-axis is not used in practice, because its determination is not as precise as for the origin, and the satellite orbits are not so sensitive with respect to its variations. Instead, the ITRS defines the convention that the rotation axis of the Earth at a given epoch (1984.0) is fixed as the Z-axis. A problem to be solved is the time evolution of the Z-axis from the reference epoch to all the other epochs. The ITRS definition says that the time evolution is ensured by using a no-net-rotation condition with regards to horizontal tectonic motions over the whole Earth (McCarthy, 2003). This cannot be done perfectly, because the whole Earth is not covered by geodetic observation stations. Even if we assume that each of the approximately 400 sites of the ITRF represents the surface motion of a 1◦ ×1◦ (111×111 km) block, we have only 1% of the Earth’s surface covered. We therefore have to extrapolate the observed point motions to larger areas. This can be done using global motion models of rigid tectonic plates, which are determined by each one geocentric rotation vector per plate, and by modelling the extended crustal deformations between the plates. These models have to be determined for the presentday surface motions and to be derived from geodetic observations for that purpose. The metric scale of the reference system is defined via the metre definition by the speed of light in vacuum. To realize it in the atmosphere, the travel time delay and refraction caused by the ionosphere and troposphere have to be reduced. As the distance measurements are performed by travel time measurements of electronic waves between emitting and receiving

5

instruments, we also have to locate the electronic reference points (phase centres) with respect to the geometric (materialized) reference points. The only way to achieve both is to improve the atmosphere models and the external calibration methods of the individual techniques.

4 Problems in the Datum Realization In practice, the realization of the datum is often not done exclusively by observations independent of the measurements of the reference frame or by conventions as described above, but it is derived from the coordinates of the reference stations itself by estimating the parameters of a similarity transformation. This methodology is applied for the International Terrestrial Reference Frame 2005 (ITRF2005), where time series of (weekly) station coordinates and Earth Orientation Parameters (EOP) are jointly combined (Altamimi et al. 2007). Similarity transformation was an adequate method of datum transfer between geodetic networks in traditional geodesy, when the accuracy of station coordinates was not as good as to be affected by network deformations. At present, however, there is the problem in principle, that terrestrial networks in different epochs are not geometrically similar at the accuracylevel of the station coordinates (mm) due to irregular (crustal) deformations. Furthermore, if the entire reference frame (station network) is moving with respect to the given datum, a similarity transformation from the new to the old positions by parameter estimation would change the datum and thus violate the definition of the reference system. All common motions of the stations of the reference network (displacements, rotations and expansion) would be transformed into the similarity parameters (shift of origin, change of orientation, scale factor). In fact, the coordinate changes caused by the station movements must go to the individual station coordinates and not to the datum. The datum parameters must be kept fixed and must not change in time in order to permit the detection of the complete coordinate changes caused by movements of the reference stations. An illustration is given in Fig. 1. The common displacements (a), rotations (b) and expansions (c) of the stations of the reference network are accumulated to

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Fig. 1 Common motions of reference stations appear in the similarity transformation parameters, if they are estimated: (a) displacements → origin shift X, Y; (b) rotations → change of orientation ω; (c) expansion. → scale factor r2 /r

et al. 1994) and geodetic (APKIM, Drewes 1998, Drewes and Meisel 2003, Drewes 2008) no-netrotation plate motion models. This must not be interpreted as a variation of the geo-centre, because the movement of the upper (lithosphere) masses is balanced by the convection in the Earth mantle. The rotational component of station motions is highly correlated with the EOP. In ITRF2005, the EOP and station velocities are for the first time estimated simultaneously. To solve the singularity and to fulfil the no-net-rotation conditions of the orientation (see above), constraints have to be introduced for the three components of the kinematic datum of velocities. In dX/dt = −10 mm/a, ITRF2005 they are realized by the NNR NUVEL-1A dY/dt = +13 mm/a, model. dZ/dt = +15 mm/a. The geologic-geophysical model NNR NUVEL1A, however, does not fulfil the no-net-rotation These numbers are nearly identical for all the condition for present-day motions. The given plate geologic-geophysical (NNR NUVEL-1A, DeMets motions represent the average over millions of an irregular deformation (d). If we estimate parameters of a similarity transformation, the movements appear as the shift of origin (), change of orientation (ω), and scale factor (r2 /r1 ). These parameters must not be mistaken for changes of the datum parameters; they are common point movements and do not change the datum. In nature there are such common displacements, rotations and expansions of points at the Earth’s surface. The tectonic motions of lithosphere plates (Fig. 2) show systematic displacements:

Reference Systems, Reference Frames, and the Geodetic Datum

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Fig. 2 Rigid lithosphere plate movements from geologic-geophysical (PB2002, for major plates identical with NNR NUVEL-1A, see Bird 2003) and geodetic (APKIM2005) models. Systematic deviations as well as the general northward trend are obvious

years. Furthermore, it does not include actual crustal deformation zones. If we integrate all the rigid plate motions and crustal deformations derived from ITRF2005 in a 1◦ × 1◦ grid over the whole Earth, we find a common rotation of

(speed of light) did not change. It is a dilatation of the network.

5 Conclusions

ωX = −0.04 mas/a ωY = +0.03 mas/a ωZ = −0.02 mas/a. This corresponds to a rotation around a pole at ϕ = −23◦ , λ = 143◦ , with velocity ω = 0.06 mas/a, or a maximum linear velocity of 1.7 mm/a. Geodetically derived kinematic models have to be used instead, e.g., the Actual Plate KInematic and crustal deformation Model (APKIM). An expansion of the Earth may occur according to the Expanding Earth Theory developed end of the 19th century, but never proved. Independently of this theory we have an expansion of the global geodetic networks caused by postglacial uplift. The distance between northern and southern Polar Regions is increasing. If we perform a seven-parameter similarity transformation between observed station coordinates at different epochs, the expansion appears as a scale factor. This must not be taken as a datum parameter, because the scale of the geodetic measurements

The described problems with the realization of the datum lead to the following conclusions:

r r r

If we use network transformations for realizing the origin, we change the definition of the origin from geocentric to the centre of the network. If we use a geological plate kinematic model for the time evolution of orientation, we violate the no-netrotation condition for the Earth crust. If we use network transformations for fixing the scale as a datum parameter, we are no longer metric; the network expansion appears as a scale.

Some of these effects may be seen in the time series of ITRF transformation parameters (Table 1, from Altamimi et al., 2002, 2007). We see a clear trend in the translation (T) and scale (D), which is caused by transferring imperfectly modelled common network movements to the datum parameters. They are thus replaced by the transformation parameters between networks. We also have drifts (“rates”) of the datum

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Table 1 Translation (T) and scale (D) parameters for transformation from ITRF2000 to other realizations (Altamimi et al. 2002, 2007) TX [m] ITRF05 ITRF00 ITRF97 ITRF96 ITRF94 ITRF93 ITRF92 ITRF91 ITRF90 ITRF89 ITRF88

−0.0001 0.0000 0.0067 0.0067 0.0067 0.0127 0.0147 0.0267 0.0247 0.0297 0.0247

dTX /dt [m/a] 0.0002 0.0000 0.0000 0.0000 0.0000 −.0029 0.0000 0.0000 0.0000 0.0000 0.0000

TY [m] 0.0008 0.0000 0.0061 0.0061 0.0061 0.0065 0.0135 0.0275 0.0235 0.0475 0.0115

dTY /dt [m/a] −.0001 0.0000 −.0006 −.0006 −.0006 −.0002 −.0006 −.0006 −.0006 −.0006 −.0006

TZ [m]

dTZ /dt [m/a]

D [ppb]

dD/dt [ppb/a]

0.0058 0.0000 −.0185 −.0185 −.0185 −.0209 −.0139 −.0199 −.0359 −.0739 −.0979

0.0018 0.0000 −.0014 −.0014 −.0014 −.0006 −.0014 −.0014 −.0014 −.0014 −0.014

−.40 0.00 1.55 1.55 1.55 1.95 0.75 2.15 2.45 5.85 8.95

−.0008 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

The datum for the station velocities (kinematic datum) has to be given by a present-day crustal motion and deformation model (e.g. APKIM) and not by is not strictly geocentric, because there are shifts of a geologic-geophysical model as an average over the origin with respect to the geo-centre; millions of years (NUVEL). is not strictly rotating with the real Earth, because a The geodetic reference system is the basis for all geological model is used as a reference; geodetic research in geodynamics and global change. If is not strictly metric, because the measurement’s the long-term stability of the datum is not guaranteed, scale is replaced by the network’s expansion; the geodetic products and statements are not reliable has no long-term stability, because the datum is and cannot be used for cogent interpretations. Motions changing with time. of the Earth’s crust as a whole, displacement, expansion or rotation cannot be detected, if the datum parameters are not fixed.

parameters, i.e., the datum is not stable in time. This leads to the consequence that the ITRF

r r r r

6 Recommendations A thorough revision of the definition and the conventions for the ITRF is required. The realization of the reference frame and the geodetic datum has exactly to follow the definitions given by the reference system, in particular with respect to the geocentric origin and the metric scale. The physical models (time dependent gravity field, atmosphere, hydrosphere, cryosphere) have to be improved for consideration of variations of the orientation through changes of the tensor of inertia, and also for modelling of the geodetic observations. The computation of future ITRF realizations has to be done without interference by the reference network. Therefore, the data processing has to start at the observation level using directly the original measurements and not derived solutions (adjusted coordinates with respect to an a priori datum). A first step toward this methodology is the use of datum free (singular) normal equations of the individual observation techniques (Angermann et al. 2005, 2007).

References Altamimi, Z., P. Sillard, C. Boucher (2002). ITRF2000: A new release of the International Terrestrial Reference Frame for Earth science applications. J. Geophys. Res., 107, B10, 2214, 19 pp, doi:10.1029/2001JB000561. Altamimi, Z., X. Collilieux, J. Legrand, B. Garayt, C. Boucher (2007). ITRF2005: A new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J. Geophys. Res., 112, B09401, 19 pp, doi:10.1029/2007JB004949. Angermann, D., H. Drewes, M. Gerstl, R. Kelm, M. Kruegel, B. Meisel (2005). ITRF combination – status and recommendations for the future Springer, IAG Symposia, 128, 3–8. Angermann, D., H. Drewes, M. Kr¨ugel, B. Meisel, (2007). Advances in terrestrial reference frame computations. Springer, IAG Symposia, 130, 595–602. Bird, P. (2003). An updated digital model of plate boundaries. G3: Geochemistry, Geophysics, Geosystems, 4, No. 3, 1027, 52 pp. DeMets, C., R. Gordon, D.F. Argus, S. Stein (1994). Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophys. Res. Lett., 21, 2191–2194.

Reference Systems, Reference Frames, and the Geodetic Datum Drewes, H. (1998). Combination of VLBI, SLR and GPS determined station velocities for actual plate kinematic and crustal deformation models. Springer, IAG Symposia, 119, 377–382. Drewes, H. (2008): The APKIM2005 – basis for a nonrotating ITRF. Springer, IAG Symposia (in press).

9 Drewes, H., B. Meisel (2003). An actual plate motion and deformation model as a kinematic terrestrial reference system. Geotechnologien Science Rep., No. 3, 40–43. McCarthy, D.D. (2003). IERS Conventions (2003). IERS Technical Note, 32, BKG, Frankfurt.

Investigations into a Dynamic Geocentric Datum Jian Wang, Jinling Wang and Craig Roberts

Abstract A “dynamic” geodetic datum is a datum that consists of coordinates and velocities of control points, which can provide users with the realistic positions of sites within the network that reflects the result of local or regional motion induced by a number of causes. A new ITRF-linked scheme is proposed and discussed in order to process available geodetic data in real-time to establish a dynamic ITRF-like datum. The authors suggest providing a variety of such datums for different applications. The noise properties of the geodetic coordinate time series are crucial to reliably estimating positions, velocities, and their stability and uncertainties. A ‘db3’ wavelet transform is used to separate the noise of the velocity series and the autocorrelation. The histogram distribution and spectrum features of the residuals indicate that the outliers impact the denoising effect. A Kalman filter-based model with the outliers being detected and removed online is investigated to obtain the “purified” velocity series for ITRF modeling. For a local scale application, interpolation methods play an important role in improving the accuracy of the datum. Six interpolation methods are investigated using a simulated velocity data set. The “local patch” model is necessary for a local datum in order to handle unexpected events.

Jian Wang The School of Surveying and Spatial Information Systems, University of New South Wales, Sydney, NSW 2052, Australia Jinling Wang The School of Surveying and Spatial Information Systems, University of New South Wales, Sydney, NSW 2052, Australia Craig Roberts The School of Surveying and Spatial Information Systems, University of New South Wales, Sydney, NSW 2052, Australia

Keywords Dynamic geocentric datum · Wavelet · Kalman filter

1 Introduction One of the ultimate goals of modern geodesy is to provide a highly stable geocentric reference datum for scientific goals, such as precise orbit determination, monitoring of sea level rise, measuring plate tectonics, etc. (Angermann et al., 2003). The International Terrestrial Reference System (ITRS) is proposed by the International Association of Geodesy (IAG) as the Terrestrial Reference System (TRS) that integrates the geometric contributions of the different geodetic techniques (Altamimi et al., 2002). The International Terrestrial Reference Frame (ITRF), a physical materialisation of the ITRS, has random errors and systematic biases and is conventionally realised by an assumption of constant velocities for a set of global tracking sites. In reality, the assumption is not consistent with the non-linear effects caused by the mass movement of planetary fluids (atmosphere, oceans, surface hydrology, etc.) relative to the solid earth, with a crust that deforms due to plate tectonic motion. A series of ITRFs were defined by the responsible ITRS Product Center (Institute Geographique National, IGN, Paris), from ITRF88 to ITRF2000 and ITRF2005 (Muller & Tesmer, 2003). Each subsequent ITRF is calculated with more data from different geodetic techniques than the previous one. Significant efforts to improve the consistency, precision and dynamic features of the ITRFs have been made over the years. The ITRF2000 solution is usually considered to be the most extensive and accurate one ever developed;

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containing about 800 stations located at about 500 sites, with better distribution over the globe compared to past ITRF versions (Altamimi et al., 2002). It includes primary core stations observed by VLBI, LLR, SLR, GPS and DORIS techniques as well as using regional GPS networks for its densification (Altamimi et al., 2002). To further improve the consistency and stability of the datum, time series of station positions and EOP from different geodetic observations is used to develop ITRF2005. The advantage of using time series of station positions is that it allows monitoring of station non-linear motion and discontinuities and to examine the temporal behavior of the frame physical parameters, namely the origin and the scale. (Altamimi et al., 2007). They do suggest using so-called “semidynamic datums” by assigning deformation models to specific geodetic datums for regional terrestrial reference frames, for instance the New Zealand Geodetic Datum 2000. Long term deformation trends and deformation triggered by unexpected events (e.g. seismic or volcanic effects) could be considered when developing the datum (Jordan et al., 2005). The philosophy of ITRS realisation is by updating the station positions and constant velocities of a set of points to achieve a higher precision ITRF. The main update work of the ITRS is to derive more precise positions while omitting any non-linear variation of the velocity. The suggested regional semi-dynamic datum realisation initiates another approach by correcting the velocities using regional deformation models. With the increasing quantity of geodetic measurements from different observation techniques, it is now possible to more accurately realise the ITRS by introducing non-linear velocity models. Also, by combining regional spatial interpolation techniques, the regional control network can be connected to the ITRF (Grant & Pearse, 1995). To realise the non-linear velocity ITRF scheme, and promote its use for regional applications, this paper will investigate a new scheme based on the available geodetic measurements that takes into account a non-linear velocity model.

2 The Scheme of the Proposed Dynamic ITRF

J. Wang et al.

a set of geodetic positions with respect to a specific epoch and a dynamic velocity model. The geodetic positions as well as the velocity time series are decided by the similar realization model of the series of ITRFs with all the available geodetic observation techniques such that accurate definitive reference station positions are given. Based on the analysis of the historic time-series of the specific station, an available model of the velocity series can be given the velocity νt of a arbitrary epoch t as: νt = f 1 (t) + f 2 (t) + f 3 (t) + f 4 (t)

(1)

Let f 1 (t), f 2 (t), f 3 (t) and f 4 (t) denote trend variation, periodic variation, the other non-linear variation and unexpected variations, respectively, with respect to epoch t. If there are more accurate models available or there is some revision to the established model, one can update the datum by only modifying the velocity model. For regional applications, the spatial velocity field at epoch t can be obtained by several interpolation models. In order to accommodate unexpected events in the real-time velocity field, a “local patch” model is suggested. The basic realisation scheme of the proposed Dynamic ITRF datum is shown in Fig. 1 in a more general sense. It is a more flexible approach to a continuous ITRF in time and spatial scales with the available data sets. The International Earth Rotation and Reference Systems Service (IERS) has been responsible for the establishment and maintenance of the ITRF since 1988. To compute the positional coordinates of the reference stations, novel data processing techniques and some standards have been developed by the IERS authorised institutes. For more information the reader is referred to Angermann et al. (2003).

3 Temporal Velocity Modeling One of the most challenging tasks in building a Dynamic ITRF is to model temporal velocity variation. There are three basic difficulties to contend with: (1) Discrimination between real geophysical signals and random noise signals, (2) Real-time outlier detection and elimination and, (3) Modeling of the velocity time series.

As distinct from the existing situation, we prefer to introduce long term and non-linear movements of This paper proposes the following approach. A the Earth’s surface into the velocity and propose a realization of the ITRS (named Dynamic ITRF) by wavelet model is used to allow the isolation of velocity

Investigations into a Dynamic Geocentric Datum

13

Fig. 1 The basic realisation scheme of the proposed Dynamic ITRF datum

Available geodetic data and combinations Fix initial epoch Velocity time series Analysis and Modeling unexpected event effect model Station position set

Refined epoch specific velocity

ITRF

variation trends from the noise signal. A Kalman filter is then used to model the velocity time series, as well as handling outliers online. According to the model, the velocities of the reference control points defining the ITRS at a given epoch are available. Such an approach is expected to realise a more accurate geocentric datum as it captures the non-linear movements of the reference points.

3.1 Possible Models for Velocity Time Series Analysis

temporal velocity model

spatial interpolation velocity model

Regional spatial velocity field.

ìlocal patch ” model

evaluating wavelet transform coefficients at different levels for different threshold value, real signals are restored by reconstruction using quantified wavelet coefficients with the thresholds. There are two methods (soft-threshold and hard-threshold) for wavelet coefficient quantification, which has been presented widely in previous literature. For more detailed information, one can refer to Mallat (1989). The ‘db3’ wavelet is chosen as an analysis function as the ‘dbN’ wavelet functions has the following advantages:

(1) The ‘dbN’ wavelet functions possess orthogonality, compacted support blocks and approximate symmetry. Many time series analysis models can be used for (2) The ‘dbN’ wavelet function can be used to perform velocity time series. For example gray system thediscrete wavelet transforms with the MALLAT fast ory, time series analysis, and more recently Wavelet decomposition and reconstruction algorithm. Theory, Kalman filter models and Neural Networks (3) In order to satisfy the need for time precision for (e.g. Kalman, 1960; Liang & Fang, 1995; Patterson outlier detection, the selected ‘dbN’ wavelet funcet al., 1993). For the analysis of the GPS velocity tion with efficient support block length of 2N−1 time series we have used daily solutions processed by should be short. SOPAC (ftp://garner.ucsd.edu/pub/). Using the COSO The ‘db3’ wavelet function is applied to de-noise site for instance, NEU velocity time series are analthe time series. Noises and the scale function, ysed for the model application, wavelet transform and wavelet function and corresponding filters are Kalman filter model. given in Fig. 2. 3.1.1 Wavelet Noise Reduction

The statistical features of the residual values are further investigated using histogram statistics, autocorreThe wavelet threshold noise reduction method is lation analysis and spectrum analysis. The situation for applied to extract the real signal. The noised signal N and E components has a similar performance. Only is firstly divided with a bandpass filter function. By N and U direction residuals are shown in Figs. 3 and 4.

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Investigations into a Dynamic Geocentric Datum

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It is noted that: (1) outliers existing in the velocity se- where D (k) and D (k) denote the covariance matriries (U) affect the wavelet noise reduction model simi- ces of the system dynamic noise and the observation lar to most of the other time series processing models; noise respectively and δk j is the (2) wavelet noise reduction models perform well; and  (3) the unexpected velocity outlier affect on the FFT1, k = j spectrum is manifested in the U direction. Kronecker function: δk j = 0, k  = j Epoch Kalman filter residuals can be given as:

3.1.2 Kalman Model for Time Series Modeling

rk = L k − Bk X (k/k − 1)

Let the linear dynamic system and the observation equation be given by (Yang et al., 2001):

(3)

where X k = k,k−1 X k−1 + k,k−1 k−1 L k = Bk X k + k

(2)

where X k , X k−1 denotes the state vector including positioning parameters and velocity parameters in three dimensions; k,k−1 and Bk denote the transition and design matrices respectively, which are assumed known; L k the observations vector specified at k; k,k−1 the noise matrix of the system from k − 1 to k; k−1 the stochastic accelaration state noise; and k denotes the observation noise. The stochastic model is:

X (k/k − 1) = k,k−1 X (k − 1/k − 1) if there are no outliers in the observations. rk has a Gaussian white noise distribution with square deviation given by: Ak = Bk D X (k/k − 1)BkT + Rk The outliers can be discriminated by the test:

E(k ) = 0, E(k ) = 0, cov(k ,  j ) = 0

H0 : E{rk } = 0, E{rk r T k} = Ak

cov(k ,  j ) = D (k)δk j , cov(k ,  j ) = D (k)δk j

H1 : E{rk } = μ, E{[rk − μ][rk − μ]T } = Ak

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Fig. 5 Outlier detection results with χ 2 test

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3.2 Purification and Prediction of Velocity Time Series A feature extraction, modeling and prediction procedure to “purify” the velocity series for the ITRF model (Fig. 6) is suggested, as the effects of the noises and Velocity time series Noise reduction Preprocessing

Outliers detection and recovery

Unexpected events model

TD has the critical values 2.706, 3.841 and 5.024 for given probability values α = 0.1, 0.05 and 0.025 Feature extraction New Simulation and respectively. One outlier detection result is shown in and modeling models Prediction Fig. 5. A local Kalman filter can be designed for each group of observations, and then an outlier detection procedure is used. At the same time, a main Kalman Physical Modeling explanation considered features filter is running. If a group of fault observations are detected, the predicted Kalman filter states are used as the estimated state to eliminate outlier effects on No Purified velocity series the unknown parameters (e.g. Chui & Chen, 1991; Qin et al., 1998). Fig. 6 Scheme for velocity series purification and prediction

Investigations into a Dynamic Geocentric Datum

some outliers in the velocity series will have an effect on the ITRF calculation. By preprocessing, feature extraction and modeling considered features, the purified velocity series can be given and used for further modeling applications.

17

are rectified if they are within the region. At the same time, the spatial velocity field is modified by appending these relevant “local patch” models. Blick et al. (2003) and Jordan et al. (2005) discuss such models.

5 Potential Improvement in the ITRF 4 Regional Velocity Field for Local Datum

One of the core problems of the associated model is how to effectively manage the increasing volume of geodetic data. A dynamic datum could be generated by 4.1 Velocity Field Interpolation a geographic information based decision-making system with a database of all the GPS/GNSS data, and Different interpolation algorithms can be used for other unexpected event field survey data, and a model regional velocity fields. Different interpolation method database. The system could provide users with an interface give different results. A velocity field consisting of 60 points is simulated as the standard test velocity to generate site velocities, baseline components at any data set to test six different interpolation algorithms, epoch, new site coordinates on the basis of the tectonic Inverse Distance to a Power, Kriging, Minimum motion model selected etc. The system also generates a different level of preciCurvature, Modified Shepard’s Method, Radial Basis sion for the Dynamic ITRF in terms of the user input Function and Local Polynomial (e.g. Journel & information for different applications. Huijbregts, 1978). The system can choose core- or user-specified sites It is shown that the local velocity field has something to do with the interpolation method. The Inverse for the ITRF and can update a dynamic datum autoDistance to a Power and Kriging methods are shown as matically, as well as provide users with a datum of a examples in Fig. 7. Some of the interpolation meth- specific precision for each reference station. All these ods have proven useful and popular in many fields. options will be researched in the future. For example Kriging is a geostatistical interpolation method. This method produces interpolation values of unknown positions from irregularly spaced data. Krig- 6 Summary ing attempts to express the trends suggested in the data, so that high points might be connected along a ridge This paper has proposed a scheme to establish a dyrather than isolated by bull’s-eye type contours. One namic geodetic datum which has such features as: (1) must choose a proper interpolation model for a specific updating the derivatives (that is, velocity) of the poregion in order to compute a high precision velocity sitions considering non-linear variations, if necessary, field. driven by user-defined modeling precision; (2) automatic velocity series “purification”; and (3) incorporating different interpolation methods to improve the local 4.2 “Local Patch” Model datum precision. A purification and prediction scheme for the velocity series is discussed in order to provide To handle unexpected events or “jumps”, it is recom- velocity series is discussed in order to provide velocity mended that a “local patch” model be introduced for series for the ITRF. Examples of wavelet noise reduceach event. The “local patch” describes a local mo- tion and Kalman outlier detection and elimination (of tion of the earth mass which only affect regional de- a velocity series) has been investigated with encouragformation. To ensure spatial continuity of the models, ing results being presented. Potential improvements of the “local patch” model would have zero velocity at its the dynamic model also include the proposition using a boundaries. By identifying which models adopted at a location based decision-making system to improve the given time and place, reference control point velocities efficiency of the proposed reference frame scheme.

18 Fig. 7 Different interpolation results of the simulated velocity field

J. Wang et al.

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Investigations into a Dynamic Geocentric Datum

References Altamimi, ZH, P, Sillard & B, Claude (2002) ITRF 2000: A new release of the international terrestrial reference frame for earth science applications. Journal of Geophysical Research, Vol. 107. 2214. Altamimi, Z, X, Collilieux, J, Legrand, B, Garayt, & C, Boucher (2007) ITRF2005: A new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters, Journal of Geophysical Research, Vol. 112, B09401, doi:10.1029/2007 JB004949. Angermann, DB, M, Meisel, H, Krugel, H, Muller, & V, Tesmer (2003) ITRF Combination-status and recommendations for the future. International Association of Geodesy symposia, IAG Symposium Sapporo, Japan, 30 June–11 July Vol. 128, pp. 3–8. Blick, G, C, Crook, D, Grant & J, Beavan (2003) Implementation of a semi-dynamic datum for New Zealand. International Association of Geodesy Symposia, IAG Symposium Sapporo, Japan, 30 June–11 July, Vol. 128, pp. 38–43. Chui, CK & G, Chen (1991) Kalman Filtering with Real-time Application 2nd Edition. Springer-Verlag Hiedelburg. Grant, DB & MB, Pearse (1995) Proposal for a dynamic national geodetic datum for New Zealand. IUGG XXI General Assembly, Boulder Colorado, July 2–14, 1995. Jordan, P, P, Denys & G, Blick (2005) Implementing localized deformation models into a semi-dynamic datum.

19 International Association of Geodesy symposia, IAG symposium Cairns, Australia 22–26 August, Vol. 130, pp. 631–637. Journel, AG & CJ, Huijbregts (1978) Mining Geostatistics, Academic Press, London. Kalman, RE (1960) A new approach to linear filtering and prediction problems, Transactions of the ASME, Journal of Basic Engineering, Vol. 82, pp. 34–45. Liang, YC & HP, Fang (1995) Predicting chaotic time series with wavelet networks. Physics D: Applied physics. Vol. 85, pp. 225–238. Mallat, SG (1989) A theory for multi-resolution signal decomposition: the wavelet representation. IEEE Trans PAMI, 117:674–693. Muller, H & V Tesmer (2003) Time evolution of the terrestrial reference frame. International Association of Geodesy symposia, IAG symposium Sapporo, Japan, 30 June–11 July, Vol. 128. pp. 9–14. Patterson DW, KH, Chan & CM, Tan (1993) Time Series Forecasting with Neural Nets: a comparative study. Proceedings of the International Conference on Neural Network Applications to Signal Processing. NNASP 1993 Singapore pp 269–274. Qin, YY, H, Zhang & SH, Wang (1998) Kalman Filter and Theory for Integration Navigation, Northwestern Polytechnical University Publishing House. Yang, Y, H, He & G, Xu (2001) Adaptively robust filtering for kinematic geodetic positioning, Journal of Geodesy, 73, pp. 109–116.

On the Strength of SLR Observations to Realize the Scale and Origin of the Terrestrial Reference System ¨ D. Angermann and H. Muller

Abstract This paper presents the results of different methods that have been applied to investigate the strength of SLR observations to realize the scale and origin of the terrestrial reference system. In this context the effect of the network geometry and the satellite altitude has been studied by analysing the covariance matrix of the datum parameters. Furthermore, the time series of the scale and translation parameters of weekly SLR network solutions were analysed from 1993 until 2007 and the effect of range bias estimation has been investigated.

reference system. However, SLR suffers from a relatively sparse distribution of tracking stations in particular on the Southern hemisphere and also from several changes in the network geometry. The SLR measurements can also be affected by technical problems of the laser ranging systems, which may require the estimation of station- and time-dependent range biases. This paper deals with investigations on the information content of SLR observations to realize the scale and origin of the terrestrial reference frame. Major issues that are addressed comprise: (1) the analysis Keywords SLR · Time series analysis · Range bias of the covariance matrix of the datum parameters to study the effect of the network configuration and of estimation · ITRF · Origin · Scale the satellite altitude; (2) the generation and analysis of weekly SLR time series solutions to investigate the stability of the network scale and translation 1 Introduction parameters; and (3) the investigation of the impact of range bias estimation on the results for the datum Satellite Laser Ranging (SLR) observations provide a parameters. direct and unambiguous measurement of the distance between space objects and terrestrial ground stations. The optical laser ranging observations are also less affected by atmospheric effects than microwave 2 SLR Station Network techniques (e.g., GPS, VLBI, DORIS). Thanks to these characteristics combined with the high quality The SLR station network along with the number of of the solutions for the orbits of the satellites, SLR normal points observed from 1993 until June 2007 is is in the unique position to provide a highly accurate shown in Fig. 1. realization of the origin and scale of the terrestrial The spatial distribution of SLR stations and the amount of SLR data observed over the past 15 years D. Angermann suffers from a lack of stations in the Southern hemiDeutsches Geod¨atisches Forschungsinstitut (DGFI), sphere. The number of SLR observation stations is in Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany, most weeks in the order of 20–25. There is a slight e-mail: [email protected] increase over the 15 years time span, which is more H. M¨uller clearly visible for the number of observed normal Deutsches Geod¨atisches Forschungsinstitut (DGFI), Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany points per week (see Figs. 2 and 3). M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

21

¨ D. Angermann and H. Muller

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Fig. 1 SLR network stations. The bars show the number of observed normal points from 1993 until 2007

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3 Datum Information of SLR Observations 3.1 Method We estimate the information of SLR observations for the realization of the geodetic datum from the covariance matrix for the datum parameters Cdatum , which is calculated by applying the following formula: −1 Cdatum = (G T Cloose G)−1

Cloose is the covariance matrix of an SLR network solution by applying loose constraints on station positions, and G is the matrix of a differential similarity transformation with seven parameters (3 translations, 3 rotations, and 1 scale). From the resulting covariance matrix Cdatum we estimate the standard deviations for the seven datum parameters and the correlations between them.

3.2 Effect of the Network Geometry

We applied this method for different station network constellations to investigate the effect of the network 0 0 728 832 936 1040 1144 1248 1352 geometry on the estimation of the datum parameters. Fig. 3 Normal points per week. The time axis is given in GPS As shown in Fig. 4, we used the three following netweeks from 1993 until 2007 work configurations for the computations:

On the Strength of SLR

23

Fig. 4 Network configurations: case A (black triangles), case B (grey circles), case C (white diamonds)

– Case A: Typical station network for Lageos computations (black triangles). – Case B: Network with homogeneously distributed stations (grey circles). – Case C: Network with stations only on the Northern hemisphere (white diamonds).

the translation parameters and the scale. We obtained similar results for the “real” network (case A) and for homogeneously distributed stations (case B). In both cases, the standard deviations for the z-component of the translation parameters are higher by a factor of 3 compared to the x- and y-component. In case C with stations located only on the Northern hemisphere the The results shown in Table 1 confirm that SLR ob- standard deviations for the translations and the scale inservations are sensitive to estimate the translation pa- crease significantly, whereas the z-component is worse rameters and the scale of the terrestrial reference frame. by a factor of 5 compared to the x- and y-component. The large standard deviations for the rotation parameThe correlations between the datum parameters are ters indicate that SLR observations do not contain in- rather small. They are below 0.4 for the network conformation to determine the orientation of the terrestrial stellations A and B. In case C with stations only located reference frame. In this specific study the rotation pa- on the Northern hemisphere the highest correlation of rameters are defined by the loose constraints only (i.e., about 0.5 is between the z-translation and the scale (see 1 m on station positions). Fig. 5). The standard deviations derived from the weekly SLR solutions are in the range of a few millimeters for

3.3 Effect of the Satellite Altitude Table 1 Standard deviations of the datum parameters derived from weekly SLR solutions based on different network configurations Standard deviations [mm] Parameter

Case A

Case B

TX TY TZ Scale Rot X RotY Rot Z

1.1 1.0 2.9 3.1 184.4 157.3 337.6

1.6 1.5 4.2 4.2 289.8 211.4 588.2

We have analysed SLR data to satellites in different altitudes to investigate the impact on the determination of the origin of the terrestrial reference frame. We apCase C plied the method described in Sect. 3.1 to estimate the 6.2 standard deviations for the translation parameters. We 7.1 obtained the largest values for Etalon, which is due to 33.3 a relatively poor tracking and due to the high altitude. 9.3 For the three other satellites the standard deviations 575.5 520.1 are in the order of 1 mm, except for the z-component 679.2 determined by LAGEOS, which is worse by a factor

¨ D. Angermann and H. Muller

24

4 SLR Time Series Solutions 4.1 Data Processing

1.0 0.8 0.6 0.4 0.2 0.0 Fig. 5 Correlation matrix of the datum parameters (tx, ty, tz, scale, rotx, roty, rotz). Network configurations: case A (upper left), case B (upper right), case C (lower figure)

of 3. For satellites with lower altitudes (e.g., Ajisai, Starlette), the accuracy of the z-component is only slightly worse compared to the x-and y-component (see Table 2). The results are consistent with theoretical considerations, as the origin in z is determined by the disturbed satellite motion due to the influence of Earth’s gravity field only, whereas the origin in x and y is additionally determined by Earth rotation (e.g., Angermann, 1991; Dietrich, 1988). Consequently, satellites with lower altitudes provide more stable results for the z-component.

Table 2 Standard deviations of the translation parameters derived from weekly SLR solutions using Laser satellites with different altitudes Etalon

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Ajisai

Starlette

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1:4.4

1:2.8

1:1.5

1:1.4

We re-analysed SLR measurements to LAGEOS 1/2 of the global tracking network from January 1993 until July 2007 with the latest version (4.08) of the DGFI software package DOGS-OC. The reference frame, conservative and non-conservative force field parameters are defined according to the IERS Conventions 2003 (McCarthy and Petit, 2004). The solved-for parameters comprise six Kepler elements for weekly accumulated orbital arcs, empirical coefficients for solar radiation and along-track accelaration, daily EOP, as well as station positions and velocities. We applied bias corrections and estimated for some weekly SLR stations range biases according to the latest ILRS table (Status: June 2007). More information on the SLR data processing can be obtained from other presentations (e.g., Angermann et al., 2002a, b; M¨uller et al., 2005). The procedure comprises the following steps: (1) In a first step we computed for the entire time span weekly normal equations for LAGEOS 1/2 independently. The free normal equations of both satellites were combined per week, and then solved by applying loose constraints (i.e., 1 m standard deviations on station positions). (2) Then we computed a multi-year SLR solution, which serves as reference for the transformation of the weekly SLR solutions into a unique frame. The weekly combined LAGEOS 1/2 normal equations were accumulated to multi-year normal equations. Within this accumulation we introduced new position and velocity parameters for stations with discontinuities according to the ILRS table. For the definition of the reference frame we applied minimum constraints. The origin of the reference frame was defined as a mean geocenter by setting the first degree and order terms to zero and the scale being defined by the speed of light and GM. The orientation was defined by six no-net-rotation (NNR)conditions to minimize the common rotation and rotation rate w.r.t. ITRF2005(SLR) station positions and velocities (Altamimi et al., 2007). (3) The loosely constrained weekly solutions were transformed to the multi-year SLR solution by applying 7 parameter similarity transformations. In order to reduce the effect of variations in the spatial

On the Strength of SLR

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distribution of the tracking stations we defined a subset of about 20 SLR core stations with dense tracking and a good global distribution on the Earth’s surface, which we used for the similarity transformations. The transformations were done iteratively by rejecting those core stations in a particular week, which have only few observations and/or large position residuals. As a result we obtained time series of station positions and transformation parameters.

4.2 Time Series of Translation and Scale Parameters To gain an insight into the strength and contribution of SLR to realize the terrestrial reference frame, we analysed the time series of translation and scale variations derived from the seven parameter similarity transformations between the weekly solutions and the multi-year reference solution. The resulting time series for the scale and translation variations from 1993 until mid of 2007 show a high long-term stability (see Figs. 6 and 7). The R.M.S. variations of the weekly estimates for the translation parameters are in the order of 5 mm for the x- and y-component and a factor of 2.5 higher for the z-component (see Table 3). The scattering of the weekly scale estimates is about 0.8 ppp. The translation and scale variations show clear annual signals. Their amplitudes and phases are given in Table 3. There is a good agreement with annual variations in the lower spherical harmonic coefficients C10 , C11 and S11 computed from SLR data by applying an analytical approach (Angermann et al., 2002b).

The results for the annual variations in the SLR network translations are also compared with geophysical model results (Chen et al., 1999; Dong et al., 1997) and with GPS results derived from a degree-one approach (Blewitt et al., 2001; Dong et al., 2003). As shown in Table 4, the SLR results of this study agree quite well with the model results, whereas the z-component of the GPS results is obviously too large by a factor of about 2. The discrepancies between the two GPS solutions are significantly larger than the estimated standard deviations, in particular for the annual amplitudes of the z-component. The differenes for the phases of the annual signals are rather large between the different solutions.

4.3 Effect of Range Bias Estimation on the SLR Results The fact, that SLR measurements can be affected by technical problems of the laser ranging systems may require the estimation of station- and time-dependent range biases. We computed a so-called test solution to assess the effect of range bias estimation on the SLR results. The solution set-up was done in a similar way as for the reference solution. The only difference was, that we estimated range biases for all stations in each weekly solution. As a result we obtained time series of weekly range biases for each SLR station. Figure 8 shows such a time series as an example. We applied standard deviations of 1 cm on the range biases parameters for three stable stations located on different continents (Graz in Austria, Yaragadee in

1993 1994 1995 1996 1997 1998 1998 2000 2001 2002 2003 2004 2005 2006 4 3 2 ppb

1 0 –1 –2 –3 –4

Fig. 6 Time series of weekly scale variations

–2000

–1000

0 JD2000 (Days)

1000

2000

¨ D. Angermann and H. Muller

26

z [cm]

y [cm]

x [cm]

1993 1994 1995 1996 1997 1998 1998 2000 2001 2002 2003 2004 2005 2006 4 3 2 1 0 –1 –2 –3 –4 4 3 2 1 0 –1 –2 –3 –4 4 3 2 1 0 –1 –2 –3 –4 –2000

–1000

0 JD2000 (Days)

1000

2000

Fig. 7 Time series of weekly translation variations

Table 3 R.M.S. variations and annual signals for the translation and scale time series derived from weekly SLR solutions over 15 years

Parameter

R.M.S. variations

Annual signals

Table 4 External comparison of the annual signals in the translation (origin) variations Source

Data

Par.

Amplitude [mm]

Phase [deg]

Amplitude

Phase

This paper

SLR

X Y Z

2.7 ± 0.6 3.8 ± 0.6 5.4 ± 1.4

220 ± 14 321 ± 10 265 ± 15

Translations X Y Z

[mm] 5.2 5.2 11.8

[mm] 2.7 ± 0.6 3.8 ± 0.6 5.4 ± 1.4

[degree] 220 ± 14 321 ± 10 265 ± 15

Dong 1997

Model

[ppb] 2.6 ± 0.9

[degree] 205 ± 28

4.2 3.2 3.5

222 339 235

Scale

[ppb] 0.79

X Y Z

Chen 1999

Model

X Y Z

2.4 2.0 4.1

244 270 228

Blewitt 2001

GPS

X Y Z

3.3 ± 0.3 4.8 ± 0.3 11.1 ± 0.2

184 ± 3 285 ± 3 214 ± 1

Dong 2003

GPS

X Y Z

2.1 ± 0.3 3.3 ± 0.3 7.1 ± 0.3

224 ± 3 297 ± 3 232 ± 3

Australia, and Monument Peak in USA) to stabilize the computations, and solved again the range biases for all stations. The resulting weekly combined LAGEOS 1/2 solutions were aligned to the multi-year reference solution by seven parameter similarity transformation as described in Sect. 4.1.

On the Strength of SLR

27

Fig. 8 Time series of weekly range biases [units are cm] for the SLR station Yaragadee (Australia)

ppb

1993 1994 1995 1996 1997 1998 1998 2000 2001 2002 2003 2004 2005 2006 4 3 2 1 0 –1 –2 –3 –4 –2000

–1000

0

1000

2000

JD2000 (Days)

Fig. 9 Time series of weekly scale variations (test solution: range biases estimated for all stations)

The time series of the weekly scale and translation variations are shown in Figs. 9 and 10. The results are comparable to those obtained from the reference solution (see Figs. 6 and 7). The R.M.S. variations for the translation and scale parameters are getting slightly higher, and the amplitudes of the annual signals are slightly reduced (see Table 5). This indicates that the solutions with all range biases estimated are more noise and that the annual signals in the transformation parameters are partly absorbed by the estimated range biases. Furthermore, the results indicate that the estimation of range biases does not change the general long-term behaviour of the time series for the translation and scale parameters.

5 Conclusions and Outlook SLR observations provide stable results to realize the origin and scale of the terrestrial reference system. The analysis of the covariance matrix of the datum parameters has shown that for LAGEOS, the standard deviations for the z-component of the network translation is worse by a factor of 3 compared to the x- and ycomponent. SLR observations to satellites with lower altitudes (e.g., Ajisai and Starlette) can determine the z-component much better, which is due to the higher sensitivity to the Earth’s gravity field. The time series for the translation and scale parameters derived

¨ D. Angermann and H. Muller

z [cm]

y [cm]

x [cm]

28

4 3 2 1 0 –1 –2 –3 –4 4 3 2 1 0 –1 –2 –3 –4 4 3 2 1 0 –1 –2 –3 –4

1993 1994 1995 1996 1997 1998 1998 2000 2001 2002 2003 2004 2005 2006

–2000

–1000

0 JD2000 (Days)

1000

2000

Fig. 10 Time series of weekly translation variations (test solution: range biases estimated for all stations)

weaker and the amplitudes of the annual signals are slightly reduced by the estimated range biases. For future realizations of the terrestrial reference R.M.S. Annual signals system it should be considered to include SLR satelvariations Parameter Amplitude Phase lites with lower altitudes, which may stabilize the Translations [mm] [mm] [degree] determination of the z-component of the origin conX 6.2 2.4 ± 0.8 226 ± 18 siderably. We are currently performing investigations Y 6.5 3.5 ± 0.8 322 ± 13 in this context and we are also going to compute Z 12.9 5.0 ± 1.6 275 ± 23 SLR time series solutions with Starlette observations, [ppb] [ppb] [degree] Scale 0.91 2.2 ± 1.1 177 ± 30 provided that the orbits can be computed with a precision compared to the high orbiting satellites. Issues in this context are the correction of atmospheric drag perturbation and the higher impact of the Earth’s from weekly SLR solutions over 15 years show a high gravity field, which should be significantly improved long-term stability. The observed annual variations in by the new GRACE models. the translation parameters are consistent with geophysical model results. It was also demonstrated that the estimation of range biases for all SLR stations does Acknowledgments Thanks are expressed to the ILRS not change the long-term behaviour of the datum pa- (Pearlman et al., 2002) and to the SLR station operators for rameter time series. However, the solutions are getting providing data of the global SLR network. Table 5 R.M.S. variations and annual signals for the translation and scale time series derived from weekly SLR solutions over 15 years (test solution: range biases estimated for all stations)

On the Strength of SLR

References Altamimi, Z., X. Collilieux, J. Legrad, B. Garayat, C. Boucher (2007). ITRF2005: A new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters, Journal of Geophysical Research, Vol. 112, B09401, doi:10.1029/2007JB004949. Angermann, D. (1991). Zur Datumsfestlegung und ShortArc Bahnverbesserung bei der Auswertung von GPSBeobachtungen. DGK Reihe C, Heft Nr. 388, M¨unchen. Angermann, D., M. Gerstl, R. Kelm, H. M¨uller, W. Seem¨uller, M. Vei (2002a). Time Evolution of an SLR Reference Frame. Advances in Space Research, Vol. 30/2, pp. 201–206, Elsevier. Angermann, D., H. M¨uller, M. Gerstl (2002b). Geocenter variations derived from SLR data to LAGEOS 1 and 2. In: Vistas for Geodesy in the New Millennium, IAG Symposia, J. Adam and K.-P. Schwarz (eds), Vol. 125, pp. 30–35. Blewitt, G. D., Lavallee, P. Clarke, K. Nurutdinov (2001). A new global mode of Earth deformation: Seasonal cycle detected. Science, 294 (5550), pp. 2342–2345. Chen, J.L., C.R. Wilson, R. J. Eanes, R. S. Nerem (1999). Geophysical interpretation of observed geocenter variations. Journal of Geophysical Research, Vol. 104, No B2, pp. 2683–2690.

29 Dietrich, R. (1988). Untersuchungen zur Nutzung k¨unstlicher Satelliten f¨ur die geod¨atische Koordinatenbestimmung. Ver¨offentlichungen des Zentralinstituts f¨ur Physik der Erde, Nr. 96, Potsdam. Dong, D., J. O. Dickey, Y. Chao, K. Cheng (1997). Geocenter variations caused by atmosphere, ocean and surface ground water. Geophysical Research Letters, Vol. 24, No. 15, pp. 1867–1870. Dong D., T. Yunck, M. Heflin (2003). Origin of the International Terrestrial Reference Frame, Journal of Geophysical Research, Vol. 108, No. B4, 2200, doi:10/1029/2002JB0022035. McCarthy, D. D. G. Petit (2004). IERS Conventions (2003), IERS Technical Note, No. 32, Verlag des Bundesamtes f¨ur Kartographie und Geod¨asie, Frankfurt am Main. M¨uller, H, D. Angermann, B. Meisel (2005). A multi-year SLR solution. In: Garate, J., J. Davilla, C. Noll, M. Pearlman (Eds.), 14th International Laser Ranging Workshop, Proceedings, Boletin ROA 5/2005, 15–19. Pearlman, M. R., J. Degnan, J. M. Bosworth (2002). The International Laser Ranging Service, Advances in Space Research, Vol. 30, No. 2, pp. 135–143, DOI:10.1016/S02731177(02)00277-6, Elsevier.

Impact of the Network Effect on the Origin and Scale: Case Study of Satellite Laser Ranging X. Collilieux and Z. Altamimi

Abstract The study of station non linear motions requires being able to discriminate between the local motion of every station and their global motion (e.g. geocenter motion). A common used approach consists in estimating Helmert parameters with respect to a secular reference frame. But if the network is small or not well distributed, the global parameters, and mainly the translation and scale parameters, will not be rigorously decorrelated from the station displacements. The distribution of the currently operating Satellite Laser Ranging (SLR) stations is one example of such a network. The understanding of the network effect needs an analysis of the station distribution over time. It appears that SLR station distribution on the X positive hemisphere of the Earth varies seasonally. We suggest here that using GPS results to constrain the estimation of SLR station displacements may improve the estimation of the Helmert parameters. These additional constraints as well as stochastic constraints are numerically tested here in a first attempt to limit the impact of the network effect. We show that the X component of the translation and the scale are the two parameters that are the most sensitive to any change in the configuration of the added constraints.

Keywords Reference frames · ITRF · Geocenter · Network effect · SLR · GPS

X. Collilieux Institut G´eographique National, LAREG, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vall´ee, France Z. Altamimi Institut G´eographique National, LAREG, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vall´ee, France

1 Introduction The Earth’s shape is constantly changing mainly due to tectonic motions, tidal effects, loading displacements or regional subsidence. To be understood, the measurements of the Earth’s crust displacements have to be referenced with respect to a known reference frame. Measurements from space geodesy provide by nature some parameters needed to define a reference frame. For example, Satellite Laser Ranging (SLR), Global Positioning System (GPS), and Doppler and Orbitography Radio positioning Integrated on Satellite system (DORIS) provide intrinsic information about the origin of the frame and the scale but none of them provide information about its orientation. Such defined frames, necessarily built thanks to additional constraints, have a time evolution with respect to the crust that we aim to monitor. This time evolution is basically analyzed thanks to Helmert parameters estimated between time series of terrestrial frames, embodied in sets of positions of stations and their covariance matrices, with respect to a piece-wise linear reference frame. Some previous studies (see Sect. 2 for references) have pointed out that Helmert parameters estimated between two frames that do not have the same shape may be biased. Indeed, it is not possible to decorrelate completely biases and motions that affect all the stations from the individual motions of the stations. This aliasing effect is a limitation of the interpretation of the Helmert parameters as physical parameters. Indeed, SLR network net translation with respect to the ITRF is desired to be analyzed as geocenter motion. We investigate here a new methodology which relies on additional constraints that could be used to limit the impact of the network effect in case of SLR data.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

31

32

Thus, the first section describes the SLR network and data that are analyzed in the scope of the study. We then propose a model which can be used to investigate and limit the network effect regarding non linear motions. Finally, first results of the new method are discussed around few relevant considerations: analysis of the estimated scale and of the annual signal as well as a comparison with a geocenter motion model.

2 Network Effect The current SLR network is composed of 87 stations mainly distributed at mid-latitude. The International Laser Ranging Service (ILRS) computes time series of positions at the weekly basis for stations tracking geodetic satellites, see Luceri and Pavlis, 2006. These positions result from the combination of five independent solutions processed by the ILRS Analysis Centres (ACs), see Pearlman et al. (2002). We use in the following ILRS weekly positions from 1996.0 to 2006.0. Since the ILRS solutions are loosely constrained, minimum constraints have been applied, following Altamimi et al. (2002). It is not possible with this finite network to completely discriminate between the global motion of every station and their individual motion. Indeed, some part of the station individual motion may leak into the Helmert parameters when those are estimated. As a consequence Helmert parameters may be biased as well as the residuals of the transformation if one wants to analyze them as station displacements. This effect has been discussed by Tregoning and van Dam (2005) and Lavall´ee et al. (2006): they conclude that the estimation of the scale parameter is a critical issue due to aliasing problem. Collilieux et al. (2007) have tried to quantify the network effect by stacking subsets of the International GNSS Service (IGS) station positions. Indeed, IGS network is much more well distributed and contains more than 200 stations from 2003. Moreover, IGS weekly sets of positions have been processed at the same sampling rate. Collilieux et al. (2007) have compared estimated Helmert parameters from different subsets of stations with the Helmert parameters estimated with the full network. They have notably used the co-located network of SLR stations as a specific subset. They have evaluated that the network effect affects the station position residuals

X. Collilieux and Z. Altamimi

at the level of 0.8 ± 0.5 mm for horizontal and 2.1 ± 0.3 mm for vertical. The Helmert parameters also differ by about 1 mm for the translations and 2 mm for the scale in terms of scatter: this confirms the results of Tregoning and van Dam (2005) and Lavall´ee et al. (2006). Following Lavall´ee et al. (2006), we have plotted the percentage of SLR stations located in the three Earth hemispheres, defined to be centered on the direction of each Cartesian axis, see Fig. 1. The proportion of SLR stations is much more important in the Northern hemisphere (Z positive axis) and some seasonal features can be detected in the X positive hemisphere. This feature is expected to be due to weather conditions (D. Coulot, personal communication). Indeed, as SLR stations track satellites when the sky is clear, range measurements may be sparse in case of bad weather conditions. The same plot realized with the GPS colocated stations to SLR systems, which observe when SLR data are available, is also presented, see Fig. 1. The distribution of the co-located GPS network reproduces quite faithfully the network availability although the number of stations is less important. Values obtained by Collilieux et al. (2007) seem consequently to be quite representative of this geometric effect. However, it should be mentioned that these values are derived taking into account the GPS internal noise processes, that are expected to be lower than noises affecting SLR positions. We suggest here to use this network jointly with the SLR network to study the network effect which occurs when a SLR data set is analyzed with Helmert parameters. The effect of adding temporal constraints will be also investigated in the following. As a first analysis, the data sets have been stacked using CATREF software, see Altamimi et al. (2007), to estimate a first set of Helmert parameters between the SLR weekly frames and ITRF2005. During that process, outliers have been identified in an iterative process. They have been listed and excluded from the ILRS sets of positions used in the following.

3 Model We suggest here to modify the basic formulation of the Helmert parameter estimation by using additional parameters. These are used to model the shape difference

Impact of the Network Effect

% of stations on positive axis

100

X axis

Fig. 1 SLR (blue crosses) and co-located GPS (red points) station network distribution over time

33

80 60 40 1996

1998

2000

2002

2004

2006

1998

2000

2002

2004

2006

1998

2000

2002

2004

2006

1998

2000 2002 Number of stations

2004

2006

Y axis

100 80 60 40 1996

Z axis

100 80 60 40 1996 30 20 10 0 1996

X i (t) = [d x i (t)dy i (t)dz i (t)]T is the displacement at the epoch t of point i and T (t), λ(t) and R(t) are the Helmert parameters. The dot notation symbolizes the derivation with respect to time. Since the X i (t) i Xm (t) = T (t) + (I + λ(t) · I + R(t)). components are less than two centimeters, the rotations [X ri (tr ) + X˙ ri (t)(t − tr ) + X i (t)], are at the level of one miliarcsecond and the scale at ∀i ∈ {1, . . . , n t } (1) the level of one part per billion (ppb), the products of these terms are at the sub-millimeter level and are i (t) i (t)]T where X m = [xmi (t)ymi (t)z m is the neglected in practice. We wish to simultaneously solve displacement weekly position of point i at the epoch t, i i i i T parameters for every available station and Helmert X r (tr ) = [xr (tr )yr (tr )zr (tr )] is the reference parameters for every week. In this model all the station solution at the epoch tr , n t is the number of points at t, between the two frames. They are called displacement parameters and are defined as follows:

34

X. Collilieux and Z. Altamimi

Table 1 List of the numerical tests Equality constraint Scaling factor of the constraint sigma Experiment

Stochastic variations

Values

LS Sto Cont1

– Applied –

Cont2

–

– – ITRF2005 IGS position residual time series ITRF2005 IGS position residual time series

positions contribute to the estimation of the Helmert parameters, as it is done in the CATREF software, see Altamimi et al. (2002). A Kalman filter has been chosen to solve this equation, see Gourieroux and Monfort (1997). Thus a time evolution law for the estimated parameters needs to be adopted by using the state equation of the Kalman filter. Random walk model, which is particularly well adapted to this formalism, has been chosen in a first approximation. We have estimated from ITRF2005 SLR residual position time series a random walk model assuming that it is measured with an additive uncorrelated noise. A random walk variance and a variance factor on the measurement noise for each position residual time series which have more than 150 points have been estimated by maximum likelihood, with a similar approach to what is done in Collilieux et al. (2007). The estimated standard deviation values of the innova√ tions fluctuate around 1 cm. yr−1 for horizontal and √ 2 cm. yr−1 for vertical with a median value of four for the variance factor, meaning that the covariance matrix of the SLR input file is probably too optimistic with regards to the estimated model. We will use these values to calibrate the Kalman filter assuming that they should not be too erroneous. Thus we use, for stations having less than 150 points, conventional values for the station displacement random walk variances which are √ √ equals to 1 cm. yr−1 for horizontal and 2 cm. yr−1 for vertical. We carried out the same analysis for the Helmert parameters. The data will be filtered forward and backward to compute smooth values of each parameter. We follow the Gelb algorithm to compute the smoothed values, following Herring et al. (1990). We use Helmert parameters values from classical least squares as a priori estimates for the Helmert parameters and non linear residual position time series from the stacking as a priori estimates for

Up

σ02D

σ0

2

– – 0.5

2

0.25

the displacements. This first experiment is called Sto in Table 1. Our second attempt is to constraint displacement parameters to external values by means of equality constraints. The Kalman filter will be reinitialized at every epoch with the given a priori values, which means that the state equation of the Kalman filter will be ignored in that investigation. The constraints are handled here using the observation equation of the Kalman filter. We use position residuals of GPS colocated sites to SLR stations computed for ITRF2005 and use their standard deviations to weight the constraints and give us the liberty to re-scale them. We apply different variance factors on the GPS standard deviations for the horizontal and vertical since GPS and SLR do not have the same sensitivity on the two components. We intentionally down-weigh the GPS horizontal precision and increase its weight on the vertical component where GPS positions have larger standard deviations – taken from the SINEX files - than SLR positions in average. The two experiments are presented in Table 1, with two different scaling factors for the GPS constraint called cont1 and cont2. The objective would be to combine this constraint with the temporal constraints of the Kalman filter when those are both controlled and understood.

3.1 Results This section sums up the results of the two parameterizations as described in the last section by means of Table 2. This table shows the Weighted Root Mean Square statistics (WRMS) of the time series and their scatter with respect to the Helmert parameters estimated by the stacking, called LS in Table 1. It also

Impact of the Network Effect

35

Table 2 WRMS of Helmert parameters and estimated annual signal computed from 2000.0. The fourth column shows the WRMS of the Helmert parameters with respect to the LS Helmert parameters. The translations are given here with the

TX

TY

TZ

Scale

convention of equation 1. A minus sign should be applied to Wu et al. (2006) geocenter time series to make them comparable. The convention for the annual signal is A cos(ωt − φ)

Method

WRMS (mm)

WRMS wrt. LS (mm)

Amplitude (mm)

Phase (deg)

LS Sto Cont1 Cont2

4.8 3.5 4.5 4.9

– 6.7 3.0 3.7

2.6 ± 0.3 1.7 ± 0.3 2.2 ± 0.3 2.3 ± 0.3

233 ± 8 258 ± 9 205 ± 9 193 ± 9

Wu et al. (2006)

–

–

1.7 ± 0.3

171 ± 11

LS Sto Cont1 Cont2

4.7 3.8 4.3 4.7

– 6.7 2.6 3.2

3.1 ± 0.3 3.0 ± 0.2 3.1 ± 0.3 3.0 ± 0.3

155 ± 9 165 ± 5 156 ± 5 156 ± 6

Wu et al. (2006)

–

–

3.8 ± 0.3

160 ± 6

LS Sto Cont1 Cont2

7.8 6.7 7.3 7.5

– 10.2 3.0 3.4

3.2 ± 0.5 2.8 ± 0.5 3.4 ± 0.5 3.5 ± 0.5

215 ± 9 210 ± 10 199 ± 8 204 ± 8

Wu et al. (2006)

–

–

4.5 ± 0.3

201 ± 4

LS Sto Cont1 Cont2

4.5 3.7 4.5 4.3

– 6.1 2.7 2.8

1.8 ± 0.3 2.7 ± 0.2 1.2 ± 0.3 1.0 ± 0.3

217 ± 4 201 ± 5 219 ± 13 243 ± 13

shows the estimated annual signals estimated in each time series simultaneously with a trend by weighted least squares. Figure 2 shows the estimated Helmert parameters in three cases with the associated standard deviations. The effect of adopting stochastic constraints is to smooth time series of parameters. Most of the high frequency variations of the displacements – as can be seen in the residuals of least squares Helmert parameters estimation – are eliminated from the displacement parameters. Indeed, the chosen random walk level does not permit high amplitude variation in a short period of time. Table 2 shows that all the Helmert parameters have lower scatters compared to the basic unconstrained least squares approach. However, it can be seen that the annual signal in the scale has increased compared to the least squares basic approach, see Table 2. The difference between the stochastic scale and the SLR scale is quite large and is at the level of 6.1 mm of scatter, see Table 2. This value is probably too large to be attributed to the network effect most especially because the annual signal would have been expected to be reduced. Adopting equality constraints to some of the displacement parameters (here, the displacement parameters of the SLR co-located stations) also changes the

estimated Helmert parameters. The WRMS decreases for most parameters but when the constraint weight is increased, the scatter decreases only in TZ compared to the unconstrained least squares results. However, the scatter can not be viewed as a criterion to measure the relevance of the method. We note that the differences between the unconstrained and constrained parameters are at the level of three millimeters, which is the range of the values expected for the network effect. It can be noted from Table 2 that the scale annual signal has reduced thanks to the equality constraints, meaning that constraining SLR displacements to co-located GPS station position residuals reduces the network effect. Indeed, no annual bias is expected to be contained in the SLR scale. Geocenter motion models are often compared by analyzing the annual signal, see Lavall´ee et al. (2006). Table 2 presents the annual signals estimated for each time series of translations that are usually interpreted in term of geocenter motion. The first comment concerns the annual signal on the X component. It can be noted that changing the parameterization or methodology has the biggest impact on that component. It is in fact not so surprising since the network has some important seasonally distribution changes on the corresponding hemisphere, see Fig. 1. The annual signal

36

X. Collilieux and Z. Altamimi

20 15

20 a)

15

10

10

5

5

0

0

–5

–5

–10

–10

–15

–15

–20 53000

53200

53400

53600

20 15

c)

–20 53000

53200

53400

53600

53200

53400

53600

0 b)

d)

10

–5

5 0

–10

–5 –10

–15

–15 –20 53000

53200

53400

53600

–20 53000

Fig. 2 Translations. (a) X component (b) Y component (c) Z component and (d) scale in millimeters. Blue: Least squares method (LS), Green: Stochastic constrains introduced (Sto), Red: Equality constrains introduced to GPS non linear residual time series (cont2)

does not change significantly on the two other components of the geocenter since the values are still enclosed in two sigma values. A comparison with the annual geocenter displacement model from Wu et al. (2006) shows interesting results. We can see that the agreement in Y and Z components is quite impressive for all the approaches in term of annual phase. However, the X component disagrees in phase about more than two months with the least squares approach. The use of stochastic constraints dephases even more the SLR geocenter annual signal on that component with respect to the geocenter model. However the use of additional constraints to the GPS displacements makes the data consistent with the model in the two sigma interval. Further investigations should be done with other geocenter models derived from geophysical data set only to confirm this result. Further investigation should be also led to improve the stochastic modeling of the displacement parameters.

4 Conclusion When a network is small, or sparse, it is delicate to discriminate the global motion of all the stations from the individual motion of each of them. The typical strategy consists in estimating Helmert parameters between a long term or secular solution, such as the ITRF, and the individual frames. The error introduced could be as large as few millimeters of scatter and does not affect all the transformation parameters in the same way. We have developed a methodology to study the network effect by introducing external information. The aim is to better define what comes from the station motion and what comes from the geocenter motion and biases. This methodology has been applied to SLR measurements. We have shown that modeling station motion and geocenter motion as random walk process can not efficiently decorrelate the Helmert parameters from the station individual motion. Additional work should be done either to better calibrate the random

Impact of the Network Effect

walk variances, for example with geodynamical model, or to find more efficient stochastic models that describe more shrewdly SLR station individual processes and geocenter motion. Constraining the SLR displacements to GPS non linear residuals has been shown to supply promising results. Indeed, the annual signal in the scale parameter decreases compared to the classical approach. The component of the geocenter motion which is the most affected by the network effect seems to be the X component since any change in the methodology has an impact on that component at the annual frequency. Indeed, we have shown that the SLR network configuration undergoes seasonal variations in the X positive hemisphere of the Earth which explains that sensitivity. We have shown that using GPS data jointly with SLR or using stochastic constraints shifts the phase of the determination of the geocenter motion in the X component as much as the models and SLR data discrepancies. Further comparisons should be done with other geocenter models coming from geophysical data only. The question of long term network effect has not been addressed here. However, it is probably the most challenging issue for the near future. Users require high quality reference frames to study very tiny effects over very long periods of time. They will be able to do so only if the reference frame that they use conserves its definition over time. This study is delicate since accurate models of the earth crust motion are needed, notably on the vertical to validate the reference frame velocities. Plag (2007), this issue, has compared the vertical velocities of a network of station with vertical velocities predicted with absolute gravimeter measurements. His results conclude that such analysis could help solving this issue.

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References Altamimi, Z., Sillard, P., and Boucher, C. (2002). ITRF2000: A new release of the International Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research, pages 2–1. Altamimi, Z., Collilieux, X., Legrand, J., and Boucher, C. (2007). ITRF2005: A new release of the international terrestrial reference frame based on time series of station positions and earth orientation parameters. Journal of Geophysical Research, 112. Collilieux, X., Altamimi, Z., Coulot, D., Ray, J., and Sillard, P. (2007). Comparison of very long baseline interferometry, GPS, and satellite laser ranging height residuals from ITRF2005 using spectral and correlation methods. Journal of Geophysical Research (Solid Earth), 112 (12403) 1–18. Gourieroux, C. and Monfort, A. (1997). Time Series and Dynamic Models. Cambridge University Press cambridge. Herring, T. A., Davis, J. L., and Shapiro, I. I. (1990). Geodesy by radio interferometry: The application of Kalman filtering to the analysis of very long baseline interferometry data. Journal of Geophysical Research (Solid Earth), 95:12561–12581. Lavall´ee, D. A., van Dam, T., Blewitt, G., and Clarke, P. J. (2006). Geocenter motions from GPS: A unified observation model. Journal of Geophysical Research (Solid Earth), 111:5405-+. Luceri, V. and Pavlis, E. (2006). The ilrs solution. Technical report, ILRS Technical Report. Pearlman, M., Degnan, J., and Bosworth, J. (2002). The international laser ranging service. Advances in Space Research, 30(2):135–143. Plag, H.-P. (2007). Collocated gps and absolute gravity measurements help to tie the geometric reference frame to the center of mass of the earth system. Proceedings of IUGG2007, Perugia. Tregoning, P. and van Dam, T. (2005). Effects of atmospheric pressure loading and seven-parameter transformations on estimates of geocenter motion and station heightsfrom space geodetic observations. Journal of Geophysical Research, 10(B03408). Wu, X., Heflin, M. B., Ivins, E. R., and Fukumori, I. (2006). Seasonal and interannual global surface mass variations from multisatellite geodetic data. 111:BO9401–.

Satellite Laser Ranging Biases and Terrestrial Reference Frame Scale Factor ´ D. Coulot, P. Berio, P. Bonnefond, P. Exertier, D. Feraudy, O. Laurain and F. Deleflie

Abstract We examine the impact of the strategy adopted regarding range biases on Satellite Laser Ranging data processing results. More especially, we test the stability of the weekly Terrestrial Reference Frame scale factors computed w.r.t. ITRF2000 according to the two different strategies which are studied in this paper. In the first approach, range biases are not considered in the data processing. In the second approach, we use a temporal de-correlation method to rigorously estimate all station range biases. The first approach provides weekly scale factors with a piece-wise behavior. More precisely, the time series exhibit a drift of −2.5±0.2 mm/yr between 2001.0 and 2006.0 and provide a RMS of 5.2 mm. The second

approach, which makes the most physical sense, really provides regular weekly scale factors. Indeed, between 2001.0 and 2006.0, these series exhibit a drift of −0.7 ± 0.2 mm/yr and the RMS of the whole time series is 3.7 mm. Moreover, less discontinuities are detected in the station position residual time series provided by the second approach than in those produced with the first approach.

D. Coulot Institut G´eographique National, LAREG, ENSG, 6-8 Avenue Blaise Pascal, Cit´e Descartes, Champs-sur-Marne, 77455 Marne la Vall´ee cedex 2, France

Among all the existing space-geodetic techniques, Satellite Laser Ranging (SLR) plays a major role for the International Terrestrial Reference System (ITRS) realization. Indeed, except for the latest realization, ITRF2005, SLR has always provided the origin and the scale (together with VLBI) of the various International Terrestrial Reference Frame (ITRF) versions since the ITRF94 computation. For the first time in the ITRF history, the ITRF Product Center (ITRF PC) has considered time series of station positions and Earth Orientation Parameters (EOPs) provided by the four spatial geodetic technique services of the International Association of Geodesy (IAG) as input for the ITRF2005 solution. The International Laser Ranging Service (ILRS) SLR scale time series estimated w.r.t. ITRF2000 has been shown to exhibit a piece-wise behavior. Moreover, the ITRF2005 analyses have shown a 1 ppb relative scale bias between SLR and VLBI w.r.t. ITRF2005. Consequently,

P. Berio UNSA/G´eosciences Azur, UMR 6526 CNRS/OCA, Avenue Nicolas Copernic, 06130 Grasse, France P. Bonnefond UNSA/G´eosciences Azur, UMR 6526 CNRS/OCA, Avenue Nicolas Copernic, 06130 Grasse, France P. Exertier UNSA/G´eosciences Azur, UMR 6526 CNRS/OCA, Avenue Nicolas Copernic, 06130 Grasse, France D. F´eraudy UNSA/G´eosciences Azur, UMR 6526 CNRS/OCA, Avenue Nicolas Copernic, 06130 Grasse, France O. Laurain UNSA/G´eosciences Azur, UMR 6526 CNRS/OCA, Avenue Nicolas Copernic, 06130 Grasse, France F. Deleflie UNSA/G´eosciences Azur, UMR 6526 CNRS/OCA, Avenue Nicolas Copernic, 06130 Grasse, France

Keywords Satellite Laser Ranging · Range biases · Terrestrial Reference Frames · Scale factor

1 Introduction

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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only the VLBI technique has provided the scale (and its rate) of this latest ITRF version, Altamimi et al. (2007). But nevertheless, the SLR technique is the satellite technique which produces the most accurate estimations of the gravitational constant GM and the ITRF scale is directly linked to this fundamental constant. Although SLR is less affected by atmospheric propagation than VLBI, GPS, and DORIS are, some residual errors affecting the station vertical components could explain this unstable scale: the poor geographic distribution of the SLR network, the dependence of SLR on weather, the physical signals which were not taken into account in the a priori models (namely, the atmospheric and hydrologic loading effects), the deficiencies in the a priori models used, the possible aliasing effects, tropospheric propagation, satellite signatures, radial residual errors, center of mass corrections, and range biases. In this paper, we only focus on these SLR biases (which seem to be the most influent errors, as shown here) and on their signatures in the Terrestrial Reference Frame (TRF) scale factors. Firstly, we explain our method and its interests for estimating range biases. To validate it, we then study two opposite strategies. In the first approach, range biases are not considered at all in the computation. In the second approach, estimated biases are applied to correct the SLR measurements during the data processing. We apply an upgraded temporal de-correlation method (in comparison with the method used in Exertier et al. (2004), Coulot (2005), and Coulot et al. (2007)) to compute these station biases per satellite. Finally, the results of these two computation strategies regarding weekly TRFs (more especially, regarding the weekly scale factor time series) are presented and discussed. We also provide an overview of the time series produced for the station positions by analyzing the particular case of the Zimmerwald’s station (7810), in Switzerland, for which the station height residual time series produced by the first approach exhibit a significant number of discontinuities. These discontinuities must certainly not be interpreted as proofs of a poor quality of this station data. On the contrary, the Zimmerwald’s station is one of the most reliable station of the SLR network. We have chosen it firstly because, over the time span of the data used, it has tracked satellites with three different “colors” (green, purple, and infrared). Secondly, it has undergone some major instrumental and data preprocessing changes over this data time span. For these

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reasons, the Zimmerwald’s station is the most eloquent example to demonstrate the efficiency of our method.

2 Different Possible Strategies 2.1 SLR Biases and Temporal De-correlation Method Range biases (systematic errors affecting the SLR measurements) can corrupt station vertical component time series if they are not estimated nor applied during SLR data processing. As an example, the Fig. 1 shows the height residual time series of the SLR station of Grasse (7835). This computation was carried out without considering nor applying any range bias (see Sect. 2.2 for more details on the data processing and the time series stacking). An offset can clearly be detected (in September 1997) in these time series. It corresponds exactly to a modification of the detection system of the station. As a consequence, we really must take range biases into account in the analysis. But, when range biases are computed simultaneously with station positions, the correlation between biases and station heights can reach 99% Furthermore, if the range biases are estimated at the same sampling rate than the station positions are, we show that some part of the station position non-linear signals can be found in the estimated biases. Figure 2 shows the weekly time series of the

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a twelve-year SLR data processing in Coulot (2005), and for a four-technique combination carried out at the measurement level in Coulot et al. (2007). It has been 20 recently improved. Indeed, the limits of the time intervals over 0 which the range biases are supposed to be constant now really correspond to station instrumental –20 changes. The corresponding epochs are found in station log files (available per station in the directory –40 ftp://cddis.gsfc.nasa.gov/pub/reports/slrlog), in SLR mails (archived in the directory ftp://cddis.gsfc.nasa. 49000 50000 51000 52000 53000 gov/reports/slrmail) and in the ILRS official eccenModified Julian Day tricity file available in the directory ftp://cddis.gsfc. Fig. 2 Weekly time series of biases computed (in mm) for the nasa.gov/slr/slrocc. SLR station of Monument Peak (7110), in United States, and the Time intervals and range bias values as well as satellite LAGEOS-1. The station positions have been computed standard deviations per station, per LAGEOS sateltogether with weekly range biases per LAGEOS satellite lite, and per wavelength are available at the Web site http://maestro.obs-azur.fr/cgi-bin/query mrb.pl. 40

Monument Peak SLR station (7110) biases computed for the LAGEOS-1 satellite. In this example, the weekly station positions have been simultaneously computed with weekly range biases per LAGEOS satellite. A significant annual signal (with an amplitude of nearly 5 mm) is detected in these bias time series. It is obvious that this unexpected annual signal makes no physical sense. Moreover, it is worth noting that estimating range biases per satellite for all SLR stations weaken the involved weekly normal systems. In this context, we have developed the so-called “temporal de-correlation method”. The main hypothesis underlying this method is the following. Range biases are directly linked to SLR station instrumentations. But, these instrumentations are not modified all the time. As a consequence, we can suppose that range biases are constant over given periods of time. We found that estimating range biases over these periods of time together with weekly station positions decreases the correlation previously mentioned up to 50% on the average. It also avoids producing unexpected signals in estimated biases. And the decorrelation method is more improved by estimating range biases per satellite and per laser wavelength for stations which track satellites with multi-color lasers. This method has already been applied: for estimating the range bias of the French Transportable Laser Ranging System (FTLRS) over a several month calibration/validation campaign in Exertier et al. (2004), over arbitrary six-month time periods for

2.2 Time Series Computation and Analysis We briefly present here the method applied to compute and to analyze the weekly station position time series. 2.2.1 SLR Data Processing SLR ranges for the two LAGEOS satellites are processed between 01/03/1993 0 h UTC and 11/18/2006 24 h UTC, the time sampling being the GPS week. In a first step, GINS software (Coulot et al. (2007)) is used to compute the orbital arcs for the two LAGEOS satellites. Then, in a second step, these orbits and the SLR data are processed by MATLO software (Coulot (2005)) to provide weekly solutions in SINEX format. In order to produce results consistent with the ILRS official combined solution provided for the ITRF2005 computation, we have used ITRF2000, Altamimi et al. (2002), as a priori TRF for station positions and EOPC04 time series, Gambis (2004), as a priori values for EOPs. All recommended models are applied for station positions and EOPs according to McCarthy and Petit (2004). As a consequence, station position time series must exhibit geocenter motion and atmospheric and hydrologic loading effects as well. In this work, we only focus on weekly station position time series but these time series are

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computed together with daily EOPs (x p , y p , UT and LOD). The Marini-Murray (1973) model is used for tropospheric propagation correction. Furthermore, MATLO software applies a semi-dynamic method to estimate residual orbital errors and uses a data weighting per station based on orbit computation statistics (see Coulot (2005)). The orbits being computed, MATLO software is used to produce the solutions according to two opposite approaches. In the first computation, range biases are not considered at all (they are either estimated nor applied); in the following, this strategy is designated by “strategy WOB” (WithOut Biases). In the second approach, range biases computed with the temporal de-correlation method (see Sect. 2.1) are applied to correct SLR measurements at the data processing step (this is the “strategy WB” - With Biases). Once the weekly TRFs are computed, they are rigorously stacked.

2.2.2 Time Series Stacking The stacking of the weekly TRFs aims to analyze the scale factor time series computed between the weekly TRFs and ITRF2000. CATREF software, which has been built for ITRF computation and recently updated for time series combination, is used for this analysis. The model implemented in CATREF software is described in several papers, the most recent being Altamimi et al. (2007). We just remind here that, as results of such a stacking, we produce a combined TRF realized in ITRF2000 (by means of minimum constraints, see Sillard and Boucher (2001)). under the form of a set of station positions and velocities estimated at a reference epoch, weekly seven transformation parameters between the individual TRFs and this combined TRF, and station position residual time series. By analyzing these residual time series, we can monitor the station non-linear motions and, in particular, the possible discontinuities. These discontinuities must be accounted for to avoid altering the combined TRF. To do so, we use a piece-wise approach consisting of estimating two station positions and two station velocities (eventually constrained to be equal) before and after the detected discontinuity. In this work, we have tested and applied several discontinuity sets according to the produced station position residuals.

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3 Results Figure 3 shows the weekly translation and scale factor time series resulting from the stacking of the TRFs computed with the strategy WOB. Figure 4 only shows the corresponding scale factor time series. We see in this figure that the three translations exhibit periodic (mainly annual) signals. Furthermore, we can see offsets in the T X time series between 1993.0 and 1997.0. As this translation is not correlated with the scale factor, these offsets do not limit the conclusions of this paper, which is exclusively dedicated to scale factor and station height time series. It is also worth noting that the T Z time series provide a drift of 0.8 ± 0.04 mm/yr. The RMS of the weekly scale factor time series (see Fig. 4) is 5.2 mm. Moreover, the Figs. 3 and 4 clearly show a piece-wise behavior: 1993.0–1996.0, 1996.0–2001.0, 2001.0–2006.0, and 2006.0–2006.9. Table 1 provides the mean values and the drifts of the scale time series over these four intervals and over the whole time interval. We note that, with this strategy WOB, the time series exhibit a drift of −2.5 mm/yr between 2001.0 and 2006.0. From all these results, we can conclude that these time series are not regular at all. Furthermore, a first stacking of the so computed TRFs only carried out with the discontinuities corresponding to Arequipa’s earthquake provides station position time series which exhibit a significant number of discontinuities, especially in the height residuals. All discontinuities applied for the ITRF2005 computations (see http://itrf.ensg.ign.fr/ITRF solutions/2005/doc/ILRSDiscontinuities.snx) are found again in this case. In order to be consistent with the ITRF2005 computations, we have carried out a second stacking for which we have only kept these “official” discontinuities. The results presented here correspond to this stacking and a significant number of station height residual time series still exhibit discontinuities. On the other hand, an approach consisting of applying all the identified discontinuities would not be relevant. Indeed, only discontinuities that really correspond to physical events should be used. As an example, Fig. 5 shows the height residual time series of the Zimmerwald’s SLR station (7810). Nine discontinuities are detected in these time series and, as shown in Table 2, seven of them can be directly linked to an instrumental change reported in

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the on-line Data Base (DB) http://maestro.obs-azur.fr/ cgi-bin/query mrb.pl. We can see that, except for the sixth discontinuity, the epochs of the instrumental changes provided in Table 2 are very close (but not

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The first value corresponds to the mean value. The second values respectively correspond to the estimated drift and its standard deviation. ∗∗∗ Bold values are significant values according to Student’s test: an estimated value xˆ with a standard deviation σ is significant if |xˆ | σ > 1.96  2. ∗∗

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value. This point is confirmed by the results provided in Table 1. The drift between 2001.0 and 2006.0 is 20 divided by 3.6 and the estimated drifts over the other time intervals are also reduced and even no more 10 significant. 0 Moreover, the stacking of the so computed weekly TRFs has been only carried out with the discontinuities –10 corresponding to Arequipa’s earthquake. Indeed, most of the discontinuities identified in the results provided –20 by the strategy WOB disappear when SLR data are corrected with the range biases computed with the tem–30 1994 1996 1998 2000 2002 2004 2006 poral de-correlation method. These discontinuities are Fig. 5 Height residual time series (in mm) of the SLR station clearly linked to range biases. of Zimmerwald (7810), in Switzerland, resulting from the stackThis is well illustrated by the time series shown on ing of the weekly solutions provided by the strategy WOB. The Fig. 8. Finally, it is worth noting that a stacking of the ITRF2005 SLR discontinuity file has been used for the stacking weekly solutions provided by the strategy WB, which has been carried out with the full ITRF2005 disconTable 2 Epochs of the nine discontinuities detected in the height tinuity file, has produced the same results than those residual time series of Fig. 5 and of the corresponding instru- presented here. mental change reported in our on-line Data Base (see text) when it exists Discontinuity number

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4 Conclusions

From all the results of this study, we can firstly conclude that SLR biases really must be rigorously taken into account in any data reduction. Indeed, the ILRS strategy adopted to derive the official combined solution for the ITRF2005 computation is close to the first approach studied here (see Luceri and ∗ The format is yr:doy. Pavlis (2006)) as weekly range biases per satellite are only estimated for a few stations over given time equal) to the epochs of the detected discontinuities periods. As a consequence, it seems that the piece-wise shown on Fig. 5. For this particular discontinuity, behavior of the ILRS solution scale factor time series it must be noted that, when an instrumental change derived w.r.t. ITRF2000 for the ITRF2005 analyses occurs, the SLR station can be out of work for a couple (Altamimi et al. (2007)) may be mainly due to an of weeks and, as a consequence, the reported epoch of inappropriate handling of these biases. We have shown that the temporal de-correlation method provides change may be inaccurate. Figure 6 shows the weekly translation and scale fac- regular scale factor time series. This method may still be improved. Indeed, it is tor time series resulting from the stacking of the TRFs obviously limited for stations for which range biases computed with the strategy WB. We note that the TZ drift is reduced (its value is provide a poor stability. In this case, the time interval 0.5 ± 0.04 mm/yr). It seems that range biases can over which biases are estimated should probably be influence not only the scale factors but also the third reduced. The use of the data of other satellites (like translations. Indeed, Fig. 7 only shows the corre- STARLETTE and STELLA, for instance) should sponding scale factor time series and these time series provide more accurate positions for these stations are clearly more regular than the previous ones (see and, as a consequence, more reliable range bias Fig. 5). The RMS of the time series is 3.7 mm, value values. Furthermore, a very small number of stations which is significantly lower than the previous 5.2 still provide position residuals with discontinuities.

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Fig. 8 Height residual time series (in mm) of the SLR station of Zimmerwald (7810), in Switzerland, resulting from the stacking of the weekly solutions provided by the strategy WB. Only Arequipa’s station discontinuities have been applied for the stacking

These discontinuities should be detected with the temporal de-correlation method applied on running variable time intervals. And, we should check that all instrumentation changes are always reported in

dedicated files. Finally, on the other hand, we should also test if all the instrumental changes reported in our on-line DB really induce significant bias variations.

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References Altamimi, Z., P. Sillard, and C. Boucher (2002). ITRF2000: A new release of the International Terrestrial Reference Frame for Earth science applications. Journal of Geophysical Research, 107, B10, doi:10.1029/2001JB000561. Altamimi, Z., X. Collilieux, J. Legrand, B. Garayt, and C. Boucher (2007). ITRF2005: A new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. Journal of Geophysical Research, 112, B09401, doi:10.1029/2007JB004949. Coulot, D. (2005). T´el´em´etrie laser sur satellites et combinaison de techniques g´eod´esiques. Contributions aux Syst`emes de R´ef´erence Terrestres et Applications. Ph.D.Thesis, Observatoire de Paris. Coulot, D., P. Berio, R. Biancale, S. Loyer, L. Soudarin, and A.-M. Gontier (2007). Towards a direct combination of space-geodetic techniques at the measurement level: methodology and main issues. Journal of Geophysical Research, 112, B05410, doi:10.1029/2006JB004336.

D. Coulot et al. Exertier, P., J. Nicolas, P. Berio, D. Coulot, P. Bonnefond, and O. Laurain (2004). The role of Laser Ranging for calibrating Jason-1: the Corsica tracking campaign. Marine Geodesy, 27, pp.1–8. Gambis, D. (2004). Monitoring Earth orientation using spacegeodetic techniques: state-of-the-art and prospective. Journal of Geodesy, 78, pp. 295–305. Luceri, C. and E. Pavlis (2006). The ILRS solution. ILRS report, http://itrf.ensg.ign.fr/ITRF solutions/2005/doc/ILRS ITRF2005 description.pdf. Marini, J.W. and C.W. Murray (1973). Correction of laser range tracking data for atmospheric refraction at elevations above 10 degrees. Report X-591-73-351, NASA GSFC. McCarthy, D. D. and G. Petit (2004). IERS Conventions. IERS Technical Note, 32, Verlag des Bundesamts f¨ur Kartographie und Geod¨asie, Frankfurt. Pearlman, M., J.J. Degnan, and J.M Bosworth (2002). The International Laser Ranging Service. Advances in Space Research, 30, 2, pp. 135–143. Sillard, P. and C. Boucher (2001). Review of algebraic constraints in Terrestrial Reference Frame datum definition. Journal of Geodesy, 75, pp. 63–73.

Status of the European Reference Frame (EUREF) Jo˜ao Agria Torres, Zuheir Altamimi, Claude Boucher, Elmar Brockmann, Carine Bruyninx, Alessandro Caporali, Werner Gurtner, Heinz Habrich, Helmut Hornik, Johannes Ihde, Ambrus Kenyeres, Jaakko M¨akinen, Hans v. d. Marel, Hermann Seeger, Jaroslav Simek, Guenter Stangl and Georg Weber

Abstract EUREF (www.euref.eu) is the IAG (International Association of Geodesy) Reference Frame Sub-commission for Europe integrated in Sub-Commission 1.3 – Regional Reference Frames, of Commission 1 – Reference Frames, of the IAG. The activities carried out by EUREF are related to the definition, realization and maintenance of the European Reference Frame. EUREF focuses on both the geo-spatial and the vertical components and has the participation of almost all the European countries. This paper presents an overview of the status and recent developments of EUREF core projects, which are aimed at upgrading European-wide geodetic reference systems to support both scientific and continental geo-referencing activities. Keywords EUREF · Regional Reference Frames · ETRS89 · GNSS permanent network · EPN · EUVN · EVRS2007 · UELN Jo˜ao Agria Torres Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Zuheir Altamimi Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Claude Boucher Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Elmar Brockmann Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Carine Bruyninx Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Alessandro Caporali Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany

1 Introduction EUREF (http://www.euref.eu) is the Reference Frame Sub-commission for Europe integrated in Sub-Commission 1.3 – Regional Reference Frames, Werner Gurtner Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Heinz Habrich Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Helmut Hornik Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany, e-mail: [email protected] Johannes Ihde Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Ambrus Kenyeres Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Jaakko M¨akinen Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Hans v. d. Marel Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Hermann Seeger Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Jaroslav Simek Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Guenter Stangl Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany Georg Weber Technical Working Group of EUREF, Secretary: DKG, Alfons-Goppel-Strasse 11, D-80539 M¨unchen, Germany

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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of Commission 1 - Reference Frames, of the IAG (International Association of Geodesy). The long-term objective of EUREF is the definition, realization and maintenance of the European Reference Frame, focusing on both the geo-spatial and the vertical components. EUREF’s mission is to provide the best possible unique and homogeneous reference system and respective realization, to be used Europe-wide in all scientific and practical activities related to precise georeferencing and navigation, Earth Sciences research and multidisciplinary applications. For that, EUREF makes use of the most accurate and reliable terrestrial and space-borne techniques available, and develops the necessary scientific background and methodology for the combination of heterogeneous measurements. The activities in support of these objectives must result in a set of high quality products and services. For this, EUREF focuses on a continuous innovation and on the developing user needs, as well as on the maintenance of an active network of people and organizations (Adam et al., 2001). Almost all the European countries participate in the activities. Furthermore, EUREF works in close cooperation with the pertinent IAG components as well as EuroGeographics, the consortium of the NMCA (National Mapping and Cadastre Agencies) in Europe. The present status of activities and the main accomplishments, in the period between the IUGG (International Union of Geodesy and Geophysics) General Assemblies held in Sapporo 2003 and Perugia 2007, are, namely: – Overview and organisation of EUREF; – Realization of the European Terrestrial Reference System 1989 (ETRS89) and the relationships with the International Terrestrial Reference System (ITRS) realizations; – The EUREF Permanent Network (EPN), its contribution to the maintenance of the European Terrestrial Reference Frame (ETRF) and the International Terrestrial Reference Frame (ITRF) and the related special projects; – Real-Time Activities and the development of the standards and the operational means to disseminate GNSS data via the Internet, in the context of the efforts within the IAG towards real-time data dissemination;

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– The European Vertical Reference System (EVRS), including the status of the Unified European Levelling Network (UELN) and of the European Unified Vertical Network Densification Action (EUVN DA); – The status of the adoption of the ETRS89 by the European countries and organisation as official systems since its definition in 1990 (EUREF, 1992); – Outreach, external liaisons and symposia; – Summary of results.

2 Overview and Organisation At the annual Symposium held in Bratislava (June 2004), the Terms of Reference (ToR) of EUREF were adopted. The ToR contain the description of EUREF, its objectives, activities, organization and the rules for membership according to the general rules expressed in the Statutes and By-laws of IUGG and, consequently, of IAG. The complete text can be found at http://www.euref iag.net/html/Overview of EUREF Terms of reference.html. The Chair represents EUREF and is assisted by a Secretary. The main tasks are the overall coordination and management, as well as facilitating the achievement of EUREF objectives. A fundamental element in the structure is the EUREF Technical Working Group (TWG), which acts as a scientific board, and has the task to govern the following activities: – to coordinate and develop the EPN; – to evaluate and classify results of GNSS campaigns as EUREF densification or extension; – to coordinate the actions for the realization of a European Height System; – to identify the actions for the continuation and development of EUREF; – to set up the working groups to run the projects defined by the plenary; – to prepare the recommendations for the EUREF plenary. At present the TWG is composed by 17 members. It met 11 times in the period 2003–2007. Information about TWG membership, meeting agendas, and various contributions are available at http://www.eurefiag.net/html/twg.html.

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3 The ETRS89

d X E Y Y – velocity vector in ETRS89 d X I Y Y – velocity vector in ITRS

3.1 Definition; Relation with the ITRS

The values for the rotation rates of the Eurasian plate since 1989 are presented in Table 1.

At the EUREF symposium held in Firenze, 1990, the following resolution was adopted (EUREF, 1992, Boucher and Altamimi, 1992): “The IAG Sub-commission for the European Reference Frame (EUREF) recommends that the system to be adopted by EUREF will be coincident with ITRS at the epoch 1989.0 and fixed to the stable part of the Eurasian Plate, and will be known as European Terrestrial Reference System 1989 (ETRS89)” (http://www.euref.eu/html/resolutions.html#Florence). As a consequence of this definition, and given its link with the ITRS, the ETRS89 realizations are characterized by a clear transformation formula between both systems. The different realizations of the ETRS89 are named ETRFyy which are linked to their corresponding ITRFyy realizations. The general relationship between the two systems is described in Boucher and Altamimi (2007) since the tectonic movement of the Eurasian plate can be described in relation to the ITRS. Both systems are related by a shift between ITRFyy and ETRFyy at epoch 1989.0 and a rotation which accounts for the motion of the stable part of the Eurasian plate since 1989. The general model is a 14 parameter transformation (Eqs. (1), (2) and (3) below), where most of the parameters are 0. For the positions: X E(tc ) = X I Y Y (tc ) + T Y Y + d R Y Y .X I Y Y (tc ). (tc − 1989.0)

(1)

X E(tc ) – position vector in ETRS89 at epoch tc X I Y Y (tc ) – position vector in ITRFyy at epoch tc T Y Y – translation vector from ITRFyy to ETRF89 tc – central epoch of observations d R Y Y – matrix containing the 3 rotation rates in ITRFyy, in the form ⎛ ⎞ 0 −d R3Y Y d R2Y Y (2) d R Y Y = ⎝ d R3Y Y 0 −d R1Y Y ⎠ −d R2Y Y d R1Y Y 0 For the velocities: d X E Y Y = d X I Y Y + d R Y Y .X I Y Y

(3)

Table 1 Rotation rates for the ITRFYY realizations YY

dR1 (mas)

dR2 (mas)

dR3 (mas)

89/90 91/92 93 94/96/97 00

0.11 0.21 0.32 0.20 0.081 ±0.021 0.054 ±0.009

0.57 0.52 0.78 0.50 0.490 ±0.008 0.518 ±0.006

−0.71 −0.68 −0.67 −0.65 −0.792 ±0.026 −0.781 ±0.011

05

The most recent values are derived from the ITRF2005 velocity field (Altamimi et al., 2007). It can be seen that for the 2000 and 2005 realizations, the accuracies of the parameters are presented as well, since they are derived from the computation of the ITRF itself. The transformation procedure is currently under discussion. A complete description of ETRS89, ETRFyy and procedures for computations in ETRS89 can be found at http://lareg.ensg.ign.fr/EUREF.

3.2 Realizations The ETRS89 was first realised by the European Terrestrial Reference Frame 1989 (ETRF89), the European sub-set of ITRF89 (SLR and VLBI sites). For the subsequent realizations, GPS campaigns were organised in order to constitute a network of high-precision geodetic reference sites. The first one was EUREF89, covering 92 sites all over Europe. For quality assessment purposes, the sites are classified as follows: – CLASS A (1 cm accuracy independent of epoch) permanent GPS observations – CLASS B (1 cm accuracy at a specific epoch) GPS campaigns since 1993 – CLASS C (5 cm accuracy at a specific epoch) The computation of the campaigns is performed according to specifications that can be summarized as follows (Boucher and Altamimi, 2007):

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The number of stations providing hourly data has increased to 84%. In addition, 42% of the EPN stations also submit data to the International GNSS Service; 10 of them contribute to the TIGA (Tide Gauge Benchmark Monitoring) Pilot Project of the IGS. The “Procedure for becoming an EPN station” has Between 2003 and 2007 the following campaigns been completely revised and has been in effect since have been validated by the TWG and accepted as class December 2006. The most important changes concern B standard, increasing to 39 the number of campaigns the new requirements to submit a commitment letter validated since 1993: guaranteeing that the station will be operated following EPN guidelines for a minimal duration of 5 years – EUREF-Slovakia-2001 campaign in Slovakia; and the fact that all new EPN stations must have an an– EUREF-Pol-2001 campaign in Poland; tenna/radome with true absolute calibrations available – EUREF-Austria-2002 campaign in Austria; from the EPN CB. – EUREF-Hungary-2002 campaign in Hungary; In addition, the ‘Guidelines for EPN Stations and – EUREF-Armenia-2002 campaign in Armenia; – EUREF-GB-2001 (re-computation of the campaign Operational Centres’ have also been reviewed. The new guidelines were issued in order to improve the in Great Britain); – EUREF-NKG-2003 campaign in the Baltic coun- data flow within the EPN and to guarantee the availtries. Points from Latvia and Lithuania included in ability of the EPN data at the regional (European) level in two regional data centres: BKG (Federal Office of the data base; – EUREF-BG-2004 campaign in Bulgaria, combined Cartography and Geodesy, Germany) and OLG (Space with the EUREF-BG92/93, previously accepted in Research Institute, Department of Satellite Geodesy Austrian Academy of Sciences, Austria). 1996. Detailed information about the EPN can be found at the web site of the EPN Central Bureau (http://epncb.oma.be/). 4 EUREF Permanent Network (EPN) – compute the coordinates in ITRS at the central epoch of the observations using the most recent ITRFyy solution; – transform the coordinates in ETRS89 at the central epoch of the observations (see Sect. 3.1).

4.1 Description A key instrument in maintaining and providing access to the ETRS89 is the EUREF Permanent Network (EPN), created in 1995. Presently the EPN covers the European continent with continuous observing dual frequency GPS and GPS/GLONASS receivers operating under welldefined standards and guidelines that guarantee the efficiency of the EPN and the long-term quality of its products (Bruyninx and Roosbeek, 2007). The EPN constitutes the European contribution to, and densification of, the International GNSS Service (IGS), and as such it strives for complete consistency with the IGS standards and models: IGS orbits and ERPs (Earth Rotation Parameters) are used for all EPN processing and the same models are used for the antenna phase centers of both satellites and receivers. The total number of EPN stations comprises now more than 200, and 37 of these stations provide both GPS and GLONASS data.

4.2 Analysis and Reference Frame Realization Sixteen Analysis Centers each process a sub-network of the EPN following the rules and guidelines set up by the EUREF Technical Working Group. The Analysis Centres submit weekly free-network coordinate solutions to the EPN Combination Centre which combines the individual sub-network solutions into one official EUREF coordinate solution including all tracking stations. This solution is tied to the same ITRFyy as the IGS orbits. Since GPS week 1303 (December 2004), the minimal constraint approach is used to tie the weekly solution to the ITRFyy. Starting in GPS week 1400, the IGS orbits are aligned to the IGS05 which is the IGS realization of the ITRF2005 (Altamimi, 2006); at the same time the IGS replaced its relative antenna phase center models with absolute models. EUREF followed these procedures and switched to absolute antenna phase center models simultaneously

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Fig. 1 Calibration types of the EPN stations: A1 – black; A2 – green; B – yellow; No calibration available – red

with the IGS. Individual antenna calibrations are used where available. The EUREF solutions, starting GPS week 1400, are tied to the IGS05. For a further discussion on the consequences see Bruyninx et al. (2006). The EPN stations are classified as follows (see Fig. 1) with respect to the antenna phase center models (Bruyninx and Roosbeek, 2007):

– Time Series monitoring Project

The general task of the Time Series Analysis Special Project (TSA SP) set up in 2000 is to promote the use of the EPN products for geophysical studies. Based on the periodically computed cumulative solution of the EPN combined weekly SINEX product, the TSA SP maintains a database of the station coordi– Type A: absolute phase center models based on nate offsets and outliers, estimates the most up-to-date coordinate and velocity solutions and performs noise robot calibrations and harmonic analysis of the time series (Kenyeres and – A1: Individual calibrations Bruyninx, 2004). – A2: Type calibrations The TSA SP contributed to the ITRF2005 by – Type B: Absolute phase centre model based on rel- providing the offset and outlier database of the EPN stations. After the release of ITRF2005 including ative model its regional densification, the regularly updated EPN coordinate and velocity solution computed by the TSA SP is considered as official for EPN stations. All 4.3 EPN Projects results are displayed at the EPNCB web-pages. – Troposphere Project To stimulate the multi-disciplinary use of the EPN, EUREF has created different projects based on the EPN infra-structure:

The project goal is to derive tropospheric (zenith total delay) parameters as part of the estimation.

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Internet. Under this project the transport protocol Ntrip (Networked Transport of RTCM via Internet Protocol) was developed. In September 2004 (Weber et al., 2005) Ntrip has been included in the standards of RTCM (Radio Technical Committee for Maritime Services). EUREF-IP established an IP address for its Ntrip Broadcaster service at http://www.euref-ip.net/home. The total number of world-wide Ntrip Broadcaster – EUREF-IP Project installations known is about 85. The total number of reference stations available via Ntrip technology Another project based on the EPN structure is amounts to approximately 1700; 52 of them are EPN EUREF-IP (IP for Internet Protocol), with the goal to stations, which is about one quarter of the EPN stations collect and disseminate GNSS data in real-time via the (see Fig. 2). The basic task within this activity is to produce a combined troposphere solution with input from the individual troposphere solutions of all Analysis Centers, which contribute to the coordinate solution. A ‘rapid’ combination derived to a given time contributes to the global IGS combined troposphere product (S¨ohne and Weber, 2004).

Fig. 2 EPN stations delivering real-time data: RTCM – green; raw data – red; RTIGS – blue

Status of the European Reference Frame (EUREF)

EPN stations use Ntrip to stream their real-time GNSS data providing full carrier phase data in raw or RTCM format; EUREF also provides the necessary tools to convert the streams to RINEX. As can be seen in Fig. 2, the RTIGS format developed by the IGS is used as well. The current EUREF-IP efforts focus on developing a Monitoring/Notification system to reach and maintain a professional level of service availability, develop Ntrip towards full HTTP compatibility, introduce UPD as an additional data transport option, and encourage more EPN station operators to participate in EUREFIP with real-time raw or RTK data. Recognizing that the project EUREF-IP reached a high level of performance and acceptance, including stations outside Europe, and the importance of realtime activities within the IAG, it was decided at the EUREF 2007 Symposium to turn this project into a routine service (EUREF, 2007). This new routine service is supported by a White Paper developed by the EUREF TWG available at http://www.epncb.oma.be/ organisation/guidelines/ EPNRT WhitePaper.pdf. In addition, guidelines have been developed for – Reference stations – NTRIP Broadcasters – High-rate RINEX Data Centers

Fig. 3 New data for UELN – situation in May 2007

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5 European Vertical Reference System The definition of the European Vertical Reference System 2000 (EVRS2000), including a European Vertical Datum and the realization procedure, is being revised, considering that the progress in global gravity models will soon make possible the realization of EVRS in relation to a World Height System. The vertical datum of the EVRS2000 was realized by one levelling pillar in the Netherlands which was related to the zero level of the Amsterdam Pile (NAP). The EVRS2007 definition considers that the datum shall be realized by various datum points distributed over Europe. These points must be distributed in such a way that the level of EVRS2000 is kept (Ihde et al., 2007). The heights are fit to the EVRS2000 solution by introducing the results of UELN 95/98 (Unified European Levelling Network) of the datum points in the free adjustment (Sacher et al., 2007). The UELN 95/98 developed in recent years by integration of the levelling networks of Estonia (1999), Latvia (1999), Romania (2000), Lithuania (2001) and Bulgaria (2002) and by replacement of old data by complete national first order levelling networks of Switzerland (2002), The Netherlands (2005), Finland (2005), Norway (2005) and Sweden (2005). The recent situation concerning the delivery of new data is shown in Fig. 3. Contacts are being established

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with Russia for the inclusion of new levelling data in height systems in Europe, related with the different tide the Baltic area. Further information about the European gauges, into a common European height reference sysVertical Reference System and UELN can be found at tem, with decimeter accuracy. The comparison between the EUVN GPS-levelling http://crs.bkg.bund.de/evrs/. derived heights and the adopted European geoid model (EGG97), showed however some non-systematic effects. 6 European Unified Vertical Network – To investigate these effects, the EUVN DensificaDensification Action tion Action (EUVN DA) was initiated in 2003. The purpose is to collect accurate and dense information In 1997 a GPS campaign consisting of 196 sites collo- on levelling and ETRS89 coordinates selected from cated at nodal points of the UELN and near tide gauges the national networks and constituting a homegenous was established. This set of points constitutes the Euro- GPS/levelling network and database. pean Vertical GPS Reference Network (EUVN97) and This information will allow for the realization of provides the link between the vertical and geo-spatial an accurate height reference surface (fitted geoid) as components of the frame. well (Kenyeres et al., 2007). Until May 2007 a total This project allowed for the computation of param- of 1434 GPS/levelling points were collected all over eters to transform the heights of the different national Europe (see Fig. 4).

Fig. 4 EUVN DA – situation in May 2007

Status of the European Reference Frame (EUREF)

7 Adoption and Promotion of the ETRS89 An important consequence of the definition and realization of the European reference system ETRS89 is its adoption by Eurocontrol and the on going process of adoption by several NMCA. Furthermore, there is a recommendation by the European Commission to adopt ETRS89 as the geodetic datum for geo-referenced information and promote the use of ETRS89 within member states. In a joint cooperation between EUREF and EuroGeographics, the description of national coordinate reference systems (CRS) in Europe and the transformation parameters between CRS and ETRS89 for practical purposes, are available at http://crs.bkg.bund.de. The published information follows the ISO 19111 Spatial Referencing by Coordinates standard (Ihde et al., 2001). In 2005 it was decided to continue to promote the use and adoption of the ETRS89 and to collect the most accurate and complete information on this subject. Consequently, a survey was conducted jointly by EUREF and EuroGeographics among 41 NMCA. From the 41 countries contacted, 28 answered the questionnaire representing about 68% of the total. The 3 different situations are as follows: – 2 will not adopt (7% of the answers); – 5 will adopt in the near future (18% of the answers); – 21 have already adopted (75% of the answers). The countries that stated that they will not adopt the ETRS89 are Luxembourg and Turkey. On the other hand, since the realization of the questionnaire, one of the countries that announced the adoption of the ETRS89 in the near future has already adopted, increasing to 22 the number of countries that announced the adoption of this system.

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and projects; its main contents are information about the EUREF structure and documentation related with the symposia and TWG meetings. The existing liaison with EuroGeographics continued through its Expert Group on Geodesy (ExGG) and was recently upgraded by the establishment of a Memorandum of Understanding (MoU) between both organizations. Another MoU was signed with EUMETNET, a network of 21 European national meteorological services, with the purpose of creating the conditions to facilitate the data exchange and to promote the increase of cooperation, for the benefit of both the meteorological and geodetic communities. There is a natural relationship between the work developed by EUREF and European initiatives related to geo-spatial information. One of the most important ongoing actions is INSPIRE, aiming at the establishment of an infrastructure for spatial information in Europe that will help to make spatial or geographical information interoperable. EUREF already registered as one of the Spatial Data Interest Communities (SDIC) and proposed experts for the drafting teams in its area of expertise. EUREF is also represented as an Associate Member at the IGC (International Committee on GNSS) of the United Nations. In the last four year period symposia took place at Bratislava (Slovakia) in June 2004, at Vienna (Austria) in June 2005, at Riga (Latvia) in June 2006 and at London (UK) in June 2007. These meetings are usually attended by about 120 participants from more than 30 countries in Europe. The EUREF web portal contains the contributions presented at the symposia, as well as the full set of resolutions of all symposia since 1990 (http://www.eurefiag.net/html/symposia.html).

9 Summary 8 Outreach and External Liaisons; Symposia Besides the web portal http://www.euref-iag.net, a new address was created in the “eu” domain, http://www.euref.eu. Both addresses coexist and give access to a portal that links to all the EUREF structures

EUREF is continuously developing activities related to the establishment, maintenance and promotion of the ETRS89 (European Terrestrial Reference System) and EVRS (European Vertical Reference System). The ETRS89 has been recommended for adoption by several European organizations and is used by the majority of the European countries.

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The realization of the EVRS2007 with new data sets and the possibility to align EVRS to a World Height System is in a very advanced stage. The UELN covers presently almost all the European countries. The EUREF Permanent Network maintains and provides access to ETRS89 using post-processing as well as real-time access. These activities reached the routine status and enable the contribution of EUREF to many projects in different fields of the geo-sciences requiring real-time GNSS data. EUREF is now-a-days a key organization for the support of a geodetic infra-structure in Europe and will certainly be an important partner in the implementation of the Global Geodetic Observing System (GGOS) thanks to the broad network of people and institutions that contribute to its work.

References Adam, J., W. Augath, C. Boucher, C. Bruyninx, A. Caporali, E. Gubler, W. Gurtner, H. Habrich, B. Harsson, H. Hornik, J. Ihde, A. Kenyeres, H. Marel, H. Seeger, J. Simek, G. Stangl, J. Torres, G. Weber, G. (2001). Status of the European Reference Frame–EUREF. In: Vistas for Geodesy in the New Millenium, Proceedings of the IAG Scientific Assembly held in Budapest, 2001. IAG Symposia Volume 125, Springer-Verlag, Berlin, 2002. Altamimi Z. (2006). ITRFmail 66, http://itrf.ensg.ign.fr/ Altamimi Z., X. Collilier, J. Legrand, B. Garayt, C. Boucher (2007). ITRF2005: A New Release of the International Terrestrial Reference Frame Based on Time Series of Stations Positions and Earth Orientation Parameters. J. Geophys. Res., 112, doi: 10.1029/2007JB004949. Boucher C, Z. Altamimi (1992). The EUREF Terrestrial Reference System and its First Realizations. EUREF Meeting, Bern, Switzerland. Boucher C, Z. Altamimi (2007). Specifications for reference frame fixing in the analysis of a EUREF GPS campaign, http://lareg.ensg.ign.fr/EUREF/memo.pdf. Bruyninx C., F. Roosbeek (2007). EPN Status and New Developments. In: Proc. of the EUREF Symposium held in London, 6–9 June 2007, Mitteilungen des BKG, Frankfurt a.M., in press.

J.A. Torres et al. Bruyninx, C., Z. Altamimi, C. Boucher, E. Brockmann, A. Caporali, W. Gurtner, H. Habrich, H. Hornik, J. Ihde, A. Kenyeres, J. M¨akinen, H. Marel, H. Seeger, W. S¨ohne, J. Simek, G. Stangl, J. Torres, G. Weber (2006). The European Reference Frame: Maintenance and Products. In: Proceedings of the GRF2006 Symposium held in Munich, 9–14 October 2006, in press. EUREF (1992). Report on the Symposium of the IAG Subcommission for Europe (EUREF) held in Florence, 28–31 May 1990. Ver¨off. d. Bayer. Komm. f. d. Int. Erdmessung; Astron-Geod. Arbeiten, Heft Nr. 52, 1992. EUREF (2007). Report on the Symposium of the IAG Subcommission for Europe (EUREF) held in London, 3–6 June 2007. Proceedings of the EUREF Symposium held in London, 6–9 June 2007, Mitteilungen des BKG, Frankfurt aM., in press. Ihde, J., C. Boucher, E. Gubler, J. Luthardt, J. Torres (2001). European Coordinate Reference Systems. Proceedings of the EUREF Symposium held in Dubrovnik, 16–18 May 2001, Mitteilungen des Bundesamtes f¨ur Kartographie und Geod¨asie, Band 23, Frankfurt a.M. Ihde, J., J. M¨akinen, M. Sacher (2007). Conventions of the European Vertical Reference System 2007. In Proceedings of the EUREF Symposium held in London, 3–6 June 2007, Mitteilungen des BKG, Frankfurt a.M., in press. Kenyeres A., C. Bruyninx (2004). Monitoring of the EPN Coordinate Time Series for Improved Reference Frame Maintenance. In: GPS Solutions, Vol. 8, No. 4:200–209. Kenyeres A., M. Sacher, J. Ihde, H. Denker, U. Marti (2007). EUVN Densification Action: Prepared for Closing. In: Proceedings of the EUREF Symposium held in London, 6–9 June 2007, Mitteilungen des BKG, Frankfurt a.M., in press. Sacher M., J. Ihde, R. Liebsch (2007). Status of EVRS2007. In: Proceedings of the EUREF Symposium held in London, 3–6 June 2007, Mitteilungen des BKG, Frankfurt a.M., in press. S¨ohne, W., G. Weber (2004). Status Report of the EPN Special Project “Troposphere Parameter Estimation”. Proceedings of the EUREF Symposium held in Bratislava, 2–5 June 2004, EUREF Publication No. 14, pp. 64–71. Mitteilungen des Bundesamtes f¨ur Kartographie und Geod¨asie, Band 23, Frankfurt a.M. Weber G., D. Dettmering, H. Gebhard (2005). Networked Transport of RTCM via Internet Protocol (NTRIP). In: Proceedings of IAG Symposia ‘A window on the Future of Geodesy’, F Sanso (ed) Vol. 128, Springer, pp. 60–64.

Reprocessing of a Regional GPS Network in Europe ¨ C. Volksen

Abstract The Bavarian Committee for International Geodesy (BEK) has been a Local Analysis Centre (LAC) for the EUREF Permanent Network (EPN) since 1995. The analysed network covers the northern part of the Mediterranean region with its diverse tectonic features. The processing strategies, the realizations of the global reference system and models for correcting different physical effects have been improved and changed during the past 10 years of analysing GPS data. All these modifications had a significant impact on the coordinate estimation. It is therefore not surprising that time series of coordinate changes always reveal such modifications and may hide tectonic events. A group from the Technische Universit¨at M¨unchen (TU Munich), Technische Universit¨at Dresden (TU Dresden) and the GeoForschungsZentrum Potsdam (GFZ Potsdam) made an effort to reprocess a global GPS network to derive consistent coordinates of stations as well as orbits and earth rotation parameters (ERPs) in the IGb00 applying absolute antenna phase centre variations (PCV) for the antennas. The reprocessed orbits and ERPs have been used at the BEK for a reprocessing of 10 years of GPS data, using a network of roughly 65 sites. This paper will focus on the improvement achieved by applying consistent products, strategies and models.

1 Introduction

The EUREF Permanent Network (EPN) has been established in the end of 1995. The main aim was and still is the realisation of the European Terrestrial Reference System, known as the ETRS89. The EPN consists today of almost 200 GNSS sites that observe the satellite signals continuously. The data of these sites are processed by more than 16 Local Analysis Centres (LAC) that estimate weekly solutions following the standards defined by the International GNSS Service (IGS). The weekly solutions then are forwarded to the EPN Combination Centre, which is currently located at the Bundesamt f¨ur Kartographie und Geod¨asie (BKG) in Frankfurt (Germany). The Combination Centre stacks the individual normal equation files that are provided by the different LACs and estimates one set of coordinates for the specific GPS week. Naturally, the standards for analysing the data have changed over the past years. As soon as new models became available and have successfully been tested they were implemented into the analysis. Just to name some examples: the elevation cut-off angle has been lowered twice, ocean loading models have been newly included, the ambiguity resolution strategies have improved and last but not least better models for the correction of the antenna phase centre became available. The later are used since GPS week 1400 in the analyKeywords Reprocessing · Regional GPS network · sis of the IGS and EPN. Above all, the impact of new realisations of the International Terrestrial Reference Velocities · ITRS · Plate tectonics System (ITRS) became visible in the coordinate estimates. Different models, different strategies and different realisations of the ITRS had their own impact C. V¨olksen on the estimated coordinate. These effects are visiBayerische Kommission f¨ur die Internationale Erdmessung ble in the time series of the EPN stations and clearly Bayerische Akademie der Wissenschaften Alfons-Goppel-Str. 11, 80539 M¨unchen, Germany show that they are not consistent. Only a homogeneous M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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and denser over the years and now consists of almost 80 stations. But only the coordinates of 65 sites officially contribute today to the weekly EPN solution. The difference between contributing and analysed sites is justified by the approach of distributed processing. The EPN central bureau assigns to each new station three to five LACs, which will process the GNSS data of this station and submit officially the coordinates. Those LACs interested in including this station in its own analysis can still analyse the data, but have to remove the coordinates from the official solution. The network analysed in this reprocessing is shown in Fig. 1. It covers the southern part of the EPN completely from the Azores to Yerevan in the Republic of Armenia. This network includes several tectonic features that characterize the Mediterranean. The main forces are a result of the plate tectonics of the African, Eurasian and Arabian plate. But there is also the rather complex tectonic structure in the eastern part of the Mediterranean with several deformation zones and the Anatolian Plate. The network used for the reprocessing agrees quite 2 The GPS Network well with the sub-network of the BEK, which is processed for the EPN analysis. But it has also been exThe BEK operates a Local Analysis Centre for the tended to the north with four additional sites: Onsala in EPN. The processed network is a part of the complete Sweden (ONSA), Brussels in Belgium (BRUS), PotsEPN network. This sub-network covers the Mediterdam in Northern Germany (POTS) and Borowiec in ranean with a focus on the Alps, Italy and the Iberian Poland (BORI). These stations have been included for Peninsula. The BEK started the analysis in the end of several reasons: 1995 with 18 sites only. The network became larger

reprocessing would allow the generation of a consistent set of coordinates and their corresponding time series. The task of reprocessing the global IGS network has been carried out by a research group from the GFZ Potsdam, the TU Dresden and the TU M¨unchen, Steigenberger et al. (2006). Data from 1994 until 2005 were processed in order to estimate orbits, ERPs and coordinates in the IGb00, which is designed to be consistent with the ITRF2000 and based on GPS observations only. This analysis has been carried out applying strictly the same standards for the entire time span. Especially the use of absolute phase centre corrections for the receiver and satellite antennas should be mentioned here. These orbits and ERPs were made available to the BEK in order to reprocess a GPS network on a regional scale. The main objective is the estimation of consistent coordinates and velocities over a time span of 10 years.

Fig. 1 The reprocessed GNSS network. Stations with a history of less than 3 years were excluded from the estimation of the site velocities

Reprocessing of a Regional GPS Network in Europe

r r r

they reach further into the stable part of the Eurasian Plate, their coordinates and velocities are well known in the IGb00 due to the long existence of the stations and they also belong to other regional networks, which makes a connection with this set of stations feasible.

Additional stations were included in the analysis, which have not been processed in the past but cover the region of interest. Finally, a few sites have been selected that do not belong to the EPN, because they are located in tectonically interesting regions like the Alps or close to plate boundaries.

3 Processing Before the reprocessing could actually start it was necessary to collect all the available data for the period 1996–2005. A large quantity of data was still available at the BEK. But since the network used in the reprocessing has been enlarged and new stations were included some data were missing on the local data storage. Therefore the missing data had to be retrieved from the two main data centres of the EPN, the BKG in Germany and the Space Research Institute, Department of Satellite Geodesy – Austrian Academy of Sciences – Austria But also the central bureau of the EPN made older data available. The main objective of the reprocessing was of course to maintain consistency. Therefore it was necessary to process the 10 years of data consistently. This can only be achieved by taking the following points into account:

r r r

apply identical and well accepted models/standards, use consistent orbits and ERPs and use the same analysis strategy for all sessions.

Obviously, orbits, ERPs and coordinates are also dependent on the models. Therefore it is not possible to estimate coordinates applying absolute antenna Phase Center Variations (PCV) while the orbits have been estimated using relative PCV. Also here consistency had to be enforced. The estimation of precise orbits and ERPs based on this regional network is of course not promising due to the small size of the network. It was therefore only

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logical to use the reprocessed orbits and ERPs based on globally distributed GPS sites, which were made available by the GFZ and TU Dresden. These datasets had also the advantage that they were estimated applying absolute PCV for the satellite and receiver antennas. The use of absolute PCV for the receiver antennas has greatly influenced the GPS community since a new method for the field calibration has been developed, Menge et al. (1998) and later improved based on an automatic robot calibration by the Institut f¨ur Erdmessung (IfE, Leibniz Universit¨at Hannover) and the GEO++ company, W¨ubbena et al. (2000). Absolute PCV for the satellite antennas have been estimated in another approach based on the absolute robot calibration results for the ground antennas, see Schmid and Rothacher (2003) and Schmid et al. (2005). The absolute PCV values used in this study are partly based on a dataset released by the IGS, which was compiled from (a) absolute robot calibration values as made available by GEO++ and IfE, (b) relative field calibrations that were converted to absolute PCV, based on the absolute PCV of the reference antenna Dorne Margolin T (AOAD/DM T), and (c) absolute satellite antenna PCV. Wherever it was possible the IGS dataset was supplemented by absolute robot calibration values for the receiver antennas as they are available from the GEO++ database. Receiver antennas with an unknown radome type have been treated as they would carry no one at all, well knowing that this causes an offset in the height component. This is specifically true for the IGS site ONSA in Onsala (Sweden), the EPN site HFLK close to Innsbruck (Austria) and several other sites. The processing itself was carried out with BERNESE 5.0 in double difference mode, see Dach et al. (2007). Phase and code observations of one day were processed with an elevation mask of 3◦ and elevation dependent weighting (w= cos2 z) was introduced. One still has to keep in mind that the elevation mask set in the GPS receivers changed in the past and was at the beginning of the EPN not as low as 3◦ . In the early years of the EPN some receivers were tracking the GPS satellites only above an elevation of 10◦ . This has only changed in the past years and today site operators are requested to set the elevation mask to 0◦ or at least 5◦ . The tropospheric zenith delay was estimated based on the wet Niell mapping function without any a priori model. These parameters are estimated for each hour, while a

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tropospheric gradient is estimated once per day. The deformation of the crust caused by ocean loading was corrected with the FES2004 model, Letellier (2004). The model parameters for each site were provided by the ocean tide loading provider, see Scherneck (1991) (http://www.oso.chalmers.se/∼loading). The ambiguity estimation was carried out in a separate process. Using the Quasi-Ionosphere-Free (QIF) ambiguity resolution strategy, suitable baselines were selected and the ambiguities resolved, Mervart (1995). For this process an elevation mask of 10◦ was selected to avoid noisier data at low elevations, which would hamper the ambiguity estimation. Other ambiguity resolution strategies were not considered since the length of the selected baseline hardly grew above 1000 km. After resolving the ambiguities the coordinates were estimated using the ionosphere free linear combination L3. The entire network was processed in clusters of 4–6 sites and later combined on the normal equation level. This procedure is advantageous since the amount of computation time is reduced significantly due to the reduced amount of memory and processing steps. For the datum definition of the final daily solution (normal equation) a minimum constraint condition condition was used (based on translations and rotations, no scale), which was based on all the IGb00 sites in the network. The normal equation system, which contained the complete information of this last run, was then stored for further processing. Finally a reduced normal equation system was generated and stored that contained only relevant information concerning the position of the sites. Applying this strategy roughly 3650 daily solutions were estimated.

4 Analysis This part of the paper deals with the coordinate and velocity estimation based on daily solutions. The reduced normal equation files of the 10 years of GPS data were used in one estimation process with the BERNESE 5.0 program ADDNEQ2. The program allows the estimation of coordinates for a certain epoch and the estimation of linear velocities. A priori information concerning antenna changes, receiver replacement, seismic events or generally bad station information causing possibly a bias in the position was not included in the first run. This first step was

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meant to detect gross errors of certain stations and to eliminate these blunders form the next estimation processes. After removing the gross errors a visual check of the position residuals was carried out. This allowed still the detection of additional outliers and as well offsets in the time series were identified. The offsets were then compared with the individual station log files, which maintain a history of the station set-up namely concerning changes in the equipment and the antenna height. In many cases offsets in the time series, especially in the height component, appeared at the same time as the antenna has been replaced. Biases caused by the replacement of receivers were not detectable. Apparently the introduction of absolute PCV does not completely overcome the problem of sudden position changes caused by antenna replacements. According to W¨ubbena et al. (2006) one has to consider a so-called near-field effect that is caused by multipath interferences induced between reflectors located in the close vicinity of the antenna and the antenna itself. There are also other electromagnetic phenomena like diffraction and antenna imaging effects that contribute to this near-field effect, which cause a systematic bias especially in the height component. Obviously this is still the reason for a number of offsets in the times series of the position changes that appear at many sites, although absolute PCV are used. Discontinuities that can be linked to the change in the elevation mask were not detected. It seems that the absolute PCV describe the antenna PCV much more realistic than the relative PCV. This is specifically true for elevations below 10◦ . The position residuals of the station Palma de Mallorca in Spain show a not yet identified event on 18th of August 2003 (see Fig. 2). A change of equipment is not recorded for this date and it is also not known that an earthquake took place in the vicinity of the station. An explanation for this sudden change of position is purely speculative. The rather horizontal position change might indicate a movement of the pillar, which might have local causes. Nevertheless, a sudden change of position is also not clearly visible. It seems rather that the movement was carried out over a longer period, as it can be seen especially form the east component. Similar patterns are known from so called slow earthquakes as they occur at several places in the world. Whenever a discontinuity in the time series of certain stations was detected a new station identifier

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61 Table 1 Comparison of coordinates and velocities between this study and the IGb00 Positions North Station BOR1 BRUS NICO ONSA POTS VILL WTZR

Fig. 2 Position residuals of the GPS station Mallorca (MALL): The velocity components on the daily positions are already removed

was assigned to the subsequent data. This allowed the estimation of a new set of coordinates for different periods of the time. In this study it was assumed that the station velocity before and after the offset was identical, therefore the velocities before and after the jump were constraint. Because of its specific characteristics, GPS is in general an interferometric technique, it is not possible to estimate precise absolute positions and velocities. In order to estimate coordinates and velocity with a relative system like GPS it was necessary to define a minimum constraint condition based on GPS-sites with well known positions and velocities. Furthermore, it was important to use sites with antennas that were well established in the IGS. Most of the relative calibration models were based on the Dorne Margolin T (AOAD/M T) antenna type. Therefore it was advantageous to use these antennas for the minimum constraint condition, because they show directly the difference between relative to absolute PCV. According to this definition only four sites were suitable for constraining the coordinates and velocities. These were the sites Nicosia (NICO),

Velocities East

Height

North

−2.7 −2.1 −0.4 −0.4 0.2 −0.7 0.3

−0.5 0.8 0.4 −0.3 −0.4 0.2 −0.2

[mm] 1.6 −2.6 −0.5 1.2 1.0 −0.4 0.6

0.1 2.1 −1.6 −1.2 −0.7 2.4 −0.2

East

Height

[mm/a] −0.3 −0.6 −0.1 −0.1 0.2 −0.1 0.2

1.2 −0.1 0.9 0.6 0.3 0.1 −0.5

Potsdam (POTS), Villafranca (VILL) and Wettzeil (WTZR). They were equipped during the entire time with AOAD/M T antennas and never changed the antenna height. This was mandatory in order not to introduce a new bias resulting from changes in the near-field of the antenna. New coordinates and linear velocities were estimated applying these constraints. Table 1 shows the difference between the a priori coordinates and velocities given in the IGb00 and newly estimated values from this reprocessing for selected sites only. The corrections for the coordinate components are in the order of 0.5–2 mm, while the corrections for the velocity components are generally better then 0.5 mm per year. The corrections for the station coordinates are in many cases much larger if the antenna used on a specific station carries a radome, which has not been corrected prior to GPS week 1400 in the IGS and EPN analysis. The transition from relative to absolute PCV might cause height changes in the order of up to 30 mm depending on the antenna type, the used radome and the satellite sky distribution, V¨olksen (2007). Typical values are in the order of 10–15 mm in the height component. The velocity component itself is not affected by the different PCV models as long as they have consistently applied and antenna changes were taken into account as offsets in the time series. This is logical since the antenna was always moving with the same bias that was linked to the antenna type.

5 Tectonics The network covers geophysical interesting areas, because three tectonic plates and several deformation zones are covered. This paper will only focus on

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the Iberian Peninsula and on an area covering the Alps, Italy and the Balkan. The eastern part of the Mediterranean is left out due to the sparsely distributed GPS sites, although knowing that this area is extremely interesting. Vertical deformations were not especially treated here since they do not show a specific pattern for certain regions. The horizontal deformation velocities were transformed from the IGb00 into the ETRS89, Boucher and Altamimi (2007). The ETRS89 is defined as connected to the stable part of the Eurasian Plate and as being identical to the International Terrestrial Reference System (ITRS) at epoch 1989.0. The concept behind this definition is based on the idea to maintain coordinates that do not change or at least show only minimal changes due to plate tectonics. In this case it has the advantage that it shows only the deformation relative to the stable Eurasian plate. This transformation works also for this study, because the stations located in the central part of the Eurasian Plate did not show any significant velocities and were clearly moving with the plate. Figure 3 shows the horizontal deformations relative to the stable Eurasian plate on the Iberian Peninsula. Confidence regions for the velocities are not shown on

purpose. The formal errors for the velocity components are of the order of 0.002–0.005 mm/a. Comparisons between the standard deviation of the positions and the residuals of the time series indicate that the formal erros are by a factor of 40 too small. It is therefore reasonable to assume standard deviation for the velocity components in the order of 0.1–0.2 mm/a. Stations velocities are usually directed to the west. The picture shows very small velocities for stations close to the Pyrenees, the eastern coast of Spain and Central Spain. Velocities of the stations A Coruna (ACOR) and Cadaques (CREU) point into the opposite direction compared to the general trend. Probably they are moving according to some still unknown local effects. The velocities of the stations on the Iberian Peninsula increase further to the south. Obviously the Iberian Peninsula moves differently compared to the stable part of the Eurasian Plate. This becomes even clearer looking at the velocities of the station in the southern part of Spain (LAGO, SFER and CEUT). They move still in westerly direction with an velocity of approximately 4–5 mm/a. This velocity is similar to the velocity of the station Rabat (RABT) on the African Plate. The white arrows in the figure describe velocities estimated

Fig. 3 Station velocities on the Iberian Peninsula given in the ETRS89

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Fig. 4 Station velocities on the Italian Peninsula and the Balkan given in the ETRS89

from the plate velocity model APKIM, Drewes (1998), relative to the Eurasian Plate motion. Of course only the velocities for the African plate are shown, since they are zero for the Eurasian plate. It appears that the motion of the sites close to the plate boundary is on both sites highly correlated and do not agree well with the plate motion model. This study rather indicates that the southern part of the Iberian Peninsula and the African Plate interact and do not move independently from each other. The velocity pattern of the Alps, Italy and the Balkan is shown in Fig. 4, again in the ETRS89. Several features can be seen. Stations in the northwestern part of this map reveal that their velocities agree quite well with the general motion of the Eurasian Plate, because they are close to zero. Also the stations LAMP on the island Lampedusa and the station NOTO in Sicily show a good agreement with the motion of the African Plate, which is indicated by the white arrows (APKIM). The motion of the Apulian block, also known as the Apulian Plate, becomes apparent. This block separates the Italian peninsula in north-south direction and reaches as far as to the north-western coast of the Balkan. The velocities of the station Dubrovnik (DUBR) and and Sarajevo (SRJV) even

indicate that the Apulian block reaches further into the mainland. The pattern of station velocities in the eastern part of the Alps is rather noisy and a clear structure cannot be seen. A very interesting deformation pattern can be seen in the southern part of the Balkan. The stations Ohrid (ORID), Sofia (SOFI) and Bucharest (BUCU) show a very similar but unexpected pattern. All three stations move southward with a rate of approximately 2.5 mm/a. This is rather surprising and indicates that there might be a yet unknown source of deformation, which needs further investigation. Similar results are reported by Noquet and Calais (2004) for the station Sofia.

6 Conclusions The analysis of 10 years of GPS data in the region of the Mediterranean has been carried out with the GPS analysis software BERNESE 5.0 in order to estimate consistent coordinates and velocities in the IGb00. While offsets caused by different realisations of the reference frame disappeared, the bias caused by the exchange of antenna types very often remained.

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Obviously the new absolute PCV as they were introduced in this analysis are not sufficient to eliminate all effects on the receiver antennas. The major candidates for introducing these offsets are near-field effects caused by multipath in the vicinity of the antenna. One has still to fear that coordinates estimated in this and all the current analysis are very often biased due to these near field effects. This will hamper of course the estimation of the true coordinates. Analysing the time series of coordinates will allow estimating the difference between two biases, but at the moment it is not possible to estimate the true unbiased coordinates. Only the estimation of the near-field effect itself would lead to the true position. The analysis of the velocities of this network revealed some new features in the Mediterranean. There is evidence that the Iberian Peninsula does not move as one unit together with the Eurasian Plate. It looks much more like a block that is locked between the central part of Europe and the African Plate and its movements are a result of this situation. It was also possible to show the motion of the so called Apulian Plate, which moves as one block towards the north-east. Surprising was the motion detected in the southern part of the Balkan, where three stations show almost an identical deformation signal. The origin is still unknown and needs further investigation. This paper describes only a small part of possible diagnosis. For the future it is planned to carry out more investigations in order to study the induced effects by different tropospheric models and also the high frequency effects by tides and other possible sources. Acknowledgments I am grateful for the reprocessed orbits and ERPs supplied by the GFZ Potsdam, Institut f¨ur Planetare Geod¨asie of the TU Dresden and the Institut f¨ur Astronomische und Physikalische Geod¨asie of the TU M¨unchen. Thanks are also expressed to Carine Bruyninx and Szabolcs Rozsa for their constructive comments in the review process. This work has been supported by the Federal Republic of Germany by the Akademienprogramm.

References Boucher, C., Z. Altamimi (2007). Memo: Specifications for reference frame fixing in the analysis of a EUREF GPS campaign. http://lareg.ensg.ign.fr/EUREF/memo2007.pdf Dach, R., U. Hugentobler, P. Fridez, M. Meindl (2007). BERNESE GPS Software Version 5.0. Astronomical Institute, University of Bern.

¨ C. Volksen Drewes, H. (1998). Combination of VLBI, SLR and GPS determined station velocities for actual plate kinematic and crustal deformation models. In: M. Feissel (Ed.): Geodynamics, IAG Symposia, Springer 1998. Kenyeres A. and Bruyninx C. (2004). Monitoring of the EPN coordinate time series for improved reference frame maintenance. GPS Solutions, Vol. 8, No 4, pp. 200–209. Letellier, T (2004). Etude des ondes de mar´ee sur les plateux continentaux. Th`ese doctorale, Universit´e de Toulouse III, Ecole Doctorale des Sciences de l’Univers, de I’Environnement et de l’Espace. Menge, F., G. Seeber, C. V¨olksen, G. W¨ubbena, and M. Schmitz (1998). Results of absolute field calibrations of GPS antenna PCV, in Proceedings of ION GPS-98, pp. 31–38, Nashville, TN. Mervart, L. (1995). Ambiguity resolution techniques in geodetic and geodynamic applications of the Global Positioning System. In Geod. Geophys. Arb. In der Schweiz, Vol. 53. Z¨urich, Switzerland. Noquet, J.-M., E. Calais (2004). Geodetic measurements of crustal deformation in the Western Mediterranean and Europe. Pure Appl. Geophysics., 161, doi: 10.1007/s00024003-2468-z. Scherneck, H-G. (1991). A parametrized solid Earth tide model and ocean loading effects for global geodetic baseline measurements. Geophys. J. Int., 106(3): 677–694. Schmid, R. and M. Rothacher (2003). Estimation of elevation dependent satellite antenna phase center variations of GPS satellites. J. Geod., 77, doi: 10.1007/s00190-003-0339-0. Schmid, R., M. Rothacher, D. Thaller, P. Steigenberger (2005). Absolute phase center corrections of satellite and receiver antennas: impact on global GPS solutions and estimation of azimuthal phase center variations of the satellite antenna. GPS Solutions, Vol. 9, No. 4, pp 283–293. Steigenberger, P., Rothacher, M., Dictrich, R., Fritsche, M., R¨ulke, A., Vey, S. (2006). Reprocessing of a global GPS network. J. Geophys. Res., Vol. 111, No. B5, B05402 10.1029/2005JB003747. V¨olksen, C. (2007). The importance of correct antenna calibration models for the EUREF Permanent Network. In: Report on the Symposium of the IAG Subcommission for Europe (EUREF), Vienna, 1.-4. June 2005. Publication No. 15, Mitteilungen des Bundesamtes f¨ur Kartographie und Geod¨asie, Vol. 38, pp. 73–78. W¨ubbena, G., M. Schmitz, F. Menge, V. B¨oder, G. Seeber (2000). Automated absolute field calibration of GPS antennas in real-time. Proceedings of the 13th International Technical Meeting of the Satellite Division of the Institute of Navigation, ION GPS-2000, Salt Lake City, Utah, September 19–22. 2000, pp. 2512–2522. W¨ubbena, G., M. Schmitz, G. Boettcher (2006). Near-field effects on GNSS sites: analysis using absolute robot calibrations and procedures to determine corrections. Submitted to Proceedings of the IGS Workshop 2006 Perspectives and Visions for 2010 and beyond, May 8–12, ESOC, Darmstadt, Germany.

Latest Enhancements in the Brazilian Active Control Network ´ Luiz P.S. Fortes, Sonia M.A. Costa, Mario A. Abreu, Alberto L. Silva, Newton J.M. Junior, Jo˜ao G. Monico, Marcelo C. Santos and Pierre T´etreault

Abstract The Brazilian Network for Continuous Monitoring of GPS – RBMC, since its foundation in December of 1996, has been playing an essential role for the maintenance and user access of the fundamental geodetic frame in the country. It provides to users a direct link to the Brazilian Geodetic System. Its role has become more relevant with the increasing use of space navigation technology in the country. Recently, Brazil adopted a new geodetic frame, SIRGAS2000, Luiz P.S. Fortes Coordenac¸a˜ o de Geod´esia, Instituto Brasileiro de Geografia e Estat´ıstica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro, RJ 21241-051 Brazil Sonia M.A. Costa Coordenac¸a˜ o de Geod´esia, Instituto Brasileiro de Geografia e Estat´ıstica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro, RJ 21241-051 Brazil Mario A. Abreu Coordenac¸a˜ o de Geod´esia, Instituto Brasileiro de Geografia e Estat´ıstica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro, RJ 21241-051 Brazil Alberto L. Silva Coordenac¸a˜ o de Geod´esia, Instituto Brasileiro de Geografia e Estat´ıstica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro, RJ 21241-051 Brazil Newton J.M. J´unior Coordenac¸a˜ o de Geod´esia, Instituto Brasileiro de Geografia e Estat´ıstica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro, RJ 21241-051 Brazil Jo˜ao G. Monico Departmento de Cartografia, Universidade Estadual Paulista, Rua Roberto Simonsen, 305, Presidente Prudente, Brazil Marcelo C. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O.Box 4400, Fredericton NB. E3B 5A3 Canada Pierre T´etreault Geodetic Survey Division, Natural Resources Canada, 615 Both St, Ottawa, ON K1A OE9 Canada

in February 2005, fully compatible with GNSS technology. The paper provides an overview of the recent modernization phases the RBMC network has undergone highlighting its future steps. From its current post-mission mode, the RBMC will evolve into a real-time network, providing real-time data and real-time correction to users. The network enhanced with modern GPS receivers and the addition of atomic clocks will be used to compute WADGPS-type corrections to be transmitted, in real time, to users in Brazil and surrounding areas. It is estimated that users will be able to achieve a horizontal accuracy around 0.5 m (1σ) in static and kinematic positioning and better for dual frequency users. The availability of the WADGPS service will allow users to tie to the new SIRGAS2000 frame in a more rapid and transparent way for positioning and navigation applications. It should be emphasized that support to post-mission static positioning, will continue to be provided to users interested in higher accuracy levels. In addition to this, a post-mission Precise Point Positioning (PPP) service will be provided based on the one currently provided by the Geodetic Survey Division of NRCan (CSRS-PPP). The modernization of the RBMC is under development based on a cooperation signed at the end of 2004 with the University of New Brunswick, supported by the Canadian International Development Agency and the Brazilian Cooperation Agency. The Geodetic Survey Division of NRCan is also participating in this modernization effort under the same project.

Keywords GNSS Wide-area DGPS

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Active

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1 Introduction The Brazilian Network for Continuous Monitoring of GPS (RBMC) (Fortes et al., 1998; IBGE, 2007a), since its foundation in December of 1996, has been playing an essential role in the maintenance and user access of the fundamental geodetic frame in the country. It provides users with a direct link to the Brazilian Geodetic System. Its role has become more relevant with the increasing use of space navigation technology in the country. Recently, Brazil adopted a new geodetic frame, SIRGAS2000, in February 2005, fully compatible with GNSS technology. Observations are collected by a network of 38 stations. These observations are daily files that are made available to users, via IBGE’s web site, within one day, and after their quality has been tested. This availability depends on the type of connection between the stations and the central storage facility, located in Rio. Most stations communicate with the central facility via the internet or by phone, allowing automatic data transfer. One station communicates via e-mail, a manual, suboptimal scheme. The stations that composed today’s RBMC have been implemented in a stepwise manner since December of 1996. Several institutions and organizations have been collaborating with IBGE either providing the location to have the equipment installed or by providing both receiver and installation. For that reason, not all stations are occupied by the same type of receiver. Figure 1 (shown at the end of the paper to be better visualized) shows the spatial distribution of RBMC stations around the country. All stations use dual-frequency receivers and choke-ring antennas. Two RBMC stations, the ones in Brasilia and Fortaleza are part of the International GNSS Service – IGS global network (IGS, 2007a). The remaining ones compose the IGS densification network in South America. These data are processed on a weekly basis by the IGS Regional Network Associate Analysis Center for the continent – IGS RNAAC SIR (Seemueller and Drewes, 2004). A few geodetic organizations in South America, among them IBGE, will take over the responsibility of being a regional analysis centre. This fact ascertains the Brazilian geodetic reference framework to be part of the global frame in a consistent and permanent way, allowing its continuous monitoring and update. RBMC observations are made available to both national and international communities via IBGE’s

L.P.S. Fortes et al.

web site. In February 2005 Brazil adopted a new geodetic geocentric reference frame, SIRGAS2000 (IBGE, 2007b). SIRGAS2000 is a densification of the ITRF2000, realized by means of a two-week long GPS campaign covering the whole of the Americas. SIRGAS2000 is realized in Brazil by most of the current RBMC stations. In September 2004, an international collaboration was established with the University of New Brunswick under the National Geospatial Framework Project (PIGN), a technology transfer project sponsored by the Canadian International Development Agency (CIDA) with the support of the Brazilian Cooperation Agency (ABC). PIGN (PIGN, 2007) project activities include technical issues, study on the impacts resulting from the adoption of the new geodetic reference frame and communication with user community. The modernization of the RBMC corresponds to PIGN Demonstration Project #7. This Demo Project aims at providing the background for the implementation of a modern reference structure that facilitates the connection to the Brazilian Geodetic System by users. Specifically Demonstration Project 7 will initiate the implementation of real-time and post-mission correction services in Brazil. Both services will provide corrections to facilitate user connection to SIRGAS2000 for positioning and navigation applications. Since the corrections will be implicity attached to SIRGAS2000, their application by the users will result in SIRGAS2000 coordinates. Users will then be directly attached to SIRGAS2000 in their positioning and navigation applications. The Geodetic Survey Division of Natural Resources Canada will contribute expertise in the development and operation of a Wide DGPS Service (CDGPS, 2007a).

2 Expansion Plan The IBGE, in partnership with the INCRA, has been working in the expansion plan of the GPS networks: RBMC and RIBaC – Community Bases Network of INCRA, managed by the INCRA (INCRA, 2007), which will provide a larger national coverage and new characteristics of operation. That plan is going to endow RBMC/RIBaC with an adequate infrastructure for collecting data from GPS and GLONASS, foreseeing the possibility of collecting GALILEO data in the future.

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This plan was put in practice in the year of 2006, with the signature of an agreement between IBGE and INCRA. After that, in the same year, 83 last generation receivers were purchased, followed by the exchange of receiver/antenna in the existing stations, and of the localities where new stations will be installed. In 2007, it was initiated the exchange of the old receivers/antenna in the existing RBMC stations by the new receivers/antennas acquired in 2006, besides the establishment of new stations.

At the moment, after the first step in the expansion plan (March – May of 2007), RBMC counts with 27 stations in operation, with data available daily in the Internet, and other 11 installed. Figure 1 shows the distribution of the station’s network. An example of the new antenna can be seen in the Fig. 2. The data of the stations, after passing for a period of tests and evaluation of its operation, can be obtained freely by the users on the IBGE website, in the RBMC area.

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Fig. 2 New antenna of the NAUS station

The second phase, programmed for the year 2007, targets the installation of 25 new receivers, extending the present network from 38 to 63 stations. The main characteristics of these new receivers are: – – – –

Network card integrated into the receiver, Online configuration via web browser; Data download by FTP (File Transfer Protocol); Memory to store up to 15 days of data at 1 Hz.

Table 1 shows the current (July 2007) receivers being used. Table 1 Type and number of receivers being used Number of receivers Receiver type

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Fig. 3 New receiver of the ONRJ station with atomic clock

The enhanced network, with modern GPS receivers and the addition of atomic clocks (Fig. 3), will be used to compute WADGPS-type corrections to be transmitted, in real time, to users in and surrounding areas. It is estimated that users will be able to achieve a horizontal accuracy around 0.5 m (1-sigma) in static and kinematic positioning and better for dual frequency users. The availability of the WADGPS service will allowusers to tie to the new SIRGAS2000 frame in a more rapid and transparent way for positioning and navigation applications. It should be emphasized that support to post-mission static positioning will continue to be provided to users interested in higher accuracy levels. This new structure, after entirely implanted, will have as main characteristics:

– Make use of remote and automatic Active Control Points (ACP); – Transfer 1-Hz real time data from ACPs to the Net3 Modernization Plan work Control Center, located in Rio de Janeiro; – Generate real-time WADGPS corrections (orbit, clocks and ionospheric); The modernization of the RBMC is under development based on a cooperation agreement signed at the end of – Make corrections available to users in Brazil (and surrounding areas) through Internet; 2004 with the University of New Brunswick, supported by the CIDA and the ABC (Fortes et al., 2006a, b). – Offer a Precise Point Positioning (PPP) service to users (T´etreault et al., 2005); The Geodetic Survey Division of NRCan is also participating in this modernization effort under the – Offer a horizontal accuracy around 0.5 m (1-sigma) in static and kinematic positioning and better for same project. From its current post-mission mode, dual frequency users; the RBMC will evolve into a real-time network, providing real-time data and real-time correction to – Collaborate with international organizations such as the IGS-IP network. users.

Latest Enhancements in the Brazilian Active Control Network

At present, the station National Observatory – ONRJ (Fig. 3), located in the Rio de Janeiro, is contributing to the IGS-IP network, has its data in real time in the following address: . A requirement to be satisfied is to have a subset of receivers connected to an atomic clock. At least 2 receivers can satisfy this requirement. One, which is located in Fortaleza, will be connected to a Hydrogen Maser. The other is located in Rio de Janeiro and will be collocated to the Brazilian Time Service at the National Observatory. The clocks will be used to help the WADGPS -type correction computations, which will

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be transmitted in real-time, to users in Brazil and surrounding areas. To offer these services, new equipment has been procured, such as 2 high-performance workstations to handle the computations, as well as 5 servers for the storage of data, up to 1.4 TB each. The software NTRIP – Networked Transport of RTCM via Internet Protocol in stream, has been tested to be used for the transmission of data and corrections. A pilot project formed by a sub-network of 6 stations (as shown in Fig. 4) will be used to test the realtime capabilities of the network.

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4 Conclusions

References

The growing use of space technology in Brazil has led to the enhancement plans for the current geodetic infrastructure of the country. These plans include a modern active network delivering not only data files for post-mission positioning but also WADGPS-type corrections to be made available to users all around the country. The RBMC network has been the fundamental geodetic infrastructure in Brazil since its inception in 1996, providing accurate connection to the Brazilian Geodetic System, in post-processing mode. Modernizations steps include new receivers; adding value with the land reform network RIBaC to use some of their stations as part of the fundamental active network: fully automatic ACPs, being some of them collocated with atomic frequency standards; the generation of correction at the Master Active Control Station – MACS; transmission of corrections through various media to users. These developments are taking place under a collaborative effort with the University of New Brunswick and with the Geodetic Survey Division of Natural Resources Canada under the National Geospatial Framework Project, supported by CIDA. The Land Reform institute (INCRA) will contribute a total of 65 new geodetic receivers. The application of WADGPS corrections will allow users to be attached to SIRGAS2000 in a direct and clear way in positioning and navigation applications. It is believed that users will be capable of real-time static and kinematic positioning with a 2D accuracy of 0.5 m (DRMS) or better, depending on the type of receiver used. For higher post-processed accuracies, a new postmission PPP service, in addition to the current ACP data files, will also be available.

CDGPS, 2007a. The Real-Time Canada-Wide DGPS Service. http://www.cdgps.com/. Fortes, L.P.S.; R.T. Luz; K.D. Pereira; S.M.A. Costa e D. Blitzkow, 1998. The Brazilian Network for Continuous Monitoring of GPS (RBMC): Operation and Products. Advances in Positioning and Reference Frames (ed) F.K. Brunner, International Association of Geodesy Symposia, Vol. 118, pp. 73–78. Fortes, L.P.S., S.M.A. Costa, M.A.A. Lima, J.A. Fazan and M.C. Santos, 2006a. Accessing the New SIRGAS2000 Reference Frame through a modernized Brazilian Active Control Network. International Association of Geodesy Symposia (Ed. C. Rizos), IAG, IAPSO and IABO Joint Assembly “Dynamic Planet”, 22–26 August 2005, Cairns, Australia, Springer, pp. 655–659. Fortes, L.P.S., S.M.A. Costa, M.A.A. Lima, J.A. Fazan, J.F.G. Monico, M.C. Santos and P. T´etreault, 2006b. Modernizing the Brazilian Active Control Network. Proceedings of the 17th International Technical Meeting of the Satellite Division of the Institute of Navigation ION GPS/GNSS 2003. 26–29th September, Fort Worth, Texas, pp. 2759–2768. IBGE, 2007a. Rede Brasileira de Monitoramento Continuo. http://www.ibge.gov.br/home/geociencias/geodesia/rbmc/ rbmc.shtm. IBGE, 2007b. Projeto Mudanc¸a do Referencial Geod´esico. http://www.ibge.gov.br/home/geociencias/geodesia/noticia sirgas.shtm. IGS, 2007a. International GNSS Service Tracking Network. http://igscb.jpl.nasa.gov/network/netindex.html. INCRA, 2007. Rede INCRA de Bases Comunit´arias do GPS. http://ribac.incra.gov.br/. PIGN, 2007. National Geospatial Framework Project. http://www.pign.org/. Seem¨uller, W. and H. Drewes, 2004. Annual Report 2002 of IGS RNAAC SIR. IGS 2001–2002 Technical Reports (eds) K. Gowey, R. Neilan e A. Moore, IGS Central Bureau, Pasadena, California, pp. 129–131. T´etreault, P., J. Kouba, P. Heroux, P. Legree, 2005. CSRSPPP: An Internet Service for GPS User Access to the Canadian Spatial Reference Frame, Geomatica, Vol. 59, No. 1, pp. 17–28.

Acknowledgments The work described here has been carried out under the scope of the Projeto de Mudanc¸a de Referencial Geod´esico and the National Geospatial Framework Project http://www.pign.org, funded by the Canadian International Development Agency (CIDA).

Searching for the Optimal Relationships Between SIRGAS2000, ´ South American Datum of 1969 and Corrego Alegre in Brazil Leonardo C. Oliveria, Marcelo C. Santos, Felipe G. Nievinski, Rodrigo F. Leandro, Sonia M.A. Costa, Marcos F. Santos, Jo˜ao Magna, Mauricio Galo, Paulo O. Camargo, Jo˜ao G. Monico, Carlos U. Silva, and Tule B. Maia

Abstract Brazil has moved towards the adoption of a geocentric system, SIRGAS2000. With the adoption of this system, starting in 2005, a great demand has been created towards transforming the current data sets from the South American Datum of 1969 (SAD69), in its two distinct realizations, and the C´orrego Alegre frames into SIRGAS2000. The fact that these four frames will co-exist until 2014 creates positive and negative situations. Due to the distortion between those frames, the relationships among them cannot be well established with Helmert transformation parameters alone. To solve this problem, five Study Groups were created to look for the optimal relationships for coordinate transformation between those frames. The approaches being investigated to augment the parameter transformation are based on: Collocation, Delaunay, Regular grids (NTv2 and Sheppard method) and Neural Networks. The research is currently going on. This paper describes the current efforts towards defining the optimal relationships among these four frames, from the mathematical point-of-view. The work described in the paper has been carried out under the scope of the National Geospatial Framework Project (www.pign.org), sponsored by the Canadian International Development Agency.

Keywords Reference frames · Coordinate transformations · Distortion modelling

Felipe G. Nievinski Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O.Box 4400, Fredericton NB., Canada E3B 5A3 Rodrigo F. Leandro Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O.Box 4400, Fredericton NB., Canada E3B 5A3 Sonia M.A. Costa Coordenacao de Geodesia, Instituto Brasileiro de Geografia e Estatistica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro, RJ, 21241-051 Brazil Marcos F. Santos Coordenacao de Geodesia, Instituto Brasileiro de Geografia e Estatistica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro, RJ, 21241-051 Brazil Jo˜ao Magna Departmento de Cartografia, Universidade Estadual Paulista, Rua Roberto Simonsen, 305, Presidente Prudente, SP, Brazil Mauricio Galo Departmento de Cartografia, Universidade Estadual Paulista, Rua Roberto Simonsen, 305, Presidente Prudente, SP, Brazil Paulo O. Camargo Departmento de Cartografia, Universidade Estadual Paulista, Rua Roberto Simonsen, 305, Presidente Prudente, SP, Brazil Jo˜ao G. Monico Departmento de Cartografia, Universidade Estadual Paulista, Rua Roberto Simonsen, 305, Presidente Prudente, SP, Brazil

Leonardo C. Oliveria Sec¸a˜ o de Ensino de Engenharia Cartogr´afica, Instituto Militar de Engenharia, Prac¸a General Tiburcio, 80-6◦ andar, Rio de Janeiro, RJ, 22290-270 Brazil

Carlos U. Silva Departmento de Engenharia de Transportes, Universidade Federal do Cear´a, Campus do Pici - Bloco 703, Fortaleza, CE, 60455-760 Brazil

Marcelo C. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O.Box 4400, Fredericton NB., Canada E3B 5A3

Tule B. Maia Departmento de Engenharia Civil, Universidade Cat´olica de Goi´as, C.P. 86. Av. Universit´aria, 1069, Goiania, GO, 74605-010 Brazil

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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1 Introduction Brazil is already user of a geodetic geocentric system since February 2005, due to the publication of a Resolution from the President of Fundac¸a˜ o Instituto Brasileiro de Geografia e Estat´ıstica – IBGE, the institution responsible for the establishment and maintenance of the Brazilian Geodetic System – SGB. In reality, until 2014 the SIRGAS System, realization 2000, will be used simultaneously with the South American Datum of 1969 (SAD 69) – for geodetic reasons – and also with the C´orrego Alegre (CA) system – this only for cartographic reasons. This 10 years time interval is called the “transition period”. IBGE and a group of institutions, among them universities and public and private companies, developed the Geodetic Reference Change Project (PMRG) and the National Geospatial Framework Project (PIGN), this one in partnership with Canadian institutions The PIGN is dedicated to study the impacts, solutions and tools used in the process of migrating from the current classical systems to the geocentric SIRGAS. Amongst these activities, there is one inherent to coordinates transformation modelling, tasks under the responsibility of Working Group 3 (WG 3) on reference conversion. Amongst the actions linked to WG 3 stands out the choice of the methodology that allows the conversion of coordinates to the new system, minimizing impacts and maximizing quality. In a way to develop a better solution for this challenge, five approaches were proposed, currently being developed through Study Groups (SG). This paper presents theoretical aspects from on-going research, as well as the complementary aspects to the conversion process.

2 Dimension of the Transformation

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(a) The adjustment process of the RGB (Brazilian Geodetic System) into either SAD69 or SIRGAS2000 were carried out on the ellipsoid (ϕ, λ), although using a 3D geodetic modeling, control and observations from GPS and besides the fact that SIRGAS2000 is a 3D system. So, there is no advantage in operating the full components coordinates transformation (ϕ, λ, h); (b) Conceptually the orthometric height is independent from the ellipsoid used in the geodetic system, being interesting and recommended that the orthometric height (H) values be kept on the stations where these values are known, not generating differences to users; (c) The planimetric network adjustment is independent from the vertical network adjustment (orthometric height). In this way, it is probable that the distortion modeling using only the components ϕ and λ minimize the eventual inconsistencies between the networks, avoiding the effect of possible problems existing on the H coordinate to affect the components ϕ and λ; (d) There is no geoidal map for C´orrego Alegre (CA). In terms of the existing geodetic and cartographic documents, there should be no risk of having different methodologies for the distortions modeling from SAD69 and CA to SIRGAS, since there are contiguous areas of mapping; (e) There are at least 3 geoidal maps already available – two to be used with SAD69 and other to be used with SIRGAS. A distortion modelling where there is the necessity of using geoidal maps could lead less experienced users to error, fact that should be avoided; (f) At the level of the SIRGAS project, the definition of the vertical system has not been reached. That being so, the production of SIRGAS heights should be avoided because this value may be altered in the near future; (g) It is necessary to design a solution that merges quality, credibility and easiness of use, since they are essential to society, as gauged by an opinion poll carried out by WG 3 on the user community independent of their familiarity with aspects related to Geodesy or applications in Cartography.

One of the most important issues debated within WG3 deals with the dimension of the coordinate transformation, i.e., identify if the conversion of the coordinates should only involve geodetic latitude (ϕ) and geodetic longitude (λ) components or all geodetic coordinates Therefore, for the stations that already have coor(including geodetic height (h)). WG 3 and the SG opted for a horizontal conversion affecting only latitude and dinates in the C´orrego Alegre or SAD 69 frames, the transformation of planimetric coordinates (ϕ and λ) longitude coordinates. This choice is based on:

Searching for the Optimal Relationships

will be done according to the methodology to be defined by WG 3. Concerning the vertical component, in the existence of:  Orthometric heights (H): they will continue with the same value on the SIRGAS2000 frame. In this way, the quality of H is kept and there is no need to calculate the orthometric height of the stations in the new system;  Geometric heights (h): this is a specific case for the SAD 69 system or alternatively for the user that has “WGS 84 coordinates.” The component h should be converted to orthometric height and after that be subject to the same treatment as outlined for orthometric heights. It is suggested that the conversion from h to H should be done using the Geoidal Map MAPGE2004, already available on http://www.ibge.gov.br/home/geociencias/ geodesy/modelo geoidal.shtm?c=15.

3 Analysis of Distortions in Horizontal Networks The present section overviews the proposal that the IBGE Study Group will submit to WG 3, along with some preliminary results, dealing with the modelling of distortions that exist among the different realizations of the Planimetric Network of the Brazilian Geodetic System. This proposal contemplates the distortions network modelling through NTv2 software – National Transformation, version 2, developed by the Geodetic Survey Division (GSD) of Natural Resource Canada. NTv2 is a set of programs that has among its functions the following tasks: evaluation of inconsistencies found between two frames, coordinate transformation between systems applying only the Helmert transformation and “grid” calculation, the latter used to represent the model of distortions derived from the materializations of the reference systems. In a first stage the file containing all planimetric stations belonging to the SGB underwent a filtering where all stations at the Northern hemisphere were removed. This procedure was necessary, as the NTv2 software do not work with the hemisphere change, or rather, during its processing it considers the latitudes absolute value, what makes impossible the network processing as a whole. Other stations also were removed from the original file, as they did not take part on

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the adjustments, either in SAD69 – realization 1996, either in SIRGAS2000. The GPS stations that were removed from the file were not adjusted due to the non-availability of the primary data for the processing and adjustement. This was a procedure defined by all GEs. The following stage consisted on the coordinate transformation between the two realizations (SAD691996 and SIRGAS2000) applying to the coordinates a conformal transformation (Helmert). The objective of this transformation is to keep the geometry of the geodetic structure, or rather, keep the relationship between the coordinates to be transformed. Therefore, given two sets of coordinates belonging to all common stations in both materializations, deriving from the SAD69-1996 and SIRGAS2000 (SAD69AJD and SIRGASADJ ) adjustments, the transformation parameters that consist of the RPR 01/2005 from the Presidency of IBGE are applied to the set of SAD69AJD coordinates, resulting in SIRGAS2000 transformed coordinates (SIRGASTRANS ). The following stage will consist of the selection of the stations that will be used to assess the performance of distortions modeling between the two sets of coordinates. For that it will be used the difference remaining (residuals) after the model application and their respective values adjusted. It is important to notice that both the magnitudes as well as the spatial distribution of the residuals are important. Generally a good model leads to small residuals. The resulting differences between SAD69TRANS and SIRGASAJD coordinates will be treated by a polynomial modeling that will represent the regional tendencies given by the correction vectors on the control stations. In this case, from the sets of SIRGASTRANS = (X1i , Y1i ) and SIRGASAJD = (X2i , Y2i ) coordinates, corrections are calculated. They will be used on the interpolation of any point that does not belong to the SIRGASTRANS and SIRGASAJD set as the formulation below. Consider: X0 = Xi 1 /np; Ui = Xi 2 − Xi 1 ; U0 = Ui /np; Sx = 1/X1max ; Su = 1/(Umax 2 − U0 2 )1/2 ; SU0 = max(Xi 2 − Xi 1 ); Zxi = Sx (X1i − X0 ); Exi = Su (Ui − U0 );

Y0 = Yi 1 /np; Vi = Yi 2 − Yi 1 ; V0 = Vi /np; SY = 1/Y1max ; Sv = 1/(Vmax 2 − V0 2 )1/2 ; SV0 = max(Y2i − Y1i ); Zyi = Sy (Y1i − Y0 ); Eyi = Sv (Vi − V0 );

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where: X0 = average latitude of the control stations; Y0 = average longitude of the control stations; Ui = differences in latitude between SIRGASTRANS and SIRGASAJD control stations; Vi = differences in longitude between SIRGASTRANS and SIRGASAJD control stations; U0 = average in latitude differences; V0 = average in longitude differences; Sx = latitude scale factor; Sy = longitude scale factor; Su = latitude differences scale factor; Sv = longitude differences scale factor; X1max = control stations maximum latitude; Y1max = control stations maximum longitude; Umax = maximum value of control stations latitude differences; Vmax = maximum value of control stations longitude differences; np = quantity of control stations (i).

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a function of the distance (di ) between the control point and the points where both the corrections and the distance (k) will be estimated for the selection of the control points for interpolation (Fig. 1). The weight is calculated according to the following formula: wi = e−(di/k2) ; di = Zxi Zyi The residuals in latitude and longitude (Rxi and Ryi ) are estimated for each control station by: Rxi = Fui + Cxi − Ui , Ryi = Fvi + Cyi − Vi , where: Fui = Ui − Exi /Su ,

Fvi = Vi − Eyi /Sv .

Prelimimary solutions have been generated and two types of cartograms generated. The first, shown in Fig. 2, contains a thematic map with the distortions The estimated corrections in any station P are based values forming a grid of 10 by 10 , using the whole on a weighted average of the estimated coordinate cor- network (classical network + GPS network) and rections for all the control points around it. The correc- a number of 25 neighbouring stations. A second tion in any latitude Cx and longitude Cy is given by: cartogram, similar to the one presented in Fig. 4, contains the displacement vectors and their direction Cxi = wi Exi / wi , Cyi = wi Eyi / wi , throughout the country. The next stage consists of the evaluation of the where Exi and Eyi are corrections on the control points networks separately, i.e., Classical Network and GPS i, according to latitude and longitude, respectively. Network, so the distortion grids with magnitudes The weight (wi ) assigned to each control point i is and residual spatial distribution inherent to each

Fig. 1 Estimated weighted average parameters of any point–ESTPM Program. [Source Junkins & Erickson, 1996]

Searching for the Optimal Relationships

Fig. 2 Thematic map with distortions values em spacing de 10 by 10

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We provide both functional and stochastic models for distortion. The functional model is made of a low order polynomial: one polynomial (function of latitude and longitude) for each of the three types of coordinates (N, E, U). This trend modeling is a requirement for the stochastic modeling to follow. The random yet correlated portion of the discrepancies, called “signal” in the collocation literature, is modeled populating the observations prior covariance matrix employing an empirical covariance function. The estimation process yields values and associated standard deviations for the functional parameters (similarity transformation and polynomial), as well as for the signal at observation points. Those estimates are then employed subsequently to predict distortion at any other point, e.g., on a regular grid for distribution to end users.

5 Coordinates Conservation Method technology can be determined. Another point that needs to be better studied is the distortion modeling on the region between the Northern and the Southern hemispheres.

4 Distortion Modelling Based on Collocation Collocation is based on a simultaneous adjustment and regression model carried out in two steps, where the covariance function is formed based on the residuals of the first step. The basic model reads: Aδˆ + B rˆ + T λˆ + w = 0, ˆ rˆ and λˆ represent, respectively the estimated where δ, solution, residual and coefficient vectors, A, B the first and second design matrices, T the Vandermonde’s matrix and w the misclosure vector. In the solution, we formulate the observation equations in terms of local Cartesian coordinates (Northing, N; Easting, E; Up, U). We believe they form the best coordinate basis for distortion modeling (i.e., the most natural directions and units), as compared to global Cartesian coordinates (X, Y, Z), geodetic coordinates (latitude, longitude, height), and arc-length coordinates (meridian arc-length, parallel arc-length, height).

The Coordinates Conservation Method (MC2) is based on Oliveira (1998). The MC2 philosophy is to try to maximize the numerical integrity of coordinates referring to the coordinate networks associated to the geodetic systems involved in the transformation process. The way found for this purpose was to associate computational geometry concepts in a single methodology, geometric transformation and numeric interpolation, using solutions by least squares as well as deterministic solutions. The methodology can be summarized as being composed of the addition of two terms, one referring to the change of system and the other related to the distortion modeling, or rather: Trf Crd. = Mud Sis + Mod Dist. The change of system (Mud Sis) is carried out by the application of the 3-parameter model, or simply through the translations between the coordinates (X, Y and Z) of the involved systems, whose mathematical model is expressed by: [X, Y, Z]T D = [X, Y, Z]T + [X, Y, Z]T O . The superscript T indicates a transposed array. The subscripts D and O represent, respectively, the final system and system of origin. This solution was chosen

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due to the fact that the systems involved are, by definition, parallel. The translation values will involved only the coordinates of Chu´a datum marker, for the SAD 69 system, and C´orrego Alegre datum marker, for the CA system, obeying the definition of the systems. The differences between the transformed coordinates and the original coordinates correspond to the distortions to be modeled. The distortion modeling (Mod Dis.) is the result of the term referring to the local distortion (Dis Loc) and the term relative to the residual distortion (Dis Res), or rather: Mod Dis = Dis Loc + Dis Res The term Dis Loc will be obtained through a set of selected stations and the application of an affine geometric transformation (AGT), quantified through its linear model and the least squares method. The set of stations will be defined with the support given by the Delaunay Triangulation (DT), or rather, the station network nearest to the station of interest and its neighbours – the stations that compose the triangulation. It must be said that to avoid inconsistencies between the inverse and direct mapping between the coordinates, the triangulation will be constructed using the coordinate averages for either the CA and SIRGAS2000, and SAD69 and SIRGAS2000 geodetic networks, guaranteeing a unique triangulation for each set. The computational system to be used to the DT is the QHULL. The linear mathematical model for the AGT is given by: xD = axD + byD + x, yD = cxD + dyD + y, where a, b, c and d are the parameters which implicitly model rotation, scale and lack of orthogonality. The residual generated by the AGT estimation will be taken as the residual distortions, last stage of the modeling process. For the Dis Res estimation two operations are fundamental: the first is with respect to the values to be used, while the second is related to the choice of the interpolator. The choice of the values occurs directly when it is identified in which triangle the point of interest is inserted. For such, it will be used the barycentric coordinates concept. As soon as the triangle is identified, i.e., the value to be used in the interpolation

process, a Hermite interpolator operator is used. This interpolator has continuity as its main characteristic. It stands out that the solution used for the term Dis Res is of deterministic nature, therefore, for the geodetic networks stations involved, it is guaranteed the maximization of the coordinate values integrity. The hypothesis is that the same integrity will also occur when modeling the distortion of stations which do not belong to the network. Some practical aspects related to the implementation still should be observed, for example, the use of normalized coordinates, and the use of binary files.

6 NN-based Distortion Modelling MODERNA is the acronym (in Portuguese) for Distortion modelling employing artifical neural networks. Artificial neural networks (NN) are an important field in Artificial Intelligence, which is an area from Computer Science. Sarle (1994) describes neural networks as being a wide class of discriminating models and non-flexible linear regression, models for data reduction and non-linear dynamic systems. Usually the NN’s are composed by several artifical neurons (or simply neurons), i.e., elements of linear (or not) computation, and often organized in interconnected layers through links (synaptic links). Among the possible applications, the NN’s can be used for data estimation and analysis. Due to the mathematical complexity of a neural model with many neurons, the adaptation which is an adjustment of the synaptic weights (or parameters) of the network is done by means of learning algorithms, usually employing known patterns (or data) sets. The learning processes are sets of well defined procedures, capable of interactively adapting the parameters of a NN. There are several tools represented by several algorithms which differ from each other in the way the parameters adjustment happens, Braga et al. (2000). The same characteristic which makes the treatment of the neural models complex, also make them extremely self-adaptable to different situations. NN’s have been used in Geodesy in several applications, as described by Leandro et al. (2005). Besides the adaptation capability, neural models usually present a high generalization capability, which refers to their capability of producing outputs (or estimates) which

Searching for the Optimal Relationships

are reasonable for the input (or observations) which have never been shown (during the learning process). This characteristic allows the solution of complex problems. The objective of the MODERNA is to use an artificial neural network to model the distortion between the Brazilian frames SAD69 and SAD69/96, and the new South American frame SIRGAS2000. During the process of development of such model, the data available will be used to obtain the adequate configuration of the neural network for this particular application. This configuration includes the type of network, the adequate topology, the algorithm and the parameters of learning process. After defining all aspects concerning to the design of a neural model, it will be validated with the data of the aforementioned geodetic frames. The final goal is having a neural model capable of estimating the distortion between the different geodetic frames used in Brazil, for any location within the country, without the need of using any additional kind of information.

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its nodes the distortion values between the different realizations. - Distortion interpolation via a regular distortion grid regular for points of interest, through known coordinates on the system of origin, and conversion coordinates on the system of arrival. The flowchart on Fig. 3 illustrates the principles of the method. Some relevant aspects can be considered in this approach. The first is related to the spacing of the distortion regular grid, adjusted to be obtained the best results, when comparing the real distortions with the modeled distortion values. The second is referred to the interpolation method used on the generation of the regular grid. The third is connected with the optimum interpolation method (in phase 3), from the regular grid already available. The third aspect, although important, is the least critical, depending on the adopted spacing of the regular grid. This way, it can be assumed that the spacing is such that a simple interpolation method could be used, as for example the bi-linear interpolation. It can

7 Distortion Modelling by Regular Grids Coordinates of point i in reference frame 1 (φ 1, λ1)i

Initially, it is important to define the concept of distortions, in the context of this approach. Considering two frames (1 and 2) and a geometric transformation T that transforms from 1 to 2 (and its inverse T−1 for the inverse case), the difference between the coordinates on the frame 2 and the on frame 2 , obtained from the transformation T, is called distortion. Although the distribution of points of the Brazilian geodetic network is not homogeneous in its extension, there are a great number of points, which permits to consider that a regular distortion grid can be generated from the distortions calculated on this irregular mesh. Once available, this regular mesh can be used for the interpolation of distortions at any point in the area covered by the mesh. This approach is based on the same principle used by NTv2. This methodology can be summarized on the following phases:

T -1

Coordinates of point i in reference frame 2 (φ 2, λ 2)i

Transformation (T)

Computed coordinates of i in reference frame 2 (without distortion modeling) (φ 2, λ2)i’

Computation of distortions (δφ ,δλ)i= (φ 2,λ 2)i-(φ 2,λ 2)i’

Generation of regular grid

Distortion Modeling

Interpolation of distortions (δ φ,δλ)i’ *

Computed coordinates of i in

Computation of error in the

reference frame 2 (after modeling process - Distortion calculation between the geodetic netdistortion modeling) (ε φ, ελ)i’= (φ 2,λ2)i-(φ 2,λ2)i’’ works through stations with known coordinates in (φ 2,λ2)i’’=(φ 2,λ2)i’+(δφ,δλ)i’ both realizations. * Can be applied to every point in the net. - Generation of a regular grid of defined spacing enclosing the whole national territory, containing in Fig. 3 Principle of distortions modeling from regular grids

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be, therefore, considered that the quality of this interpolation will be strongly dependent from the first step (regular grid spacing). The second aspect, referring to the distortions interpolation is critic, due to the irregular distribution of the sample points. For this reason, it was considered in this approach the interpolation principles used by Shepard (1968), which is based on the use of a weight function which considers both the distance as well as the direction. Experiments with grids of several spacing are being done, with the objective to choose the dimension of the regular grid which permits the distortion modeling between the SIRGAS 2000 and SAD 69 reference, that allow to obtain results with EMQ smallest values for the distortions components ϕ and λ Fig. 4.

Fig. 4 Detailed North-Eastern view of map of distortions between SIRGAS2000 and SAD69

8 Choice of the Most Appropriate Methodology The process of choosing a methodology among the proposed ones will happen in a particular event, programmed to occur in October 2007, in Rio de Janeiro. Some tests with the implemented methodologies will be carried out. Among them, it can be quoted: (a) a test involving the geodetic stations not used in the modeling (playing the role of test points), with the objective of quantifying an indicator of external quality;

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(b) testing the computational efficiency of the modeling technique; (c) the assessment of the smoothness and continuity of the solution; (d) testing the maintenance of the integrity of the coordinates integrity, i.e., to verify the quality of the transformation to and from two frames; and, (e) other tests as necessary.

9 Conclusions Brazil is going through a process which will culminate in the adoption of a new geocentric frame, SIRGAS2000. Several Study Groups have been formed, under the umbrella of the National Geospatial Framework Project, to investigate among several methodologies dealing with the modelling the distortions of geodetic networks, in order to come up with the optimum methodology (or combination of them). This paper presents a summary of each one of them. These methodologies are based in different methods. For example, one is based on Neural Networks, one on LeastSquares Collocation, a third one is based on Delaunay Triangulation and the last two make use of NTv2 program, with slight modifications. A Workshop will be held in Rio de Janeiro in October 2007 in order to discuss on the characteristics of each one of those methodologies, but also to choose the most efficient and accurate among them. The inter-relation between the classical networks and SIRGAS2000 becomes even more important if we realize the size of the classical networks, composed of nearly 7,000 points (whereas reference GPS networks in Brazil amount to less than 100 points). Historically, all mapping, notably the ones in cadastral scale, have been done (even recently) attached to classical networks. For topographical scales, similarity transformation by itself would be sufficient. But, when one considers the cadastral applications, the distortions inherent to the classical networks must be taken into consideration. Several institutions in Brazil have invested in the search of a solution to the coordinate’s conversion and the products associated to them – such solution will allow minimizing the impacts cause by changing the geodetic framework. More than a consistent and necessary geodetic commitment, the project is based on

Searching for the Optimal Relationships

a wider commitment for the country, playing the role of intermediaries to its social developement, without excluding the consequent technological and economic development. Acknowledgments The work described in this paper has been carried out under the scope of the Projeto de Mudanc¸a do Referencial Geod´esico and the National Geospatial Framework Project http://www.pign.org, funded by the Canadian International Development Agency (CIDA). NTv2 was developed by the Geodetic Survey Division of NRCan.

References Braga, A. P., A.P.L.F. Carvalho and T.B. Ludermir (2000). Redes Neurais Artificiais: Teoria e Aplicac¸o˜ es. Livro T´ecnico e Cientifico, Rio de Janeiro. Costa, S. M. A. (1999). Integrac¸a˜ o da Rede Geod´esica Brasileira aos Sistemas de Referˆencia Terrestres, Doctoral Dissertation. Curso de P´os-Graduac¸a˜ o em Ciˆencias Geod´esicas, Universidade Federal do Paran´a. Haykin, S. (1999). Neural Networks – A Comprehensive Foundation, Second edition. Prentice Hall. New Jersey.

79 IBGE (2005). Resoluc¸a˜ o do Presidente 1/2005 – Altera a caracterizac¸a˜ o do Sistema Geod´esico Brasileiro. Rio de Janeiro, Fundac¸a˜ o Instituto Brasileiro de Geografia e Estatistica, 25/02/2005. Available in ftp://geoftp.ibge.gov.br/ documentos/geodesia/pmrg/legislacao/RPR 01 25fev2005.pdf Junkins, D. (1998). “NTv2 – Procedures for the Development of a Grid Shift File.” Geodetic Survey Division – Geomatics Canada. Leandro, R. F., C. A. U. Silva and M. C. Santos (2005). “Feeding neural network models with GPS observations: a challenging task.” International Association of Geodesy Symposia (Ed. C. Rizos), IAG, IAPSO and IABO Joint Assembly “Dynamic Planet” Cairns, Australia, 22–26 August 2005, Springer, pp. 186–193. Oliveira, L. C. (1988). Realizac¸o˜ es do Sistema Geod´esico Brasileiro Associadas ao SAD 69: uma proposta metodol´ogica de transformac¸a˜ o. Doctoral Dissertation, Departmento de Engenharia de Transportes, Universidade de S˜ao Paulo. S˜ao Paulo, 225 pp. Availabe in http://www.teses.usp.br/teses/Disponiveis/3/3138/ tde-08112005-081539 Sarle, W. S. (1994). “Neural networks and statistical models.” Proceedings of the Nineteeth Annaul SAS User Group International Conference, Cary, N.C., USA. Sheppard, D. (1968). “A two-dimensional interpolation function for irregularly spaced data.” 23rd National Conference of the ACM, ACM Press, pages 517–524.

The Permanent Tide in Height Systems Jaako M¨akinen and Johannes Ihde

Abstract We describe the treatment of the permanent tide in various geodetic quantities with an emphasis on systems of gravity-related heights. We review the historical development leading to the present situation, and discuss possible scenarios for the future, especially in view of the adoption of a World Height System.

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Keywords Permanent tide · Heights · Levelling · Geoid · Geopotential · Reference systems · Gravity · Sea level · Altimetry.

1 Introduction The time-averages of the tide-generating potentials of the Sun and the Moon are not zero. At the surface of the Earth both their magnitude and their range are a few parts of 10−8 of the potential of the Earth: To deal with the permanent deformation they cause, there are two concepts that are available for the 3-D shape of the Earth (also called crust or topography) and three concepts for the gravity field.

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In the non-tidal or tide-free system, the permanent deformation is eliminated from the shape of the Earth. From the potential field quantities (gravity, geoid etc) both the tide-generating potential, and the deformation potential of the Earth (the indirect effect) are eliminated. This corresponds to physically

Jaako M¨akinen Finnish Geodetic Institute Geodeetinrinne 2, FIN-02430 Masala, Finland Johannes Ihde Federal Agency for Cartography and Geodesy Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany

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removing the Sun and the Moon to infinity. Typically the permanent deformation is however treated using the same Love numbers h and k (and Shida number ) as for the time-dependent tidal effects (conventional tide-free system), instead of estimates for secular (fluid) Love numbers. In the mean system, the permanent effect is not removed from the shape of the Earth. The shape therefore corresponds to the long-time average under tidal forcing. The potential field retains the potential of this average Earth, and also the time-average of the tide-generating potential (though it is not due to the masses of the Earth). For the potential field quantities, a “middle alternative”, the zero system is available. It eliminates the tide-generating potential but retains its indirect effect, i.e., the potential of the permanent deformation of the Earth. Thus its partner is the mean system for 3-D shape; because of this it is often said that for the 3-D shape we have zero≡mean. In this alternative, the gravity field is generated only by the masses of the Earth (plus the centrifugal force).

The terminology above has become more or less standard since Ekman (1989). In older literature sometimes the same terms were used with different definitions. The current IAG resolution (1983, see Tcherning 1984) requires the zero system for the gravity field and the mean system for the 3-D Earth. Current practice is to use the zero system for gravity, tide-free for 3-D positions (e.g., ITRFxx), and mixed (overwhelmingly mean) for potential differences determined with precise levelling. Geopotential models are nowadays often provided both in conventional tide-free and in zero versions.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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The permanent tide in height systems cannot be discussed separately from the permanent tide in other geodetic quantities. The purpose of this paper is to give an overview of the present situation and to discuss different scenarios for the future.

2 Basic Relations Zadro and Marussi (1973) derived formulas for the time-averages of the lunar and solar tide-generating potential. For the present discussion we only note that rounded to 1 mm we have at the (say) GRS80 ellipsoid (Heikkinen 1978, Ekman 1982) W2 /g = −0.296 sin2 ϕ + 0.099 [m]

(1)

Here W2 is the sum of the time averages of the above mentioned potentials, g is the acceleration of gravity, and ϕ is the geodetic latitude. Formulas given in the literature with more digits differ in the coefficients at the 0.1 mm level. This will be discussed elsewhere. In a century the obliquity of the ecliptic changes roughly by 1 which changes the coefficients in Eq. (1) by 4 parts in 104 . Figure 1 illustrates schematically the differences between the various tidal concepts for the geoid and for the topography (crust). The geoids are represented by the solid lines and the crusts by the dashed lines. The differences are in the flattening. Using the zero geoid as a reference we enumerate four geoids, from largest flattening to smallest, and their height relative to this reference (the solid line):

Fig. 1 Schematic illustration of different tidal concepts for the crust/topography (dashed lines) and the geoid (solid lines), as sections in a meridional plane. The crusts are from largest to smallest flattening: mean, conventional tide-free, fluid (“realistic”) tide-free. The geoids are from largest to smallest flattening: mean, zero, conventional tide-free, fluid (“realistic”) tide-free. The relative differences in flattening are correctly scaled while the flattening itself obviously is not. More explanations in text

r r r

mean crust at 0 W2 conventional tide-free crust at − g ∂∂ϕ ,  ≈ 0.08 W2 fluid tide-free crust at − g ∂∂ϕ ,  ≈ 0.22

In the east-west direction there are no differences between the tidal systems. The heights of various crusts above various geoids are collected in Table 1. They are taken relative to the r mean geoid at +W2 /g height of the mean crust above the zero geoid. Note r zero geoid at 0 that in calculating numerically these heights, compatir conventional tide-free geoid at −kW2 /g, k ≈ 0.3 ble coordinates must be used for the crust and for the r fluid tide-free geoid at −kW2 /g, k ≈ 0.93 geoid. E.g., if the crust is given in the ITRFxx, then the geoid surface in 3-D space must also be given in Similarly, using the mean crust (topography) as a the ITRFxx, independently of whether we are using the reference, we enumerate three crusts from largest flat- mean, zero, or tide-free geoid. tening to the smallest (the dashed lines):

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mean crust at 0 conventional tide-free crust at −hW2 /g, h ≈ 0.6 fluid tide-free crust at −hW2 /g, h ≈ 1.93

For completeness we remark that the tangential (or transverse) difference between the various crusts is not illustrated. In the north direction we have

Table 1 Crust above the geoid: all combinations of tidal concepts. They are taken relative to the height of the mean crust above the zero geoid Geoid Crust

Mean

Tide-free

Mean Zero Tide-free

−W2 /g 0 kW2 /g

(−h + k)W2 /g −hW2 /g (−h + 1)W2 /g

The Permanent Tide in Height Systems

3 Present Situation

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˚ RH 2000 adopted in Sweden in 2005 (Agren and Svensson 2007) and the N2000 adopted in Finland in The current IAG resolution (1983) recommends the 2007 (Saaranen et al. 2007). Regarding continental and global height systems, zero system for the gravity field quantities and the the European Vertical Reference System (EVRS2000; mean (≡zero) system for the 3-D positions. Ihde and Augath 2001) is zero-tide by definition, but Current practice for 3-D positioning is tide-free. due to the mixed systems (mostly mean, some nonCoordinates in the International Terrestrial Reference tidal) of the national observations going into the United Frame ITRFxx, and through it all modern continenEuropean Levelling Network UELN-95/98, the curtal and national reference coordinates (e.g., those given rent realization EVRF2000 is mixed. The Conventional in the European Terrestrial Reference system ETRS89) Vertical Reference System (CVRS) proposed by the are in the conventional tide-free system. Current practice for gravity is zero-tide. It appears IAG ICP1.2 “Vertical Reference Frame” is zero-tide that all modern reference gravity values since about (Ihde et al. 2007). See also Heck (1993, 2004). During the adoption of the new height systems, 1988 are in the zero-tide system. there has been no renewed debate about the advanGeopotential models (EGMs) are nowadays usually tages and disadvantages of the different tidal systems. provided both in conventional tide-free and in zero-tide The attitude has been mostly “we do as the IAG recversions as far as the potential coefficients are conommends”. The current recommendations are the IAG cerned. The difference is only in the J2 or C20 term. resolutions adopted at the XVIII General Assembly of Note that the EGMs are to be evaluated at (conventhe IUGG in Hamburg in 1983 and they are as follows tional) tide-free coordinates (Pavlis 1998). Regional gravimetric geoid models in the past often (see Tcherning 1984): Resolution No. 9 inherited their tidal systems implicitly from the EGM The International Association of Geodesy used, but nowadays are usually explicitly stated to be recognizing the high level of accuracy of both absolute in the zero-tide system. and relative gravity measurements recently attained In satellite altimetry of the oceans, the Geophysical considering the necessity to adopt standard corrections to Data Records (GDRs) by the operating agencies typigravity observation in order to allow intercomparisons becally provide the range from the satellite to the instantween measurements at different epochs of time taneous sea surface, the height of the satellite above a recommends reference ellipsoid, tidal corrections that do not remove 1. that the tidal correction applied to the gravity obthe permanent tide, and geoid heights of an EGM in servations follow the final recommendations of the the mean tidal system, relative to the field of the referStandard Earth Tide Committee as presented at the XVIII IUGG General Assembly [the Committee ence ellipsoid (see e.g., AVISO/Altimetry 1996). Calrecommended zero tidal gravity (Rapp 1983), note culations with the quantities in the GDR thus give the added by the authors] height of the sea surface relative to a mean geoid, as Resolution No. 16 recommended by Rapp et al. (1991). Models of mean The International Association of Geodesy sea surface, constructed from multi-mission altimetry recognizing the need for the uniform treatment of tidal data, refer the surface either to an ellipsoid, or to an corrections for various geodetic quantities such as gravity and station positions, and EGM geoid in the mean tidal system, though the tidal considering the reports of the Standard Earth Tide Comsystem of the geoid is not always explicitly stated in mittee and S.S.G. 2.55, Predictive Methods for Space the papers. Techniques, presented at the XVIII General Assembly, National height systems are overwhelmingly in the recommends that: mean-tide system, i.e. mean crust over the mean geoid. 3. the indirect effect due to the permanent yielding of This has usually not been by design: if no “luni-solar the Earth be not removed. correction” is applied to precise levelling, then the mean system emerges automatically. The luni-solar correction when applied has almost invariably (and 4 Historical Development unwittingly) resulted in a conventional tide-free system (tide-free crust over tide-free geoid). Zero-tide For understanding the present situation it is useful to systems (mean crust over zero geoid) are very recent: review the development that lead us here.

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Tidal corrections were introduced to improve the precision of geodetic measurements. In many cases their implications for the reference systems and reference frames dawned on geodesists much later. There is no “automatic split” of a tidal correction into a timevariable and permanent part, e.g. when simple tidal corrections are calculated on the basis of ephemerides. Then the permanent part must be evaluated separately and restored by design. Therefore at the first stage of tidal corrections, the conventional tide-free systems were the almost inevitable outcome.

4.1 Precise Levelling Tidal corrections appeared in the 1940s. The early paper by Jensen (1949) pointed out the significance of the time-independent part and introduced the concepts of mean geoid and tide-free geoid (he called the latter “zero” geoid). However, the levellers doing the “luni-solar correction” usually did not distinguish between the permanent and the time-independent part. This led to the conventional tide-free system. Most countries did not make any correction at all and ended with the mean tidal system.

J. M¨akinen and J. Ihde

has subsequently been taken for granted by most writers on the subject. Thus almost all authors who have advanced the adoption of mean-tidal gravity have simultaneously proposed the adoption of a normal gravity field containing the mean tide, such that the mean-tide contribution will be eliminated from free-air anomalies. How such a normal field is implemented is shown e.g. by Vermeer and Poutanen (1997). On the other hand, Yurkina et al. (1986) demonstrate that Stokes’ formula can be modified to allow the input of gravity anomalies containing the mean tide. The discussion that was started by Heikkinen’s (1979) paper led to the first IAG resolutions on the subject, adopted at the XVII General Assembly of the IUGG in Canberra (1979). The IAG in its Resolution No. 15 (see Mueller, 1980). resolves that the tidal effect be removed completely from all geodetic observations, without restoring the permanent deformation, and that, consequently, the so-called Honkasalo correction should not be applied to observed gravity.

This (conventional) tide-free system was considered the simplest way to solve the problem caused by the Honkasalo correction to Stokes’ formula, by obtaining an anomalous potential that is harmonic outside the masses of the Earth. Instructions how to eliminate the Honkasalo correction from the IGSN71 values were subsequently published by the International Gravity Bureau’s Working Group for World Gravity Standards 4.2 Acceleration of Gravity (Uotila 1980). The problems of the tide-free system were discussed Tidal corrections to gravity measurements became im- by Ekman (1979, 1981) and Groten (1980) who indeportant when the accuracy of spring gravimeters im- pendently proposed zero tidal gravity. The IAG then proved in the 1950s. The corrections led to nontidal revised its position and in 1983 recommended the zero gravity. system (see Chap. 3). It is note-worthy that the StanHonkasalo (1964) found that tide-free gravity does dard Earth Tide Committee (Rapp 1983) did not disnot represent the time-average of the acceleration of cuss the merits of the zero-tide system, only how to the free fall. He introduced a correction (“Honkasalo implement it technically. The zero system was quickly correction”) to remedy this, i.e., to obtain mean tidal adopted for absolute gravity measurements through the gravity. The correction was applied in the Interna- IAGBN processing standards (Boedecker 1988). Subtional Gravity Standardization Net IGSN71 (Morelli sequently, all absolute gravity work and all new naet al. 1974) which is thus in the mean tidal system. tional gravity systems are in the zero system. Then Heikkinen (1979) pointed out that the permanent tide-generating force is due to masses external to the geoid. If it is retained in the free air gravity anomalies, the direct application of Stokes’ formula will lead 4.3 Positioning to erroneous results. This viewpoint (the importance of the simple appli- The tide-free approach seems to have entered the 3-D cation of Stokes’ formula without ad hoc corrections) reference frames more or less by accident (not by

The Permanent Tide in Height Systems

design), through the processing programs of the observations (VLBI, SLR, GPS). That apparently happened despite the recommendations of the IERS processing standards (McCarthy 1992, 1996; McCarthy and Petit 2004) for the mean (≡ zero) system. The situation was pointed out by Poutanen et al. (1995). In fact, attempts were made in the 1990s to go over to the mean system for the global 3-D reference frames. That however provoked strong protests from the users, as it would have abruptly changed the established coordinates of the stations by 0.1 m and more. The attempts were abandoned.

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Zero system for potential. Potential differences (or gravity-related heights) are needed to describe how water flows. Potential differences in the zero system do not fulfill this task, as the water does not care where the potential comes from and settles according to potential in the mean system. Similarly, clock rates (general relativity) do not distinguish between sources of potential. This will be discussed in the next chapters.

6 Considerations for the Future 5 Some Pros and Cons of Tidal Systems A thorough review of the advantages and disadvantages of various tidal systems was given by Ekman (1996). Here we only take up some additional points.

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Realistic tide-free system (fluid Love numbers). Apparently this has not been proposed by anyone. But many authors have considered that one of the main arguments against tide-free systems in general is that the secular or fluid Love numbers are not known or cannot be known well enough. Implicitly they are thus saying that if these numbers were known, a realistic tide-free system might be a desirable alternative. However, a realistic tide-free system would take us even further from the actual average Earth than the conventional tide-free system. Conventional tide-free system. For the 3-D positioning the advantage is that it is the present realization, i.e., we have continuity of coordinates. For the potential field quantities there does not seem to be any “standalone” advantages, but it would bring them in line with the 3-D positions. Such proposals have recently been made. Zero system for gravity acceleration. The deviation (−30. . . + 60μ Gal) of the time average of the acceleration of the free fall from the acceleration of gravity in the zero system has not turned out to be a disadvantage. Any user (say in metrology) who requires the free-fall value with this accuracy will probably need the instantaneous value. When the time variation is calculated; the permanent contribution can be restored at the same time.

A frequently evoked principle in geodetic theory and practice is the “consistency of all geodetic quantities”. It is actually used in two different meanings:

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(1) Beauty of theory; in the current context it usually boils down to the requirement that Stokes’ formula should be applicable without adhoc corrections (2) Simplicity for the user: simple relations between the commonly used geodetic quantities (in the present context, say normal heights, height anomalies, ellipsoidal heights) should hold without the user needing to worry about their tidal systems

In view of the different tidal systems of quantities currently in practical use, it seems that (2) which might be perhaps termed “the age of innocence” is not at present achievable whatever we do with the tidal system of the heights. As noted by Groten (2000), different applications may in fact require different tidal systems. Some of the major uses of a World Height System (WHS) are expected to be in the ocean sciences and in hydrographic mapping. If/when geodesists adopt a WHS in the zerotide system, a parallel system in the mean-tide system (or alternatively a set of adhoc corrections) would need to be maintained for oceanography. E.g., to start presenting the global mean sea surface (say, from altimetry) relative to a zero geoid could be misleading and lead to confusion. Similar considerations could hold for clocks (frequency standards). This is taken up briefly in the next chapter.

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7 Clocks and the Permanent Tide According to general relativity, the frequency of clocks depends on potential by df dW =− 2 f c

(2)

where f is frequency, W potential, and c velocity of light. Thus dW = −1 m2 s−2 , which corresponds e.g. to a height change of +0.1 m causes a relative frequency change d f / f = 1.1 × 10−17 . Clock frequency has been proposed as a method of determining potential differences (Bjerhammar 1975, 1985; Vermeer 1983). Bjerhammar (1985) even points at the possibility of defining the geoid as a surface of constant clock rate. Clock stability of the order of 10−17 has recently been demonstrated and 10−18 is claimed to be within sight. Now, the permanent tide-generating potential W2 influences f as the clocks do not care where the potential comes from. Thus the mean tidal system for potential is relevant for clocks. The W2 corresponds to about 3 × 10−17 in d f / f . If the zero tide system for potential is adopted, then the “reference frequency” of a stationary clock will different from its frequency averaged over time (i.e., tides) in exactly the same way as gravity in the current zero system is different from the time-average of the acceleration of the free fall. This has not been a problem for gravity; could it be a problem for clocks? This is obviously not merely a technical question but goes right into the heart of the definition and realization of Terrestrial Time.

8 Discussion A very large number of 3-D reference coordinates both for scientific and practical applications are currently given in a conventional tide-free system. Similarly the zero tidal system is well established in gravimetry. Thus it seems that a uniformity of the tidal systems in geodetic practice cannot be attained on a short term, whichever tidal system is chosen for gravity-related heights. A world height system (WHS) when implemented will probably be adopted for practical applications that we only in part can foresee now. If/when geodesists

adopt a WHS that is zero-tide, it appears that a parallel system in mean-tide system (or ad-hoc corrections) would need to be maintained at least for oceanography and perhaps in the future for clock rates. Can we at this stage identify any other WHS user group that would rather use mean-tide heights (potentials) than zero-tide heights? Similarly, are there any users outside of geodesy who emphatically require zero-tide heights? Depending on the answers to the previous questions, would it be reasonable for the geodesists to insist that their zero-tide WHS be the principal system in practical applications and the mean-tide WHS a shadow system (or a set of ad-hoc corrections)? Geodesists inputting data to boundary value problems would obviously need first to do transformations to their quantities (gravity anomalies etc.) if they are computed in a mean-tide WHS. Would this be only an aesthetic problem? These questions need to be debated in the work towards the adoption of a world height system. Acknowledgments The authors would like to thank two anonymous reviewers whose comments helped to improve the paper.

References ˚ Agren J. and R. Svensson (2007). Postglacial land uplift model and system definition for the new Swedish height system RH 2000. LMV-Raport 2007:4. Lantm¨ateriet, G¨avle, 124 p. AVISO/Altimetry (1996). AVISO user handbook for merged TOPEX/POSEIDON products. AVI-NT-02-101. Edition 3.0, 201 p. Bjerhammar A. (1975). Discrete approaches to the solution of the boundary value problem in physical geodesy. Boll. Geod. Sci. Aff. XXXIV:2, 185–240. Bjerhammar A. (1985). On a relativistic geodesy. Bull. G´eod. 59, 207–220. Boedecker, G. (1988). International Absolute Gravity Basestation Network (IAGBN). Absolute gravity observations data processing standards & station documentation. BGI Bull. Inf. 63, 51–57. Ekman M. (1979). The stationary effect of the Sun and the Moon upon the gravity of the Earth, and some aspects of the definition of gravity. Uppsala University, Geodetic Institute, Report 5. Ekman M. (1981). On the definition of gravity. Bull. G´eod. 55, 167–168. Ekman M. (1989). Impacts of geodynamic phenomena on systems for height and gravity. Bull. G´eod. 63, 281–296. Ekman M. (1996). The permanent problem of the permanent tide. What to do with it in geodetic reference systems? Bull. Inf. Mar´ees Terrestres 125, 9508–9513.

The Permanent Tide in Height Systems Groten E. (1980). A remark on M. Heikkinen’s paper “On the Honkasalo term in tidal corrections to gravimetric observation”. Bull. G´eod. 54, 221–223. Groten E. (2000). Report of Special Commission 3 of IAG. In: K.J. Johnston, D.D. McCarthy, B.J. Luzum, G.H. Kaplan (eds), Towards models and constants for submicroarcsecond astrometry. Proceedings of IAU Colloquium 180, U.S. Naval Observatory, Washington, DC, USA, 27–30 March 2000, Washington, DC: U.S. Naval Observatory, 2000, pp. 337–352. Heck, B. (1993). Tidal corrections in geodetic height determination. In: Linkwitz, K.; Eisele, V.; M¨onicke, H.-J. (eds): International Symposium. on Applications of Geodesy to Engineering, Stuttgart 1991. IAG Symposia 108, pp. 11–24. Springer. Heck, B. (2004). Problems in the definition of vertical reference frames. In: Sans`o, F. (ed); V Hotine-Marussi Symp. on Mathematical Geodesy, Matera 2003. IAG Symposia 127, pp: 164–173. Springer, Heikkinen M. (1978). On the tide-generating forces. Publ. Finn. Geod Inst. 85, 150 p. Heikkinen M. (1979). On the Honkasalo term in tidal corrections to gravimetric observations. Bull. G´eod. 53. 239–245. Honkasalo T. (1964). On the tidal gravity correction. Boll. Geofis. Teor. Applicata. VI:21, 32–36. Ihde J. and W. Augath (2001). The vertical reference system for Europe. In: J.A. Torres and H. Hornik (eds), Report on the Symposium of the IAG Subcommission for Europe (EUREF) held in Troms¨o, 22–24 June 2000. Ver¨offentlichungen der Bayerischen Kommission f¨ur die Internationale Erdmessung, Astronomisch-Geod¨atische Arbeiten 61, 99–101. Ihde J. et al (2007). Conventions for the definition and realization of a conventional vertical reference system (CVRS). IAG Inter-Commission Project ICP1.2 Vertical Reference Frames. Presented at the XXIV General Assembly of the IUGG, Perugia, Italy, July 2–13, 2007, 25 p. Jensen H. (1949). Formulas for the astronomical correction to the precise levelling. Geodætisk Institut, København, Meddelelse 23. McCarthy D.D., ed. (1992). IERS Standards. IERS Technical Note 13, Observatoire de Paris, Paris. McCarthy D.D., ed. (1996). IERS Standards. IERS Technical Note 21, Observatoire de Paris, Paris. McCarthy D.D. and G. Petit, eds. (2004). IERS Conventions (2003). IERS Technical Note 32. Verlag des Bundesamts fur Kartographie und Geod¨asie, Frankfurt am Main.

87 Morelli C, C. Gantar, T. Honkasalo, R.K. McConnell, J.G. Tanner, B. Szabo, U. Uotila and C.T. Whalen (1974). The International Gravity Standardization Net 1971 (I.G.S.N.71). International Association of Geodesy, Publ. Spec. No. 4. Paris. Mueller I.I. (ed.) (1980). The Geodesist’s Handbook. Bull. G´eod. 54:3. Pavlis E.C. (1998). On the reference frames inherent in the recent geopotential models. In: M. Vermeer and J. Adam (Eds), Proceedings of the Second Continental Workshop on the Geoid, Hungary, Budapest, March 10–14, 1998. Rep. Finn. Geod. Inst. 98:4, pp. 29–40. Poutanen M. M. Vermeer and J. M¨akinen (1995). The permanent tide in GPS positioning. J. Geodesy 70, 499–504. Rapp R.H. (1983). Tidal gravity computations based on recommendations of the Standard Earth Tide Committee. Bull. Inf. Mar´ees Terrestres 89, 5814–5819. Rapp R.H., R.S. Nerem, C.K. Shum, S.M. Klosko and R.G. Williamsson (1991). Consideration of permanent tidal deformation in the orbit determination and data analysisi for the Topex/Poseidon mission. NASA Technical Memorandum 100775, 11 p. Saaranen V., P. Lehmuskoski, P. Rouhiainen, M. Takalo and J. M¨akinen (2007). The new Finnish height system N2000. Proceeding of the EUREF Symposium, Riga, 14 –17 June 2006. Mitteilungen des BKG, Frankfurt a/M, in press. Tcherning C.C. (ed) (1984). The Geodesist’s Handbook. Bull. G´eod. 58:3. Uotila U.A. (1980). Note to the users of International Gravity Standardization Net 1971. Bull. G´eod. 54, 407–408. Vermeer M. (1983). Chronometric levelling. Rep. Fin. Geod. Inst. 82:2, 7 p. Vermeer M. and M. Poutanen (1997). A modified GRS-80 normal field including permanent tide and atmosphere. In: J. Segawa, H. Fujimoto, S. Okubo (eds), Gravity, Geoid and Marine Geodesy. International Symposium, Tokyo, Japan, September 30–October 5, 1996. IAG Symposia 117, pp. 515–522. Springer. ˇ Yurkina M.I., Z. Simon and A. Zeman (1986). Constant part of the Earth tides in the Earth figure theory. Bull G´eod. 60, 339–343. Zadro M.B. and A. Marussi (1973). On the static effect of Moon and Sun on the shape of the Earth. Boll. Geod. Sci. Aff. XXXII:4, 253–260.

Combined Adjustment of National Astro-Geodetic Network and National GPS2000 Geodetic Network Cheng Pengfei, Cheng Yingyan and Mi Jinzhong

Abstract The combined adjustment between the National Astro-Geodetic Network and the National GPS2000 Geodetic network in China is described. In this paper, measurements of astro-geodetic network had been taken for nearly 60 years from early 1920’ to 1980’. Some astro-geodetic points may have moved during such a long time period, so coordinate analysis for points co-located in the two networks is very important. The factors which influence combined adjustment, such as earth’s crustal movement, the change of astronomic system, geoid improvement, etc., must be analyzed in advance. In this paper, the measurements and their accuracies are also introduced together with mathematical and stochastic model used in the adjustmen. Due to the huge number of unknowns (about 150 thousands), the solution strategy for large scale equation using PC is also described. Satisfied results are obtained after the combined adjustment.

Keywords Astro-geodetic network · 2000 GPS network · Combined adjustment · Covariance component estimation

Cheng Pengfei Chinese Academy of Surveying and Mapping, 16 Beitaiping Road, 100039, Beijing, People’s Republic of China Cheng Yingyan Chinese Academy of Surveying and Mapping, 16 Beitaiping Road, 100039, Beijing, People’s Republic of China

1 Introduction Both 1954 Beijing Coordinate System (BJ54) and 1980 Xi’an Coordinate System (Xi’an80) have been used as national coordinate systems in China for more than 50 and 20 years, respectively. The BJ54 was directly introduced from the coordinate system used in the former Soviet Union, which was based on Krasovsky’s ellipsoid. The Xi’an80 was established through adjustment of the National Astro-Geodetic Network (NAGN) in 1982, which was based on IAG75 ellipsoid and the sum of squared height anomalies was minimized over mainland China. The NAGN consists of nearly 50,000 control points, its accuracy is only about 10−5 –10−6 due to lack of external control. Obviously, such an accuracy can not meet current demands of various applications. Also, with the development of modern surveying technologies, control points in geocentric coordinate system are needed for many applications such as high-precision aerial photography. In 1990’s three national GPS networks were established by the State Bureau of Surveying and Mapping (SBSM), the Military Surveying and China earthquake administration, respectively. In 2003, the three GPS networks were adjusted in ITRF2000 and resulted in the so-called national GPS2000 network with a baseline accuracy of 10−7 –10−8 . The purpose of re-adjustment of the astrogeodetic network is to tie the triangulation points with ITRF2000 and to enhance the accuracy of the network.

Mi Jinzhong Chinese Academy of Surveying and Mapping, 16 Beitaiping Road, 100039, Beijing, People’s Republic of China

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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2 Data Analysis 2.1 Data Description The combined adjustment concerned two networks, i.e. GPS2000 and NAGN. The coordinate components and their variance-covariance of the co-located points from the GPS2000 are used, meanwhile, raw measurements from NAGN such as astronomical azimuths, directions and EDM distances, zero-order traverses are used together with the baselines between GPS and triangulation points. New measurements of 1,688 points from five sub-networks of order II are used to replace the old ones of 1,375 points. All 326,902 measurements for 50,970 points were analyzed among which 48,919 points were finally chosen including 336 co-located points between NAGN and GPS2000. 314,976 directions, 2,146 EDM distances, 1,064 astro-azimuths, 72 zero-order traverses and 1,022 baselines were processed during the adjustment. The number of observations is shown in Table 1. Table 1 Observation type in combined adjustment Observation type

Observations analyzed

Observations adopted

Astronomical azimuths Directions EDM distances Zero-order traverses Baselines between GPS and triangulation points

1,086 326,902 2,159 72 1,022

1,064 314,976 2,146 72 1,022

considered. The influences of deflections of the vertical to the ground measurements were considered. Analysis shown that the conversion of deflections of the vertical from Xi’an80 ellipsoid into ITRF2000 is needed, because the origin and orientation differs a lot between the two systems.

3 Mathematical Model Helmert’s block adjustment is used for combining NAGN with GPS2000 network. All measurements are reduced to WGS84 ellipsoid.

3.1 Observation Equation (1) Distance observation equation For end points i, j with longitude and latitude L i , Bi , L j , B j , we use the following symbols: 1 b = B j − Bi , B = (Bi + B j ),  2 t = tg B, c = a 1 + e 2 , η2 = e 2 cos2 B, V 2 = 1 + η2 , N = c/V, l = L j − L i , m = l cos B Where, a is semi-major axis of the ellipsoid and e is the second eccentricity. Then observation equation yields [Gu Dansheng, 1997]. Vsi j = ai j d Bi + bi j d L i + ci j d B j

2.2 Factors to be Considered The following factors must be considered before the adjustment. Firstly the influence of the plate movement was analyzed, it was neglected because better results were attained without this correction, and it is impractical to convert the change of position into raw measurements while the boundary between neighboring plates are not accurately defined. Secondly, the influences to the astronomical measurements due to the change from the fourth fundamental Catalogue (FK4) to the fifth fundamental Catalogue (FK5) were estimated. Comparisons shown that the influence of datum conversion is quite small for astronomic latitude, longi-tude, and astro-azimuth. Therefore, only the correction of the earth pole (JYD 1968.0 for NAGN) to CIO(IRP) is

+ di j d L j + Si j × 10−6 S + li j

(1)

with Psi j as weight for distance, and  N2 t 2 bm 2

2 − m 1 + 3t b + ai j = − 2 12 30ρ

V 4 S  t 2 − 3b − m 2 η2 2

N2 b2 t 2 2 b2 t 2 bi j = − m cos B 1 − − m − 30ρ

S 12 6 4  N2 t ci j = b − m2

4 2 30ρ V S

  2 bm t 2 − 1 + 3t 2 + 3b − m 2 η2 12 2 di j = −bi j

Combined Adjustment of National Astro-Geodetic Network

Where ΔS is the scale factor between the ground network and GPS network. One scale is estimated for the whole network. (2) Azimuth observation equation V Ai j = pi j d Bi + qi j d L i + ri j d B j + ti j d L j + A + l Ai j

(2)

With p Ai j as weight for azimuth observation. In which,    m b2 tb

m 2 q 1− − 1 + 2η − p 12 2 4    1 l2 b cos B sin B = q 1+ − 300 p 6 2     1 b2 tb

m m 2 = q 1− + 1 + 2η + 300 p 12 2 4

b2 − m 2 l2 b2 =1 + η2 + + p 12 24  m2

− 2 3b2 − m 2 η4 p   ⎧

   ⎨ Ai j − z i − Ni j ρ

Ai j − z i − Ni j  < 1 =

     ⎩ A − z − N + 2π ρ

A − z − N  ≥ 1

1 pi j = 300 qi j ri j q

li j



ij

i

ij

ij

i

ij

Where B, m, t have the same meaning as in equation (1), p = b2 + m 2 , Δ A is orientation parameter of astro-geodetic network to GPS2000 network. The approximate azimuth Ai j between point i to j can be obtained using Bessel backward equation from their approximate coordinates, Z i is approximate orientation with positive sign, Ni j is the direction observation. (3) Direction observation equation Vi j = −dz i + pi j d Bi + qi j d L i +ri j d B j + ti j d L j +li j (3) With p Di as the weight for direction observation.

91

where X 1 refers to unknown parameters of inner points (points within block), Y is unknown vector of common points (points on dividing line between neighboring blocks and GPS points). The number of unknowns is more than 150 thousands. Thus, the key problem of software design is to save memory and to seek for effective algorithm. The strategies are described as follows: (1) To reduce the number of unknown parameters by eliminating orientation angle in each group of directions within blocks, and to keep orientation angles of directions for points on dividing line. (2) A11 is stored in one dimension arrays and in lower triangle matrix with variable bandwidth. (3) Only nonzero elements in A21 are stored in one dimensional array with the method suggested by (Gaustavson, 1972). The method takes the advantage of less memory and is convenient for matrices manipulations, such as summa-tion, multiplication etc. The basic idea is that, for a sparse matrix, two index arrays is designed to store row number and column number information of nonzero elements in matrix A21 respectively. Figure 1 gives an example: There are 5, 2, 2 nonzero elements in row 1, 2, 3 respectively, IA is row index and JA is column index, so the index is IP = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 . . . . . . AN = 11 14 17 18 19 22 28 33 34 41 54 55 79 81 ...... IA = 1 6 8 10 11 13 13 14 16 . . . . . . JA = 1 4 7 8 9 2 8 3 4 1 4 5 9 1 . . . . . . Index IA indicates first IP of each row in AN array, so the number of nonzero elements in i t h row is IA(I+1)IA(I).

11 ∗

∗

41

3.2 Solution of Normal Equation

A11 . . . .. A21 A22

  i

X1 Y





D1 = D2

∗

22

∗

∗

∗



14 ∗

33 34 ∗

∗

∗ ∗

17 18 19 28 ∗

∗ ∗ ∗ ∗ ∗ ∗

∗

∗ ∗ ∗

∗

∗

∗

∗ ∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗ ∗ ∗

∗

∗

∗

∗

∗

∗

∗ ∗ ∗

∗

∗

∗

79

81 82

For ith block, the normal equation reads: 

∗

∗

∗

54 55

93 94

∗ ∗ ∗

∗

∗ ∗ ∗

∗

∗

∗

(4) Fig. 1 Sparse elements in A21 for demonstration

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Part elements inversion can be obtained through (4) Combined with outer memory Only matrix A22 stay in memory while A21 and A11 block decomposition or column decomposition. The are written into files when reduced matrix correspond- formula for matrix inversion is ing to that block is formed, it is helpful to save memory.      A11 A12 M11 M12 I11 0 (5) To solve the normal equation and to estimate rms = (8) 0 I22 A21 A22 M21 M22 of unknowns with Cholesky factorization instead of inverting a huge matrix. 

T −1 T T −1 A11 A12 M12 = −A12 A11 I11 A22 − A12 From Eq. (4), the reduced equation is



 T −1 A11 A12 M22 = I22 (9) A22 − A12 −1 A22 − A21 A−1 A − A A D (5) Y = D 2 21 11 1 11 12

T T −1 T M12 M11 = A−1 11 I11 − A12 A11 I11 supposed: A−1 11 A12 = C then A11 C = A12 The matrix A12 could be considered as n 2 colThe formula for part column elements inversion of T and Q is (Liu changxue, 1990): umn vectors, so the corresponding column vector Q 12 22 C = (C1 , C2 , C3 . . . ..) can be solved as normal equaT T ˜ T T ˜ tion with Cholesky factorization. The final parameter M12 Q 22 = I˜22 M22 Q 12 = I˜22 vector Y can be solved from the following equation: where I˜11 , I˜22 are unit vectors related to column which ˜T,M ˜ T refer to (A22 − A21 C)Y = D2 − C T D1 (6) inversion need to be determined, and M 12 22 corresponding inversion of column matrices. The block parameters X i can be solved through backThe position accuracy of ith point can be estimated ward solution. by Mp =

3.3 Normal Root Mean Square The normal root mean square of the network can be computed by σˆ 2 =

Vˆ T P Vˆ + Y2T PY 2 Y2 m − tx

(7)

 1 μ × Q Bi Bi (Mi /ρ)2 300 2 1/2 +Q L i L i (N j cos B j )/ρ

(10)

Where μ is rms of unit weight, N j is radius of curvature in prime vertical, Mi is radius of curvature in meridian. The rms of side and azimuth can be obtained from observation equation and variance-covariance of two end points according to the rule of error propagation.

where Vˆ T P Vˆ is the sum of weighted square of residuals and Y2T PY 2 Y2 is the sum of weighted square of coordinate corrections of GPS points, m is the number of observations, t x is the number of unknown parameters. 3.5 Helmert Covariance Component

Estimation

The ground network concerned multi-type observations, including direction, traverse, astro-azimuth, etc., with different orders of accuracy for each. The best Because no matrix inversion is done in solving the weight ratios of GPS network to ground network, equation, the variance and covariance of all parame- weights for different observations, and for different ters could not be obtained as byproducts. Coordinate orders of the same kind of observation are estimated. accuracy can be estimated at points appointed beforeFor m observations, rigorous Helmert covariance hand or at all points by changing right hand vector of component can be estimated as follows: normal equation.

3.4 Accuracy of Unknowns

Combined Adjustment of National Astro-Geodetic Network θˆ = Wθ m×1 ⎡     2  n 1 − 2tr N −1 N1 + tr N −1 N1 tr N −1 N1 N −1 N2  −1   −1 2 ⎢ S=⎣ n 2 − 2tr N N2 + tr N N2 ...............   ⎤ . . . .. tr  N −1 N1 N −1 Nm  −1 −1 ⎥ .... tr N N2 N Nm ⎥ ⎦ . . . .. . . . . . . ..    2 . . . .. n m − 2tr N −1 Nm + tr N −1 Nm S

m×m m×1

Wθ

m×1

= V1T P1 V1

V2T P2 V2

V3T P3 V3

 VmT Pm Vm

T 2 2 2 2 σˆ 02 σˆ 03 . . . ..σˆ 0m θˆ = σˆ 01

. . . . . . ..

m×1

(11)

93

2 tr N −1 Ni = ti A + λ2Ai jk ti j A +

+ . . . + 2 ti B + λ2Bi j ti j B  + λ2Bi jk ti jk B + . . . . 

λ Ai j ti j A + tr N −1 Ni = ti A +

+ . . . + 2 ti B + λ Bi j ti j B  + λ Bi jk ti jk B + . . . . 

tr N −1 Ni N −1 N j = λ Ai j λ A ji ti j A + 2λ Bi j λ B ji ti j B + +

Regarding to directions of triangulation network, if m directions of different orders were observed, then the triangulation points can be flagged as order one, order two, order three, and so on. The observation equations with different weight flag is (Cui xizhang, 1992):

V2 =A22 Z 2 + . . . . . . A21 Z 12 + A23 Z 13 + . . . . . . + A213 Z 123 + . . . .. + B22 X 2 + . . . .. + B21 X 12 + . . . + B213 X 123 + . . . . . . − L 2

(12)

λ Ai jk ti jk A

λ Ai jk λ Ai jk A ti jk A

2λ Bi jk λ B jik ti jk B + . . . . . .

Where the subscript A and B are orientation angle parameters and coordinate unknowns respectively, ti is number of points with flag i, ti j is the number of points with weight flags i and j, ti jk is the number of points with weight flags i j and k. The coefficients are expressed as:

V1 =A11 Z 1 + . . . . . . A12 Z 12 + A13 Z 13 + . . . . . . + A123 Z 123 + . . . .. + B11 X 1 + . . . .. + B12 X 12 + . . . + B123 X 123 + . . . . . . − L 1

λ2Ai jk ti jk A

λ Ai j =

N Ai j N Ai j + N A ji

λAji =

N A ji N Ai j + N A ji

λ Ai j + λ A ji = 1 N Bi j λ Bi j = N Bi j + N B ji

N B ji λAji = in which, Ai j is coefficient matrix of orientation angle N Bi j + N B ji parameters, Bi j is coefficient matrix of coordinate unλ Bi j + λ B ji = 1 knowns X , and Z i is orientation angle unknowns with ......... only one weight flag, Z i j is orientation angle unknowns with two weight flags i and j, X i refers to coordinate unknown with flag i, and X i j refers to coordinate unknown with flag i and j. Equation (12) can be rewritten 4 Adjustment and Solution in matrix form as: V = AZ + B X − L

4.1 Data Distribution (13)

Suppose the weights of m different orders be: P = The whole network is divided into 16 blocks (numberdiag[P1 P2 P3 P4 . . . . . . . . . Pm ], and omitted the off- ing from east to west), 1,822 points located on dividing lines. Detailed information is listed in Table 2: diagonal elements, the normal matrix is: N = diag[N A11 N A22 . . . .N A12 + N A21 . . . ..N A123 + N A213 + N A312 . . . ..

4.2 Stochastic Model and Weight (14) Determination

N B11 N B22 . . . ..N B12 + N B21 . . . ..N B123 + N B213 + N B312 . . . . . . . The traces of matrices can be obtained:

(1) Unknown parameters Coordinate unknowns are set as follows: two for points in blocks, i.e. (B, L), three for co-located points, i.e.

94

C. Pengfei et al. Table 2 The statistics of data distribution for different blocks Reg

Total Num

Point in Reg

Point in Ln

GPS Num

Direct

Side

AZ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Sum

2210 3193 2337 3891 4657 4347 4577 4508 3839 3258 2735 3199 2448 1972 1578 170 48, 919

2116 3058 2216 3670 4404 4126 4347 4313 3709 3109 2636 3096 2347 1902 1543 169 46, 761

83 121 107 199 216 195 198 170 106 123 83 80 76 45 20 0 1822

11 14 14 22 37 26 32 25 24 26 16 23 25 25 15 1 336

14809 21379 15759 25637 31634 29418 31566 30938 24986 20237 16531 18373 13705 10168 8813 1023 314, 976

39 20 21 40 104 82 79 131 72 98 237 290 268 434 227 4 2146

74 38 44 66 116 60 56 94 78 74 74 88 66 80 52 4 1064

(B, L, H ), additional observation equations for leveling and triangulation height are added. (2) Stochastic model The weights for direction, EDM and azimuth measurement are listed in Tables 3 and 4 respectively, which are based on post variance obtained from the individual adjustment of the NAGN in early 1980’s (Technical report, 1982), the rms of unit weight is taken as 1.0. (3) Stochastic model of GPS2000 network The absolute position accuracies of GPS2000 are at centimeter level in ITRF2000, and relative accuracy reaches millimeter level. Baseline accuracies are better than 10−7 . Prior weight in stochastic model of GPS2000 network adopted that for “national GPS 2000 adjustment”. (4) scale ratio of NAGN to GPS2000 is 1.0

4.3 Accuracy Estimation

Position accuracies of total 48,919 points, accuracies of 2,146 distance and 1,046 azimuth are estimated: (1) Position accuracy of NAGN Position accuracies of 48,583 ground points (except for 336 co-location points) are estimated, the statistics show that rms of 30,803 points are within 0.1 m (about 63.4% of points), 46,375 points within 0.3 m (about 95.5%), 658 points are over 0.5 m (about 1.3%). Average accuracy is 0.11 m. The maximum rms is 1.45 for a point located in Basin Talimu in Xinjiang province (B = 39◦ 31 , L = 80◦ 55 ) where only one traverse crossing the basin and no GPS points nearby. (2) Position accuracy of GPS points Position rms of 336 co-located points are generally at centimeter level, among which 233 points are within 0.01 m (69.4%), 314 points are within 0.03 m (93.6%), Table 3 A prior weight of direction observations 335 points are within 0.05 m (99.7%), only one point Chain 1 Order 1, Chain 2 Order 3, Patch 2, with rms being 0.06 m. The average position accuracy traverse 1 traverse 2 traverse 3 for GPS points is 0.01 m. m 0.50

0.65

0.82

1.19

1.46

(3) Accuracy of azimuths wt 4.00 2.37 1.49 0.71 0.47 Rms of total 1,064 azimuths are estimated, the distribution is shown in Table 5. The average rms is 0.33

, with a maximum of 0.56

. (4) Relative accuracy of distance measurements for Table 4 A prior weight of azimuth, baseline observation NAGN type rms weight The relative accuracies of total 2,146 distance meaazimuth 0.8(sec) 1.6 2 4 4 2 baseline Si /70 × 10 (m) (70 × 10 ) /Si surements are estimated. The distribution is shown in Baseline extending Si /23 × 104 (m) (23 × 104 )2 /Si 2 Table 6. The average rms is 4.275 ppm.

Combined Adjustment of National Astro-Geodetic Network

95

Table 5 The distribution of azimuth rms (unit: arc second) Number %

≤ 0.1

0.1 ∼ 0.2

0.2 ∼ 0.3

0.3 ∼ 0.4

0.4 ∼ 0.5

0.5 ∼ 0.6

> 0.6

0 0

91 8.6

337 31.7

391 36.7

214 20.1

31 2.9

0 0

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Savg (m)

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Side Num

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Baseline extending Basclinc, zero order traverse NASM-2A CBB-1 NASM-4D MRA/CW DI/50, DM-20 AGA-8 AGA-600 K&E-III Average

0.358 0.169 0.296 0.482 0.616 0.825 1.030 0.222

20531.083 10932.807 14117.149 10312.644 12679.795 14901.689 17759.790 22756.462 16997.503

1.869 1.563 2.798 4.826 5.447 6.162 6.010 1.025 4.275

1/570000 1/650000 1/480000 1/210000 1/210000 1/180000 1/170000 1/1030000 1/280000

261 39 270 12 29 905 287 343

(5) Relative position accuracy of NAGN Relative position accuracies of 4,552 pairs of points are shown in Fig. 2 which shows average relative position accuracies of different length. (6) Estimation of variance components The variances of different orders of observation are estimated with Helmert’s covariance component

estimation, see column 5 in Table 7, rms of directions is listed in column 6. From Table 7 one can see that the posterior rms of directions is nearly the same as aprior one. There is a little difference for weight flag 3, standing for old basic triangulation chains of order two. Flag 6 indicates items related with azimuth. After combined

Fig. 2 The avarage relative rms of indirect points of territory network

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Table 7 Aprior and post rms of directions (unit:s) ! 2 Aprior rms Obs number PVi σ20i

Post rms σi

1 2 3 4 5 6

0.50 0.65 0.82 1.19 1.46 0.80

0.46 0.63 0.75 1.18 1.41 0.84

25,993 128,672 7,695 104,603 48,013 1,064

14,290.528 61,260.697 3,979.165 53,123.399 25,258.279 993.278

0.85 0.92 0.83 0.98 0.93 1.12

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adjustment, remarkable improvements are for achieved for the NAGN, see Table 8.

5 Concluding Remark Measurements of 310,000 directions, 3,000 distances and 1,064 azimuths are involved in the combined adjustment with 157,892 unknown parameters. The ratio of aprior rms of unit weight to the posterior one is 1.0025, which is quite close to 1. The coordinates of 48,919 points are obtained in ITRF2000 with position aceuracies better than 0.1 m for most of the points. The results show that the accumulation errors are greatly reduced after combined adjustment, and accuracy is obviously improved for distance, orientation, etc.. With nearly 50,000 of control points in ITRF2000, transformation between ITRF2000 and Xi’an80 can be easily done with high accuracy for different applications.

References Cui Xizhang et. al, (1992). Generalized Adjustment (edition 2) [M], Beijing: publication of surveying and mapping. Gu Dansheng (1997). A Group of High Accuracy and Practical formulae for computer on ellipsoid [J]. Bulletin of Surveying and Mapping.

Azimuth RMS

Gaustavson, F.G. (1972): Some basic techniques for solving sparse systems of linear equations, In Rose and Willoughby. Huang Weibing (1992). Modern Adjustment Theory and Application [M], Beijing: publication of PLA. Liu Changxue (1990). The Computation Method of Super Large Scale Sparse Matrix [M]. ShangHai Scientific and Technical Publishers. Tian Yuanping etc. (1998). A Simplified Algorithm of Helmert’s Variance Components Estimation [J], Heilongjiang: Engineering of Surveying and Mapping. Technical report (1982). “nationwide GPS 2000 adjustment” Wang Zhizhong (1998). Maximum Posterior Estimation of Variance Components [J], Heilongjiang: Engineering of Surveying and Mapping. Wang Songgui (1999). Improved Estimators of Variance Components [J], Beijing: ACTA Mathematicae Applicatae SINICA. Xiong Jie (1988). Geodesy on Ellipsoid [M]. PLA Publication, Yu Zhongtao etc. (1993). The Symposium of Error Theory and Application of Adjustment Model [D], Beijing: publication of surveying and mapping.

Symposium GS002

Gravity Field Conveners: C. Jekeli, U. Marti, S. Okubo, N. Sneeuw, I. Tziavos, G. Vergos, M. Vermeer, P. Visser

Improved Resolution of a GRACE Gravity Field Model by Regional Refinements A. Eicker, T. Mayer-Guerr and K.H. Ilk

Abstract The available gravity field models derived from data of the GRACE mission (Tapley and Reigber (2001)) have provided us with an unprecedented accuracy in gravity field determination. Nevertheless the projected GRACE baseline accuracy has not been achieved yet. One reason out of many could be the insufficient modelling of the satellite data by a global representation by means of spherical harmonics. To extract the signal information present in the satellite and sensor data to full content, it seems reasonable to improve global solutions by regional recovery strategies. Especially in the higher frequency part of the spectrum the gravity field features differ in different geographical areas. Therefore the recovery procedure should be adapted according to the characteristics in the respective area. In the approach presented here in a first step a global gravity field represented by a spherical harmonic expansion up to a moderate degree has to be derived. It is then refined by regionally adapted high resolution refinements being parameterized by splines as space localizing base functions. In this context a special attention is paid to the signal to noise ratio in different geographical areas. The approach is demonstrated by examples based on the analysis of the original GRACE Level 1B data. A. Eicker Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, D-53115 Bonn, Germany, e-mail: [email protected] T. Mayer-Guerr Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, D-53115 Bonn, Germany K.H. Ilk Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, D-53115 Bonn, Germany

Keywords GRACE · Regional gravity field recovery · Space localizing basis functions · Regularization

1 Introduction The regional gravity field recovery strategy presented here is based on a global reference field. The global solution is parameterized by spherical harmonic coefficients and the regional solutions are represented by spherical splines as space localizing base functions. This procedure provides several advantages. The regional approach allows to exploit the individual signal content in the observations and a tailored regularization for regions with different gravity field characteristics. The advantage compared to a uniform global regularization is that the regularization factor is selected for each region individually such that there is no overall filtering of the observations which would lead to a mean dampening of the global gravity field features. Even within a recovery region different regularization areas with individual regularization parameters can be assigned to take care of the varying signal content as described in Sect. 3. This way it is possible to extract more information out of the given data than would be possible with a global gravity field determination. Areas with a smooth gravity field signal for example can be regularized more strongly without dampening the signal, and in areas with strong high frequency signal no unneccessary dampening has to be performed. Furthermore, besides a high resolution static gravity field, GRACE enables the recovery of temporal gravity variations as well. These temporal variations are often regional phenomena, thus a regional representation seems reasonable. First results are presented in Sect. 5.2.

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The parameterization of the potential is performed in terms of spherical harmonics for the global solution, in case of a regional gravity field recovery the potential In the following the observation equations that estabhas to be parameterized by space localizing base funclish the relationship between the unknown gravity field tions. It can be modeled as a sum of base functions as functionals and the observations provided by GRACE follows data are to be derived. This functional model is a type I of Fredholm integral equation of the first kind, which V (r P ) = ai (r P , r Q i ), (5) represents a solution of a boundary value problem to i=1 Newton’s equation of motion formulated for short arcs of the satellite orbit as applied to GRACE data by with the field parameters a arranged in a column mai Mayer-Guerr (2006). The boundary values of the short trix and the base functions arc are denoted by r A and r B and the integral equation # " N can be formulated according to R E n+1 (r P , r Q i ) = kn Pn (r P , r Q i ). (6) r n=0 r(τ ) = (1 − τ )r A + τ r B The coefficients kn are the degree variances of the 1 gravity field spectrum to be determined, 2

−T K (τ, τ ) f(τ ) dτ . (1)

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τ =0

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kn =

2 2 (cnm + snm ),

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),

, τ ≤ τ τ (1 − τ preted as isotropic and homogeneous harmonic spline (3) K (τ, τ ) =

functions (Freeden et al. (1998)). The nodal points are τ (1 − τ ), τ ≤ τ, generated on a grid by a uniform subdivision of an icosahedron of twenty equal-area spherical triangles. The force function f indicates the specific force In this way the global pattern of spline nodal points Q i function acting on the satellite. It contains among shows approximately uniform nodal point distribution. different disturbing forces the unknown Earth gravity The observation equations are established for short field. The observations delivered by GRACE are arcs over the selected regional recovery area, while precise intersatellite measurematical model for those the coverage with short arcs should be slightly larger observations can be derived by projecting the relative than the recovery region itself to protect the solution vector between the two satellites onto the line-of-sight from geographical truncation effects. Every short arc connection. This can be formulated for the case of the builds a partial normal equation. To consider differintersatellite ranges by ent accuracies of the short arcs these normal equations are combined by estimating a variance factor for evρ(τ ) = e12 (τ ) · (r2 (τ ) − r1 (τ )), (4) ery arc by means of variance component estimation as described by Koch and Kusche (2001). In case of with r2 (τ ) and r1 (τ ) denoting the positions of the two a regularization the regularization factor can be deterGRACE satellites and e12 being the unit vector in line- mined by this same procedure of variance component estimation as well. of-sight direction. τ=

t − tA T

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3 Regionally Adapted Regularization Due to the ill-posedness of the downward continuation process the solution has to be regularized. Here the Tikhonov regularization has been applied and the regularization parameter has been determined by means of variance component estimation as proposed by Koch and Kusche (2001). The process of variance component estimation delivers the optimal regularization parameter under consideration of the given signal to noise ratio. In case of a regional gravity field determination this results in one regularization parameter optimally tailored to the respective region. This is an improvement in comparison to a global gravity field parameterization which allows only one regularization factor for the complete Earth resulting in an overall mean damping of the gravity field features. But even within smaller geographical areas the gravity field features may vary significantly. Therefore it seems reasonable to further adapt the regularization procedure. The proposed approach takes not only one regularization matrix with one associated regularization parameter per region into account but allows several matrices with respective parameters: N=

1 T 1 1 A PA + 2 R1 + · · · + 2 Rn . 2 σ σx1 σxn

(8)

Fig. 1 Different spline points belonging to different regularization areas

each unknown parameter is related to a particular geographical location as exemplarily illustrated in Fig. 1. This is an inevitable premise when the elements of the regularization matrix are supposed to be assigned to a certain region.

4 Regional Gravity Field Recovery

As mentioned in Sect. 1 the regional solutions are calculated as refinements to a global gravity field. This global reference field is commonly represented by spherical harmonics and the regional solution, represented by space localizing splines, models the residual gravity field. Presented in this paper are two  different calculations. In the first one the EGM96 1 for j inside i R( j, j) = (9) (Lemoine et al. (1998)) was chosen as global reference 0 for j outside i field and regional refinements have been calculated from one month of GRACE data. The calculations of The original identity matrix as applied in the monthly solutions has the advantage that a solution deTikhonov regularization process has thus been divided termined from more data (and thus assumably higher into single diagonal matrices according to the mem- accuracy) can serve as comparison and accuracy bership of the respective unknown parameters to the measures can be obtained. For this one-month solution different regularization groups. the resolution as defined by the grid of spline nodal points corresponds to a spherical harmonic degree R1 + · · · + Rn = I (10) of N = 140, which results in approximately 153 km nodal point distance. The spline kernel according to The possibility of adapting the regularization proce- Eq. (6) was expanded up to this maximum degree as dure in this particular way is a unique feature of a field well. The results are described in Sect. 5.1. In the secparameterization by space localizing basis functions, as ond calculation the global gravity field ITG-Grace02s For each regional regularization group i the regularization matrix Ri is a diagonal matrix that contains a one for each regional spline parameter located inside this particular region and a zero for parameters belonging to basis functions outside the regularization group.

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(Mayer-Guerr (2006)) served as reference field and a high resolution regional refinement was determined from 4,5 years of GRACE data. Its resolution can be compared to a maximum spherical harmonic degree of N = 170, the spline point distance averages about 126 km. In the same calculation step a lower resolution time variable regional gravity field was calculated from the same data set. Therefore a second set of spline kernels was introduced, the shape of the kernel and the grid pattern corresponding to N = 60. To represent the time variabilities the spline coefficients ai in Eq. (5) were assumed to be time variable and a trend and a yearly period was estimated. Results can be found in Sect. 5.2.

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Figures 2 and 3 show the differences between the regional solution calculated from one month of data for the area of the Andes compared to the ITG-Grace02s. The global gravity field ITG-Grace02s was calculated from 3 years of GRACE data, therefore it is expected 250° 10°

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to be of superior accuracy and is used as reference solution. In Fig. 2 one uniform regularization parameter was determined for the region and in Fig. 3 the regularization procedure was adapted by calculating two different paramters, one for the ocean area and one for the continental areas. The introduction of the additional regularization area improves the RMS significantly from 16.51 to 12.86 cm. In case of the displayed area of the Andes region the choice of continental areas and ocean areas as the two regularization regions proves to be reasonable, as here the signal on the ocean is significantly less rough compared to the Andes themselves. In other areas having strong high frequency gravity signal on the oceans due to deep ocean trenches etc. a different choice of the regularization regions might be neccessary. To evaluate the quality of the one-month regional spline solution, it was compared to the same month of the GFZ-RL04 monthly gravity fields. The EigenGL04C was used as reference solution and the differences to the two monthly solutions compared to the Eigen-GL04C are displayed in Figs. 4 and 5. The comparison reveals that the regionally refined spline solution matches the Eigen-GL04 significantly better than the monthly GFZ-RL04 solution.

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For several applications a global gravity field model represented by spherical harmonics without losing the details of a regional zoom-in proves to be useful. The spherical harmonic coefficients can be calculated in a stable computation step, in principle up to an arbitrarily high degree, by means of quadrature methods. In our approach we make use of the Gauss-LegendreQuadrature (see for example Sneeuw (1994)). More details concerning the combination of regional gravity field solutions can be found in Eicker et al. (2006). The result of the combined monthly global GRACE solution is displayed as differences compared to the ITG-Grace02s in Fig. 6.

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5.2 Time Variable Gravity Field As described in Sect. 4 in one simultaneous recovery step the high resolution stationary gravity field was estimated together with a lower resolution time variable gravity field from 4,5 years of GRACE data. The calculations were performed for the Mekong delta area. Figure 7 shows the yearly amplitude of the gravity field signal in this area. Figure 8 illustrates the gravitational effect of the Sumatra Andaman Earthquake in December 2004. Displayed are the differences between a gravity field calculated from 2 years of data before the Earthquake and a gravity field originating from data of 2 years after the incident. The differences in the gravity signal due to mass displacements in the area of the Earthquake become obvious.

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Acknowledgments The support by BMBF (Bundesministerium fuer Bildung und Forschung) and DFG (Deutsche Forschungsgemeinschaft) within the frame of the Geotechnologien-Programm is gratefully acknowledged.

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References

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gravity field as well as for the task of calculating gravitational time variabilities. In this case the individual adaption of the regularization process has to be pointed out as it allows a tailored filtering according to the regional gravity field features and therefore improves the regional refinements significantly. If neccessary a global spherical harmonic solution can be derived as well, in this context the combination of regional solutions to a global gravity field solution seems to be a reasonable alternative to deriving a global gravity field solution directly. It was shown that time variablities can be detected by regional recovery methods as well. As especially many of the temporal variations take place on a regional scale, a regional modelling of those phenomena seems particularly reasonable.

0.6

Fig. 8 Differences in the gravity field before and after the Sumatra - Andaman Earthquake

6 Conclusions The regional recovery approach presents very promising results, in the case of a static high resolution

Eicker A, Mayer-Guerr T, Ilk KH (2006) An Integrated Global/Regional Gravity Field Determination Approach based on GOCE Observations. Observation of the Earth System from Space, Springer, Berlin - Heidelberg. Freeden W, Gervens T, Schreiner M (1998) Constructive Approximation on the Sphere. Oxford University Press, Oxford. Koch KR, Kusche J (2001) Regularization of geopotential determination from satellite data by variance components. J Geodesy 76 (5):259–268. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. NASA/TP-1998-206861, Goddard Space Flight Center, Grenbelt, MD. Mayer-Guerr T, (2006) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnboegen am Beispiel der Satellitenmissioncn CHAMP und GRACE, Dissertation University of Bonn, http://hss.ulb.uni-bonn.de/diss-online/landw fak/ 2006/mayer-guerr-torsten. Sneeuw N, (1994) Global spherical harmonic analysis by least squares and numerical quadrature methods in historical perspective. Geophys. J. Int., 118: 707–716. Tapley BD, Reigber C (2001) The GRACE mission: status and future plans. Report for mission selection, in The four candidate Earth explorer core missions, Eos Trans AGU 82 (47), Fall Meet. Suppl., G41 C-02.

The Role of the Atmosphere for Satellite Gravity Field Missions Th. Gruber, Th. Peters and L. Zenner

Abstract Traditionally, in satellite derived static global gravity field models, the total mass of the Earth is considered to be the sum of the solid Earth mass and the water masses in land (hydrology), oceans, ice (continental ice and sea ice) and atmosphere. The total mass is reflected in the GM value (gravity constant times Earth mass), which usually is a parameter of geodetic reference systems. As the whole system Earth is mass conserving in principle GM should be constant over time. Any time dependency of GM could only be caused by wrong estimates for GM (or the corresponding zero degree coefficient of the spherical harmonic series). For analysis of the time variable gravity field the situation is more complex, because masses are fluctuating at various temporal and spatial scales and are exchanged between components of the Earth system. Disregarding mass variations inside the solid Earth (e.g. by earthquakes, mantle convection, etc.) the water cycle represents the major source of mass variations, of which the atmosphere plays the most prominent role (by pressure field variations, by forcing ocean circulation, by precipitation and evaporation). In order to take into account such temporal variations for gravity

Th. Gruber Institut f¨ur Astronomische und Physikalische Geod¨asie, Technische Universit¨at M¨unchen, Arcisstrasse 21, 80333 M¨unchen, Germany Th. Peters Institut f¨ur Astronomische und Physikalische Geod¨asie, Technische Universit¨at M¨unchen, Arcisstrasse 21, 80333 M¨unchen, Germany L. Zenner Institut f¨ur Astronomische und Physikalische Geod¨asie, Technische Universit¨at M¨unchen, Arcisstrasse 21, 80333 M¨unchen, Germany

field analysis the sampling pattern of the satellite mission and its sensors in addition play a crucial role. This means that during gravity field recovery the known part of mass variations with respect to the mean value have to be modelled independently and corrected for in order to avoid aliasing of temporal signals into the resulting fields. In contrast, unknown time variable mass variations have to be estimated. Both approaches are applied nowadays in the GRACE data analysis simultaneously. The paper introduces the problem of atmospheric mass variability for gravity field determination with satellite data. In the main part of the paper estimates for the total mass of the atmosphere and its variability in time as well as an analysis of the impact of the vertical structure and of potential errors in the atmospheric fields are provided. Results indicate, that the total atmospheric mass during the last 20 years slightly increased, what could be a hint to global warming. Results of error analysis also show, that for reaching the ultimate performance with GRACE data analysis, the vertical structure of the atmosphere as well as potential errors in the atmospheric fields are crucial. Keywords Gravity field · Atmosphere · GRACE · De-aliasing

1 Introduction Gravity field related satellite observations like those from GRACE (inter-satellite range measurements) or in future from GOCE (gravity gradients) reflect the instantaneous distribution of mass in the system Earth. This includes all solid, liquid and atmosphere particles

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H

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Fig. 1 Problem description atmosphereic de-aliasing

in the Earth and its envelope. The situation is reflected in Fig. 1 below. As the observation system (satellite) and the masses are moving in time it is not guaranteed, that by a sufficient length of observation time, the mass variation effects are canceled out, because of the mean operator. There is a strong dependency on the sampling of the satellite orbit (Sat. in Fig. 1) and the spatial-temporal behavior of the mass variations in the different subsystems (like low and high atmospheric pressure systems as denoted by L and H in Fig. 1). De-aliasing then means, that instantaneous mass variations with respect to a static mean (in case of the atmosphere this is a mean or standard atmosphere) are taken into account during the pre-processing of observations or during the estimation procedure in the gravity field determination process. In analogy to the atmosphere, this holds for all other mass variation phenomena in the system Earth. Specifically for the atmosphere, there is also a variation of the center of mass of the atmospheric column (black dots for high and low pressure systems in Fig. 1), which has influence on the satellite observations, because the distance of the attracting mass is varying. This is addressed in more detail in Chap. 3. The situation is slightly different when we are observing gravity field quantities on the Earth surface (like gravity observation in a point P in Fig. 1). In this case the attracting mass of the atmosphere is above the measurement system and therefore has to be addressed in a different way. First, the direct attraction of the atmospheric mass and second, the indirect effect by deformation of the Earth crust due to atmospheric loading has to be taken into account. As for the satellite data, again, the mean atmosphere is taken into account via a

standard atmosphere model. This is a well known process in gravimetry and described in detail in geodetic literature like Torge (1989). When combining surface data with satellite gravimetry another problem is encountered. Potential theory requires that all masses are inside the boundary surface (e.g. the geoid for Stokes theory or the Earth surface for Molodenskii theory). As atmospheric masses are always above the boundary surface the boundary condition is not anymore fulfilled. For this reason a correction has to be applied to gravity anomalies, which are related to the boundary surface and determined from surface gravimeter observations. This correction again makes use of a standard atmosphere. Details on the theory behind and the computation of the correction are described in Moritz (1980).

2 Total Mass of the Atmosphere In satellite geodesy the total mass of the Earth always is the sum of the mass of all sub-systems (solid, liquid and aeriform). There is no possibility to separate the individual components from gravity field observations. Various authors estimated the total mass of the atmosphere by analyzing the mean global surface pressure field. Pressure represents nothing else as the weight of the atmosphere above the Earth surface. Table 1 provides a selection of estimates published during the last

Table 1 Estimates for mass of the atmosphere Author

Year

Value [1018 kg]

Ekholm IAG (GRS67)

1902 1967

Ecker & Mittermayer

1969

Wenzel Trenberth & Guillemot NIMA (WGS84) Trenberth & Smith This Study (see below)

1985 1994

5.16 × 1018 5.3163 (for CIRA) 5.2924 (for US Standard Atmosphere) 5.296059 (for US Standard Atmosphere) 5.320921 (for CIRA 1961) 5.31694 5.1441

1997 2004 2007

5.24 with 0.15 uncertainty 5.1480 ECMWF Op. Analysis: 1.1.95-03.04.95: 5.12396 4.4.95-31.03.98: 5.14762 1.4.98-31.12.06: 5.15139 ECMWF ERA-40 1.9.57-31.08.02: 5.16156

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century. The last entry in this table provides the results The following formula is applied in order to convert pressure fields to potential spherical harmonic series: for our own analysis described in the sequel. The estimates between the different authors vary 

 significantly. There seems to be a different estimaa 2 (1 + kn ) C = PS − PSRef nm tion technique with different results in geodesy (see (2n + 1)Mg results from IAG 1967, Ecker & Mittermayer, 1969, Ear th Wenzel 1985, NIMA 1997) and in atmospheric sciPnm (cos θ ) cos mλd S (1) ences (see results from Ekholm 1902, Trenberth & 

 a 2 (1 + kn ) Guillemot 1994, Trenberth & Smith 2004). Our own PS − PSRef Snm = (2n + 1)Mg study is based on the analysis of surface pressure data Ear th from the European Center for Medium Weather Range Pnm (cos θ ) sin mλd S Forecast (ECMWF). Here we distinguish between results from the operational analysis and the 40 years re-analysis (ERA-40). The mass of the atmosphere is where estimated from the degree and order zero coefficient of the spherical harmonic series of the atmospheric potenC,S: spherical harmonic coefficients, tial. This means we convert surface pressure to gravity n: degree, potential and perform a spherical harmonic analysis m: order, up to degree and order 180. In order to deal with a: semi major axis of reference ellipsoid, smaller quantities for the time-series analysis a referkn : loading Love numbers, ence surface pressure is introduced, which in principle M: mass, represents an assumption about the time-invariant surg: gravity constant, face pressure field. In our case we have selected the PS : surface pressure, first time step of the time series as reference surface PS ref : reference surface pressure, pressure. For estimating the total mass of the atmoPnm : associated Legendre polynomials, sphere the reference surface pressure field is set to zero. θ, λ: geographical co-latitude, longitude. 1995 6E– 09

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2006. The results indicate a change of the geocenter due to atmosphere of 2.6 mm in x-, 0.8 mm in y- and 6.7 mm in z-direction within this period. The change in the 0◦ term corresponds to an increase of atmospheric mass of 4.7 × 1013 kg per year. As atmospheric mass variations mainly are caused by variations of its we part, there is a strong indicator that atmosphere must have become warmer and more humid in this period.

Degree 0 and degree 1 terms are connected with the total mass and the center of mass of the atmosphere respectively. Figure 2 below shows the results for some of the coefficients from the operational analysis (for period 1995–2006). The coefficient time series shows some interesting results.

r r

r

r

Seasonal variation: For all three coefficients a clear seasonal signal becomes visible. Degree 0 term: There are significant jumps in the time series, which are related to changes in the operational model of ECMWF. Here specifically changes of the orography (average land height) used in the operational model are the reason for the jumps. A change in height means that depending on the sign of this change either more or less atmosphere mass is above the point. In case the mean height over land varies then also the total mass of the atmosphere changes. The amplitude spectrum of the time series (not shown) exhibits a clear signal at daily and annual frequency for all coefficients. Degree 1 terms show significant signals for frequencies between 1 and 20 cycles per year. In order to exclude the jumps a regression line was estimated for the period from May 1998 to Dec.

Coefficient relative to 1.9.1957 00h

5E–10

Fig. 3 Time series of degree 0 coefficients from ECMWF re-analysis surface pressure data and regression line for complete and selected period

1960

The same computations have been repeated for the surface pressure data of the ECMWF ERA-40 re-analysis covering a 45 years period (Sept. 1957 – Aug. 2002). Results are shown in Fig. 3. From the re-analysis coefficient time series the following conclusions can be drawn:

r

r

Until 1979 the fluctuations are much stronger than after. This probably has to do with a much better quality and number of observations (availability of satellite data, etc.) starting in the 1980’s. Therefore we analyzed the whole 45 years time series as well as a 22 years time series from Sept. 1980 to Aug. 2002. From the full period we get a negative change for the degree 0 coefficient, which corresponds to a decrease of atmospheric mass of about 1.73 × 1013 kg per year. Geocenter variations due to atmosphere

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0

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–5E–10

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–1.5E– 09

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The Role of the Atmosphere

r

would correspond to −1.3 mm in x-, 1.4 mm in yand 2.5 mm in z-direction. From the selected period over 22 years we get an increase of atmospheric mass of about 4.6×1013 kg per year, which agrees very well to the results from the operational analysis (see Fig. 2). Geocenter variations due to atmosphere would correspond to −1.2 mm in x-, 1.5 mm in y- and 2.6 mm in z-direction. The latter results better agree with the full re-analysis period than with the operational analysis.

3 Impact of Vertical Structure of the Atmosphere

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The sequence of equations has to be evaluated as follows. First, the pressure of the model levels is computed for each grid point and each level applying formulas Eq. (2). In addition the geopotential height of the model levels is computed by Eq. (3) using the pressure of the model levels computed before. Then within Eq. (4) the vertical integration is performed by the inner integral by simple numerical integration for the pressure differences of neighbouring levels. After subtraction of the reference field the spherical harmonic analysis is performed similar to Eq. (1). Because there is a degree dependent factor in the inner integral the process of vertical integration has to be repeated for each degree of the spherical harmonic series. Tν = (1 + 0.608S)T ; Pk+1/2 = ak+1/2 + bk+1/2 PS (2)

N As introduced in Chap. 1 for satellite gravimetry there P j+1/2 1 level  =  + Rdr y Tv ln (3) k+1/2 S is a dependency on the center of mass of the atmoγ P j−1/2 j=k+1 spheric column. This means, that due to the deviation of the location of the center of mass of the atmosphere from the Earth surface (where the surface pressure is  a 2 (1 + kn ) given), the attraction of the atmospheric mass acting on Cnm = − the satellite is different, because the distance between (2n + 1)Mg Ear th the satellite and the attracting mass has changed. This ⎤ ⎛⎡ ⎞ 0 #n+4  " has to be taken into account when processing satellite ξ a ⎥ ⎜⎢ ⎟ gravimetry data as it is for example done for GRACE d P ⎦ − PIRef + ⎝⎣ nt ⎠ a −  a (for details see Flechtner, 2007). PS In order to compute the center of mass of the atmo· Pnm (cos θ ) cos mλ sin θdθdλ spheric column a set of atmospheric parameter profiles (4) is needed. Atmospheric models discretise the vertical   2 structure by introducing a number of pressure or model a (1 + kn ) Snm = − levels. Since 2006 the ECMWF operational model pro(2n + 1)Mg vides data on 91 model levels, which are distributed inEar th ⎤ ⎛⎡ ⎞ homogeneous in distance from the Earth surface and 0 #n+4  " ξ a ⎥ ⎜⎢ ⎟ which are reflecting the density structure of the atmod P ⎦ − PIRef + ⎝⎣ nt ⎠ a− a sphere (i.e. distance between model layers is smaller PS at lower altitudes where more mass is present). The · Pnm (cos θ ) sin mλ sin θ dθdλ data needed for determining the center of mass are specific humidity and temperature at the model levels, and the pressure and geopotential height (height of the where (additional to Eq. (1)): geopotential surface above the geoid) at the Earth surface. These parameters are introduced in the sequence S: specific humidity at model level, of formulas provided below in order to convert them T : temperature at model level, to gravity potential spherical harmonic series (similar Tv : virtual temperature at model level, to Eq. (1) for surface pressure). A full derivation of ak+1/2 , bk+1/2 : coefficients for computing pressure these formulas and their application can be found in on model levels, Flechtner (2007). Pk+1/2 : pressure at model level,

110 Signal Atmosphere Vertical Integration (VI) Atmosphere Signal Difference SP-VI Error EIGEN-GL04S1 (Actual GRACE Field) Error Prediction GRACE

10–2 Degree Standard Deviation in Geoid Height [m]

Fig. 4 Square root of degree variances in terms of geoid heights of pressure signal, difference between surface pressure and vertical integration (for 1.1.2004 00 h) and of GRACE errors

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Rdry : gas constant for dry air, γ : standard gravitational acceleration,  S : geopotential height at Earth surface, k+1/2 : geopotential height at model level, ξ : geoid height, PI nt Ref : vertically integrated reference pressure,

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look differently for other periods. By processing several time steps in different periods of the year (not shown here) the signal to error structure as shown in Fig. 4 can be confirmed.

4 Impact of Errors of Atmosphere Fields In order to identify the impact of the vertical integration, compared to the surface pressure approach on the resulting spherical harmonic series, for one time step both were computed and compared against each other and against the GRACE error curve. Results of this test case are shown in Fig. 4 in terms of square root of degree variances for a specific time step. Results indicate that for the current GRACE error level (e.g. for the EIGEN-GL04S1 model from GFZ) the differences between the surface pressure approach (Eq. (1)) and the vertical integration (Eqs. 2, 3 and 4) are not significant. Nevertheless, for further improvements in GRACE processing targeting to the originally predicted error level for the long wavelengths up to degree 20, the vertical integration has to be applied. As this is a result for a specific time step results could

The whole procedure of atmospheric de-aliasing relies on the assumption of error-free atmospheric parameter fields. It is well known from various studies (e.g. Velicogna et al, 2001) that data from atmospheric models have large uncertainties for not well observed areas like Antarctica and the Southern oceans. As it is difficult to estimate error levels for the atmospheric parameters involved (see Eqs. 1, 2, 3, 4) we use the difference between the ECMWF and the NCEP (National Center for Environmental Prediction) model results as error measure. We are well aware that the real error could look quite differently, because both atmospheric models rely on the same or similar input data. In conclusion, the difference between both models can be regarded as modelling error instead as absolute error

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Fig. 5 Square root of degree variances in terms of geoid heights of ECMWF and NCEP pressure signals, of difference between ECMWF and NCEP signals (for January 2006) and of GRACE errors

Degree Standard Deviation in Geoid Height [m]

of the atmospheric parameters. Currently there is running an interdisciplinary project within the “German Research Foundation Priority Programme” on Mass Transport and Mass Distribution, which involves scientists from geodesy and atmospheric sciences. The goal of this project is to determine representative error fields for the atmospheric parameters involved. As results from this project are not yet available, we stick to the model differences as explained above and estimate the impact of them to the derived atmospheric gravity potential. Figure 5 shows the mean differences of atmospheric potential for January 2006 between the ECMWF and NCEP surface pressure analysis in terms of square root of degree variances compared to the predicted GRACE errors. First, it is obvious, that the NCEP analysis exhibits a smaller signal for higher degrees than the results based on the ECMWF data. This probably is caused by a different resolution of the atmospheric models. Second, comparing the difference curve with the predicted error one clearly can see that there is a significant impact on GRACE for the long wavelengths up to degree 30. This implies that further investigations about the quality of the models have to be performed

in order to realistically estimate the uncertainties of the atmospheric potential caused by the error-free assumption. Optimally, for each time step of the atmospheric analysis, errors can be used to estimate also uncertainties of the spherical harmonic series of the atmospheric potential by a rigorous error propagation. Such uncertainties could be further used during the GRACE data analysis in order to exploit the GRACE observations together with the underlying background fields. This is part of the work to be performed in the project mentioned before.

5 Conclusions With the launch of the GRACE mission and its resulting gravity field time series a new level of accuracy has been reached. Therefore, the atmosphere and specifically the time variability of its mass became an important parameter during the data analysis. The paper investigated three dedicated topics about the atmosphere and its variability in context with the GRACE mission. From these investigations the following conclusions can be drawn:

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r

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The total mass of the atmosphere can be estimated from the global surface pressure as it is delivered from global atmospheric circulation models. The results show well agreement between all studies based on this data. Estimates based on geodetic techniques provide larger values for the mass. The reason for this is not obvious. Our estimation of the atmospheric mass based on the operational and re-analyzed ECMWF fields shows that there is a strong dependency on the orographic (topographic) model applied in the atmospheric model. Any change in the orography (as it happened for the operational model) causes jumps in the atmospheric mass. Analyzing the time series of the degree 0 term for a consistent time period of the operational atmospheric data shows an increase of mass over the period from 1998 to 2006 of about 4.7 × 1013 kg per year. This could be caused by a temperature increase of the atmosphere corresponding to more humidity (water) in the atmosphere. The analysis of the 45 years of re-analyzed ECMWF surface pressure shows different results when analyzing the full period or the more reliable period starting at 1980. The latter period over 22 years provides similar results than the operational analysis over 8 years, while the full period provides a quite different result with an atmospheric mass decrease. Obviously there is a strong dependency on the atmospheric model and the data used in the atmospheric analysis. One can argue that starting with the 1980’s, the quality of the model output became better due to the availability of much more data from satellites. This is supported by the much smoother time series of the degree 0 coefficient (see Fig. 3). The center of mass of the atmospheric column (vertical integration) has to be taken into account in order to reach the predicted error level with GRACE. Error parameters provided with most recent GRACE gravity field solutions indicate no dependency on the vertical structure of the atmosphere. The signal of the atmospheric potential computed from differences between ECMWF and NCEP atmospheric models is significant above the predicted error for GRACE (and also above the error estimates from recent GRACE models). This implies,

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that further studies about the impact of potential errors in the atmospheric parameter fields have to be performed. Currently, together with atmospheric scientists there is work ongoing to identify error parameters for the atmospheric fields applied in the de-aliasing process in order to take them into account in the estimation process. We expect a much deeper insight into the problem when results of the project will become available. Generally, one can conclude that the mass of the atmosphere and its temporal variations became an important parameter in satellite gravimetry. The atmospheric models applied as background models become more and more important. Therefore, a deeper insight into the modelling of the atmosphere as well as the error parameters is needed.

References Ecker E. & Mittermayer E. (1969). Gravity Corrections for the Influence of the Atmosphere. Bolletino die Geofisica Teorica ed Applicata, Vol. 11, pp. 70–80. ¨ Ekholm N. (1902). Uber die H¨ohe der homogenen Atmosph¨are und die Masse der Atmosph¨are. Meteorologische Zeitschrift Vol. 19, p. 249–260. Flechtner F. (2007). AOD1B Product Description Document for Product Release 01–04, Document GRACE 327-750, 3.1, 13.4.2007. IAG International Association of Geodesy (1967). Geodetic Reference System 1967. Bureau Central de l’ Association Internationale de G´eod´esie. Moritz H. (1980). Advanced Physical Geodesy. Herbert Wichmann Verlag Karlsruhe, Abacus Press Tunbridge Wells Kent. NIMA National Imagery and Mapping Agency (1997). World Geodetic System 1984: Its Definition and Relationship with local geodetic Systems. NIMA TR 8350.2 Third Edition. Torge W. (1989). Gravimetry. Walter de Gruyter & Co, Berlin. Trenberth K.E. & Guillemot C.J. (1994). The Total Mass of the Atmosphere. Journal of Geophysical Research, Vol. 99, D11, pp. 23079–23088. Trenberth K.E. & Smith L. (2005). The Mass of the Atmosphere: A Constraint on Global Analyses. Journal of Climate, Vol. 18, 6, pp. 864–875. Velicogna I., Wahr J. & Van den Dool H. (2001). Can Surface Pressure be Used to Remove Atmospheric Contributions from GRACE Data with Sufficient Accuracy to Recover Hydrological Signals? Journal of Geophysical Research, Vol. 106, B8, pp. 16415–16434.

Assessment of GPS-only Observables for Gravity Field Recovery from GRACE A. J¨aggi, G. Beutler, L. Prange, R. Dach and L. Mervart

Abstract Kinematic positions of individual low Earth orbiting satellites equipped with spaceborne GPS receivers have been used in the past to determine the long wavelength static part of the Earth’s gravity field. In the near future GPS-derived relative kinematic positions of present and upcoming formation flying satellites like COSMIC and SWARM could be used in addition to perform and improve the long wavelength static part of the gravity field also with non-dedicated satellites. Since space baselines between satellites can be determined more precisely from GPS than the individual positions, a corresponding improvement of the estimated gravity field coefficients is commonly expected. We review and extend the principles of gravity field determination from kinematic positions of single satellites and apply them to kinematic baseline data. Simulated as well as real data from the GRACE GPS receivers are used to evaluate our procedures and to assess the impact of different GPS observables and processing strategies on the quality of the estimated gravity field coefficients. A. J¨aggi Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland, e-mail: [email protected] G. Beutler Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland L. Prange Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland R. Dach Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland L. Mervart Institute of Advanced Geodesy, Czech Technical University, Thakurova 7, 16629 Prague, Czech Republic

Keywords GRACE · Kinematic orbit determination · Ambiguity-fixing · Gravity field recovery

1 Introduction The Gravity Recovery And Climate Experiment (GRACE) mission, launched on March 17, 2002, is the first pair of formation flying low Earth orbiters (LEOs) equipped with on-board geodetic-type receivers for the Global Positioning System (GPS). The main purpose of the GRACE mission is the determination of the time-varying part of the Earth’s gravity field with unprecedented accuracy from GPS, accelerometer, and inter-satellite K-band range observations (Tapley et al., 2004). Concerning the determination of the long wavelength static part of the Earth’s gravity field from GPS data only, the GRACE formation offers the possibility to assess the impact of different types of undifferenced or differenced GPS observables on orbit determination and on the subsequent estimation of gravity field coefficients. Due to the interferometric nature of differential GPS the relative positions between the two GRACE satellites can be determined with much higher precision than the absolute positions. It is thus commonly expected that the most precise knowledge of the 3-D inter-satellite vector would significantly strengthen gravity field recovery, which would in turn be attractive for future LEO constellations such as SWARM (Gerlach et al., 2006). This study focuses on the assessment of different types of precise kinematic GRACE A and B positions and baseline estimates established by J¨aggi (2006) and their application to gravity field determination. In order to validate the gravity field recovery, the estimated gravity field coefficients are compared with

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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the official and superior GRACE gravity field models mainly based on K-band observations. No use of accelerometer data is made in this study, implying that empirical parameters have to compensate in particular for the unmodeled non-gravitational perturbations.

2 Single Satellite Approach This section briefly introduces the principles of a threestep procedure applied to gravity field recovery. In a first step GPS observations are processed to derive kinematic LEO positions together with epochwise covariance information (see Sect. 2.1). In a second step the kinematic LEO positions are weighted according to the epochwise covariance information and serve as pseudo-observations to set up normal equations for the unknown gravity field coefficients on a daily basis in a generalized orbit determination problem (see Sect. 2.2). In a third step the obtained normal equations may be modified on request, and are then accumulated into weekly, monthly, and annual systems (see Sect. 2.3).

2.1 Step I: Kinematic Orbit Determination The geometric strength and the high density of GPS observations allows for a purely geometrical approach to determine kinematic LEO positions at the observaˇ tion epochs by precise point positioning (Svehla and Rothacher, 2004). The ephemerides are represented by a time series of three coordinates per epoch, which are determined in a standard least-squares adjustment process of GPS observations together with all other relevant parameters without using any information on LEO dynamics.

2.2 Step II: Generalized Orbit Determination

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are the six Keplerian elements pertaining to epoch t0 . q1 , . . . , qd denote d additional dynamical parameters considered as unknowns, e.g., gravity field coefficients and other orbit parameters which describe the perturbing acceleration acting on the LEO. Let us assume that an a priori orbit r 0 (t) is available, e.g., realized by a fit of the LEO pseudo-observations with few dynamical orbit parameters only. Gravity field recovery may then be set up as an orbit improvement process, i.e., the actual orbit r(t) is expressed as a truncated Taylor series with respect to the unknown orbit parameters pi about the a priori orbit, which is represented by the parameter values pi 0: n

r(t) = r 0 (t) + i=1

∂ r 0 (t) · Δpi , ∂ pi

(2)

. . where Δpi = pi − pi 0 denote the n = 6 + d corrections to be estimated. Numerical integration techniques are applied to solve the so-called variational equations (Beutler, 2005) to obtain the partial derivatives of the a priori orbit r 0 (t) with respect to the parameters pi , which allow to set up observation equations in order to form the daily normal equations for all corrections Δpi in a standard least-squares adjustment of the LEO pseudo-observations.

2.2.1 Gravity Field Parameters The static part of the Earth’s gravitational potential is modeled in our analysis with a series of normalized spherical harmonic (SH) coefficients (Heiskanen and Moritz, 1967) from degree 2 up to the maximum degree of 90. One and the same set of SH coefficients is set up for all daily arcs without applying any regularization.

2.2.2 Arc-Specific Parameters

Apart from the six Keplerian elements, arc-specific empirical parameters such as scaling factors and pseudo-stochastic parameters are set up in order to compensate for orbit modeling deficiencies (J¨aggi r r¨ = −G M 3 + f 1 (t, r, r˙ , q1 , . . . , qd ) (1) et al., 2006). The estimation of scaling factors for conr stant and once-per-revolution empirical accelerations = is primarily motivated by compensating for the main with a set of initial conditions r (k) (t0 ) r (k) (E 1 , . . . , E 6 ; t0 ), k = 0, 1 where E 1 , . . . , E 6 part of the unmodeled non-gravitational perturbations

The equation of motion of a low Earth orbiting satellite including all perturbations reads in the inertial frame as

Assessment of GPS

in the radial, along-track, and cross-track directions. Remaining deficiencies have to be further reduced by setting up low-order polynomials for the along-track accelerations, and in particular by estimating additional pseudo-stochastic pulses (instantaneous velocity changes) at predefined epochs for all three directions. Pseudo-stochastic pulses are well suited for the task of gravity field recovery, because they do not affect the LEO trajectory in-between the pulse epochs. A short spacing of 5 min is used between subsequent pulse epochs for setting up the daily normal equations.

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gravity field coefficients. In a third step, identical with the single satellite approach, the obtained normal equations are further modified.

3.1 Step I: Kinematic Baseline Determination The quality of the GRACE GPS observations allows for a reconstruction of the space baseline with sub-mm precision for a reduced-dynamic orbit determination and with few mm precision for a kinematic orbit determination, both in an extended Kalman-filter environment and in a batch least-squares environment (Kroes, 2006; J¨aggi, 2006). The most convenient method to reconstruct the space baseline is to process the GRACE data alone without data from the GPS ground tracking network by fixing the positions of one satellite to a previously established reference trajectory of high quality, e.g., to a reduced-dynamic orbit obtained from undifferenced GPS observations (J¨aggi et al., 2007). Note that such a procedure does not affect the relative kinematic positions as long as the quality of the reference trajectory is better than about 10 cm. Formally, ephemerides for both satellites are thus available, but only ephemerides differences are meaningful for gravity field recovery.

The normal equation level is attractive for parameter modifications, because they usually may be performed efficiently (compared to a full re-computation of the daily normal equation systems). Standard modifications include, e.g., constraining or removing of selected model parameters. Whereas no use is made of the former option in this study, the latter option is conveniently used in this study to generate normal equations with a spacing of pseudo-stochastic pulses of multiples of 5 min, e.g., with a spacing of 15 min. Before accumulating the modified daily normal equations into weekly, monthly, and annual systems, the arc-specific parameters are pre-eliminated. Consequently, the corrections of the arc-specific parameters are no longer available, but their impact on the SH coefficients is correctly taken into account. 3.2 Step II: Generalized Orbit Eventually, the accumulated normal equation system Determination is inverted in order to obtain the corrections of the SH coefficients and the associated full covariance information. Due to the availability of ephemerides for both satellites, observation equations for both satellites including all corrections Δpi can formally be established for each day in full analogy to Sect. 2.2. Note that normal equa3 Baseline Approach tions must not be formed for the individual satellites, however, because the “absolute” ephemerides are influThe three-step procedure of the baseline approach enced by the selected reference trajectory as mentioned is closely related to the single satellite approach de- in Sect. 3.1. Therefore, differences between the obserscribed in Sect. 2. In a first, different step differenced vation equations have to be formed before constructing GPS observations are processed to derive LEO posi- the daily normal equations. Similar to Sect. 3.1, it is tions together with epoch-wise covariance information necessary to fix the orbit parameters for one satellite for a relative use. In a second, slightly modified step to the a priori values obtained from the fit of the LEO the positions serve as weighted pseudo-observations pseudo-observations with few dynamical orbit paramto set up daily normal equations for the unknown eters (see Sect. 2.2).

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Numerical experiments confirm that gravity field recovery from the outlined procedure is independent of the selected reference trajectory. The impact of fixing the orbit parameters for one of the two satellites to the a priori values obtained from the dynamical fit of the LEO pseudo-observations can be critical, however, and will be discussed in Sect. 5.3.

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4.1 Validation of L1/L2 and LA/L2 Solutions

Difference degree amplitude

The availability of state-of-the-art GRACE gravity field models, mainly based on K-band observations, allows for an almost independent validation of models derived from GPS data only. Figure 1 shows the difference degree amplitudes with respect to the superior gravity field model EIGEN-GL04C (F¨orste et al., 2006) for different monthly and an4 Processing Undifferenced Data nual solutions obtained from GRACE A. Compared to the L1/L2 solutions the solutions derived from The final GPS orbit solutions and the 30 s high-rate kinematic positions based on the LA and L2 carrier satellite clock corrections of the year 2003 from the phase observations exhibit a better agreement for the Center for Orbit Determination in Europe (CODE) are high degrees by almost a factor of 2 with respect to used to derive two different one-year series of kine- EIGEN-GL04C due to the smaller LA carrier phase matic position for both GRACE satellites from undif- noise. The agreement for the low degrees up to about ferenced (zero difference) ionosphere-free GPS carrier 25 remains, however, unchanged by using the more phase data at 30 s intervals (J¨aggi, 2006). For the first accurate LA carrier phase observations. This either series, the ionosphere-free observations are formed us- means that EIGEN-GL04C “sees” something different ing the L1 and the L2 GPS carrier phase data, i.e., using for the low degrees, which is unlikely, or that our the accompanying carrier phase observations of the P1 solutions are limited by systematic errors which show and P2 codes. For the second series, the ionosphere- up more clearly for the low degrees when processing free observations are formed using the LA and the L2 longer data spans. One might tend to observe a small GPS carrier phase data, i.e., using the accompanying dislocation of this “limiting degree” from about 20 carrier phase observations of the C/A and P2 codes. for the monthly solutions to about 25 for the annual Note that the LA observations are taken on the same solutions. frequency as the L1 observations, but that they exhibit a lower noise for JPL BlackJack receivers (Dunn et al., 2003). Gravity field recovery from degree 2 up to the maximum degree of 90 is performed with both series of 10–5 EIGEN−GL04C kinematic positions for both GRACE satellites according to Sect. 2 without taking accelerometer data into 10–6 account. In analogy to Prange et al., (2008), showing for CHAMP that SH coefficients are almost unaffected 10–7 by different background models due to the estimation of pseudo-stochastic pulses, a spacing of 15 min between subsequent pulses is used. The pulse spacing is, 10–8 however, not a critical parameter for GRACE due to the higher orbital altitude of the twin satellites. Solid GRACE A 10–9 Earth tides are modeled according to the IERS 2000 L1/L2 LA/L2 conventions, ocean tides according to the CSR model 10–10 from Schwiderski. The gravity field model EIGEN-2 0 10 20 30 40 50 60 70 80 90 Degree of spherical harmonics (Reigber et al., 2003) serves as a priori model for solving the generalized orbit determination problem. Note Fig. 1 Difference degree amplitudes of monthly and annual rethat the final gravity field results are independent of the coveries from GRACE A kinematic positions based on L1/L2 or LA/L2 GPS carrier phase observations a priori gravity field model (Prange et al., 2008).

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4.2 Orbit Validation with K-band Data

Allan deviation (mm/s)

approaches the L1/L2 solution for increasing interval lengths, which could be an indication that parts of The K-band ranging system aboard GRACE provides the systematic errors found in Fig. 1 for the low the unique opportunity to directly validate the dis- degrees are already introduced in the kinematic orbit tance between the twin satellites. Therefore, K-band determination. Remember that K-band measurements range residuals obtained from distances between the cannot reveal the position errors common to both kinematic GRACE A and B positions were analyzed GRACE satellites, which additionally propagate into to reveal systematic errors introduced in the kinematic the gravity field recovery and thus also show up in orbit determination, and to separate them from system- Fig. 1, but not in Fig. 2. atic errors introduced later during gravity field recovery. K-band measurements are, however, insensitive to common errors of both GRACE satellites. 4.3 Comparison with CHAMP Solutions In order to characterize the noise of K-band range residuals, it is instructive to compute Allan deviations for different sampling intervals as it is commonly Table 1 and Fig. 3 show geoid height differences done for the characterization of precision clocks and and difference degree amplitudes with respect to oscillators (Allan, 1987). 10 days of K-band range EIGEN-GL04C for various annual solutions. The residuals of the L1/L2 and LA/L2 solutions are thus GRACE A solution based on the LA/L2 GPS carrier selected to compute Allan deviations for sampling phase observations from Fig. 1 is included, as well intervals ranging from 30 s to 24 h. Figure 2 shows as a corresponding solution obtained from GRACE that both solutions are separated by about a factor of B. It can be recognized that the GRACE B solution 2 for short interval lengths due to the different carrier exhibits in general a better agreement with respect phase noise. Up to an interval length of about 10 min to EIGEN-GL04C due to the better quality of the the slope of the L1/L2 solution is almost −1, which GRACE B GPS carrier phase observations. Although would represent a white noise process according to the number of GRACE B GPS observations is smaller Allan (1987). The LA/L2 solution, however, does not than those from GRACE A, the former exhibit a exhibit a purely white noise process for short interval smaller carrier phase noise (Kroes, 2006). For comparison purposes CHAMP one-year solulengths. It can be recognized that the Allan deviation tions are included in Table 1 and Fig. 3, as well. The solution denoted as “CHAMP” is based on the same or1 bit parametrization as the two GRACE solutions. Our LA /L2 official model AIUB-CHAMP01S shows, however, L1/L2 that a further improvement is possible by combining solutions with different spacings of pseudostochastic 0.1 pulses on the level of normal equations (Prange et al., 2008), which yields a solution comparable to the model ITG-CHAMP01S (Mayer-G¨urr et al., 2005). 0.01 Note that all CHAMP solutions are based on the LA GPS carrier phase observable (denoted as L1 in 0.001

Table 1 RMS of cumulative geoid height differences (cm) up to different degrees of annual recoveries with respect to EIGENGL04C (1◦ × 1◦ grid including area-weighting) 0.0001 10

100

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Solution

D&O 30

D&O 50

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GRACE A GRACE B CHAMP AIUB-CHAMP01S ITG-CHAMP01S

1.70 1.71 1.43 1.35 1.27

5.72 5.26 5.73 5.19 5.53

31.21 27.76 24.42 22.20 26.42

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Fig. 3 Difference degree amplitudes of annual recoveries from GRACE A, GRACE B, or CHAMP kinematic positions based on LA/L2 GPS carrier phase observations

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the CHAMP observation files (K¨ohler, 2001)). All CHAMP solutions cover the time period from day 071/2002 to 070/2003, which is different from the time period of the GRACE solutions. It can be recognized that the agreement of both GRACE solutions with respect to EIGEN-GL04C is not so good as the agreement of the CHAMP solutions, especially for very low and very high degrees. Whereas the effect related to the high degrees is explained by the higher orbital altitude of the GRACE satellites, no sizeable effect is expected for the lower degrees. Actually, the low degree SH coefficients are formally even better determined from GRACE than from CHAMP due to the better quality of the GRACE GPS carrier phase observations (Kroes, 2006). Further investigations based on identical observation periods and the inclusion of accelerometer data are necessary to resolve this issue.

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the GRACE A positions to a reduced-dynamic reference trajectory previously established by a zero difference solution (see Sect. 3.1). Note that the doubly differenced ionosphere-free observations are formed using the L1 and the L2 GPS carrier phase observables. The carrier phase ambiguities either remain unresolved (float), or they are resolved (fixed) to their integer values during the baseline determination.

5.1 Validation of Baseline Solutions

Figure 4 shows the difference degree amplitudes with respect to EIGEN-GL04C for different solutions based on either undifferenced or on differenced GPS data from the 55 days test period. For comparison, the solution denoted as “ZD A” is based on zero difference GPS data of GRACE A only, whereas the solution denoted as “ZD A-B” is based on vector differences which are constructed from zero difference kinematic 5 Processing Doubly Differenced Data positions of GRACE and GRACE B. The solution “ZD A-B” exhibits only a slightly better agreement 55 days of data (243–297, 2003) have been selected with respect to EIGEN-GL04C for the very high to determine the kinematic space baseline alone with- degrees, but considerably larger differences for the out data from the ground tracking network by fixing very low degrees due to the processing of differential

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Fig. 4 Difference degree amplitudes of recoveries from 55 days of GRACE kinematic single satellite or baseline data

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positions and, additionally, due to the fixing of the orbit parameters of GRACE A to the a priori values obtained from the dynamical fit of the GRACE A pseudo-observations (see Sect. 5.3). The solution denoted as “DD A-B” in Fig. 4 is based on vector differences which are obtained from double difference kinematic baseline solutions with unresolved carrier phase ambiguities. It can be recognized that the solution “DD A-B” exhibits differences with respect to EIGEN-GL04C for the very low degrees of the same order as the solution “ZD A-B”, but shows a significantly better agreement for the medium to high degrees due to the use of doubly differenced GPS carrier phase observations. For a fair comparsion, the solution has to be compared, however, with the solution denoted as “ZD A&B”, representing the combination on the normal equation level of the solutions “ZD A” and “ZD B” (single satellite solutions obtained from GRACE A and B, respectively). The benefit of the solution “DD A-B” is restricted to the very high degrees only. The elimination of systematic errors of high frequency common to both GRACE satellites (see Sect. 5.2) are responsible for the observed small improvement of the solution “DD A-B”. At least one year of differenced GPS data should be processed, however, to decide how much the determination of the very high

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degrees would improve by processing doubly differenced GPS data.

5.2 Impact of Ambiguity Resolution Figure 7 shows the daily K-band range RMS errors obtained from distances between the (fixed) reduceddynamic GRACE A orbit and the estimated kinematic GRACE B positions from the ambiguity-fixed (top) and the ambiguity-float (bottom) solutions. Correctly resolved ambiguities considerably strengthen the kinematic baseline estimates, which in turn improves the overall K-band range RMS error from 15.9 mm for the ambiguity-float solution to 4.4 mm for the ambiguityfixed solution. For comparison, the K-band range RMS error obtained from kinematic GRACE A and B positions derived from undifferenced GPS data is 20.5 mm. Figure 5 shows, however, that the difference degree amplitudes with respect to EIGEN-GL04C are almost the same for solutions based on vector differences obtained from double difference kinematic baseline solutions with either resolved or unresolved carrier phase ambiguities. Instead of an expected reduction of the difference degree amplitudes of the ambiguityfixed solution by a factor of 15.9 : 4.4 ≈ 3.6, only

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Fig. 5 Difference degree amplitudes of recoveries from vector differences from the ambiguity-fixed or the ambiguity-float kinematic GRACE baseline

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minor benefits are observed for the SH coefficients up to degree 10. In order to explain Fig. 5, it is instructive to characterize the noise of the K-band range residuals of the ambiguity-float and the ambiguity-fixed solution by Allan deviations for different sampling intervals. Figure 6 shows that both the ambiguity-fixed and the ambiguity-float solutions exhibit a slope of exactly −1 for short intervals and coincide up to an interval length of about 5 min. For longer sampling intervals, however, clearly better noise characteristics are observed for the ambiguity-fixed solution. Due to the correctly resolved carrier phase ambiguities, the estimated kinematic positions are not affected by additional systematic errors of low frequency. According to Fig. 7 such errors can dominate the kinematic baseline determination, but the corresponding frequencies are too low to affect gravity field recovery beyond degree 10 (see Fig. 5). Note that the ambiguity-fixed solution represents a purely white noise process. Comparing the slope for the very short intervals in Fig. 6 with the corresponding part for the L1/L2 solution in Fig. 2, the reduction of high frequent systematic errors in doubly differenced solutions may be recognized.

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5.3 Simulation Study Dynamic GRACE A and B orbits for 20 days in a true gravity field, defined by the gravity field model EIGEN-2 up to degree 90, serve as true orbits to gen-

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erate different sets of GRACE A and B positions. Only random errors, but no modeling errors, are considered in this simulation study. Figure 8 shows the difference degree amplitudes with respect to the true gravity field for different 20-day solutions. Although not necessary for the applied simulation scenario, pulses are set up for all solutions of Fig. 8 with a spacing of 30 min. For comparison, the solution denoted as “S , 1 cm” is based on GRACE A positions affected by white noise

10–5

of 1 cm RMS error, whereas the solution denoted as “D , 1 cm” is based on vector differences which are constructed from GRACE A andGRACE B positions, both affected by an independent white noise process of 1 cm RMS error. It can be recognized that in the absence of systematic errors the solution “D , 1 cm” does not offer any advantages with respect to the solution “S , 1 cm”. Both solutions are almost identical, apart from a small degradation of the solution “D , 1 cm” for the very low degrees due to the processing

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of differential positions. This degradation is, however, much smaller than the degradation observed for the solutions based on real GRACE data (see Fig. 4). The solution denoted as “D , 1 mm” is based as well on vector differences constructed from GRACE A and GRACE B positions. The applied white noise processes of 1 cm RMS error are, however, not independent from each other in order to obtain vector differences affected by an RMS error of about 1 mm. Due to the fact that the relative positions between two satellites can be determined with a much higher precision than the individual positions, a similar approach is also applied by Gerlach et al. (2006) to study the performance of gravity field recovery for SWARM from inter-satellite range data. Figure 8 shows that the difference degree amplitudes of the solution “D , 1 mm” are reduced (shifted) almost perfectly by one order of magnitude with respect to the solution “D , 1 cm” due to the smaller noise assumed. The results with real GRACE data reveal, however, that such an assumption is wrong (see Sect. 5.2). In addition, the error-spectra of SH coefficients will differ for intersatellite range data and for vector differences. The solution denoted as “D∗ , 1 mm” is based on the same vector differences as the solution “D , 1 mm”. As opposed to all previously discussed solutions, a slightly wrong gravity field model up to degree 30 instead of the true gravity field model is used as a priori model for solving the generalized orbit determination problem. In fact gravity field recovery results are independent from the a priori gravity field model, but the fixing of the orbit parameters for one satellite to the a priori values obtained from the dynamical fit of the LEO pseudoobservations (see Sect. 3.2) can be critical when using vector differences for gravity field recovery. Figure 8 shows that the difference degree amplitudes up to degree 20 are considerably degraded with respect to the solution “D , 1 mm” by fixing to a trajectory which does not represent the LEO pseudo-observations at a level better than 10 cm (which is the case for the white noise scenario of solution “D , 1 mm”). Therefore, a similar degradation also results for solutions based on real GRACE data as it is observed in Fig. 4. In order to avoid this problem the fit of the LEO pseudoobservations has to be improved, e.g., by solving for pseudo-stochastic orbit parameters in addition to the few dynamical parameters.

A. J¨aggi et al.

6 Conclusions We reviewed and extended the principles of gravity field determination from kinematic positions of single satellites and applied them to kinematic baseline data. The estimated gravity field models derived from one year of undifferenced GRACE and CHAMP GPS data and solutions derived from a 55 days test period (days 243–297, 2003) of doubly differenced GRACE GPS data are compared with the official and superior GRACE gravity field models mainly based on K-band observations. No use of accelerometer data is made in this study. We observe a better agreement of the estimated SH coefficients with respect to the official GRACE models when using kinematic positions based on the LA/L2 instead of the L1/L2 carrier phase GPS observations due to the smaller LA carrier phase noise. The quality of the estimated low degree SH coefficients is unchanged, however, when using the more accurate LA carrier phase observations due to systematic errors either present in the kinematic positions or in the observation models used for gravity field determination. The analysis of Allan deviations of K-band residuals of kinematic positions indicates that parts of the suspected errors are introduced already in the kinematic orbit determination. The agreement with respect to the official GRACE models of the estimated SH coefficients from GRACE A or B for the very low and the very high degrees is not as good as for CHAMP. Whereas an inferior quality is expected from GRACE for the very high degrees due to the higher orbital altitude of the twin satellites, further investigations are necessary to explain the effect related to the very low degrees. The inclusion of accelerometer data might help to further reduce a possible degradation due to non-gravitational perturbations not entirely captured by the pseudo-stochastic orbit modeling. Compared to a combination of solutions derived from undifferenced GRACE A and B data, solutions based on vector differences derived from doubly differenced GRACE GPS data show a slightly better agreement for the very high degrees with respect to the official GRACE models, but they show a considerably worse agreement for the very low degrees. The analysis of Allan deviations of K-band residuals of the kinematic baselines as well as the analysis of simulated

Assessment of GPS

data indicates that the former effect is related to a reduction of systematic errors of high frequency in doubly differenced GPS data. The degradation of the very low degrees for the solutions based on real GRACE data is mainly explained by fixing one of the two satellite orbits to dynamical trajectories of insufficient quality, and only partly by processing differenced GPS data. The resolution of the carrier phase ambiguities to their integer values turns out to have only a small impact on the subsequent gravity field recovery. The analysis of Allan deviations of K-band residuals of the kinematic baselines with unresolved and resolved ambiguities, respectively, showed that the different treatment of the ambiguities cannot affect the kinematic baselines on intervals shorter than about 5 min. Therefore, small benefits for the ambiguity-fixed solution can only be expected up to about degree 10. Acknowledgments The authors are grateful to the Center for Orbit Determination in Europe (CODE) for using GPS orbit solutions and clock corrections. CODE is a joint venture of the Astronomical Institute of the University of Bern (AIUB, Switzerland), the Federal Office of Topography (swisstopo, Switzerland), and the Federal Agency of Cartography and Geodesy (BKG, Frankfurt).

References Allan DW (1987) Time and frequency (time-domain) characterization, estimation, and prediction of precision clocks and oscillators. In: IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 34, 647–654 Beutier G (2005) Methods of celestial mechanics. Springer, Berlin Heidelberg New York Dunn C, Bertiger W, Bar-Sever Y, Desai S, Haines B, Kuang D, Frankline G, Harris I, Kruizinga G, Meehan T, Nandi S, Nguyen D, Rogstad T, Thomas JB, Tien J, Romans L, Watkins M, Wu SC, Bettadpur S, Kim J (2003) Instrument of GRACE. GPS World 14: 17–28

123 F¨orste C, Flechtner F, Schmidt R, K¨onig R, Meyer U, Stubenvoll R, Rothacher M, Barthelmes F, Neumayer KH, Biancale R, Bruinsma S, Lemoine JM, Loyer S (2006) A mean global gravity field model from the combination of satellite mission and altimetry/gravimetry surface gravity data – EIGENGL04C. Geophysical Research Abstracts, Vol. 8, European Geosciences Union. Gerlach C, Visser PNAM (2006) SWARM and gravity: possibilities and expectations for gravity field recovery. In: Proceedings of First International Science Meeting, SWARM, Nantes, France Heiskanen WA, Moritz H (1967) Physical Geodesy. Freeman J¨aggi A (2006) Pseudo-stochastic orbit modeling of low earth satellites using the global positioning system. Ph.D. Thesis, Astronomical Institute, University of Bern, Switzerland J¨aggi A, Hugentobler U, Beutler G (2006) Pseudo-stochastic orbit modeling techniques for low-Earth orbiters. J Geod 80: 47–60 J¨aggi A, Hugentobler U, Bock H, Beutler G (2007) Precise orbit determination for GRACE using undifferenced or doubly differenced GPS data. Adv Space Res 39: 1612–1619 K¨ohler W (2001) CHAMP RINEX format observable extensions for CHAMP SST Data. CH-RINEX-EXT, GFZ Potsdam, Germany Kroes R (2006) Precise Relative Positioning of Formation Flying Spacecraft using GPS. Publications on Geodesy, Nederlandse Commissie voor Geodesie Mayer-G¨urr T, Ilk KH, Eicker A, Feuchtinger M (2005) ITGCHAMP01: a CHAMP gravity field model from short kinematic arcs over a one-year observation period. J Geod 78: 462–480 Prange L, J¨aggi A, Beutler G, Mervart L, Dach R (2008) Gravity field determination at the AIUB - the celestial mechanics approach. This Volume. Reigber C, Schwintzer P, Neumayer KH, Barthelmes F, K¨onig R, F¨orste C, Balmino G, Biancale R, Lemoine JM, Loyer S, Bruinsma S, Perosanz F, Fayard T (2003) The CHAMPonly Earth Gravity Field Model EIGEN-2. Adv Space Res 31: 1883–1888 ˇ Svehla D, Rothacher M (2004) Kinematic precise orbit determination for gravity field determination. In: Sans`o F (Ed) A Window on the Future of Geodesy. Springer, Berlin Heidelberg New York, 181–188 Tapley BD, Bettadput S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 305(5683)

Detecting the Baltic Sea Level Surface with GPS-Measurements and Comparing it with the Local Geoid Model ¨ Aive Liibusk and Harli Jurgenson

Abstract Gravimetric geoid NKG04 (Forsberg et al., NKG Geoid Meeting, Copenhagen) is derived by KMS using all available gravimetric data from the region. Some areas, also close to the Estonia are not completely covered by gravity measurements, example the eastern part of the Gulf of Finland. Baltic Sea is measured by airborn gravimetry with accuracy probably 2 mGal. Our idea was to compare the geoid on the sea areas against the independent method like GPSlevelling on the mainland. Main problem have been of course how to remove water tilt during the campaign. The existent GPS device on board a ship stores data every second and determines the heights with an accuracy of a few centimetres (using the kinematic method with post-processing, several base stations close to the ferry line). As a result, it is possible to observe the present relative water level profile in reference to the ellipsoid. If we take into account the tilt of the water level at the moment of measurement, we can observe the relative change of the geoid using independent methodology, which serves as a comparison to the gravimetric geoid solution. With this method we explored some areas on the Baltic Sea covered with regular ferry lines where the geoid profile changes faster. One such area lies about 30 km north of the island of Hiiumaa, where the geoid has a “lump”: the separation of the geoid from the ellipsoid changes by 1 m over a 70-km distance starting from Paldiski; further towards Sweden the original

separation is restored. In addition, we analyzed all the Estonian and Swedish profiles, also using GPS data from Swedish base stations. We also performed the same kind of measurements on ferries running the regular line TallinnSt. Petersburg-Helsinki-Tallinn. Those measurements were of particular interest as there was no gravimetric data available for the eastern part of the Gulf of Finland. As a third track, measurements were performed on liners running between Sillam¨ae and Kotka. The results show that the 150-km geoid NKG04 profile close to Hiiumaa did not differ any more than 15 cm from the GPS-measured level surface. The influence of the water tilt was more or less eliminated using the available tide gauge data. The profile between Tallinn and St. Petersburg manifested similar difference. Most of the measurements were repeated several times on the same profile.

1 Introduction

The gravimetric geoid NKG04 was derived by KMS using all the gravimetric data available from the region. Unfortunately, the data set does not originate from one and the same time period, the coverage is not homogeneous, and the quality varies from region to region. Some areas, including those close to Estonia, such as Aive Liibusk the eastern part of the Gulf of Finland, are almost Department of Geomatics, Estonian University of Life Sciences, uncovered by gravity measurements. The Baltic Sea Kreutzwaldi 5, Tartu 51014, Estonia, was measured by airborne gravimetry with the accue-mail: [email protected] racy of probably 2 mGal. Our idea was to compare the Harli J¨urgenson Department of Geomatics, Estonian University of Life Sciences, geoid for the sea areas with an independent method. We used GPS-measurements and ferries on regular lines to Kreutzwaldi 5, Tartu 51014, Estonia M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Fig. 1 Tracks measured by GPS on ferry. Background is Geoid Model NKG04; contour interval is 0.2 m (500,000 m responses to longitude 24◦ )

for water tilt estimation. After processing the GPS data, the water tilt correction was added. The wave effect was eliminated using the trend line of real stored points. Estonian tide gauges provide a water level relative to the Baltic height system (BK77) and Finnish ones relative to the mean sea level. The mean sea level values of tide gauges were recalculated into the N60 2 Method height system to eliminate the land uplift effect. The formula presented by The Finnish Institute of Marine A GPS receiver was placed on a ferry, using kinematic Research (http://www.fimr.fi) was used. The Finnish N60 height system differs from technology (or real-time kinematic). The GPS receiver stores the position and the height every second. Post- the BK77 system by about 3 cm (J¨urgenson and processing was performed using several base stations Kall, 2004). Furthermore, the Estonian height system located close to the measured line. The GPS receiver has been affected by relative land uplift – about 6 cm was placed on the open deck of the ferry (Fig. 2) and during the 30 years. However, the Estonian tide gauges was static during the measuring period. GPS receivers used in the campaign are situated often in the region Trimble 5700 and 5800 were used. We only wanted to with similar land uplift. Hence, all the water level determine the relative change of the level surface; the heights were computed to the Baltic height system absolute height from the sea level was not important. In BK77 before they were used. The mean sea topography was small in the tested some instances, two GPS receivers were used simultaregions, about 4 cm between Tallinn and St. Petersneously on the ferry (Sillam¨ae – Kotka ferry line). During the measurements, the present water level burg, (Ekman and M¨akinen, 1996) and probably heights were obtained from the closest tide gauges 10 cm between Paldiski and Stockholm (Putanen and cover areas on the Baltic Sea. The good coverage of the GPS permanent base stations made the measurements easier. How to eliminate water tilt during the campaign was the main hurdle to negotiate.

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Kakkuri, 1999) (Fig. 3). The measuring lines were mostly shorter, as a result of which the influence was smaller and was not taken into account here. The accuracy of the GPS measurements is a few cm depending on the actual atmosphere, the number of satellites and the PDOP as well as the vector length (for example 1 ppm makes 4 cm per 40 km). The number of satellites in particular may become a problem due to long distances from base stations. Planning the timing of the measurement is difficult because we are depending on the ferry schedule. Taking into account also the water tilt elimination problems, the level surface accuracy is 5–15 cm according to our estimation. Yet this method is estimation to the geoid, and actual geoid changes are usually 10 times bigger along the measuring route. This is of special interest in the case of areas for which gravimetric data is missing or has serious quality problems.

3 Test Measurements

Fig. 2 GPS Trimble 5700 on board

Fig. 3 The mean sea surface topography on the Baltic Sea (Poutanen, Kakkuri 1999 a)

The first test measurements took place in 2004 between Paldiski and Kapellsk¨are. The method was tested in

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Fig. 4 GPS measurement on ferry to determine relative geoid change. Background is the gravimetric geoid NKG04 with a contour line interval of 0.2 m

areas of faster geoid changes. One of the areas lies about 30 km to the north of the island of Hiiumaa (North-West Estonia). The area is especially interesting because the geoid surface is placed a little higher compared to the surrounding area (Fig. 4). Continuous RTK measurements were performed on board ferries running the regular line between Paldiski, Estonia, and Kapellsk¨are, Sweden. Two base stations were used for RTK (in Dirhami and in the north of Hiiumaa) in order to ensure that the baseline length does not exceed 34 km. A GSM connection was used for transmission of corrections. Ship speed was 30 km/h, thus the half distance (70 km) was measured during 3 h. The westernmost part of the line was measured using the kinematic method with post-processing from the Hiiumaa base station. The storing interval of the GPS measurements was 1 and 2 s. Trimble 5700 GPS devices were used. The RTK fixed solution was obtained during the measurements. Wind speed was generally less than 6 km/h. The ferry’s up-and-down movement was normally less than 20 cm (Fig. 5, scale of the figures is a little distorted, while only east or north component is used). The tide gauge data did not show any remarkable changes in water level during the measurements

(less than 3 cm). However, that day’s water level at the tide gauges of Ristna and Dirhami (Fig. 4) differed from the mean sea level by +3 and +17 cm respectively. This indicated that water was tilted by 14 cm between the tide gauges (over 80 km in the East–West direction). The Paldiski tide gauge provided a reading similar to that of Dirhami. Unfortunately, there were no more tide gauges available in the region. From the results, a smooth trend line was generated to eliminate local sea level change caused by wind effect. The gravimetric geoid difference from Paldiski to the maximum point was 92 cm (Figs. 4 and 5) while the RTK GPS measurements showed a relative change of 77 cm (Fig. 5, the value was obtained from the trend line) along the same line. If we take into account the tilt of the water level (about 7 cm per 70 km, Table 1), the results agree with about 8-cm accuracy: 92–7 = 85 cm, 85–77 = 8 cm. So, a comparison of the relative change using the gravimetric solution with the ship GPS solution showed that the preliminary results agreed within an 8-cm accuracy range for the eastern part of the test line. The middle part (measured from the Tahku base station using RTK, 410–430 km in the East–West direction) was a bit

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Fig. 5 Water level relative heights by kinematic GPS between 370 and 502 km along the East coordinate (500 km responses to longitude 24◦ )

Table 1 Tide gauges at Ristna, Dirhami and Paldiski Station/time 29.06.2004 Ristna Ristna Ristna Paldiski Paldsiki Dirhami Dirhami 30.06.2004 Ristna Ristna Paldiski Paldiski Ristna Dirhami Dirhami

Water level BK77 (cm) 08.00 14.00 20.00 08.00 20.00 08.00 20.00

+4 +1 +3 +17 +12 +17 +14

08.00 14.00 08.00 20.00 20.00 08.00 20.00

+3 +5 +16 +26 +5 +17 +23

biased (ca 15 cm, Fig. 5). The last 30 km of the western part of the line were measured using kinematic measurement with post-processing. In the western part, the water difference measured with GPS was 1.08 m. If we subtract the water tilt correction 8 cm, the water profile difference is 1 m. Similarly, the west part shows a clearly negative trend until East coordinate 370 km, as does the gravimetric geoid: by 1.1 m respectively (Figs. 5 and 6). We can see that the tilt-corrected water (sea) level change agrees with the NKG04 model in the region within the range of 10 cm. To improve reliability the measurements were repeated on 28 and 29 June, 2004. The entire dis-

tance from Paldiski to Sweden was measured using kinematic GPS. Only the kinematic method with postprocessing was used. A base station was established in the north of Hiiumaa (Lehtma, Fig. 1). Additionally, data from a second base station at Hanko was used. Figure 6 shows the same distance as Fig. 5. Unfortunately, that time the west part of the distance was measured under a stronger wind. The water level was even more stable than the first time; the western part of the profile was about 10 cm lower; consequently, the tilt was 10 cm per 130 km (Table 2). The results are about the same as those obtained in the first round. From Paldiski to the maximum point of the gravimetric geoid the height increased by 84 cm, tilt-corrected water level 94 cm (correction applied +4 cm) (Table 2). As well, the western part coincided within the range of 14 cm.

3.1 Measurements between Estonia and Sweden on 6 and 7 November 2004 Figure 1 presents the measured lines between Paldiski and Kapellsk¨are. The aim was to measure a geoid low situated south of Marinehamn. Again, a ferry running the regular line was used; the results are presented in Figs. 7 and 8. Additionally, data from base stations Godby and Stavsnas was used for post-processing. The stations are

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Fig. 6 Water level relative heights by kinematic GPS over 132 km, 6 November 2004 (500 km responses to longitude 24◦ ) Table 2 Tide gauge data from some stations in Estonia and Sweden (6 and 7 November 2004) Station/time 06.11.2004 Ristna Ristna Ristna Lehtma Lehtma Lehtma Dirhami Dirhami Paldiski Paldiski Stockholm Stockholm Stockholm Stockholm Marviken Marviken Marviken Marviken 07.11.2004 Ristna Lehtma Lehtma Dirhami Paldiski Stockholm Stockholm Stockholm Stockholm Stockholm Stockholm Marviken Marviken Marviken Marviken Marviken Marviken

Water level BK77 (cm) 08.00 14.00 20.00 08.00 14.00 20.00 08.00 20.00 08.00 20.00 14.00 18.00 20.00 23.00 14.00 18.00 20.00 23.00

−1 −1 +2 +7 0 +5 +7 +4 +12 +4 0 +3 +2 −1 −5 −3 0 −4

08.00 01.00 07.00 08.00 08.00 00.00 01.00 03.00 05.00 07.00 08.00 00.00 01.00 03.00 05.00 07.00 08.00

−1 0 +7 +5 +5 −5 −3 −8 −7 −7 −7 −6 −6 −10 −7 −4 −5

a part of the RTK network in Finland and Sweden respectively. As well, tide gauge data from Stockholm and Marviken was used (Table 2). All the tide gauge data were converted to the Baltic height system. As the water was almost untilted during the measurements (Table 2) we did not apply any water tilt correction. The changes in the geoid and the measured water profile coincide with each other within the range of 10 cm (Figs. 7 and 8) (Morozova, 2005).

3.2 Measurements between Estonia and St. Petersburg on 13 and 14 December 2004 Figure 1 presents the measured lines between TallinnSt. Petersburg-Helsinki. The regular ferry was used. The results are presented in Fig. 9. The base stations at Pedassaare and Virolahti were used for post-processing. The Virolahti base is a part of the Finnish RTK network. A dual-frequency GPS receiver Geotracer 3220 was stationed in Pedassaare (Estonian coast). Tide gauge data from Hamina, St. Petersburg, Toila, Kunda, Loksa and Narva-J˜oesuu was used (Table 3). The tide gauge data was converted to the Baltic height system. The profiles coincide within the accuracy of a few cm-s between Tallinn and St. Petersburg (Fig. 9) and about 10 cm between St. Petersburg and Helsinki.

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Fig. 7 Water level relative heights by kinematic GPS, 6 November 2004 (300 km responses to longitude ∼ 20.5◦ )

Fig. 8 Water level relative heights by kinematic GPS, 7 November 2004 (300 km responses to longitude ∼ 20.5◦ )

3.3 Measurements Between Estonia and Finland on 16 and 18 May 2006

situated in Toila, Estonia, were used in post-processing. Southern part of the route is post-processed from Toila and North part of the route from Pernaja base station (Fig. 10). The measurement track between Estonia and Finland Tide gauge data were collected from Hamina and is presented in Fig. 1. Two dual-frequency GPSHelsinki in Finland and from Toila, Narva-J˜oesuu receivers set up on board a ferry were used for the and Kunda in Estonia. As the tide gauge data varmeasurements (Trimble R8 and Trimble 5800). The ied very little, by a maximum of 5 cm, during the receivers were placed on the right and left boards of the measurements, no water tilt corrections were added to ferry. The use of two receivers simultaneously allowed the GPS measurement results. the comparison and estimation of the measurement Figure 10 presents the chart of the Kotka-Sillam¨ae results. track. To improve the readability of the chart, 17 m was Data from the Pernaja GPS base station, a part of subtracted from the heights measured by the Trimble the Finnish network of GPS permanent base stations, 5800 receiver and 17.5 m from those measured by the and from the dual-frequency Geotracer 3220 receiver

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Fig. 9 Water level relative heights by kinematic GPS between Tallinn and St. Petersburg, 13 December 2004. (700 km responses to longitude ∼ 27.5◦ )

Table 3 Tide gauge data from some Estonian, Finnish and Russia stations (13 and 14 December 2004) Station/time

Water level BK77 (cm) 1

13.12.2004 Loksa Kunda Toila Narva-J˜oesuu St. Peterburg Hamina Helsinki Hanko 14.12.2004 Loksa Kunda Toila Narva-J˜oesuu Narva-J˜oesuu St. Peterburg St. Peterburg St. Peterburg St. Peterburg St. Peterburg St. Peterburg

2 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00

+17 +19 +22 +21 +35 – – –

08.00 08.00 08.00 08.00 20.00 08.00 19.00 20.00 21.00 22.00 00.00

+21 +21 +24 +24 +24 +44 +42 +45 +47 +48 +55

theless, it follows the same trend as in all the earlier measurements. The lack of data in Fig. 10 is due to the kinematic data processing software ambiguity solutions yielding no solution in the middle of the Gulf of Finland. The data (heights) from the two GPS receivers used on the ferry show identical sharp protrusions, which invite a conclusion that they have to do with sea surface peculiarities or external disturbances in these particular areas rather than with GPS measurement errors. The protrusions were observed in exactly the same places irrespective of whether the ferry was bound for Estonia or Finland. The sharp changes in height (10–40 cm) occurred over short distances (1–6 km).

3.4 The Accuracy of the GPS Measurements

We calculated some profiles from different base stations. An example is given in Fig. 11. Between Estonia and Sweden, the different GPS stations yielded results that were close, with the vertical R8 receiver. As for the most part the Sillam¨ae–Kotka component remaining within the range of 10 cm. The ferry line runs parallel to geoid NKG04 contours distance between the base stations was in some cases (Fig. 1), the height change is not so drastic; never- more than 80 km.

Detecting the Baltic Sea Level Surface

133

Fig. 10 Water level relative heights by kinematic GPS between Sillam¨ae and Kotka, 18 May 2006. (6655 km responses to latitude ∼ 60◦ ). (Scale is slightly distorted, while only north component is used)

Fig. 11 Water profiles calculated from different GPS base stations (Hanko, Godby, Stavsn¨as, Lehtma) (Liibusk, 2005)

4 Conclusion

of 15 cm. Consequently, we did not discover any big or remarkable differences between the measured Using GPS kinematic measurements, some water level surface and the NKG04, even in areas for level profiles were measured on the sea. The water which the gravity data was of low quality or missing tilt effect was eliminated inasmuch as possible using altogether. different tide gauge data. The GPS measurements yielded results with the accuracy of about 6 cm; the Acknowledgments We are thankful to the companies Geotrim regional water tilt effect could not be completely OY in Finland and SWEPOS in Sweden, who supplied us with removed. However, the results showed a profile similar GPS base stations data, and to the Estonian Science Foundation, to that of Geoid Model NKG04 within the bounds who financed the research under Grant 5731.

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References Ekman, M., M¨akinen J. (1996): Mean sea surface topography in the Baltic Sea and its transition area to the Northern Sea: a geodetic solution and comparisons with oceanographic models. J. Geoph. Res. 101, C5, pp 11993–11999. Forsberg, R., Strykowski, G., Bilker, M. (2004): NKG-2004 Geoid Model – most recent model. NKG Geoid Meeting, Copenhagen. J¨urgenson, H. (2003): Determination of Estonian Precision Geoid. PhD thesis. Estonian Agricultural University, Tartu. J¨urgenson, H., Kall, T. (2004): The difference between the N60 and BK77 height systems. Nord J Surv Real Estate Res, Volume1, Number 1, pp 72–85.

¨ A. Liibusk and H. Jurgenson Liibusk, A. (2005): Detecting the Baltic Sea level surface with GPS-measurements and Comparing it with the Nordic Geoid Model NKG04. Poster. Metobs 140 Geophysics Meeting, Tartu. Morozova, N. (2005): Comparison of the gravimetric geoid NKG04 against the geoid surface tracked by GPS on some areas of the Baltic Sea. MSc thesis. Estonian Agricultural University, pp 94. Poutanen, M., Kakkuri, J. (1999 a): Final Results of the Baltic Sea Level 1997 GPS Campaign. Research works ot the SSC 8.1 of the International Association of Geodecy. Kirkkonummi, pp 192.

Strengthening the Vertical Reference in the Southern Baltic Sea by Airborne Gravimetry Henriette Skourup, Rene Forsberg, Sofie Louise Sandberg Sørensen, Christian Jermin Andersen, Uwe Sch¨afer, Gunter Liebsch, Johannes Ihde and Uwe Schirmer

Abstract In order to improve the gravity field and geoid modelling in the border area between Germany and Denmark, and over the Baltic Sea, an airborne gravity campaign BalGRACE 2006 (Baltic Airborne GRAvity Campaign and Elevation Estimation) was carried out in October 2006. The area covered was bounded approx. by 53.5–55.5◦ N, 8–15◦ E, with about 10,000 km of track data. A main purpose of this endeavour was to improve the gravity data base and geoid modelling in this area, hence, to give a substantial contribution towards comparison, verification and improvement of the vertical reference. Using the newly obtained airborne data in connection with existing terrestrial and satellite observations, Henriette Skourup Danish National Space Center, Department of Geodynamics, Juliane Maries Vej 30, 2100 Copenhagen Oe, Denmark Rene Forsberg Danish National Space Center, Department of Geodynamics, Juliane Maries Vej 30, 2100 Copenhagen Oe, Denmark Sofie Louise Sandberg Sørensen Danish National Space Center, Department of Geodynamics, Juliane Maries Vej 30, 2100 Copenhagen Oe, Denmark Christian Jermin Andersen Danish National Space Center, Department of Geodynamics, Juliane Maries Vej 30, 2100 Copenhagen Oe, Denmark Uwe Sch¨afer Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany Gunter Liebsch Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany Johannes Ihde Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany Uwe Schirmer Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany

the combined data set allows various methodological investigations into gravity field modelling to be carried out. First numerical results of modelling the gravity field and the regional geoid are presented applying different geoid determination techniques. Keywords Airborne gravimetry · Quasigeoid · Vertical reference

1 Introduction The border region between Germany and Denmark and the adjacent Baltic and North Sea areas are characterized by a quite inhomogeneous spatial distribution of gravity points and of their measurement epochs. Compared to the land regions, the sea areas contain less observations with poorer data quality. In some areas there are no observations at all, e.g. in some of the mud flats, lagoons, along coastlines, and around the numerous islands (cf. Fig. 1). By their inherent nature the different existing data (land gravity, marine gravity, and reliable satellite altimetry gravity) have practically no over-lapping zones in the near-coastal areas. Neither by terrestrial observations nor by ship-borne measurements alone, is it possible to obtain an appropriate, homogeneous data coverage along and across the coastlines. Due to the last years improvement in positioning airborne gravimetry has become a major tool for gravity data acquisition in such areas. Acknowledging this, and recognizing that reliable gravity measurements carried out in a well-defined gravity reference system are a necessary precondition for advanced gravity field models on land as well as in

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Fig. 1 Data coverage in the south-western Baltic Sea before BalGRACE 2006 (land, marine, sea-bottom, and from KMS02 altimetry model derived gravity data)1

the adjacent sea areas, the Bundesamt f¨ur Kartographie und Geod¨asie (BKG) and the Danish National Space Center (DNSC) agreed in 2006 to have jointly two airborne gravity campaigns (one to cover the Baltic Sea and the other to cover the North Sea), to improve in that way the current gravity data situation in the German-Danish border region. This will strengthen the control of the height reference systems used in Denmark and Germany and in that way will contribute to the creation of a panEuropean height reference frame. The first of the campaigns – BalGRACE 2006 – was carried out October, 16–25, 2007 in the south-western Baltic Sea using a Beech King Air B200 airplane. In this study are given first results of using the airborne gravity survey data (i) to fill-up various existing data gaps; (ii) to evaluate the so far available gravity observations, and (iii) to substitute or eliminate older, less reliable data, and, (iv) for improved geoid modelling. It is expected, that from the BalGRACE 2006 survey also other projects will have a special benefit, like e.g. the establishment of a fixed connection

1

A colour version of all figures can be found on the IAG Commission 2 website.

between Denmark and Germany across the Fehmarn belt, coastal engineering and protection, etc.

2 Method 2.1 Measurement Setup The survey was carried out using a King Air B200 airplane with pressure cabin owned by COWI A/S, Denmark. The equipment installed in the airplane included an airborne gravimeter, a laser scanner, 4 geodetic GPS receivers and an inertial navigation system (INS). The instruments were powered by 28 V DC of the aircraft, converted into 12, 24 and 220 V AC by a specially designed power/conditioner system (including backup batteries), designed by Greenwood Engineering, Denmark. A LaCoste & Romberg gravimeter (Serial no.: S-38) specially developed for air and marine applications was used. This was the first time DNSC used the S-38 gravimeter in an airborne survey, but an identical type of instrument (S-99) has been used to measure gravity in various surveys worldwide, e.g. Greenland, Malaysia, and Mongolia, showing very good results. Prior to the BalGRACE project the S-38 was tested in

Strengthening the Vertical Reference

a brief marine survey for one week in the Skagerrak, with good results.

2.2 Basic Principle The free-air gravity anomalies at aircraft level are obtained from g = f z − f z0 − h

+ δge¨otv¨os + δgtilt + g0 − γ0

∂γ ∂ 2γ 2 − (h − N ) + 2 (h − N ) ∂h ∂h where f z is the gravimeter observation, f z0 the apron base reading, h

the GPS-derived vertical acceleration, δge¨otv¨os the E¨otv¨os correction computed by the formulas of Harlan (1968), g0 the apron gravity value, γ0 normal gravity at the GRS80 ellipsoid, h the GPS ellipsoidal height and N the geoid undulation. The platform off-level correction δgtilt is based on a platform modelling approach as described in Olesen (2003). All data were filtered with a symmetric second order Butterworth filter with a half power point at 200 seconds, corresponding to a resolution of 5.5 km (halfwavelength).

3 Results and Evaluation of the Air-Borne Data The gravity data was processed using the standard DNSC processing setup, as explained in details in Olesen (2003). Free-air anomalies were calculated for each track individually, and no cross-over adjustment was used to modify the single-line results, which are in principle given as free-air data (and associated filtered gravity value) at altitude. Unfortunately a couple of problems became apparent during the processing of the data. Due to varying pressure surface and a relative low flight elevation height (approx. 1,000 ft), the aircraft was forced by the air traffic authorities to change height at several locations along the lines. Even small changes in height provide excessive signals in the gravimeter, resulting in gaps in the processed gravity data. Another more serious problem became clear during the processing. The gravimeter S-38 seems to be

137

sensitive to turbulence, which was not the case for earlier DNSC surveys employing the identical type of instrument, S-99. Due to excessive turbulence on some flight legs, airborne gravity results seem to increase erroneously by 5–10 mgal (and even up to 20 mgal) during turbulent periods as compared to surface data (over much of the region there is relatively good surface data coverage). Subsequent discussions with the S-38 manufacturer ZLS indicate that the sensor bearings might have wear, but also that the smaller serial-number S-meters tend to have poorer mechanical air-dampers for the beam. Due to the sensitivity of the S-38 gravimeter to turbulence, the data set was divided into three categories, classified as (i) little or no turbulence (bold lines), (ii) medium turbulence (thin lines), (iii) heavy turbulence as judged by the unfiltered vertical accelerations from the gravimeter and GPS (cf. Fig. 2 (a)). Data showing large vertical accelerations due to changes in flight altitude as well as data classified as heavy turbulence were discarded from the data set. All data were used, however, to provide a complete data set for geoid predictions, using least-squares collocation with a priori error estimates in the airborne data dependent on turbulence level, as well as surface data on the ground (cf. Fig. 2 (b)). This data set was of course not used for the validation experiments outlined in the sequel. The overall internal accuracy of BalGRACE 2006 has been estimated using cross-over points. The result of the cross over analysis in the best turbulence case gives an rms deviation of 1.79 mgal (cf. Table 1), implying a white noise along track error of 1.3 mgal, which is a very good result, in accordance with earlier DNSC airborne surveys in the central Baltic Sea (Forsberg et al., 2001). A comparison of selected BalGRACE 2006 airborne gravity anomaly values with terrestrial free-air anomalies from the BKG data base indicate a quite good reproduction of anomalies over land, whereas larger deviations occur over the adjacent sea areas – mainly in the mud flats at the North Sea coast and along the coastlines around the numberous islands in the Baltic Sea. Figure 3 illustrates this behaviour, showing the difference of the free-air anomaly pattern before BalGRACE with the pattern when the BalGRACE data are incorporated. Since in the mentioned areas we had less or no observations at all, or sometimes

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(b)

55°

54°

8°

10°

12°

14°

Fig. 2 (a) Track segments and their qualification (bold lines represent data with little or no turbulence) (b) Free-air anomalies from collocation-filtered air-borne and surface gravity data

Along both tracks the observed air-borne anomalies have been compared to the free-air anomalies obtained Cross over # of crossings Max. difference RMS from a regional model (Liebsch et al., 2006) based on analysis (mgal) (mgal) terrestrial-only data. Little or no 26 2.77 1.79 The central and western segment of track R12 is turbulence (i) mainly over ground (land territory) where we have Medium 34 4.62 2.13 good terrestrial data coverage. It is obvious (cf. also turbulence (i) + (ii) Fig. 4 (c)) that here the airborne data reproduce to a large extend the terrestrial data. Contrary to that, the eastern segment of track R12 crosses the Oderhaff laonly less-quality shipborne data available, it actually goon – an inlet of the Baltic Sea between Poland and confirms, that the greatest impact from the airborne Germany – without nearly any gravity data at all. Here, data is given here. The locations of two typical tracks up to now the existing terrestrial gravity model (grey are presented in Fig. 4(a): the first track R12 (lower, curve) was based only on extrapolated data, and, theresouthern track) was flown under conditions with fore, the relevant deviations to the recently observed relatively low turbulences, and a second one F12 (up- airborne data (black curve) exemplify the information per, northern track), that had substantial turbulences. input coming from the airborne gravity data. Table 1 Cross-over analysis (the numbers are in accordance to the classification given above)

Fig. 3 Absolute deviations of free-air anomalies (FA) with / without BalGRACE 2006 airborne gravity data [mgal]

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Fig. 4 (a) Absolute deviations between observed and modelled free-air anomalies along two typical tracks. F12 (northern) and R12 (southern) [mgal]; (b) free-air anomalies along track F12; (c) free-air anomalies along track R12 airborne data (black curve), terrestrial gravity model (grey curve)

Along track F12 we had medium turbulences. Compared to track R12 this result in a larger bias and greater deviations between the airborne-derived anomalies and the anomalies predicted with the model that is based on terrestrial-only data (see Fig. 4 (b)). Making various checks with the BalGRACE 2006 gravity data against existing data sources (cf. Fig. 5 (a), (b), (c)), we were able to identify marine gravity data that appear to have a remarkable systematic offset. For example, in the border area between the Danish island Falster and the German island Hiddensee there have been recognized segments of ship tracks/older marine data (Plaumann, 1979) that had a systematic bias of about 5–10 mgal to the airborne data, but also to other data sets, like satellite altimetry data and seabottom gravity data. We used in this investigation the KMS02 (Andersen & Knudsen, 1998) satellite altimetry gravimetry data selected only in the more open sea areas in the eastern part of the region. Therefore, according to these comparisons some older shipborne data should be considered with care and might be considered to be excluded from the data base for future geoid computations. Also minor biases at the 1 mgal level could point to inconsistencies in the application of the atmospheric correction. Although the gravity anomalies derived from the KMS02 altimetry model perform surprisingly well in the shallow coastal water regions of the Baltic Sea, there are also some

problematic zones, as for instance, the northeast corner of the tested area shown in Fig. 5 (b).

4 Preliminary Geoid Modelling To estimate the effect of the BalGRACE data on the geoid computation, we compared for the border area between Denmark and Germany the former German quasigeoid solution with a newly computed quasigeoid. For this first and preliminary model we added about 7,000 selected airborne data from BalGRACE 2006 to approx. 73,000 terrestrial free-air anomalies and 135 German GPS/levelling data that had been used also in earlier / previous computations. The result is given in Fig. 6. The model has been derived by using the removecompute-restore approach in frame of a point mass approximation for the gravitational functionals. For removing the long-wavelength part the global model Eigen-CG01C has been used, whereas the topography has been modelled by a digital terrain model with 50 m spacing complemented by GTOPO30 data (Global 30 Arc-Second Elevation Data Set, Earth Resources Observation and Science (EROS)). Comparing both gravity models to each other (i.e. without and with incorporating the BalGRACE 2006

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airborne data); one obtains resulting changes in the German quasigeoid solution of up to about +/ − 4 cm, located mainly at the margins, where especially in the northern part the airborne data give new substantial contributions.

5 Summary / Conclusion The BalGRACE airborne data provide in many places new and substantial information for improved gravity and geoid modelling playing in some areas a key role for identifying less reliable data. It thus illustrates the usefulness of airborne gravity data in unified land and marine gravimetry. The obtained data material clearly indicates the important impact that high-grade airborne data give on (i) improving the gravity data coverage in the land-sea crossing zones of the German-Danish border region and on (ii) a better modelling of the geoid, i.e. of the vertical reference in the southern Baltic Sea. Table 2 Statistics of comparing different data sets in the test area shown in Fig. 5 Differences

Bias [mgal]

RMS [mgal]

(a) marine gravity minus airborne gravity (b) KMS02 altimetry/gravity minus airborne gravity (c) sea-bottom gravity minus airborne gravity

−3.8

2.8

−0.6

2.6

−1.5

1.7

References

Fig. 5 Absolute deviations of free-air anomalies in [mgal] (a) between marine and airborne gravity (upper plot) (b) between altimetry-KMS02 and airborne gravity (middle plot), and (c) between sea-bottom and airborne gravity (lower plot). All illustrations in same grey scale

Andersen, O. B., P. Knudsen (1998). Global marine gravity field from the ERS-1 and GEOSAT geodetic mission altimetry. In J. Geophys. Res. Vol. 103, No. C4, pp. 8129–8137. Forsberg, R., A. V. Olesen, K. Keller, M. Møller, A. Gidskehaug, D. Solheim (2001). Airborne gravity and geoid surveys in the Arctic and Baltic Seas In: Proceedings of International Symposium on Kinematic Systems in Geodesy, Geomaties and Navigation (KIS-2001), Banff, June 2001, pp. 586–593. Harlan, R. B. (1968). E¨otv¨os corrections for airborne gravimetry. In: J. Geophys. Res., 3, pp. 4675–4679. Ihde, J., U. Schirmer, F. Stefani, F. T¨oppe (1998). Geoid modelling with point masses. In: Proceedings of the Second Continental Workshop on the Geoid in Europe, Reports of the Finnish Geodetic Institute, Masala, Finland, pp. 199–204. Liebsch, G., U. Schirmer, J. Ihde, H. Denker, J. M¨uller (2006). Quasigeoidbestimmung f¨ur Deutschland. In: DVW Schriftenreihe, Band 49.

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Fig. 6 German quasigeoid model improvement due to incorporating BalGRACE 2006 data (absolute changes in [cm])

Olesen, A. V. (2003). Improved airborne scalar gravimetry for regional gravity field mapping and geoid determination. In: Kort og Matrikelstyrelsen, Technical Report No. 24, 55 pp.

Plaumann, S. (1979). Eine seegravimetrische Vermessung in der westlichen Ostsee. In: Geologisches Jahrbuch, Reihe E, Heft 14, pp. 25–42.

Downward Continuation of Airborne Gravimetry and Gradiometry Data Using Space Localizing Spline Functions ¨ and A.A. Makhloof F. Mueller, T. Mayer-Gurr

Abstract Airborne gravimetry is capable of filling the gap between the long wavelength parts of the gravity field provided by the satellite missions such as CHAMP or GRACE and the short wavelength parts derived from terrestrial data. Furthermore, airborne gravimetry techniques are not restricted to continental areas and are not subjected to certain limitations in contrast to terrestrial data. Considering optimal conditions the measurement accuracy varies around 1–2 mGal for a spatial resolution of approximately 2 km. In order to achieve this accuracy the downward continuation process in connection with the representation of the gravity field functionals becomes of special importance. In this paper we would like to present the use of space localizing spline functions and their effect on the downward continuation process. The use of the presented approach is demonstrated on simulated airborne gravimetry data as well as on airborne gradiometry data. After all the spline-method is applied to a real gravity field composed of upward continued terrestrial gravity data. Keywords Downward continuation · Space localizing base function · Airborne gravimetry · Airborne gradiometry

1 Introduction Airborne gravimetry as well as airborne gradiometry contribute a significant part with respect to the determination of the earth’s gravity field. On the one hand these techniques can resolve wavelength which can not be resolved by satellite missions (e.g. CHAMP, GRACE or GOCE) and on the other hand gravity values measured using airborne techniques are more economic than terrestrial methods with comparable resolutions. The basic measuring principle is shown in Fig. 1. In case of (scalar) airborne gravimetry the aircraft observes the total acceleration f, which is composed of the earth’s gravity field g and other disturbing accelerations a, caused by the aircraft’s motion. The major problem is to separate the desired

f

g a

F. Mueller Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany, e-mail: [email protected] T. Mayer-G¨urr Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany A.A. Makhloof Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany

Fig. 1 Measuring principle of airborne gravimetry

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gravity signal from the disturbing acceleration. With the help of kinematic differential GPS the position of the aircraft can be obtained and the disturbing accelerations can be removed at every point. To account for the still remaining noise, the data is low-pass filtered, resulting in band-limited gravity disturbances at flight altitude. In case of airborne gradiometry the gravimeter is replaced by a gradiometer, which observes the gravity gradient. As a result the disturbing accelerations are eliminated by this method. Furthermore airborne gradiometry is capable of resolving very short wavelength, which makes it also very interesting for mineral prospecting. However, in physical geodesy a main object is to derive the geoid heights. For that reason any observation at flight altitude has to be referred to sea level, which requires downward continuation. This ill-posed problem is presented in the following by using space localizing spline functions.

2 Theory

F. Mueller et al.

2.1 Gravity Disturbance in Terms of Space Localizing Spline Functions A general form of the spline function can be given as (Freeden et al., 1998): ∞

ϕi (Q) =

kn Pn (Q, Si ).

(1)

n=0

The coefficients kn define the shape of the spline function. In this presentation degree variances are used. The points Si represent the grid, on which each spline is located. The disturbance potential in terms of spline functions at geoid level can be formulated as follows: ∞

N

T (Q) =

kn Pn (Q, Si ).

ai i=1

(2)

n=0

The gravity disturbances at flight level can be expressed in terms of an upward continuation with the Neumann function: δg(P) =

R 4π



K (P, Q) N eumann T (Q)d (3) In this chapter the space localizing spline functions are  derived for the case of (scalar) airborne gravimetry and for the case of airborne gradiometry. In the following the point P refers to the flight altiSubstituting the approximation approach Eq. (2) in tude and the Q refers to geoid level. Eq. (3) results in:

Fig. 2 A general spline function

Downward Continuation of Airborne



1 δg(P) = 4π

145

∞

which can be calculated by chain rule:

(2n + 1)(n + 1)

"

n=0



R R+H

×

 kn Pn (Q, Si ) d.

ai i=1

ϕαβ =

Pn (P, Q)

∞

N

∂ϕ · ∂r ∂ 2ϕ + ∂r ∂t



#n+2

After applying the addition theorem of spherical harmonics this equation reads: (n+1)

ai i=1

"

∞

N

n=0

∂r ∂r ∂α ∂β . 2 ∂ ϕ ∂t ∂t ∂t 2 ∂α ∂β (9) ·

n=0

(4)

δg b (P) =

∂ 2r ∂ϕ ∂ 2t ∂ 2ϕ + · + 2 ∂α∂β ∂t ∂α∂β ∂r 2 ∂r ∂t ∂ ϕ ∂r ∂t · + · + ∂α ∂β ∂r ∂t ∂β ∂α

R R+H

#n+2 kn Pn (P, Si ).

(5) A major advantage of this representation arises from the fact, that Eq. (3) is not approximated in terms of block mean values such as integral methods actually do. For a comparison of integral versus spline representation the reader is referred to M¨uller and MayerG¨urr (2003).

The second derivatives with respect to r and t read: " #n+2 −(n + 1) R Pn (t) kn R r n=0 " #n+2 ∞ ∂ Pn (t) R kn = r dt n=0 " #n+3 ∞ (n + 2)(n + 1) R = kn Pn (t). (10) 2 r R n=0 " #n+2 ∞ (n + 1) ∂ Pn (t) R = kn R r dt n=0 " #n+2 2 ∞ ∂ Pn (t) R = kn r dt 2

∂ϕ = ∂r ∂ϕ ∂t ∂ 2ϕ ∂r 2 ∂ 2ϕ ∂r ∂t ∂ 2ϕ ∂t 2

∞

n=0

2.2 Gravity Gradient in Terms of Spacelocalizing Spline Functions 3 Numerical Tests Regarding airborne gradiometry the gravity gradient serves as the observation. In the following the functional model for this type of oberservation will be shown. The gravity gradient is calculated as second derivatives of the gravitational potential. In terms of spline functions it can be given as:

All calculations except for the example are based on a simulated data set given by a spherical harmonic expansion up to degree and order n = 2160. The spherical harmonic coefficients are based on the extended GPM3E97A by Wenzel (1998), see Fig. 3. Based on this geopotential model gravity disturN bances and tensor components are calculated for dif∇∇T (P) = ai ∇∇ϕ(P, Si ), (6) ferent altitudes at a sampling of 2 arcmin. i=1 With respect to the gravity disturbances an observawith tion error of σδg = ±2 mgal is assumed. The observation error for the tensor components is assumed to be t(P, Si ) = cos(ψ). (7) σ = ±1 E¨otv¨os. For H = 0 m an error-free data-set Tii is calculated, that will be considered as the “true” data The ∇∇ϕ in Eq. (6) are the set of the second derivaset. The contribution of EGM96 (Lemoine et al., 1998) tives of the base functions: of degree n = 2-360 has been removed in advance.   Since the gravity disturbance and gravity gradient ∂ 2ϕ are linear functionals of the unknown spline parame∇∇ϕ = ters x = ai we can express them as a matrix vector ∂α∂β (8) αβ product. In case of gravity disturbances this relationship is given as: with α, β = x, y, z

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Fig. 3 Degree variances of the synthetic model

⎛ ⎜ ⎝

⎞ ⎛ ⎞ δg(Pi ) ϕ(P1 , S1 ) · · · ϕ(P1 , Si ) ⎟ ⎜ ⎟ .. .. . .. .. ⎠ = Ax with A = ⎝ ⎠ . . . δg(PN ) ϕ(PN , S1 ) · · · ϕ(PN , Si ) N ×i (11)

we have to deal with an ill-posed problem regularization is necessary. The regularization parameter is calculated by variance components estimation (Koch and Kusche 2003), see Fig. 4.

In case of airborne gradiometry the left side of Eq. (11) contains the tensor components and the right side contains the second derivatives of the base dunctions. 3.1 Results-Gravity Disturbances The matrix A represents the design matrix. The simulation covers an area of 3◦ × 3◦ resulting in N = Figures 5, 6, 7, 8 and 9 show the differences between 8281 observations and i = 1369 unknowns. Since the error-free data-set at H = 0 m and the calculated Iteration Iteration

Startvalues Startvalues

2 σˆ Data

σ xx2

2

2

2 σ Data Data

r = nobs −r

ˆ ˆ = ePe r

(

1

tr= A T AN −1

2 σ Data

Regularization Regularization xˆ Txˆxˆ

σˆ x2 = r rx

Normalequation Normalequation

x

Nx ˆˆ = b b xˆ =

Fig. 4 Applied calculation schema

1 22 σ Data

ATA +

1

σ 2x

N

−1

I

x 1 2 σ Data b

rx = u −

N AT I

σ

solving

xˆ

1

σ x2

ˆx

(N −1 ) rtr=N

)

Downward Continuation of Airborne

Fig. 5 Simulated gravity disturbances at H = 4 km

Fig. 6 Error for H = 2 km

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Fig. 7 Error for H = 4 km

Fig. 8 Error for H = 6 km

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149

Fig. 9 Error for H = 8 km

In order to test the effect of the space localizing disturbing potential at H = 0 m derived from gravity disturbances at flight level. All errors are expressed in spline functions on the downward continuation process, the gravity values at ground were upward conterms of geoid noise. tinued to an altitude of H = 6 km, as shown in Fig. 12. Table 1 Error statistics for different altitudes [m] In a second step the upward continued gravity valAltitude Min Max Mean RMS ues were calculated to their origin at the surface ac2000 −0.0390 0.0280 0.0003 0.0070 cording to Eq. (11). 4000 −0.0391 0.0395 0.0004 0.0082 The following two Figs. 13 and 14 show the dif6000 −0.0381 0.0494 0.0003 0.0093 ference between the downward continued data and the 8000 −0.0418 0.0394 −0.0004 0.0114 surface data shown in Fig. 11.

3.2 Results – Gravity Gradients Figure 10 shows the simulated tensor components at H = 4 km. The results for the different flight altitudes are summarized in Table 2. The calculated tensor components are compared with the error-free data-set in analogy to the gravity disturbances.

4 Conclusions

This paper has investigated the use of spline functions as a representation of gravity functionals. In this context the emphasis was put on the downward continuation process, which is a major aspect in airborne gravimetry. The results for gravity disturbances and tensor components have shown, that this method is well suited for this inverse problem. In case of grav3.3 Example – Upward Continued High ity disturbances the error in terms of geoid hights does Resolution Gravity Data not exceed 1 cm up to a flight altitude of 6 km (see Table 1). In case of the gravity gradient, each tensor Since the prior calculations are based on a synthetical component could be reproduced with an rms smaller data set, the following results are based on real terres- than 1 E up to an altitude of H = 4 km. It should be mentioned that these results are based on a simulated trial gravity data as shown in Fig. 11.

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Fig. 10 Tensor components at H = 4 km

Table 2 Error statistics for different altitudes [1/s2 ] Altitude/T Min 2000 Txx Txy Txz Tyy Tyz Tzz 4000 Txx Txy Txz Tyy Tyz Tzz

Max

Mean

RMS

−1.2E − 09 −1.2E − 09 −1.7E − 09 −2.5E − 09 −1.9E − 09 −2.5E − 09

1.4E − 09 1.2E − 09 1.4E − 09 2.3E − 09 2.1E − 09 3.2E − 09

−1.2E − 11 2.8E − 11 −2.1E − 12 1.9E − 11 −6.4E − 11 −6.8E − 12

4.3E − 10 3.3E − 10 5.3E − 10 7.0E − 10 7.2E − 10 8.7E − 10

−2.0E − 09 −1.6E − 09 −2.9E − 09 −4.1E − 09 −3.4E − 09 −4.3E − 09

2.0E − 09 2.3E − 09 2.0E − 09 3.4E − 09 3.8E − 09 5.4E − 09

−1.9E − 11 3.4E − 11 −8.9E − 12 1.6E − 11 −8.3E − 11 2.6E − 12

6.3E − 10 4.4E − 10 7.0E − 10 8.2E − 10 9.3E − 10 9.9E − 10

scenario. In order to investigate space localizing spline functions on a more real scenario, terrestrial gravity data were upward continued and afterwards downward continued to their origin. The comparison shows an accuracy of ±1.6 mgal (Table 3). For future research

Table 2 (continued) Altitude/T Min 6000 Txx Txy Txz Tyy Tyz Tzz 8000 Txx Txy Txz Tyy Tyz Tzz

Max

Mean

RMS

−3.6E − 09 −2.6E − 09 −4.6E − 09 −6.9E − 09 −5.9E − 09 −8.4E − 09

3.7E − 09 3.4E − 09 3.7E − 09 5.9E − 09 7.0E − 09 9.3E − 09

−2.6E − 11 2.7E − 11 −7.2E − 12 2.4E − 11 −9.3E − 11 2.0E − 12

9.3E − 10 8.0E − 10 9.8E − 10 1.1E − 09 1.5E − 09 2.0E − 09

−6.8E − 09 −3.9E − 09 −7.4E − 09 −1.1E − 08 −1.1E − 08 −1.5E − 08

7.4E − 09 4.0E − 09 7.4E − 09 9.4E − 09 1.2E − 08 1.4E − 08

2.9E − 11 7.6E − 12 −2.7E − 11 −5.6E − 11 −7.3E − 11 2.8E − 11

1.9E − 09 1.3E − 09 2.0E − 09 3.2E − 09 3.6E − 09 4.5E − 09

we would like to include aspects of topography in our calculations, since this could improve the downward continuation step. Furthermore we would like to apply space localizing spline functions to a real airborne data set.

Downward Continuation of Airborne

Fig. 11 Gravity disturbances based on 9842 ground stations at the surface [mGal]

Fig. 12 Gravity disturbances upward continued to H = 6 km [mGal]

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Fig. 13 Error – downward continued data compared to surface data [mGal]

Fig. 14 Error (zoomed) – downward continued data compared to surface data [mGal]

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Table 3 Error statistics based on Fig. 14 [mGal] Ground data Error

Min

Max

Mean

RMS

−59.3 −6.1

170.3 5.3

34.4 0.002

42.9 1.6

References Freeden, W., Gervens, T. Schreiner, M. (1998) Constructive Approximation on the Sphere. Oxford University Press, Oxford. Heiskanen WA, Moritz H (1967) Physical Geodesy, WH Freeman and Company, San Francisco Koch KR, Kusche J (2003) Regularization of geopotential determination from satellite data by variance components estimation

Lemoine FG and 14 authors (1998) The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP1998-206861, NASA-GSFC, Greenbelt MD M¨uller, F., Mayer-G¨urr, T. (2003) Comparison of downward continuation methods of airborne gravimetry data, in: F. Sans`o (ed.) A Window on the Future of Geodesy, IUGG General Assembly 2003, Sapporo, Japan, International Association of Geodesy Symposia, Vol. 128, pp. 254–258, Springer Wenzel G (1998) Ultra-high degree geopotential models GPM98 A, B and C to degree 1800. Proc of the joint meeting Int. Gravity Commission and Int Geoid Commission, Trieste, 7– 12 September (www.gik.uni-karlsruhe.de)

The Earth’s Gravity Field Components of the Differences Between Gravity Disturbances and Gravity Anomalies R. Tenzer, A. Ellmann, P. Nov´ak and P. Vajda

Abstract The differences between gravity anomalies and gravity disturbances are numerically investigated in this study. In particular, we demonstrate how the topography, atmosphere and mass irregularities enclosed within the geoid contribute to these differences. The numerical analysis is conducted at the area of study in the Canadian Rocky Mountains. Results reveal that the dominant contributions due to the topography and mass irregularities within the geoid are similar in magnitude but opposite in sign to the extent that their combined contribution is reduced significantly. Keywords Atmosphere · Geoid · Gravity anomaly · Gravity disturbance · Topography

1 Introduction

GPS receivers in land, seaborne and airborne gravimetry and dedicated satellite gravimetry missions. For investigations involving the geological structure as well as geophysical and geodetic applications (such as geophysical explorations, solving the geodetic boundary value problems, etc.), gravity disturbances can be used along with gravity anomalies. The principal differences between theoretical definitions of the gravity anomaly and the gravity disturbance were discussed for instance by Hackney and Featherstone (2003). In this study, the differences between gravity anomalies and gravity disturbances are investigated numerically.

2 Relation Between the Gravity Anomaly and the Gravity Disturbance

To date, gravity anomalies have been the most widely For a point at the Earth surface, the difference between available gravimetric data type. Nowadays, on the the gravity anomaly g and the gravity disturbance δg other hand, gravity disturbances can be delivered by is equivalent to the change in normal gravity through modern measurement techniques supplied with the a shift of the height anomaly ζ . When taking into account only the linear approximation of the normal gravity gradient, ∂γ /∂h, we can write the following relaR. Tenzer tion (see e.g. Heiskanen and Moritz 1967, Eq. 2-147e) Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands A. Ellmann Department of Civil Engineering, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia P. Nov´ak Research Institute of Geodesy, Topography and Cartography, Ondˇrejov 244, 251 65, Czech Republic P. Vajda Geophysical Institute, Slovak Academy of Sciences, Dubravska cesta 9, Bratislava 845 28, Slovak Republic

g(rt , ) − δg(rt , ) ∼ =

∂γ (r, φ) ζ (). ∂h

(1)

With reference to the Bruns (1878) formula, the height anomaly ζ is defined as the disturbing gravity potential T at the Earth’s surface divided by the normal gravity γ evaluated for a point on the telluroid (Molodensky et al. 1960).

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T (rt ,  ζ () ∼ . = γ (rn , φ)

gravitational potentials V t and V a from the disturbing gravity potential T . Alternatively, the computation of the gravity field contribution generated the mass irregThe geocentric latitude φ and longitude λ in the ularities within the geoid can be realized solving the above equations represent the geocentric direction , boundary-value problems which are commonly used and r is the geocentric radius (rt , rn and r g specify the in geoid/quasigeoid modeling. Hence, the detailed geocentric radii of points stipulated at the Earth’s sur- description of used numerical methods is not provided face, telluroid and geoid). in this study. Seitz et al. (1994) showed that the linearization applied in the term on the right-hand side of Eq. (1) through the linear approximation of the normal gravity gradient and the height anomaly is accurate to better 3 Numerical Study than ∼ 10 μGal. To investigate the influence of the topography, at- The numerical investigation of the differences between mosphere and mass irregularities enclosed within the gravity anomalies and gravity disturbances was congeoid to the differences between gravity anomalies and ducted at the area of study bounded by the parallels gravity disturbances, the disturbing gravity potential T of 48 and 61◦ arc-deg northern geodetic latitude and in Eq. (1) is rewritten as (Van´ıcˇ ek et al. 2004) the meridians of 228 and 250◦ arc-deg eastern geodetic longitude. This area comprises the rough part of the g t a T (rt , ) = T (rt , ) + V (rt , ) + V (rt , ), (3) Canadian Rocky Mountains, surrounding flat regions and the small part of the Pacific Ocean. To demonwhere V t and V a are the gravitational potentials gen- strate the contributions of the topography, atmosphere erated by the topography and atmosphere respectively, and mass irregularities enclosed within the geoid, the and T g is the geoid-generated disturbing gravity poten- computation was realized individually for the compotial. Substituting from Eqs. (2) and (3) back to Eq. (1) nents on the right-hand side of Eq. (4). The secondary indirect effects of topography and and applying the approximation γ −1 ∂γ /∂ h ≈ −2r −1 , atmosphere on the gravity anomalies are shown in we finally get Figs. 1 and 2 respectively. The former is dominant with the secondary indirect effect of atmosphere being 2 g(rt , ) − δg(rt , ) ∼ = − T g (rt , ) approximately two orders of magnitude smaller. At rt the area of study, the secondary indirect effect of 2 t 2 a − V (rt , ) − V (rt , ). (4) topography 2R−1 V t ranges from 102.5 to 137.5 mGal. r r t

(2)

t

From Eq. (4), the difference between the gravity anomaly and the gravity disturbance consists of the residual component 2r −1 T g associated with the gravity field contribution generated by the mass irregularities within the geoid, plus the secondary indirect effects 2r −1 V t and 2r −1 V a of topography and atmosphere respectively. The gravity field contributions generated by the topography and atmosphere are evaluated by Newton’s integration using existing density and terrain models. The contribution due to the mass irregularities within the geoid can be readily evaluated so that the disturbing gravity potential T is first computed from the known value of the height anomaly ζ according to the Bruns formula (Eq. (2)). The corresponding geoid-generated disturbing gravity potential T g is then obtained by subtracting the Fig. 1 Secondary indirect effect of topography (units in mGal)

The Earth’s Gravity Field Components

Fig. 2 Secondary indirect effect of atmosphere (units in mGal)

The secondary indirect effect of atmosphere 2R−1 V a varies between 1.7 and 1.8 mGal. As seen in Fig. 1, the secondary indirect effect of topography is strongly correlated with the topography. The maxima are situated in the mountains while minima over the ocean and flat regions. A strong correlation of the secondary indirect effect of atmosphere with the topography is also recognized, see Fig. 2. Opposite to the effect of topography, minima of the secondary indirect atmospheric effect are located in the mountains. In our numerical realization, the lateral model of topographical mass density distribution discretised for a regular geographical grid of 30 × 30 arc-sec was used. The detailed 3 × 3 arc-sec digital terrain model (DTM) was used for the numerical integration at the vicinity (inner zone) of the computation point (ψ < 150 arcsec), while 30 × 30 arc-sec DTM up to the spherical distance ψ of 57.5 arc-min. The 5 × 5 arc-min mean topographic heights were used for the reminder of the 5 × 5 arc-deg near zone, and 30 × 30 arc-min mean heights for the far zone. The analytical expression for the gravitational potential generated by a rectangular prism (Nagy et al. 2000) was utilized for the integration within the inner zone. For the rest of the integration area, the Newton-Cotes numerical integration was implemented for the surface integration domain, adopting the closed analytical expression for the radial component of Newton’s integral (Gradshteyn and Ryzhik 1980). For modelling the atmospheric effect on potential, parameters of the United States Standard Atmosphere model were adopted in which the lateral atmospheric

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mass density variation and temporal atmospheric mass redistribution are not taken into account. The contribution due to mass irregularities within the geoid 2R−1 T g is shown in Fig. 3. It varies between −111.1 and −143.6 mGal. Comparison of the results in Figs. 1 and 3 reveals that the effects of topography and mass irregularities within the geoid have similar trends and magnitudes but of opposite sign. As consequence, their complete contribution is much smaller than the individual components. As shown in Figs. 4 and 5, the gravity anomalies range from −139.7 to 125.7 mGal, and the gravity disturbances vary between −144.6 and 121.3 mGal.

Fig. 3 Effect of mass irregularities within the geoid to the difference between gravity anomalies and gravity disturbances (units in mGal)

Fig. 4 Gravity anomalies (units in mGal)

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8

differences between gravity anomalies and gravity disturbances is mainly due to the errors of the input gravity data, the errors of digital terrain models, and the errors within the numerical methods used for a computation. Moreover, large errors can especially be expected due to uncertainties within the topographical mass density distribution. Disregarding the water bodies, the variation of the actual topographical mass density is mostly within ±300 kg m−3 of the commonly adopted mean value of density 2670 kg m−3 . The effect of anomalous topographical mass density distribution then amounts to about 10% of the total contribution of topography on the gravitational potential except for some parts of the world, where larger density variaFig. 5 Gravity disturbances (units in mGal) tions were documented (up to 20–30%). It can be thus empirically estimated, that the errors in estimation of The differences between the gravity anomalies and the secondary indirect effect due to uncertainties within the gravity disturbances are finally shown in Fig. 6. The the actual topographical mass density distribution may differences vary between −0.5 and 10.0 mGal. Over reach up to 10 mGal or even more. However, the correthe particular computation area the correlation of these sponding errors in computing the differences between differences with the topography is only slightly notice- gravity anomalies and gravity disturbances are reduced able as there is not any significant presence of the high- significantly. The reason is that these systematic errors frequency signal. On the other hand, the presence of propagate with opposite sign into the contribution due the long-wavelength signal indicates the existence of to the mass irregularities within the geoid. large irregularities of mass density distribution within the Earth’s mantle as it is the case for the geoid (cf. Pick et al. 1973). 4 Summary and Conclusions In our numerical results large errors can be assumed. Decimetre errors in geoid/guasigeoid modelling correspond to the errors in differences between gravity The differences between gravity anomalies and gravanomalies and gravity disturbances up to a few hun- ity disturbances were treated individually for the comdreds of microgals. The inaccuracy of computing the ponents associated with the gravity field contributions generated by the topography, atmosphere and mass ir10 regularities enclosed within the geoid. The results are 60 summarized in Tables 1 and 2. The fist two components 9 0

8

58

7

2

6

8

56

5

4

6

54

4

4

3

52

2

Table 1 Effect of topography, atmosphere and mass irregularities within the geoid to the differences between gravity anomalies and gravity disturbances (units in mGal) Effect

Min

Max

Atmosphere Topography Geoid mass irregularities

1.7 102.5 −111.1

1.8 137.5 −143.6

6

50 48

1

6

6

0

Table 2 Gravity anomalies, gravity disturbances and differences between them (units in mGal)

-1

230

235

240

245

250

Fig. 6 Difference between gravity anomalies and gravity disturbances (units in mGal)

Gravity anomalies Gravity disturbances Differences

Min

Max

−139.7 −144.6 −0.5

125.7 121.3 10.0

The Earth’s Gravity Field Components

represent the secondary indirect effects of topography and atmosphere. The numerical investigation reveals that the main contributions are due to the secondary indirect topographical effect and the component generated by the mass irregularities within the geoid. These effects are two orders of magnitude larger than the secondary indirect atmospheric effect. However, the dominant contributions are approximately equal in magnitude but opposite in sign to the extent that their combined contribution is reduced significantly. As consequence, the secondary indirect atmospheric effect contributes to the differences between gravity anomalies and gravity disturbances for about 15–20%. In the area of study, the differences between gravity anomalies and gravity disturbances range between −0.5 and 10.0 mGal. It is a well-known fact that in the global context these differences can reach up to ±31 mGal corresponding to maximum values of the height anomalies of ±100 m. It represents the variations of the Earth’s disturbing gravity potential at the level of about ±987.5 m2 s−2 . Acknowledgments The data used for numerical realization were kindly supplied by Geodetic Survey Division of Natural Resources Canada.

References Bruns, H., (1878). Die Figur der Erde. Publisher Preuss Geod Inst, Berlin. Cruz, J.Y., (1985). Disturbance vector in space from surface gravity anomalies using complementary models. Report, 366, Dept of Geodetic Science and Surveying, The Ohio State University, Columbus, OH.

159 Gradshteyn, I.S., and I.M. Ryzhik (1980). Tables of Integrals, Series and Products. Translated by Jeffrey, A., Academic Press, New York. Hackney, R.I., and W.E. Featherstone (2003). Geodetic versus geophysical perspectives of the gravity anomaly. Geophysical Journal International, 154(1), pp. 35–43. Heiskanen, W.H., and H. Moritz (1967). Physical Geodesy. San Francisco, W.H., Freeman and Co. Molodensky, M.S., V.F. Yeremeev, and M.I. Yurkina (1960). Methods for Study of the External Gravitational Field and Figure of the Earth. TRUDY Ts NIIGAiK, 131, Moscow, Geodezizdat; English translation: Israel Program for Scientific Translation, Jerusalem 1962. Nagy, D., G. Papp, and J. Benedek (2000). The gravitational potential and its derivatives for the prism. Journal of Geodesy, 74, pp. 552–560. Pick, M., J. P´ıcha, and V. Vyskoˇcil (1973). Theory of the Earth’s Gravity Field. Elsevier, Amsterdam. Seitz, K., B. Schramm, and B. Heck (1994). Non-linear effects in the scalar free geodetic boundary value problem based on reference fields of various degrees. Manuscripta Geodaetica, 19, pp. 327–338. Van´ıcˇ ek, P., R. Tenzer, L.E. Sj¨oberg, Z. Martinec, E.W. Featherstone (2004). New views of the spherical Bouguer gravity anomaly. Geophysical Journal International, 159(2), pp. 460–472; doi:10.1111/j.1365-246X.2004.02435.x.

Research on the Calibration of Onboard Accelerometer by Dynamic Method Zou Xiancai, Li Jiancheng, Jiang Weiping, Xu Xinyu and Chu Yonghai

Abstract For the new generation of satellite gravity missions, the observations of the onboard accelerometer can be calibrated by the dynamic method. In this paper, the strategy to calibrate the onboard accelerometer is studied in detail using the simulated precise orbit, and the impacts of the reference earth gravity model on the accelerometer calibration are discussed specifically. To evaluate the impacts of different strategies on the accelerometer calibration, the satellite orbit is integrated again with the calibrated accelerometer observations and the related dynamic models, and compared with the precise orbit. When the reference earth gravity model is fixed in the accelerometer calibration, the standard deviation of the differences between the re-integrated orbit and the simulated may be up to the order of meters, which shows that the reference earth gravity model has a significant effect on the accelerometer calibration. When the geopotential coefficients and the accelerometer calibration parameters are estimated simultaneously (simultaneous solution method in short), the accelerometer Zou Xiancai School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China, e-mail: [email protected] Li Jiancheng School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Jiang Weiping School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Xu Xinyu School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Chu Yonghai School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

calibration parameters are independent of the reference earth gravity model. The results in this paper show that the simultaneous solution method is suitable for the calibration of the onboard accelerometer. It is also illustrated that to determinate the earth gravity model and the calibration parameters more reliably and accurately, we should choose an optimal maximum degree of the recovered earth gravity model. Keywords Dynamic method calibration · Satellite gravity

·

Accelerometer

1 Motivation and Background The high-accuracy onboard accelerometer (ACC) is one of the key instruments for modern satellite gravity mapping missions such as CHAMP (Reigber et al. 2002b) and GRACE (Tapley et al. 2004). The main function of an onboard ACC is to provide an efficient tool to measure the non-conservative forces acting on the satellite precisely, instead of the complex surface force modeling. The basic principle of the operation of the ACC is to keep a proof mass in balance position by the control and feedback system, from which the non-conservative force can be derived (Bernard 1992; Touboul 2001). During the operation of the satellite, its internal and external environment changes continuously and the characteristics of the ACC will be influenced. As a result, the ACC readings need to be calibrated before the subsequent data processing. The ACC calibration can be performed by estimating the scale and bias factors along its three axes respectively (Bettadpur 2003). Generally, there are two

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where ( f x , f y , f z ) is the non-conservative force acting on the satellite in the earth-centered inertial (ECI) frame, ( f xacc , f yacc , f zacc ) is the ACC readings, which is given in the satellite body fixed frame (SBF). (k x , bx ), (k y , b y ) and (k z , bz ) are the scale and bias factors of the ACC along the x, y, and z axis respectively. R is the rotation matrix which transforms the ACC readings from SBF into ECI. When calibrating the ACC with the precise orbit r, the observation model reads:

schemes used widely to calibrate the ACC. One is the energy balance method (energy method in short hereafter), and the other is the dynamic method. As for the energy method, two techniques can be adopted (Gerlach et al. 2003). One is to calibrate the ACC using an existing earth gravity model (EGM). When an a priori EGM is selected, the disturbing potential along the determined satellite orbit can be computed analytically. Starting from the energy balance principle, the in-situ disturbing potential can also be computed with the satellite orbit and the ACC observations (Jekeli 1999). The difference between the two disturbing potential sets can be compensated by the scale and bias factors of the ACC. Thus, the calibration parameters can be established. It must be noted that there may exist a problem of downward continuation to compute the crossover differences because the radial distances of the ascending and descending arcs on crossover points are different. The basic problem of the aforementioned methods is whether the reference EGM would affect the ACC calibration. For the downward continuation, the remove-restore method needs a reference EGM cannot be distinguished clearly. To magnify the effects, we study this subject through the orbit comparison, which is more sensitive to the force models.

 r = r r0 , v0 , β

(2)

where r0 and v0 are the initial position and velocity of the satellite, and β is the parameter vector of the background models, including the ACC scale and bias factors. The linearization of the above observation equation is a custom process for the dynamic method (Reigber 1989; Oliver and Eberhard 2001). If the geopotential coefficients are fixed in the estimation process, the input data for the ACC calibration are the same as those used in the energy method. Therefore, the dynamic method can be used to assess the strategy used in the energy method. Furthermore, if we estimate the geopotential coefficients too, the calibration of the ACC and the refinement of EGM can be accomplished simulataneously.

2 Theory 3 Numerical Results

According to the general criterion of the dynamic method, the scale and bias parameters of the ACC can be introduced into the force models of satellite and estimated like the other unknowns (Reigber 1989). Specifically speaking, the calibration of ACC can be realized with the following force model (Visser and Ijssel 2003):

The dynamic models used to simulate the precise orbit are GGM02S truncated to the degree and order 90 (Tapley et al. 2005) and the randomly prechosen scale and bias factors, which are listed in Table 1. The Level1B ACC products (Case et al. 2004) are used as the raw ACC measurements (i.e., ACC readings in SBF) in Eq. (1). Through the massive tests, the are length is configured as 12 h, i.e. the scale and bias parameters of ACC are estimated once per 12 h, and the sampling interval of the precise orbit is 10 s.

⎛

⎛ ⎞ ⎞ k x f xacc + bx fx ⎝ f y ⎠ = R ⎝ k y f yacc + b y ⎠ fz k z f zacc + bz

(1)

Table 1 The simulated scale and bias factors of the accelerometer Scale X 1.000300002

Scale Y 1.100000010

Scale Z

Bias X/(m/s−2 )

Bias Y/(m/s−2 )

Bias Z/(m/s−2 )

0.920000001

2.13 × 10−10

−1.10 × 10−9

5.82 × 10−10

Research on the Calibration

163

We adopt two methods to assess the calibration strategies in the paper. One is to compare the calibration results with the pre-chosen scale and bias values, the other is to compare the orbits (i.e. integrating the satellite orbit with the refined dynamic models, including the calibrated ACC observations, and comparing it with the precise orbit). When the reference EGM is fixed in the calibration, four EGMs are chosen for the comparison, GGM02S (CASE A, i.e. the model used to simulate the precise orbit), EIGEN-IS (Reigber et al. 2002a) (CASE B), EGM96 (Lemoine et al. 1998) (CASE C) and OSU91 (Rapp et al. 1991) (CASE D). The initial values 1 and 0 ms−2 are selected for the scale and bias factors respectively to bootstrap the iteration. The simulated orbit spans 24 h, so there are two groups of ACC calibration results (see Tables 2 and 3, No 1 and No 2 denote the two arcs) according to the configuration of the arc length. From the results of CASE A, the reliability of the software developed by the research group can be validated. Comparing the calibration results with

their true values used in the orbit simulation, it can be concluded that the reference EGM has significant impacts on the ACC calibration, which can be verified further by the orbit comparison. Re-integrating the orbit and comparing it with the simulated orbit, we find that the maximum radial difference can be up to 4.58 m (see Tables 4 and 5, CASE D). It can be concluded that to calibrate the ACC, a state-of the-art gravity field model should be used. Otherwise the error of the a priori EGM will be aliased into the ACC scale and bias factors. To demonstrate it more clearly, we can evaluate the effects of the different EGMs on the satellite orbit. When we integrate the orbit with the OSU91 and the GGM02S as the background model respectively, the maximum difference between the two sets of the integrated satellite orbit can be up to the order of 100 m in the position and 90 mm/s in the velocity (see Fig. 1). After the calibration of the ACC with the geopotential coefficients fixed, the radial difference is reduced sharply (see Tables 4 and 5, CASE D). Most of the contribution of the earth

Table 2 The calibration results of accelerometer with various earth gravity models as the reference field (No 1) CASE A CASE B CASE C CASE D

Scale X

Scale Y

Scale Z

Bias (X/ms−2 )

Bias (Y/ms−2 )

Bias (Z/ms−2 )

1.000300002 1.000453932 1.002412821 1.007908614

1.100000010 1.100144320 1.101651619 1.106296895

0.920000001 0.919851205 0.918146376 0.912827509

2.13 × 10−10 2.72 × 10−9 5.00 × 10−8 1.51 × 10−7

−1.10×10−9 −4.92×10−9 −3.29×10−9 4.29×10−8

5.82×10−10 −3.08×10−9 −1.29×10−8 1.46×10−7

Table 3 The calibration results of accelerometer with various earth gravity models as the reference field (No 2) CASE A CASE B CASE C CASE D

Scale X

Scale Y

Scale Z

Bias (X/ms−2 )

Bias (Y/ms−2 )

Bias (Z/ms−2 )

1.000300002 1.000385853 1.000644125 1.007213139

1.100000010 1.100078848 1.099487294 1.103674106

0.920000001 0.919900678 0.920118335 0.914544029

2.13 × 10−10 6.87 × 10−9 3.84 × 10−8 5.73 × 10−7

−1.10×10−9 6.81×10−9 −6.29×10−9 7.86×10−8

5.82×10−10 1.49×10−9 −1.86×10−8 8.93×10−7

Table 4 Statistics of radial differences between the orbit integrated with calibration results and the simulated orbit (/m, No 1) CASE A CASE B CASE C CASE D

Min

Max

Mean

STD

−7.36 × 10−8 −4.34 × 10−2 −9.92 × 10−1 −4.58

1.75 × 10−7 8.12 × 10−2 8.81 × 10−1 3.11

2.13 × 10−8 1.22 × 10−2 1.28 × 10−2 −4.46 × 10−2

5.45 × 10−8 3.03 × 10−2 3.92 × 10−1 1.33

Table 5 Statistics of radial differences between the orbit integrated with calibration results and the simulated orbit (/m, No 2) CASE A CASE B CASE C CASE D

Min

Max

Mean

STD

−1.29 × 10−7 −5.14 × 10−2 −8.73 × 10−1 −2.49

2.42 × 10−8 6.82 × 10−2 8.40 × 10−1 3.20

−7.36 × 10−8 1.65 × 10−2 −2.07 × 10−2 −1.51 × 10−2

8.18 × 10−8 3.16 × 10−2 3.85 × 10−1 1.36

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Fig. 1 Effects of different Earth gravity models on the satellite orbits (GGM02S v.s. OSU91)

gravity to the residual observations, i.e. the difference between the simulated orbit and the integrated orbit is aliased into the scale and bias factors of the ACC. When we use the calibrated accelerometer observations in the following data processing directly, it is equivalent that the observation model is contaminated. To the authors’ view, this effect is hard to be quantified and eliminated. From the above analysis, a preliminary deduction can be made that to reduce the effects of a priori EGM error and improve the reliability and accuracy of the ACC calibration, the geopotential coefficients should

be estimated simultaneously (we name it simultaneous solution method in this article). It must be noted that in order to estimate the geopotential coefficients, globally distributed data are necessary. Therefore multiple arcs are simulated in this run, and the orbit spans 15 days, 30 arcs in total. The calibration results and comparisons are listed in Tables 6 and 7, for simplicity, only the statistics of the scale and bias factors’ differences from the true values are given. The last records in Tables 6 and 7 are the radial differences (129600 epochs in total). The OSU91 is selected as the initial EGM.

Table 6 Statistics of differences between accelerometer calibration results by simultaneous solution method and the true values (maximum degree is 60) Min Scale X Scale Y Scale Z Bias X(/ms−2 ) Bias Y(/ms−2 ) Bias Z(/ms−2 ) Radial difference (/m)

−1.99×10−10

−2.30×10−10 −1.60×10−10 −1.36×10−13 2.29×10−12 1.56×10−13 −6.36×10−5

Max 2.13×10−10

1.25×10−10 1.49×10−10 −1.21×10−13 2.31×10−12 1.68×10−13 9.86×10−5

Mean 1.41×10−11

−1.43×10−11 −3.63×10−11 −1.30×10−13 2.30×10−12 1.61×10−13 2.70×10−9

STD 1.04×10−10 8.00×10−11 8.48×10−11 1.30×10−13 2.30×10−12 1.61×10−13 1.92×10−6

Table 7 Statistics of differences between accelerometer calibration results by simultaneous solution method and the true values (maximum degree is 90) Scale X Scale Y Scale Z Bias X(/ms−2 ) Bias Y(/ms−2 ) Bias Z(/ms−2 ) Radial difference (/m)

Min

Max

−7.45×10−8

1.13×10−6

−6.98×10−8 −1.06×10−7 −1.38×10−11 −1.04×10−11 −2.20×10−12 −2.57×10−2

9.86×10−8 7.32×10−8 1.65×10−11 9.44×10−12 2.08×10−12 3.89×10−2

Mean 7.01×10−9

6.31×10−9 7.62×10−9 2.20×10−12 7.98×10−14 5.80×10−15 2.74×10−6

STD 4.45×10−8 4.01×10−8 4.13×10−8 7.94×10−12 5.49×10−13 9.27×10−13 5.92×10−4

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Fig. 2 Distribution of geoid error per degree of the recovered gravity model by simultaneous solution method (maximum degree is 60)

More specifically, when the estimated maximum degree of the geopotential coefficients is 60 (3717 unknowns), the numerical truncation error can be ignored from the geoid error of the refined EGM (see Fig. 2). Comparing the radial differences in Tables 4 and 5, CASE D and the same item in Table 6, it can be seen that the radial differences are reduced from 1.33 m and 1.36 to 1.92 × 10−6 m, which implies that the ACC calibration results are independent of the a prior EGM when the simultaneous solution method is employed. We also test the impact of the selection of the maximum degree on the determination of the EGM with the simultaneous solution method. When the estimated maximum degree of the geopotential coefficients is 90 (8277 unknowns), the comparisons are listed in Table 7. From the comparisons in Table 7, it can be seen that the ACC calibration is reliable too in this case, while we must point out that the accuracy of the recovered EGM is deteriorated seriously now (See Figs. 2 and 3). Although the simulation is error free, the choice of the maximum degree has significant effect on the accuracy of the recovered EGM. When we process the in-situ mission data, it must be considered carefully, or adopt the regularization methods (Kusche and Kless 2002). This subject is another technical

one, so we will not extend the discussion on it in this article.

4 Discussions and Conclusions In this paper, the calibration of an onboard accelerometer using the dynamic method is studied by the numerical simulations. The tests show that the choice of the reference EGM has significant impact on the calibration, which can be seen more directly and clearly through the orbit comparison. The maximum radial difference reaches 4.58 m in the tests. It is recommended that an update-to-date gravity model should be used when the earth model is fixed in the ACC calibration, otherwise the difference between the reference EGM and the real gravity field will be aliased into the subsequent estimation of EGM. However, when the geopotential coefficients and the calibration parameters are estimated simultaneously with the dynamic method, the effects can be reduced substantially. We conclude that the more stable and reliable way to calibrate the accelerometer is to refine the EGM simultaneously. Moreover, the calibration with the dynamic method needs not to interpolate the velocity from the

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Fig. 3 Distribution of geoid error per degree of the recovered gravity model by simultaneous solution method (maximum degree is 90) ˇ satellite orbit, while the interpolation error is one of the Gerlach, C., N. Sneeuw, P. Visser, D. Svehla (2003). CHAMP gravity field recovery using the energy balance approach Aderror sources in the application of the energy method. vances in Geosciences, 1.73–80. The trouble for the dynamic method is that many test Jekeli, C. (1999). The determination of gravitational potential runs are needed to determine the basic configurations differences from satellite-to-satellite tracking. Celestial Mefor a better result, including the determination on the chanics and Dynamical Astronomy, 75:85–101. maximum degree of the EGM to be estimated, the arc Kusche J., R. Kless (2002). Regularization of gravity field estimation from satellite gravity gradients. Journal of Geodesy, length, the data spanning and the sampling interval. Acknowledgments The study is supported partly by the National Natural Science Foundation of China (No. 40637034 and 40704003) and the Open Fund of Key Laboratory of Geospace Environment & Geodesy Ministry of Education, China, under contract No. 06-09. The authors are grateful for the financial support from the IUGG2007 General Assembly. We are also very grateful to the two anonymous reviewers for their extremely valuable comments and suggestions for the improvement of the original manuscript.

References Bernard, A. (1992). A three-axis ultrasensitive accelerometer for space. ONERA TP no. 1992–208, pp11. Bettadpur, S. (2003). UTCSR Level-2 processing standards document for Level-2 product release 0001, CSR-GR-03-03, Center for Space Research, the University of Texas at Austin. Case K., G. Kruizinga, and S. Wu, (2004). GRACE Level 1B data product user handbook. JPL Publication D-22027.

76:359–368 Lemoine, F. G., S. C. Kenyon, J.K. Factor, R.G. Trimmer, N. K. Pavlis, D. S. Chinn, C. M. Cox, S. M. Klosko, S. B. Luthcke, M. H. Torrence, Y. M. Wang, R. G. Williamson, E. C. Pavlis, R. H. Rapp and T. R. Olson (1998). The Development of the Joint NASA GSFC and the NIMA Geopotential Model EGM96, NASA TP-1998206861, NASA Goddard Space Flight Center. Oliver, M., G. Eberhard (2001). Satellite orbits – models methods, applications, Springer Verlag, Berlin. Rapp, R. H., Wang, Y. M., and Pavlis, N. (1991). The Ohio State 1991 geopotential and sea surface topography harmonic coefficient models. Rep.410, Dept. of Geod. Sci. and Surv., Ohio State University, Columbus, Ohio. Reigber, C. (1989). Gravity field recovery from satellite tracking data. Theory of Satellite Geodesy and Gravity Field Determination. Edited by R. Rummel, Lecture Notes in Earth Sciences, Vol. 25, pp 197–234. Reigber, C., G. Balmino, P. Schwintzer, R. Biancale, A. Bode, J.-M. Lemoine, R. K¨onig, S. Loyer, H. Neumayer, J.-C. Marty, F. Barthelmes, F. Perosanz, S. Y. Zhu (2002a). A High-Quality Global Gravity Field Model from CHAMP GPS Tracking Data and Accelerometry (EIGEN-IS). Geophysical Research Letter, 29:1692–1695.

Research on the Calibration Reigber, H., C. Ltihr, P. Schwintzer (2002b). CHAMP mission status. Advances in Space Research, 30(2):129–134. Tapley, B. D., J. Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, P. Nagel, R. Pastor, T. Pekker, S. Poole, F. Wang (2005). GGM02 an improved earth gravity field model from GRACE. Journal of Geodesy, 79:467–478. Tapley, B. D.,S. Bettadpur,M. Watkins,C. Reigber (2004). The gravity recovery and climate experiment: mission overview and early results. Geophysical Research Letter, 31:9607–9610.

167 Touboul, P. (2001). Space accelerometers: present status. Gyros, Clocks, Interferometers: Testing Relativistic Gravity in Space. Edited by C. L¨ammerzahl, C.W.F. Everitt, F.W. Hehl, Lecture Notes in Physics, Vol. 562, pp 273 Visser, P. N. A. M., J. van. den. Ijssel (2003). Verification of CHAMP accelerometer observations. Advances in Space Research, 31:1905–1910.

Current Status of Gravity Measurements in the Republic of Croatia with the Fundamental Gravity Network Finalization Project I. Grgi´c, M. Luˇci´c, M. Liker, T. Baˇsi´c, B. Bariˇsi´c and M. Repani´c

Abstract The Fundamental Gravity Network (FGN) is foundation for all national gravity measurements. This network consists of 42 points: 6 absolute gravity points (0. Order Gravity Network) and 36 relative gravity points (I. Order Gravity Network). Further densification of FGN will be carried out by the lower order networks (II. Order Gravity Network). Fundamental gravity points should be homogeneously spaced over the whole state. In some large countries the distance between gravity points can be more than few hundred kilometres. Levelling connection of the absolute and first order gravity points at the national levelling network benchmarks was not performed. Also, the position of gravity points in respect to existing geodetic network stayed unknown. Quality positional and height definition of gravity points is necessary for calculating different corrections, which are needed for processing gravimetric measurements. By implementation of the FGN Finalization Project, along with already I. Grgi´c Croatian Geodetic Institute, Savska c. 41/XVI, Zagreb 10144, p.p.19, Croatia, e-mail: [email protected] M. Luˇci´c Croatian Geodetic Institute, Savska c. 41/XVI, Zagreb 10144, p.p.19, Croatia, e-mail: [email protected] M. Liker Croatian Geodetic Institute, Savska c. 41/XVI, Zagreb 10144, p.p.19, Croatia, e-mail: [email protected] T. Baˇsi´c Croatian Geodetic Institute, Savska c. 41/XVI, Zagreb 10144, p.p.19, Croatia, e-mail: [email protected] B. Bariˇsi´c Croatian Geodetic Institute, Savska c. 41/XVI, Zagreb 10144, p.p.19, Croatia, e-mail: [email protected] M. Repani´c Croatian Geodetic Institute, Savska c. 41/XVI, Zagreb 10144, p.p.19, Croatia, e-mail: [email protected]

finalized projects, the modern gravity foundation for the Republic of Croatia will be established.

1 Introduction The primary task of geodesy is the measurement and mapping of the Earth’s surface. The classical definition has to be extended with physical aspect because Earth’s gravity field influences on mapping quality of the Earth’s surface (Torge 2003). Gravity values are required in the modern height systems for the geoid determination and different tasks in geophysics. Earth’s gravity field data enables more precisely defined ellipsoid and geoid relations, which provides more precisely determined height benchmarks. For scientific researches of Earth’s physical shape (geoid determination) gravity networks are founded and provide frame for realization of global, regional or local gravity measurements. For gravity determination there are absolute and relative methods. For global measurements International Gravity Standardization Net 1971 (IGSN71) standard is used. Today that standard can be realized using absolute gravity-meters independently of global reference system. Global absolute gravity network is realized for observations of gravity’s temporal variations. Gravity point’s position and height is determined with meter precision and millimetre to centimetre precision, respectively (Torge 1989).

2 Gravity Network in the Republic of Croatia Like the other geodetic networks in the Republic of Croatia, gravity network also followed the trend

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of connecting at the European level. The result of this trend was UNIGRACE project. Six absolute gravity points have been established (0. Order Gravity Network) during two phases of UNIGRACE project. Those points provided reference for all other gravity works in Croatia. After the finalization of UNIGRACE project the main preconditions for gravity network of the first order, which leans on reestablished six absolute stations, were accomplished. In the year 2000 revision of inherited gravity network of the first order was started. The revision showed that the 25 old points could be used for the new, I. order gravity network. With 11 new I. order gravity points the frame for the Fundamental Gravity Network has been founded, that comprises 36 points. The gravity measurements have been performed during year 2003 (Baˇsi´c et al. 2004). After measurement processing and adjustment, the gravity value at each point has been obtained. Standard deviations for each point of absolute and relative gravity network are shown in Tables 1 and 2: Gravity points’ position is defined on several different ways:

r r r

Coordinates of 6 points are determined graphically using Croatian Base Map (1:5000), Coordinates of 5 points are determined graphically using Topographic Map (1:50,000), Coordinates of 11 points are determined using manual GPS device.

Heights are also defined using different methods, so one can only evaluate positional and height precision of I. order gravity points as inadequate. Existing first order gravity points are defined during project “Fundamental Gravity Network of the Republic of Croatia” realised by Faculty of Geodesy for the purposes of State Geodetic Administration. After absolute gravity point revision it was established that absolute gravity point in Makarska has been destroyed. Because absolute point in Makarska was planned to be final point of calibrating base, the damage is even greater. During absolute gravity measurements in Makarska, eccentric point in Institute “Planine i more” had been observed, therefore that point was included in BGN r Coordinates of 14 points are taken from old descrip- test measurements. Levelling connection of fundamental and first order gravity points with benchmarks of tion lists, state height network was not performed, as well as positional determination of gravity points regarding to existing network. Table 1 Standard deviations for absolute gravity points Quality positional and height definition of gravity (CGI 2002) points is necessary for calculating different corrections, AP σg (10−8 ms−2 ) which are needed for adjusting gravimetric measureAGT01 – Osijek 2.8; 1.3∗ ments. A number first order gravity points have good AGT02 – Zagreb (Maksimir) 3,701 quality stabilization, so except gravity measurements, AGT03 – Zagreb (Puntijarka) 1.002 quality positional and height determination is required AGT04 – Pula 1.954 AGT05 – Makarska 2.657 to obtain future multiple uses in different professional AGT06 – Dubrovnik 2.4; 7.5∗∗ and scientific assignments. Table 2 Standard deviations for fundamental gravity points (Bosiljevac 2006) GP I. order

σ R (10−8 ms−2 )

GP I.order

σ R (10−8 ms−2 )

GP I.order

σ R (10−8 ms−2 )

GT101 GT102 GT103 GT104 GT105 GT106 GT107 GT108 GT109 GT110 GT111 GT112

6.58 4.75 5.67 4.91 4.07 4.30 3.60 4.77 4.95 4.64 4.47 4.03

GT113 GT114 GT115 GT116 GT117 GT118 GT119 GT120 GT121 GT122 GT123 GT124

3.77 4.42 4.57 5.85 3.80 5.53 4.41 5.34 3.87 4.56 4.40 5.77

GT125 GT126 GT127 GT128 GT129 GT130 GT131 GT132 GT133 GT134 GT135 GT136

5.07 6.01 5.86 5.85 6.78 6.07 5.87 6.08 5.78 5.53 5.18 5.79

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Fig. 1 Fundamental and first order gravity points

GT112

GT114 GT113

GT110

46 GT117 GT120 GT121

GT106

GT111 GT115

GT118

GT122

GT104

GT108

AGT03 AGT02

GT119

AGT01 GT102

GT107

GT101

GT109

GT116

GT105 GT103

GT124

45

GT123 AGT04

GT125 GT126

GT127 GT128

GT129

44 GT130 GT131 GT132

GT133 EKS-MM GT134

43 GT135

GT136 AGT06

42 13

14

15

16

17

18

19

2.1 Measurement Methods According to project properties and field conditions, double step, star and profile methods are used. Double step method was mainly used, star method up to certain extent, while profile method was used only if it was necessary. Step method – it is economical method and presents Fig. 3 Star method combination of single and star measurement. Drift is defined using multiple occupation of point, Fig. 2. non economical method because of frequent reoccupyStar method – this method is homogeneous and ap- ing the central point, Fig. 3. propriate for determining local basic networks, but also Profile method – double line measurement with daily returning on origin point eliminates unilateral errors and controls gravity drift slips, Fig. 4.

Fig. 2 Step method

Fig. 4 Profile method

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3 Stabilization of Gravity Points

stabilization of gravity point: concrete pillar, point on object etc. Due to reduction of expenses during establishment First order gravity network includes 36 points. Most of of BGN and for obtaining larger precision, gravity them (25) are taken from BGN of former Yugoslavia, points are going to be connected with state height netand 11 points are established afterwards. Old points work benchmarks of upper orders if it is possible, and are stabilized with concrete pillar (dimensions: position will depend on fact if field works will be per0.5 × 0.5 × 0.95-1 m), rock with engraved square formed simultaneously or separately. (dimensions: 0.5 × 0.5 m), and with letters “GT” carved on upper side of pillar. New stabilization is made of concrete cylinder (dimensions: 0.5 ×1 m), 4 Project Plan Fig. 5 (right). During establishment of BGN and EUVN DA Rasis for gravity network establishment or parts of netproject, it was noticed that some gravity points are works are office concept designs supplemented with slightly damaged while some heavily (CGI 2006). All field recognition. It is very important to create concept old accepted points did not have materialized centre design of network establishment that shows all connecof point, so for future multiple use they had to be tions to network higher order. Base for concept design extra stabilized. Extra stabilization of existing gravity was database created in Croatian Geodetic Institute for points will be performed before terrain measurements purposes of Department of basic geodetic works, drafts to obtain position and height of gravity points, Fig. 5 of permanent geodetic points, point positional descrip(left). Extra stabilization of gravity points is provided tions etc. After terrain recognition any possible deviawith benchmark (0.05 − 0.07 m length), depending on tions from notional plan have to be implemented in the final project plan.

4.1 Positional Determination of Gravity Points Quality positional and height defining of gravity points are necessary for calculating different corrections, which are needed for adjusting gravimetric measurements. Because BGN do not lean on existing network that has complete positional determination, it is necessary to define all gravity points that have not been determined earlier. Necessarily requirement for GPS measurements for determination of gravity points is that they are measured with at least two reference stations. New points are going to be observed using static method to obtain relative accuracy of +1 cm +2 ppm horizontal, and +2 cm +2 ppm vertical.

4.2 Positional Determination During Simultaneous Measurements Fig. 5 Stabilization of old gravity point with benchmark (botom left) and new gravity point (botom right)

Average distance between closest benchmark and gravity point is about 800 m; therefore levelling

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measurements will be obtained for at least 2 h by r Use 15 s registration interval during processing, point. Assuming that gravity points are stabilized r Use standard troposphere models, in advance it would be useful to perform GNSS r Adjust network using several IGS reference stations measurements to obtain better accuracy of posiand permanent GNSS stations in Dubrovnik and tional determination. Positional determination will be Osijek. obtained using static method during height determiAfter adjustment unique transformation model nation. This simultaneous method of positional and “T7D” will be used along whole state area to obtain height determination is going to be used anywhere geodetic data transformation from historic reference where it will be possible. To obtain required GNSS system to new official geodetic reference system measurements is necessary to: (HTRS96/HDKS). r use dual-frequency instruments, Final result will be ellipsoid coordinates and normal r use same antenna type, orthometric heights and plane Gauss- Kr¨uger coordir use GNSS permanent stations in Osijek and nates for all gravity points (Baˇsi´c et al. 2006). Dubrovnik or GPS second order points, Gravity point standard deviations are shown in the r use IGS points (permanent stations) with known co- Table 3. ordinates, r use 15 s interval registration, Table 3 Standard deviation values r use elevation mask above 10◦ , Position Height r obtain GPS measurements around 2 h per point. First order gravity points 25 mm 35 mm Coordinates will be presented in HTRS96 (or ETRS89) or ITRFyy reference system with epoch information. For every gravity points with accomplished GPS measurements is necessary to prepare:

Differences between twice determined coordinates should not be above half of standard deviation value.

r r

4.3 Gravity Point Height Determination

GPS data form, Original measurements and RINEX format data.

If some of gravity points are not adequate for GPS measurement, they will be defined using conventional geodetic methods from previously determined points. Using that field method it will be possible to measure horizontal and vertical angles and distances from determined to unknown points, or to obtain traverse from known points to gravity point.

Fundamental gravity points are going to be connected to the closest benchmark using geometric levelling method. To reduce expenses and gain better accuracy, points will be attached to the closest state height network benchmarks, and height data will be obtained using precise levelling method. It is planned, depending on terrain conditions, to use levelling line between two benchmarks and gravity point in the middle.

4.2.1 Data Processing 4.3.1 Data Processing During processing it should be conducted by following principles:

r r r r

Data will be processed and adjusted using least square adjustment method. Measurements will be obtained in GNSS observations should be processed using some two directions with same accuracy (one observer, same of commercial or scientific software, instrument and weather conditions). Use IGS final orbits and IGS files with pole motion According to the Geometric levelling regulations information, (Klak et al., 1993): Process measurements using actual reference ITRF r Largest discrepancy in precise double line frame, ◦ levelling is: Elevation mask above 10 ,

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√ d1 = ±4 s mm.

r

Largest discrepancies between final values and fixed height differences in precise levelling are:

5 Concept of Croatian Fundamental Gravity Network Finalization

In last few years Republic of Croatia has defined 0. order gravity network (absolute gravity network) and I. √ order gravity network used for basic measurements. d2 = ±(2.5 + 4 s) mm. Absolute gravity points were determined by absolute gravity meters (FG-5), and I. order gravity points According to the Geometric levelling regulations were measured with modern Scintrex relative gravity (Klak et al., 1993) it should be considered: meters CG-3M and CG-5 (CGI 2002). During project realization extra attention will be r Levelling line in two directions on different weather given to measurement method. Fundamental gravity conditions (one direction in morning and other in network consists of 34 points, and in future will be afternoon); r Use invar staff, change plates, staff rods, parasol Table 4 Basic data for gravity point height determination etc.; distance: 374,45 m r Levelling staffs should be vertically calibrated ev- Total Average distance: 951 m ery two months; Min. distance: 0m r Largest observation distance is 40 m. Max. distance: 9062 m

Fig. 6 Example of gravity point connection to GPS network and closest benchmark

Current Status of Gravity Measurements

expanded with II. order gravity points and absolute gravity eccentric points with the intention of obtaining network datum and scale. Because of fact that height is very important for gravity measurement interpretation, it is necessary to connect gravity points to closest benchmark into common Croatian height system, Table 4. Gravity points with poor positional determination should be also measured using GNSS technology. Before project realization, field recognition will be obtained to screen points on field, and if necessary to renovate damaged points.

175

Gravity Network, Croatia will gain good foundation of points with defined gravity, position and height that will provide their application in different geodetic works. After finalization of Fundamental Gravity Network and result interpretation, need for realization of fundamental gravity network will be satisfied, and will serve as foundation for densification with second order gravity network, and approach to European standards. Besides field works needed for Fundamental Gravity Network finalization, Croatian second order gravity network project is planned with systematic and professional approach. After project realization, and with earlier gravity network projects, foundation for modern Croatian 6 Conclusion gravity network will be obtained. However, for final project finalization in next few years it will be Defining, obtaining, processing and adjusting GPS necessary to perform some important works: and levelling measurements for Croatian Fundamental

Fig. 7 Example of gravity point connection to GPS network and closest benchmark

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Re-measure absolute gravity points, establish new Baˇsi´c, T., Markovinovi´c, D., Rezo, M. (2006): Mikrogravimetrijske mreˇze i projekt gravimetrijske mreˇze II. reda Republike point in Plitvice as gravity base point and restore Hrvatske, SGA: “Izvjeˇsc´ a o znanstveno-struˇcnim projektima site in Makarska, or establish new absolute gravity iz 2004.-2005. godine”, Zagreb 2006 point in Split; Bosiljevac, M. (2006): SGA personal correspondence Continue with second order gravity points measure- Croatian Geodetic Institute (2002): Izvjeˇsc´ e o podacima apments, by phases, until network completion. solutne gravimetrijske mreˇze Republike Hrvatske, Zagreb,

References Baˇsi´c, T., Markovinovi´c, D., Rezo, M. (2004): Tehniˇcko izvjeˇsc´ e o obavljenim radovima na projektu Osnovna gravimetrijska mreˇza Republike Hrvatske. Project for SGA, Faculty of Geodesy, Zagreb.

December 2002. Croatian Geodetic Institute (2006): Izvjeˇsc´ e o obavljenim radovima-Projekt proguˇsc´ enja EUVN mreˇze (EUVN DA), Zagreb, July 2006. Feil, L. (1990): Teorija pogreˇsaka i raˇcun izjednaˇcenja - drugi dio, Faculty of Geodesy, Zagreb. Klak, S., Feil, L. Roˇzi´c, N. (1993): Pravilnik o radovima geometrijskog nivelmana. Proposal, Zagreb. Torge, W. (1989): Gravimetry. Walter de Gruyter, Berlin-New York. Torge, W. (2003): Geod¨asie. Walter de Gruyter, Berlin-New York.

The Development of the European Gravimetric Geoid Model EGG07 H. Denker, J.-P. Barriot, R. Barzaghi, D. Fairhead, R. Forsberg, J. Ihde, A. Kenyeres, U. Marti, M. Sarrailh and I.N. Tziavos

Abstract The European Gravity and Geoid Project (EGGP) is a project within IAG Commission 2, reporting to Sub-commission 2.4. The main goal of the project is to compute an improved European geoid and quasigeoid model based on new and improved data sets which have become available since the last computation in 1997 (EGG97). The improvements include better global geopotential models from the CHAMP H. Denker Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany; e-mail: [email protected] J.-P. Barriot Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany R. Barzaghi Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany D. Fairhead Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany R. Forsberg Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany J. Ihde Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany A. Kenyeres Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany U. Marti Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany M. Sarrailh Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany I.N. Tziavos Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany

and GRACE missions, better digital elevation models (DEMs) in some regions (e.g., new national DEMs, SRTM3, GTOPO30), updated gravity data sets for selected areas, updated ship and altimetric gravity data, improved procedures for the merging of ship and altimetric data, the use of GPS/levelling data, as well as refined computation techniques. This contribution describes the progress made during the 4-year term from 2003 to 2007, including the development of a new geoid and quasigeoid model EGG07 for entire Europe. First, the status of the gravity and terrain data sets as well as the development of the EGG07 model by the spectral combination approach is described. Then, the EGG07 and other models are evaluated by independent GPS and levelling data, showing that the use of GRACE geopotential models as well as upgraded gravity and terrain data leads to significant improvements compared to EGG97 (in total by 25 – 65%). The results indicate an accuracy potential of the EGG07 model in the order of 0.03 – 0.05 m at continental scales and 0.01 – 0.02 m over shorter distances up to a few 100 km, provided that high quality and resolution input data are available. Keywords Geoid · Quasigeoid · Gravity field modelling · GPS/levelling · EGGP · CHAMP · GRACE

1 Introduction The last published high-resolution European geoid and quasigeoid model EGG97 dates back to 1997 and was computed at the Institut f¨ur Erdmessung (IfE), Leibniz Universit¨at Hannover, as an IAG enterprise (cf. Denker

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and Torge, 1998). EGG97 was based on the global geopotential model EGG96 (Lemoine et al., 1998) and high-resolution gravity and terrain data available at that time. The evaluation of EGG97 by GPS and levelling data revealed the existence of long wavelength errors at the level of 0.1 to 1 ppm, while the agreement over distances up to about 100 km was at the level of 0.01 – 0.02 m in many areas with a good quality and coverage of the input data (Denker and Torge, 1998; Denker, 1998). However, since the development of EGG97, significant new or improved data sets have become available, including strongly improved global geopotential models from the CHAMP and GRACE missions, new national and global terrain data sets, new or updated gravity data sets, improved altimetric results, as well as new GPS and levelling results. Last but not least, also the gravity field modelling techniques improved. Considering all these advancements, a complete re-computation of the European geoid and quasigeoid was considered appropriate and promised significantly improved accuracies, especially at long wavelengths. Therefore, after the IUGG General Assembly in Sapporo in 2003, it was decided to support this task in the form of an IAG Commission 2 Project, named “CP2.1 – European Gravity and Geoid Project (EGGP)”. The EGGP is reporting to Sub-commission 2.4 and has strong connections to the International Gravity Field Service (IGFS), with its centres Bureau Gravim´etrique International (BGI), International Geoid Service (IGeS), National Geospatial-Intelligence Agency (NGA), and GeoForschungsZentrum Potsdam (GFZ), as well as to several other IAG bodies, e.g., EUREF. The project is organised by a steering committee (H. Denker (Chair), J.-P. Barriot, R. Barzaghi, R. Forsberg, J. Ihde, A. Kenyeres, U. Marti, I.N. Tziavos) and has about 50 national delegates (project members) from most of the countries in Europe. Due to the confidentiality of many data sets, only one data and computation centre exists at the Institut f¨ur Erdmessung (IfE), complemented by a second confidential gravity data centre at Bureau Gravim´etrique International (BGI). Further details on the project can be found in Denker et al. (2005), and the terms of reference are given in EGGP (2003). Within the framework of the EGGP, interim results and status reports were provided roughly on an annual basis in Porto, 2004 (Denker et al., 2005), Austin, 2005 (Denker, 2005c) and Istanbul, 2006 (Denker

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et al., 2007). The present contribution summarizes the progress made within the EGGP during the 4-year term from 2003 to 2007, including the development of a completely updated geoid and quasigeoid model EGG07. Moreover, the new model is evaluated by GPS and levelling data and the progress with regard to the previous EGG97 model is outlined.

2 Gravity and Terrain Data Since the start of the project, significant improvements of the gravity data base were made, including new or revised data sets for nearly all European countries. New gravity data sets were supplied for Austria, Belgium, Croatia, Cyprus, Denmark, Estonia, Finland, France, Germany, Greece, Italy, Latvia, Luxemburg, Netherlands, Norway, Portugal, Serbia, Slovenia, Spain, Sweden, Switzerland, and Turkey. Moreover, a few smaller updates are still pending and may be included in the near future. Significant progress was also made in the collection and reprocessing of marine gravity data. All marine gravity data collected until 2003 were edited and crossover adjusted (cf. Denker and Roland, 2005), which lead to significant data improvements. The comparisons with independent altimetric gravity anomalies from, e.g., the KMS02 model (Andersen et al., 2005), showed a RMS difference of 18.0 mgal for the original data set, 10.2 mgal for the edited data set, and 7.8 mgal for the edited and crossover adjusted data set, respectively, which proves the effectiveness of the entire processing scheme (cf. Denker and Roland, 2005). In addition, also after 2003, significant new marine gravity data sets became available, originating mainly from the authorities of the Scandinavian countries (coverage: Baltic Sea, North Sea, North Atlantic), France (coverage: western parts of the Mediterranean Sea, Atlantic) as well as the National GeospatialIntelligence Agency (NGA), U.S.A. (coverage: central and eastern parts of the Mediterranean Sea). Moreover, also some airborne data sets were provided by the Scandinavian authorities, covering mainly the Baltic Sea and parts of the North Atlantic and Greenland coastal waters. The afore mentioned new gravity data sets were not crossover adjusted together with the other marine gravity data sources, mainly because all

The Development of the EGGP Model

of them are of high quality and also due to a lack of time. In addition to this, the public domain data from the Arctic Gravity Project ArcGP (Forsberg and Kenyon, 2004) were integrated in the project data base. Finally, data from the EGG97 data base were utilized for some areas (e.g., Eastern Europe and Africa). All EGGP gravity data sets were carefully checked regarding the underlying reference systems, and if necessary, transformations were done to the target systems, being ETRS (European Terrestrial Reference System), EVRS (European Vertical Reference System) and absolute gravity. In the merging process of all data sources, emphasis was put on a careful check with respect to systematic and gross errors, which was one of the most time-consuming steps. The progress in the collection of gravity data is documented for selected examples in Fig. 1. The left part of the figure shows the old status in 1997 (EGG97; Denker and Torge, 1998) and the right part shows the new status as of July 2007 (EGG07) for entire Europe (top), Scandinavia (middle) and the Mediterranean Sea (bottom, Morelli ship data excluded). In this context it should be noted that with-in the EGGP the Morelli ship gravity data for the Mediterranean Sea (e.g., Morelli et al., 1975) was completely excluded as comparisons with newer data sources revealed significant systematic discrepancies. Comparable progress was made in the collection of high-resolution digital elevation models (DEMs). For the EGG97 computation, digital elevation models (DEMs) with a resolution of about 200 m were only available for Central and Western Europe, while coarser grids with a resolution of 0.5 km to 10 km had to be used for the remaining parts of Europe. As of 1997, only Germany released a very high resolution DEM with a grid size of 1 × 1 (approx. 30 m), but meanwhile also Switzerland and Austria provided 1 × 1 DEMs for the EGGP. At present, high-resolution national DEMs do not exist or are confidential for large parts of Eastern Europe. Hence, in all areas not covered by high-resolution national DEMs, fill-ins from public domain data sets had to be utilized. However, compared to EGG97, significantly improved fill-ins are available now, e.g., from the Shuttle Radar Topography Mission (SRTM) with a resolution of 3 ×3 (SRTM3; JPL, 2007) or the global public domain model GTOPO30 with a resolution of 30 × 30 (USGS, 2007). As the SRTM3 model

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covers only the latitudes between 60◦ N and 54◦ S, the GTOPO30 model had to be used for the regions in the far North. All available DEMs were merged into a new European DEM with a common grid size of 3 × 3 , covering the area 25◦ N − 85◦ N and 50◦ W − 70◦ E. Furthermore, for the area of Germany, Austria and Switzerland, a corresponding 1 × 1 DEM was created. The 3 × 3 and 1 × 1 DEMs comprise about 6.6 and 1.7 billion elevations, respectively. In the merging process, the highest priority was given to the national DEMs, followed by the SRTM3 and GTOPO30 data. For testing purposes, a second 3 × 3 European DEM was created using only the public domain data sets SRTM3 and GTOPO30. Within the merging process, the SRTM3 and GTOPO30 DEMs were also evaluated by comparisons with the high-resolution national DEMs. In Germany, the differences between the national and SRTM3 DEMs showed a standard deviation of 7.9 m and maximum values up to about 300 m. The largest differences were located in opencast mining areas and resulted from the different epochs of the data. Histograms of the differences showed a clear deviation from the normal distribution with a long tail towards too high SRTM3 elevations, which is expected due to the fact that SRTM is a “first return system”, providing elevations of whatever the radar has bounced off from, and in many instances this is above the actual ground level (cf. Denker, 2005a). The evaluation of the GTOPO30 model by national and SRTM3 DEMs demonstrated that in large parts of Europe the longitudes of GTOPO30 should be increased by 30 (one block). In Central Europe, the longitude shift reduced the standard deviation of the differences to the national and SRTM3 models by roughly 75% to about 10 m. Altogether, the national DEMs augmented by the SRTM3 and GTOPO30 data provide a significantly improved European DEM, as compared to EGG97.

3 Development of the EGG07 Model Several updated geoid and quasigeoid computations were carried out up to now within the EGGP (e.g., Denker et al., 2005; Denker, 2005c). At first, the gravity and terrain data sets were taken from the EGG97 computation, but the global geopotential

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Fig. 1 Locations of terrestrial gravity data for entire Europe (top), Scandinavia (middle) and the Mediterranean Sea excluding the Morelli data (bottom). The left part shows the status in 1997 (EGG97) and the right part shows the status of July 2007 (EGG07). Grey shaded areas indicate ArcGP and KMS2002 data

model EGM96 was replaced by CHAMP and GRACE models. Secondly, all gravity and terrain data sets were completely updated and combined with the CHAMP and GRACE geopotential models. In contrast to the

previous interim results, the new EGG07 model is based on completely updated gravity and terrain data sets for entire Europe, as described in the previous section.

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The computation strategy is outlined in Denker et al. (2005). The primary gravity field quantity to be computed is the height anomaly or the quasigeoid undulation, with the advantage that only gravity field observations at the Earth’s surface and in its exterior enter into the calculations, avoiding assumptions about the Earth’s interior gravity field. A geoid model can then be derived by introducing a density hypothesis, which should be identical to the one used for the computation of corresponding orthometric heights (e.g., Helmert heights). The remove-restore technique is used, considering high-resolution terrestrial gravity and terrain data in combination with a state-of-the-art global geopotential model. Terrain reductions according to the residual terrain model (RTM) technique (Forsberg and Tscherning, 1981) are applied to smooth the data and to avoid aliasing effects. The gravity field modelling at IfE is based on the spectral combination technique (e.g., Wenzel, 1982) with integral formulas, which can be efficiently evaluated by ID FFT. In this method, the combination of terrestrial gravity data and a global geopotential model is done by means of spectral weights, which depend on the accuracy of the input data sets. Due to the high accuracy of the global models at long wavelengths, the terrestrial data mainly contribute the shorter wavelength components. So far, time has not allowed testing other modelling techniques, e.g., standard least-squares collocation, wavelets, or the fast collocation approach (e.g., Sans`o and Tscherning, 2003), but this is envisaged for the future. The final gravity data set used for the computation of the EGG07 model consisted of 5,354,653 observations from 709 sources. In addition, 195,840 gravity values from the ArcGP project (Forsberg and Kenyon, 2004) as well as 951,251 altimetric anomalies from the KMS2002 data set (Andersen et al., 2005) were utilized. Thus, in comparison with EGG97, the amount of gravity data approximately doubled (see also Fig. 1). All gravity observations were RTM reduced, resulting in a significant data smoothing. The required reference topography was computed by a moving average   filter over 30 × 45 blocks. All terrain effect computations were done by exact prism formulas in combination with detailed and coarse grids for the outer zones; mainly to speed up the computations.

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With regard to the global geopotential model used in combination with gravity and terrain data, previous studies have shown that the GRACE based models lead to substantial improvements as compared to EGM96 (e.g., Denker et al., 2005b). Different satellite-only and combined GRACE models were employed (e.g., from GeoForschungszentrum Potsdam – GFZ, Center of Space Research at the University of Texas – CSR, and Jet Propulsion Laboratory – JPL), and it turned out that it is advantageous to use a high-degree model with a maximum degree lmax = 360, as this leads to smaller residual quantities and reduced effects of approximation errors in the mathematical modelling. Hence, the pure GRACE satellite-only geopotential models were blended with the EGM96 coefficients from degree 90 onwards in corresponding test computations. The evaluation of all GRACE based quasigeoid models showed only small differences, where the newer models associated with longer observation periods gave slightly better results (cf. Denker, 2005b,c). For the computation of EGG07, the latest highdegree geopotential model EIGEN-GL04C (lmax = 360) from GFZ Potsdam (F¨orste and Flechtner, 2007; see also F¨orste et al., 2005) was used, being a combination of CHAMP, GRACE and terrestrial data. It should be noted that the EGG07 model can also serve as a typical example for all other GRACE based solutions. The spectral weights used in connection with the EIGENGL04C model are shown in Fig. 2, together with corresponding values for a recent CHAMP model and the EGM96 model. A correlated noise model with an error

Fig. 2 Spectral weights in connection with the geopotential model EGM96, a recent CHAMP model, and the GRACE based model EIGEN-GL04C

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standard deviation of 1 mgal was used for the terrestrial gravity data in the derivation of the spectral weights. The computation area for the EGG07 model is 25◦ N − 85◦ N and 50◦ W − 70◦ E. The grid spacing is 1 × 1 , yielding 3,600 × 7,200 = 28,080,000 grid points. The GRS80 constants, zero degree undulation terms, and the zero-tide system were used throughout all computations.

4 Evaluation of the EGG07 Model The EGG07 quasigeoid model and all other test solutions were evaluated by independent national and European GPS and levelling data sets (all levelled heights were normal heights). In order to demonstrate the progress resulting from improved gravity and terrain data on the one hand as well as the GRACE based global geopotential models on the other hand, also the previous European quasigeoid model EGG97 (based on the geopotential model EGM96; denoted as EGG97/EGM96 below; Denker and Torge, 1998) and the solution EGG97/GRACE (EGG97 gravity and terrain data, EIGEN-GL04C geopotential model based on GRACE) are addressed. Table 1 shows the statistics of the differences between the above mentioned quasigeoid models and GPS and levelling data from selected national and continental campaigns. In all cases, a constant bias was subtracted from the differences in order to account for different height system levels and very long wavelength errors in all data sets involved (GPS, levelling, quasigeoid). Results are provided for Germany (GPS/levelling data from Bundesamt f¨ur Kartographie und Geod¨asie, BKG, Frankfurt; e.g., Liebsch et al., 2006), Netherlands, Austria, Switzerland, a French traverse from Marseille to Dunkerque (1,100 km long) with new levelling data (NIREF), Russia (Demianov and Majorov, 2004), and the EUVN DA initiative (e.g., Kenyeres et al., 2007). The EUVN DA project aimed at a densification of the previous EUVN campaign (Ihde et al., 2000) by collecting high quality GPS and levelling data from participating European countries. At present, about 1,300 points are available with interstation distances ranging from about 50 – 100 km. Within the EUVN DA project, the normal heights were derived in part directly from geopotential numbers of UELN

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(United European Levelling Network) stations and partially by simple transformations with up to three parameters. For the EUVN DA and the Russian GPS and levelling data set, the differences with respect to EGG97 and EGG07 are also depicted in Fig. 3. The results in Fig. 3 and Table 1 clearly show that the new EGG07 model performs significantly better than EGG97. The improvements result from the updated gravity and terrain data as well as from the utilization of a GRACE based global geopotential model. Table 1 shows that solely through the introduction of a GRACE based global geopotential model, the RMS differences reduce by 27% (Switzerland) to 64% (Germany), when going from the EGG97/-EGN96 to the EGG97/GRACE solution. However, also the update and re-processing of the gravity and terrain data leads to substantial improvements in the GPS and levelling comparisons in most cases. The additional improvements from the upgraded terrestrial data range from 0% (Germany) to 23% (Netherlands); the improvements are in particular high in those areas where the data basis was significantly extended, e.g., in Belgium, Netherlands, and Austria. The overall improvement of EGG07 over EGG97 ranges from about 25 – 65% (see Table 1). Also long wavelength discrepancies are significantly reduced from 0.1 to 1.0 ppm for the EGG97 model to typically below 0.1 ppm for all GRACE based solutions (cf. also Denker et al., 2005; Denker, 2005b,c). Of special interest are the results from the comparisons with the EUVN DA GPS and levelling data set. Figure 3 (top) shows that the EGG07 model performs quite well over most parts of Europe, the two exceptions being Great Britain and Italy. Regarding Great Britain, the levelling heights are suspected to contain systematic errors (e.g., Hipkin et al., 2004); this is also confirmed by removing a north-south and eastwest trend in the comparisons, which reduces the RMS difference from about 0.15 m (bias case) to 0.05 m (bias and tilt case). On the other hand, the situation for Italy is less transparent, as the differences exhibit wavy structures and not a clear trend. Furthermore, the older EGG97 model appears to fit better to the Italian GPS and levelling data than EGG07, but nevertheless it is not believed that EGG97 is more accurate in Italy than EGG07. In fact, the differences between EGG07 and EGG97 in Italy are mainly a result of replacing the EGM96 geopotential model by a GRACE based

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Table 1 Comparisons of GPS and levelling data with EGG97 (named EGG97/EGM96), EGG97/GRACE (EGG97 gravity and terrain data, GRACE geopotential model), and EGG07 (named EGG07/GRACE). A constant bias is subtracted. Units are m GPS/Levelling

Quasigeoid

#

RMS [m]

Min [m]

Max [m]

Improvement versus EGG97

Germany

EGG97/EGM96 EGG97/GRACE EGG07/GRACE EGG97/EGM96 EGG97/GRACE EGG07/GRACE EGG97/EGM96 EGG97/GRACE EGG07/GRACE EGG97/EGM96 EGG97/GRACE EGG07/GRACE EGG97/EGM96 EGG97/GRACE EGG07/GRACE EGG97/EGM96 EGG97/GRACE EGG07/GRACE EGG97/EGM96 EGG97/GRACE EGG07/GRACE EGG97/EGM96 EGG97/GRACE EGG07/GRACE

907 907 907 84 84 84 106 106 106 188 188 188 16 16 16 48 48 48 1,285 1,285 1,285 908 908 908

0.099 0.036 0.036 0.035 0.023 0.015 0.108 0.072 0.054 0.081 0.059 0.061 0.086 0.034 0.039 0.253 0.127 0.089 0.257 0.217 0.207 0.158 0.097 0.079

–0.192 –0.092 –0.152 –0.062 –0.046 –0.044 –0.181 –0.130 –0.145 –0.129 –0.200 –0.204 –0.175 –0.063 –0.067 –0.760 –0.245 –0.139 –0.928 –0.737 –0.693 –0.669 –0.392 –0.518

+0.338 +0.120 +0.075 +0.120 +0.058 +0.047 +0.247 +0.188 +0.096 +0.258 +0.294 +0.152 +0.128 +0.070 +0.086 +0.693 +0.295 +0.192 +0.716 +0.0602 +0.587 +0.641 +0.521 +0.549

– 64% 64% – 34% 57% – 33% 50% – 27% 25% – 60% 55% – 50% 65% – 16% 19% – 39% 50%

Netherlands

Austria

Switzerland

France (Traverse with new levelling) Russia

EUVN DA (all)

EUVN DA (excl. Great Britain and Italy)

model, which leads to long wavelength changes over Italy and the Central and Western Mediterranean Sea. At present, the situation is further analysed with the help of Italian colleagues. For the EUVN DA data set, the RMS difference reduces from 0.16 m (EGG97) to 0.08 m (EGG07) when excluding Great Britain and Italy (see Table 1). The differences show small long wavelength structures (see Fig. 3 top right) and include error components from all data sets involved, i.e. GPS, levelling and quasigeoid. On the whole, the EUVN DA results are considered as quite satisfactory, as the data set is covering a very large area from the Iberian Peninsula to Northern Scandinavia, the Baltic States, Poland and Bulgaria. Significant improvements were also obtained over Russia (see Table 1 and Fig. 3). The RMS difference reduced from 0.25 m for EGG97 to 0.09 m for EGG07 (65% improvement), which is remarkable as this is just the result of reprocessing the supplied mean Bouguer anomalies with new terrain models from the SRTM mission. For the French traverse from Marseille to Dunkerque (1,100 km long), the RMS difference

is 0.039 m for EGG07, which is a 55% reduction versus EGG97. It is also interesting to note that the RMS difference with the IGN69 (old) levelling heights is only 0.075 m for EGG07, which clearly proves that the new levelling is better than the old one. Regarding the other national GPS and levelling data sets, the RMS differences for EGG07 range from about 0.01 to 0.06 m, the higher values being associated with Austria and Switzerland (high mountain areas). In smaller regions with extensions up to a few 100 km, the agreement is even better and in the order of 0.01 – 0.02 m. In this context, it should be noted again that the differences include error contributions from all data sets involved, i.e. GPS, levelling, and gravimetric quasigeoid. Furthermore, the RMS differences between the gravimetric quasigeoid and GPS and levelling data are also in a reasonable agreement with the internal error estimates of about 0.02 – 0.03 m for the GRACE based solutions. The accuracies achieved for large parts of Europe are about the optimum one can expect at present with up-to-date gravity, terrain, and GRACE data; further improvements are mainly anticipated from the GOCE mission, as the terrestrial

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Fig. 3 Comparison of GPS and levelling data from the EUVN DA initiative (top) and Russia (bottom) with the quasigeoid solutions EGG97 (left part) and EGG07 (right part). A constant bias is subtracted

entire Europe. The evaluation of this model by independent GPS and levelling data showed that the use of GRACE geopotential models as well as upgraded gravity and terrain data leads to significant improve5 Conclusions ments compared to EGG97 (in total by 25 – 65%). In addition, the long wavelength errors, existing in Significant progress was made within the framework EGG97, were substantially reduced to typically below of the European Gravity and Geoid Project EGGP 0.1 ppm. The results indicate an accuracy potential regarding the collection and homogenization of of the gravimetric quasigeoid models in the order of high-resolution gravity and terrain data. A completely 0.03 – 0.05 m at continental scales and 0.01 – 0.02 m updated quasigeoid model EGG07 was derived for over shorter distances up to a few 100 km, provided data sets can hardly be improved in many regions of Europe.

The Development of the EGGP Model

that high quality and resolution input data are available in the area of interest. The data base may still be improved in coastal zones and sea areas as well as over large parts of Eastern Europe. Furthermore, modelling problems may exist in the high mountains which have to be studied further. Nevertheless, the present accuracy potential allows a number of interesting applications, including accurate estimates of the potential W0 (reference geopotential of the vertical datum) and vertical datum unifications. Finally, it is a pleasure to thank all persons and agencies that supported the EGGP with data and expert know-how.

References Andersen, O.B., P. Knudsen, R. Trimmer (2005). Improved high resolution altimetric gravity field mapping (KMS2002 Global Marine Gravity Field). IAG Symposia 128:326–331, Springer Verlag. Denker, H. (1998). Evaluation and improvement of the EGG97 quasigeoid model for Europe by GPS and leveling data. Reports of the Finnish Geodetic Institute, 98:4, 53–61, Masala, 1998. Denker, H. (2005a). Evaluation of SRTM3 and GTOPO30 terrain data in Germany. IAG Symposia 129:218–223, Springer Verlag. Denker, H. (2005b). Improved European geoid models based on CHAMP and GRACE results. IAG Internat. Symp. Gravity, Geoid and Space Missions - GGSM2004, Porto, Aug. 30–Sept. 3, 2004, CD-ROM Proceed. Denker, H. (2005c). Improved modeling of the geoid in Europe based on GRACE data. GRACE Science Team Meeting The Univ. of Texas at Austin, Texas, USA., Oct. 13–14, 2005, CD-ROM Proceed. Denker, H., W. Torge (1998). The European gravimetric quasigeoid EGG97 – An IAG supported continental enterprise. IAG Symposia 119:249–254, Springer Verlag. Denker, H., M. Roland (2005). Compilation and evaluation of a consistent marine gravity data set surrounding Europe. IAG Symposia, Vol. 128:248–253, Springer Verlag. Denker, H., J.-P. Barriot, R. Barzaghi, R. Forsberg, J. Ihde, A. Kenyeres, U. Marti, I.N. Tziavos (2005). Status of the European Gravity and Geoid Project EGGP. IAG Symposia 129:125–130, Springer Verlag. Denker, H., J.-P. Barriot, R. Barzaghi, R. Forsberg, J. Ihde, A. Kenyeres, U. Marti, I.N. Tziavos (2007). The European Gravity and Geoid Project EGGP. Gravity Field of the Earth, 1st Internat, Symp. of the Internat. Gravity Field Service, 28 Aug. – 1 Sept., 2006, Instanbul, IAG Symposia, Springer Verlag, in press. Demianov, G.V., A.N. Majorov (2004). On the definition of a common world normal height system. In: Physical Geodesy, Scientific and Technical Reports on Geodesy, Photogramme-

185 try and Cartography, Federal Office of Geodesy and Cartography (Roskartographia), ZNIIGAiK, Moscow, 168–182 (in Russian). EGGP (2003). Commission Project 2.1, European Gravity and Geoid, Terms of Reference and Objectives, http://www.ceegs.ohio-state.edu/ iag-commission2/cp2.1.htm. Forsberg, R., C.C. Tscherning (1981). The use of height data in gravity field approximation by collocation. J. Geophys. Res. 86: 7843–7854. Forsberg, R., S. Kenyon (2004). Gravity and geoid in the Arctic Region – The northern GOCE polar gap filled. Proceed. 2nd Internat. GOCE Workshop, Esrin, March 8–10, 2004, CD-ROM Proceed. F¨orste, C., F. Flechtner (2007). Combined Gravity Field Model EIGEN-GL04C. GFZ Potsdam, http://www. gfzpotsdam.de/pbl/op/grace/index GRACE.html. F¨orste, C., and 13 others (2005). A new high resolution global gravity field model derived from combination of GRACE and CHAMP mission and altimetry/gravimetry surface gravity data. Poster, EGU General Assembly 2005, Vienna, 24–29 April 2005. Hipkin, R., K. Haines, C. Beggan, R. Bingley, F. Hernandez, J. Holt, T. Baker (2004). The geoid EDIN2000 and mean sea surface topography around the British Isles. Geophys. J. Int. 157, 565–577. Ihde, J. et al. (2000). The height solution of the European Vertical Reference Network (EUVN). Ver¨off. Bayer. Komm. f¨ur die Internat. Erdmessung, Astronom. Geod. Arb., Nr. 61: 132–145, M¨unchen. JPL (2007). SRTM – The mission to map the world. Jet Propulsion Laboratory, California Inst. of Techn., http://www2.jpl.nasa.gov/srtm/index html. Kenyeres, A., M. Sacher, J. Ihde, H. Denker, U. Marti (2007). EUVN DA: Establishment of a European continental GPS/leveling network. Gravity Field of the Earth, 1st Internat. Symp. of the Internat. Gravity Field Service, 28 Aug. – 1 Sept., 2006, Istanbul, IAG Symposia, Springer Verlag, in press. Lemoine, F.G., and 14 others (1998). The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. NASA/TP1998-206861, Greenbelt, MD., USA. Liebsch G., U. Schirmer, J. Ihde, H. Denker, J. M¨uller (2006). Quasigeoidbestimmung f¨ur Deutschland. DVWSchriftenreihe, No. 49, 127–146. Morelli, C., M. Pisani, C. Gantar (1975). Geophysical anomalies and tectonics in the Western Mediterranean. Bolletino di Geofisica Teorica ed Applicata, Vol. XVIII, No. 67, Trieste. Sans`o, F., C.C. Tscherning (2003). Fast spherical collocation: theory and examples. J. Geod. 77:101–112. USGS (2007). Global 30 arc-second elevation data set GTOPO30. U.S. Geological Survey, Center for Earth Resources Observation and Science (EROS), http://edc.usgs.gov/products/elevation/gtopo30/gtopo30.html. Wenzel, H.-G. (1982). Geoid computation by least squares spectral combination using integral kernels. Proceed. IAG General Meet., 438–453, Tokyo.

An Attempt for an Amazon Geoid Model Using Helmert Gravity Anomaly D. Blitzkow, A.C.O.C. de Matos, I.O. Campos, A. Ellmann, P. Van´ıcˇ ek and M.C. Santos

Abstract This paper describes the computation of a geoid model for the Amazon Basin (GEOAMA) limited by 5◦ N and 10◦ S in latitude and 70◦ W and 50◦ W in longitude. The software package SHGEO developed by the University of New Brunswick, Canada, was used for the calculation. The geoid model was derived by using the following data: digital terrain model SRTM3 (Shuttle Recovery Topography Mission) version 2.0 with 3” grid, the geopotential model EIGEN-GL04S1, degree and order 150, derived from GRACE satellite, and terrestrial gravity data basically observed along the rivers. For GEOAMA validation the longitudinal profiles of some rivers over the basin derived from three geoid models (EGM96, D. Blitzkow Laboratory of Topography and Geodesy, Department of Transportation, University of S˜ao Paulo, EPUSP-PTR, Postal Code 61548, CEP:05424-970, S˜ao Paulo, S˜ao Paulo, Brazil

MAPGEO2004 and EIGEN-GL04C) combined with geodetic heights from 28 GPS stations close to the tide gage stations were used. The results show that GEOAMA is in good agreement with the EGM96, MAPGEO2004 and EIGEN-GL04C profiles and with the average of the main rivers (Solim˜oes and Amazonas) gradient (20 mm/km). MAPGEO2004 has been the official geoid model in Brazil since 2004. It was developed by Brazilian Institute of Geography and Statistics (IBGE) and Surveying and Geodesy Laboratory of the University of S˜ao Paulo (LTG/USP).

Keywords Geoid modeling· GPS· Height· DTM

1 Introduction

A.C.O.C. de Matos Laboratory of Topography and Geodesy, Department of Transportation, University of S˜ao Paulo, EPUSP-PTR, Postal Code 61548, CEP:05424-970, S˜ao Paulo, S˜ao Paulo, Brazil

The Amazon rainforest has been a challenge for scientists that study the dynamics of changes in this area. It is considered the largest hydrographic basin in the world with 6, 110, 000 km2 , extending from the Andes I.O. Campos Laboratory of Topography and Geodesy, Department of to the Atlantic Ocean, with 6 countries involved and Transportation, University of S˜ao Paulo, EPUSP-PTR, Postal ∼4.7 million km2 (68%) in Brazil (Campos, 2004). Code 61548, CEP:05424-970, S˜ao Paulo, S˜ao Paulo, Brazil The main river has two denominations: from the borA. Ellmann der of Peru to Manaus, it is called Solim˜oes, and from Department of Transportation, Faculty of Civil Engineering, Manaus to the Atlantic Ocean, it is called Amazonas. Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, In general, the Amazon region is very flat and for this Estonia reason the river gradient is very small. In Manaus, for P. Van´ıcˇ ek Department of Transportation, Faculty of Civil Engineering, example, the height of the water level during the dry Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, season is only about 12 m above the mean sea level. Estonia The vertical reference network in Amazon region is esM.C. Santos Department of Geodesy and Geomatics Engineering, University sential for hydrological and hydrodynamical studies, to compile a regional Digital Terrain Model (DTM) and to of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 construct topographic maps. M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Nevertheless, due to the lack of roads and the forest density, it is impossible to carry out spirit levelling for establishing a vertical network in the region. An alternative is to use satellite geodesy associated with a geoid model. This paper presents a particular effort to establish the geoid model GEOAMA for the Amazon Basin, limited by 5◦ N and 10◦ S in latitude and 70◦ W and 50◦ W in longitude. GEOAMA was computed using EIGEN-GL04S1 up to degree and order 60 as the reference field. The reduced Helmert mean gravity anomalies in blocks of 5 were estimated in continental areas. For the ocean KMS-2002 satellite altimetry model was used (Andersen and Knudsen, 1998). The digital terrain model SRTM3, version 2, was chosen for computing the new geoid. The processing of the modified Stokes integral was carried out using the Stokes-Helmert’s geoid software (SHGEO), which is a precise scientific package for gravimetric geoid determination based on the Stokes-Helmert approach (Ellmann and Van´ıcˇ ek, 2007). The present effort is involved in the context, of a major project of LTG/USP with GeoForschungsZentrun (GFZ), Germany, and Institut de R´echerche pour le D´ev´eloppement (IRD), France, for hydrological and gravity variations studies in Amazon. The GEOAMA is an advanced version with respect to MAPGEO2004 (IBGE,2004; Lobianco et al., 2005). In fact, MAPGEO2004 was computed using EGM96 up to degree and order 180 as the reference field. The reduced Helmert mean gravity anomalies were estimated in blocks of 10’. For the ocean KMS99 satellite altimetry model was used (Andersen and Knudsen, 1998). The DTM used obtained from the digitalization of topographic maps, combined with GLOBE model (Hasting and Dunbar, 1999), where topographic maps were unavailable had a resolution of 1 × 1 . The processing of the modified Stokes integral was carried out using FFT technique.

2 Data Set The following data set was used for the gravimetric geoid computation and its validation over the Amazon area: (1) free-air gravity anomalies; (2) geopotential model for computing the long wavelength component of the geoid and of the gravity anomaly; (3) a highprecision DTM for the computation of terrain correction and other topographic effects on geoid modeling;

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and (4) a set of GPS geodetic heights on the levelling network as control points. This data set is reviewed in the sequel.

2.1 Terrestrial Gravity Data The terrestrial gravity data used in this study has been provided by Petrobras (Brazilian oil company). The observations, a total of 98,176 points, were taken basically along-side the Amazon, Solim˜oes and Madeira rivers in the 1960’s. The majority of the data was measured by LaCoste&Romberg gravitymeter with 0.1 mGal accuracy (Lobianco, 2005). Figure 1 shows the spatial distribution of the points within our target area. Note that due to the rainforest there are vast gravity data gaps.

Fig. 1 Gravity data for the Amazon region

2.2 Geopotential Model The “satellite-only” model EIGEN-GL04S1, developed up to degree 150, was used to generate the long wavelength contribution of the geoid. On the other hand, EIGEN-GL04C was used to compute the gravity anomaly in the gaps. The latter is a combination of GRACE (Gravity Recovery and Climate Experiment) and LAGEOS satellite missions plus 0.5◦ × 0.5◦ grid of gravimetry and altimetry surface data and it is complete to degree and order 360 in terms of spherical harmonic coefficients (F¨orste et al., 2006).

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2.3 Digital Terrain Model

3 Geoid Computations

Gravimetric geoid computation by the Stokes integral requires gravity data referred to the sea level. To ensure the harmonicity of the quantities to be downward continued from the topographic surface to the geoid level Helmert’s second condensation method is often applied. A DTM is used to compute the terrain correction and the indirect effect on the geoid. The data gridding is also an important issue. The mean free-air gravity anomalies used in this study were obtained via mean complete Bouguer anomalies by an approach discussed in Jan´ak and Van´ıcˇ ek (2005). For the present study, we have a suitable gridded topography with a grid size of 3 × 3 (approximately 90 × 90 m) from SRTM3 version 2. The SRTM heights are referred to the EGM96-derived geoid and the coordinates in the WGS84 reference ellipsoid (Lemoine et al., 1998a; Lemoine et al., 1998b; Hensley et al., 2001; JPL, 2004; Farr et al., 2007). Over the areas with no SRTM3 information available the 30 × 30 . DTM2002 topographic model (Saleh and Pavlis, 2002) was used instead. The DTM2002 combines data from GLOBE (Global Land One-kilometer Base Elevation), version 1.0, constructed by NOAA/NGDC (Hasting and Dunbar, 1999), and ACE (Altimeter Corrected Elevation), from Earth and Planetary Remote Sensing Laboratory, University of Montfort, UK. The global ACE model is derived from altimetry data (Johnson et al., 2001). The heights are referred to the Mean Sea Level. The DTM2002 data was thinned down to 90 × 90 m, i.e. the same spatial resolution as adopted by the SRTM3 model.

The SHGEO precise geoid determination software was employed to compute the GEOAMA geoid model. This package has been developed under the leadership of Professor Petr Van´ıcˇ ek at the Department of Geodesy and Geomatics Engineering, University of New Brunswick (UNB), Canada. This software (Ellmann, 2005a,b) uses StokesHelmert method. The gravity anomalies over the entire Earth are required for the geoid determination by the original Stokes formula. In practice, the area of integration is limited to some domain around the computation point, usually circular. The Stokes equation used to compute the geoidal heights (Ellmann and Van´ıcˇ ek, 2007) is

2.4 Control Points In the framework of the Hydrology and Geochemistry of the Amazon Basin (HiBAm) international research program, there was an attempt to determine the heights at various control points (zero of limnimeters scale) along the Amazon region rivers, with reference to a consistent origin (geoid) (Kosuth and Cazenave, 2002; Campos, 2004). A total of 28 stations were selected to carry out GPS observations on Bench Marks (BM) established for this purpose as close as possible to the limnimeters. The GPS/levelling data have been connected to the control points by spirit levelling.

N (Ω) =  R S M (ψ0 , ψ(Ω, Ω  ))Δg(r g , Ω)dΩ  4π γ0 (φ) Ωψ0 R  2 Δg h (r g , Ω) 2γ0 (φ) n−1 n M

+

n=2

δV t (r g , Ω) δV a (r g , Ω) + + γ 0(φ) γ 0(φ)

(1)

where  Δg(r g , Ω) = Δg (r g , Ω) − h

M 

 Δgnh (r g , Ω)

(2)

n=2

The geocentric position (r, Ω) of any point can be represented by the geocentric radius r and a pair of geocentric coordinates Ω = (φ, λ), where φ and λ are the geocentric spherical coordinates; R is the mean radius of the Earth. The modified Stokes kernel S M (ψ0 , ψ(Ω, Ω  )) can be computed according to Van´ıcˇ ek and Kleusberg (1987), where ψ(Ω, Ω  ) is the spatial geocentric angle between the computation and integration points; dΩ  is the area of the integration element. The UNB approach works to minimize the truncation bias, i.e., uses the low-frequency part of the geoid described by a global geopotential model and a spheroid of degree M as a new reference surface (Van´ıcˇ ek and Sj¨oberg, 1991) instead of the Somigliana-Pizzeti reference ellipsoid. The upper limit (M) for the modified Stokes kernel (geopotential

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model) used in this case was set to 60. This option showed good agreement with GPS/levelling data. In the right-hand side of Eq. (1) the first term is the Helmert residual co-geoid. Since the low-degree reference gravity field is removed from the anomalies before the Stokes integration (Eq. (2)), the longwavelength contribution to the geoidal height (Heiskanen and Moritz, 1967), i.e., the reference spheroid must be added to the residual geoid (the second term on the right-hand side of Eq. (1)). The sum of the first and second terms results with Helmert co-geoid. The third term on the right-hand side of Eq. (1) is the primary indirect topographical effect (Martinec, 1993) and the last term is the primary indirect atmospheric effect on the geoidal heights (Nov´ak, 2000). Accounting for the indirect effects is needed to transform the Helmert geoidal heights back into the real space. The gravity anomaly referred to the geoid surface of Eq. (2) (Δg h (r g, Ω)) is obtained by downward continuation. It is solved using the Poisson integral equation (Heiskanen and Moritz, 1967). This equation had originally been designed as a formula for the upward continuation of harmonic quantities. It can be written as (Kellogg, 1929) Δg h (rt , Ω)  R = K  Δg h (r g , Ω  )dΩ  4πrt (Ω) Ω  ∈Ω0

corrections to the Helmert gravity anomaly, such as the direct atmospheric effect, the secondary indirect atmospheric effect, ellipsoidal correction for the gravity disturbance and ellipsoidal correction for the spherical approximation. The 5 × 5 grid of the mean free-air gravity anomalies (Fig. 2) was computed from point gravity data. The EIGEN-GL04C derived anomalies were used to fill gaps with no terrestrial gravity data. Over the ocean, the satellite altimetry-derived gravity anomalies (Global Marine Gravity, 2 arcminute) were used. This model was produced by the Geodetic Division of Kort og Matrikelstyrelsen (KMS2002), the National Survey and Cadastre of Denmark, processing Geosat and ERS-1 satellite altimetry data (Andersen and Knudsen, 1998). The computed Amazon geoid model is shown in Fig. 3.

(3)

where K  = K [rt (Ω), ψ(Ω, Ω  ), R] is the spherical Poisson integral kernel (Sun and Van´ıcˇ ek, 1998). Downward continuation is an inverse problem to the original Poisson integral. The term Δg h (rt , Ω), on the left-hand side of Eq. (3), is the Helmert gravity anomaly referred to the Earth’s surface; it can be obtained by (Van´ıcˇ ek et al., 1999)

Fig. 2 Mean free-air gravity anomalies

Δg h (rt , Ω) = Δg(rt , Ω) + δ At (rt , Ω) 2 + δV t (rt , Ω) + δ Aa (rt , Ω) rt (Ω) 2 δV a (rt , Ω) + εδg (rt , Ω) − εn (rt , Ω) (4) + rt (Ω) The first term on the right-hand of the Eq. (4) is freeair anomaly; the second and third terms are the direct and secondary indirect topographic effects on the gravitational attraction. Other terms are non-topographical Fig. 3 Geoid heights in the Amazon region

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4 Geoid Validation

Table 2 Statistics among GEOAMA (D) and EGM96 (A), MAPGEO2004 (B) and EIGEN-GL04C (C)

It has been used the 28 control points (limnimeters) available in Amazon (§2.4) with geodetic coordinates (φ, λ, h) referred to the zero of the limnimeter scale. In a first step of the validation, the geoidal height (height anomaly) has been estimated in these points using EGM96 (A) (Lemoine et al., 1998a,b), MAPGEO2004 (B) (Lobianco et al., 2005) and EIGENGL04C (C) (F¨orste et al., 2006) for comparison with GEOAMA (D). Table 1 shows the 28 control points and their respective geoidal height differences among GEOAMA and the three referred models (D-A, D-B and D-C). The statistics are presented in the Table 2. The biggest differences between GEOAMA and EGM96 are close to the Amazonas estuary; Tabatinga and Santo Antonio do Ic¸a in Solim˜oes River (Fig. 4); and Vista Alegre, Novo Aripuan˜a and Humaita in Madeira River (Fig. 5). Looking to the RMS (Root Mean Square) difference, the column D-C shows a smaller value. It means that EIGEN-GL04C fits much

Mean RMS Maximum Minimum

D-A (m)

D-B (m)

D-C (m)

−0.28 0.52 0.75 −1.32

−0.02 0.61 1.12 −0.91

0.19 0.38 1.11 −0.57

better than the others to GEOAMA (Figs. 4 and 5). A special attention was addressed to the comparison of GEOAMA with MAPGEO2004. Both models employ the same terrestrial gravity data, but different geopotential models, respectively EIGEN-GL04S1 and EGM96, as well as different upper limit for the reference field, degree and order 180 for MAPGEO2004 and 60 in the case of GEOAMA. MAPGEO2004 used “remove-restore” technique together with the modification of Stokes kernel proposed by Van´ıcˇ ek and Kleusberg (1987), in Fast Fourier Transform (FFT) approach. The GEOAMA has a finer spatial resolution (5’ grid) than the MAPGEO2004 (10’ grid).

Table 1 The 28 control points and their respective geoidal height differences, water levels (NA) for Solim˜oes, Amazonas and Madeira Rivers (June 26, 1999) and estuary distance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Station

D-A (m)

D-B (m)

D-C (m)

NA (m)

Dist. (km)

Tabatinga S˜ao Paulo de Olivenc¸a Santo Antonio do Ic¸a Fonte Boa Comunidade das Miss˜oes Itapeu´a Manacapuru Porto Trapiche 15 Itacoatiara Parinfins ´ Obidos Santar´em Prainha Almerim Porto de Moz Igarape Aruan˜a Gurup´a Porto de Santana Vila Urucurituba Nova Olinda do Norte Borba Vista Alegre Novo Aripuan˜a Manicor´e Vila Carara Humait´a Conceic¸a˜ o da Galera Porto Velho

0.75 −0.10 −0.55 −0.43 −0.34 −0.19 −0.37 −0.21 −0.21 −0.21 −0.02 −0.05 −0.24 −1.32 −1.05 −0.92 −1.04 −0.99 −0.36 −0.46 −0.38 −0.69 −0.59 0.18 0.45 0.74 0.37 0.28

1.12 0.76 0.14 0.09 0.08 0.07 −0.29 −0.13 −0.27 −0.18 −0.35 −0.41 −0.16 −0.82 −0.91 −0.90 −0.81 −0.81 −0.24 −0.19 0.01 −0.28 −0.29 0.83 1.09 0.87 0.79 0.74

1.11 0.62 0.43 0.27 0.00 0.20 0.16 0.03 −0.03 0.33 0.37 0.63 0.81 −0.07 0.18 0.34 0.08 −0.26 −0.20 −0.29 0.11 0.02 −0.08 0.26 0.69 0.45 −0.38 −0.57

12.33 14.23 14.09 22.2 15.53 17.34 20.01 29.30 – 8.75 7.69 7.20 4.05 – 3.48 – – 0.00 – 18.86 18.70 17.59 17.91 18.10 – 14.66 – 7.73

3182.7 2932.5 2785.0 2436.3 1915.6 1710.3 1395.9 1319.8 1109.3 897.7 733.2 585.5 408.4 303.5 265.1 210.2 214.3 39.0 28.7 77.7 164.4 254.8 307.6 454.5 683.8 801.6 906.9 1042.3

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Fig. 4 Geoidal height differences among GEOAMA and EGM96, MAPGEO2004 and EIGEN-GL04C for Solim˜oes and Amazonas rivers

A second step in the validation of GEOAMA comThe biggest differences between the models are close to the Amazonas estuary, at the Solim˜oes River close prises the estimation of the orthometric height at the to Brazilian border with Peru and Manicor´e to Porto limnimeters H Z E R O (zero of the scale) for 21 of 28 Velho at Madeira River. stations. The equations used to compute H Z E R O are

Fig. 5 Geoidal height differences among GEOAMA and EGM96, MAPGEO2004 and EIGEN-GL04C for Madeira river

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Fig. 6 Water level for Amazonas, Solim˜oes and Madeira Rivers – MAPGEO2004 and GEOAMA (June 26, 1999)

HG P S−ST A ≈ h − N

Figure 6 shows height of the water surface (z) versus estuary distance for the main rivers and Madeira river H Z E R O = HG P S−ST A ± dn (6) with respect to GEOAMA and MAPGEO2004 at June 26, 1999. where dn is the spirit leveling height difference between GPS point and the control point. The consistency of the models can be checked 5 Conclusion against the analysis of the river longitudinal profile both at high or low water level season. The choice The smallest mean difference is between GEOAMA was for the high level. Table 1 shows water levels and MAPGEO (−2 cm), whereas the smallest spread is (NA) for Solim˜oes, Amazonas and Madeira rivers between GEOAMA and EIGEN-GL04C (38 cm). The at June 26, 1999, derived from 21 control points, most likely reasons for the control points differences as a function of the estuary distance (Table 1, last between GEOAMA and MAPGEO2004 stem from discolumn), Atlantic Ocean for Amazonas/Solim˜oes and crepancies between their reference fields (EGM96 for Amazonas for Madeira. The level of the water in the MAPGEO2004 and EIGEN-GL04C for GEOAMA) rivers, in particular in Amazon where the slope of the and the resolution of GEOAMA (5 grid). rivers is small, is a natural source of comparison of the Due to the fact that Amazon is a very flat region, the height estimation using the different geoid models. rivers slope is very low. So, the analysis of the height of The value of NA at the specific date, added to the water surface (z) is a very important information for H Z E R O , provides the height of the water surface, the validation of the geoid model. The different geoid (5)

models analyzed fit quite well to the height of the water (7) surface (z) of the three rivers (Solim˜oes, Amazonas and Madeira), at June 26, 1999. But, the GEOAMA has the best approximation to the historical data of the main The average gradient of the main rivers (Solim˜oes rivers (∼20 mm/km). and Amazonas) is, from Tabatinga to the ocean, approximately equal 20mm/km (CPRM, 1999). Using Acknowledgments The authors acknowledge Agˆencia Na´ the different geoid models related to this paper (MAP- cional de Aguas (ANA) for the control points data, BASE Aerofotogrametria e Projetos S.A. for the computer donation, GEO2004, EIGEN-GL04C and GEOAMA), the gradiCNPq and CAPES/COFECUB for supporting this research. ent was estimated for each interval of the 21 control The activity is undertaken with the financial support of Governpoints and averaged for the total distance. The values ment of Canada provided through the Canadian International derived are 23.09, 23.24, 22.52 mm/km, respectively. Development Agency (CIDA). z = HZ E R O + N A

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Combination of Gravimetry, Altimetry and GOCE Data for Geoid Determination in the Mediterranean: Evaluation by Simulation R. Barzaghi, A. Maggi, N. Tselfes, D. Tsoulis, I.N. Tziavos and G.S. Vergos

Abstract Local geoid determination is traditionally carried out on land and at sea areas using gravity anomaly and altimetry data. This determination can be aided and improved by the data of missions such as GOCE. In order to assess the performance of the combination of heterogeneous data for local geoid determination, simulated data for the area of the central Mediterranean Sea are analyzed. These data include gravity anomaly, altimetry, and GOCE observations processed with the space-wise approach. The results show that GOCE data improve the results for areas not well covered with other data types, while also accounting for any long wavelength errors of the adopted reference model. Even when the ground gravity data are dense, data from GOCE improve the error standard deviation and eliminate biases. At sea, the altimetry data give the dominant geoid information. However the geoid accuracy is sensitive to orbit calibration errors and the unmodelled mean R. Barzaghi DIIAR - Politecnico di Milano - Piazza Leonardo da Vinci, 32 - 20133 Milano, Italy A. Maggi DIIAR - Politecnico di Milano - Piazza Leonardo da Vinci, 32 - 20133 Milano, Italy N. Tselfes DIIAR - Politecnico di Milano, Polo Regionale di Como - Via Valleggio, 11 - 22100 Como, Italy D. Tsoulis Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box440, 54124, Thessaloniki, Greece I.N. Tziavos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box440, 54124, Thessaloniki, Greece G.S. Vergos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box440, 54124, Thessaloniki, Greece

sea surface topography. If such effects are present the GOCE data can account for them. Keywords Mediterranean geoid · GOCE mission · Gravity · Altimetry

1 Introduction GOCE (Gravity field and steady-state Ocean Circulation Explorer) is a satellite mission (ESA, 1999) designed by ESA (European Space Agency), which will be launched in 2008. The goal of this mission is the determination of the stationary part of the gravity field to a high degree of accuracy and spatial resolution. The main instrument on board the satellite will be the “gradiometer”, composed by six accelerometers, and measuring the second derivatives of the potential (the full tensor) along the satellite orbit (the so-called gradients). Additional information on the gravity field will be derived from the tracking of the satellite orbit, by means of a GPS receiver, and the accelerometers measurements of the non-gravitational forces. Three different approaches will be applied for the determination of the global gravity field models from GOCE: the direct approach (Bruinsma et al., 2004), the time-wise approach (Pail et al., 2005) and the spacewise approach (Migliaccio et al., 2004). In a previous study (Maggi et al., 2007), the contribution of GOCE filtered data from the space-wise approach and a GOCE geo-potential model, were evaluated for local geoid determination in a combination scheme with terrestrial data employing least squares collocation in a simulation. It was found that the benefit from the GOCE long wave-length information

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will be very significant. The present paper, refers to a larger region incorporating satellite altimetry data as well. No topographic information (Arabelos and Tscherning, 1990) or bathymetric information (Vergos and Sideris, 2003) will be used for data reduction, even though this is possible with real data.

2 Simulation of Data In the frame of the EGG-C (European GOCE Gravity Consortium) (Balmino, 2001) activities for the preparation of the GOCE mission, full simulation solutions are computed so that methodology and software efficiency are ensured (Migliaccio et al., 2006). The latest simulated data set available by EGG-C is used and the data are processed. These data include 60 days (note that at least 1 year is expected) of: gradients with inflight calibration noise (including instrumental errors, satellite errors, etc.), signal simulated from EGM96 (Lemoine et al., 1998), orbit positions and velocities, common mode accelerations, rotations and attitude information. The orbit positions and velocities are used to obtain (pseudo-) observations of the disturbing potential along the orbit (Jekeli, 1999). These data

Fig. 1 The simulated geoid of the central Mediterranean, i.e. the synthesis from EGM96 and GPM98 referenced to EIGEN g104c

R. Barzaghi et al.

are processed with the space-wise approach. This is a multi-step collocation procedure that consists of the Wiener filter, gridding by least squares collocation, harmonic analysis of the grids, and iterations made in order to account for the non-optimality of the Wiener filter. For details of these steps and related results with these data, see (Migliaccio et al., 2007). For the present study a filtered grid of potential T and a grid of second order radial derivatives Trr are used. In the latitude interval −83 to 83 degrees these grids have an error Standard Deviation (STD) of 0.022 m2 s−2 and 0.6110−12 s−2 (milli-E¨otv¨os) respectively. The high quality of the T grid is also due to noise-free positions used. The accuracy is much smaller over the poles because of the GOCE orbit inclination (i = 96.5◦ ). Note that the gradients are measured in the gradiometer reference frame (x,y,z) but through the filtering and gridding they are transformed into the Local Orbital Reference Frame (ξ, η, r) (LORF), where ξ is almost along-track, η cross-track and r radial. Ground gravity anomalies Δg are simulated with the signal from EGM96 up to degree and order 360 and white noise of 5 mgal. A set of true available

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Fig. 2 The simulated MDSST

observations is used for the longitude and latitude. The EGM96 reference sphere is used as radius. The altimetry simulation is based on the data from the Geodetic Mission (phase E) of ERSI as provided by AVISO (1998) in the merged T/P & ERSI corrected Sea Surface Heights (CORSSHs) products. This set of ERSI SSHs refers to reprocessed data which have improved orbit accuracy. The procedure followed was based on fitting the original ERS data (NOAA JGM-2 orbits) to the more precise TOPEX/Poseidon data (JGM-3 orbits) using a global minimization principle of the ERSI-T/P dual crossover differences (Le Traon et al., 1995; Le Traon and Ogor, 1998). Based on this adjustment, improved ERS orbits with an accuracy similar to T/P ones (2 cm rms) was obtained. Observations of the geoid N are generated from EGM96. A generous error is added to them: track-wise bias and tilt, plus white noise of total 4 cm RMS (Root of Mean Square). A “bow tie” error (Schrama, 1989) of 3.5 cm RMS is also simulated, but it is not added unless specified so. In order to have a more detailed geoid in the simulation, the GPM98 model (Wenzel, 1998) is added to all the data, from degree 361 to 720 and order 0 to 720. For the collocation solution, EIGEN g104c

(F¨orste et al., 2007) is subtracted, up to degree and order 360, from all the data (Fig. 1). The differences of this model to EGM96 are interpreted as long wavelength errors. This does not imply a judgement by the authors about the quality of these models. These differences are similar to the nominal EGM96 errors (Maggi et al., 2007) and this is useful to perform simulations where a model with long wavelength errors is used as reference. A Mean Dynamic Sea Surface Topography (MDSST) is also simulated. The model determined by Rio (2004) (Fig. 2) is taken and interpolated to the altimetry points. The MDSST is not present in the altimetry observations, unless specified so.

3 Local Geoid Determination 3.1 Collocation with Ground Gravity and Altimetry Data A local geoid determination is often made using the well-known method of least squares collocation (Moritz, 1980).

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The predicted geoid undulation based on gravity and undulation is given by:

3.2 Utilization of GOCE Data

The functions (Eqs. (4), (6) and (7)) are those suitable for including gridded potential in the geoid prediction. where y is the vector of observed Δg and N , Css is a For the gridded second radial derivatives, the following matrix computed from the Δg and N covariances and functions are also needed: the covariance function of cross-covariances, C N s is a matrix computed from the Trr and its cross covariance functions with T , Δg and cross-covariances of the predicted N with the observed N . Δg and N and Cvv describes the Δg and N noise.  +3 max 2 μ2  R In spherical approximation, the covariance and crossβ 2 σ˜ 2 P (t), C Trr P ,Trr Q = 6 covariance functions to be used in Eq. (1) are obtained rPrQ R =min by: (10)  +2       max +3 +1 max μ2  R R μ2  R2 β σ˜ 2 P (t), C Trr P ,TQ = 4 C g P , g Q = 4 α 2 σ˜ 2 P (t), rP rQ R rPrQ R ˆ = C N s (Css + Cvv )−1 y, N

(1)

=min

=min

(2)

(11)

 +2  +3 max 1 μ2  R R (3) C C g P ,TQ , 2 = δ σ ˜ P (t),

g P ,Trr Q   γ¯ 5 rP rQ R = min  +2  +1 max μ2  R R (12) C g P ,TQ = 3 α σ˜ 2 P (t), rP rQ R 1 =min (13) C Trr P ,N Q = C Trr P ,TQ , (4) γ¯ 1 (5) where, β = ( + 1)( + 2) and δ = α β . C N P ,N Q = 2 C TP ,TQ , γ¯ The covariance functions needed for Trr are easy to 1 C TP ,N Q = C TP ,TQ , (6) compute because the radial direction of LORF coinγ¯ cides with the up direction of an East-North-Up (ENU)  +1  max 2 2  frame. If other derivatives along directions not coinμ R C TP ,TQ = 2 σ˜ 2 P (t), (7) ciding with ENU were to be used then all the covarirPrQ R =min ance functions would have to be computed and linear combinations would have to be made, according to where t = cos(ψ), ψ is the spherical distance, P (t) the rotations between the used directions and the ENU are Legendre polynomials of degree , R is the mean (Tscherning, 1993). This procedure is made inside the earth radius, r P , r Q are the radii of points P and Q, μ is space-wise approach, because the other two directions the gravitational constant times the earth mass, σ˜ 2 are of LORF differ from the East and North. However the some adapted signal degree variances and a = (−1). gridded second radial derivatives produced with the The quantity γ¯ is the mean normal gravity used space-wise approach are easy to handle in collocation. throughout the simulations. If all four data types are used, vector y (Eq. (1)) now The prediction error covariance matrix is computed includes, in addition to Δg and N , the gridded T and by (Moritz, 1980): Trr . The signal and error matrices (Eqs. (4) and (8))  include the covariances of Trr and T and their cross T −1 Ce = C N N − Cs N (Css + Cvv ) Cs N , (8) covariances with Δg and N . C g P ,N Q =

where, the matrix C N N is computed from the N covariance function. For comparisons with the actual er- 3.3 A simulation of Geoid Prediction rors computed from simulations, the point-wise STD is used: The simulated geoid over the central Mediterranean

σi = Ce (i, i). (9) is examined. The considered area is bounded between

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35◦ <ϕ<47◦ and 2◦ <λ<20◦ and divided in 24 cells of 3 by 3 degrees. The geoid is predicted by least squares collocation on a regular spherical grid of 0.2 degrees. A separate collocation is made for every cell. A limit for the number of data is set: 1500 gravity anomalies Δg per cell / 2000 for 3 cells that cover the Alps, 1500 altimetry observations N per cell, 1100 gridded Trr and 350 gridded T . Δg and N are taken with overlap of 1◦ around each cell, while GOCE gridded data are taken with 2◦ overlap. The degree variances σ˜ 2 are based on considering a-priori knowledge of the degree variances of the differences between EGM96 and EIGEN g104c, up to degree 360, and the signal degree variances of GMP98 from degree 361 to 720. These degree variances are scaled to fit the gravity anomaly variance locally. Consistency between the true errors and the predicted errors (Eq. (9)) justify this simple choice. For optimal results a simultaneous fit to Trr and Δg could be made (Knudsen, 1987) and for real data closed covariance expressions up to infinite degree (Tscherning and Rapp, 1974) can be used. A more thorough study about the degree variances estimation for the combination of heterogeneous data types is a very interesting and important topic on its own, though it is this is not addressed here. Fortunately the collocation results are robust to some variations of the covariance function (Sans`o et al., 1999). For the 24 cells of the whole area the addition of GOCE data to the already available Δg and N data leads to significant improvement (Table 1). Edge effects, data gaps and distribution problems are resolved with GOCE. The effect of heterogeneous data combination is better seen if smaller areas are examined. Therefore, four “zoom-in” cases, from the twenty four examined, are presented herein. All refer to areas included in a single 3 by 3 cell, so that no errors of cells treated separately are mixed. The first zoom is made in the area between 44◦ <ϕ<47◦ and 6◦ <λ<8◦ . This is inside a cell with

Table 2 Geoid prediction errors, in cm, for an area in the Alps for 2000 Δg, 1100 Trr and 350 T (case alps 1.) and for 1000 Δg and 650 Trr (alps 2) for different data combinations Case

Data used

STD

RMS

Alps 1 Alps 1 Alps 1 Alps 2

g

g, Trr

g, T, Trr

g,

9.7 4.8 4.7 6.1

15.2 5.1 4.7 6.1

2000 Δg that covers mostly the Alps. It is seen that GOCE improves the error STD but improves much more the RMS (Table 2) because the Δg only solution has a mean value problem. This problem is present in almost all the cells, regardless of the number and distribution of Δg. The solution for this cell is repeated with 1000 points of Δg and 650 points of Trr . It is seen that is better to add some GOCE data than to increase the number of Δg observations. The second “zoom” is made in an area between 42◦ <ϕ<44◦ and 11◦ <λ<13◦ . There, the mean value problem solution with the use of T is seen (Table 3). The third “zoom” is made at a cell bounded between 41◦ <ϕ<43◦ and 5◦ <λ<8◦ degrees. The dense altimetry data give a very good solution (Table 4). If the MDSST is added this quality is degraded. This effect is present at every area where the MDSST is high. If GOCE data are added most of these problems are fixed. From the fourth “zoom”, referring to the area between 35◦ <ϕ<38◦ and 17◦ <λ<20◦ degrees, a similar result is obtained. When the bow tie error is added to altimetry data, long wave-length errors arise, but they are corrected significantly when GOCE data are added (Table 5). An important conclusion can be drawn based on the aforementioned results, i.e., that the altimetric information dominates, but unmodelled MDSST and Table 3 Geoid prediction errors, in cm, for an area in central Italy for 1500 Δg and 350 T (case centre 1) Case

Data used

STD

RMS

centre 1 centre 1

g

g, T

5.4 3.1

12.9 3.1

Table 1 Geoid prediction errors, in cm, for the whole area (case whole.) and different data combinations

Table 4 Geoid prediction errors, in cm, for an area of sea, (case sea 1.) and the same area but with altimetry observations containing the MDSST (case sea 2)

Case

Data used

STD

RMS

Case

Data used

STD

RMS

Whole Whole Whole Whole

g

g, N

g, N, Trr

g, N, T, Trr

16.5 9.9 5.1 4.8

17.1 9.9 5.0 4.8

Sea 1. Sea 2. Sea 2. Sea 2.

N N N, T, Trr N, g, T, Trr

1.5 3.4 3.7 3.0

1.5 12.4 3.8 3.0

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Table 5 Geoid prediction errors, in cm, for an area of sea, (case sea 1.) and the same area but with altimetry observations with “bow tie” error (case sea 2) Case

Data used

STD

RMS

Sea 1. Sea 2. Sea 2. Sea 2.

N N N, T, Trr N, g, T, Trr

1.6 2.7 3.3 2.8

1.7 8.8 3.5 3.0

possible “bow tie” errors can be identified and consequently corrected with the information of GOCE. A final test for a large area between 35◦ <ϕ<47◦ and 0◦ <λ<20◦ is performed. 6000 Δg are used, distributed on a grid and with a 3 mgal white noise, thus simulating an interpolated grid. The estimate was predicted at 2145 stations in the area 37◦ <ϕ<45◦ and 2◦ <λ<18◦ . Then, the impact of using 6000 of the gridded T or Trr values, distributed on the same grid used for Δg, is tested. The decisive contribution of these data when combined with gravity anomalies is verified, even for a large area with good distribution of Δg and with lower noise (Table 6). Table 6 Geoid prediction errors, in cm, for a large test area with 6000 Δg, Trr and T (case large) Case

Data used

STD

RMS

Large. Large. Large. Large. Large.

g Trr T

g, Trr

g, T

7.0 13.5 15.1 3.8 3.6

8.0 13.5 15.1 3.8 3.6

4 Conclusions Both, the GOCE data and the gravity field model derived from GOCE data, are expected to present good long wavelength information. This will be very useful for local gravity field modelling. The results presented in this paper show that the use of processed GOCE data significantly increases the accuracy of local geoid estimation with gravity data on ground or at sea and the use of GOCE potential estimates resolves mean value problems. In marine regions the altimetric information dominates and spans the entire geoid spectrum. However if MDSST effects and/or errors like the “bow tie” exist, other long wave-length information, just like that

of GOCE, is needed. The latter points out the utility of GOCE for MDSST estimation, as well as the possible use of GOCE data for calibration of other data types. Acknowledgments This work was performed under ESA contract No. 18308/04/NL/NM (GOCE HPF) and in the frame of the 3rd Community Support Program (Opp. Supp. Progr. 2000 2006), Measure 4.3, Action 4.3.6, Sub-Action 4.3.6.1 (International Scientific and Technological Co-operation), bilateral cooperation between Greece and Italy.

References Arabelos; D. and C. C. Tscherning (1990). Simulation of regional gravity field recovery from satellite gradiometer data using collocation and FFT. Bulletin Geodesique. 64, pp. 363–382. AVISO (1998) AVISO User Handbook – Corrected Sea Surface Heights (CORSSHs), AVI-NT-011-311-CN, Ed 3.1. Balmino, G. (2001). The European GOCE Gravity Consortium (EGG-C). In: Proceedings of 1st International GOCE User Workshop. ESA-ESTEC, Noordwijk, The Netherlands, April 23–24 2001. Bruinsma, S., J.C. Marty and G. Balmino (2004). Numerical simulation of the gravity field recovery from GOCE mission data. In: Proceedings of 2nd International GOCE User Workshop. Frascati, Italy, March 8–10 2004. ESA (1999). Gravity Field and Steady-State Ocean Circulation Mission. ESA SP-1233 (1). ESA Publication Division, c/o ESTEC, Noordwijk, The Netherlands. F¨orste Ch., F. Flechtner, R. Schmidt, R. K¨onig, U. Meyer, R. Stubenvoll, M. Rothacher, F. Barthelmes, H. Neumayer, R. Biancale, S. Bruinsma, J. M. Lemoine, and S. Loyer (2007). Global mean gravity field models from combination of satellite mission and altimetry / gravimetry surface data Proceedings of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Italy. Jekeli, C. (1999). The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75, pp. 85–101. Knudsen, P. (1987). Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data. Bulletin Geodesique. 61, pp. 145–160. Le Traon PY and Ogor F (1998) ERS-1/2 orbit improvement using Topex/Pos´eidon: The 2 cm challenge. J Geophys Res(103)C4: 8045–8057. Le Traon PY, Gaspar P, Bouyssel F, and Makhmara H (1995) Using TOPEX/POSEIDON data to enhance ERS-1 orbit. J Atm Ocean Tech 12: 161–170. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp and T.R. Olson (1998). The development of the joint NASA GSFC and NIMA geopotential model

Combination of Gravimetry EGM96. NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, Maryland. Maggi A., F. Migliaccio, M. Reguzzoni, and N. Tselfes (2007). Combination of Ground Gravimetry and GOCE Data for Local Geoid Determination: a Simulation Study. Proceedings of the 1st International Symposium of the International Gravity Field Service, August 28 – September 01, 2006, Istanbul, Turkey, In print. Migliaccio, F., M. Reguzzoni, and F. Sans`o (2004). Spacewise approach to satellite gravity field determination in the presence of coloured noise. J Geod 78, pp. 304–313. Migliaccio, F., M. Reguzzoni, and N. Tselfes (2006). GOCE: a full-gradient solution in the space-wise approach. Proc. of the IAG2005 – Scientific Assembly, 22–26 August 2005, Cairns, Australia. P. Tregoning and C. Rizos (eds), Vol. 130, Springer, Berlin, pp. 383–390. Migliaccio, F., M. Reguzzoni M, Sans`o F., Tselfes N., Tscherning C. C., and Veicherts M. (2007). The latest of the space-wise approach for GOCE data analysis. Proceedings of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Italy. Moritz, H. (1980). Advanced Physical Geodesy. Wichmann, Karlsruhe. Pail, R., W. D. Schuh and M. Wermuth (2005). GOCE gravity field processing. In: International Association of Geodesy Symposia, ‘Gravity, Geoid and Space Missions’, C. Jekeli, L. Bastos and J. Fernandes (eds), vol. 129, Springer-Verlag, Berlin, pp. 36–41. Rio M.-H. (2004) A mean dynamic topography of the Mediterranean Sea estimated from the combined use of altimetry,

201 In-Situ Measurements and a General Circulation Model. Geoph Res Let Vol. 6, 03626. Sans`o, F., G. Venuti and C. C. Tscherning (1999). A theorem of insensitivity of the collocation solution to variations of the metric of the interpolation space. In: International Association of Geodesy Symposia, vol. 121, K.P. Schwarz ed., Springer-Verlag, Berlin, pp. 233–240. Schrama, E. J. O. (1989): The role of orbit errors in processing of satellite altimeter data. Report No. 33, Netherlands Geodetic Commision. Publications on geodesy, New series. Delft, 1989. Tscherning, C. C. (1993). Computation of covariances of derivatives of the anomalous gravity potential in a rotated reference frame. Manuscripta Geodaetica, 18, pp. 115–123. Tscherning, C. C. and R. H. Rapp (1974): Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degree-variance models. Reports of the Department of Geodetic Science No. 208, The Ohio State University, Columbus, Ohio. Vergos, G. S. and M. G. Sideris (2003): Altimetry-derived marine gravity field estimation from multi-satellite data. Proceedings of the 3rd Meeting of the International Gravity and Geoid Commission, August 26–30, 2002, Thessaloniki, Greece, I.N. Tziavos (ed), Ziti editions, pp. 314–319. Wenzel, G. (1998). Ultra high degree geopotential models GPM98A, B and C to degree 1800. Proceedings Joint Meeting of the International Gravity Commission and International Geoid Commission, September 7–12, Trieste 1998. Bollettino di Geofisica teorica ed applicata.

An Attempt Towards an Optimum Combination of Gravity Field Wavelengths in Geoid Computation ¨ Hussein A. Abd-Elmotaal and Nobert Kuhtreiber

Abstract The optimum combination of gravity field wavelengths in the framework of geoid computation remains always a hot research topic. Different approaches for such a combination of the wavelengths exist. The window technique (Abd-Elmotaal and K¨uhtreiber, Journal of Geodesy, 77, 77–85) has been suggested to get rid of the double consideration of the topographic-isostatic masses within the data window in the framework of the remove-restore technique. The modified Stokes’ kernel has been suggested to possibly combine the local data signals with the global geopotential earth models. Both techniques have been used in computing a gravimetric geoid for Austria. The available gravity, height and GPS data for the current research are described. The EGM96 geopotential model has been used. A broad comparison between modified Stokes’ kernel and window techniques has been carried out within this investigation in the framework of the geoid computation. The comparison is made on two different levels; the residual gravity anomalies after the remove step and the computed geoid signals before and after scaling to the GPS/levelling geoid. The results proved that the reduced gravity anomalies using the window technique are the smoothest, un-biased and have the smallest range. The modified Stokes’ kernel technique gives the best fit to the GPS/levelling derived geoid. The window technique gives, however, fairly better results than the Stokes’ un-modified kernel technique.

Hussein A. Abd-Elmotaal Civil Engineering Department, Faculty of Engineering, Minia University, Minia 61111, Egypt Nobert K¨uhtreiber Institute of Navigation and Satellite Geodesy, TU Graz, Steyrergasse 30, A-8010 Graz, Austria

Keywords Modified Stokes’ kernel · Window technique · Remove-restore technique · Geoid determination

1 Introduction The optimum combination of gravity field wavelengths in the framework of geoid computation still remains a hot research topic. There are different approaches for such a combination of the wavelengths. The current investigation considers a comparison of three approaches, namely the traditional un-modified Stokes’ kernel technique, the modified Stokes’ kernel technique and the window technique (Abd-Elmotaal and K¨uhtreiber, 2003). The used data sets are described. The Stokes’ technique of geoid determination, within the removerestore scheme, with un-modified and modified Stokes’ kernel, after Meissl (1971), is described. The window technique (Abd-Elmotaal and K¨uhtreiber, 2003) within the remove-restore scheme is outlined. The harmonic analysis of the topographic-isostatic potential is then given. The reduced gravity anomalies using both techniques under investigations are then computed and compared. A gravimetric geoid for Austria has been computed by the three different approaches considered within the current investigation. A wide comparison among these approaches has been carried out in the framework of the geoid computation. The comparison is made on two different levels; the residual gravity anomalies after the remove step and the computed geoid signals. It should be noted that many scholars have suggested different modifications of the Stokes’ kernel

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204

and have studied the topic of the optimum combination of gravity field wavelengths. The reader may refer, e.g., to Sj¨oberg and Hunegnaw (2000); Nov´ak et al. (2001); Sj¨oberg (2003a, b, 2004); Van´ıcˇ ek and Featherstone (1998); Featherstone (1999; 2003); Huang et al., (2000); Silva et al., (2002).

49

48

47

2 The Data 46

2.1 Gravity Data

9

10

11

12

13

14

15

16

17

18

Fig. 2 Used GPS benchmarks with known orthometric heights

The gravity data set for this investigation is a set of free-air gravity anomalies at 5796 stations in Austria and neighbouring countries (Fig. 1). Figure 1 shows, more or less, a homogeneous data distribution within Austria. The gravity data outside Austria have been included to correct the edge effect in the computed gravimetric geoid. The gravity data covers the window (45.7◦ N ≤ φ ≤ 49.7◦ N and 8.5◦ E ≤ λ ≤ 18.2◦ E).

49

48

in the latitude and the longitude directions, respectively, and a fine model of 11.25 × 18.75 resolution. The fine DHM covers the window 44.75◦ N ≤ φ ≤ 50.25◦ N; 7.75◦ E ≤ λ ≤ 19.25◦ E. The coarse DHM covers the window 40◦ N ≤ φ ≤ 52◦ N; 5◦ E ≤ λ ≤ 22◦ E. The coarse DHM has been created by integrating the Austrian fine DHM with GTOPO30 (30 ×30 ) (Gesch and Larson, 1996) and global bathymetry model provided by the Naval Oceanographic Office (1 × 1 ). Figure 3 shows the coarse digital height model used for this investigation. It shows the high mountainous structure of the Alps.

47

4000 3500 50

46 9

10

11

12

13

14

15

16

17

18

2500 2000 48

Fig. 1 Distribution of the used gravity data set 46

2.2 GPS Benchmarks

3000

44

Figure 2 shows the distribution of the available GPS 42 benchmarks (referred to ITRF96) with known orthometric heights (referred to UELN98) in Austria. It 6 8 10 12 14 16 18 20 shows that most of the stations are located in the eastern part of Austria. Only few stations are located at the Fig. 3 The coarse (90 × 150 ) digital height model mountainous western part of Austria.

1500 1000 500 0 –500 –1000 –1500 –2000 –2500 –3000 –3500

2.3 Digital Height Models

3 Traditional Remove-Restore Technique

Two different Digital Height Models are available. A coarse model of 90 × 150 resolution

Within the well-known remove-restore technique, the effect of the topographic-isostatic masses is removed

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205

It is believed that using classical un-modified from the source gravitational data and then restored to the resulting geoidal heights. For example, in the case Stokes’ kernel in the remove-restore technique implies of gravity data, the reduced gravity anomalies in the a wrong combination of gravity field wavelengths. framework of the remove-restore technique are computed by

gr ed = g F − gG M − gh ,

(1)

5 Stokes’ Integral with Stokes’ Modified Kernel

where g F stands for the free-air anomalies, gh is The contribution of the reduced gravity anomalies N g the effect of topography and its compensation on the can be given by gravity anomalies, and gG M is the effect of the refer ence field on the gravity anomalies. Thus the final comR =

gr ed S M E (ψ)dσ, (6) N puted geoid N within the remove-restore technique can

g 4πγ be expressed by: σ N = NG M + N g + Nh ,

(2)

where N G M gives the contribution of the reference field, N g gives the contribution of the reduced gravity anomalies, and Nh gives the contribution of the topography and its compensation (the indirect effect).

where S M E (ψ) is the modified stokes’ kernel after Meissl (1971) given by S M E (ψ) = S(ψ) − S(ψo ) for(0 < ψ ≤ ψo ), =0

for(ψ > ψo ),

(7)

where the optimal cap size ψo is empirically determined through the comparison of the computed gravimetric geoid to the GPS/levelling geoid.

4 Stokes’ Integral with Stokes’ Un-modified Kernel 6 The Window Technique The contribution of the reduced gravity anomalies N g can be given by Stokes’ integral (Heiskanen and Moritz, 1967, p. 94)

The conventional way of removing the effect of the topographic-isostatic masses faces a theoretical problem. A part of the influence of the topographic-isostatic  R N g =

gr ed S(ψ)dσ, (3) masses is removed twice as it is already included in 4πγ the global reference field. This leads to some double σ consideration of the part of the topographic-isostatic where γ is the normal gravity, R is the mean earth’s ra- masses. Figure 4 shows schematically the conventional dius and S(ψ) stands for the Stokes’ un-modified ker- gravity reduction for the effect of the topographicisostatic masses. The short-wavelength part depending nel given by (Heiskanen and Moritz, 1967., p. 94) on the topographic-isostatic masses is computed for a point P for the masses inside the circle. Removing the 1 S(ψ) = − 4 − 6s + 10s 2 − (3 − 6s 2 ) ln(s + s 2 ) (4) effect of the long-wavelength part by a global earth’s s gravitational potential reference field normally implies removing the influence of the global topographicwith isostatic masses, shown as a rectangle in Fig. 4. The double consideration of the topographic-isostatic ψ s = sin , (5) masses inside the circle (double hatched) is seen. 2 A possible way to overcome this difficulty is and ψ is the spherical distance between the computa- to adapt the used reference field to the effect of the topographic-isostatic masses for a fixed data tional point P and the running point Q.

¨ H. A. Abd-Elmotaal and N. Kuhtreiber

206

of the topography and its compensation (the indirect effect) Nh is computed using the Stokes’ integral Eq. (3) using the un-modified Stokes’ kernel given by Eq. (4). P

TI EGM96

7 Harmonic Analysis of the Topographic-Isostatic Potential The harmonic coefficients of the topography and its isostatic compensation as well as the harmonic series expansion of the topographic-isostatic potential can be expressed by (Abd-Elmotaal and K¨uhtreiber, 2003, pp. 78–79; Hanafy, 1987, P. 80):

Fig. 4 The traditional remove-restore technique

TI P

data area adapted EGM96

 n ∞  GM  R n  ¯ ¯ TT I (P) = Tnm Rnm (P), r P r P m=−n n=0 (10) where

Fig. 5 The window remove-restore technique

T¯nm =

window. Figure 5 shows the advantage of the window remove-restore technique schematically. Consider a measurement at point P, the short-wavelength part depending on the topographic-isostatic masses is now computed by using the masses of the whole data area (small rectangle). The adapted reference field is created by subtracting the effect of the topographic-isostatic masses of the data window, in terms of potential coefficients, from the reference field coefficients. Thus, removing the long-wavelength part by using this adapted reference field does not lead to a double consideration of a part of the topographic-isostatic masses (no double hatched area in Fig. 5). The remove step of the window remove-restore technique can then mathematically be written as

gr ed = g F − gG M Adapt − gh ,

(8)

where gG M Adapt gives the contribution of the adapted reference field. The restore step of the window remove-restore technique can be written as N = N G M Adapt + N g + Nh ,



 H Q n+3 ρQ 1+ −1 R σ 

  n+3  tQ To n+3 1− + ρQ 1 − −1 R R − To R3 M(2n + 1)(n + 3)

 

× R¯ nm (Q)dσQ,

(11)

where To is the normal crustal thickness, H is the topographic height, t is the compensating root/antiroot and M denotes the mass of the earth, given by M=

4πR 3 ρM , 3

(12)

where ρ M denotes the mean earth’s density and ρ is given by ρ = ρo

for H ≥ 0,

ρ = ρo − ρw

for H < 0,

(13)

where ρo denotes the density of the topography and ρw is the density of ocean’s water. The density contrast ρ is given by

(9)

ρ = ρ1 − ρo , where NG M Adapt gives the contribution of the adapted reference field. It should be noted that the contribution where ρ1 is the density of the upper mantle.

(14)

An Attempt Towards an Optimum Combination

207

8 Gravity Reduction

9 Geoid Computation

The following parameter set has been used during the gravity reduction and the geoid determination as they empirically proved to fit the Austrian gravity field to a good extent (K¨uhtreiber and Abd-Elmotaal, 2001):

The contribution of the gravity anomalies on the geoid undulation N g has been computed by 1D-FFT technique using the FFTGEOID program by Sideris and Li (1993). Three methods are used in the current investigation to compute a gravimetric geoid for Austria. They are:

To = 30 km,

(15)

ρo = 2.67 g/cm ,

(16)

ρ = 0.20 g/cm .

(17)

3 3

The EGM96 geopotential model has been used for the traditional remove-restore technique. An adapted reference field has been created by subtracting the potential coefficients of the topographic-isostatic masses of the data window (40◦ N ≤ φ ≤ 52◦ N; 5◦ E ≤ λ ≤ 22◦ E) computed by Eq. (11) from the EGM96 coefficients. This adapted reference field has been used for the window remove-restore technique. Table 1 shows the statistics of the gravity reduction after each reduction step for the traditional and window remove-restore techniques. It should be noted that the reduced anomalies for Stokes’ integral with modified Stokes’ kernel are the same as those for the Stokes’ kernel with un-modified Stokes’ integral (the upper part of Table 1). Table 1 shows that using the window technique gives the best reduced gravity anomalies. The range has dropped by its one-third and the standard deviation drops by about 20%. Also the reduced anomalies are perfectly centered (un-biased). This property makes the window-technique reduced anomalies best suited for interpolation and other geodetic purposes.

r r r

Stokes’ integral using classical un-modified Stokes’ kernel (classical Stokes’ geoid), Stokes’ integral using Meissl’s modified Stokes’ kernel (Stokes/Meissl geoid), Stokes’ integral using window technique (Window geoid).

All computed geoids are compared to the GPS/levelling geoid. Figure 6 shows the absolute geoid differences between the classical Stokes’ geoid and the GPS/levelling geoid. Figure 6 shows a high-order polynomial structure of the differences. The range of the differences is quite large (about 2 m). As mentioned earlier, the optimum cap size ψo can be empirically determined. This is achieved by comparing the computed Stokes/Meissl geoid to the GPS/levelling geoid. Table 2 shows the statistics 4.2

49

4 3.8 3.6

48

3.4 3.2 3 2.8

47

2.6 2.4 2.2

10

11

12

13

14

15

16

17

Fig. 6 Absolute geoid differences between classical Stokes’ geoid and GPS/levelling geoid. Contour interval: 5 cm

Table 1 Statistics of the reduced gravity after each reduction step

Statistical parameters Min.

Max.

Average

St. dev.

Reduced gravity

mgal

mgal

mgal

mgal

g F

g F

g F

g F

g F

−154.16 −210.72 −123.66 −194.55 −62.39

187.15 132.27 81.97 204.99 71.60

9.70 −12.91 −20.09 −1.46 0.23

− gG M − gG M − gT I − gG M Adapt − gG M Adapt − gT I W in

42.16 37.60 25.88 44.43 20.32

¨ H. A. Abd-Elmotaal and N. Kuhtreiber

208 Table 2 Statistics of the empirical tests for the cap size ψo for the Stokes/Meissl geoid

49

0.2

Differences to GPS/levelling geoid

Cap size ψo ψo ψo ψo ψo

= 0.5◦ = 1.0◦ = 1.5◦ = 1.7◦ = 2.0◦

0

Min.

Max.

Average

St. dev.

m

m

m

m

−4.57 −2.87 −1.74 −1.43 −1.01

−1.31 −1.21 −1.04 −0.85 −0.36

−2.87 −2.06 −1.32 −1.07 −0.72

0.85 0.44 0.16 0.11 0.15

48

–0.2 –0.4

47

–0.6 –0.8

10

11

12

13

14

15

16

17

Fig. 8 Absolute geoid differences between window geoid and GPS/levelling geoid. Contour interval: 5 cm

of the empirical tests for the cap size ψo for the Stokes/Meissl geoid. It shows that ψo = 1.7◦ gives the optimum cap size in view of the standard deviation of the absolute differences to the GPS/levelling geoid. Figure 7 shows the absolute geoid differences between the Stokes/Meissl geoid (cap size ψo = 1.7◦ ) and the GPS/levelling. Figure 7 shows better polynomial structure of the differences than that in the case of Stokes’ geoid. The range of the differences drops to about 58 cm. Figure 8 shows the absolute geoid differences between the window and the GPS/levelling geoids. Figure 8 shows better polynomial structure of the differences than that in the case of Stokes’ geoid. The range of the differences drops to about 1 m. Table 3 illustrates the statistics of the absolute geoid differences between the computed geoids within the current investigation and the GPS/levelling geoid. Table 3 shows that the Stokes’ geoid has the worst differences to the GPS/levelling. This confirms what has been stated earlier that using the classical un-modified Stokes’ kernel in the remove-restore technique implies a wrong combination of gravity field

–0.9

49

–1 –1.1

48

–1.2 –1.3

47 –1.4 –1.5

10

11

12

13

14

15

16

17

Fig. 7 Absolute geoid differences between Stokes/Meissl geoid (cap size ψo = 1.7◦ ) and GPS/levelling geoid. Contour interval: 5 cm

wavelengths. Table 3 shows also that either using the window technique or the modified Stokes’ kernel gives better differences to the GPS/levelling geoid (in terms of either the mean difference or the range/standard deviation). Table 3 Statistics of the absolute geoid differences between the computed geoids and the GPS/levelling geoid Statistical parameters Geoid technique Stokes Stokes/Meissl Window

Min.

Max.

Average

St. dev.

m

m

m

m

2.32 −1.43 −0.79

4.29 −0.85 0.27

3.47 −1.07 −0.32

0.49 0.11 0.28

10 Conclusions Stokes’ technique, within the remove-restore scheme, with un-modified Stokes’ kernel cannot correctly handle the combination of the geoid wavelengths. A modification of the kernel or a new technique should be introduced. Both the modified Stokes’ kernel and the window technique can handle the combination of the geoid wavelengths within the remove-restore scheme. The modified Stokes’ kernel technique gives the best fit to the GPS/levelling derived geoid. The reduced gravity anomalies using the window technique are the smoothest (20% less in the standard deviation), unbiased and have the smallest range (one-third less). This property makes the window-technique reduced anomalies suite best for interpolation and other geodetic purposes.

An Attempt Towards an Optimum Combination

References Abd-Elmotaal, H., and K¨uhtreiber, N. (2003) Geoid Determination Using Adapted Reference Field, Seismic Moho Depths and Variable Density Contrast, Journal of Geodesy, 77, 77–85. Featherstone, W. (1999) A Comparison of Gravimetric Geoid Models over Western Australia, Computed using Modified Forms of Stokes Integral, Journal of the Royal Society of Western Australia, 82, 137–145. Featherstone, W. (2003) Software for Computing Five Existing Types of Deterministically Modified Integration Kernel for Gravimetric Geoid Determination, Computers & Geosciences, 29, 183–193. Gesch, D.B., and Larson, K.S. (1996) Techniques for Development of Global 1-Kilometer Digital Elevation Models, In: Pecora Thirteen, Human Interactions with the Environment - Perspectives from Space, Sioux Falls, South Dakota, August 20–22, 1996. Hanafy, M. (1987) Gravity Field Data Reduction Using Height, Density, and Moho Information, Ph.D. Dissertation, Institute of Theoretical Geodesy, Graz University of Technology, Graz. Heiskanen, W.A., and Moritz, H. (1967) Physical Geodesy. Freeman, San Francisco. Huang, J., Van´ıcˇ ek, P., and Nov´ak, P. (2000) An Alternative Algorithm to FFT for the Numerical Evaluation of Stokes’s Integral, Studia Geophysica et Geodaetica, 44, 374–380. K¨uhtreiber, N., and Abd-Elmotaal, H. (2001) Gravimetric Geoid Computation for Austria Using Seismic Moho Data, In Sideris, M., edt. (2001) Gravity, Geoid and Geodynamics 2000, International Association of Geodesy Symposia, Banff, Canada, July 31 – August 4, 2000, 123, 311–316.

209 Meissl, P. (1971) A Study of Covariance Functions Related to the Earth’s Disturbing Potential, Department of Geodetic Science, The Ohio State University, Columbus, Ohio, 151. Nov´ak, P., Van´ıcˇ ek, P., V´eronneau, M., Holmes, S., and Featherstone, W. (2001) On the Accuracy of the Modified Stokes’s Integration in High-Frequency Gravimetric Geoid Determination, Journal of Geodesy, 74, 644–654. Sideris, M.G. and Li, Y.C. (1993) Gravity Field Convolutions Without Windowing and Edge-Effects, Bulletin Geodesique, 67, 107–118. Silva, M.A., Blitzkow, D., and Lobianco, M.C.B. (2002) A Case Study of Different Modified Kernel Applications in QuasiGeoid Determination, Proceedings of the 3rd Meeting of the International Gravity and Geoid Commission, Thessaloniki, Greece, August 26–30, 2002, 138–143. Sj¨oberg, L.E. (2003a) A Computational Scheme to Model the Geoid by the Modified Stokes’ Formula Without Gravity Reductions, Journal of Geodesy, 77, 423–432. Sj¨oberg, L.E. (2003b) A General Model for Modifying Stokes Formula and its Least-Squares Solution, Journal of Geodesy, 77, 459–464. Sj¨oberg, L.E. (2004) A Spherical Harmonic Representation of the Ellipsoidal Correction to the Modified Stokes’ Formula, Journal of Geodesy, 78, 180–186. Sj¨oberg, L.E., and Hunegnaw, A. (2000) Some Modifications of Stokes’ Formula that Account for Truncation and Potential Coefficient Errors, Journal of Geodesy, 74, 232–238. Van´ıcˇ ek, P., and Featherstone, W. (1998) Performance of Three Types of Stokes’s Kernel in the Combined Solution for the Geoid, Journal of Geodesy, 72, 684–697.

BVP, Global Models and Residual Terrain Correction M. Elhabiby, D. Sampietro, F. Sanso` and M.G. Sideris

Abstract The integral of Stokes or similar formulae, used to estimate global models of the gravity fields, are based on the solution of the external boundary value problem (BVP) for the disturbing potential T . This approach presupposes the disturbing potential T to be harmonic on the geoid, which implies that there are no masses outside the geoid (Van´ıcˇ ek and Krakiwsky, Geodesy: the concepts, 1986; Bajracharya, Terrain effects on geoid determination, 2003). When solving this problem with the remove-restore procedure, we have to compute the long wavelengths provided by a set of spherical harmonic coefficients (global model) and the short wavelengths calculated using topographic heights and residual terrain correction (RTC) (Forsberg, International School for the Determination and Use of the Geoid, 2005). When performing this operation two inconsistencies arises: first the reference surface used is not a continuos surface, but a set of horizontal planes; second this set of surface is different from the one implied in the global model. This paper suggests a new procedure to better harmonize the various steps involved in solving the geodetic boundary value problem. First, the difference between using a block or a moving average operator as M. Elhabiby University of Calgary, 2500 University Drive NW Calgary, Alberta, Canada D. Sampietro Politecnico di Milano, Polo regionale di Como, via Valleggio 11, 22100 Como, Italy, e-mail: [email protected] F. Sans`o Politecnico di Milano, Polo regionale di Como, via Valleggio 11, 22100 Como, Italy M.G. Sideris University of Calgary, 2500 University Drive NW Calgary, Alberta, Canada

reference surface for smoothing g is evaluated. Second, we shall look more closely to the procedure of reducing data to the reference surface. Particularly a new sequence that allows moving data always in free air is proposed. Numerical tests proved that there are significant differences between choosing the reference surface either computed from block or moving average operator. The residual field estimated using the second approach seems to be smoother than the one obtained with the first one. Moreover, a possible influence on the procedure of computing global models, due to the new method described in the paper, has been proved. Keywords Boundary value problem · Remove-restore procedure · Terrain correction · Moving average operator

1 Motivation Global models of the gravity field are usually employed in the computation of high resolution-high accuracy local geoids in a remove-restore procedure. For the purpose of this work we consider a simplified problem and we do not deal with the effect of atmospheric masses, even if it is known that this plays an important role (Sj¨oberg, 1999). Typically free air gravity anomalies g are first reduced by a global model component g M (the long wavelength component) and then by a so called residual terrain correction term g RT C (the short wavelength component), thus arriving at a smother residual field:

gr = g − g M − g RT C

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

(1) 211

212

which is further transformed into a residual anomalous potential either by integration or by collocation. At the end, the contributions subtracted in terms of g, are added back in terms of T . The method is justified by a large number of successful and tested solutions and research done in the geodetic community (for example Barzaghi et al., 1995; Sideris, 1995; Bl`azquez et al., 2003). Nevertheless, a certain inconsistency arises from the fact that, when g RT C is computed, a different reference (smooth) surface is used rather than the BVP solution surface leading ultimately to the estimation of the global model. It is true that when dealing with the processing of the residual term gr → Tr we can invoke the downward continuation idea, supported by the Runge-Krarup theorem (Borre, 2006). Such an approximation can approach the correct solution in a faster or slower way and certainly the presence of an irregular body (due to the differences between the two reference surfaces), which is ±2.67 gcm−3 density hanging in the air, should not constitute a favorable condition for the proposed solution. We remark as well that the inconsistence of this approach ultimately derives from the fact that, while g M (and every other quantities related to TM ) is separeted from the rest in terms of frequency analysis with respect to a spherical harmonic expansion, g RT C is in fact described geometry-wise by a Newton integral. In this paper, we propose to better harmonize the various steps by a procedure in which: 1. data are reduced by a residual terrain correction defined between the actual Earth surface S and a suitably smoothed surface Ss ; 2. then observed values are moved to the surface Ss ; 3. and finally the block averages of g, that we shall denote gs , are formed directly on Ss with the same resolution used to produce Ss itself. The combination of Eqs. (1) and (2) is proposed with a more convenient sequence, so that data are moved in free air only. It is proposed to estimate the global model as an approximate solution of the B.V.P. with Ss as a boundary and gs as the boundary data. In this way, any further local computation can be performed knowing explicitly the body that is not accounted for by the global model, namely the layer L(S, Ss ) between the true surface known to the local modeller, and the smooth surface

M. Elhabiby et al.

Ss known to all community as it has to be produced together with the global model. We shall also look more closely to the question of whether block averages of gs (currently used) or moving average values, suitably under-sampled, should be used, because of their smooth pattern.

2 Smoothing g The smoothing of g to construct global models is performed by computing the RTC with respect to a horizontal plane at height equal to the average height of measurement points entering into a block of assigned size (Nowell, 1999). In other words, the attraction due to a layer with top surface equal to S = {h(Q)} (cfr. Fig. 1) and a smoothed bottom surface S¯ = {h¯ Ai j } formed by the upper tops of prisms with base Ai j and height equal to the mean value of h(Q)∀Q ∈ Ai j is evaluated. One of the objectives of this research study to investigate the effectiveness and the differences between using block (h¯ Ai j ) and moving average (h˜ Ai j ) applied to h. The difference between the two concepts is highlighted in Fig. 1. For the sake of efficiency, we introduce the following notation: we call B(·) the “classical” Bouguer operator B(h(P)) = kh(P) − q(h(P))  1 [h(P) − h(Q)]2 q(h(P)) = G ρ d S0 (Q) 2 r P30 Q 0 S0

(2) (3)

wher e k = 2π G ρ = 0.1119 mGal · m−1 .

Fig. 1 Differences between observed heights, block average and moving average

BVP, Global Models

213

Fig. 2 Geometry and notation used

B(·) is considered as a non linear operator acting on the “height function” h(P). Some symbols in formulae Eqs. (2) and (3) are clarified in Fig. 2 where it can be seen that S0 is the zero level reference surface approximated, as is customary, by a plane; P, Q are points on S and P0 , Q 0 are their projections on S0 . It can be proved that when we subtract the Bouguer TC of the actual surface S with that of the reference surface Ss the part out of the integral terms cancels and the integral terms reproduce, with a good approximation, the RTC computed starting from a spherical surface. What we obtain is that with the operator B(·) we can form also the “residual” terrain correction, g RT C , i.e., the ¯ only by: attraction due to the layer L(S, S)

g RT C = B(h P ) − B(h¯ P ).

(4)

Such a formula is correct if we accept the approximation involved in Eqs. (2) and (3). Even in the case of having one these approximations, namely to consider S0 as a plane, which has a larger impact on Eq. (2), will have an attenuated effect in Eq. (4). More refined formulae can indeed be used, but this one is certainly sufficient for the purpose of this paper (e.g. Biagi and Sans`o, 2001; Forsberg 1985). We are in fact interested in comparing Eq. (4) with a residual terrain correction ˜

g˜ RT C , where the residual body is the layer L(S, S), i.e.:



g˜ RT C = B(h P ) − B h˜ P .

(5)

More precisely, we are interested in block averages of differences of the two quantities, i.e.:

g¯ T A = M A ( g RT C − g˜ RT C );

(6)

with g¯ RT C = B(h P ) − B(h¯ P ) and M A (·) = 1 ¯ T A will show the significance |A| A (·)d A. In fact g of the differences when the corrected free anomalies ¯ Using Eqs. (2), (4) are averaged with respect to S˜ or S. and (5) we find:  

g¯ T A = M A B h˜ P − B(h¯ P ) =     k M A h˜ P − h¯ A − M A q h˜ P − q(h¯ P ) . (7) Let us consider the term 



1 q h˜ P = Gρ 2



 S0

h˜ P − h˜ Q r P30 Q 0

2 d S0

(8)

and put in this formula h˜ P = h¯ P + δ h˜ P ; we obtain then:

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M. Elhabiby et al.

  q h˜ P = q δ h˜ P + q(h¯ P )   δ h˜ P − δ h˜ Q (h¯ P − h¯ Q ) + Gρ d S0 . r P30 Q 0 S0

(9)

If we closely examine the third term in the R.H.S. of Eq. (9), it can be seen immediately that the block A, to which P belongs, does not contribute to the integral because h¯ Q = h¯ P when both P, Q ∈ A, while the rest of ˜ h¯ are the integral has a fairly small value because δ h, already smoothed surfaces and r −3 is a weight dropping quickly to zero. Some numerical experiment can Fig. 3 Covariance functions and notation used confirm such an approximation even in rough mountainous areas (the contribution of this term is less than √ 0.01 mGal). So we can put in Eq. (7): seems to be smoother than the other one ( C(0) is 10−5 L     ¯ Our conclusion is that the for S˜ and 1.6 · 10−5 for S).

g¯ T A = k M A h˜ P − h¯ A − M A q δ h˜ P . difference of the two procedures is generally not very (10) large (less than 1 mGal), in flat area case. However, in The formula has been tried on different areas of in- areas of most varied topography, like Alps, Himalaya, creasing roughness (some statistics on the DTMs used Andes, Rocky Mountains, etc. the difference should be are shown in Table (1)), averaging the DTM, which has taken into account (see Table 2). In particular, results a resolution of 100×100 m, over blocks of 10 ×10 km. obtained using S˜ seems to be better than that obtained The results are shown in Table (2). Other tests with with S¯ (see Fig. 3). flatter areas have showed us that in such areas we can REMARK: In computing our examples of RTC, we put g¯ T A = 0 and neglect the problem. Moreover, we asked ourselves whether the computation of terms like have computed, for the Mt. Blanc area, gs using as M A (h P ) for sufficiently large blocks could be substireference surface both S¯ and S˜ and we have estimated tuted by a statistical reasoning, where h P could be an empirical index of smoothness obtained from the considered as a random field, intrinsically stationary covariance √ function of gs (the index has been com- (Wackernagel 2003) with a variogram: puted as C(0) L , where C(0) is the covariance function for distance equal 0 and L is the semicorrelation length. 1  γ (r P0 Q 0 ) = E (h P − h Q )2 . (11) The notation used is shown in Fig. 3). The result is that 2 ˜ the residual field obtained using as reference surface S Table 1 Statistics on the DTMs used for numerical tests DTM zone

Mean [m]

RMS [m]

Como area Mt. Etna Mt. Blanc Rocky Mountains

973 1873 730 1623

±780 ±497 ±914 ±470

Table 2 Mean value of the difference g˜ RT C − g¯ RT C computed using Eq. (10) in different areas of increasing roughness DTM zone Como area Mt. Etna Mt. Blanc Rocky Mountains

g˜ RT C − g¯ RT C [mGal] 0.67 −1.36 3.34 4.07

Substituting this in Eq. (3) and identifying M A (·) with E(·), we get then:  E(q(h P )) = G ρ

S0

γ (r P0 Q 0 ) d S0 (Q). r3

(12)

Accordingly, the g RT C has been computed by substituting q(h P ) with E(q(h P )) given by Eq. (12). This formula has been implemented in our examples producing the results presented in Table (3). So the significance of the approximated formula Eq. (12) is proved to be at least as effective as the use of the block averages, in addition to the reduction of the computational effort.

BVP, Global Models

215

Table 3 Difference between the quadratic term computed by variogram and using M A (q(h¯ A )) DTM zone Variogram −q(h¯ A ) [mGal] Como area Mt. Etna Mt. Blanc Rocky Mountains

−3.41 0.27 −6.85 1.27

P+

* h0

P*h-

h+

3 Reducing Data to Ss We want to look more closely now at the process of __ forming block averages and at its interrelation with the application of the RTC. We shall not further discuss here the alternative of Fig. 4 Geometry and notation used ¯ but we shall rather consider reducing data to S˜ or to S, the practice of taking directly the average of g(P) upward derivative  ∂ gs (h¯ A ) is significantly dif∂h −  ∂ values over some block A, after the RTC has been ap gs (h¯ A ) + , ferent from the downward derivative, ∂h plied. In practice the following operation is performed: applied to points P+ , lying above the top of the prism. ⎧ Therefore, the idea that the linear term has zero mean ⎪ ⎨ gs (P) = g(P) − g RT C (P) might not be very well verified. (13) M A ( gs (P)) ∼ = gs ( P¯ A ) The remedy that we propose for such a problem is ⎪ ⎩¯ to always move data in free air. This can be achieved by PA = center o f A applying the RTC according to the following scheme: where the index s stands for “smoothed”. 1. move in free air, possibly with the aid of a prior Although Eq. (13) can give an approximate picture global model g M , g from all measure points P− , of reality, having a deep look two types of problems where h − < h¯ A , to P0− lying on the reference surcan be found. In fact let us consider Eq. (13) as the face: result of the application of the block average M A (·) to the relation:

g(P0− ) ∼ = g(P− )+ ∂ g M (P− ) ¯

gs (P0 , h P ) = gs (P0 , h¯ A )+ (15) (h A − h − ); ∂h ∂ (14) (h P − h¯ A ) gs (P0 , h¯ A )+ ∂h 2. now all the data points are either P+ (with h + > 2 1 ∂ 2 h¯ A ) or P0− (with h 0− = h¯ A ), so we can safely ap gs (P0 , h¯ A ) + . . . (h P − h¯ A ) 2 ∂h 2 ply RT C formulae i.e.:  Then Eq. (13) is derived under the assumption that (h P − h Q ) ∂ ¯ RT C (P) = G d S−

g (P , h ) is basically constant because

g is ρ s 0 A s ∂h r P3 Q S smooth and because it is computed at a constant al (16) (h P − h Q ) ¯ titude. This yields M A (h P − h¯ A ) = 0, i.e. the elimGρ dS r P3 Q ination of the linear term. If furthermore we believe S¯ that the quadratic term is much smaller, we finally come to Eq. (13). These assumptions, however, could without drowning any measure point into masses, be questioned (geometry and symbols are shown in 3. finally considering that in P+ we have: gs (P+ ) = ¯ is removed one Fig. 4) because once the layer L(S, S)

g(P+ ) − g RT C (P+ ) and in P0− we indeed alpart of the points, call them P− , with heights h P− < ready have gs (P0− ) = g(P0− ) − g RT C (P0− ), ¯ havwe can see that all what is required is to move (in h¯ A are embedded into a body with boundary S, free air) gs (P+ ) and take it down by the formula: ing the usual density of 2.67 gcm−3 . Accordingly the

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M. Elhabiby et al.

gs (P0+ ) ∼ = gs (P+ )− ∂ g M (P+ ) (h + − h¯ A ). ∂h

(17)

Consequently, we have a number of shifted observations, all lying on the smooth reference surface S¯ with block height h¯ A . In this way moving the average of such quantities does not mix with the gravitational signal due to internal masses, the gravitational signal of ¯ and the signal due to the fact that the the layer L(S, S) measurement points lie on a rugged surface like S. When the restore procedure has to be applied (the residual corrections are in terms of T ) the steps (1)–(3) have to be reverted; namely first points P0+ have to be moved to P+ , then the RTC has to be applied and finally P0− has to be move back to P− . One objection to this proposal could be that when we move up P− we have not yet smoothed g; however, this is compensated by the fact that the upward continuation operator is already a self smoother, contrary to the downward continuation operator. Only after these steps have been performed, we can form block averages of all the values gs (P0k− , h¯ A ), gs (P0k+ , h¯ A ) and consider this average as approximately equal to gs ( P¯ A ). Differences between smoothing g using Eqs. (13), (14), (15) and (16) have been computed on the mt. Blanc area. Results are shown in Fig. 5 and are of the order of few mGals. REMARK: when we are at the point described ˜ and above, we already have a smooth surface S¯ (or S)

   a gs P¯ (or gs P˜ ) defined on it. We have the impression that any attempt of restoring at this point an “average topographic” effect can only worsen the smoothness of gs . In other words, we think that the global model should be computed using this gs and S¯ by one of the approximation techniques for the solution of the geodetic B.V.P., with the finite basis of ellipsoidal harmonics up to some maximum degree. We believe that if S¯ was made available to the users, ¯ then they could always recover the effect of L(S, S) and therefore, remove it from the data, and only after this, subtract the long wavelength component implied by the global model, which has been consistently ¯ estimated excluding the same effect of L(S, S).

4 Conclusion In this paper, a number of minor problems arising in the computation of R.T.C. have been studied, some solutions proposed and the difference with respect to actual common practice has been evaluated. In particular, it has been shown that the use of a moving aver˜ instead of a block-averaged age topographic surface S, one, gives differences in g RT C of the order of a few milligals in mountainous areas. In the first case the residual field seems to be smoother than in the second one, as shown by the different values of the index √ C0 L . The same difference is found when we exploit an approximative statistical formula. Moreover, a procedure to compute g RT C on a smooth reference surface by moving observation points only in free-air has been proposed and a possible influence on the procedure of computing global models has been discussed.

References

Fig. 5 Difference between the two methods to smooth g

S. Bajracharya. Terrain effects on geoid determination. MSc. Thesis, Department of Geomatics Engineering, Calgary, 2003. R. Barzaghi, M.A. Brovelli, G. Sona, A. Manzino, and D. Sguerso. The new italian quasigeoid: Italgeo95. Bulletin ` dInformation no 77, IGeS Bulletin no 4, pages 137–152, 1995. L. Biagi and F. Sans`o. Telight: a new technique for fast rtc computation. IAG Symposia, 123:61–66, 2001.

BVP, Global Models E.B. Bl`azquez, A.J. Gil, G. Rodr`ıguez-Caderot, M.C. De Lacy, and J.J. Ruiz. Andalusgeoid2002: the new gravimetric geoid model of andalusia (southern spain). Stud. Geophys. Geod., 47:511–520, 2003. K. Borre. Mathematical foundation of geodesy, selected paper of Torben Krarup. Springer, Berlin, 2006. R. Forsberg. Gravity field terrain effect computation by fft. Bull. Geod., 59:342–360, 1985. R. Forsberg. Terrain effects in geoid computations. International School for the Determination and Use of the Geoid, Lecture Notes:V1-V29, 2005.

217 D.A.G. Nowell. Gravity terrain correction - an overview. J. Applied Geod. 42:117–134, 1999. M.G. Sideris. Fft geoid computations in canada. Bulletin ´ dInformation no 77, IGeS Bulletin no 4, pages 37–52, 1995. L.E. Sj¨oberg. The iag approach to the atmospheric geoid correction in stokes formula and a new strategy. J. Geod., 73:362–366, 1999. P. Van´ıcˇ ek and E. Krakiwsky. Geodesy: the concepts. Elsevier science B. V., Amsterdam, 1986. Hans Wackernagel. Multivariate geostatistics: an introduction with applications. Springer, Berlin, 2003.

Domain Transformation, Boundary Problems and Optimization Concepts in the Combination of Terrestrial and Satellite Gravity Field Data P. Holota and O. Nesvadba

Abstract The purpose of this paper is to further develop a method of combining terrestrial and satellite gravity field data, which in an optimal way exploits gravity field information contained in the solution of boundary problems in a close neighborhood of the Earth. The method is considerably generalized in this paper. In particular effects caused by the topography of the Earth are taken into account. The starting point is a transformation of the boundary problems considered under a small modification of curvilinear coordinates. In the new coordinates the boundary and also the boundary condition is simpler, but topography dependent coefficients appear in the structure of Laplace’s operator. Effects caused by the topography of the Earth are treated as perturbations and refinements of the solution are expressed as corrections constructed by means of successive approximations. As a consequence of the transformation a spherical mathematical apparatus may be applied at each iteration step. Especially Green’s function is constructed for the problem solved in a domain bounded by two concentric spheres. The structure of an iteration step connects the paper with earlier results. Within the iteration step they provide an applicable treatment. The continuation of the solution is discussed with a particular view to its harmonic branch and regularity at infinity. The reasoning leads to optimization concepts considered in the paper.

P. Holota Research Institute of Geodesy, Topography and Cartography, 250 66 Zdiby 98, Praha-v´ychod, Czech Republic, e-mail: [email protected] O. Nesvadba Land Survey Office, Pod S´ıdliˇstˇem 9, 182 11 Praha 8, Czech Republic, e-mail: [email protected]

Keywords Modelling of the Earth’s gravity field · Geodetic boundary-value problems · Overdetermined problems · Terrestrial and satellite data · Optimization

1 Introduction The combination of terrestrial and satellite data is one of the challenges in gravity field modelling today. Data coming or expected from satellite missions (like e.g. GOCE) are often treated within the so-called spacewise approach, see Migliaccio et al. (2004) and, as a rule, typical combination schemes result in overdetermined problems, see also Sacerdote and Sans`o (1985). This paper ties to Holota (1995, 2007), Holota and Kern (2005) and Holota and Nesvadba (2006, 2007), where in the first step the combination problem has been solved in a domain bounded by two spheres and then, in an optimal way, the solution was extended harmonically to an unbounded domain. This means some simplification. Therefore, our aim is to generalize the approach. The fact that in reality the lower boundary of the solution domain essentially differs from a sphere is taken into consideration. Several combination schemes are considered in the contributions above. In Holota and Nesvadba (2006, 2007) in particular: (1) the combination of terrestrial gravity measurements with a satellite-only Earth’s gravity field model (EGM) and (2) with data coming from satellite missions (like e.g. GOCE within the space-wise approach) is discussed. In this paper we will confine ourselves to the first problem only in order to show the generalization of the approach in more detail. Nevertheless the methodology can be applied

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

219

220

P. Holota and O. Nesvadba

also to other combination schemes, possibly with some modifications. In the sequel ΩTOPO means a domain bounded by two surfaces, from above by a geocentric sphere of radius Re and from below by a surface ΓTOPO , which is star-shaped at the origin of a system of rectangular Cartesian coordinates xi , i = 1, 2, 3 and about as irregular as the surface of the Earth. We assume the origin is at the center of gravity of the Earth and that the x3 - axis coincides with the rotation axis of the Earth. Finally, let h be the height of the boundary point x ∈ ΓTOPO above a geocentric sphere of radius Ri . (Obviously, we suppose that h < Re − Ri for all x ∈ ΓTOPO .)

for any values of the variables considered, ω(Ri ) = 1,

dω 1 (Ri ) = dr Ri

(9)

and ω(r ) = 0

for r ∈ [Re − ε, Re ]

(10)

where ε is a positive number, but essentially smaller than Re − Ri . Moreover, supposing that ∂|x| dω =1+ h>0 ∂r dr

(11)

2 Gravimetry and a Satellite-Only Model

we can see that our transformation is a one-to-one mapping between ΩTOPO and a domain Ω, Ω ≡ {x; x ∈ In this section we will look for T such that R3 , Ri < |x| < Re }, i.e. a layer bounded by two spheres of radius Rl and Re . Clearly, the transformation (1) carries the boundary ΓTOPO described by the placement ΔT = 0 in ΩTOPO vector of length Ri + h into a sphere of radius Ri in the ∂T |x| + 2T = −|x|Δg for x ∈ ΓTOPO (2) space of curvilinear coordinates r, ϕ, λ. ∂|x| The transformation Eqs. (4), (5) and (6) is a modiT = t for |x| = Re (3) fied version of a map applied in studies on Molodensky’s problem, i.e. in an unbounded domain in contrast Here Δ is Laplace’s operator, Δg is the usual grav- to our case, see Holota (1985, 1989, 1991). Nevertheity anomaly and t means the input (a disturbing part) less, it acts identically if applied to the boundary conobtained from an available satellite-only Note dition given by Eq. (2). Thus, considering Eqs. (7), (9) 3 EGM. that x ≡ (x 1 , x2 , x3 ) and that |x| = ( i=1 xi2 )1/2 . and (11), we easily deduce that condition Eq. (2) turns Our starting point is a transformation (mapping) of into the domain ΩTOPO ⊂ R3 given by   h −1 ∂ T +2T = −(Ri +h)Δg (12) (4) (Ri +h) 1 + x1 = [r + ω(r )h(ϕ, λ)] cos ϕ cos λ Ri ∂r x2 = [r + ω(r )h(ϕ, λ)] cos ϕ sin λ (5) x3 = [r + ω(r )h(ϕ, λ)] sin ϕ

(6) for r = Ri , which subsequently yields

Here R3 means 3-dimensional Euclidean space. The curvilinear coordinates ϕ (geocentric latitude), λ (geocentric longitude) are the usual spherical coordinates of the point x ∈ ΩTOPO and r is defined by with |x| = r + ω(r )h(ϕ, λ)

(7)

2 ∂T T = f + ∂r Ri



for r = Ri

h f =− 1+ Ri

(13)

 Δg

(14)

where ω(r ) is a suitably chosen and twice differentiable auxiliary function of r ∈ [Ri , Re ], such that Note. We can immediately see that in condition Eq. (13) the left hand side has a simple Stokesian ω(r )h(ϕ, λ) > −r (8) structure.

Domain Transformation, Boundary Problems and Optimization Concepts

minant g = |gi j |, see Sokolnikoff (1971). Hence, after some algebra, we obtain

3 Laplacian A somewhat more complicated problem is to express Laplace’s operator of T in terms of the curvilinear coordinates r, ϕ, λ, which do not form an orthogonal system. We will use tensor calculus. Putting (for purely technical reasons) y1 = r , y2 = ϕ, y3 = λ we easily deduce that the Jacobian      ∂ xi  dω   D= h (r + ωh)2 cos ϕ = 1+ ∂yj  dr

gi j ( y) =

∂ xk ∂ xk · ∂ yi ∂ y j

g12 g13 g22 g23 g33



 ∂T ∂yj √ 2 ∂ T 1 ∂ g · gi j ∂ T = gi j +√ ∂ yi ∂ y j g ∂ yi ∂yj √

−2

[Δ S T − δ(T, h, ω)]

(24)

where 2 ∂T 1 ∂2T ∂2T + 2 + 2 2 r ∂r ∂r r ∂ϕ sin ϕ ∂ T 1 ∂2T − 2 + 2 r cos ϕ ∂ϕ r cos2 ϕ ∂λ2

∂T ∂2T + A2 2 ∂r ∂r 2 1 ∂ T 1 ∂2T + A3 + A4 r ∂r ∂ϕ r cos ϕ ∂r ∂λ

(17) (18) (19) (20) (21) (22)

g · gi j

(23)

(25)

δ(T, h, ω) = A1

(16)

so that we are in the position to express Laplace’s operator of T . In terms of the curvilinear coordinates yi it has the following general form 1 ∂ ΔT = √ g ∂ yi

h ΔT = 1 + ω r

(15)

in the coordinates yi . Going back to our original notation, we have   dω 2 = 1+ h dr   dω ∂h =− 1+ h ω dr ∂ϕ   dω ∂h = 1+ h ω dr ∂λ  2 2 2 ∂h = (r + ωh) + ω ∂ϕ ∂h ∂h = −ω2 · ∂ϕ ∂λ  2 ∂h = (r + ωh)2 cos2 ϕ + ω2 ∂λ



ΔS T =

of the transformation is positive (apart from zero values for ϕ = −π/2 and π/2) in view of Eqs. (8) and (11). Moreover, trough an elementary calculation we obtain components of the metric tensor

g11

221

(26)

and       dω −1 ω dω ω h h − + 2 Δ1 h 2 − A1 = 1 + dr dr r r r  −2 dω ω dω −2 1+ (27) h |gr ad 1 h|2 dr r 2 dr   dω −3 + 1+ h dr 

 2 h 2  ω 2 2 d ω × 1+ω + |gr ad 1 h| h r r dr 2      dω −2 dω ω A2 = 1 + h − h (28) 2 dr dr r   

dω 2  ω −2 2  ω 2 2 + − |gr ad 1 h| h − dr r r   dω −1 ω ∂h A3 = 2 1 + h (29) dr r ∂ϕ   dω −1 ω ∂h A4 = 2 1 + h (30) dr r cos ϕ ∂λ are topography-dependent coefficients with

where g i j = G i j /g is the associated (or conjugate) metric tensor and G i j is the cofactor of gi j in the deter- and

 |gr ad 1 h|2 =

∂h ∂ϕ

2 +

1 cos2 ϕ



∂h ∂λ

2 (31)

222

Δ1 h =

P. Holota and O. Nesvadba

1 ∂ cos ϕ ∂ϕ

 cos ϕ

∂h ∂ϕ

 +

1 ∂ 2h cos2 ϕ ∂λ2

(32)

Δ S T ∗ = 0 in Ω ∂T ∗ ∂ T ∗∗ 2 ∗ 2 ∗∗ T = f − T + − ∂r Ri ∂r Ri

(39) for r = Ri

(40) see Holota (1989, 1990). Recall that |gr ad 1 h|2 and ∗ ∗∗ for r = Re (41) Δ1 h are known as the first and the second Beltrami’s T = t − T differential operator on the unit sphere, respectively. Note finally that Δ S T has the well-know structure of Using the fundamental properties of Newton’s integral, Laplace’s operator for r , ϕ, λ interpreted as spherical cf. Kellogg (1953) or Courant and Hilbert (1962), for the particular solution we can take, e.g., coordinates.  g 1 T ∗∗ = − dV (42) 4π l Ω

4 Problem in Curvilinear Coordinates The solution T has to be a harmonic function in the original domain ΩTOPO , it has to meet Eq. (1). In coordinates r , ϕ, λ this means that ΔT = 0 in Ω

(33)

Ω ≡ {x(r, ϕ, λ); x ∈ R3 , Ri < r < Re }, which results in

where l 2 = r 2 +r 2p −2rr p cos ψ, provided that the volume integral converges. The quantities with and without the subscript p are referred to the computation point and the variable point of the integration, respectively. Similarly, ψ is the angle between the placement vector of the computation point and the variable point of the integration. Nevertheless there is another choice, which even has better properties. In this case T ∗∗ = −

Δ S T = δ(T, h, ω)

in Ω

(34)

1 4π

 gG(r p , ψ, r )d V

(43)

Ω

where G is Green’s function related to the problem in view of Eq. (24) and inequality, Eq. (8). Hence, the given by Eqs. (35), (36) and (37). In particular for T ∗∗ problem given by Eqs. (1), (2) and (3) and expressed given by Eq. (43) we have now in terms of the curvilinear coordinates r , ϕ, λ attains the following form ∂ T ∗∗ 2 ∗∗ T = 0 for r = Ri (44) + ∂r Ri Δ S T = g in Ω (35) and ∂T 2 T = f for r = Ri (36) + ∂r Ri (45) T ∗∗ = 0 for r = Re T = t for r = Re (37) This will be clear from the construction of G in the in the next section. where g = δ(T, h, ω) and f is given by Eq. (14). As is well-known we can split the solution T , we are looking for, into two parts, i.e.,

5 Green’s Function ∗

T =T +T

∗∗

(38)

where T ∗∗ is a particular solution of Poisson’s equation Δ S T ∗∗ = g in Ω and T ∗ has to be sought as a solution of the following problem

Following general principles in constructing Green’s functions, we start with the fundamental solution J = J (r, ψ, r p ) =

1 l

(46)

Domain Transformation, Boundary Problems and Optimization Concepts

of Laplace’s equation. For r < r p we have ∞  rn

J=

n=0

The insertion into Eqs. (52) and (53) then yields

Pn (cos ψ)

r n+1 p

¯ n(e) ¯ n(i) − (n + 2)q n H (n − 1) H n (n + 2)Ri =− Pn (cos ψ) r n+1 p rn ¯ n(e) = p Pn (cos ψ) ¯ n(i) + H q n+1 H Ren+1

(47)

where Pn is the usual Legendre polynomial of degree n. For r = Rl < r p we put

F

(i)



∂J 2J = + ∂r Ri

 r =Ri

=

∞  (n + 2)R n−1 i

r n+1 p

n=0

223

Pn (cos ψ)

q=

(48)

∞  r np n=0

r n+1

Pn (cos ψ)

∞  r np n=0

Ren+1

Pn (cos ψ)

(51) for r = Ri

for r = Re

∞  

(52)

¯ n(e) = Dn H Dn

(61)

for any individual n, where  Dn(i)

= −(n+2)

Rin r n+1 p

−

r np Ren+1

 q

n

Pn (cos ψ) (62)

and Dn(e)

=

(n − 1)r np Ren+1

+

(n + 2)Rin r n+1 p

q

n+1

Pn (cos ψ) (63)

(54) where 

¯ n(i) and H ¯ n(e) are surface spherical hamonics. and H

(e)

and

Hence, inserting into Eq. (54), we can see that

 Hn(i) (r, ϕ, λ) + Hn(e) (r, ϕ, λ)

 Ri n+1 ¯ (i) Hn (ϕ, λ) = r  n r ¯ n(e) (ϕ, λ) Hn(e) (r, ϕ, λ) = H Re

¯ n(i) = Dn H Dn

(53)

n=0

Hn(i) (r, ϕ, λ)

(i)

(50)

Because Ω is bounded H can be expressed as H (r, ϕ, λ) =

(60)

Considering the fact that in our studies q is positive and definitely greater than 0.5, we can easily deduce that Dn always differs from zero. Following Cramer’s rule, we thus immediately have that

Δ S H = 0 in Ω

H = F (e)

(59)

Dn = (n + 2)(1 + q 2n+1 ) − 3

(49)

In the next step we will look for a function H (r, ϕ, λ) such that

2 ∂H H = F (i) + ∂r Ri

Ri Re

Equations (57) and (58) form a linear system for the ¯ n(i) and H ¯ n(e) . Its determinant is unknows H

and for r = Re > r p we put F (e) = [J ]r =Re =

(58)

where

Similarly, for r p < r we have J=

(57)

(55)

H = H (r, ψ, r p ) ⎡  n+1   ∞  1 ⎣ n + 2 Ri2 n − 1 rr p n − =− Dn Ri rr p Re Re2 n=0   n n r r p +q 2n+1 (n + 2) n+1 + n+1 Pn (cos ψ) r rp (64)

(56) Consequently, we are able to define a function G = G(r p , ψ, r ) = J − H

(65)

224

P. Holota and O. Nesvadba

The function is Green’s function for the problem given 6 Iteration Process by Eqs. (35), (36), and (37) and, provided that Ri < r p < Re , we obviously have In reality, however, g depends on T and T ∗∗ is a value of an integro-differential operator K applied to T , i.e., ∂G 2 G = 0 for r = Ri (66) + ∂r Ri T ∗∗ = K T (70) Within this interpretation Eq. (38) turns into

and G=0

for r = Re

(67)

T = T∗ + KT

(71)

in view of its construction. G makes it also possible to and represents an integro-differential equation for T . express the solution of the problem given by Eqs. (35), Eq. (71) has a formal structure of Fredholm’s equation (36), and (37). The natural point of departure is the of the second kind. Our aim is to solve it by means of (slightly modified) Green’s third identity the method of successive approximations.  1 T =− G(r p , ψ, r )ΔT d V (72) T = lim Tn , Tn = T ∗ + K Tn−1 4π n Ω    ∂G(r p , ψ, r ) ∂T 1 −T d S where T0 is a starting approximation, e.g. T0 = T ∗ . G(r p , ψ, r ) − 4π ∂r ∂r The crucial point is to show that K is a contraction r =Ri    mapping, which guarantees the convergence of the ∂G(r p , ψ, r ) ∂T 1 −T d S iteration process Eq. (72) on the basis of Banach’s G(r p , ψ, r ) + 4π ∂r ∂r fixed point theorem. The contractivity of K has been r =Re (68) discussed already in Holota (1985, 1986 and 1989), though for G constructed for Molodensky’s problem. Nevertheless, the reasoning can be fully applied also Taking then into consideration Eqs. (66) and (67) and here. Therefore, we confine ourselves just to main the fact that T represents the solution of the boundary points. The application of Banach’s fixed point theoproblem in question, we easily deduce that indeed T = rem is considered in Sobolev’s weight space W2(2) (Ω) ∗ ∗∗ T + T as in Eq. (38) and that equipped by the norm   1 1 ∂G ⎡   T∗ = − f G dS − t d S (69) 4π 4π ∂r ⎣ ||u||2,2 = |u|2 + Ri2 |gr ad u|2 r =Ri r =Re Ω

Note that Green’s identity above gives just one of the possible interpretations of Eq. (38). The function T ∗ can be expressed also in other way, e.g. by means of series of spherical harmonics. We will use it later. Finally, provided that Ri < r < Re , we can see that in analogy to Eqs. (66) and (67) also ∂G/∂r p + 2G/Ri = 0 for r p = Ri and G = 0 for r p = Re in view of the symmetry of Green’s function. Clearly, this justifies Eqs. (44) and (45). Therefore, T ∗∗ , a particular solution of Poisson’s equation given by Eq. (43), does not affect the boundary values.

⎤1/2  2 ⎞ 3   2   ∂ u  ⎠ +Ri4   dV ⎦  ∂ xi ∂ y j 

(73)

i, j

The most intricated step is the estimate of the second order derivatives of K u. For this purpose a special case of the Calderon-Zykmund inequality has been used. The inequality belongs to the concept of the so-called L p estimates for Poisson’s equation (an analogue to Schauder’s theory in H¨older’s spaces), see Gilbarg and Trudinger (1983).

Domain Transformation, Boundary Problems and Optimization Concepts

Banach’s theorem has a constructive nature and the constant α such hat ||K u − K v||2,2 ≤ α||u − v||2,2

(74)

225

with Q = a1

(2)

∂T ∂2T + A2 2 ∂r ∂r    (79)  dω −1 ω ∂T + 1+ h Δ S hr dr r ∂r

for all u, v ∈ W2 (Ω) depends on the coefficients Ai , i.e., it essentially depends on the topography of the Formally, Eq. (78) is a differential equation for g, boundary of the original solution domain ΩTOPO . In but we know its solution. Nevertheless the equation particular α derived in Holota (1986) depends on esmakes it possible to express g in terms of Q. We will sential supreme values confine ourselves to a linear theory in h. In this case we (l) supess |Ai | (75) have g = Q (T, h, ω), where Ω



(l)

 dω h ∂T = r + 2r − 2ω 2 2 dr dr r ∂r     dω ω h ∂2T ∂T +2 r + −ω Δ hr S dr r ∂r 2 r ∂r (80) 2d

2ω

Q of Ai , i = 1, 2, 3, 4 and the inequality α < 1 (which is sufficient for K to be a contractive mapping) has been proved for the topography of a realistic range of heights and relatively gentle slopes. This means some restriction. Nevertheless, we share an opinion expressed in a communication by Sans`o (2007) that the restriction might be mitigated by using different function spaces, Obviously, in the linear theory Eq. (71) turns into ( p) T (l) = T ∗ + K (l) T (l) and the iteration process e.g. W2 (Ω) with p > 2. For practical applications it is convenient to give the (l) term K T a somewhat different form and to display (81) T (l) = lim Tn(l) , Tn(l) = T ∗ + K Tn−1 n its anharmonic nature (with respect to Δ S ) more distinctly. Therefore, in the representation of K T we try (l) is a linear analogue to Eq. (72). Here T0 is a starting to transform the volume integration into the surface in(l) approximation, e.g. T0 = T ∗ . tegration. In the next step the integration by parts and the Starting with δ(T, h, ω) given by Eqs. (26), (27), Green’s second identity are the main tools used in (28), (29) and (30), we immediately see that transforming the integral ∂T ∂2T g = δ(T, h, ω) = a1 (76) + A2 2 ∂r ∂r   dω −1 ω ∂ + 1+ h (T Δ1 h + 2gr ad 1 T, gr ad 1 h) dr r 2 ∂r

K (l) T = −

dω h a1 = A 1 − 1 + dr

−1

ω Δ1 h r2

K (l) T =



  dω −1 ω h h g 1+2 1+ dr r   dω −1 ∂g + 1+ ωh h =Q dr ∂r (78)

Q (l) G(r p , ψ, r ) d V

(82)

Ω

∂T 1 ωh + ∂r 4π Ri

(77)

Following now Holota (1989), we arrive at



Referring to Holota (1989) again, we obtain

where ,  means the scalar product and 

1 4π

1 + 4π

 r =Ri

 h f G dS r =Ri

∂f h G dS ∂r

(83)

with f extended for r > Ri according to 2 ∂T T = f + ∂r Ri cf. Eq. (13). It is remarkable that

(84)

226

P. Holota and O. Nesvadba

 ΔS

∂T ωh ∂r



= Q (l)

(85)

The function T˜ ∗ is harmonic in the domain Ω and obviously, it solves the following problem Δ S T˜ ∗ = 0 in Ω ∂ T˜ ∗ 2 ˜∗ $ + T = −Δg ∂r Ri T˜ ∗ = t for r = Re

can be deduced from Eqs. (82) and (83) since the integrals on the right hand side of Eq. (83) represent harmonic functions (with respect to Δ S ). Now Eq. (83) combined with T ∗ immediately yields T =

1 ∂T ωh − ∂r 4π

1 − 4π



 r =Ri

 t r =Re

h 1− Ri

∂G dS ∂r



 ∂f h G dS f − ∂r

Tn =

∂ Tn−1 1 ωh − ∂r 4π

1 + 4π





r =Ri

t r =Re

(86)

∂G dS ∂r

 ∂(Δg)n−1 h G dS Δg − ∂r

for r = Ri

(94) (95)

As regards Tn we easily deduce that

Finally, going back to our iteration process, we can see that 

(93)

Δ S Tn = Δ S T˜ ∗∗ = Q (l) (Tn−1 , h, ω)

(96)

in view of Eq. (93) and an analogue to Eq. (85) valid for the (n − 1)th iteration. Hence, Δ S T − Q (l) (T, h, ω) = 0

(97)

since (87)

Δ S T − Q (l) (T, h, ω) = lim[Δ S Tn − Q (l) (Tn−1 , h, ω)]

(98)

n

in terms of the norm  · 2,2 . Moreover, in view of Eq. (24) also ΔT = 0 within the accuracy of our linear theory. Note also that by character Eq. (92) corre∂ Tn−1 2 Tn−1 (88) sponds to the decomposition in Eq. (38). − (Δg)n−1 = − ∂r Ri Return now to Eq. (89). It is remarkable that $ resembles the “remove formally the structure of Δg step” usually applied to boundary data in the solution 7 N-th Iteration Step of geodetic boundary-value problems. Note, however, that only an effect caused by the geometry of the The structure, which represents the function Tn in boundary is taken into account here. Similarly, the effect of the term T˜ ∗∗ represents what can be considEq. (87), has an instructive value. Indeed, putting ered a “restore step” in Eq. (92). In particular for the ∂(Δg)n−1 computation point belonging to the sphere of radius $ = Δg − Δg h (89) Ri we have ω(Ri ) = 1 and T˜ ∗∗ is of structure, which  ∂r  1 ∂G 1 $ G d S (90) is rather well known. t dS + Δg T˜ ∗ = − 4π ∂r 4π with

r =Re

T˜ ∗∗

r =Ri

∂ Tn−1 = ωh ∂r

(91)

we can write that Tn = T˜ ∗ + T˜ ∗∗

(92)

$ T˜ ∗ and T˜ ∗∗ depend on Tn−1 , Remark. Clearly, Δg, but we skip the index n − 1 for simplicity reasons.

8 Optimization and Final Comments For r ∈ [Re − , Re ] we obviously have Δ S Tn = 0 in view of Eq. (10) and the smoothness of ω(r ). It is also clear in this case that Tn and T˜ ∗ represent a part of the same branch of a harmonic function. The function Tn can be also harmonically extended for r > Re . Recall that the extension is analytic and

Domain Transformation, Boundary Problems and Optimization Concepts

thus also unique. Moreover, using for r > Re an elementary continuation of the auxiliary function ω (which obviously is given by zero), we can see that Tn extended in the way as above represents a solution of Δ S Tn = Q (l) (Tn−1 , h, ω) in the whole exterior of the sphere of radius Ri . Unfortunately, the extension of Tn (apart from very exceptional and unlikely cases) is of little use, since it lacks physical meaning. The problem is that in general there is no guarantee that the analytic extension of Tn as above is regular at infinity, in other words, there is no guarantee that for r → ∞ the extension of Tn decreases as c/r (c is a constant) or faster. However, to see it we have to represent T˜ ∗ in other way, since formula (90), which resulted from the use of the second Green’s identity, does not offer a suitable apparatus for this purpose. The alternative is the representation by means of a series of spherical harmonics. Because Ω is bounded, T˜ ∗ can be expressed as T˜ ∗ =

∞ 



k=0



+

r Re

Ri r k

k+1

It is now clear from Eq. (99) that at infinity the regularity of T˜ ∗ is violated by “internal” terms ∗(e) (r/Re )k T˜k . They can be considered a consequence of a contamination of the input data by measurement errors. Clearly, the boundary values given for Ri are enough to determine a harmonic function in Ωext ≡ {x ∈ R3 ; r > Ri } and thus in Ω ⊂ Ωext , so that the data for r = Re have the nature of excess data. In addition, recalling that Tn = T˜ ∗ for r > Re − ε, we can see that the “internal” terms may violate also the regularity of Tn . Finally, following the ideas as in Holota and Kern (2005), Holota (2007) and Holota and Nesvadba (2006, 2007), we can solve the overdetermined problem above by minimizing the functional  Φ(u) =

(u − T˜ ∗ )2 d V

Ω

(99)

 (u, v) =

uv Ωext

where T˜k(i) and T˜k(e) are the respective Laplace’s surface spherical harmonics. They can be obtained for any individual k from the following system ∗(i) ∗(e) $k (k − 1)T˜k − (k + 2)q k T˜k = Ri Δg

q

k+1

∗(i) T˜k

+

∗(e) T˜k

= tk

(104)

in the space H2 (Ωext ) of harmonic functions in the domain Ωext , which is endowed with inner product

T˜k∗(i) (ϕ, λ)

∗(e) T˜k (ϕ, λ)

227

(100) (101)

1 dV r2

(105)

Note in particular that the functions from H2 (Ωext ) are regular at infinity. The functional Φ is coercive and attains its minimum in H2 (Ωext ). Thus, the function u ∈ H2 (Ωext ) that minimizes the functional Φ can be obtained from   uv d V = T˜ ∗ v d V (106)

which results from the insertion into Eqs. (94) and (95) Ω Ω and the orthogonality of spherical harmonics. Recall $ k and tk are Laplace’s sur- valid for all v ∈ H2 (Ωext ). In our case Eq. (106) reprethat q = Ri /Re , while Δg $ and sents Euler’s necessary condition for Φ to have a minface spherical harmonics in the development of Δg imum at the point u. t, respectively, i.e. in Nevertheless, our aim is not to develop the op∞ timization idea in detail. In the preceding sections $ $ k (ϕ, λ) Δg(ϕ, λ) = (102) Δg k=0 a tie was created to combination problems solved for domains of spherical boundaries. Therefore, we and rather refer to the contributions mentioned above, ∞ where the procedure is discussed at length. Now tk (ϕ, λ) (103) it may be applied on the level of an iteration step. t (ϕ, λ) = k=0 In these contributions the solution is expressed in The determinant Dk = (k + 2)(1 +q 2k+1 ) − 3 of the the spectral as well as space domain and also the system coincides with Dn in Eq. (60). Thus it always respective Green’s integral kernels are shown. The discussion is added numerical tests and simulations differs from zero.

228

P. Holota and O. Nesvadba

for data derived from the EGM96 and parameters close Holota P (1991) On the iteration solution of the geodetic boundary-value problem and some model refinements. Conto the orbit of the expected GOCE mission. tribution to Geodetic Theory and Methodology. XXth Genl. Note finally, that the function u is a compromise beAssembly of the IUGG, IAG-Sect. IV, Vienna, 1991. Potween the gravity field information coming from the litecnico di Milano, 1991, 31–60; also in: Travaux de l’AIG, ground and the satellite level. It also offers a modified Tome 29, Paris, 1992, 260–289 opt outcome Tn = u + T˜ ∗∗ of the n-the iteration step Holota P (1995) Boundary and initial value problems in airborne gravimetry. In: Proc. IAG Symp. on Airborne Gravity Field in Eq. (92). A discussion on relation to Tn and to the Determination, IUGG XXI Gen. Assembly, Boulder, Coliteration procedure considered will be amplified in the orado, USA, July 2–14, 1995, Special report No. 60010 of next paper on this topic. Acknowledgments The work on this paper was supported by the Grant Agency of the Czech Republic through Grant No. 205/06/1330. This support is gratefully acknowledged. Thanks go also to an anonymous reviewer and to Prof. F. Sans´o for valuable comments.

References Courant R and Hilbert D (1962) Methods of Mathematical Physics Volume. II: Partial Differential Equations (by Courant R), Springer Vlg., New York and London Gilbarg D and Trudinger NS (1983) Elliptic Partial Differential Equations of Second Order. Springer, Berlin etc. Holota P (1985) A new approach to iteration solutions in solving geodetic boundary value problems for real topography. Proc. 5th Int. Symp. Geod. and Phys. of the Earth, GDR, Magdeburg, Sept. 23rd–29th, 1984, Part II. Veroff. d. Zentr. Inst. f. Phys. d. Erde, Nr. 81, Teil II, 4–15 Holota P (1986) Boundary value problems in physical geodesy: Present state, boundary perturbation and the Green-Stokes representation. Proc. 1st Hotine-Marussi Symp. on Math. Geodesy, Rome, 3–5 June 1985, Vol. 2. Politecnico di Milano, 529–558 Holota P (1989) Laplacian versus topography in the solution of the Molodensky problem by means of successive approximations. In: Kejlso E, Poder K and Tscherning CC (Eds.) Festschrift to Torben Krarun Geodaetisk Inst., Meddelelse No. 58, Kobenhavn, 213–227 Holota P (1990) Christoffel symbols and the Laplacian in detailed studies of the Earth’s gravity field with emphasis on topography and eccentricity effects. In: S¨unkel H and Baker T (Eds.) Sea Surface Topography and the Geoid. IAG Symposia, Vol. 104, Springer, New York etc., 167–177

the Dept. of Geomatics Engineering at The University of Calgary, Calgary, 67–71 Holota P (2007) On the combination of terrestrial gravity data with satellite gradiometry and airborne gravimetry treated in terms of boundary-value problems. In: Tregoning P and Rizos C (Eds.) Dynamic Planet. IAG Symp., Cairns, Australia, 22–26 Aug 2005. IAG Symposia, Vol. 130, Springer, Berlin etc., Chap. 3, 362–369 Holota P and Kern M (2005) A study on two-boundary problems in airborne gravimetry and satellite gradiometry. In: Jekeli C, Bastos L and Fernandes J (Eds.) Gravity, Geoid and Space Missions. GGSM 2004, IAG Intl. Symp., Porto, Portugal, Aug 30–Sept 3, 2004. IAG Symposia, Vol. 129, Springer, Berlin etc., 173–178 Holota P and Nesvadba O (2006) Optimized solution and a numerical treatment of two-boundary problems in combining terrestrial and satellite data. Proc. 1st Intl. Symp. of the Intl. Gravity Filed Service ‘Gravity Filed of the Earth’, 28 Aug–1 Sept, 2006, Istanbul, Turkey. Spec. Issue: 18, Genl. Command of Mapping, Ankara, June 2007, 25–30 Holota P and Nesvadba O (2007) A regularized solution of boundary problems in combining terrestrial and satellite gravity field data [CD-ROM]. Proc. ‘The 3rd International GOCE User Workshop’, ESA-ESRIN Frascati, Italy, 6–8 Nov 2006 (ESA SP-627, Jan 2007), 121–126 Kellogg OD (1953) Foundations of potential theory. Dover Publications, New York Migliaccio F, Raguzzoni M and Sans`o F (2004) Space-wise approach to satellite gravity field determination in the presence of coloured noise. J. of Geod., Vol. 78, 304–313 Sacerdote F and Sans`o F (1985) Overdetermined boundaryvalue problems in physical geodesy. Manuscripta geodaetice, Vol. 10, No. 3, 195–207 Sokolnikoff IS (1971) Tensor analysis, Theory and applications to geometry and mechanics of continua. Nauka Publishers, Moscow (in Russian)

Numerical Solution of the Fixed Altimetry-Gravimetry BVP Using the Direct BEM Formulation ˇ R. Cunderl´ ık and K. Mikula

Abstract A direct formulation of the boundary element method (BEM) for the Laplace equation leads to an integral relationship that represents a superposition of the single-layer and the double-layer potential. Such boundary integral equation (BIE) can be applied to the altimetry-gravimetry boundary value problem of the third kind (AGBVP-III) that is formulated for mixed boundary conditions (BC), i.e., (i) the Dirichlet BC at oceans/seas where the disturbing potential can be evaluated on a mean sea surface (MSS) using the global geopotential models (GGMs), and (ii) the oblique derivative BC on lands in the form of surface gravity disturbances that can be obtained from terrestrial gravimetric measurements located by means of precise satellite positioning. Their 3D positions together with MSS models available from altimetry determine the Earth’s surface as a fixed boundary. In our approach the collocation technique with linear basis functions is used to discretize BIEs and transform them to a linear system of equations. The input surface gravity disturbances at collocation points on lands are simulated from GGMs. Large-scale computations performed on parallel computers result in numerical solutions directly on the complicated Earth’s surface. The surface gravity disturbances at oceans/seas and the disturbing potential on lands are the numerical results. They are compared with values evaluated directly from GGM using spherical ˇ R. Cunderl´ ık Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinsk´eho 11, 831 01 Bratislava, Slovakia, e-mail: [email protected] K. Mikula Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinsk´eho 11, 831 01 Bratislava, Slovakia, e-mail: [email protected]

harmonics. The differences indicate a discrepancy between the normal and actual gravity field. An additional constant is added to the Dirichlet BC in order to find an agreement between the gravity disturbances at oceans/seas computed by our approach and the ones induced by the prescribed Dirichlet BC. Keywords Altimetry-gravimetry boundary value problem III · Boundary element method · Method of weighted residuals · Collocation method · Piecewise linear basis functions · Parallel computing · Global gravity field modelling

1 Introduction The global gravity field modelling is usually formulated in terms of the geodetic boundary-value problems (BVPs) for the Laplace equation. During last decades, (at least) three kinds of the altimetry-gravimetry boundary-value problem (AGBVP) have been defined according to the type of input data. They have been discussed in a number of papers. Let us mention some of them that deal with the definition and well-posedness of the problem as well as with the uniqueness and existence of the solution, e.g. Mather, (1974), Holota, (1980, 1983a,b), Sans´o, (1983), Svensson, (1983), Bjerhammar and Svensson, (1983), Sacerdote and Sans´o, (1987), Lehmann, (1999) and Grebenitcharsky and Sideris, (2005). At present, the role of AGBVPs becomes more important as it is essentially related to the vertical datum problem, see e.g. Rummel and Teunissen (1988), Lehmann (2000), Sacerdote and Sans´o (2003). Thanks to satellite altimetry we know 3D position of the mean

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

229

ˇ R. Cunderl´ ık and K. Mikula

230

sea surface (MSS), where we are able to evaluate the gravity potential using global geopotential models (GGMs) and spherical harmonics. Hence, we can define the Dirichlet BC for AGBVP-III in areas of oceans and seas. In areas of lands, terrestrial gravimetry combined with precise satellite positioning yields surface gravity disturbances that represent oblique derivative BC. At the same time the Earth’s surface as a fixed boundary is given. Although a density of such data is so far not sufficient, surface gravity disturbances have one striking advantage. They are globally homogeneous and independent from any local (national) vertical datum. This can be challenging for the future investigation, especially for solving the vertical datum problem. In this paper we discus a numerical solution of the linear AGBVP-III as the scalar-fixed exterior BVP with mixed BC. In our approach we use a direct formulation of the boundary element method (BEM), where the boundary integral equations (BIEs) can be derived through the application of Green’s third identity or through the method of weighted residuals (Brebbia et al. 1984). Both methods lead to the same integral relation. In this paper we use the weighted residuals technique, which has also a natural tie to a weak formulation of the problem. First application of BEM to the gravity field modelling was investigated in Klees (1992). This approach based on the indirect BEM formulation and the Galerkin method was subsequently extended, see e.g. Klees et al. (2001), and also applied to the mixed geodetic BVPs (Klees et al. 1997). The direct BEM formulation applied to the linear fixed gravimetric ˇ BVP was published by Cunderl´ ık et al. (2007). This approach is also suitable for the mixed BVPs and can be applied to the linear AGBVP-III. In this paper we use the same model, where the collocation method with linear basis functions is applied to discretize BIEs ˇ and transform them into the linear system (Cunderl´ ık et al. 2007). Here we briefly outline the mathematical model while focusing on the numerical results of practical experiments. The formulation of AGBVP implicitly involves the normal gravity field. Its parameters define the value U0 of the normal gravity potential on the surface of the reference ellipsoid. The value W0 of the actual gravity potential on the geoid represents a fundamental vertical datum parameter. Since W0 is not precisely known, the value U0 is not necessarily equal to W0 . Consequently,

there is a discrepancy between the actual and normal gravity field, which affects input data of the mixed BC. Hence, there is a discrepancy between the disturbing potential and its derivative, both presented in BIE of the direct BEM formulation. Here we present an experimental way how to perturb the linear AGBVP-III in order to get a correspondence between the gravity disturbances at oceans/seas computed by our approach and the ones induced by the prescribed and perturbed Dirichlet BC. This perturbation has a direct connection to a difference between U0 and W0 , cf. (Lehmann, 2000) or (Sacerdote and Sans´o, 2003). The presented approach utilizes a fact that the perturbation does not affect components of the stiffness matrix, since they are implicitly given by geometry of the doˇ main (Cunderl´ ık et al. 2007). Considering the Earth’s surface as a fixed boundary, these components are invariable w.r.t. the mixed BC and its perturbations.

2 Direct BEM Formulation for AGBVP-III The linear AGBVP-III represents the scalar-fixed exterior BVP for the Laplace equation with mixed BC, cf. e.g. (Lehmann, 1999), ΔT (x) = 0, T (x) = T

MMS

(x) + δW,

∇T (x), s(x) = −δg(x), T = O(|x|

−1

x ∈ R 3 − Ω,

(1)

x ∈ Γs (sea),

(2)

x ∈ Γ L (land),

(3)

) as x → ∞.

(4)

Note that T , T (x) = W (x) − U (x),

x ∈ R3,

(5)

is the disturbing potential, where W is the actual and U the normal gravity potential at any point x, the domain Ω is the body of the Earth with its boundary Γ = Γ S ∪ Γ L (the Earth’s surface), ,  represents the inner product of two vectors and s(x) = −∇U (x)/γ (x),

x ∈ Γ.

(6)

The surface gravity disturbance δg in Eq. (3) compares the magnitudes of the actual gravity g = |∇W | and the normal gravity γ = |∇U | at the same point x

Numerical Solution of the Fixed Altimetry-Gravimetry BVP Using the Direct BEM Formulation

δg(x) = g(x) − γ (x),

231

x ∈ R3.

(7) variant w.r.t. the input BC. Then a perturbation of the Dirichlet BC in Eq. (2) allows us to find a corresponM M S represents the prescribed value dence of the Neumann data at oceans/seas computed In Eq. (2), T of the disturbing potential on MSS evaluated by GGM by our approach and ones induced by the prescribed and δW is a perturbation of the Dirichlet BC related to Dirichlet BC. the difference of U0 and W0 , i.e. δW = W0 − U0 .

(8)

Equation (3) represents the oblique derivative BC as the normal to the Earth’s surface Γ does not coincide with the vector s defined by Eq. (6) (the surface deflection of the vertical is neglected). Let us apply the direct BEM formulation to the linear AGBVP-III in Eqs. (1), (2), (3) and (4). The main objective is to replace the Laplace Eq. (1) that governs the solution in the exterior domain R 3 − Ω by an equation that gives the solution on the boundary Γ only (see e.g. (Brebbia et al. 1984) and (Schatz et al. 1990)). Such integral equations can be derived using Green’s third identity, where a harmonic function T is represented as a superposition of the single-layer and double-layer potential. The same integral relation can be obtained through the method of weighted residˇ ual. A detailed derivation is discussed in (Cunderl´ ık et al. 2007). Here we only present the final BIE,  1 ∂G (x, y)d y S = T (x) + T ( y) 2 ∂n    ∂T ( y)G(x, y)d y S, x ∈ Γ, = ∂n Γ

(9)

Γ

where dS is the area element and the kernel function G is the fundamental solution of the Laplace equation, G(x, y) = (4π|x- y|)−1 , x, y ∈ R 3

(10)

In the numerical experiments presented in this paper we use the same numerical technique as discussed ˇ in Cunderl´ ık et al. (2007), i.e. the collocation method with linear basis functions. In such a way we get a discrete form of BIE Eq. (9) that subsequently yields a linear system of equations. Rearranging this system according to different types of BC leads to a dense nonsymmetric stiffness matrix. Since both the kernel functions in BIE Eq. (9) depend on direct distances only, components of the stiffness matrix are given by the geometry of the fixed Earth’s surface. Hence, they are in-

3 Numerical Experiments Presented numerical experiments deals with a numerical solution of the linearized fixed AGBVP-III. At first we model the fixed Earth’s surface using the following datasets: (i) on lands we add the global topography model SRTM30 PLUS V1.0 (Becker and Sandwell, 2003) to EGM-96 (Lemoine et al., 1998) in order to get geocentric positions, (ii) at oceans/seas we use MSS model KMS04 (Andersen et al. 2005). Then, we approximate the Earth’s topography obtained in the way described above by a triangulated surface ˇ (Cunderl´ ık et al. 2007). Vertices of this triangulation represent collocation points were we need to insert input data as the mixed BC. We generate the disturbing potential at oceans/seas and the surface gravity disturbances on lands from the geopotential coefficients of EIGEN-GL04C (F¨orste et al., 2006) using the Fortran program GEOPOT97.V0.4.F (developed by Dru Smith in 2006). The parameters of the normal gravity field are given by Geodetic reference system 1980 (GRS80) (Moritz, 1992). A mesh size of the triangular elements is 0.75◦ (in latitude) in all presented numerical experiments. It represents 86 402 collocation points regularly ˇ distributed over the whole Earth’s surface (Cunderl´ ık et al. 2007). The corresponding dense stiffness matrix requires approximately 64 GB of the internal memory and parallel computing is inevitable. Therefore the standard MPI (Message Passing Interface) subroutines are implemented for the code parallelization (Aoyama and Nakano 1999). In order to solve such large-scale linear system of equations, we use the nonstationary iterative methods BiConjugate Gradient Stabilized (BiCGSTAB) method (Barrett et al. 1994), which is suitable for dense and nonsymetric matrices. ˇ For more details see (Cunderl´ ık et al. 2007). Final large-scale computations were accomplished on the high-speed parallel computer IBM SP5 at the HPC Centre CINECA (see the acknowledgment). As mentioned in the Introduction, the difference between U0 and W0 can yield a discrepancy between

232

ˇ R. Cunderl´ ık and K. Mikula

Fig. 1 Residuals between the numerical solution and EIGEN-GL04C for the surface gravity disturbances at oceans/seas

the Dirichlet and the Neumann terms in BIE Eq. (9). In our first experiment we assume that U0 is equal to W0 , i.e. δW = 0m 2 .s −2 . The obtained numerical solution of such AGBVP-III is compared with values evaluated directly from EIGEN-GL04C using spherical harmonics. Residuals between both the solutions are

Fig. 2 Residuals between the numerical solution and EIGEN-GL04C for the disturbing potential on lands

depicted in Fig. 1 (for the surface gravity disturbances at oceans/seas) and Fig. 2 (for the disturbing potential on lands). Basic statistical characteristics of the residuals are presented in Table 1 (the first row). They show an overall shift of the obtained numerical solution w.r.t. EIGEN-GL04C. Consequently, we perturb the

Numerical Solution of the Fixed Altimetry-Gravimetry BVP Using the Direct BEM Formulation

233

Table 1 Statistical characteristics of residuals between the numerical solution and EIGEN-GL04C both perturbed by different values δW δW [m2 .s−2 ]

Mean(δg) [mGal]

0 −2 −2.896 −4 −6 −8 −10 −12 −15

0.06 0.02 0.00 −0.02 −0.07 −0.11 −0.15 −0.19 −0.26

SD(δg) [mGal] 5.05 5.04 5.04 5.04 5.04 5.04 5.04 5.04 5.04

Max(δg) [mGal]

Min(δg) [mGal]

Mean(T) [m2 .s−2 ]

66.02 65.98 65.96 65.94 65.91 65.87 65.84 65.80 65.75

−63.62 −63.65 −63.67 −63.69 −63.72 −63.76 −63.80 −63.83 −63.89

−1.96 −1.69 −1.56 −1.41 −1.13 −0.85 −0.58 −0.30 0.12

Dirichlet BC in Eq. (2) by different values of δW . The recomputed numerical solutions are compared with “perturbed” EIGEN-GL04C, i.e. (i) with the gravity disturbances of EIGEN-GL04C at oceans/seas and (ii) with the disturbing potential of EIGEN-GL04C on lands, but perturbed by the corresponding δW (the same like in Eq. (2). The statistical characterristics of such residuals are presented in Table 1. Figure 3 shows how the mean values of residuals for both, the surface gravity disturbances at oceans/seas and the disturbing potential on lands, are changing for the different δW . The mean values of residuals in different continents are depicted in Fig. 4 and the corresponding statistics are in Table 2.

SD(T) [m2 .s−2 ] 2.96 2.91 2.89 2.87 2.83 2.81 2.79 2.78 2.79

Max(T) [m2 .s−2 ]

Min(T) [m2 .s−2 ]

9.75 9.80 9.82 9.86 9.91 9.97 10.02 10.07 10.16

−29.83 −29.38 −29.18 −28.93 −28.48 −28.03 −27.57 −27.12 −26.45

4 Results and Discussion The numerical solution of the linear AGBVP-III using the direct BEM formulation shows a good agreement with EIGEN-GL04C. Since the mixed BC are simulated from EIGEN-GL04C, this agreement experimentally confirms a convergence of our approach that is based on the integration over the complicated Earth’s surface. Geometry of the fixed Earth’s surface determines the stiffness matrix, whose components represent relations between elements given by the kernel function G in Eq. (10) and its normal derivative, respectively (Brebbia et al. 1984). Then an overall shift

[m2.s–2] [m.s–2]

δW = –2.896 m2.s–2

0.5

Mean value of residuals

0

Fig. 3 The overall mean values of residuals between the numerical solution and EIGEN-GL04C both perturbed by different values δW (for the disturbing potential)

δgrez = δg BEM - δg GGM

–0.5

–1

Trez = T BEM – TGGM –1.5

–2 –15 –14 –13 –12 –11 –10 –9 –8 –7 –6 δW = W0 – U0

–5

–4

–3

–2

–1

0 [m2.s–2]

ˇ R. Cunderl´ ık and K. Mikula

234 Fig. 4 The mean values of residuals between the numerical solution and EIGEN-GL04C in different continents both perturbed by different values δW (for the disturbing potential)

[m2.s–2]

2

–2

δW = –2.896 m .s

2

Mean value of residuals

1

0 Australia Antarctica Africa

–1

North America

–2

Europe South America

–3

Asia

–4 –15 –14 –13 –12 –11 –10 –9

–8

–7

–6

–5

–4

–3

–2

–1

2 –2 0 [m .s ]

δW = W0 – U0 Table 2 Statisitcal characteristics of residuals between the numerical solution and EIGEN-GL04C in different continents (for the disturbing potential [m2 .s−2 ]) North America

South America

Europe

δW

Ø

SD

Ø

SD

Ø

SD

Ø

Africa SD

Ø

SD

Ø

SD

Ø

SD

0 −2 −2.896 −4 −6 −8 −10 −12 −15

−1.29 −1.08 −0.99 −0.88 −0.67 −0.47 −0.26 −0.06 0.25

1.90 1.92 1.93 1.94 1.97 2.00 2.05 2.09 2.17

−2.12 −1.87 −1.76 −1.62 −1.37 −1.11 −0.86 −0.61 −0.23

2.78 2.78 2.79 2.79 2.81 2.83 2.85 2.88 2.92

−1.92 −1.67 −1.56 −1.42 −1.17 −0.92 −0.67 −0.42 −0.05

2.21 2.17 2.15 2.14 2.12 2.11 2.12 2.13 2.17

−0.97 −0.64 −0.50 −0.31 0.01 0.34 0.66 0.99 1.48

1.34 1.34 1.34 1.35 1.37 1.40 1.45 1.50 1.59

−3.89 −3.52 −3.36 −3.16 −2.79 −2.42 −2.05 −1.69 −1.14

4.00 3.91 3.88 3.83 3.75 3.68 3.62 3.56 3.49

−0.28 −0.11 −0.03 0.07 0.24 0.41 0.58 0.76 1.02

1.20 1.21 1.22 1.22 1.24 1.26 1.29 1.31 1.36

−0.70 −0.49 −0.40 −0.29 −0.09 0.12 0.32 0.52 0.82

2.03 2.03 2.04 2.04 2.06 2.07 2.09 2.12 2.16

of the residuals between our numerical solution (for δW = 0 m−2 .s−2 ) and EIGEN-GL04C (Table 1) indicates a discrepancy between the Dirichlet and the oblique derivative BC. It is partly due to an additional discrepancy between the normal and actual gravity field (the difference between U0 and W0 ) and partly due to the accuracy of the numerical solution as an approximate solution. Table 1 and Fig. 3 show that the perturbation of the Dirichlet BC can lead to a correspondence of the gravity disturbances computed by our approach and the ones induced by the prescribed Dirichlet BC. However, linear tendencies of mean values in Fig. 3 indicate that a good correspondence for the surface gravity disturbances at oceans/seas and for the disturbing potential on lands is achieved for different δW . In addition, mean

Asia

Australia

Antarctica

values evaluated for different continents have similar slope but they are shifted (Fig. 4, Table 2). Therefore our criterion for a choice of the optimal perturbation is the zero mean value for the residuals of the surface gravity disturbances at oceans/seas. Further reasons, why we prefer this criterion, are: (i) accuracy of the 3D position of KMS04 is higher than for the topography model SRTM30 PLUS V1.0 + EGM-96, (ii) the problem of the oblique derivative almost vanishes at oceans/seas, (iii) the normal gravity field generated by the reference ellipsoid does not consider masses in its exterior, then the free air gradient of the normal gravity differs from the gradient of the actual gravity above the reference ellipsoid considerably. The last point is probably a main reason for different mean values in different continents (Fig. 4, Table 2).

Numerical Solution of the Fixed Altimetry-Gravimetry BVP Using the Direct BEM Formulation

Considering our criterion, the optimal value of the perturbation δW is −2.896 m2 .s−2 (Fig. 3). In this case the overall mean value for the residuals of the disturbing potential on lands is −1.564 m2 .s−2 and the standard deviation is 2.888 m2 .s−2 (Table 1). Corresponding mean values in different continents are depicted in Fig. 4. The mean values for both, the disturbing potential and gravity disturbances, are in a good agreement in Australia (Fig. 4, Table 2). High residuals in mountainous regions (Himalayas and Andes) (Fig. 2) strongly influence an overall mean value on lands as well as the continental ones in Asia and South America (Fig. 4), where the standard deviations are the highest (Table 2). The overall standard deviation for the gravity disturbances at oceans/seas is 5.044 mGal (Table 1). The highest residuals are along edges of the lithospheric plates (Fig. 1). Here the used global triangulation is not refined enough to model such abrupt change in gravity data. This can be improved by a further refinement of the discretization.

5 Conclusions The boundary element method (BEM) applied to the linear altimetry-gravimetry BVP of the third kind gives a numerical solution directly on the fixed Earth’s surface. The direct BEM formulation that represents the superposition of the single-layer and the double-layer potential is suitable for the mixed geodetic BVPs. The collocation technique with linear basis functions appears to be efficient to obtain precise numerical results. In the presented approach, components of the stiffness matrix, given by the geometry of the fixed Earth’s surface, are invariant w.r.t. input boundary conditions (BC). Consequently, a perturbation of the Dirichlet BC needs to be applied in order to find a correspondence of gravity disturbances computed by our approach and the ones induced by the prescribed Dirichlet BC. This perturbation has a direct connection to a difference between the normal gravity potential on the reference ellipsoid U0 and the gravity potential W0 on the geoid as a fundamental vertical datum parameter. Acknowledgments The work has been performed under the Project HPC-EUROPA (RII3-CT-2003-506079), with the support of the European Community – Research Infrastructure Action under the FP6 “Structuring the European Research Area” Programme. Special thanks go to the High Performance Cen-

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tre CINECA (Consorzio Interuniversitario) in Bologna, Italy, namely to Giovanni Erbacci, Gerardo Ballabio and Fiorella Sgallari. The work has been partly supported by the grant VEGA 1/3321/06 and the project APVV-LPP-0216-06.

References Andersen OB, Knudsen P, Trimmer RG (2005) Improving high resolution altimetric gravity field mapping. In F. Sanso (ed) A Window on the Future of Geodesy, IAG Symposia volume 128, Springer, Berlin Heldelberg New York, pp 326–331 Aoyama Y, Nakano J (1999) RS/6000 SP: Practical MPI Programming. IBM, Poughkeepsie, New York Barrett R, Berry M, Chan TF, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. http://www.netlib.org/ templates/Templates.html. Becker J, Sandwell D (2003) Accuracy and Resolution of Shuttle Radar Topography Mission Data. Geophys. Res. Lett., 30 (9) Bjerhammar A, Svensson L (1983) On the geodetic boundaryvalue problem for a fixed boundary surface – satellite approach. Bull. G´eod., 57: 382–393 Brebbia CA, Telles JCF, Wrobel LC (1984) Boundary Element Techniques, Theory and Applications in Engineering. Springer-Verlag, New York. ˇ Cunderl´ ık R, Mikula K, Mojzeˇs M (2007) Numerical solution of the linearized fixed gravimetric boundary-value problem. J. Geod. (first online) F¨orste CF, Flechtner R, Schmidt R, K¨onig U, Meyer R, Stubenvoll M, Rothacher F, Barthelmes H, Neumayer R, Biancale S, Bruinsma JM, Lemoine FG, Loyer S (2006) A mean global gravity field model from the combination of satellite mission and altimetry/gravimetry surface data: EIGEN-GL04C. Geophys. Res. Abstr., 8: 03462 Grebenitcharsky RS, Sideris MG (2005) The compatibility conditions in altimetry-gravimetry boundary value problems. J. Geod., 78: 626–636 Holota P (1980) The altimetry-gravimetry boundary value problem. Paper presented at the International Scientific Conference of Sec.6 Intercosmos, Albena, Bulgaria Holota P (1983a) The altimetry gravimetry boundary value problem I: linearization, Friedrich’s inequality. Boll. Geod. Sc. Afini., XLII: 13–32 Holota P (1983b) The altimetry gravimetry boundary value problem II: weak solution, V-ellipticity. Boll. Geod. Sci. Afini., XLII: 69–84 Klees R (1992) Loesung des fixen geodaetischen Randwertproblems mit Hilfe der Randelementmethode. DGK, Reihe C, Nr. 382, Muenchen Klees R, Ritter S, Lehmann R (1997) Integral equation formulations for geodetic mixed boundary value problems. Progress Letters of the Delft Institute for Earth Oriented Space Research (DEOS), 97.1, Delft University of Technology, pp 1–8

236 Klees R, Van Gelderen M, Lage C, Schwab C (2001) Fast numerical solution of the linearized Molodensky problem. J. Geod., 75: 349–362 Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) EGM-96 – The Development of the NASA GSFC and NIMA Joint Geopotential Model. NASA Technical Report TP-1998-206861 Lehmann R (1999) Boundary-value problems in the complex world of geodetic measurements. J. Geod., 73: 491–500 Lehmann R (2000) Altimetry-gravimetry problems with free vertical datum. J. Geod., 74: 327–334 Mather RS (1974) On the solution of the geodetic boundary value problem for the determination of sea surface topography. Geophys. J. R. Astr. Soc., 39: 87–109 Moritz H (1992) Geodetic reference system 1980. Bull. Geod., 66 (2): 187–192

ˇ R. Cunderl´ ık and K. Mikula Rummel R, Teunissen P (1988) Height datum definition, height datum connection and the role of the geodetic boundary value problem. Bull. Geod., 62: 477–498 Sacerdote F, Sans´o F (1987) Further remarks on the altimetrygravimetry problems. Bull. Geod., 61: 65–82 Sacerdote F, Sans´o F (2003) Remarks on the role of height datum in altimetry-gravimetry boundary-value problems. Space Sci. Rev., 108: 253–260 Sans´o F (1983) A discussion on the altimetry-gravimetry problems. In: Geodesy in Transition, University of Calgary, Calgary pp 71–107 Schatz AH, Thom´ee V, Wendland WL (1990) Mathematical Theory of Finite and Boundary Element Methods. Birkh¨auser Verlag, Basel · Boston · Berlin Svensson L (1983) Solution of the altimetry–gravimetry problem. Bull. Geod., 57: 332–353

On the Combination of Gravimetric Quasi-Geoids and GPS-Levelling Data R. Klees and I. Prutkin

Abstract The subject of the paper is an analysis of an approach for the combination of gravity anomalies and GPS-levelling data for regional gravity field modelling recently proposed (Prutkin and Klees, J Geod 2007). The well-known incompatibility of the two data sets is explained as the effect of non-uniqueness of regional gravity field modelling from a regional set of gravity anomalies. Hence, the proper mathematical formulation for the combination of the two data sets is a Cauchy boundary value problem. The differences between gravimetric quasi-geoid and GPS-levelling data are used as boundary data. We present an alternative interpretation of the approach. Moreover, we analyze the behaviour of the solution of the Cauchy boundary value problem if the size of the target area increases. The solution is expected to converge to zero if the size of the target area increases and in the limit covers the globe. This is confirmed by numerical tests. Finally, we investigate the effect of data noise on the solution. We show that when information about the data noise standard deviation is properly exploited in the course of the numerical solution of the Cauchy boundary value problem, we obtain a stable solution, which outperforms any gravity-only solution. Of particular interest for practical applications is the ability of the approach to correct for any lack of gravity field information inside and outside the target area, which is limited only by the spatial density and quality of the GPS-levelling data. R. Klees Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands I. Prutkin Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands

Keywords Local gravity field modelling · Non-uniqueness · Cauchy problem for the Laplace equation · GPS/levelling · Quasi-geoid

1 Introduction The combination of gravity anomalies with GPSlevelling data becomes increasingly important for regional geoid modelling as the number of stations of national height networks observed with GPS increases continuously. An example are the Netherlands, where this number has increased from about 20 stations in the nineteens to almost 500 stations today, which corresponds to a density of 1 point per 85 km2 . The difference between a GPS ellipsoidal height and a normal height is equivalent with a point height anomaly or a point disturbing potential value. From this point of view, the combination of gravity anomalies and GPS-levelling data reduces to the joint processing of gravity anomalies and disturbing potential values, which could be dealt with by least-squares collocation (e.g., Tscherning 1986) or by least-squares techniques in combination with an appropriate parameterization of the regional gravity field in terms of linear combination of a finite number of basis functions (e.g., Klees and Wittwer 2007). What makes this problem challenging is the fact that the two data sets do not contain the same information about the quasi-geoid (e.g., de Min 1996, Featherstone 1998, Fotopoulos 2005). This is attributed to a number of factors, such as the lack of a global coverage with high-resolution gravity anomalies and systematic errors in gravity anomaly data sets, GPS data, and national height networks. A straightforward way to deal with this problem is the

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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corrector-surface approach (e.g., Bruijne et al. 1997, Grebenitcharsky et al. 2005, Featherstone 2000, Nahavandchi and Soltanpour 2006). That is, first a gravimetric quasi-geoid is computed and evaluated at the GPS-levelling points. Then, from the differences the parameters of a corrector surface are estimated, and subsequently used to correct the gravimetric height anomalies. The problem with this approach is among others that the corrector surface is not the trace of a harmonic function, i.e., it is incompatible with the gravity anomalies. Therefore, the corrected height anomalies are not treated anymore as representing the quasi-geoid, but more as a tool to facilitate the transformation of GPS ellipsoidal heights into physical heights. Prutkin and Klees (2007) have proposed an alternative approach for the combination of gravity anomalies and GPS-levelling data. They formulate a Cauchy boundary value problem for the Laplace equation and solve this problem numerically treating the observed differences between gravimetric height anomalies and GPS-levelling data as boundary data. The result is a function (they call it ‘innovation function’), which has to be added to the gravimetric height anomalies in order to get the final quasi-geoid. The innovation function could also be interpreted as a corrector surface, but has the particular property that it is fully compatible with the gravity anomalies, because it is the trace of a harmonic function. Another advantage of this approach compared with a corrector surface is that the user does not need to decide what parameterization is the most appropriate one. The subject of this paper is to analyze in more detail the approach by Prutkin and Klees (2007). In particular, we want to investigate how the solution of the Cauchy boundary value problem behaves if the size of the target area increases. This question is related to the fact that the Cauchy boundary value problem should have the zero solution if the coverage with gravity anomalies is global. Moreover, we want to investigate how the approach behaves in the presence of noise in gravimetric height anomalies and GPS-levelling data. This question is important from a practical point of view, because the Cauchy problem is formulated as a deterministic boundary value problem. Noisy input data could yield a wrong innovation function, similar to the problem of over-fit and under-fit when using the corrector surface approach. The paper is organized as follows: in Sect. 2, we give a brief summary of the method by Prutkin and

R. Klees and I. Prutkin

Klees (2007). The main results are contained in Sect. 3. We present and discuss the results of extensive numerical experiments devoted to the behaviour of the innovation function as function of the size of the area and the effect of noisy data on the quality of the innovation function. The main conclusions of this study are addressed in Sect. 4.

2 A Summary of the Approach by Prutkin and Klees Prutkin and Klees (2007) use the difference between gravimetric height anomalies (i.e. computed from gravity anomalies) and geometric height anomalies (i.e. derived from GPS-levelling data) inside the target area as boundary data of a Cauchy boundary value problem:

T˜ = 0, ∂ T˜ | S = 0, ∂r T˜ | S = f,

in D, (1)

where S is a part of the telluroid surface, corresponding to the target area, D ⊂ R3 is a suitably chosen vicinity of the target area S, f is equal to the differences between gravimetric and geometric height anomalies when transformed into disturbing potentials using Bruns’ theorem (Heiskanen and Moritz 1967), and T˜ is the ‘innovation function’. Once T˜ has been computed, it has to be added to the disturbing potential computed from gravity anomalies. The result represents the (final) disturbing potential computed from gravity anomalies and GPS-levelling data. Using Bruns’ theorem, the final disturbing potential can be transformed into height anomalies inside the target area. Notice that the innovation function T˜ is the trivial function if there are no differences between gravimetric and geometric height anomalies, i.e. if the function f is zero inside the target area S. The same solution can be derived in a different way, which gives more insight into the problem. Suppose gravity anomalies and GPS-levelling data are given inside the target area. Then, the disturbing potential T satisfies the boundary conditions −

∂T 2 |S − T |S = f1, ∂r r

T |S = f2,

(2)

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where S is a part of the telluroid surface, corresponding to the target area, f 1 represents the gravity anomaly data, and f 2 represents the GPS-levelling data after transformation into disturbing potential values using Bruns’ theorem. For the time being, we assume that gravity anomalies and GPS-levelling data are noise-free. We will discuss the noisy case in Sect. 3.2. The problem of finding a harmonic function T with boundary data Eq. (2) constitutes a Cauchy boundary value problem for the Laplace equation and not a classical boundary value problem of potential theory, like the Robin problem. In practice, we have many more gravity anomaly data at our disposal than GPS-levelling data. Therefore, it is reasonable to consider first the gravity anomaly data set and to find a disturbing potential Tgrav that satisfies the boundary condition −

∂ Tgrav 2 | S − Tgrav | S = f 1 . ∂r r

(3)

Any method of regional gravity field modelling from gravity anomalies can be used to compute Tgrav , e.g. least-squares collocation, Stokes’ integral approach, or the recently proposed SRBF approach of Klees and Wittwer (2007). In the second step, we solve a Cauchy problem for the Laplace equation:

T˜ = 0, in D, ∂ T˜ | S = 0, ∂r T˜ | S = f 2 − Tgrav | S .

(4)

Problem Eq. (4) is exactly alike problem Eq. (1) when we set f = f 2 − Tgrav | S . The existence and uniqueness of a solution of problem Eq. (4) in the vicinity of the local area S is guaranteed by the Cauchy-Kovalevskaya theorem (Ebenfelt and Shapiro 1995). Finally, we obtain an approximation to the disturbing potential T as Tˆ = Tgrav + T˜ . The function Tˆ satisfies the GPSlevelling data,

  ∂ Tgrav 2 | S − Tgrav | S

gTˆ = − ∂r r 2˜ ∂ T˜ |S − T |S . − ∂r r

(6)

The first term on the right-hand side of Eq. (6) is equal to f 1 (cf. Eq. (3)). The second term is zero according to Eq. (4). Hence,

gTˆ = f 1 + δ f,

(7)

2 δ f := − T˜ | S  = 0. r

(8)

with

Since gT = f 1  = gTˆ , the functions Tˆ and T do not generate the same gravity anomalies inside the target area S. The difference δ f is proportional to the differences between gravimetric and geometric height anomalies. In the presence of noise, Tˆ and T can be considered the same if δ f is below the noise level of the gravity anomalies. For instance, suppose the differences between gravimetric and geometric height anomalies are below 1 m. Then, δ f does not exceed 0.3 mGal, which is below the noise level of most gravity anomaly data sets used in local gravity field modelling. Then, the function Tˆ satisfies sufficiently well the gravity anomaly data and the GPS-levelling data. Hence, it provides a solution to problem Eq. (2). To make δ f vanishing, we have to replace the Cauchy problem Eq. (4) by the Cauchy problem

Tˆ = 0, in D, ∂ T˜ 2 | S + T˜ | S = 0, ∂r r T˜ | S = f 2 − Tgrav | S .

(9)

Problem Eq. (9) differs from problem Eq. (4) with respect to the first boundary condition. The correct boundary condition of problem Eq. (9) involves the gravity anomaly operator, whereas the simplified boundary condition of problem Eq. (4) involves the gravity disturbing operator. The reason for introducing this simplification is that problem Eq. (4) can be solved with one of the numerical methods proposed (5) Tˆ | S = Tgrav | S + ( f 2 − Tgrav | S ) = f 2 by Prutkin and Klees (2007), whereas problem Eq. (9) cannot. To the knowledge of the authors, a numeras T does. At the same time, the gravity anomalies ical method to solve problem Eq. (9) has not been developed yet. computed from Tˆ are

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3 Analysis of the Behaviour of T˜

Latitude

been analyzed yet, but is important in practical applications. To investigate the behaviour of differences between gravimetric and geometric height anomalies If the target area is the globe, a unique solution (up as function of the size of the target area, as well as to terms of spherical harmonic degree 1) is obtained the stability of our solution against the noise in inifrom a global coverage with gravity anomalies. Then, tial data, extensive numerical tests have been underGPS-levelling data do not contain complementary in- taken. The initial data set includes 132,923 terrestrial formation, but redundant information, and the prob- gravity anomalies in the USA, which are distributed lem becomes an overdetermined problem. Moreover, over an area of 10 × 10 deg (cf. Fig. 1). The global the function f in problem Eq. (1) is zero and T˜ zero, gravity model EIGEN-GL04C (F¨orste et al. 2007) was too. This allows for another interpretation of the prob- removed from the data. A spherical radial basis funclem of non-uniqueness as formulated by Prutkin and tion (SRBF) model of the disturbing potential was esKlees (2007): the function f represents the contribu- timated from the gravity anomaly data set according tion of gravity anomalies outside the target area to the to the procedure by Klees and Wittwer (2007). The disturbing potential inside the target area, when the lat- SRBF model, which comprises 2600 SRBFs, is then ter is computed from gravity anomalies inside the tar- used as the true disturbing potential. Gravity anomaget area. A simple experiment to verify this is to let the lies inside the SRBF model. Figure 2 shows the gravtarget area increase and observe the amplitude of the ity anomalies and disturbing potentials of the SRBF functions f and T˜ . This amplitude should decrease, model. with increasing size of the target area, although definitely not monotonically. Another aspect of high relevance for practical applications is the problem of data noise. So far, we as45 sumed that the function f in Eq. (1) is noise-free, but 44 in reality it is not. The error budget of f consists of 43 systematic and random errors. The main contributor 42 are commission and omission errors in the gravimetric 41 height anomalies and errors in GPS ellipsoidal heights. 40 The contribution of errors in the normal heights is ex39 pected to be significantly smaller than the other two 38 error sources, although systematic errors in national 37 height networks may be significant for some target 36 areas. Systematic errors in the function f propagate 35 ˆ directly into the solution T . To remove them, addi250 251 252 253 254 255 256 257 258 259 260 Longitude tional information is necessary. In this paper, we want to concentrate on random errors. The effect on ran- Fig. 1 Distribution of the 132,923 gravity anomalies dom errors on the solution of problem Eq. (4) has not

Fig. 2 Gravity anomalies (left) in mGal and disturbing potentials (right) in m2 /s2 of the SRBF model used in the simulations. The white areas correspond to data gaps (see Fig. 1)

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241

The first target area is a 2 × 2◦ area centred at the 10 × 10◦ area. A gravimetric quasi-geoid was computed using gravity anomalies inside the target area and following the approach by Klees and Wittwer (2007). Next, the size of the target area was subsequently increased to 4 × 4◦ , 6 × 6◦ , 8 × 8◦ , and finally, to 10 × 10◦ . Each time a gravimetric quasi-geoid was computed using the gravity anomalies inside the target area. The gravimetric solution was compared with the SRBF model. The gravity anomalies, the disturbT −T ing potential Tgrav , and the error gravγ , where γ is the normal gravity at the telluroid are shown in Fig. 3

for the 2×2◦ , 4×4◦ , and 8×8◦ target areas. The peakto-peak error decreases, from 1.7 m (2 × 2◦ target area) to 0.32 m (4 × 4◦ ), 0.18 m (6 × 6◦ ), 0.23 m (8 × 8◦ ), and 0.19 m (10 × 10◦ ). The decrease is not monotonic, as expected, but the overall trend is that the differences between gravimetric and geometric height anomalies decrease with increasing target area. The influence of the lack of gravity anomaly data outside the target area on the computed height anomalies is clearly visible in the 3 plots at the right column of Fig. 3. The range of differences between gravimetric and geometric height anomalies and their patterns differ significantly for the 2 × 2◦ , 4 × 4◦ , and 8 × 8◦ target areas. For the 2 × 2◦ centre area, which is common to all target areas, we computed an innovation function T˜ using height anomalies at 65 evenly distributed points, which serve as GPS-levelling points. After correction

Fig. 3 Gravity anomalies in mGal (left column), disturbing potential Tgrav in m2 /s2 (middle column), and error in Tgrav transformed into height anomaly errors in m (right column) for the

(from top to bottom) 2 × 2◦ , 4 × 4◦ , and 8 × 8◦ target areas. The figures in the middle and right column show only the 2 × 2◦ centre part that is common to all target area

3.1 Differences Between Gravimetric and Geometric Height Anomalies as Function of the Size of the Target Area

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R. Klees and I. Prutkin

Table 1 Error standard deviation (i.e., mean error subtracted), minimum error, and maximum error of Tgrav (first row) and T˜ (second row) for various target areas. In each case, the innovation function T˜ was computed using 65 simulated GPS-levelling points located inside the 2 × 2◦ centre area, which is common to all target areas Target area

σ [cm]

Min [cm]

Max [cm]

2 × 2◦

33.6 0.7 8.6 0.7 3.6 0.5

−129.3 −4.6 −44.0 −1.0 36.3 −1.4

46.7 3.0 −11.6 1.4 55.4 2.1

4 × 4◦ 8 × 8◦

of the gravimetric height anomalies using the innovation function, the differences shown in right column of Fig. 3 reduce significantly (cf. Table 1). This shows that GPS-levelling data inside the target area are very well suited to correct for the influence of missing gravity anomaly data outside the target area.

where K is the Poisson kernel, and Sh is a surface parallel but below the telluroid S at a distance h. The parameter h is a free parameter, which controls the stability of the solution. Note that without a significant effect on the solution, S and Sh can be considered as part of a sphere with radius R and R − h, respectively, where R is the radius of a mean Earth sphere. According to Prutkin and Klees (2007), the integral equation u = K F is solved as Fˆ = (K + α I )−1 u,

(11)

where I is the identity operator and K is the Poisson operator. There are many methods known in literature to solve integral equations of the first kind. A very simple approach is obtained if the integral equation is discretized on a grid. If the data u are interpolated at the same grid points, the (approximate) solution of Eq. (11) is Fˆ = (K + αI)−1 u

(12)

3.2 T˜ and Noisy Data

where α is a regularization parameter. Fˆ can be computed iteratively without the need to store the comTo address the issue of noisy data, we considered plete matrix K in memory. This is very efficient from the 2 × 2◦ target area. We select 65 points, evenly a numerical point of view, in particular if the number distributed over the target area, which serve as GPS- of grid points is large. The regularization parameter α levelling points. We compute the (geometric) height controls the stability of the solution as h does. Instead anomaly at these points using the SRBF model and of searching for the optimal pair (α, h), Prutkin and differences between geometric and gravimetric height Klees (2007) fix the regularization parameter α to some anomalies (function f at 65 GPS-levelling points). minimum value. Then, for a range of values h, the funcˆ These differences were superimposed by Gaussian tion Fˆ = F(h), the innovation function T˜ = T˜ (h), and white noise with zero mean and a standard deviation the standard deviation of the residuals T˜ (h) − f ε at the of 1, 2, and 3 cm, respectively. We obtain ‘observed’ GPS-levelling points are computed. The latter is given differences f ε = f + ε at the 65 GPS-levelling points. as Moreover, true height anomalies were also computed % & at 1445 evenly distributed control points using the I  2 &1  SRBF model. They are used to evaluate the quality of T˜h,i − f ε,i , σh = ' (13) I the estimated disturbing potential Tˆ . i=1 According to Prutkin and Klees (2007), the noise is considered indirectly, by choosing an appropriate where I is the number of GPS-levelling points (I = height of the downward and upward continuation, 65 in our simulations). The selected value of h is the which is performed when solving problem Eq. (4) one for which the standard deviation of the residuals is numerically. More specifically, the solution of problem close to the standard deviation of the data noise: Eq. (4) involves the solution of a first-kind integral σh ≈ σε . (14) equation  u(x) =

K (x, y)F(y)d S(y), sh

x ∈ S,

(10)

The choice of the parameter h controls the smoothˆ ness of F(h) and T˜ (h) (see Prutkin and Klees, 2007),

On the Combination of Gravimetric Quasi-Geoids

243

and the magnitude of σh . The smaller h, the smaller σh and vice versa. If in the presence of data noise, the parameter h is chosen too small, the innovation function T˜ (h) will be too rough, i.e., it assimilates a part of the data noise (overfit). Vice versa, if the parameter h is chosen too large, the innovation function T˜ (h) is too smooth and does not represent well the data f (underfit). Therefore, the challenge is to choose the parameter h such that the innovation function does not assimilate data noise. Eq. (14) is used as an indicator of a proper choice of the parameter h. In order to investigate how the parameter h controls noise and whether the choice Eq. (14) is appropriate, we have computed a series of solutions with different parameters h for various noise standard deviations σε . Mis-modelling of noise was simulated by choosing σh significantly smaller than σε . The results are summarized in Table 2. When we compare the first row with the second row in Table 2, we see that the noise filtering works fine: increasing the data noise from σε = 1 cm to σε = 2 cm has almost no effect on the quality of the innovation function if σh ≈ σε is chosen. If σh is chosen too small, e.g. σh = 1 cm compared with σε = 2 cm (third row in Table 2), the quality of the innovation function T˜ becomes worse as the error statistics at the 65 GPS-levelling points and the 1445 control points reveal. Then, part of the noise is interpreted as signal and the innovation function is too rough. This conclusion is confirmed by the numerical tests with a standard deviation σε = 3 cm (fifth and sixth column in Table 2). Therefore, in practical applications it is necessary to know a good estimate of the data noise. This is typical for all approximation problems involving noisy data and not a particularity of the proposed methodology. Table 2 Fit of innovation function T˜ to the initial differences f in the presence of white Gaussian data noise ε. Data noise standard deviation: σε ; standard deviation of the residuals at the at the 65 GPS-levelling 65 GPS-levelling ( points: σh; RMS error 2 1 65 ˜ points: σh,1 = 65 i=1 Th,i − f i ; RMS error at the 1445

4 Conclusions The method by Prutkin and Klees (2007) is a novel approach to the combination of gravity anomalies and GPS-levelling data. We have given an alternative interpretation to the method, and have analyzed the behaviour of this method as function of the size of the target area and the data noise. The main conclusions are: 1. The innovation function corrects for the unmodelled contribution to the quasi-geoid of gravity anomalies outside the target area, which have not been used when computing the gravimetric quasi-geoid. The necessary information is provided by GPS-levelling data inside the target area. 2. The approach by Prutkin and Klees (2007) can also be used to correct for any un-modelled gravity signal (independently whether inside or outside the target area) using GPS-levelling data inside the target area. The limit to what extent that can be done depends only on the density and quality of the GPSlevelling data. In the same way, any global gravity field model can be replaced by GPS-levelling data inside the target area. Only if the quality of the global model is superior to the quality of the GPSlevelling data, a combined approach which takes the stochastic properties of the individual data sets into account would provide a better solution. 3. If the size of the target area increases, the amplitude of the innovation function decreases although not monotonically. In the limit, i.e., when the size of the target area is the globe, the innovation function is zero (maybe up to terms of spherical harmonic degree 1). ( control points: σh,2 =

1 1445

1445  j=1

T˜h, j − f j

2 . For com-

parison, also the maximum noise superimposed to the height anomaly differences at the 65 GPS-levelling points is given in the second column

GPS-levelling points

Control points

σε [cm]

Max ε [cm]

σh [cm]

σh,1 [cm]

Min [cm]

Max [cm]

σh,2 [cm]

Min [cm]

Max[cm]

1 2

2.2 7.0

3

7.3

1 2 1 3 1

0.7 0.9 1.4 1.4 2.0

−1.1 −1.6 −2.4 −3.4 −4.2

2.0 2.4 4.8 4.5 4.7

0.8 1.3 1.7 1.7 2.3

−2.1 −3.9 −4.6 −5.3 −6.6

4.4 5.8 5.2 5.9 6.4

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4. Data noise can properly be accounted for by carefully choosing the depth of upward and downward continuation in the numerical solution of the Cauchy boundary value problem. This requires a priori information about the noise in gravimetric and geometric height anomalies. So far, only information about the noise standard deviation can be dealt with; moreover, the noise standard deviation must be uniform. Noise correlations or non-homogeneous noise over the target area cannot be taken into account yet. If the noise is nonhomogeneous, a better solution could be obtained if the target area is subdivided into sub-areas, and the approach is applied to each sub-area.

References Bruijne AJT de, Haagmans RHN, Min EJ de (1997) A preliminary North Sea Geoid model GEONZ97. Rep MDGAP9735, Directoraat-Generaal Rijkswaterstaat. Meetkundige Dienst Ebenfelt P, Shapiro HS (1995) The Cauchy-Kowalevskaya theorem and generalizations. Commun Partial Diff Equ 20: 939–960 Featherstone WE (1998) Do we need a gravimetric geoid or a model of the base of the Australian Height Datum to transform GPS heights? Aust Surv 43: 273–280 Featherstone WE (2000) Refinement of a gravimetric geoid using GPS and levelling data. J Surv Eng 126: 27–56

R. Klees and I. Prutkin F¨orste C, Schmidt R, Stubenvoll R, Flechtner F, Meyer U. K¨onig R, Neumayer H, Biancale R, Lemoine JM, Bruinsma S, Loyer S, Barthelmes F, Esselborn S (2007) The GeoForschungsZentrum Potsdam/Groupe de Recherche de G`eod´esie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C. J Geod, doi: 10.1007/s00190-007-0183-8 Fotopoulos G (2005) Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data. J Geod 79: 111–123 Grebenitcharsky RS, Rangelova EV, Sideris MG (2005) Transformation between gravimetric and GPS/levelling-derived geoids using additional gravity information. J Geodyn 39: 527–544 Heiskanen WA, Moritz H (1967) Physical Geodesy. WH Freeman and Company Editor. San Francisco Klees R, Wittwer T (2007) Local gravity field modelling with multi-pole wavelets. In: Tregoning P, Rizos C (eds) Dynamic Planet – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools. IAG Symp Vol 130, Springer, Berlin Heidelberg, pp 303–308 Min E de (1996) De geoide voor Nederland. Nederlandse Commissie voor Geodesie, Groene serie Vol 34, Delft Nahavandschi N, Soltanpour A (2006) Improved determination of heights using a conversion surface by combining gravimetric quasi-geoid/geoid and GPS-levelling height differences. Stud Geopys Geod 50: 165–180 Prutkin I, Klees R (2007) On the non-uniqueness of local quasigeoids computed from terrestrial gravity anomalies. J Geod, doi: 10.1007/s00190-007-0161-1 Tscherning CC (1986) Functional methods for gravity field approximation. In: S¨unkel H (ed) Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol 7, Springer, Berlin Heidelberg, pp 3–47

An Alternative Approach for the Determination of Orthometric Heights Using a Circular-Arc Approximation for the Plumbline G. Manoussakis, D. Delikaraoglou and G. Ferentinos

Abstract The orthometric height is the distance, measured positive outwards along the plumbline, from the geoid to a point of interest usually situated on the Earth’s topographic surface. According to its “classical definition”, it can be computed from the geopotential number of a point, using the mean value of the Earth’s gravity acceleration along the plumbline within the topography (i.e. between the geoid and the Earth’s surface). Hence, the main problem in the rigorous definition of an orthometric height reduces to the accurate evaluation of the mean value of the Earth’s gravity acceleration along the plumbline. Alternatively, recent efforts concentrate on the determination of orthometric heights from GPS derived geodetic heights (above the ellipsoid) and geoid undulations derived from detailed local geoid models using the familiar Stokes integration or FFT techniques. In this paper, we seek to determine the orthometric height from the knowledge of the geodetic (ellipsoidal) height and a representation for the gravity field at the surface point and without any information about the topographic mass distribution. We show that an approximate determination of the orthometric height of a point on the Earth’s surface can be made by a methodology which relates the orthometric height with the G. Manoussakis Department of Surveying Engineering, National Technical University of Athens, Iroon Polytechneiou 9, Zografos 157 80, Greece D. Delikaraoglou Department of Surveying Engineering, National Technical University of Athens, Iroon Polytechneiou 9, Zografos 157 80, Greece G. Ferentinos Liontos & Associates S.A., Lambrou Katsoni 36, Ampelokipi, Athens, Greece

geopotential number C, the magnitude of the gravity vector g , and the curvature k of the plumbline, all determined at the point of interest on the physical surface. The required geopotential number C is computed through the evaluation of the Earth’s gravity potential W from one of the available Global Geopotential Models (GGMs) in spherical harmonics, while g and k are computed by suitable analytical formulae which use the first and second partial derivatives of the disturbing potential T (E¨otvos components) and the normal potential U accordingly. An overview is given of the steps involved in the computational process and the assumptions made. The proposed approach was tested using different GGMs and an extensive GPS/Leveling dataset on benchmarks in the USA. Results from these comparisons are presented for the larger part of the conterminous United States in non-mountainous areas. They demonstrate that generally the differences between these determined orthometric heights and actual orthometric heights from geodetic leveling typically range from a few centimeters and up to 3 decimeters, thus showing the viability of the methodology and its future promise as new and continually improving geopotential models from the CHAMP/GRACE and GOCE missions become available to be used for this purpose. Plans for future work will also be given. Keywords Orthometric height · Curvature of the plumbline · Geopotential models · GPS/Leveling

1 Introduction The orthometric height of a point P on the Earth’s physical surface is the length of the plumbline passing

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through P and the geoid. This length can be determined from the quotient of the geopotential number of P and the mean value of the measure of the gravity vector along the plumbline, or from the subtraction of geoid undulation from the geometric height. In mathematical terms, we have HP =

W0 − W P   g M 

(1)

which, when downward-continued, provides the necessary geoid height, offering also the advantage that the method is insensitive to systematic errors due to the GPS (or any other space-based positioning technique), the geoid model or the orthometric height system used.

2 The Analytical Model for Approximating Orthometric Heights

or For the purposes of this work, in order to compute the orthometric height of various points on the Earth’s physical surface we start from a formulation relating In Eqs. (1) and (2) H P , h P and N P stand respec- the geometry of an equipotential surface and the pointtively for the orthometric height, the geodetic (ellip- wise determination of the curvature of the plumbline. soidal) height and the geoid undulation of the point P, Beginning with the vector equation S¯ W P of the equipowhile the symbols W0 and W P are the values of the tential surface W (X, Y, Z ) = W P around the point P gravity potential   on the geoid and the point P respec- in the form tively, and g M  denotes the mean value of the magnitude of the gravity vector g for the segment of the (3) S¯ W P (X, Y ) = (X, Y, Z (X, Y )) plumbline between the point P and the geoid. When using Eq. (2) the curvature of the plumbline, as well as the angle between the vertical on the ellipsoid pass- Manoussakis (2005) has determined the Gauss curvaing through P and the tangent line to the plumbline at ture K G of an equipotential surface around a point P the point P is not taken into account (Sj¨oberg, 2006). from  M    In Eq. (1) the mean value g has to be determined L N − M2 and, in contrast to Eq. (2), where one considers the KG = (4) EG − F2 plumbline as a straight line that is coincident with the vertical on the ellipsoid, no additional hypothesis is made for the geometry of the plumbline. where E, F, G are the elements of the first fundaUp to now, in practice, the usual approach to the mental form of the equipotential surface and respecproblem of determining orthometric heights is to fre- tively L, M, N are the elements of the second fundaquently use Helmert orthometric heights which, how- mental form of the equipotential surface, all of which ever, are limited in their accuracy due to the impre- can be determined respectively in terms of the first cise topographic correction required to be applied to and second order partials of the Earth’s gravity pothe mean gravity values used. Attempting to improve tential W at the point of interest P, with respect to this situation, recently Tenzer et al. (2005) have pro- the geocentric Cartesian coordinates X , Y , Z of the posed the use of a rigorous formulation of the topo- point (Manoussakis (2005)). This formulation eventugraphic correction in order to improve the determina- ally leads to Eq. (5) shown in Table 1 which allows detion of mean gravity. In addition, Sj¨oberg (2006) has termining the Gauss curvature of an equipotential surproposed a new method that uses the known geode- face at a surface point P with known coordinates from tic height of a point, in combination with the Geopo- the representation of the gravity field at that point. The tential number and some gravity data in the surround- main role of the fundamental elements E, F, G and L, ing area of the point of interest to determine the geoid M, N is to provide the necessary information concernheight, and in turn to obtain the orthometric height ing the geometry of the equipotential surface which, in by applying Eq. (2). His approach is based on us- turn, affects the evaluation of the orthometric height. ing the known normal gravity field to determine the Similarly, the curvature of the plumbline k at the height anomaly of a point with known geodetic height point P is given as HP = h P − N P

(2)

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Table 1 Equation expressing the Gauss curvature K G of an equipotential surface      ∂2W ∂ W 2 ∂2W ∂ W 2 ∂2W ∂ W ∂ W − KG = − +2 ∂ X∂ Z ∂ X ∂ Z ∂ X2 ∂ Z ∂ Z2 ∂ X      2 2 2 ∂W ∂2W ∂ W 2 ∂ W ∂W ∂W ∂ W − +2 · − ∂Z ∂Y ∂ Z ∂Y ∂ Z ∂Y 2 ∂ Z 2 ∂Y  

2 ⎫ ∂ W ∂ W ∂2W 1 ∂ W ∂2W ∂ W ∂2W ∂2W ∂ W ∂ W ⎬ · + − − −  ∂Z ∂Y ∂ X ∂ Z ∂ X ∂Y ∂ Z ∂ Z ∂ X ∂Y ∂ Z 2 ∂ Z ∂Y ⎭ ∂W 2 4 |g | ∂Z

(5)

  ,   d a  d a 3 d 2 a     k= × 2  dt   dt dt 

Using this approach for the purposes of this work (6) we seek to estimate the orthometric height of a point P with coordinates (ϕ P , λ P , h P ) on the Earth’s physical surface from the analytical approximation of the where a = a (t) = (X (t), Y (t), Z (t)) is, in the global formula Cartesian frame, the vector equation of the plumbline parameterized with the parameter t. t P  Taking into account that H P = ds = |((gradW ) ◦ a ) (t)| dt dX ∂ W dY ∂W d Z ∂W = , = , = , dt ∂ X dt ∂Y dt ∂Z

a¯

0

(9)

t P

(7)

|(g ◦ a ) (t)| dt

= 0

from which the second derivatives d 2 X/dt 2 , d 2 Y /dt 2 and d 2 Z /dt 2 can be evaluated as functions of the first where (g ◦ a ) denotes the gravity vector along the and second order partials of the gravitational potential plumbline of the point to the geoid. W with respect to the Cartesian coordinates X , Y , Z , it The actual computations for the evaluation of can be shown that after some manipulation of Eq (6), Eq. (9) require the determination of the partial derivathe curvature of the plumbline k at the point P can be tives of the gravitational potential W in Cartesian obtained from Eq. (8) shown in Table 2. Again, this coordinates (X , Y , Z ) from the partial derivatives of equation has a special significance because it makes W in spherical coordinates (r, ϕ, λ), as well as to split possible to determine the curvature of the plumbline as usual the earth’s gravity potential W in two parts, passing through a surface point P, if its coordinates i.e. the normal part U and the disturbing part T , so that (φ, λ, h) P are known and without knowing the vector T = W − U . In that way, for the first order partials of W , the following relations hold, equation of the plumbline. Table 2 Equation expressing the curvature of the plumbline ⎧     2 ⎨ ∂W ∂2W ∂ W ∂2W ∂ W ∂ W ∂2W ∂ W ∂2W ∂ W ∂2W ∂ W ∂2W ∂ W |P + + − − − kP = ⎩ ∂Y ∂ X ∂ Z ∂ X ∂Y ∂ Z ∂Y ∂ Z ∂ Z 2 ∂Y ∂ X ∂Y ∂ X ∂Y ∂ Z ∂ Z ∂Y 2 ∂Y     2 ∂2W ∂ W ∂2W ∂ W ∂2W ∂W ∂2W ∂ W ∂W ∂W ∂2W ∂ W ∂2W ∂ W |P + + − + + + + ∂ Z ∂ X2 ∂ X ∂ X ∂Y ∂Y ∂ X∂ Z ∂ Z ∂ X ∂ X∂ Z ∂X ∂Y ∂ Z ∂Y ∂ Z2 ∂ Z + ·

∂W ∂X



∂2W

∂W ∂W ∂W ∂W + + + ∂ X ∂Y ∂ X ∂Y ∂ Z ∂ Z ∂Y 2 ∂Y ∂ X2 ∂ X ∂2W

∂2W

1 

∂W ∂X

2

 +

∂W ∂Y

2

 +

∂W ∂Z

2

3 2

∂2W

 −

∂W ∂Y



∂2W

∂2W

∂W ∂W − ∂ X ∂Y ∂Y ∂ X∂ Z ∂ Z

 2 |P

⎫1 ⎬2 ⎭ (8)

|P

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⎡

∂W ∂U ∂T ∂W ∂U ∂T = + , = + , and ∂X ∂X ∂ X ∂Y ∂Y ∂Y ∂W ∂U ∂T = + , (10) ∂Z ∂Z ∂Z

whereas similar relations can be derived for the second order partial derivatives of W . For the first order partial derivatives of the disturbing potential T we use Eq. (11), shown in Table 3, where the matrix in the right hand side is invertible and furthermore the partial derivatives of T with respect to r , ϕ, λ can be determined from the finite series of a spherical harmonics expansion of any recent GGM. Hence, the first order partial derivatives of T with respect to X , Y , Z at P can be determined from Eq. (11). Similarly, the first order partials of the normal potential U with respect to X , Y , Z are determined from the relations Eq. (12), also shown in Table 3, in terms of the ellipsoidal coordinates (u, β, E) of the point of interest P, with u being the semi-minor axis of the ellipsoid whose surface passes through point P, β being the reduced latitude and E is the linear eccentricity (obeying the relation E = ae, where a is the semi-major axis of the Earth’s ellipsoid and E is its first eccentricity). The second order partials of T with respect to X , Y , Z at P can be determined from the following equation

∂2T ⎢ ∂ X2 ⎢ 2 ⎢ ∂ T ⎢ ⎢ ∂Y 2 ⎢ ⎢ ∂2T ⎢ ⎢ ⎢ ∂ Z2 ⎢ ∂2T ⎢ ⎢ ⎢ ∂ X ∂Y ⎢ ∂2T ⎢ ⎢ ⎢ ∂ X∂ Z ⎣ ∂2T ∂Y ∂ Z

⎤

∂2T ⎥ ⎢ ∂x2 ⎥ ⎢ 2 ⎥ ⎢ ∂ T ⎥ ⎢ ⎥ ⎢ ∂ y2 ⎥ ⎢ 2 ⎥ ⎢ ∂ T ⎥ ⎢ ⎥ ⎢ ⎥ = M(Φ, Λ) ⎢ ∂z 2 ⎥ ⎢ ∂2T ⎥ ⎢ ⎥ ⎢ ∂ x∂ y ⎥ ⎢ ⎥ ⎢ ∂2T ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ x∂z ⎦ ⎣ ∂2T ∂ y∂z

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(13)

as functions of the astronomic latitude Φ and longitude Λ of point P, and the E¨otvos components (Tx x , Tx y , Tx z , Tyy , Tyz , Tzz ) of the disturbing potential T in a Local North Oriented Frame (LNOF) which is a right-handed North-East-Up frame with the x-axis pointing North, the y-axis pointing East and the z-axis pointing Up. The full form of matrix M(Φ, Λ) is shown in Table 4. The required astronomic coordinates Φ, Λ can be obtained from the geodetic coordinates (ϕ, λ) of the point of interest P and the components of the deviation of the vertical ξ , and η which, in turn, together with the E¨otvos components are determined from any of the latest GGM spherical harmonics expansions. The corresponding formulae

Table 3 Equations used for the computation of the required first order partials of W (T and U) ⎡ ⎡ ⎤ ⎤ ∂T ∂T ⎤⎢ ∂X ⎥ ⎢ ∂r ⎥ ⎡ cos φ cos λ cos φ sin λ sin φ ⎢ ⎢ ⎥ ⎥ ⎢ ∂T ⎥ ⎢ ∂T ⎥ ⎣ ⎢ ⎥ = −r sin φ cos λ −r sin φ sin λ r cos φ ⎦ ⎢ ⎥ ⎢ ∂Y ⎥ ⎢ ∂φ ⎥ r cos φ sin λ r cos φ cos λ 0 ⎣ ∂T ⎦ ⎣ ∂T ⎦ ∂λ ∂Z √ ⎡ √ 2 ⎤ ⎡ ⎤ u u + E2 u2 + E 2 sin λ ∂U ⎡

cos β cos λ − sin β cos λ − ⎢ ⎥ 2 2 u 2√ + E 2 sin2 β u 2 + E 2 cos β ⎥ ⎢ ∂ X ⎥ ⎢ u2 + √ E sin β ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ∂U ⎥ ⎢ u u 2 + E 2 u2 + E 2 ⎥⎢ cos λ ⎢ ⎥⎢ ⎥=⎢ cos β sin λ − sin β sin λ − √ ⎢ ∂Y ⎥ ⎢ u 2 + E 2 sin2 β ⎣ u 2 + E 2 sin2 β u 2 + E 2 cos β ⎥ ⎥ ⎣ ∂U ⎦ ⎢ ⎣ ⎦ u2 + E 2 u sin β cos β 0 ∂Z u 2 + E 2 sin2 β u 2 + E 2 sin2 β Table 4 The form of the matrix M(, ) involved in Eq. (13) ⎡ sin2 Λ cos2 Λ 0 sin 2Λ 2 2 2 2Φ ⎢ sin2 Φ cos2 Λ sin Φ sin Λ cos sin Φ sin 2Λ ⎢ ⎢ cos2 Φ cos2 Λ cos2 Φ sin2 Λ sin2 Φ cos2 Φ sin 2Λ ⎢ 1 ⎢ sin Φ sin 2Λ − 1 sin Φ sin 2Λ 0 − sin Φ cos 2Λ ⎢ 2 2 ⎣ − 1 cos Φ sin 2Λ − 1 cos Φ sin 2Λ 0 cos Φ cos 2Λ 2 2 1 1 1 2 2 2 sin 2Φ cos Λ − 2 sin Φ sin Λ sin Φ cos Φ − 2 sin 2Φ sin 2Λ / 01 M(Φ,Λ)

⎡

0 − sin 2Φ cos Λ sin 2Φ cos Λ − cos Φ sin Λ − sin Φ sin Λ cos 2Φ cos Λ

(11)

∂U ∂u ∂U ∂β 0

⎤ 0 sin 2Φ sin Λ ⎥ ⎥ sin 2Φ sin Λ ⎥ ⎥ cos Φ sin Λ ⎥ ⎥ sin Φ sin Λ ⎦ cos 2Φ sin Λ 2

⎤ ⎥ ⎥ ⎥ ⎦

(12)

An Alternative Approach for the Determination of Orthometric Heights.

for the second order partials of U with respect to X , Y , Z are much more complicated and can be derived from Eq. (12) by differentiating once or they can be found in Manoussakis (2005). Therefore, we omit the details of these computations due to the space restriction of this paper. Once the curvature of the plumbline is determined from Eq. (8), in order to compute the orthometric height at any point on the Earth’s physical surface one can use the following analytical approximation HP ∼ = s 3     W0 − W P 2 1 W0 − W P 4 2 (14) + kP = |g P | 4 |g P | or equivalently in terms of the Gauss curvature  HP =

 ∂W ∂Z

(W0 − W P )2 K G (P) 2   |P L N − M2 |P

(W0 − W P )4 1 k 2P + K G2 (P)  4   ∂W 4 2 2| | L N − M P ∂Z P

1/2

(15) where the magnitude |g P | of the gravity vector is being calculated as 3       ∂W 2 ∂W 2 ∂W 2 | | |P |g P | = P + P+ ∂X ∂Y ∂Z (16) Equations (14) and (15), given here without derivation, result from the assumption that the curved segment of the plumbline between the point P at the earth’s surface and the corresponding point P  along the plumbline on the geoid is part of a circular arc of radius R P , such as RP =

1 kp

HP =

249 n 

S P ∼ = co sseg , co  +

(18)

p=1 n→+∞

(termed as Model 2), where each of the plumbline segments sseg is computed from an expression similar to Model 1 (Eq. 14). As a last remark on the computational process just outlined, it should be noted that the magnitude |g P | of the gravity vector also depends on the first order partial derivatives of the disturbing potential T , or equivalently on the gravity disturbance δg, the deviations of the vertical ξ , and η, and the disturbing potential T , all of which can be obtained in terms of spherical harmonics using any of the latest GGMs. Therefore, the sought orthometric height of any point of interest P depends on the parameters φ P , λ P , h P , δg , T, ξ, η, a .

3 Test Results We have tested the methodology just outlined with an extensive dataset of GPS/Levelling Benchmarks from the conterminous United States, cf. Fig. 1. The dataset, which is available on the Web (US National Geodetic Survey, 2003), contains a total of 14,308 benchmark points in the conterminous USA (including 579 points in Canada along the northern border), which were selected by NGS so that to have an orthometric height in the NAVD 88 datum established by geodetic leveling, and also to have an ellipsoidal height in the NAD 83

(17)

where k p typically, even for rugged terrain, should be expected to be less than 10−5 (cf. Featherstone, 1999). For a full derivation of Eqs. (14) and (15) one should consult Manoussakis (2005). Equation (14) for the purposes of this paper is hereby termed as Model 1. The analytical approximation of H P can also be computed in a piecewise Fig. 1 Distribution of the benchmark in the dataset used fashion as

250

datum established by GPS surveys at a level of 10 ppm horizontal accuracy (or better). A detailed description of the dataset’s components, the reference systems involved, the retrieval parameters, and the data cleansing process can be found in Milbert (2003). For out tests we used a sub-set of 11,671 of these benchmark points which were located east of the meridian with λ = 258◦ East up to longitude λ = 300◦ East, cf. Fig. 1. The reason for this selection was that we wanted to test first our methodology in non-mountainous areas, i.e. at points with orthometric heights not exceeding 1200 m. This resulted in excluding from these tests the area of the western part of the United States. In this data subset, the orthometric heights of the benchmark points were less than 400 m in 10,338 of the points (88.4%), in the range 400–800 m in 1225 points (10.5%) and in the range 800–1200 m in 108 (0.1%) points. Respectively, the geoid over the area covered by this subset of points is quite smooth with geoid heights ranging from −8.6 m up to −41.3 m, with 97% (11,339) of the points having geoid heights between −37.7 and −23.2 m. For the numerical tests that follow, all necessary parameters of the gravity field, i.e. N , δg, ξ , η, and the E¨otvos components Tx x , Tyy , Tzz , Tx y , Tx z , Tyz which are needed for the above computations, were obtained using G. Wenzel’s Geopgrid software, while for all the first and second partials of the normal potential U , and the disturbing potential T , the magnitude of the gravity vector g , the curvature of the plumbline k and our modeled orthometric height H P (using Model 1 above) we developed and used our own program CalHgt, which for W0 uses the value 62, 636, 860.85 m2 /s2 (i.e. that of the GRS80). For that purpose we used the following GGMs: EGM96 to degree and order (n,m) = 360, GPM98C (n,m = 720), the as well the more recent models EIGEN-CG03C (n,m = 150) and EIGEN-GL04C (n,m = 360). The CG03C model is a combined CHAMP, GRACE and surface gravity data model, whereas GL04C has been derived from a combination of data from the GRACE and LAGEOS mission and 0.5 × 0.5 degrees gravimetry and altimetry surface data. Statistical results of the differences dH = HNGS − HCalHgt between the orthometric heights from the NGS leveling dataset and the corresponding orthometric heights HCalHgt obtained from our previously outlined method (Model 1) are shown in Fig. 2. Using the EGM96 geopotential model we obtained a mean difference in the aforementioned

G. Manoussakis et al.

orthometric heights of −0.31 m with a standard deviation of 0.55 m (cf. Fig. 2(a)). The corresponding results when we used the EIGEN-GL04C geopotential model were slightly better exhibiting a mean difference of −0.24 m with a standard deviation of the mean of 0.51 m (cf. Fig. 2(b)). Similar statistics were obtained using the CG03C geopotential model (mean difference −0.23 m, St. Dev. 0.56 m) and the GPM98C model (mean difference −0.30 m, St. Dev. 0.55 m). The noted differences dH in these tests are consistent with the expected short wavelength limitations of the used geopotential model(s), a notion which was supported by the fact that the corresponding geoid height differences dN between the geoid heights from the used geopotential models and the NGS (GPS/Leveling) data exhibited a similar behavior, both in statistical terms as well as the noted spatial trends of these differences. For example, the geoid height differences dN obtained from the GL04C model and the NGS GPS/Leveling data showed for the 11,671 benchmark points a mean geoid height difference of 0.32 m with a standard deviation of 0.53 m. Therefore, based on these findings, it is reasonable to assume that in the near future the availability of new CHAMP-GRACE-GOCE based Gravity Earth Models with better resolution in the short wavelengths of the gravity field would potentially provide even better results in the computation of orthometric heights using our tested Model 1. To test further the premise of “What if” we had a geopotential model closer, e.g. to the detailed NGS GPS/Levelling-based geoid model – perhaps a future satellite GGM model of similar resolution, we have repeated the previous tests using the GL04C model in order to compute for each benchmark point all the necessary parameters (δg, ξ , η, E¨otvos components) for the computations of HCalHgt , except for the disturbing potential T and the gravity disturbances δg (on the geoid) which were indirectly obtained from the NGS’s GPS/Leveling data, i.e. using the fundamental Bruns formula T = γo NGPS/Levelling where γo is the normal gravity on the reference ellipsoid and respectively the known relation δg = −

∂T ∂T ≈ gGL04C − NGPS/Levelling ∂h ∂h

(19)

Furthermore, the same tests were repeated using the piecewise Model 2, i.e. Eq. (18). The corresponding

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Fig. 2 Statistics of the orthometric height differences d H = HNGS − HCalHgt : (a) using the EGM96, and (b) using the EIGEN-GL04C geopotential models respectively

Table 5 Statistical results on the differences d H = HNGS − HCalHgt using T and δg values from the NGS (GPS/Leveling) data Model 1 Number Mean (m) St Dev (m) Min (m) Max (m) dH’s (m) [−0.2, 0.2]

Model 2 11,671 −0.004 0.126 −1.336 1.818 11,156

Number Mean (m) St Dev (m) Min (m) Max (m) dH’s (m) [−0.03, 0.03]

11,671 −0.001 0.018 −0.087 0.148 10,888

statistical results are shown in Table 5. Using Model 1 we obtained a mean difference between the HCalHgt and the NGS orthometric heights of −0.004 m with a standard deviation of 0.12 m, with the HCalHgt values in 11,156 out the total 11,671 (i.e. 95.6%) of the tested benchmark points agreeing with the NGS orthometric heights at below the 20 cm level. Respectively, using the piecewise Model 2, we obtained a mean difference of −0.001 m with a standard deviation at the level of 0.02 m, with the HCalHgt values in 10,888 out of the total 11,671 (i.e. 93.3%) benchmark points agreeing with the NGS orthometric heights at below the 3 cm level.

4 Conclusions We have outlined a viable alternative method for obtaining orthometric heights through an analytical circular-arc approximation of the curvature of the plumbline and from the knowledge of the geodetic height (or ellipsoidal height) and a representation for the gravity field at the computation point. The results just outlined demonstrate that using the proposed new methodology to computing approximate values for orthometric heights is a viable approach that enables us to take advantage of the GPS-derived positions on the Earth’s surface and the continually improving satellite-based GGMs. The results (Model 1) with current GGMs demonstrated the validity of this approach through the agreement of the so-derived orthometric heights with actual leveling-based orthometric heights at a level largely limited only by the information content (i.e. short wavelength resolution) of the Geopotential models used. The results with the piecewise Model 2 support the premise that the method will give even

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better results closer to the anticipated better resolution of the future (CHAMP/GRACE/COGE) GGMs, at which time will be possible to study other influencing factors such as the plumbline torsion, spherical approximation etc.

References Featherstone, W.E. (1999) – “The use and abuse of vertical deflections”, Proceedings of the 40th Australian Surveyors Congress, Fremantle, Australia, November, 85–97. Manoussakis G. (2005) – “The determination of the orthometric height without the use of boundary value problems”, PhD Dissertation, Department of Surveying Engineering, National Technical University of Athens.

G. Manoussakis et al. Milbert (2003) – “Documentation for the GPS Benchmark Data Set of 23-July-98” IGeS Bulletin N.8, International Geoid Service, Milan, pp. 29–42, available online in http://www. ngs.noaa.gov/GEOID/GPSonBM96/gpsbmdoc.html (last accessed in August 14, 2007). Sj¨oberg, L.E. (2006) – “A new technique to determine geoid and orthometric heights from satellite positioning and Geopotential numbers”, J. Geod. 80, pp. 304–312. Tenzer, R., Vanicek P., Santos M., Featherstone W.E and Kuhn M. (2005) – “The rigorous determination of orthometric heights”, J. Geod. 79, pp. 82–89. US National Geodetic Survey (2003) – “GPS on Bench Marks for GEOID03”, available online in http://www. ngs.noaa.gov/GEOID/GPSonBM03/index.html (last accessed in August 14, 2007).

On Evaluation of the Mean Gravity Gradient Within the Topography R. Tenzer and A. Ellmann

Abstract To reduce the absolute gravity measurements onto the geoid, the vertical gravity gradient is usually applied. Alternatively, the reduction can be realized by applying the mean vertical gravity gradient. In this study, the mean gravity gradient along the plumbline within the topography is investigated. We demonstrate how the mean gravity gradient is related with the mass density distribution within the topography, the shape of the Earth and the vertical change of gravity gradient with the depth, whereas the effect of atmosphere is negligible. We also show how the errors due to uncertainties in topographical mass density distribution propagate to the errors in a prediction of the mean gravity gradient. The numerical analysis is conducted at the area of study in the Canadian Rocky Mountains.

vertical gravity gradient was used recently by Csap´o and Papp (2000), and R´ozsa and T´oth (2005). In this study we utilize a similar methodology as invented in evaluation of the mean gravity in Tenzer et al. (2005) for a prediction of the mean gravity gradient within the topography. We compute the mean gravity gradient as the difference of the gravity values evaluated on the Earth’s surface and on the geoid and divided by the orthometric height. The ground gravity anomalies, the terrain and density (atmospheric and topographical) models are used for a computation. The inaccuracy of prediction of the mean gravity gradient within the topography is mainly due to uncertainties within the topographical mass density distribution, the errors of a digital terrain model (DTM), and the errors of numerical methods used for a computation.

Keywords Density · Gravity · Mean gravity gradient · Topography

2 Mean Gravity Gradient 1 Introduction The fundamental principles for a prediction of the gravity inside the topography from the surface gravity measurements are given by the Poincare-Prey’s gravity gradient. A more accurate method of computing the

R. Tenzer Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands A. Ellmann Department of Civil Engineering, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia

The mean vertical gravity gradient W hh along the plumbline within the topography is defined as

1 W hh () = o H ()

Ho ()

h=0

∂ g(h, ) dh, ∂h

(1)

where H o is the orthometric height, and ∂ g/∂h the vertical gravity gradient. Disregarding the deflection of the vertical from the geocentric radial direction, the mean vertical gravity gradient W hh can be computed approximately as the mean radial gravity gradient W rr from

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1 W rr () = o H ()

o R+H  ()

r =R

∂ g(r, ) dr, ∂r

3 Mean Gravitational Gradients (2) by Topography and Atmosphere

The mean radial gravitational gradients due to the towhere ∂ g/∂r is the radial gravity gradient. The geocen- pography and atmosphere are evaluated by the Newton tric latitude φ and longitude λ represent the geocentric integration. The mean radial gravitational gradient by t direction , and r is the geocentric radius. In Eq. (2), topography W rr is computed from the mean radius of the Earth R is adopted in a spherical approximation of the geoidal surface. Solving the o  integral on the right-hand side of Eq. (2), we arrive at   R+H  ( ) G t W rr () ∼ ρ t (r  ,  ) = o H () g(rt , ) − g(r g , )  ∈o r  =R W rr () = . (3)    H o () ∂−1 r g , ; r  ,  × ∂r From Eq. (3), the mean radial gravity gradient W rr   within the topography is evaluated as the difference of ∂−1 rt , ; r  ,  − r 2 dr  d , (6) the gravity values computed at the Earth’s surface and ∂r on the geoidal surface and divided by the orthometric height H o . The geocentric radii of the Earth’s surface where ρ t is the topographical mass density distribution and of the geoidal surface are denoted as rt and r g re- function,  the Euclidean spatial distance,  the total o spectively. solid angle, and d = cos φ  dφ  dλ the infinitesimal The actual gravity g can be treated separately for surface element on the unit sphere. the components g t and g a generated by the topograThe numerical prediction of the mean gravity gradiphy and atmosphere, and the geoid-generated gravity ent within the topography was conducted at the rough g g (cf. Tenzer 2004). Moreover, the geoid-generated part of the Canadian Rocky Mountains bounded by gravity g g is sub-divided into the normal gravity γ and the parallels of 49◦ and 54◦ arc-deg northern geodethe geoid-generated gravity disturbance δg g . The fol- tic latitude and the meridians of 241◦ and 246◦ arclowing decomposition of the gravity g is then written deg eastern geodetic longitude. Over the area of the study the orthometric heights range from 460 m up to 2708 m, see Fig. 1. The lateral topographical mass den−3 t a + g (r, ) + g (r, ). (4) sity distribution varies from 2490 to 2900 kg m , see Fig. 2. The lateral model of topographical mass density disApplying the above decomposition of the gravity to tribution discretised for a regular geographical grid of Eq. (3), the mean radial gravity gradient W rr is com30 × 30 arc-sec was used. The detailed 3 × 3 arc-sec puted as a sum of the mean normal gravity gradient DTM was used for the numerical integration at the U rr , the mean radial gradient of the geoid-generated g g g vicinity (inner zone) of the computation point (ψ < gravity disturbance T rr = W rr − U rr (W rr denotes 150 arc-sec), while 30 × 30 arc-sec DTM up to the the mean radial gradient of the geoid-generated gravt spherical distance ψ of 57.5 arc-min. The 5 × 5 arcity) and the mean radial gravitational gradients W rr a min mean topographic heights were used for the reand W rr due to the topography and atmosphere respecminder of the 5 × 5 arc-deg near zone, and 30 × 30 tively; i.e., arc-min mean heights for the far zone. The analytical expression for the gravitational attraction generated by g t a W rr () ∼ = U rr + T rr () + W rr () + W rr (). (5) a rectangular prism (Nagy et al. 2000) was utilized for the integration within the inner zone. For the rest of Methods of computing the components of mean ra- the integration area, the Newton-Cotes numerical indial gravity gradient with the numerical examples are tegration was implemented for the surface integration domain, adopting the closed analytical expression for given in the next two sections. g(r, ) = γ (r, φ) + δg g (r, )

On Evaluation of the Mean Gravity Fig. 1 Topography (units in m)

Fig. 2 The lateral topographical mass density distribution (units in kg m−3 )

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the radial component of Newton’s integral (Gradshteyn and Ryzhik 1980). The computation of the mean radial gravitational gradient due to the topography was realized individually for the commonly adopted mean value 2670 kg m−3 of topographical mass density and for the laterally varying anomalous topographical mass density. The mean radial gravitational gradient due to the topography for the constant topographical mass density is shown in Fig. 3. We estimated that it varies between 0.149 and 0.238 mGal m−1 , and it is strongly correlated with the local topography. The effect of anomalous topographical density on the mean radial gravity gradient is shown in Fig. 4. From a numerical estimation, this effect varies from −0.013 to 0.014 mGal m−1 . It represents approximately 7–8% of the total topographic effect on the mean radial gravity gradient. The complete effect of topography on the mean radial gravity gradient is shown in Fig. 5. It varies from 0.147 to 0.237 mGal m−1 . Disregarding the water bodies, the variation of the actual topographical mass density is mostly within ±300 kg m−3 of the commonly adopted mean value of density 2670 kg m−3 . The effect of anomalous topographical mass density then amounts to about 10% of

Fig. 3 The mean radial gravitational gradient due to the topography computed using the constant topographical mass density (units in mGal m−1 )

R. Tenzer and A. Ellmann

the total contribution of topography on the mean gravity gradient except for some parts of the world, where larger density variations were documented (up to 20–30%). It can be thus empirically estimated, that the errors of the mean gravity gradient due to uncertainties within the actual topographical mass density distribution may reach at most up to a few μGal m−1 . With reference to the Poincare-Prey’s gravity gradient (cf., Heiskanen and Moritz 1967, p. 164), the contribution of topography on the vertical gravity gradient is about 0.2238 mGal m−1 disregarding the terrain correction. From Fig. 3, this value coincides with accuracy better than 15 μGal m−1 with our upper estimation (0.238 mGal m−1 ) of the contribution of topography on the mean radial gravity gradient computed for the mean topographical density. As follows from the corresponding lower estimation (0.149 mGal m−1 ), the spherical terrain correction has to be taken into account in computing the vertical gravity gradient and its mean value. Otherwise, errors due to disregarding the terrain can reach 70 μGal m−1 or more depending on the local topography. The mean radial gravitational gradient by atmoa sphere W rr is given by

On Evaluation of the Mean Gravity Fig. 4 The effect of laterally varying anomalous topographical mass density distribution on the radial gravity gradient (units in mGal m−1 )

Fig. 5 The total effect of topography on the radial gravity gradient (units in mGal m−1 )

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a W rr ()

G ∼ = o H () 



r A

There are two principal approaches for computing the mean radial geoid-generated gravity disturbance gradient depending on the available type of gravity data at the Earth’s surface. Specifically, we distinguish two methods which are formulated for the gravity anomalies and the gravity disturbances as the input data:

ρ a (r  ,  )

 ∈or  =R+H o ( )

∂−1 (r g , ; r  ,  ) ∂r  ∂−1 (rt , ; r  ,  ) 2   − r dr d , ∂r

×

(7)

where ρ a is the atmospheric mass density distribution function, and r A the upper limit of the atmosphere. The American Standard Atmosphere density model was used for a computation. We estimated that the effect of atmosphere on the mean radial gravity gradient is at least 10−3 smaller than the effect of the topography and thus completely negligible.

4 Mean Geoid-Generated Gravity Gradient As stated previously, the mean radial geoid-generated gravity gradient can be computed as the sum of the two components, namely the mean normal gravity gradient U rr and the mean radial gradient of the geoidg generated gravity disturbance T rr . The mean normal gravity gradient within the topography can be computed approximately as the mean normal gravity gradient along the normal to the surface of the reference ellipsoid at the interval between the telluroid and the ellipsoidal surface. The difference between the orthometric and normal heights is disregarded as well as the change of the normal gravity gradient within the geoidal height. The mean normal gravity gradient can be computed explicitly according to the SomiglianaPizzetti’s theory of the normal gravity field of the ellipsoid of revolution (Pizzetti 1911, Somigliana 1929). Alternatively, the value −0.3086 mGal m−1 of the normal linear gravity gradient can be adopted, neglecting a very small inaccuracy due to the variation of the normal gravity gradient with height. The mean radial geoid-generated gravity disturg bance gradient T rr reads T rr () ∼ = g

δg g (rt , ) − δg g (r g , ) H o ()

1. If the gravity disturbances are available at the Earth’s surface, the computation is realized in three successive steps: (i) the direct effects of the topography and atmosphere (g t and g a ) are subtracted from the gravity disturbances δg at the Earth’s surface to obtain the geoid-generated gravity disturbances δg g ; (ii) the geoid-generated gravity disturbances are downward continued onto the geoidal surface (Eq. (10)); and (iii) the mean radial geoid-generated gravity disturbance gradient is computed according to Eq. (8). 2. If the gravity anomalies are available at the Earth’s surface, the computation is realized again in three successive steps: (i) the direct and secondary indirect effects of the topography and atmosphere (g t − 2r −1 V t and g a − 2r −1 V a , where V t and V a denote the gravitational potentials of the topography and atmosphere respectively) are subtracted from the gravity anomalies g at the Earth’s surface; (ii) the geoid-generated gravity anomalies

g g are downward continued onto the geoidal surface (Eq. (9)); and (iii) the differences between the geoid-generated gravity disturbances on the geoidal surface and the Earth’s surface in the nominator on the right-hand side of Eq. (8) are computed by the integral convolution of the radial derivative of the Stokes kernels and the geoid-generated gravity anomalies on the geoid (Eq. (11)). The discrete downward continuation of gravity anomalies g g is given by (Bjerhammar 1962; see also Bjerhammar 1963, 1987) −1 ∼ R gnz Rt (gt − gfz g =FP t ),

(9)

where gt is the vector of gravity anomalies g g (xi ) observed at the points {xi : i = 1 . . . I } at the Earth’s surface. The vector gfz t of the far-zone contributions to gravity anomalies is subtracted from gt . The solution of Eq. (9) provides the vector gnz g of the nearzone contributions to gravity anomalies g g (yi ) which are parameterized at the geoidal points {yi : i = (8) 1 . . . I }. We assume that the horizontal positions of x i

On Evaluation of the Mean Gravity

259

and yi are identical, and these points form the coarse grid with the equal-angular step size. The matrix of discretised Poisson integral equations is denoted as P; Rt is the diagonal matrix of the geocentric radii {|xi | : i = 1 . . . I }; i.e., {Rt }i, j = δi, j |xi |, where δi, j = 1 and δi, j = 0 otherwise; and F is the diagonal filter matrix of the solution. To unify the size of the near-zone contribution for each equation in the system of Eq. (9), zero values are assigned to all coefficients of the matrix P which extend the near-zone integration sub-domain, i.e., {P}i, j = 0 for all ψi, j > ψo , where ψo is the maximal spherical angle of the near-zone integration sub-domain. The diagonal elements of the filter matrix F are formed so that the matrix-vector product of F and P−1 Rt (g t − g fz t ) provides the solution only for the values of the vector g nz g which comprise entirely the near-zone contribution. By analogy with Eq. (9), the gravity disturbances δg g are downward continued according to −1 ∼ R δgnz Rt (δgt − δgfz g =FP t ),

(10)

where δgt is the vector of the gravity disturbances δg g (xi ) from which the vector of the far-zone

Fig. 6 The mean radial geoid-generated gravity disturbance gradient (units in mGal m−1 )

nz contributions δgfz t is subtracted, and δgg is the vector of the near-zone contributions to gravity disturbances δg g (yi ) on the geoid. Data interpolation and advanced numerical integration are indispensable in forming the diagonal and near-diagonal elements of the matrix P. Since the downward continuation is an ill-posed problem in sense of large high-frequency uncertainties in results due to errors within input gravity data, the regularization should be applied. For details see for instance Tikhonov and Arsenin (1977) or alternatively Martinec (1996). The mean radial geoid-generated gravity disg turbance gradient T rr is computed as the integral convolution of the geoid-generated gravity anomalies on the geoid and the difference between the radial derivatives of the Stokes and extended Stokes functions ∂S(ψ)/∂r and ∂S(r, ψ)/∂r

1 R g T rr () ∼ = o 4π H () × g

g

 

∂S(Ψ ) ∂S(r, Ψ ) − ∂r ∂r

 ∈O (r g ,  )d .



(11)

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In out test, the mean gravity disturbances were distributed at the regular geographical grid of 5 × 5 arcmin. The near-zone sub-domain up to 2 arc-deg of the spherical distance was considered. From the numerical estimation, the mean radial geoid-generated gravity disturbance gradient varies between −0.015 and 0.015 mGal m−1 , see Fig. 6. Comparing Figs. 5 and 6, the contribution to the mean radial gravity gradient due to the mass irregularities within the geoid is much smaller than the corresponding contribution due to the topography.

Table 1 The components of the mean radial gravity gradient (units in mGal m−1 )

The variations of the mean radial gravity gradient due to the contributions of topography for the constant mass density of 2670 kg m−3 , the anomalous lateral topographical mass density distribution and the mass irregularities enclosed within the geoid are summarized in Table 1. From our predictions, the dominant contribution is due to the topography (of the mean density), while the contributions due to the anomalous topographical density and mass irregularities within the

The components of the mean gravity gradient due to:

Max Min Mean STD

Fig. 7 The mean gravity gradient (units in mGal m−1 )

5 Conclusions

Topography of mean density

Anomalous topographical Complete effect density of topography

Mass irregularities within the geoid

Mean radial gravity gradient

0.238 0.149 0.212 0.010

0.014 −0.013 −0.007 0.005

0.015 −0.015 0.001 0.004

−0.066 −0.166 −0.102 0.0114

0.237 0.147 0.205 0.009

On Evaluation of the Mean Gravity

geoid are less than a few μGal m−1 . Moreover, the effect of atmosphere on the mean gravity gradient is completely negligible. Results of the numerical investigation at the area of study in the Canadian Rocky Mountains reveal that the mean gravity gradient varies between −0.166 mGal m−1 and −0.066 mGal m−1 (Fig. 7). From a comparison of our prediction with the Poincare-Prey’s gravity gradient (−0.0848 mGal m−1 ), large discrepancies can be recognised especially in mountainous regions. As discussed in Sect. 3, these discrepancies are mostly due to the disregarding the terrain correction in a definition of the Poincare-Prey’s gravity gradient. Concluding from our results, the shape of the Earth (terrain) contributes mainly to the differences between the actual gravity gradient and the normal gravity gradient. In contrary, contributions of the topographical mass density variations and of the mass irregularities within the geoid are much smaller to the extent that their total contribution on the gravity gradient is less than a few μGal m−1 . Acknowledgments The gravity data and digital terrain and density models used in this study were kindly supplied by the Geodetic Survey Division of Natural Resources Canada.

References Bjerhammar, A., (1962). Gravity reductions to a spherical surface. Royal Institute of Technology, Division of Geodesy, Stockholm.

261 Bjerhammar, A., (1963). A new theory of gravimetric geodesy. Royal Institute of Technology, Division of Geodesy, Stockholm. Bjerhammar, A., (1987). Discrete physical geodesy, Report, 380, Dept. of Geodetic Science and Surveying. The Ohio State University, Columbus. Csap´o, G., and G. Papp (2000). Measuring and modelling of the vertical gradient of gravity – Hungarian examples. Geomatikai K¨ozlem´enyek, III, pp. 109–129. Gradshteyn, I.S., and I.M. Ryzhik (1980). Tables of Integrals, Series and Products. translated by Jeffrey A, Academic Press, New York. Heiskanen, W.H., and H. Moritz (1967). Physical geodesy. San Francisco, W.H., Freeman and Co. Martinec, Z., (1996). Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. J Geod, 70, pp. 805–828. Nagy, D., G. Papp, and J. Benedek (2000). The gravitational potential and its derivatives for the prism. J Geod, 74, pp. 552–560. Pizzetti, P., (1911). Sopra il calcolo teorico delle deviazioni del geoide dall ellissoide. Atti R Accad Sci, Torino, V. 46. R´ozsa, S., and G. T´oth (2005). Prediction of vertical gravity gradients using gravity and elevation data. IAG Symposia series 128, A window on the Future of Geodesy, pp. 344–349. Somigliana, C., (1929). Teoria Generale del Campo Gravitazionale dell’Ellisoide di Rotazione. Memoire della Societa Astronomica Italiana, IV. Milano. Tenzer, R., (2004). Discussion of mean gravity along the plumbline. Stud. Geophys. Geod., 48 (2), pp. 309–330. Tenzer, R., P. Van´ıcˇ ek, M. Santos, W.E. Featherstone, M. Kuhn (2005). The rigorous determination of orthometric heights. J Geod 79, pp. 82–92. Tikhonov, A.N., and V.Y. Arsenin (1977). Solutions of ill-posed problems, V.H., Vinston, Washington, D.C.

Comparison of Techniques for the Computation of a Height Reference Surface from Gravity and GPS-Levelling Data R. Tenzer, R. Klees, I. Prutkin, T. Wittwer, B. Alberts, U. Schirmer, J. Ihde, G. Liebsch and U. Sch¨afer

Abstract Three different techniques for quasigeoid modelling from gravity and GPS-levelling data are compared: (i) a penalized least-squares technique using spherical radial basis functions provides a gravimetric quasigeoid solution; the combination with R. Tenzer Delft Institute of Earth Observation and Space Systems (DEOS), TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands R. Klees Delft Institute of Earth Observation and Space Systems (DEOS), TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands I. Prutkin Delft Institute of Earth Observation and Space Systems (DEOS), TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands T. Wittwer Delft Institute of Earth Observation and Space Systems (DEOS), TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands B. Alberts Delft Institute of Earth Observation and Space Systems (DEOS), TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands U. Schirmer Bundesamt f¨ur Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, 60598 Frankfurt am, Main, Germany J. Ihde Bundesamt f¨ur Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, 60598 Frankfurt am, Main, Germany G. Liebsch Bundesamt f¨ur Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, 60598 Frankfurt am, Main, Germany U. Sch¨afer Bundesamt f¨ur Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, 60598 Frankfurt am, Main, Germany

GPS-levelling data is formulated as the solution of a Cauchy boundary-value problem for the Laplace operator. This solution when added to gravimetric solution yields the final quasigeoid; (ii) a direct least-squares solution using gravimetric and GPS-levelling data as observations and point masses as parameterization of the disturbing potential. The inconsistency between GPS-levelling data and gravimetric data is treated assigning high weights to GPS-levelling data in the least-squares adjustment; (iii) a least-squares collocation technique for computing a gravimetric quasigeoid. The combination with GPS-levelling data is realized using a low-degree polynomial corrector surface estimated from the differences between gravimetric height anomalies and height anomalies from GPS-levelling data. The three methods are compared using real data for an area in Germany. The results reveal a very similar performance of these methods if gravity data and GPS-levelling data are combined, whereas gravimetric quasi-geoid solutions differ significantly.

Keywords Corrector surface · GPS-levelling · Least-squares collocation · Quasigeoid · Spherical radial basis functions

1 Objective The performance of three approaches for the computation of a height reference surface from regional terrestrial gravity data and GPS-levelling data is investigated: (i) a penalized least-squares technique using spherical radial basis functions (SRBFs) and an innovation function; (ii) a combined adjustment of gravity

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and GPS-levelling data using point masses; (iii) a leastsquares collocation approach and a corrector surface. The first approach is realized in two successive steps: firstly, gravity data are used to estimate the coefficients of a SRBF representation of the disturbance potential; secondly, an innovation function is estimated from differences between gravimetric height anomalies and GPS-levelling data, and added to the gravimetric solution. In the stand-alone gravimetric solution, a data-driven strategy is implemented to select automatically the horizontal positions and depths of the SRBFs using Generalized Cross Validation (GCV), see e.g. Wahba (1977). Variance Component Estimation (VCE) techniques are utilized for the observation group weighting and the selection of the optimal regularization parameter (Kusche 2003). For more details of this technique see Klees et al. (2007). After finding the gravimetric solution, the combination with GPS-levelling data is formulated in terms of a Cauchy boundary-value problem for the Laplace operator, which is solved numerically (Prutkin and Klees 2007). The solution, which we call the innovation function, is used to correct the gravimetric height anomalies similar to what a corrector surface does. The second approach performs a least-squares adjustment of gravity data and GPS-levelling data simultaneously (Liebsch et al. 2006). The parameterization of the gravity field is realized by two layers of point masses at predefined 3D-locations. The solution is forced to fit the GPS-levelling data by assigning high weights to them in the least-squares adjustment. The third approach uses a least-squares collocation to process the gravity data. After finding the gravimetric solution, the combination with GPS-levelling data is performed using a 6-parameter corrector surface, which models the systematic differences between gravimetric quasigeoid heights and geometric quasigeoid heights provided by the GPS-levelling data.

R. Tenzer et al.

GPS-levelling points. The data have been reduced by the contribution of EIGEN-CG03C (F¨orste et al. 2005) and the effect of topography, resulting in residual gravity anomalies and residual geometric height anomalies from GPS-levelling data. Within the computation area, the residual gravity anomalies vary from −48.3 to 57.4 mGal; the mean value is 2.9 mGal. The residual geometric height anomalies vary from −1.21 to −0.23 m; the mean value is −0.61 m.

3 Numerical Experiment

Penalized least-squares using SRBFs: The data-adaptive SRBF algorithm (Klees et al. 2007) selected 10,000 (coarse grid) Poisson-kernel type SRBFs on an equal-angular grid with 3.3 km grid spacing and 92 additional (local refinement) SRBFs located below data points. This is about 17% of the number of observations. The optimal depth 6 km of the coarse grid SRBFs was found by GCV. This corresponds to a 3-km correlation length of the kernel, which is obtained when the gravity anomaly operator is applied to the Poisson kernel. This is much less than the 19 km correlation length of the empirical gravity anomaly auto-covariance function. The depths of the local refinement SRBFs were selected individually within the interval from 0.1 to 6.0 km using GCV. The gravity anomalies over Eastern and Western Germany data were assigned to different observation groups as it was expected that the accuracy is different. From VCE, we obtained a standard deviation of 0.47 mGal for the Eastern Germany data set and 0.45 mGal for the Western Germany data set. Hence, both data sets have about the same quality. These estimates were then adopted as the a-priori standard deviations in a stand-alone processing of the residual gravity anomalies by means of a penalized least-squares technique. VCE was used to obtain the optimal regularization parameter. The regularization matrix was the identity 2 Data Sets matrix. Figure 1 shows the histogram of the least-squares The comparison of the three techniques is done for an residuals. The residuals vary between −4.5 and area in Germany, bounded by the parallels of 51 and 4.3 mGal; the mean value equals 0.0 mGal; the RMS 53◦ Northern spherical latitude and 11 and 13◦ Eastern of the residuals is 0.49 mGal. The differences between the geometric height spherical longitude. The data area extends from 50.5 to 53.5◦ and 10.5 to 13.5◦ respectively. The data set com- anomalies and the gravimetric height anomalies at the prises 59351 point surface gravity anomalies and 154 GPS-levelling points (cf. Fig. 2) were taken as input

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Fig. 1 Histogram of the SRBF least-squares residuals (min = −4.5 mGal, max = 4.3 mGal, mean = 0.0 mGal, RMS = 0.49 mGal)

Fig. 2 Differences between geometric (from GPS-levelling data) and gravimetric height anomalies (min = 0.43 m, max = 0.77 m, mean = 0.56 m)

data to estimate an innovation function according to the approach of Prutkin and Klees (2007) (cf. Fig. 3). This innovation function represents a correction to be applied to the gravimetric height anomalies in order to obtain the final quasi-geoid heights. The histogram of the differences between observed and modelled height anomaly differences at the GPS-levelling points is shown in Fig 4. The RMS

difference between the innovation function and the observed differences between gravimetric and geometric height anomalies is 0.7 cm; the minimum difference is −2.2 cm, the maximum difference is 2.1 cm. This is slightly better than the expected accuracy of the observed differences, which may imply that the innovation function models to some extent also data noise.

266 Fig. 3 Innovation function (min = 0.43 m, max = 0.77 m, mean = 0.56 m. RMS fit to the data is 0.007 m)

Fig. 4 Histogram of the differences between observed and modelled height anomaly differences at the GPS-levelling points (min = −2.2 cm, max = 2.1 cm, mean = 0.0 cm, RMS = 0.7 cm)

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Comparison of Techniques

Combined point mass approach: In the combined point mass approach, the parameterization of the residual disturbance potential is realized by two spherical layers of point masses at the depths of 5–30 km respectively. Two spherical grids have been used: 2 × 3 at the depth 5 km (5400 point masses) and 12 × 18 at the depth 30 km (150 point masses). The 5550 point masses form the complete parameterization of the residual disturbance potential. This is only 9% of the total number of observations; hence the smallest number of unknowns among the three methods. For the combined least-squares adjustment of gravity data and GPS-levelling data, the a priori standard deviations 1 mGal and 0.15 cm were chosen for gravity anomalies and height anomalies from GPS-levelling data, respectively. In comparison to the a posteriori standard deviations, the gravity data are down-weighted by a factor 2, and the GPS-levelling data are up-weighted by a factor 10. In this way, the quasigeoid model fits quite well to the geometric height anomalies derived from GPS-levelling data. This procedure can be seen as an alternative to the two step approaches consisting in gravimetric geoid modelling and the estimation of a corrector surface. Figure 5 shows the histogram of the gravity anomaly residuals of the least-squares adjustment. The residuals range from −4.1 to 5.2 mGal; the mean value is 0.0 mGal; the RMS of the residuals is

Fig. 5 Histogram of the gravity anomaly residuals of the combined least-squares adjustment of gravity data and GPS-levelling data (min = −4.1 mGal, max = 5.2 mGal, mean = 0.0 mGal, RMS = 0.73 mGal)

267

0.73 mGal. The larger RMS value is due to the weighting for the gravity anomalies and the height anomalies mentioned above. Hence, the least-squares solution fits the GPS-levelling data very well, which in turn leads to larger gravity anomaly residuals. Although this approach lacks a sound mathematical-statistical basis, the quality of the quasi-geoid is not worse than the solutions obtained with the other two approaches. We also did a stand-alone adjustment of the gravity anomalies using the same parameterization. The histogram of the gravity anomaly residuals after the stand-alone least-squares adjustment is shown in Fig. 6. The RMS of the gravity anomaly residuals reduces to 0.34 mGal. This is even smaller then the RMS of the SRBF approach with 0.49 mGal and indicates that the parameterization has been chosen appropriately. Figure 7 shows the histogram of the residuals at the GPS/leveling points. They range from −2.6 to 2.4 cm; the mean value is 0.0 cm; the RMS is 0.7 cm. Least-Squares Collocation (LSC): We used the GRAVSOFT software package (Tscherning et al. 1994) to compute a least-squares collocation solution using the gravity anomaly data set. The empirical and the analytical gravity anomaly auto-covariance functions are shown in Fig. 8. The correlation length of the gravity anomaly auto-covariance functions is about 19 km.

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Fig. 6 Histogram of the gravity anomaly residuals of the stand-alone least-squares adjustment (min = −4.4 mGal, max = 4.9 mGal, mean = 0.0 mGal, RMS = 0.34 mGal)

Fig. 7 Histogram of the height anomaly residuals at GPS-levelling points after the combined least-squares adjustment (min = −2.6 cm, max = 2.4 cm, mean = 0.0 cm, RMS = 0.7 cm)

The histogram of the differences between observed and predicted gravity anomalies is shown in Fig. 9. The residuals range from −3.8 to 4.2 mGal; the mean is 0.0 mGal; the RMS of the residuals is 0.53 mGal. We note that only 5316 observations were used in the LSC solution, which were selected randomly from the data set. Different realizations of the random selection or selecting more data have a negligible effect on the gravity field solution.

To combine the GPS-levelling data with the gravity anomaly data, a corrector surface was fitted through the differences between the predicted height anomalies at the GPS-levelling points and the geometric height anomalies being derived from GPS-levelling data (cf. Fig. 10). This can be seen as a simplistic approach compared with LSC with parameters. An algebraic polynomial of degree two in latitude and longitude has been fitted through the differences using

Comparison of Techniques

269

Fig. 8 The empirical and analytical gravity anomaly auto-covariance functions

least-squares. Figure 11 shows the corrector surface. The height anomaly residuals at the GPS-levelling points range from −3.5 to 3.5 cm; the mean is 0.0 cm; the RMS of the residuals is 1.1 cm; the histogram of the height anomaly residuals at the GPS-levelling points is shown in Fig. 12. Alternatively, we computed an innovation function using the differences at the GPS-levelling points as input data (cf. Fig. 13). The accuracy of approximation

Fig. 9 Histogram of the LSC residuals (min = −3.8 mGal, max = 4.2 mGal, mean = 0.0 mGal, RMS = 0.53 mGal)

was again chosen 0.7 cm. The histogram of the differences between innovation function and observed differences is shown in Fig. 14. A comparison of the histograms of Figs. 12 and 14 reveals a better fit of the innovation function to the observed height anomaly differences at the GPS-levelling points compared with the 6-parameter corrector surface.

270 Fig. 10 Differences between geometric and gravimetric height anomalies (min = 0.49 m, max = 0.56 m, mean = 0.53 m)

Fig. 11 The 6-parameter corrector surface (min = 0.51 m, max = 0.56 m, mean = 0.53 m)

R. Tenzer et al.

Comparison of Techniques Fig. 12 Histogram of the residuals of a 6-parameter corrector surface fitted through the differences between gravimetric height anomalies (from LSC) and geometric height anomalies at the GPS-levelling points (min = −3.5 cm, max = 3.5 cm, mean = 0.0 cm, RMS = 1.1 cm)

Fig. 13 Innovation function (min = 0.49 m, max = 0.56 m, mean = 0.53 m)

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Fig. 14 Histogram of the differences between innovation function and observed height anomaly differences at the GPS-levelling points (min = −1.6 cm, max = 1.6 cm, mean = 0.0 cm, RMS = 0.7 cm)

4 Summary and Conclusions

vide almost the same RMS of about 0.7 cm, while the accuracy of the LSC/corrector surface approach is Table 1 shows the RMS of the gravity anomaly resid- 1.1 cm. If the gravimetric height anomalies of the LSC uals for the three methods. It is about the same for solution are combined with GPS-levelling data using the penalized least-squares technique using SRBFs and an innovation function, the RMS can be reduced to the LSC method. The larger RMS for the combined 0.7 cm by construction. Tables 3 and 4 show the differences between the point mass approach (RMS of gravity anomaly residuals about 0.7 mGal) is due to the assignment of high three solutions obtained at the GPS-levelling points weights to GPS-levelling data in the combined least- and at the equal-angular grid. The differences are squares adjustment. The RMS for a gravity only solu- mostly random, without any systematic trend. The main conclusion is that the penalized leasttion of the point mass approach is significantly smaller squares technique using SRBFs in combination with (0.34 mGal). an innovation function, the combined point mass Table 2 shows the RMS of the height anomaly difapproach, and a LSC method in combination with a ferences at the GPS-levelling points. The innovation 6-parameter corrector surface provide quasi-geoids, function and the combined point mass approach proTable 1 Comparison of accuracy in terms of RMS of gravity anomaly residuals Technique

Min (mGal)

Max (mGal)

Mean (mGal)

RMS (mGal)

Penalized least-squares using SRBFs Combined point mass approach Gravity only solution of point mass approach Least-squares Collocation

−4.5 −4.1 −4.4 −3.8

4.3 5.2 4.9 4.2

0.0 0.0 0.0 0.0

0.49 0.73 0.34 0.53

Table 2 Comparison of accuracy in terms of RMS of differences between observed and computed residual height anomalies Technique

Min (cm)

Max (cm)

Mean (cm)

RMS (cm)

Innovation function Combined point mass approach Corrector surface

−2.2 −2.6 −3.5

2.1 2.4 3.5

0.0 0.0 0.0

0.7 0.7 1.1

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Table 3 Height anomaly differences between the 3 approaches at the GPS-levelling points Differences

Min (cm)

Max (cm)

Mean (cm)

RMS (cm)

Innovation function – combined processing Innovation function – corrector surface Combined processing – corrector surface

−2.5 −2.8 −2.2

2.1 3.4 2.0

0.0 0.0 0.0

0.6 1.1 0.8

Table 4 Height anomaly differences between the 3 approaches at the equal-angular grid Differences

Min (cm)

Max (cm)

Mean (cm)

RMS (cm)

Innovation function – combined processing Innovation function – corrector surface Combined processing – corrector surface

−2.8 −3.0 −2.5

2.1 3.6 2.2

-0.1 0.2 0.1

0.7 1.3 1.0

which are statistically indifferent. The differences between the 6-parameter corrector surface and the innovation function are within the uncertainty of the input data. The advantage of the innovation function approach is that it takes into account the mathematical nature of the observed differences between a gravimetric quasi-geoid and GPS-levelling data (cf. Prutkin and Klees 2007). Moreover, there is no need to select a proper parameterization as any corrector surface requires. The code implementation of the least-squares approach using SRBFs in combination with an innovation function offers an advantage in such a way, that it provides an adaptive and convenient choice of the parameters, in accordance with the distribution and quality of the input data. Hence this approach can be used easily for different areas and data sources.

References F¨orste, C., F. Flechtner, R. Schmidt, U. Meyer, R. Stubenvoll, F. Barthelmes, R. K¨oning, K.H. Neumayer, M. Rothacher, Ch. Reigber, R. Biancale, S. Bruinsma, J.M. Lemoine, J.C. Raimondo (2005). A new high resolution global gravity field model derived from Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data. Poster Presented at EGU General Assembly 2005, Vienna, Austria, 24–29, April 2005.

Klees, R., R. Tenzer, I. Prutkin, T. Wittwer (2007). A data-driven approach to local gravity field modelling using spherical radial basis functions. Journal of Geodesy (submitted). Kusche, J., (2003). A Monte-Carlo technique for weight estimation in satellite geodesy. Journal of Geodesy, 76, pp. 641–652. Liebsch, G., U. Schirmer, J. Ihde, H. Denker, J. M¨uller (2006) Quasigeoidbestimmung f¨ur Deutschland. DVW Schriftenreihe, Band 49. Prutkin, I., and R. Klees (2007). On the non-uniqueness of local quasi-geoids computed from terrestrial gravity anomalies. Journal of Geodesy, DOI: 10.1007/s00190-007-0161-1. Tscherning, C.C., P. Knudsen, and R. Forsberg (1994). Description of the GRAVSOFT package. Geophysical Institute, University of Copenhagen, Technical Report, 1991, 2nd Ed. 1992, 3rd Ed. 1993, 4th Ed. 1994. Wahba, G., (1977). A survey of some smoothing problems and the method of generalized cross-validation for solving them. Applications of Statistics, P.R. Krishnaiah, Ed. Amsterdam, The Netherlands: North-Holland, pp. 507–523.

The Inversion of Poisson’s Integral in the Wavelet Domain M. Elhabiby and M.G. Sideris

Abstract A wavelet transform algorithm combined with a conjugate gradient method is used for the inversion of Poisson’s integral (downward continuation), used in airborne gravimetry applications. The wavelet approximation is dependent on orthogonal wavelet base functions. The integrals are approximated in finite multiresolution analysis subspaces. Mallat’s algorithm is used in the multiresolution analysis of the kernel and the data. The full solution with all equations requires large computer memory, therefore, the multiresolution properties of the wavelet transform are used to divide the full solution into parts at different levels of wavelet multiresolution decomposition. Global wavelet thresholding is used for the compression of the kernel and because of the fast decrease of the kernel towards zero, high compression levels are reached without significant loss of accuracy. Hard thresholding is used in the compression of the kernel wavelet coefficients matrices. A new thresholding technique is introduced. A first-order Tikhonov regularization method combined with the L-curve is used for the regularization of this problem. First, Poisson’s integral is inverted numerically with the full matrix without any thresholding. The solution is obtained using the conjugate gradient method after 28 iteration steps with a root mean square error equal to 5.58 mGal in comparison to the reference data. Second, the global M. Elhabiby Department of Geomatics Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada, T2N 1N4 M.G. Sideris Department of Geomatics Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada, T2N 1N4

hard thresholding solution achieved a 94.5% compression level with less than 0.1 mGal loss in accuracy. These high compression levels lead to large savings in computer memory and the ability to work with sparse matrices, which increases the computational speed.

Keywords Wavelet multiresolution analysis · Hard thresholding · Inverse problem · Downward continuation · Airborne gravimetry · Conjugate gradient method

1 Introduction From the mid 1980s until now, wavelet techniques have been used in many applications that involve signal processing, such as image processing, medical diagnostics, geophysics, pattern recognition, electromagnetic wave scattering, boundary value problems, and so on (Goswami and Chan, 1999). The wavelet transform is a very efficient algorithm for decomposing and reconstructing signals (Chan, 1995). It is a very powerful tool for evaluating singular geodetic integrals because of its localization and compression properties (Gilbert and Keller, 2000). Kernels with singularity decay from the singular point rapidly and smoothly (Van´ıcek and Christou, 1994). The wavelet transform of such kernels leads to a significant number of small value wavelet coefficients. Thus, high compression levels of the kernels can be achieved. These properties were the main reason behind testing the wavelet technique in the downward continuation problem for airborne gravimetry applications.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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The inversion of Poisson’s integral for downward continuation was evaluated by two-dimensional FFT by Bl´aha et al. (1996) and Wu (1996). Forsberg (2002) tested different operational methods based on collocation and FFT, and applied these to the downward continuation of airborne surveys over Greenland and Svalbard. These approaches rely on stationary assumptions. The main advantage of the wavelet approach is its ability to deal with non-stationary environment. Keller (1995) introduced a theoretical harmonic downward continuation technique using a Haar Wavelet. In this paper, the main objective is the numerical inversion of Poisson’s integral in the wavelet domain, using the new wavelet-optimization combined algorithm developed by Elhabiby (2007). The algorithm in this study is a combination of the orthogonal wavelet transform by Mallat’s algorithm (Mallat, 1997), hard thresholding, a conjugate gradient algorithm, and Tikhonov regularization.

2 Poisson’s Integral

dσ =

R(r 2p − R 2 )



1 Rδg D (ϕ, λ)dσ 4πr p l3 (1) where R is the radius of the sphere, r p is the radial distance of the computational point, dσ is the surface element on the sphere, and l is the spatial distance between the data point and the computational point. Equation (1) is presented in the planar approximation, which is a special representation of the spherical case on a plane, as follows (Bl´aha et al., 1996): δgu (ϕ p , λ p ) =

1 δgu (x p , y p , z 0 ) = ... 2π  ... z 0 [(x p − x)2 + (y p − y)2 + z 02 ]−3/2 . . . (5) . . . δg D (x, y, 0)dxdy where δgu is the upward continued gravity disturbance at the flight elevation z 0 , δg D is the gravity disturbance at level zero. The downward continuation is evaluated through the inversion of Eq. (5), where δg D will be the unknown and δgu will be the observation. The inversion of Poisson’s integral in the wavelet domain will be done through an optimization procedure (gradient method). For gridded data with equal spacing ( x and y), Poisson’s kernel is as follows:

3 The Wavelet Transform The discrete wavelet transform (DWT) coefficients ω j,k of a signal or a function f (x) are computed by the following inner product (Chui et al., 1994): ω j,k =  f (x), ψ j,k (x)

(7)

where ψ j,k is the wavelet expansion function and both j and k are integer indices for the scale and translation of the wavelet function, respectively. The inverse wavelet transform is used for the reconstruction of the signal from the wavelet coefficients ω j,k by (Mallat, 1997) f (x) =



ω j,k ψ j,k (x)

(8)

k

Equations (7) and (8) are named as analysis and synthesis, respectively. In this research, the dyadic l = [(x p − x)2 + (y p − y)2 + z 02 ]1/2 , (3) wavelet algorithm (decreasing the number of coefficients per level to half) developed by Daubechies is used (Daubechies, 1992). ≈ 2z 0 R,

and

z0 ( x y)[(x p −x)2 +(y p −y)2 +z 02 ]−3/2 (6) 2π

j

r 2 − R 2 = (r − R)(r + R) = z 0 (r + R)

(4)

Therefore, in planar approximation Poisson’s integral is

Kp = The Poisson integral is used for the upward continuation problem and is inverted for the downward continuation problem. It is based on the first boundary value problem of physical geodesy, Dirichlet’s problem. The Poisson integral is applied for the harmonic continuation of the gravity disturbance as follows (Heiskanen and Mortiz, 1967):

1 dx dy. R2

(2)

The Inversion of Poisson’s Integral Fig. 1 Daubechies with 4 vanishing moments, scaling function to the left and wavelet function to the right

277

1

1 0.5

0.5

0 0

–0.5

0

2

4

The new implementation mainly relies on the extension of the 1D–2D wavelet transform. For the 2D wavelet transform, the Mallat algorithm will go through a tensor product of two different directional 1D wavelet transforms (Chui et al., 1994):

6

0

2

4

6

The thresholding value is chosen as follows: δ = median(|ωl,k |)

(14)

If it is equal to zero, then ϕ(x, y) = ϕ(x).ϕ(y)

(9)

ψ H (x, y) = ψ(x).ψ(y)

(10)

ψ (x, y) = ϕ(x).ψ(y)

(11)

ψ D (x, y) = ψ(x).ψ(y)

(12)

V

δ=

1 max(|ωl,k |) 20

(15)

For more details, see Donoho and Johnstone (1994). A new approach is tested in this study in order to Daubechies (db4) wavelets are used in this research optimize the compression levels achieved with an ac(Fig. 1). ceptable practical accuracy. This new approach relies The choice of the wavelet family in this study is on increasing the thresholding value estimated from done base on the properties required for the efficient Eqs. (14) or (15) four times, each time by one order wavelet representation of the inversion of the Poisson of magnitude (Elhabiby, 2007). integral. The Beylkin approach (Beylkin et al., 1991) requires orthogonal wavelets. Mallat’s algorithm for the multi-resolution analysis is useful only if the wavelet filters h and g are finite and compact support. 3.2 Multiresolution Representation of the The ideal choice for fulfilling the three properties— Poisson Integral orthogonal, finite, and compact support—is the Daubechies wavelets (Fig. 1) (Elhabiby, 2007). The Poisson integral is represented in the wavelet domain using multiresolution analysis (four levels of decomposition are used here). 3.1 Wavelet Thresholding First the 2D wavelet transform of the gravity disturbances is introduced as follows: Wavelet thresholding is a technique used to compress  the wavelet coefficients of the Poisson kernel. Wavelet appr. αδg (x, y) = δg D ϕ(x)ϕ(y) dx dy coefficients (absolute) larger than certain specified   threshold δ are the ones that should be included in H (x, y) = δg D ϕ(x)ψ(y) dx dy αδg reconstruction (Ogden, 1997). Hard thresholding is   (16) used in this paper, as follows: V δg D ψ(x)ϕ(y) dx dy αδg (x, y) =    ω j,k if |ω j,k | ≥ δ D ωˆ j,k = (13) (x, y) = δg D ψ(x)ψ(y) dx dy αδg 0 otherwise

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Fig. 2 Poisson kernel in the wavelet domain (db4) at four levels of decompositions

The kernel wavelet representation is substituted in Then, the wavelet transform of the Poisson kernel is the Poisson integral Eq. (5), to obtain the following computed: equation:      appr. appr. β K p (x, y) = K p ϕ(x)ϕ(y) dx dy δgu (x p , y p , z 0 ) = βK p hk hs s  k   H K p ϕ(x)ψ(y) dxdy β KHp (x, y) = + β K p h k gs + β KV p gk h s (18)  (17) s s k k

K p ψ(x)ϕ(y) dxdy β KV p (x, y) =  + β KDp gk gs δg D (x, y, 0)dxdy  D s k K p ψ(x)ψ(y) dxdy β K p (x, y) = δgu (x p , y p , z 0 ) is the output from the integration at one computational point. The Poisson kernel in the wavelet domain with four Then, the wavelet transform representation of the levels of decomposition is shown in Fig. 2. gravity disturbances δg D is substituted in Eq. (18). By Using Beylkin’s (1993) algorithm (which mainly re- interchanging the order of integration and summation lies on using orthogonal wavelets), the kernel and the and subsequently integrating, the solution will be as gravity disturbances are represented on wavelet basis follows: using the wavelet decomposition coefficients, h for the   appr. appr. approximation decomposition and g for the detail deδgu (x p , y p , z 0 ) = αδg β K p composition. The summation uses dyadic intervals to    H H V V D D αδg β K p + αδg βK p + αδg βK p avoid redundancy and decrease the computational ef- + fort (Mallat, 1997).

(19)

The Inversion of Poisson’s Integral

279

(20)

where Aij

i j bj

ai

is a matrix containing the wavelet coefficients of the Poisson kernels; each line is corresponding to the wavelet coefficients of one kernel (corresponding to each computational point) in a row vector format. is the number of computational points. is the number of the wavelet coefficients of the kernel. is a column vector containing the unknown wavelet coefficients of the gravity disturbances δg D . is a column vector containing the gravity disturbances δgu .

The procedure used in solving Eq. (20), i.e. inversion of the Poisson integral in the wavelet domain, is a combination of Beylkin’s (1993) non-standard algorithm for fast wavelet computations of linear operators, Mallat’s algorithm (Mallat, 1997), and the conjugate gradient method, which is used to invert a system of linear equations in the form of Eq. (20) (Barrett et al., 1994).

4 Numerical Results

4.2 Numerical Inversion in the Wavelet Domain The Poisson integral was inverted numerically with the full matrix without any thresholding (the full matrix contained 26,190,775 elements). The solution was achieved through the formation of the normal equation. The iteration diverged (no solution) with a tolerance value equal to 1 × 10−6 . The difference between the diverged solution and the reference data is shown in Fig. 3. 51.5 Latitude [deg]

Aij b j = ai

the North-South direction and 60 spacing in the EastWest direction. The data is downward continued from 7288 to 4370 m altitude. The data at 4370 m is used as reference data for the evaluation of the downward continuation integral.

51.4 RMSE = 69.38 mGal 51.3 51.2 51.1

243.4

243.6 243.8 Longitude [deg]

244

51.5 Latitude [deg]

Due to the orthogonal properties of the wavelet used, the wavelet coefficients multiplications in Eq. (19) lead directly to the solution (gravity disturbances at the computational points). In other words, the inverse wavelet transform step is done implicitly. Consequently, using this algorithm decreases the computational effort by 30% compared to standard algorithms, where the inverse wavelet transform is required for the output (Keller, 2004). For the inversion of the Poisson integral, the problem is formulated in the form of a set of linear equations and solved using the conjugate gradient method:

51.4 RMSE = 5.58 mGal 51.3 51.2 51.1

243.4

–150

–100

243.6 243.8 Longitude [deg] –50

0 mGal

244

50

100

150

4.1 Data Used

Fig. 3 The wavelet full matrix inversion of the Poisson integral (downward continuation) before regularization (top) and after Tikhonov regularization (bottom)

The data are gravity disturbances created from upward continued ground data in the Kananaskis region (Glennie and Schwarz, 1997), in Alberta, Canada. The data was created at two altitudes: 4370 m and 7288 m. They are on the same grid with 30 spacing in

The condition number of the normal matrix (full matrix) was computed for the determination of the instability of the problem and was equal to 7.02 × 1028 , which was huge. The singular value decomposition was done for the analysis of the normal matrix.

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M. Elhabiby and M.G. Sideris

Fig. 4 Regularization parameters (Optimal regularization parameter in red)

L- curve

5

10

1.5869e– 012

5.602e–014

4.4953e– 011

solution norm || x2||

1.2734e – 009

3.6072e– 008 4

10

1.0218e–006 2.8945e–005 0.00081993 0.023227

3

10 –12 10

–10

10

–8

10

–6

10

–4

10

–2

10

0

10

2

10

4

10

residual norm|| Ax-b||2

Tikhonov regularization was used to stabilize this problem. The regularization parameter was determined from the L-curve between the residual and solution norms of the linear system of equations (based on the singular value decomposition of the normal matrix). This curve provides all possible values that can help in regularizing this problem using Tikhonov regularization. The value that achieved minimum RMSE (root mean square error) of the solution in comparison with the reference data was equal to 0.023, which is almost the lowest point of the curve that achieved solution (Fig. 4). The regularization parameter was added to the diagonal of the normal matrix, and the solution was repeated. The solution converged after the regularization with 28 steps. The difference between the regularized wavelet full matrix solution and the reference data is shown in Fig. 3, with an RMSE equal to 5.58 mGal.

the regularized numerical inversion introduced in Sect. 4.2.; see Table 1, Global thresholding case. With a 76% compression level, there was almost no loss in accuracy in comparison to the full matrix regularized solution (same RMSE of 5.58 mGal in comparison to the reference data). The thresholding value estimated from Eqs. (14) or (15) was increased four times, each time by one order of magnitude, as shown in Table 1. These four global thresholding values were used for the inversion of the Poisson integral. The four cases converged after 28 iterations with a tolerance value equal to 1 × 10−6 . The same regularization parameter (Tikhonov) was used in the four cases. The four solutions converged with RMSE 0.001, 0.01, 0.11, and 0.37 mGal for 79.9, 89.7, 94.5, and 97%, respectively. The target accuracy is 0.1 mGal and this is satisfied by cases (a), (b) and (c). For case (d), a sudden degradation in the accuracy occurred to 0.37 mGal, because the thresholding affected the main energy of the kernel.

4.3 Combined Tikhonov Regularization and Wavelet Thresholding 5 Conclusions The regularized wavelet inversion solution has the problem that the design matrix Aij requires a large computer memory. The wavelet thresholding technique introduced in Sect. 3.1 (Eqs. (13) (14), (15)) was applied to compress the design matrix used in

A new combined wavelet-optimization technique is used for the inversion of Poisson’s integral. Orthogonal wavelets are essential for this algorithm. Matrix storage can be reduced by compression, through the

The Inversion of Poisson’s Integral

281

Table 1 Wavelet global thresholding versus wavelet full matrix solution for the evaluation of the Poisson integral Hard threshold

Full matrix

Global

(a)

(b)

(c)

(d)

Threshold value Storage (MB) RMSE (mGal) Comp% No. of iterations

– 186 – 0% 27

6 × 10−8 72.6 0.000 76% 27

6 × 10−7 60.41 0.001 79.8% 28

6 × 10−6 30.94 0.01 89.7% 28

6 × 10−5 16.5 0.11 94.5% 28

6 × 10−4 9.14 0.37 97% 28

hard thresholding technique. This compression can be optimized by the new wavelet thresholding technique introduced in this study. The Poisson integral was inverted numerically. The solution by the wavelet full matrix approach diverged. Tikhonov regularization was used to stabilize this solution. The conjugate gradient method converged to the solution after 28 iterations with an RMS accuracy equal to 5.58 mGal in comparison to the reference data, which is the expected accuracy for this type of application and data. The global fixed thresholding led to a 94.5% compression level with 0.1 mGal loss in the accuracy in comparison to the full matrix solution. This algorithm can be used in other applications such as the inversion of the Stokes integral (satellite altimetry). Generally, this technique can help in geocomputations and data compression, to decrease the required data storage size, especially in realtime applications such as airborne gravimetry and terrain correction. Acknowledgments Prof. Dr. Wolfgang Keller, Institute of Geodesy, University of Stuttgart. Prof. Dr. Klaus-Peter Schwarz, Dept. of Geomatics Engineering, University of Calgary.

References Barrett R., M. Berry, and T. F. Chan (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM. Beylkin G. (1993), Wavelets and Fast Numerical Algorithms, Lecture Notes for short course, AMS-93, Proceedings of Symposia in Applied Mathematics, v.47: pp. 89–117, 1993. Beylkin G., R. Coifman, and V. Rokhlin (1991). Fast wavelet transforms and numerical algorithms I., Communications in Pure and Applied Mathematics, 44, pp. 141–183. Bl´aha T., M. Hirsch, W. Keller, and M. Scheinert (1996). Application of a spherical FFT approach in airborne gravimetry. Journal of Geodesy 70, pp. 663–672, Springer-Verlag.

Burrus C. S., R. A. Gopinath, and H. Guo (1998). Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice Hall, New Jersey. Chan Y. T. (1995). Wavelet basics. Kluwer Academic, Dordrecht. Chui C. K., L. Monefusco, and L. Puccio (1994). Wavelets: theory, algorithms, and applications. Wavelet analysis and its applications, Vol. 5, Academic Press, INC. Daubechies I. (1992). Ten lectures on wavelets. Society for Industrial and Applied Mathematics. Donoho D. L., and I. Johnstone (1994). Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika 81: pp. 425–455. Elhabiby M. (2007). Wavelet representation of geodetic operators. UCGE reports, No. 20250, PhD thesis, Department of Geomatics Engineering, University of Calgary Forsberg R. (2002). Downward continuation of airborne gravity. 3rd Meeting of the International Gravity and Geoid Commission Gravity and Geoid 2002 proceedings. Gilbert A., and W. Keller (2000), Deconvolution with wavelets and vaguelettes, J Geod 74: pp. 306–320. Glennie C. and K. P. Schwarz (1997). Airborne gravity by strapdown INS/DGPS in a 100 km by 100 km Area of the Rocky Mountains. In: Proceedings of International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation (KIS97), Banff Canada, June 3–6, pp. 619–624. Goswami J. C. and A. K. Chan (1999). Fundamentals of wavelets: Theory, algorithms, and applications. Wiley series in microwave and optical engineering, Wiley-Interscience, New York, USA. Heiskanen W. A., and H. Mortiz (1967). Physical Geodesy. W. H. Freeman and Company, San Fransisco. Keller W. (1995). Harmonic downward continuation using a Haar wavelet Frame. XXI General Assembly of IUGG, Boulder, Colorado, USA. Keller W. (2004). Wavelets in geodesy and geodynamics. Walter de Gruyter. Mallat S. G. (1997), A wavelet tour of signal processing, Academic Press, san Diego. Ogden R. T. (1997). Essential Wavelets for Statistical Applications and Data Analysis. Birkhauser, Boston, USA. Van´ıcek P. and N. Christou (1994). Geoid and its geophysical interpretations. CRC Press, Boca Raton, Florida, USA. Wu L. (1996). Spectral methods for post processing of airborne vector gravity data. M.Sc. Thesis, UCGE Report #20104, Department of Geomatics Engineering, University of Calgary, Calgary, Canada.

On the Principal Difficulties and Ways to Their Solution in the Theory of Gravitational Condensation of Infinitely Distributed Dust Substance A. M. Krot

Abstract The gravitational condensation problem of infinitely distributed substance is directly connected with gravitational instability problem. The linearized theory of gravitational instability leads to the wellknown Jeans’ criterion. The main difficulty of Jeans’ theory is connected with a gravitational paradox: for an infinite homogeneous substance there exists no potential of gravitational field in accord with the Poisson equation. Since the classic gravitational theory did not give good solution of the gravitational condensation problem, the statistical theory for a cosmological body forming (so-called the spheroidal body model) has been proposed. In this model the forming cosmological bodies are shown to have fuzzy contours and are represented by spheroidal forms. This work considers a gas-dust protoplanetary cloud as a rotating and gravitating spheroidal body. The distribution functions and the mass densities for an immovable spheroidal body as well as rotating one have been derived. This work explains a slowly evolving process of gravitational condensation of a spheroidal body from an infinitely distributed substance. The equation for initial evolution of distribution function of a gas-dust protoplanetary cloud is derived. Because of the specific angular momentums are averaged during a conglomeration process, the specific angular momentum for a protoplanet in the protoplanetary cloud is found in this paper. As a result, a new law for planetary distances gives a very good estimation of observable planetary distances for our solar system.

A.M. Krot Laboratory of Self-Organization System Modeling, United Institute of Informatics Problems of National Academy of Sciences of Belarus, 6, Surganov Str., Minsk, 220012, Belarus

Keywords Gravitational condensation · Statistical model · Distribution functions · Gravitating spheroidal bodies · Gravitational field potentials · Planetary distances law

1 Introduction There are principal difficulties in the theory of gravitational condensation and gravitational instability of an infinitely distributed gas-dust substance in space. The gravitational condensation problem of an infinitely distributed substance is directly connected with gravitational instability problem, see for example Jeans (1919, 1929), Weizsaecker (1943, 1947), Schmidt (1944, 1962), ter Haar and Cameron (1963), Safronov (1969), Goldreich and Ward (1973). The linearized theory of gravitational instability leads to the well-known Jeans’ criterion. The main difficulty of Jeans’ theory is connected with a gravitational paradox: for an infinite homogeneous substance there exists no gravitational field potential in accord with the Poisson equation, see Safronov (1969). Since the classic gravitational theory did not give any explanation on the gravitational condensation problem, the statistical theory for a cosmological body forming (so-called the spheroidal body model) has been proposed by Krot (1996 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006). Within the framework of this theory, gravitating cosmological bodies have fuzzy outlines, and they are represented by means of spheroidal forms. A slowly evolving in time process of a gravitational compression of a spheroidal body (close to an unstable mechanical equilibrium state) has been investigated by Krot (2000). The pro-

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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posed theory follows from the conception for forming a spheroidal body from a gas-dust protoplanetary nebula; it permits to derive the form of distribution functions for an immovable spheroidal body and rotating one as well as their mass density, potential and strength of gravitational field and also to find the distribution function of specific angular momentum for a rotating spheroidal body (Krot, 2004, 2005, 2006). It was shown by Jeans (1919, 1929), the important law of statistical mechanics can be obtained from equation for evolution of distribution function of a dust-like substance. However, the main problem of self-condensation of an infinitely distributed substance was not solved by Jeans’ theory. This work explains a slowly evolving process of gravitational condensation of a spheroidal body from an infinitely distributed substance. The equation for an initial evolution of distribution function of a gas-dust cloud is derived here. It is found this equation coincides completely with the analogous equation for a slowly compressed spheroidal body, see Krot (2000, 2001). The proposed theory is also applied to exploring formation of planets in our solar system. As a result, the obtained law for the solar system planetary distances generalizes the well-known Schmidt law (1944). The new law gives a very good estimation of real planetary distances in the solar system (the relative error of estimation is 0%; absolute error of estimation is 1.4% besides its maximal value is equal to 11% for Earth, see Krot (2006).

2 The Gravitational Condensation Problems of Infinitely Distributed Substance The gravitational condensation problem of an infinitely distributed substance is directly connected with gravitational instability problem, see for example Jeans (1919, 1929), Schmidt (1962), Safronov (1969). The linearized theory of gravitational instability leads to the well-known Jeans’ criterion: ω2 ≤ 4π γρ,

(1)

where γ is the Newton gravitational constant, ω is a frequency of oscillating disturbances, ρ is a mass density of an infinitely distributed substance. Nevertheless,

some works, in particular Lifschitz (1946) and Bonnor (1957), point to that infinite homogeneous nonrotating substance can not be in an equilibrium state, therefore, the small disturbances do not manage to form any dense bunches. However, process of planet forming is very long in the time, in this connection the Newtonian consideration of locally equilibrium system becomes preferable, see Safronov (1969). The main difficulty of Jeans’ theory is connected with a gravitational paradox: for an infinite homogeneous substance there exists no gravitational field potential ϕg in accord with the Poisson equation, see Safronov (1969):

∇ 2 ϕg ≡

∂ 2 ϕg ∂ 2 ϕg ∂ 2 ϕg + + = 4π γρ. ∂x2 ∂ y2 ∂z 2

(2)

It is seen from Eq. (2) if a mass density value ρ  = 0 then both gravitational potential ϕg and, therefore, gravitational force Fg grow unlimited with a distance. This difficulty is reduced within framework of Jeans’ theory and its next modifications by means of supposition that the Poisson equation can not be applied to an infinite substance in whole but to disturbances δρ from a mean value ρ only. It is also supposed there is no gravitational force in an infinite homogeneous immovable substance because a gradient of acceleration and pressure is absent. Otherwise, it could not be in rest, see Safronov (1969). It has been shown by Jeans (1919, 1929), the important law of statistical mechanics can be obtained from equation for evolution of distribution function of a gas-dust substance. Evidently, a system of gas and dust particles (forming a gas-dust protoplanetary cloud) has relaxation time longer than for a laboratory system of gas molecules. In this connection if influence of inner forces in this gas-dust cloud can be ignored then this cloud will not change form and sizes with time noticeably. Such state is a mechanic equilibrium state, see Parenago (1954). Fluctuation interactions of subsystems of this system can violate this equilibrium permanently though rarely. In other words, gas-dust cloud can accept a mechanic equilibrium state in a time moment, i.e. evolution of a gas-dust cloud consists in replacement of equilibrium states, see Krot (2000). Choosing frame of reference into a gas-dust cloud let us introduce a distribution function of coordinates (x, y, z) = r and velocities (u, v, w) = ν as follows:

On the Principal Difficulties

(x, y, z, u, v, w, t)dxdydzdudvdwdt. Now let us consider an ensemble of particles whose number is defined by a distribution function Φ multiplied by N (N is a general number of particles into a proto-planetary cloud). These particles have coordinates x,y,z and velocity components u,v,w in a time moment t. After time interval dt these particles have new coordinates x +udt, y +vdt, z +wdt and new velocity ∂ϕ ∂ϕ ∂ϕ components u + ∂ xg dt, v + ∂ yg dt, w + ∂zg dt. Since we investigate the same ensemble of particles the following equality is true:

285

– difficulties occur to finding a general (not particular) solution of Jeans Eq. (4) in view of impossibility determining an analytical expression for gravitational potential of a gas-dust cloud.

Let us investigate an evolution of statistical distribution function describing a state of gas-dust cloud in space and time. First of all, let us consider an immovable gas-dust cloud in a state of instable mechanic equilibrium. Let Φ(x, y, z, t) is a function of probability density for location of particles in an immovable gas-dust cloud in some time moment t = t0 . Let us suppose that an initial evolution of gas-dust cloud is caused by a slow process of gravitational tightening its local parts exclusively, i.e. by means of chang∂ϕg ing function Φ(x + Δx, y + Δy, z + Δz, t0 + Δt) N · (x + udt, . . . , u + dt + . . . , t + dt) ∂x near point (x, y, z, t0 ). Since function Φ describes a d xd ydzdudvdwdt = physical process of gravitational condensation then it = N · Φ(x, y, z, u, v, w, t)d xd ydzdudvdwdt. (3) is differentiable. That is why this function can be represented by means of Taylor series in vicinity of the Now if we represent function Φ by series in Eq. (3) point (x, y, z, t0 ): then we obtain the Jeans equation (1915): Φ(x + Δx, y + Δy, z + Δz, t0 + Δt) = Φ(x, y, z, t0 ) ∂Φ ∂Φ ∂Φ ∂Φ ∂Φ ∂Φ ∂ϕg + |(x,y,z,t0 ) · Δx + |(x,y,z,t0 ) · Δy u +v +w + · ∂x ∂y ∂x ∂y ∂z ∂u ∂ x ∂ 2Φ ∂Φ ∂Φ ∂ϕg ∂Φ ∂ϕg ∂Φ |(x,y,z,t0 ) · (Δx)2 |(x,y,z,t0 ) · Δz + + + · + · + =0 (4) ∂z ∂x2 ∂v ∂ y ∂w ∂z ∂t ∂ 2Φ ∂ 2Φ + |(x,y,z,t0 ) · (Δy)2 + |(x,y,z,t0 ) · (Δz)2 2 However, the main problem of self-condensation of ∂y ∂z 2 an infinitely distributed substance was not solved by ∂Φ + ... + (5) |(x,y,z,t0 ) · Δt + . . . Jeans’ theory. In this connection this work explains a ∂t slowly evolving process of gravitational condensation of a spheroidal body, see Krot (1996, 1999, 2000), from Let us put the point (x,y,z) to the origin of cooran infinitely distributed substance. The equation for an dinates (0,0,0) and investigate its space-time behavinitial evolution of a distribution function of a gas-dust ior on a

small distance from the origin of coordinates protoplanetary cloud is derived as below. Δr = (Δx)2 + (Δy)2 + (Δz)2 . We also suppose that (Δx)2 ≈ (Δy)2 ≈ (Δz)2 , i.e. (Δx)2 ≈ (Δr )2 /3, and the function has extremum in the origin point:

3 Equations for Evolution of Distribution Function of a Dust-like Substance

Problems of gravitational condensation of a gas-dust cloud within framework of the Jeans’ theory are the following, see Parenago (1954):

∂Φ ∂Φ ∂Φ |(0,0,0,t0 ) = |(0,0,0,t0 ) = |(0,0,0,t0 ) = 0. ∂x ∂y ∂z (6) Taking into account Eq. (6) the relation Eq. (5) becomes: √ √ √ Φ(Δr/ 3, Δr/ 3, Δr/ 3, t0 + Δt) − Φ(0, 0, 0, t0 ) = 1 ∂Φ | + (Δr )2 ∇ 2 Φ|(0,0,0,t0 ) + . . . (7) = Δt ∂t (0,0,0,t0 ) 3

– a model of gravitational condensation of an infinitely distributed substance has not been developed up to now; – a gravitational potential for an infinitely distributed homogeneous immovable substance is not defined As far as (Δr )2 << 1 and Δt << 1, the right part analytically; of Eq. (7) can be limited by two terms only. Moreover,

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4 The Derivation of Function of Particle Distribution in Space Based on Statistical Model of Gravitating √ √ √ Spheroidal Body Φ(Δr/ 3, Δr/ 3, Δr/ 3, t0 +Δt) = Φ(0, 0, 0, t0 ). since we consider the same ensemble of particles then

(8) Let us consider the statistical theory of cosmological body formation (or initial gravitational condensation) beginning from the derivation of distribution function of particles in a space filled in a homogeneous and isotropic dust-like nebula, see Krot (1996 1997, 1998, = 0. 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006b). (9) First of all, we represent a gas-dust body consisting of N similar particles of mass m 0 in an isotropic space. Inside it, we choose some space coordinates and a direction in space. Let a radius-vector r with coordi(10) nates (x,y,z) be chosen in a three-dimensional space. If in the dust-like body there are N particles, then dN x have coordinates in the interval [x, x + dx], dN y and dN z have coordinates in the intervals [y, y + dy] and [z, z + dz] respectively at a given moment of time. The probabilities of any particle having coordinates (11) in the above mentioned intervals are equal to:

With provision of Eq. (7) the relation Eq. (8) becomes: ∂Φ (Δr )2 2 |(0,0,0,t0 ) + limΔr →0 · ∇ Φ|(0,0,0,t0 ) ∂t 3Δt Δt→0 If there exists a limit:

limΔr →0 · Δt→0

(Δr )2 = G, 3Δt

then equation (9) becomes the following: ∂Φ = −G∇ 2 Φ ∂t

because of an arbitrary rule for choice of the origin of coordinates and the beginning time reference. We consider Eq. (11) as an antidiffusion equation (or equation of slow-flowing gravitational tightening, see Krot (2000, 2001)). The Eq. (11) describing the Gauss-Markov process results from the Chapman-Kolmogorov equation in its limiting case when “between neighboring points there occur very quick jumps (Δt → 0) for very small distances ( r → 0)” (see Nicolis and Prigogine (1977)), the antidiffusion coefficient being define by Eq. (10). Evidently, the derived Eq. (11) as well as the Jeans Eq. (4) describes the different stages of evolution of a gas-dust cloud. Consequently, there exists a more general evolutionary equation which generalizes both of them:

dN x = ϕ(x)dx; N dN y dN z dp y = = ϕ(y)dy; dpz = = ϕ(z)dz, (13) N N

dpx =

where ϕ(x), ϕ(y) and ϕ(z) are one-dimensional probability densities, i.e. shares of particles whose coordinates belong to the elementary intervals close to x, y and z respectively. Let us introduce a threedimensional distribution characteristic, i.e. a volume density of probability: dpx,y,z =

dN x,y,z = (x, y, z)dxdydz. N

(14)

Taking into account the condition of homogeneous and isotropic space with a dust-like body, the volume density of probability has to depend on the value of ∂Φ ∂Φ r 2 + ν · ∇Φ + · ∇ϕg + G∇ Φ = 0. (12) = |r | only, i.e. (x, y, z) = (r ), where r = ∂t ∂ ν x 2 + y 2 + z 2 . On the other hand, a particle has all the three given coordinates independent from each other. Obviously, if ν = 0 then Eq. (12) becomes the ini- Then, according to the theorem of complex event probtial evolution Eq. (11), on the contrary, if ν  = 0 then ability we have: Eq. (12) can be transformed to the Jeans’ Eq. (4) be   dpx,y,z = dpx dp y dpz = ϕ(x)ϕ(y)ϕ(z)dxdydz. (15) cause |G∇ 2 Φ| << ν ∇Φ + ∂Φ . · ∇ϕ g ∂ ν

On the Principal Difficulties

287

Comparing Eq. (14) with Eq. (15) we obtain the factorization rule for a probability volume density function: ϕ(x)ϕ(y)ϕ(z) = Φ(r ).

(16)

Differentiating Φ(r ) as a composite function with respect to x and then to y and z, let us represent the functional Eq. (16) as a differential one, see Krot (1996, 1999): ˙ r (r ) ϕ˙ y (y) ϕ˙ x (x) ϕ˙ z (z) Φ = = = = −α, xϕ(x) yϕ(y) zϕ(z) r Φ(r )

(17)

when α is some constant. It follows from (17) if a parameter α > 0 then (  α 3/2 −α 2 α −αx 2 e 2r . e 2 and Φ(r ) = ϕ(x) = 2π 2π (18) Taking into account Eq. (18) let us write Eq. (14) in the spherical coordinates (r, θ, ε): dN r,θ,ε 2 = (α/2π )3/2 e−αr /2r 2 dr sin θ dθdε. N

(19)

in all directions at the same distance from mass center if a rotation is absent, see Krot (1996, 1999). The greatest density is concentrated in the interval [0, r∗ ], where √ r∗ = 1/ α is a point of the mass density bending, outside of which it decreases quickly. It follows from Eq. (22), the iso-surface of mass density (isostera) for the spheroidal body is a sphere. Let us note because of r∗ ∼ R, where R is a mean radius of gas-dust protoplanetary cloud, then α ∼ 1/R 2 is a very small positive parameter. There is a threshold value αg that if α ≥ αg then gravitational field arises in the spheroidal body. According to Eq. (22) we will seek a spherically symmetric solution ϕg depending only on r, therefore the Poisson Eq. (2) becomes: 1 r2



∂ ∂r

  α 2 2 ∂ϕg (r ) r = 4π γρ0 e 2 r . ∂r

Since ϕg is a function of r alone, one obtains (Krot, 1996, 1999): 4π γρ0 ϕg (r ) = − αr

r

α 2

e− 2 r dr.

(23)

0

Let us transform Eq. (19) into: dN r,θ,ε /dV = N (α/2π )3/2 e−αr

2 /2

Obviously, for large r the last expression turns into .

(20) ϕg (r ) = −

γM . r

where dV = is an elementary volume in spherical coordinates. Since the value dN r,θ,ε /dV = This relation, as is known (see Landau and Lifsnr,θ,ε is a local concentration of particles, we have chitz 1951), describes the gravitational potential of a (Krot, 1996, 1999): field produced by one particle (or a spherical body) of 2 mass M. ρ(r ) = M(α/2π )3/2 e−αr /2 (21) r 2 sin θ dθ dεdr

where ρ(r ) = m 0 n(r ) is a mass density of the substance, M = m 0 N is a mass of the body composed of the particles. By denoting ρ0 = M(α/2π )3/2 the expression Eq. (21) is written down as follows: ρ(r ) = ρ0 e−αr

2 /2

,

5 The Equation of a Slowly Evolving Process of Initial Gravitational Condensation of a Spheroidal Body from an Infinitely Distributed (22) Substance

where α is a parameter of gravitational compression (Krot, 2000). As appears from Eq. (22) under influence of interactions of particles, there arises a substance mass density inhomogeneous along the radial coordinate r. In such a way, a great number of particles form a spheroidal body whose density is uniform

Let us find the differential equation describing the process of gravitational condensation of a spheroidal body in the vicinity of mechanic equilibrium. We consider from now on that parameter α > 0 is a slowly changing function of time starting from some moment t0 , i.e.

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A.M. Krot

α = α(t) at t > t0 . Moreover, function α = α(t) is a positively defined monotonically increasing time function. We consider the mass density ρ as a function of two variables r and t. Since the dependence on t is expressed by a complex function, then partial derivative ρ with respect to t is ∂ρ ∂ρ dα = ∂t ∂α dt

initial gravitational compression (α = t 2 /G 2 ) of a spheroidal body, see Krot (2000). Let us note that Eq. (29) coincides completely with the analogous Eq. (11) for a distribution function of a dust-like substance because ρ(r ) = MΦ(r ) in accord with Eqs. (18), (21). This means that spheroidal body can be considered as a model of cosmic infinitely dis(24) tributed dust-like substance.

As follows from Eqs. (24), it is necessary first to calculate derivative ρ with respect to α, applying relation 6 The derivation of Function of Particle Eq. (21) for that purpose: Distribution in Space Based on ∂ρ Me−αr /2 α 1/2 = (3 − αr 2 ) ∂α (2π )3/2 2 2

(25)

Statistical Model of Rotating and Gravitating Spheroidal Body

Let us calculate now the Laplacian operator of ρ:

Let us consider a spheroidal body consisting of N particles with mass m 0 and rotating uniformly with an an   gular velocity Ω. We choose a plane of rotation as (x,y) ∂ρ(r, α) 1 ∂ r2 ∇ 2ρ = 2 and z as an axis of rotation of the spheroidal body. To ∂r r ∂r  α 3/2 obtain a distribution function for particles in space we 2 = −α M e−αr /2 (3 − αr 2 ). (26) use a cylindrical frame of reference (h, ε, z): 2π

Comparing Eqs. (25) and (26) one can see that these relations coincide to the accuracy of multiplier −1/2α 2 . Then substituting Eqs. (25), (26) into (24), one obtains:   ∂ρ 1 dα (27) =− ∇ 2 ρ. ∂t 2α 2 dt The obtained relation Eq. (27) reminds, in form, the antidiffusion equation because an initial density disturbance does not decrease but increase. Denoting through the gravitational compression coefficient, see Krot (2000, 2001):

x = h · cos ε; y = h · sin ε; z = z,

(30)

where 0 ≤ h < ∞, −∞ ≤ z ≤ ∞, 0 ≤ ε ≤ 2π . The choice of the cylindrical rotating frame of reference gives us an advantage because the rotating spheroidal body remains relatively immovable in this system. Analogously to the mentioned case Eq. (16), in cylindrical rotating frame of reference (h, ε, z) the probability volume density function is equal to: (r ) = ϕ(h)ϕ(ε)ϕ(z),

(31)

where ϕ(h), ϕ(ε), ϕ(z) are one-dimensional probability densities. Differentiating  as a composite function (28) with respect to h,z and ε, let us represent (31) in a differential equation form, see Krot (2004, 2005, 2006): we write down the equation of the spheroidal body’s gravitational compression in the form: ϕ˙ h (h)/ hϕ(h) = ϕ˙ z (z)/zϕ(z) (32) = (1/ h 2 )ϕ˙ ε (ε)/εϕ(ε) = −α, ∂ρ 2 (29) = −G(t)∇ ρ ∂t where α is the parameter of gravitational compression Since α(t) is a monotonically increasing function, and obviously α > 0. Integrating Eq. (32) and taking then G(t) > 0. Two particular cases of Eq. (29) into account the normalization conditions we obtain a have been considered: that of quasi-equilibrium probability volume density function in the cylindrical gravitational compression (α = t/2G 1 ) and that of coordinate system (Krot, 2004, 2006): 1 dα G(t) = · , 2α 2 dt

On the Principal Difficulties

289

Φ(h, z) = (α/2π )3/2 (1 − ε02 )e−α(h

2 (1−ε 2 )+z 2 )/2 0

(33a)

lar momentum λn and λn+1 respectively. According to the mentioned above we have that

and in the spherical coordinate system: Φ(r, θ ) = (α/2π )

3/2

μn = (λn + λn+1 )/2.

2 2 2 (1−ε02 )e−αr (1−ε0 sin θ)/2

(33b)

By analogy with the non-rotating case we can find the mass density for rotating and gravitating spheroidal body in the spherical system coordinate, for example (Krot, 2004, 2005, 2006): ρ(r, θ ) = ρ0 (1 − ε02 )e−αr

2 (1−ε 2 sin2 θ)/2 0

.

(36)

As a result of a conglomeration process, the specific angular momentums are averaged, therefore, the specific angular momentum for a protoplanet is equal to the ratio, see Schmidt (1962): μn λn = (

(34)

μn λ f (λ)dλ)/(

μn−1

f (λ)dλ)

(37)

μn−1

Obviously, the iso-surfaces of the mass density are If we use the linear approximation of the function flattened ellipsoidal ones, and ε02 is a parameter of f (λ) then after all simplifications the general solution their flatness (a squared eccentricity of ellipse), i.e. of the difference Eq. (37) has the form, see Krot (2006): 0 ≤ ε02 ≤ 1. The distribution function of a specific angular λn = A + B · n + (3λ)n/(n+1) ⎤ ⎡ momentum value for an uniform rotating spheroidal ent[n/2]  body are obtained simply using the relation for the ⎣ A0 + 2 Ak cos(2π nk/(n + 1))⎦ (38) probability volume density Eq. (33a) in the cylindrical k=1 coordinate system. As it has been shown in the work (Krot, 2006) a desired probability density function where λ is a mean value of specific angular momenf (λ) is tum (it has been taken into consideration that coefficients Ak = Ak−n , Bk = −Bn−k under derivation of 2 f (λ) = (α(1 − ε02 )/2)e−α(1−ε0 )λ/2 , (35) Eq. (38)). Let us use the formula for relation of the specific angular momentum λn with the square root of where λ is a value of specific angular momentum. radius Rn for orbit of n-th planet (under condition of almost circular character of planetary orbit):

7 A Model of Evolution for a Rotating and Gravitating Spheroidal Body and its Application to the Solar System Forming We consider a gas-dust protoplanetary cloud as a rotating and gravitating spheroidal body with the specific angular momentum distribution function f (λ). According to Schmidt’s hypothesis (1944) let us suppose that a domain boundary is determined by the value of specific angular momentum whose magnitude is equidistant from values of specific angular momentums for two neighboring bunches (or protoplanets). Let μn be a value of specific angular momentum corresponding to the boundary between domains of nth and (n + 1)-th protoplanets (or bunches in a gas-dust protoplanetary cloud) with values of the specific angu-

λn =



2γ M Rn .

(39)

Taking into account Eq. (39) this solution permits us to obtain a new law for square root of planetary distances in the form:

Rn = a + b · n + (3λ)n/(n+1) ⎤ ent[n/2]  ⎣a0 + 2 ak cos(2π nk/(n + 1)⎦ ⎡

(40)

k=1

where a, b, a0 , . . . , a[n/2] are coefficients to be sought for a planetary system in dependence on λ. Obviously, the proposed law for planetary distances Eq. (40) generalizes the well-known Schmidt’s law. According to Hoyle’s theory (see Hoyle (1960, 1963), Nieto (1972)) we consider a mean value of angular momentum of the proto-Sun cloud being equal to

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Table 1 Comparison of planetary distances laws for the solar system Solar system planets Mercury Venus Earth Mars (Zerera) Jupiter Saturn Uranus Neptune

Observable distances

Titius-Bode law

Titius-Bode law modified by Wurm

0.3871 0.7233 1.0000 1.5237 – 5.2028 9.580 19.141 30.198

0.4 0.7 1.0 1.6 2.8 5.2 10.0 19.6 38.8

0.387 0.68 0.973 1.559 2.731 5.075 9.763 19.139 37.891

Empirical Formulas Blagg

Richardson

Schmidt law

Proposed law

0.387 0.723 1.000 1.524 2.67 5.200 9.550 19.23 30.13

0.3869 0.7240 0.9994 1.5252 2.8695 5.1935 9.5053 19.2104 30.3005

0.3844 0.6724 1.0404 1.4884 – 5.1984 10.7584 18.3184 27.8784

0.3871 0.7231 1.1124 1.5237 – 5.2028 9.580 19.1410 30.1980

λ = 2.012 1014 (m2 /s). Let us apply the proposed law Eq. (40) to the planetary distance estimation in our solar system. Supposing the case n = 0 corresponds to the planet of Mercury, n = 1 conforms to Venus, . . ., n = 7 corresponds to Neptune. In Table 1, a comparative analysis for different laws is presented. For comparison, the Table 1 presents also both the well-known Titius-Bode law and its modification of Wurm as well as the empirical formulas of Blagg and Richardson and also the Schmidt law (see Schmidt (1962), Nieto (1972), Vityazev e.a. (1990)). As it follows from the Table 1, the proposed law gives the best results in prediction of planetary distances for the solar system: the mean error of its estimation of planetary distances in the solar system is equal to 1.4%. For all planets of the solar system excepting Earth and Pluto the absolute estimation error and, naturally, relative one are equal to 0% (for Earth the absolute estimation error is 11% and relative one is 10%; moreover, the proposed law shows that Pluto is not a result of the solar system forming). Let us note this exception is not lack of the proposed theory because the theories of Weizsaecker (1943, 1947), ter Haar and Cameron (1963) also revealed an exceptional peculiarity for the planets of Earth and Pluto. Thus, the proposed law and Weizsaecker’s and terHaar-Cameron’s theories lead to the similar result.

8 Conclusion There are the theories for exploring the solar system formation: electromagnetic theories (Birkeland (1912), Alfven (1942), Alfven and Arrhenius (1976), e.a.), gravitational theories (Schmidt (1944, 1962), Gurevich and Lebedinsky (1950), Woolfson (1964), Safronov (1969), Dole (1970), Vityazev e.a. (1990), e.a.), nebular theories (Weizsaecker (1943, 1947), Berlage (1948), Kuiper (1949, 1950), Hoyle (1960, 1963), ter Haar (1963, 1972), Nakano (1970), Cameron (1963, 1988), e.a.). In spite of great number of work aimed to exploring formation of the solar system, however, the mentioned theories were not able to explain all phenomena. In this connection in 1995 the statistical theory for a cosmological body forming (so-called the spheroidal body model) has been proposed, see Krot (1996 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006). The proposed theory starts from the conception for forming a spheroidal body as a protoplanetary cloud from a gas-dust protoplanetary nebula; it permits to derive the form of distribution functions both for an immovable spheroidal body and rotating spheroidal body as well as their mass density, gravitational potentials and strengths. This work explains a slowly evolving process of gravitational

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condensation of spheroidal body from an infinitely Krot, A.M. (1999). Use of the statistical model of gravity for analysis of nonhomogeneity in earth surface. In Proc. of distributed substance (see Eqs. (11) and (29)). SPIE’s 13th Annual Intern. Symp. “AeroSense”, Vol. 3710, A gas-dust protoplanetary cloud as a gravitating and Orlando, Florida, USA, April 5–9, 1999, pp. 1248–1259. rotating spheroidal body with the specific angular mo- Krot, A.M. (2000). Statistical description of gravitational field: a mentum distribution function is considered in this panew approach. In: Proc. of SPIE’s 14th Annual Intern. Symp. per. Because of specific angular momentums are av“AeroSense”, Vol. 4038, Orlando, Florida, USA, April 24– 28, pp. 1318–1329. eraged during a conglomeration process, the specific angular momentum for a protoplanet in the protoplan- Krot, A.M. (2001). Gravidynamical equations for a thweakly gravitating spheroidal body. In: Proc. of SPIE’s 15 Annual Inetary cloud is found in this paper. As a result, a new tern. Symposium “AeroSense”, Vol. 4394, Orlando, Florida, law for the solar system planetary distances is derived USA, April 16–20, pp. 1271–1282. here, and it generalizes the well-known Schmidt law. Krot, A.M. (2002). A quantum mechanical approach to description of gravitating body. In: Proc. of 34th Scientific AssemMoreover, unlike the known planetary distance laws, bly of the Committee on Space Research (COSPAR)-The 2nd the proposed law is established by a physical depenWorld Space Congress (WSC-2002), Houston, Texas, USA, dence of planetary distances from the value of the speOctober 10–19. cific angular momentum for the solar system. This new Krot, A.M. (2003). The equations of movement of rotating and law gives a very good estimation of real planetary disgravitating spheroidal body. In: Proc. of 54th Intern. Astronautical Congress (IAC), Bremen, Germany, September 29– tances in the solar system.

References Alfven, H. (1942). On the cosmogony of the solar system. Pts.I,II, and III, Stockholms Obs. Ann., 14(2). Alfven, H., and G. Arrhenius (1976). Evolution of the solar system. NASA, Washington. Berlage, H.P.,Jr. (1948). The disc theory of the origin of the solar system. Proc. Koninkl. Ned. Akad. Wetenschap.(Amsterdam), 51, pp. 796–806. Birkeland, O.K. (1912). Sur l’origine des planetes et de leurs satellites. Akad. Sci.,Paris,Compt.Rend., 155, pp. 892–895. Bonnor, W B. (1957). Jeans’ formula for gravitational instability. Monthly Notices Roy. Astron. Soc., 117, p. 104. Cameron, A. G. W. (1988). Origin of the solar system. Annual Review of Astronomy and Astrophysics, 26, pp. 441–472. Dole, S.H. (1970). Computer simulation of the formation of planetary systems. Icarus, 13, pp. 494–508. Goldreich, P., and W.R. Ward (1973). The formation of planetesimals. Astrophys. J., 183, pp. 1051–1062. Gurevich, L.E., and A.I. Lebedinsky (1950). Gravitational condensation of dust cloud. Dokl. Akad. Nauk SSSR, 74(4), pp. 673–675 (in Russian). Hoyle, F. (1960). On the origin of the solar nebula. Quart. J. R. Astron. Soc., 1, pp. 28–55. Hoyle, F. (1963). Formation of the planets. In: Origin of the solar system. Jastrow R., Cameron A.G.W. (eds.), Academic Press, New York, p. 63. Jeans, J. (1919). Problems of Astronomy and Stellar Dynamics, University Press, Cambridge. Jeans, J. (1929). Astronomy and Cosmogony, University Press, Cambridge. Krot, A.M. (1996). The statistical model of gravitational interaction of particles. Uspekhi Sovremenno¨ı Radioelektroniki (Special issue “Cosmic Radiophysics”, Moscow), 8, pp. 66–81 (in Russian).

October 3, Preprint IAC-03-J.1.08. Krot, A.M. (2004). The statistical model of rotating and gravitating spheroidal body with the point of view of general relativity. In: Proc. of 35th COSPAR Scientific Assembly, Paris, France, July 18–25, Abstract-Nr. COSPAR 04-A-00162. Krot, A. (2005). A statistical model of beginning rotation of gravitating body. In: Proc. of 2nd European Geoscinces Union (EGU) General Assembly, Vienna, Austria, April 24–29, Geophysical Research Abstracts, vol. 7, EGU05-A04550, 2005; SRef-ID: 1607-7962/gra/. Krot, A. (2006). The statistical approach to exploring formation of solar system. In: Proc. of European Geoscinces Union (EGU) General Assembly, Vienna, Austria, April 02–07, Geophysical Research Abstracts, vol. 8, EGU06- A-00216, 2006; SRef-ID: 1607-7962/gra/. Kuiper, G. P. (1949). The law of planetary and satellite distances. Astrophys. J., 109, pp. 308–313. Kuiper, G. P. (1951). On the origin of the solar system. Proc.Natl. Acad. Sci. USA, 37, pp. 1–14. Landau, L.D., and E.M. Lifschitz (1951). Classical Theory of Fields. Addison–Wesley Publishing Co., Reading, Massachusetts. Lifschitz, E.M. (1946). On the gravitational stability of the expanding Universe. Zh.E.Th.F., 16, pp. 587–602 (in Russian); (1946) reprinted by Journ. of Phys., 10, p. 116. Nakano, T. (1970). Origin of the solar system. Progr. Theor. Phys., 44, pp. 77–98. Nicolis, G., and I. Prigogine (1977). Self-organization in Nonequilibrium systems: From Dissipative Structures to Order through Fluctuation, John Willey and Sons, New York etc. Nieto, M.M. (1972). The Titius-Bode Law of Planetary Distances: Its History and Theory, Pergamon, Oxford, New York. Parenago, P.P. (1954). Lectures on Stellar Astronomy, Gosizdat, Moscow (in Russian).

292 Safronov, V.S. (1969). Evolution of Protoplanetary Cloud and Formation of Earth and Planets, Nauka, Moscow; (1972) reprinted by NASA Tech. Transl. F-677, Washington, D.C. Schmidt, O.Yu. (1944). Meteorite theory of origin of Earth and planets. Dokl. Akad. Nauk SSSR, 45 (6), pp. 245–249 (in Russian). Schmidt, O.Yu. (1962). The Origin of Earth and Planets, Acad. of Sci. USSR Press, Moscow (in Russian). ter Haar, D., and A.G.W. Cameron (1963). Historical review of theories of the origin of the solar system. In: Origin of the Solar System. Jastrow R, Cameron A.G.W. (eds), Academic Press, New York, pp. 1–37.

A.M. Krot ter Haar, D. (1972). Some remarks on solar nebula types theories of the origin of the solar system. In: Origin of the Solar System. CNRS, Paris, pp. 71–79. Vityazev, A.V., Pechernikova, G.V. and V.S. Safronov (1990). The Terrestrial Planets: Origin and Early Evolution, Moscow, Nauka (in Russian). Weizsaecker, C.F. (1943). Uber die Enstehung des Planetensystems. Z. fur Astrophys., 22, pp. 319–355. Weizsaecker, C.F. (1947). Zur Kosmogonie. Z. fur Astrophys., 24, pp. 181–206. Woolfson, M.M. (1964). A capture theory of the origin of the solar system. Proc. Roy. Soc. (London), A282, pp. 485–507.

Representation of Regional Gravity Fields by Radial Base Functions M. Antoni, W. Keller and M. Weigelt

Abstract The research aims at an investigation of the optimal choice of local base functions, to derive a regional solution of the gravity field. Therefore, the representation of the gravity field is separated into a global and a residual signal, which includes the regional details. To detect these details, a superposition of localizing radial base functions is used. The base functions are developed from one mother function, and modified by four parameters. These arguments can be separated into two coordinates, one scale factor and a shape parameter. The observations of a few residual gravity fields are simulated by orbit integration and the energybalance technique, in order to test the current approach. After selecting a region of interest, the parameters of the base functions are estimated. In order to get the optimal positions, two searching algorithms are compared. In the first algorithm the scale factors are estimated, while the positions and shape parameters are fixed. This method requires no initial values, because of the linear, but ill-posed and maybe ill-conditioned problem, but usually a regularization is necessary. The second algorithm searches possible positions for one base function in each step, until a termination condition is fulfilled, and improves the positions and scale factors in one adjustment. The results in the second case are better and faster for the test fields, but they depend M. Antoni Geod¨atisches Institut der Universit¨at Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany W. Keller Geod¨atisches Institut der Universit¨at Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany M. Weigelt Geod¨atisches Institut der Universit¨at Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany

on the initial values, the number of iterations and an assumption of an approximate constant orbit height. Keywords Local base function · Radial base function · Regional gravity field · Energy-balance technique · Searching algorithms

1 Introduction In this article the development of a function in an infinite and a finite series of spherical harmonics is described. The truncation causes an omission error, which still contains a part of the high frequency signal. In order to find this signal, a part of the error can be modeled by localizing base functions. For the modeling the radial base functions are used, which are mainly depending on the spherical distance to the centre of the base, a shape parameter and a scale factor. In order to test this approach, some residual fields are simulated by using the orbit integration and the energy balance technique. By using two different methods, the scale parameters and positions of the base functions are estimated. In the first algorithm a searching grid is used in the adjustment, and the positions and shape parameters are fixed. Though it is easy and very common to compute, the grid causes some problems, which might be reduced by a regularization (Schmidt et al., 2007). On the other hand, an iterative algorithm is tested, which produces better results. However, the computations are more complicated and also include some additional assumptions. The aim of this study and the future work is to develop an adjustment with a minimum number of

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part of the omission error as well, as this information about the reference field is ignored in this case. The basic idea of modelling regional fields is to separate the function, here the disturbing potential, into a global and a residual representation. For the residual 2 Residual Gravity Field signal, a reference field in the form of the limited sum of spherical harmonics in formula Eq. (2) is subtracted Like every other square integrable function on the from the function, so that the signal itself is equal to sphere, the disturbing potential T (λ, ϑ, r ) can be T (λ, ϑ, r ). Although the signal T (λ, ϑ, r ) is called an error in the global representation, it contains useful developed into a series of spherical harmonics information about the regional gravity field.   n ∞ In order to model this information, the field can be GM  R n+1  ¯ m Pn (cos ϑ) T (λ, ϑ, r ) = separated into regional parts. In the region of interest, R r n=0 m=0 the signal can be modeled by a superposition of a sec·( C nm cos mλ + S nm sin mλ) (1) ond system of base functions "b (λ, ϑ, r, ψb ) as self-adaptive base functions, so that it might be possible to avoid any regularization.

In this series { C nm , S nm } are the spherical harmonic coefficients, which describe the potential in the spectral domain (Moritz, 1980). The product of the trigonometric functions {sin mλ, cos mλ} and the assom ciated Legendre-functions P n (cos ϑ) creates the base  R n+1 enables the harmonic confunctions. The term r tinuation into the exterior of the sphere with the radius R. If all coefficients are well known and error free, the potential will be completely described. As this is not the case, the infinite series is cut off at the degree and order N. The truncation causes an omission error

T (λ, ϑ, r ) by neglecting the terms with higher degree and order, which are needed to describe local and regional details:

T (λ, ϑ, r ) =

B 

ηb "b (λ, ϑ, r, ψb ) + δT (λ, ϑ, r ),

b=1

(3)

which produces a smaller residual error δT (λ, ϑ, r ) for this region. In contrast to the spherical harmonics, the localizing base functions "b (λ, ϑ, r, ψb ) contain a small set ψb of free parameters, that will be explained in the next paragraphs. The functions should rapidly tend to zero, so that using them in one region has only a small influence on other regions. Another desirable property is an easy transformation to coefficients of spherical harmonics, because in this case the values can be added and all calculations could be done in the usual way. Here, the radial base functions "b are chosen as the localizing system. In contrast to the series Eq. (2), n N   GM  R n+1  m where the coefficients determine the scale of the spheriT (λ, ϑ, r ) = P n (cos ϑ) R r cal harmonics, the scale factors ηb can be used as an arn=0 m=0 gument of the radial base functions. They form a fam·( C nm cos mλ + S nm sin mλ) ily of simple functions, which are mainly depending + T (λ, ϑ, r ) (2) on the spherical distance #b between the measuring points (λ, ϑ) and the centre (λb , ϑb ) of the base funcBesides the smoothing of details by the truncation, tion. The centre of a radial base function, with a posspherical harmonic functions also produce a peri- itive scale factor, is identical with the position of its odic signal, because their maxima and minima are maximum value. A variable shape parameter σb is indistributed over the whole sphere. In order to avoid cluded to modify the functions (Freeden et al., 1998). this, the superposition of all base functions up to an Neglecting the rotation of the base function and the infinite degree is necessary again. If the series is finite, movement of the (simulated) satellite, which is equal the omission error includes this periodic behavior. In to the choice λb = ϑb = 0 and #b = ϑ, the mother the application of residual fields, a reference signal function ψ with the parameters η = ηb and σ = σb can is subtracted from the observations, to calculate the always be described by a sum of (unnormalized) Legomission error. The errors of the coefficients become endre polynomials Pn (cos ϑ) up to certain degree N˜ .

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This degree is not necessarily identical with the degree the measuring point. The index b is introduced for all N of the spherical reference field: parameters as a distinction from the mother function: ψ(#, λb , ϑb , σb , ηb ) = ψ(ϑ, 0, 0, σ, η) =η

N˜ 

σ n Pn (cos ϑ)

ψb (#b , λb , ϑb , σb , ηb ) = ηb (4)

N˜ 

σbn Pn (cos #b ) (5)

n=0

In the further text the abbreviation ψb stands for one particular base function, with a full set of parameters The influence of the shape parameter is demon- (λb , ϑb , σb , ηb ) and a maximum degree N˜ , which is strated in the Fig. 1. If the parameter is close to one, fixed for all base functions. A potential-function with a here σ = 0.999, the base function has a good localiz- harmonic continuation into the orbit can be written by ing structure and tends to zero outside of the region. In this special case the values are smaller than 10% of the "b (λ, ϑ, r, ψb ) maximum, if the spherical distance is greater than 5◦ . N˜   GM  R n+1 n For a smaller shape parameter (σ = 0.888) the base = σb Pn (cos #b ), (6) ηb R r function has the effect of a long wavelength in the ren=0 gional gravity field and the values are obviously greater than zero for all points. The shape parameter must not where the position (λ, ϑ, r ) might be depending on the be greater than one, in order to avoid problems with time. Using spherical coordinates for the centre of the convergence of the radial base function. This behavior base (λb , ϑb ) and the measuring point (λ, ϑ) the cosine will necessitate to constrain the estimation of the shape of the spherical distance cos #b can be expressed by parameters in future studies (Antoni et al., 2007). the scalar-product of the unit-vectors, which leads to To change the centre from the pole to another point, either a direct rotation of the coordinate system can be cos #b = sin ϑb sin ϑ cos(λb − λ) + cos ϑb cos ϑ. used or the transform can be calculated by the Wignerd-functions, which allows direct rotations of spherical The next step is to estimate the parameters of the loharmonics (Kostelec and Rockmore, 2003). The intro- calizing functions for a particular region. In this work, duction of the Wigner-coefficients will also be helpful, only the scale parameters ηb and the positions of some in order to calculate the gradients in any direction. base functions are estimated, while the shape parameIn the general formula of radial base functions the ters are fixed. The reason is the correlation of the pacolatitude has to be replaced by the spherical distance rameters, which demands more effort to estimate all of #b = # (λb , ϑb , λ, ϑ) between the base centre and them, especially for the shape parameter. n=0

Fig. 1 Radial base functions with different shape parameter σ at the pol λb = ϑb = 0. Both functions are developed until degree N˜ = 120 and normalized to a maximum value of one

(a) σ = 0.999

(b) σ = 0.888

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3 Simulation of the Observations In order to test the approach, some residual fields are generated by buried masses (Barthelmes, 1986). Therefore, the EGM96 is chosen as the representation of the global field. To get a local signal, the spherical harmonic coefficients of a small number of extra masses in the test region are calculated and added to the EGM96. The positions and velocities of a CHAMP-like satellite are simulated by orbit integration over the modified field for one month. By using the energy-balance technique (Weigelt, 2007) T (λ, ϑ, r ) = E kin (λ, ϑ, r ) − U (λ, ϑ, r ) − Z (λ, ϑ, r ),

function, which is exactly the parameter of the simulation. This shape parameters are chosen, so that the values of the base functions become smaller than 10% of the maximum for a spherical distance #b ≥ 5◦ . The aim is to optimize the other parameters, especially the positions. The adjustment and the downward continuation of gravity fields by some (radial) base functions is an ill-posed and often ill-conditioned problem, which usually requires a regularization. The aim of this study is to avoid a regularization, which can modify the results by its additional information or assumptions. In order to achieve this, the number of the unknows has to be by reduced, which can be done by optimizing the positions and shape parameters of a few base functions instead of using a fixed searching grid.

(7)

synthetic potential observations are created along the orbit. In this formula E kin (λ, ϑ, r ) is the kinetic energy, the term U (λ, ϑ, r ) describes the normal potential and Z (λ, ϑ, r ) is the centrifugal potential. For this simulation only two buried masses are chosen, with the positions (λ1 , ϑ1 ) = (281.5◦ , 38.5◦ ) and (λ2 , ϑ2 ) = (287.5◦ , 44.5◦ ). Both of them have the same, ideal shape parameter σ = 0.9999999999 and a scale-factor of η1,2 = ±2.200 · 10−7 . These parameters cause a residual potential on the ground of about 2 ±12.3 ms2 next to the centre of the base functions. After the transformation into spherical harmonic coefficients up to degree N˜ = 120, the positions and velocities are calculated by an orbit integration over the modified gravity field up to degree N = 150. The energy-balance technique delivers the potential values at the calculated positions. A reference is calculated by a pointwise harmonic synthesis up to degree N = 150 over the ‘normal’ EGM96 for the same positions. By subtraction of the potential with and without the buried masses a residual signal is produced. To reduce the data set and to simplify the parameter estimation, a region of interest is selected around the buried masses in North America.

4 Methodology and Results In the following estimations the shape parameters σb are fixed to the value σ = 0.99999999 for every base

Fig. 2 Simulated potential values of the residual field after downward continuation on the ground. In this field two masses are buried with ψ1 = (#1 , 281.5◦ , 38.5◦ , 0.9999999999) and ψ2 = (#2 , 287.5◦ , 44.5◦ , 0.9999999999). The scale factors have the same size, but different signs, with η1 = 2.200 · 10−7 and η2 = −η1

4.1 Searching Grid (SG) The common method of working with radial base functions is to choose an array of possible positions in the region of interest, either on a regular or irregular grid. A good spatial resolution demands a high number of base functions in a narrow grid, but the problem becomes very ill-conditioned. In theory the base functions, which are close to the buried masses, get a high scale factor, whereas the other scales should become very small. Because of the difference of the searching positions and the locations of the buried masses, and the down-

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regularization of the problem, and will be investigated in the future. The calculating time for the searching grid is nearly linear dependent on the number of base functions, which is visualized in Fig. 4. For every base function the resolution needs around 1 min plus at least 10 min for preparation. In Fig. 3 the grid includes 182 base functions, and it is calculated in circa 200 min.

4.2 Iterative Search Algorithm (ISA) Fig. 3 Modulus of the differences between the simulated and the estimated potential on the ground after using a regular searching grid with 182 base functions. The distance between the base functions in the grid is 1.5◦ in co-latitude and longitude and all have the same shape parameter σ = 0.99999999. The real positions of the buried masses are not included in the grid

Another method, which is shown in Fig. 5, is to find the positions of the centres of the base functions in the data. Therefore, an iterative search algorithm is developed, to find a single base function in every step. A first initial position can be located by searching the extrema in the residual signal. These positions still includes some errors, especially if the variation of the orbit height is not taken into account. For lower orbits the modulus of the values obviously increases and so the extrema are found in these tracks. An upwarddownward continuation to a mean orbit height fails,

Fig. 4 Calculating time in the searching grid for a some grid sizes. In the first approximation the time seems to be linear dependent on the number of base functions with an offset of around 10 min

ward continuation of the signal this is not the case. Another reason are the numerical problems, which occur in very narrow grids, which cause nearly linear dependent columns in the adjustment. Every function next to the optimal positions gets some high scale, greater than the hidden masses allow, and in order to adjust this to the data, their scale factors are oscillating. This causes some artificial effects, especially far away of the buried masses in the boundary of the region. In Fig. 3 the differences of the real residual field minus the estimated field is shown, and the artificial effects appear as distinct pattern in the region. They might be reduced by a Fig. 5 Workflow of the iterative search algorithm

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because the initial parameters of the coefficients for the radial derivatives are not known well enough. The assumption, that this first position is not totally wrong, can be used to reduce the problem of different heights. By selecting a number of data points around these localizations, the real place should be included. As the selected region is small enough, the potential could be approximated by a second order harmonic polynomial P(X, Y, Z ) = a11 X 2 + a22 Y 2 + a33 Z 2 +a12 XY + a13 XZ + a23 YZ +a10 X + a30 Z + a20 Y + a00 (8) in a local coordinate system (X,Y,Z). After estimating the coefficients, the polynomial can be used to transform the observation locally into constant orbit height. For this, a mean orbit height r of the selected data points and a local grid (λjk , ϑjk ) with a spherical distance to the initial position less than 3◦ is calculated. These positions are transformed into local coordinates (X jk , Yjk , Z jk ), and by searching the extrema of the polynomial, an improved position can be found and transformed into the global Earth-fixed coordinates (λ0b , ϑb0 , R). Now, the potential of this single base function is calculated using the chosen shape parameter and an adjustment for the scale factor. In the next step, this potential values are subtracted from the residual field and the searching algorithm is repeated with the new residuum. The process continues until a termination condition is satisfied. Possible conditions are the number of iterations, or the quotient of the scale factors. Additionally the positions of the located base functions are checked, in order to avoid small distances, which make the adjustment instable. The first condition permits a limitation of the number of base functions, in order to reduce the estimated parameters. The other conditions are used, to stabilise the solution. Finally, an adjustment improves the results by using the gradients of all positions and scale factors together. It can be recognized from Fig. 6 that the potential differences trend to zero, and the exceptions to this trend are near the buried masses. The small differences in the position still cause an error, but the error is just half as big as in the searching grid case (Fig. 3). In this particular example the search positions are extremely good and have a maximum difference of about 0.06◦ in longitude or co-latitude from the real location

Fig. 6 Modulus of the differences between the simulated and the estimated potential after using ISA and the downward continuation on the ground. The maximal disagreement in the positions is about 0.06◦ , the difference in the scale factors occurs at the third digit

of the masses. In the iterative search algorithm only 2 base functions are fixed, and the method is three times faster than the searching grid, with 40 min instead of 200 min.

5 Conclusion The study demonstrates two methods of finding the positions of radial base functions in the observations. The most common way is to fix the positions and shape parameters at the beginning, and to estimate only the scale factors. Because of the ill-conditioned problem a regularization would be necessary, which is neglected here, to show the benefit of the iterative algorithm. The ISA-method searches possible positions in a successive way, but it also requires an approximate shape parameter at the beginning. In both cases the errors of the residual signal minus the estimated field become smaller than 10% of the signal, but the iterative search algorithm performs better. The differences between the simulated and the estimated potential on the ground decrease by a factor of 2–2.5 using the ISA-approach. Reasons for the good results with the two methods are the use of the exact shape parameter and the possible separation of test masses in the orbit and on the ground. The iterative method reduces the artificial effects produced by ‘wrong localization’, which are the

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main problem in the searching grid. By using the distance criterion in the termination conditions, an ill-conditioned matrix can be avoided in the ISA, which is also not guaranteed for narrow grids. On the other hand, the iterative algorithm might produce extra base functions, if the position or shape parameters are not correct. To reduce this effect, it would be a good idea to find a group of bases in one step. In future work the calculation time must also be improved, especially for more base functions. More investigations are necessary in order to estimate the shape parameters as well, but the problem is highly non-linear due to closeness of all parameters.

References Antoni, M., W. Keller and M. Weigelt (2007). Regionale Schwerefeldmodellierung durch Slepian- und radiale Basisfunktionen Geodetic week, Leipzig Barthelmes, F. (1986). Untersuchungen zur Approximation des a¨ ußeren Gravitations-feldes der Erde durch Punktmassen mit optimierten Positionen Ver¨offentlichungen des Zentralinstituts f¨ur Physik der Erde, Nr 92, Potsdam

299 Freeden, W., T. Gervens and M. Schreiner (1998). Constructive Approximation on the Sphere Clarendon Press, Oxford Kostelec, P. J. and D. N. Rockmore (2003). FFTs on the Rotation Group http://www.cs.dartmouth.edu/ geelong/soft/0311-060.pdf, Moritz H. (1980). Adv. Phys Geod Wichmann, H. Karlsruhe Schmidt, M., M. Fengler, T. Mayer-G¨urr, A. Eicker, J. Kusche, L. S´anchez and S.-C. Han (2007). Regional gravity modeling in terms of spherical base functions J. Geod., 81:17–38 Weigelt, M. (2007). Global and Local Gravity Field Recovery from Satellite-to-Satellite Tracking PhD thesis, University of Calgary

Precise Gravity Time Series and Instrumental Properties from Combination of Superconducting and Absolute Gravity Measurements H. Wziontek, R. Falk, H. Wilmes and P. Wolf

Abstract Precise monitoring and a conclusive interpretation of temporal gravity variations at a given station is based upon accurate knowledge about the properties of the used instruments. Only the combination of concurrent sets of superconducting and absolute gravity measurements allow both. Whereas absolute gravimeters provide the scale and reference level, superconducting gravimeters enable to determine gravity variations with high sensitivity and temporal resolution. A method is proposed here to derive the scale factor and zero drift function of the superconducting gravimeter as well as a reliable survey of the instrumental stability of absolute meters with high precision without the need of gravity reductions. In this way it is possible to separate between geophysical signals and instrumental effects in the time series. Results for the stations Bad Homburg, Wettzell in Germany are presented, demonstrating the potential of the technique. Keywords Absolute gravity · Calibration · Superconducting gravimeter · Regional comparison site

H. Wziontek Bundesamt f¨u Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, D-60958 Frankfurt/Main, Germany R. Falk Bundesamt f¨u Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, D-60958 Frankfurt/Main, Germany H. Wilmes Bundesamt f¨u Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, D-60958 Frankfurt/Main, Germany P. Wolf Bundesamt f¨u Kartographie und Geod¨asie (BKG), Richard-Strauss-Allee 11, D-60958 Frankfurt/Main, Germany

1 Motivation For a conclusive interpretation of temporal variations in gravity obtained from superconducting gravimeters as well as for establishing the gravity standard with absolute gravimeters, the instrumental properties of both meter types are of fundamental importance. While the sensor of the superconducting gravimeters (SG) shows a low but very stable drift rate, which must be precisely estimated to expose long-term changes in gravity, absolute gravimeters (AG) are based on direct realization of physical standards for length and time, which are regularly controlled. Thus drift can be excluded, but regular supervision of the instruments is necessary and – especially after maintenance – checks for possible offsets are essential. In contrast, the SG must be precisely calibrated due to its relative measuring principle. A combination of the measurements of both gravimeter types allows to determine instrumental properties and results in a reliable and precise continuous gravity time-series with long-term stability. In classical combination approaches, instrumental scale and drift for the SG were treated separately and no conclusion on absolute meters was drawn so far. The estimation of the instrumental scale is usually carried out over relatively short periods, typically for several AG measurement epochs which are processed independently on each other. The instrumental drift is obtained afterwards from mean absolute values, which implies the removal of the main time-dependent constituents of the time varying gravity field (tides, atmosphere). This reduction step is a possible source of inconsistencies, since different models or parameters may be applied or even the use of different algorithms may lead to different results.

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Alternatively, for a comparison with highest precision, drift and scale function for the SG and offset checks for the AG can be carried out simultaneously, based on primary observations, the single drop experiments. Such an approach overcomes the problem of different, possibly inconsistent reductions. The large amount of parallel gravity observations with different absolute gravimeters at SG stations which is available at BKG allows an examination of this procedure.

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cision levels – a single drop has a standard deviation of about 100 nms−2 , while the SG measurements reach a noise level below 0.1 nms−2 in short timescales – only the observational error of the absolute gravity values is taken into account, whereas the SG data are considered as error free parameters. This leads to the observation equations gAG + v = n

nE D   i i ˜ g0 + ei t di t + USG + O j , 0

2 Functional Relationships As explained in our previous work (Wziontek et al., 2006, 2007), the instrumental properties of SG are characterized by a scale function f E (t) and a function for the systems zero drift f D (t) of time t. Usually, the drift can be treated as linear function. More complex functional relationships, such as piecewise defined low-order polynomials, polynomials of higher order or an exponential function (Van Camp & Francis, 2007), can be introduced and proved, if a sufficient amount of data covering a time span of several years can not be fitted adequately. The scale of the SG can be considered as constant, if neither changes in the sensors magnetic field nor modifications of the electronics occurred. Absolute gravity values are derived from direct measurements of time and distance by a procedure described in Niebauer et al. (1995). Each single drop experiment results in a gravity value gAG which is used further in this study instead of the original observations. Outliers in the drop data are assumed to be detected by suitable criteria and removed during preprocessing. Despite their absolute characteristics, changes caused by maintenance of equipment, mechanical wear or misalignment may result in offsets with respect to previous observations or to other instruments. For predefined intervals [t j , t j+1 ] (e.g. maintenance intervals), respective parameters will be estimated and their significance checked. Since the collocated sensors are sensitive to the same time dependent gravity signal and are recorded with sufficient high sampling rates, the resulting gravity value of each single drop experiment from the AG can directly be associated with a value from SG registration by means of a simple interpolation without applying any reductions. Because of the distinct pre-

(1)

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with n D = 1, n E = 0 for linear drift and constant scale, the preprocessed SG data U˜ SG and the AG offsets O j . The constant g0 describes the difference between the level of the SG time series and a mean absolute gravity value1 over all AG measurements. The removal of steps in the SG time series remains subject of the preprocessing, because normally the steps are sharp and well to identify and can be corrected with a precision of some tenth of nm/s2 . An exception are problematic steps after strong disturbances in conjunction with data gaps, which can be solved reliably only with help of AG measurements. In such a case, the treatment is equivalent to AG offsets. In the functional model (1), the residuals v are min imized in the least squares sense ( v 2 → Min) in order to calculate the parameters di , ei and O j . In case of parallel operation of AG this equation has to be extended by an additional offset parameter Mk , describing the gravity difference for monument k. Because the AG measurements do not represent a uniform precision level, a suitable weighting scheme depending on the drop and set scatter should be used to attenuate less reliable observation sets.

3 Extension of the Approach Equation (1) is not necessarily restricted to a single SG time series. For dual-sphere gravimeters, both sensors can be jointly compared with the same set of absolute measurements. Further, different SG stations, covered with a common subset of absolute instruments over a 1 Gravity differences between AG observations at different monuments are estimated with an additional parameter within the adjustment.

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comparable time span, can be treated the same way. Different parameters for each SG sensor are estimated, but if an instrumental offset of an AG exists, it must be detectable for the same periods at all stations and for all sensors. Therefore, the same offset constants for the AG in question are checked for all SG sensors at all stations. This improves consistency of the offset tests and allows for a decoupling from the SG drift parameters. The stations Wettzell and Bad Homburg are both equipped with dual-sphere SG (SG-29, SG-30) providing four SG time series and are covered with regular absolute measurements. At fundamental station Wettzell, during the period 2000–2006, a total number of ≈ 73, 000 drops are available, resulting from two observation epochs per year, whereas at the regional comparison site Bad Homburg regular absolute gravity measurements with various meters are carried out at least once per month (Wilmes & Falk, 2006). During the period 2002–2006, observations from eight different FG5 (FG5-101, FG5-227, FG5-301 of BKG and five instruments of visiting groups) exist, yielding in more than 380,000 drops. In about 60% of the 173 measurement epochs, instruments were operating side by side. The simultaneous adjustment procedure for both stations, Bad Homburg and Wettzell, results in a reliable estimation of instrumental parameters for SG30 and SG29 as well as for AG offsets constants. The residual curves based on these results for the lower sensor, respectively2 , are given in Fig. 1. The scale factors obtained for SG30 agree well with previous estimations using AG measurements at selected epochs (Harnisch et al., 2002) and (yet unpublished) results from calibration experiments with artificial accelerations with the Frankfurt Calibration Platform (Richter et al., 1995). Remarkably, a discrepancy between the scale factors of lower and upper sphere of about 0.6 nm/s2 /V, which was indicated by tidal analysis results, could now be solved. For SG-29, the scale factors are systematically too low for both sensors (≈ 1.3 nm/s2 /V), a fact which is currently under examination. Due to the huge number of observations, the formal error from the adjustment ±0.2 nm/s2 /V suggests a more precise estimation than in previous studies. However, the larger scatter, which becomes visible from individual calculations for each measurement 2 Except from a slight difference in the drift parameter, the results for the upper sphere look similar.

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epoch clearly shows a larger uncertainty. The applied weighting scheme, damping less reliable AG observations, may lead to a smoother estimation of the mean scale for the SG, but the outcome of the error propagation seems too optimistic. The drift rates for both sensors of SG30 deviate barely significant from zero and it can be concluded, that both sensors have a very low drift (Table 1). The rates for SG29 are with around 10 nm/s2 /y slightly larger, but still low as well. For the AG, periods for offset checks were predefined in accordance with maintenance intervals or instrumental modifications. The estimation of these parameters was stable and for most of the periods, significant offsets were obtained, not exceeding a magnitude of ±30 nm/s2 . The differences between AG observations and SG30-1 before and after applying these offsets are shown in Fig. 2. Even before an offset correction, most of the observations are within the boundaries given by the instrumental accuracy of ±20 nm/s2 . This is also demonstrated with the histogram in Fig. 3. Nevertheless, groups of absolute measurements showing a systematic deviation can be identified easily. For instrument FG5-101 this becomes most clear. During the year 2004 and early 2005, this instrument obviously shows a systematic positive difference with respect to the time series of SG30. Six offset periods were specified (Table 2), but only for two of them the offset amount changes significantly. During the periods 3–5, an offset of 30 nm/s2 is detected, but in period 6 the instrument returns to its initial level. This result is consistent with all SG times series at both stations an therewith mutual artefacts from the SG can be excluded. This demonstrates the stability of the absolute gravimeter over a timespan of several years and the efficiency of the proposed method. In a next step, stable periods will be merged in order to reduce the number of parameters and to improve the separation between offset and drift determination of the SG. In general, the consistency of the absolute observations could be enhanced significantly. Before offset corrections, the differences to the SG time series verifies with a standard deviation of ±22 nm/s2 the high accuracy of the FG5 instruments, although a certain number of differences exceed a normal distribution. As shown in Fig. 3, the use of offset corrections results in a further improvement, the standard deviation reduces to ±14 nm/s2 and the distribution looks almost normal.

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Fig. 1 Gravity values after removal of main parts of the time dependent gravity signal for (a) Bad Homburg (b) Wettzell based on results of the simultaneous adjustment for both stations. The black and grey curves show the residual SG time series (lower sensors only) before and after drift correction, symbols indicate mean absolute gravity values for the observation epochs after offset correction. Vertical lines designate significant offset periods

Table 1 Refined instrumental parameters for double sphere gravimeter SG30 from combination with different AG over the period 2002–06

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Fig. 2 Differences between AG measurements and SG30-1 (a) before (b) after applying offset corrections. The scatter is clearly decreased and the consistency enhanced

Table 2 Offset check results for AG FG5-101 over the period 2000–2006. The stability between different periods is clearly visible and the instrument returns to its initial level in April 2004

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Fig. 3 Histogram of the differences between AG measurements and SG30-1 before (large pattern) and after (narrow pattern) applying offset corrections. The standard deviation reduces from 22 to 14 nm/s2

4 Conclusions

References

The proposed method of a strict combination of SG and AG measurements based on most primary observations without gravity reduction applied, allows a simultaneous estimation of instrumental parameters for both types of instruments with reasonable results. For SG, a constant scale and a linear drift can be resolved well, while for AG, plausible offsets for fixed periods are obtained. The approach could be extended to a simultaneous adjustment for several stations and SG time series, common subsets of absolute instruments supposed. The stability of the AG could be confirmed and further improved. In addition to the determination of instrumental parameters, the procedure shows another important benefit. The combination leads to a reliable, homogenous, continuous time series at the respective combination site.

Harnisch, M., Harnisch, G., & Falk, R. (2002). Improved scale factors of the BKG superconducting gravimeters, derived from comparisons with absolute gravity measurements. Bulletin d’Information des Mar´ees terrestres, 135, 10627– 10638. Niebauer, T., Sasagawa, G., Faller, J., Hilt, R., & Klopping, F. (1995). A new generation of absolute gravimeters. Metrologia, 32, 159–180. Richter, B., Wilmes, H., & Nowak, I. (1995). The Frankfurt calibration system for relative gravimeters. Metrologia, 32, 217– 223. Van Camp, M. & Francis, O. (2007). Is the instrumental drift of superconducting gravimeters a linear or exponential function of time? Journal of Geodesy, 81, 337–344. Wilmes, H. & Falk, R. (2006). Bad Homburg – a regional comparison site for absolute gravity meters. In Francis, O. & van Dam, T. (Eds.), International Comparison of Absolute Gravimeters in Walferdange (Luxembourg) of November 2003, volume 26 of Cahiers du Centre Europ´een de G´eodynamique et de Seismologie (EGCS), (pp. 29–30)., Luxembourg. Wziontek, H., Falk, R., Wilmes, H., & Wolf, P. (2006). Rigorous combination of superconducting and absolute gravity measurements with respect to instrumental properties. Bulletin d’Information des Mar´ees terrestres, 142, 11417–11422. Wziontek, H., Falk, R., Wilmes, H., & Wolf, P. (2007). Combination of superconducting and absolute gravity measurements to determine instrumental properties. In Dergisi, H. (Ed.), Proceedings of the 1st Symposium of the International Gravity Field Service (IGFS) “Gravity Field of the Earth”, 28 August–1 September 2006, Istanbul, Turkey.

Acknowledgments The authors thank for providing the raw data of measurements: M. van Camp, Royal Observatory of Belgium, Bruxelles, Belgium (FG5-202), J. Kostelecky and V. Palinkas, Research Institute of Geodesy, Topography and Cartography, Praha, Czech Republic (FG5-215), L. Timmen and O. Gitlein, Institute f¨ur Erdmessung, Leibniz University Hannover, Germany (FG5-220), B. R. Pettersen, Norwegian University of life Sciences, As, Norway (FG5-226) and A. Pachuta and T. Olszak, Institute of Geodesy and Geodetic Astronomy, Warsaw University of Technology, Poland (FG5-230).

On a Feasibility of Modeling Temporal Gravity Field Variations from Orbits of Non-dedicated Satellites P. Ditmar, A. Bezdek, X. Liu and Q. Zhao

Abstract A numerical study has been conducted in order to analyze under what conditions temporal variations of the Earth’s gravity field can be recovered from the orbit of a satellite that is not equipped with an accelerometer. In the study, the motion of one of GRACE satellites was reproduced. A variant of the acceleration approach was used to process the simulated data. It turned out that the presence of non-gravitational satellite accelerations does not obscure temporal gravity field variations, provided that the satellite orbit (and, consequently, the set of observed satellite accelerations) is noise-free. Therefore, inaccuracies of the orbit determination is the only reason why a satellite orbit cannot be used to monitor temporal gravity field variations. It is shown that these inaccuracies should be reduced 30–100 times with respect to the level reached currently for the CHAMP and GRACE mission. Of course, a simultaneous utilization of data from several satellites can make this requirement less stringent.

P. Ditmar Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology Kluyverweg 1, 2629 HS Delft, The Netherlands A. Bezdek Astronomical Institute, Academy of Sciences of the Czech Republic. Fricova 298, 25165 Ondrejov, Czech Republic X. Liu Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology Kluyverweg 1, 2629 HS Delft, The Netherlands Q. Zhao Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology Kluyverweg 1, 2629 HS Delft, The Netherlands

Keywords Gravity field · GRACE · GPS · Nongravitational accelerations · Satellite navigation

1 Introduction Nowadays, temporal variations of the Earth’s gravity field attract a lot of attention of researchers. Models of this phenomenon allow numerous natural processes to be studied, including accumulation and ablation of the continental water stocks, melting of ice caps, non-steric sea level rise, etc. (see e.g. Flury et al., 2006). Unfortunately, gravity field variations at a regional scale could be observed so far with only one satellite mission, GRACE (Tapley et al., 2004). We consider this fact as very annoying because: (i) delivery of such data will be suspended after the end of the GRACE mission (a follow-on mission is not planned yet); (ii) a failure of a GRACE payload may worsen the data quality (or even terminate the data acquisition) at any moment; and (iii) temporal resolution of the GRACE mission is limited to the re-visit period (about 1 month). All that is the motivation to pose the following question: is it possible to monitor temporal gravity field variations at a regional scale with other (possibly, non-dedicated) satellites? In what follows, we presume that such a satellite is equipped with a GPS receiver of geodetic quality. This assumption is not unrealistic, since GPS is becoming a routine tool for satellite navigation. Furthermore, we assume the presence of a high-quality attitude determination sub-system, so that the satellite’s center of mass can be continuously traced. Finally, the satellite orbit is assumed to be sufficiently low (the satellite altitude of the order of 500 km is considered). No-

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tice that the presence of an on-board accelerometer to measure non-gravitational satellite accelerations is not requested. The posed question is triggered by the research of Ditmar et al. (2007), who showed that the absence of an accelerometer on board the CHAMP satellite would not worsen the quality of gravity field modeling from the data collected. More specifically, the objectives of our study are as follows:

erations of the satellite for the same time interval are presented in Fig. 2. Analysis of the results obtained allows us to conclude that behavior of surface forces on the day and night side of the Earth is indeed very different. In the Earth’s shadow, they show little variations, especially at the cross-track and the radial component, which are not sensitive to the atmospheric drag (see also Table 1).

r

Table 1 Average standard deviations of GRACE nongravitational accelerations w.r.t. to the mean values estimated per day/night passage, in nm/s2 (outliers are removed)

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r r

to understand better the behavior of nongravitational satellite accelerations; to confirm that a satellite orbit can be used to monitor temporal gravity field variations, even in the absence of information about non-gravitational accelerations, provided that orbit determination errors are neglected; to investigate the optimal data processing strategy for monitoring temporal gravity field variations; to analyze how orbit determination errors propagate into the gravity field and define the acceptable level of these errors needed for monitoring temporal gravity field variations.

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In general, simulated accelerations are in a good agreement with observations. Nevertheless, variations of non-gravitational accelerations in the Earth’s shadow are underestimated in the simulations. Most probably, this is because not all the effects are taken into account. Therefore, it was decided to consider real non-gravitational GRACE accelerations, and not simulated ones, in the rest of the study, having removed the observed outliers.

2 Non-gravitational Accelerations It is well-known that the cause of non-gravitational ac- 3 Gravity Field Recovery from Data celerations of a satellite at a low orbit is the surface Contaminated by Non-gravitational forces due to:

Accelerations (Semi-numerical Study)

1. 2. 3. 4.

atmospheric drag; direct solar radiation pressure; Earth’s albedo; Earth’s infra-red radiation.

For a satellite at a 500-km altitude, the atmospheric drag is dominant at the along-track component. Furthermore, only the atmospheric drag and the infrared radiation play a role when the satellite is in the Earth’s shadow.

2.1 Simulations To demonstrate the behavior of surface forces, nongravitational accelerations of a GRACE satellite have been simulated (Fig. 1). Real non-gravitational accel-

3.1 Study Set-up To compute parameters of the gravity field, we have exploited a variant of the acceleration approach (Ditmar and van Eck van der Sluijs, 2004), which has already demonstrated a good performance in CHAMP data processing (Ditmar et al., 2006, 2007). A 31-day set of satellite accelerations (30-s sampling rate) was simulated along a reference orbit, which was defined as the real orbit of a GRACE satellite in Aug. 2003. Furthermore, real accelerometer data were used to define non-gravitational satellite accelerations. The gravity signal was simulated on the basis of temporal gravity field variations derived from the GLDAS model of continental hydrology (Rodell et al., 2004), see Fig. 3. No random noise was added to the data at this stage.

Feasibility of Modeling Temporal Gravity Field Variations

Fig. 1 Simulated non-gravitational accelerations of a GRACE satellite during one satellite revolution on Aug. 13, 2003: the along-track (top), cross-track (middle), and radial component (bottom). The contributors considered are: atmospheric drag (squares), direct solar radiation pressure (asterisks), Earth’s

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albedo (crosses), and Earth’s infra-red radiation (triangles). The total accelerations are shown as solid grey lines. The feature in the central part of the plots corresponds to the passage through the Earth’s shadow

Fig. 2 Real GRACE accelerometer data for the same time interval as in Fig. 1. The outliers presumably correspond to thruster events

The gravity field in data processing was parameterized with spherical harmonics up to degree and order 70. Two data processing strategies were tried. Essentially, the difference between them was in the assumption about noise in the data. The first strategy was tuned for the type of noise that is really present in the simulated data. In other words, for noise that consists of non-

gravitational accelerations only. More specifically, we estimated and subtracted a data bias per day/night passage per component. In total, this gave rise to 2571 additional unknown parameters. Furthermore, the weights assigned to the data were different for day and night orbit segments as well as for different components. These weights were based on standard

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Fig. 3 Geoid height variations in Aug. 2003 caused by continental hydrology (GLDAS model). Gaussian filter of 400-km radius is applied

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Results of simulated data processing are shown in Figs. 5 and 6. In spite of contamination of data with non-gravitational accelerations, both processing 1e–05 strategies managed to recover temporal gravity field variations quite well. This fact is remarkable, since 1e–06 a common opinion is that inaccurately modeled or 1e–05 0.0001 0.001 0.01 measured non-gravitational accelerations are one of Frequency (Hz) the major problems in high-precision gravity field Fig. 4 Typical square-root power spectral density of noise in modeling from satellite data. Surprisingly, the secsatellite accelerations estimated on the basis of GPS data ond processing strategy (tuned for realistic noise in satellite accelerations) has yielded better results than the first one (specifically tuned for the presence of non-gravitational accelerations only). This is probably deviations of non-gravitational accelerations from the due to a too large number of additional parameters mean value at the corresponding orbit segment, see estimated in the first case. Therefore, only the second Table 1. The second strategy was tuned for noise that processing strategy was used in the rest of the study. can be expected in observed satellite accelerations in reality. Data bias was estimated once per a 10-day interval per component (in total, 12 additional unknown parameters). Furthermore, frequency-depended data 4 Gravity Field Recovery from Data weighting was applied. In doing so, we made use of Contaminated by Non-gravitational power spectral density of noise in observed acceleraAccelerations and Random Noise tions of the CHAMP satellite (Ditmar et al., 2007), see Fig. 4. Our experience tells that noise in CHAMP and GRACE acceleration data is very similar. Furthermore, Finally, data contaminated by both non-gravitational the coefficient C20 in resulting models of gravity field accelerations and random noise were considered. variations was set equal to 0, when the second strategy The noise generated was stationary and frequencywas applied. It is important to notice that Gaussian dependent; its power spectral density coincided with filter of 400-km radius was applied to all the results that shown in Fig. 4. The gravity field model obtained obtained. Thus, the figures shown below are consistent in this case contains nothing but noise (Fig. 7), even though the second data processing strategy, which was with Fig. 3, where the “true” signal is presented. 1e–04

Feasibility of Modeling Temporal Gravity Field Variations Fig. 5 Geoid height variations estimated with the first processing strategy; the rms error is 0.57 mm

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applied, is tuned for the random noise added. In fact, this is not surprising: nobody could expect a satisfactory recovery of temporal gravity field variations from the observed orbit of a single satellite like CHAMP or GRACE. Therefore, the question to answer is how much random noise should be reduced in order to enable a detection of temporal gravity field variations. In order to answer this question, we have repeated the numerical experiment several times with a reduced level of random noise. The results presented in Figs. 8 and 9 have been obtained after reducing the noise level 30 and 100 times, respectively. Obviously, such a reduction makes temporal gravity field variations visible again.

5 Discussion and Conclusions The conducted study allows us to conclude that the presence of unknown non-gravitational accelerations does not prevent a low satellite orbit (and, consequently, observed satellite accelerations) from being used for monitoring temporal gravity field variations. However, the presence of random noise in an orbit derived from GPS data makes temporal gravity field variations not observable; reduction of the noise level 30–100 times is required. At the first glance, such a reduction is impossible. However, we cannot exclude that such a goal will be reached in a foreseeable future thanks to better data processing strategies and a higher quality of GPS technology. Utilization of additional

312 Fig. 7 Geoid height variations estimated after adding realistic noise to satellite accelerations; the rms error is 24 mm

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GNSS systems, like GLONASS and GALILEO may References also be helpful. Finally, one should not forget that our estimations have been obtained for data from a P. Ditmar and A. A. van Eck van der Sluijs. A technique for single satellite. Obviously, increasing the number of Earth’s gravity field modeling on the basis of satellite accelerations. Journal of Geodesy, 78:12–33, 2004. satellites under consideration N times will reduce √ P. Ditmar, V. Kuznetsov, A. A. van Eck van der Sluijs, E. errors in recovered gravity field models N times, Schrama, and R. Klees. ‘DEOS CHAMP-01C 70’: a model provided that noise in orbits of different satellites is of the Earth’s gravity field computed from accelerations not correlated. of the CHAMP satellite. Journal of Geodesy, 79:586– Acknowledgments Computations were done on the SGI Altix 3700 super-computer in the framework of the grant SG027, which was provided by “Stichting Nationale Computerfaciliteiten” (NCF).

601, 2006. P. Ditmar, R. Klees, and X. Liu. Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise. Journal of Geodesy, 81:81–96, 2007. J. Flury, R. Rummel, C. Reigber, M. Rothacher, G. Boedeker, and U. Schreiber, editors. Observation of the Earth System

Feasibility of Modeling Temporal Gravity Field Variations Fig. 9 Geoid height variations estimated after reducing the added realistic noise 100 times; the rms error is 0.58 mm

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from Space. Springer, Berlin Heidelberg New York, 2006. M. Rodell, P. R. Houser, U. Jambor, J. Gottschalck, K. Mitchell, C.-J. Meng, K. Arsenault, B. Cosgrove, J. Radakovich, M. Bosilovich, J. K. Entin, J. P. Walker, D. Lohmann, and D. Toll. The global land data assimilation system. Bulletin of the American Meteorological Society, 85(3):381–394, 2004. B. D. Tapley, S. Bettadpur, M. Watkins, and C. Reigber. The gravity recovery and climate experiment: mission overview and early results. Geophysical Research Letters, 31, 2004. L09607, doi: 10.1029/2004GL019920.

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EOF Analysis on the Variations of Continental Water Storage from GRACE in China W. Hanjiang, Z. Guangbin, C. Pengfei and C. Xiaotao

Abstract The GRACE satellite mission has provided the Earth’s gravity field models for more than four years by the end of October, 2006, which is in form of spherical harmonic coefficients. They are useful to infer variation of continental water storage in large scale. In this paper, the Empirical orthogonal functions (EOF) are used to analyze the time series of J2 inferred from GRACE and SLR respectively to see the characteristics and the differences between the two time series. With 50 monthly models of the Earth’s gravity field (from August, 2002 to October, 2006) from Center for Space Research, University of Texas at Austin (UTCSR), the variations of continental water storage in China are computed. Meanwhile, EOF is used to analyze the water storage anomalies to get the characteristics of variation. The annual amplitudes in Yangtze River basin and Yellow River basin are derived with EOF reconstruction, respectively.

Keywords GRACE · Water storage · EOF · SLR

W. Hanjiang Chinese Academy of Surveying and Mapping, Beijing, 100039, P.R. China Z. Guangbin School of Geodesy and Geomatics, Wuhan University, Wuhan, 430079, P.R. China C. Pengfei Chinese Academy of Surveying and Mapping, Beijing, 100039, P.R. China C. Xiaotao Chinese Academy of Surveying and Mapping, Beijing, 100039, P.R. China

1 Introduction The Earth is a complex dynamic system. The redistribution of the Earth’s mass will lead to the change of gravity field in different time scale. Gravity observations with high precision and resolution may provide valuable information about the movement and exchange of the Earth’s mass. The variation of continental water storage plays an important role in the research of global climate change. The interrelated information got from many techniques are of great values in economy and sociality for the application of water resources including evaluation and improvement of climate model, the understand of large scale hydrology process, and the monitoring of water consume in agriculture, etc. The implement of GRACE satellite mission provides the possibility to estimate the variation of continental water storage in large scale. Methods for inferring the variation of water storage have been developed for the past several years (e.g., Wahr and Molenaar, 1998; Swenson et al., 2002). Meanwhile, the variations of water storage of representative area in the world have been investigated and many valuable achievements have been got (e.g., Swenson, 2003; Tapley et al., 2004; Ramilliena et al., 2005; Ries and Bettadpur, 2005). In this paper, the variation of continental water storage in China is analyzed using EOF, and the annual characteristics of water storage variation in main river basins of China are derived by EOF reconstruction. Meanwhile, the seasonal variation of J2 from monthly GRACE gravity models is compared to that from Satellite Laser Ranging (SLR).

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2 EOF Analysis of J2 Time Series Empirical Orthogonal Function is also called Principal Components Analysis (PCA), which is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences. With the reconstruction from the main high-variance components, the data may be filtered optimally. The process of EOF analysis is as follows (Lindsay, 2002):

of lesser significance. With lose of information, but if the eigenvalues with respect to the ignored components are small, you don’t lose much. Then, by taking the eigenvectors that you want to keep and forming a matrix with these eigenvectors in the columns, a feature vector VF , which is just a fancy name for a matrix of vectors, is constructed: VF = (ξ1

ξ2

ξ3 . . .

ξi )

(2)

where ξ1 , ξ2 , . . . ξi are the first, the second, . . .and (1) Getting the data the i the eigenvectors, i ≤ n. The data may be one-dimensional or multi- (5) The acquiring of principal component of the dimensional, and the intervals between neighbordata set ing elements should keep consistent. Once the components (eigenvectors) to be kept (2) Subtracting the mean have been chosen and a feature vector is formed, For EOF to work properly, the mean from each dithe principal component D F of the data can be acmension must be subtracted. quired by multiplying the transpose of the feature (3) The calculation of covariance matrix as well as vector on the left of the original data set D, transeigenvectors and eigenvalues of the covariance posed: matrix For one-dimensional data, the technique may be D F = VFT ∗D T (3) visualized as sliding a window of width M down a where D is the residual data after subtracting the series d of length N and determining the orthogomean values. nal patterns which best capture the variance in the views of the series thus obtained. These “empirical orthogonal functions” are eigenvectors of the M × M lag-covariance matrix C D (Myles, 1997): 2.1 EOF Analysis of GRACE J2 Time Series

(C D )ij =

1 kmax

DikT Dkj =

kmax 1 

kmax

di+k−1 d j+k−1

k=1

(1) where kmax = N −M +1 and D is the “augmented” dataset, Dkj = d j+k−1 . For multi-dimensional data, these “empirical orthogonal functions” (EOF) are just the eigenvectors of the covariance matrix among each dimension. (4) Choosing components and forming a feature vector These eigenvectors (EOF) provide us the patterns in the data. The first eigenvector with respect to the big eigenvalue is the most important component. The second eigenvector gives us the other, less important, pattern in the data. In general, once eigenvectors are found from the covariance matrix, the next step is to order them by eigenvalue, from big to small. This gives the components in order of significance. Now, one can ignore the components

The Earth’s anomalous gravitational potential V is written in a series expansion with spherical harmonics (e.g. Guenter Seeber, 2003): GM V = r

 1+

l  ∞   ae l=1 m=0

r

(Clm cos mλ + Slm sin mλ)Plm (cos ϑ)) The harmonic coefficients Clm , Slm are integrals of the mass and describe the mass distribution within the central body, ae is the equatorial radius and Plm are the so-called associated Legendre functions or Legendre polynomials. In satellite geodesy it is usual to replace Cl0 as Cl0 = −Jl . For l = 2, C20 exceeds all other potential coefficients by a factor of 103 . The monthly GRACE spherical harmonic coefficients infer the movement and exchange of the Earth’s mass, among which J2 plays a more important role. In order to evaluate the

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Fig. 2 The leading EOFs of J2 from GRACE Fig. 1 The time series of J2 from GRACE and SLR

quality of GRACE J2 and to get its characteristics, EOF is applied to J2 time series (see Fig. 1 (a)), derived from UTCSR Release-01 GSM gravity models (Srinivas, 2007). The time span is from August, 2002 to October, 2006 (coefficients for June, 2003 are interpolated with cubic polynomial). The 4 leading EOFs are shown in Fig. 2, and the power spectrums of these EOFs are shown in Fig. 3. As shown in Figs. 2 and 3, EOF1 and EOF2 contain signals of period of one-year, while EOF3 and EOF4 show four-month period. Although GRACE J2 time series contains signals of period of one-year, four-month, the energy of each signal is almost equal. The seasonal variation of GRACE J2 time series is not clear.

2.2 EOF Analysis of SLR J2 Time Series UTCSR provides a monthly J2 estimates from the analysis of SLR data by tracking five geodetic satellites: Lageos-1 and 2, Starlette, Stella and Ajisai. For getting different characteristics of J2 between GRACE and SLR techniques, EOF is also used to analyze the Fig. 3 The power spectrum of leading EOFs of J2 from GRACE

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Fig. 4 The leading EOFs of J2 from SLR

time series of J2 from SLR (see Fig. 1 (b)). The time span of SLR is selected according to that of GRACE. The leading EOFs and their power spectrum of J2 from SLR are shown in Figs. 4 and 5. Figures 4 and 5 show that SLR J2 time series contains strong one-year period signals. And the energy of one-year period signal is very large, which contains more than 80% of total Energy. The seasonal variation of SLR J2 time series is also very clear. So, it can be concluded that the seasonal variation of J2 time series from SLR is clearer than that from GRACE. It is customary among users of Release-01 gravity field models to remove the J2 harmonics from the GSM product files before using the remaining harmonics for geophysical analyses. Now, replacing the values from GSM product files by J2 estimates obtained from SLR may be another effective approach.

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Fig. 5 The power spectrum of the leading EOFs of J2 from SLR

estimate regional variation of water storage in a few hundred kilometers. For the variations of regional water storage, regional spatial filters are designed to minimize both the satellite measurement error and the leakage error. The surface mass variability over a region σ˜ region is approximated by the following equation (Wahr and Molenaar, 1998):

σ˜ region =

lt  l 2l + 1 2aρa π  Wl . 3 1 + kl l=0 m=0

(4)

P lm (cos θ )( C lm cos mλ + S lm sin mλ)

where θ is colatitude, λ is east longitude; a is the mean radius of the Earth, l and m are degree and order of spherical expansion, ρa is the average density of the Earth, kl is load Love number of degree l, C lm and

S lm are the changes of dimensionless Stokes coeffi3 EOF Analysis of the Variation of cients, P lm (cos θ ) are normalized associated Legendre Continental Water Storage in China functions, and Wl represents the Gaussian averaging coefficients. The coefficients Wl can be computed with recursion Swenson and Wahr (2002, 2003a,b) introduced an averaging method based on a simple Gaussian filter to relations:

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1 2π

1 1 + e−2b 1 − W1 = 2π 1 − e−2b b

due to its relatively low precision, because the orbit shape of GRACE is not sensitive to J2 of the Earth’s gravity field (Wahr et al., 2004; Chen et al., 2004). And (5) the value in June, 2003 is got by averaging the neighboring monthly values. 2l + 1 EOF is then used to analyze the time series of the Wl+1 = − Wl + Wl−1 b water storage variations. The three leading EOFs are shown in Fig. 7 together with their power spectrum. ln 2 , r is the radius of Gaussian avwhere b = (1−cos(r/a)) The first EOF shows strong one-hundred-day peeraging function. riod, and other periods are not very clear. The secFigure 6 gives the weighting functions with respect ond EOF indicates clear ten-month period signal, and to different radius. The weight decreases rapidly with the third one contains two main periods: one-year and degree l for a larger radius. three-month. In this study, 50 monthly Release-01 GSM gravity field models from UTCSR, which are represented by the full-normalized coefficients of a spherical har- 4 EOF Reconstruction of the Variation of Continental Water Storage in Yangtze monic and estimated from only GRACE data, are used to get the variations of continental water storage in River Basin and Yellow River Basin China. The maximum degree and order of the monthly estimates is 120. Each model is estimated from apThe Yangtze and Yellow are the two longest rivers in proximately 30-day GRACE data, and the tide effects China. In the study, EOF is used to analyze the averincluding ocean-tides, body-tides and pole-tides along aged variation of continental water storage in Yangtze with non-tidal atmosphere and ocean effects are taken River basin (105◦ ∼ 120◦ E, 25◦ ∼ 35◦ N ) and Yellow out in the processing. The time span is from August, River basin (103◦ ∼ 118◦ E, 33◦ ∼ 42◦ N ), and the 2002 to October, 2006. Gaussian averaging radius annual signal is shown. of 800 km is used. And, the gravity field models are In the EOF reconstruction, 83.06 and 83.18% of truncated to 20. The average is subtracted from each total variance is shown in Yangtze and Yellow River data set. basin, respectively. See Fig. 8. The variations of continental water storage in China From this figure, one can see clear seasonal variafrom August, 2002 to October, 2006 are calculated by tions of continental water storage. And, the annual amEq. (4). The component of J2 is replaced with SLR J2 plitudes after EOF reconstruction in Yellow River basin are about 4 cm, while those in Yangtze River basin are 5 cm or so. W0 =

5 Summary

Fig. 6 Gaussian weighting functions for different radius

Based on EOF analysis, the J2 time series from GRACE and SLR are analyzed. The seasonal variation of J2 time series from SLR is clearer than that from GRACE. The variations of continental water storage in China are analyzed with EOF. The variations of continental water storage in China show regional imbalance and seasonal change. Besides, the annual amplitudes of regional averaged water storage variations in Yangtze River basin and Yellow River basin are reconstructed using EOF. The annual amplitudes in Yellow River basin are about 4 cm, while those in Yangtze River basin are 5 cm or so.

320 Fig. 7 The leading EOFs of water storage variations

Fig. 8 The variations of water storage in Yellow River basin and Yangtze River basin

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References Myles R. Allen (1997) Optimal filtering in singular spectrum analysis. Phys. Lett. 234(6), 419–428 Srinivas Bettadpur (2007) Product Specification Document. Center for Space Research, The University of Texas at Austin Chen J. L., Wilson C. R., Tapley B. D., et al. (2004) Low degree gravitational changes from GRACE: validation and interpretation. Geophys. Res. Let. 31, L22607 G. Ramilliena, F. Frapparta, A. Cazenavea, A. Guntner (2005) Time variations of land water storage from an inversion of 2 years of GRACE geoids. Earth Planetary Sci. Lett. 235, 283–301 J. Ries, S. Bettadpur (2003) Report of GRACE Science Team Meeting. Center for Space Center, The University of Texas at Austin J. Ries, S. Bettadpur (2005) Report of GRACE Science Team Meeting. Center for Space Center, The University of Texas at Austin Guenter Seeber (2003) Satellite geodesy, 2nd edition, ISBN 3-11-017549-5 Lindsay I Smith (2002) A tutorial on Principal Components Analysis

321 Sean Swenson, John Wahr (2002) Methods for inferring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity. J. Geophys. Res. 107, B 9,2193 Sean Swenson, John Wahr (2003a) Estimated accuracies of regional water storage variations inferred from the Gravity Recovery and Climate Experiment (GRACE). Water Resource Res. 39, 8, 1223 Sean Swenson, John Wahr (2003b) Monitoring changes in continental water storage with GRACE. Space Science Reviews. 108, 345–354 Byron D. Tapley, Srinivas Bettadpur, John C. Ries, Paul F. Thompson, Michael M. Watkins (2004) GRACE measurements of mass variability in the Earth system. Science. 305 John Wahr, Mery Molenaar (1998) Time variability of the Earth’s gravity field: Hydrological and oceanic effects and their possible detection using GRACE. J. Geophys. Res. 103(B12): 30, 205–30, 229 John Wahr, S. Swenson, V Zlotnicki, I. Velicogna (2004) Timevariable gravity from GRACE: First results. Geophys. Res. Let. 31, L 11501

Influence of Hydrology-Related Temporal Aliasing on the Quality of Monthly Models Derived from GRACE Satellite Gravimetric Data J. Encarnac¸a˜ o, R. Klees, E. Zapreeva, P. Ditmar and J. Kusche

Abstract Temporal variations of the Earth’s gravity field are traditionally represented with a set of spherical harmonic coefficients derived once per month. In practice, however, the gravity field changes continuously (e.g. because of on-going hydrological processes). This discrepancy causes temporal aliasing which manifests itself as errors in the obtained gravity field models. The goal of this study is to quantify the influence of temporal aliasing caused by hydrological processes onto the quality of computed monthly gravity field models; the Zambezi river basin is used as the test area. The GRACE observations are simulated along the actual GRACE orbits as gravitational potential differences. The mass variations obtained by the Least Square adjustment are compared with the “true” values used for data simulations. It is shown that the optimal estimation period is determined by the trade-off between the J. Encarnac¸a˜ o Physical and Space Geodesy, Department of Earth Observation and Space Systems, Faculty of Aerospace Engineering, T.U. Delft R. Klees Physical and Space Geodesy, Department of Earth Observation and Space Systems, Faculty of Aerospace Engineering, T.U. Delft E. Zapreeva Physical and Space Geodesy, Department of Earth Observation and Space Systems, Faculty of Aerospace Engineering, T.U. Delft P. Ditmar Physical and Space Geodesy, Department of Earth Observation and Space Systems, Faculty of Aerospace Engineering, T.U. Delft J. Kusche Physical and Space Geodesy, Department of Earth Observation and Space Systems, Faculty of Aerospace Engineering, T.U. Delft

two conflicting effects of ever-decreasing propagated noise and ever-increasing aliasing. Catchments smaller than roughly 2000 km and estimation periods up to a month produce solutions with no noticeable aliasing. A catchment 3300 km wide has an optimal estimation period of 15 days and the one that is 4400 km large is most accurate if estimated for not more than 11 days. This is under the assumption that aliasing does not corrupt the solution significantly if it is less than half the propagated noise.

1 Introduction Satellite gravimetric data acquired by the GRACE mission are extensively used to retrieve temporal variations of the Earth’s gravity field on a monthly basis. Considering the accuracy achievable by GRACE, the measured mass variations can be large enough to change significantly within one month. This is especially true for hydrological mass transport processes due to their high variability over relatively short periods of time. A constant parametrization in time is often used for the gravity field inversion. Since GRACE re-visits each particular area once in a while, the collected “snapshot” will not be representative of the monthly average. This discrepancy is called aliasing and it is expected to play a dominant role under certain conditions. This paper aims at identifying these conditions considering the cell size and the estimation period. This implies that the aliasing discussed in this work is solely due to non-homogeneous distribution over time of the observations.

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2 Theory

3 Experiment Setup

This report considers that the mass distribution in a river basin is averaged in space so that only one unknown needs to be estimated. However, since hydrological processes are inherently distributed on the surface of the Earth, this unknown is discretized into a grid of n point-masses, thus simulating a layer of water with constant thickness. There are p potentialdifference observations that are recorded at orbital positions of satellites. Each of these observations is generated according to:

The experiment is based on real GRACE orbits provided by Data Process and Access (2007). The simulated signal (equation 1) is generated from the Lumped Elementary Watershed (LEW) monthly hydrological model over the Zambezi river basin (Winsemius et al. 2006), which is taken as the truth. In Fig. 1 an example of the water height distribution is shown. Considering the distributed water storages as a layer with constant thickness, the mass distribution in the Zambezi river basin due to hydrological processes can be represented as in Fig. 2.

Vi = G



 mj

1 l Aji

j

−

1 l Bji

 (1)

where:

V : potential difference (numbering p); G : universal gravitational constant; m : point-mass (numbering n); l : distance from point-mass j to satellite A or B at the position of observation i. The design matrix (which reduces to a column vector) is built accordingly: Ai = G





1 l Aji

j

−

1



l Bji

(2)

The least-squares estimation is then performed in the usual way (notice that the solution is a scalar due to the way the inversion is formulated): m ∗ = (A T A)−1 A T V

Fig. 1 The lumped elementary watershed model for the Zambezi river basin in 11-2004 (color bar in mm of water height)

(3)

The error between the estimation and a given model is: ε = m ∗ − m model

(4)

In order to retrieve the propagated noise, the Normal matrix is scaled with the appropriate noise magnitude and inverted, thus providing the Covariance matrix, which includes the variance of the point-masses in its diagonal: εnoise =

1 n2

 A T A.

1 σ2

−1 (5)

Fig. 2 Point-mass representation of the water storage (color bar in mm of water height)

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The point masses are represented as dots, disposed in a 10×10 grid. Their magnitude is given by the water height distribution according to LEW, averaged over the whole basin and scaled considering the total basin area and the density of water:

mj =

1 Aρw h w n

(6)

where: A : area of the Zambezi river basin (4.9 × 106 km2 ); ρw : density of water (1000 kg/m3 ); h w : basin-wide average water height (time dependent). The data region, i.e. the region of the globe where GRACE observations are considered is three times larger than the basin and is the part of the Earth’s surface shown in the plot above. The simulations were performed for the data referring to the period between January 1st, 2003 and December 31st, 2005. Not only monthly solutions but also submonthly solutions will be considered. The latter are computed taking into account observations from day 1 up until day x. This shows the evolution of the reached accuracy throughout the month and, therefore, allows to infer conclusions regarding the optimal estimation period.

Fig. 3 Example of the evolution of the solution throughout the month, data of 11-2004

3.1.1 Noise The noise standard deviation assumed in this report was chosen to be 0.0038 m2 /s2 . The link between the accuracy of the range-rates and the accuracy of the potential differences is given by the formula (Jekeli 1999): std( V ) ≈ |x˙ |std(ρ12 ˙ )

(7)

where: |x˙ | : orbital velocity of satellites (assumed to be 3.1 Decoupling Aliasing and Noise 7610 m/s); ρ12 ˙ ˙ : noise associated with ρ12; The errors associated with noise and aliasing are alρ12 ˙ : inter-satellite range-rate. ways jointly present in a solution computed from realThe chosen noise level corresponds to a noise stanworld measurements. In the conducted simulations, dard deviation in the range-rates of 0.5 μm/s. aliasing and propagated noise were considered sepaAnother parameter relevant to the noise propagation rately. Aliasing is computed as the differences between is the observation sampling rate, which is equal to 5 s. the estimated water heights and the mean value of the time-dependent water-height evolution for a particular (sub-) monthly interval, as given by equation 4. Therefore, aliasing is the difference between the two lines 4 Results seen in Fig. 3, left-hand side. The water height is assumed to evolve following a linear trend connecting the The evolution of noise and aliasing in time is shown end-of-month water height values provided by LEW. in Fig. 4. One can see that the more observations are Propagated noise is calculated from analytically taken into account (larger estimation period, i.e. movpropagated noise of potential differences. An example ing away from zero in the horizontal axis), the lower of its profile over a month is shown in Fig. 3, right- the error associated with noise and the larger the error hand side. associated with aliasing.

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between the way the aliasing and noise errors changes over the months.

4.1 Propagated Noise and Cell Size The question now is to determine at which cell sizes does aliasing become dominant over random noise. Several cell sizes were considered, ranging between roughly 1100 and 4400 km, as depicted in Fig. 6. The water height per unit area is kept constant. Thus, different cell sizes reflect different magnitudes in Fig. 4 Comparison between aliasing and propagated noise for sub-monthly solutions (RMS over 36 months)

Figure 4 takes into account the error considering all months. Also it is interesting to observe how aliasing and propagated noise behave at half through and at the end of each month, thus allowing to see in which months does the former become dominant over the latter. To that end, Fig. 5 is shown. There is a large variability in the accuracy throughout the months. It is suggested that it is due to the simultaneous influence of spatial aliasing (which is dependent on the ground track pattern) and the magnitude of the point-mass (which changes from one month to the other, according to the hydrological model) and therefore influences the temporal aliasing too. It is also Fig. 6 Cell sizes that were considered for determining the deworthwhile to notice that there is no clear correlation pendency between time and space resolution

Fig. 5 Monthly comparison between aliasing and propagated noise for monthly and 15-day solutions

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error is quantified in terms of the water height per unit area. For simplicity, it is reported as a single line in other plots.

4.2 Trade-off Between Spatial and Temporal Resolution

Fig. 7 Monthly comparison between aliasing and propagated noise of different cell sizes, monthly solutions

the gravitational influence of the water mass. Figure 7 presents the results. Clearly, propagated noise increases with decreasing cell size due to the downward propagation effect. The monthly solutions are not dominated by aliasing if a cell size of 1100 km, or smaller, is considered. It could also be argued that a cell size of 2200 km is in the same condition, with rare exceptions. Solutions for larger cell sizes, due to their much higher accuracy, are corrupted with aliasing more often. Notice that aliasing is similar for all cell sizes, as seen in Fig. 8, because the

Fig. 8 Monthly comparison between aliasing of different cell sizes, monthly solutions

In the previous section, the cell size at which aliasing becomes dominant over propagated noise was investigated. Referring to Fig. 4, it was suggested that there is a dependency between the cell size and the optimal estimation period. Inverting the gravity field over longer periods results in an aliasing-corrupted solution, which is undesirable. Fig. 9 relates the cell size with the optimal estimation period. Although never becoming dominant, aliasing is as large as propagated noise for the larger cell sizes. It is now considered that aliasing does not corrupt significantly the solution when it is less than half the propagated noise. This factor is matter for debate but such a conservative value is assumed in this report. Therefore, the estimation periods now presented should be regarded as a sensible upper limit. Table 1 roughly summarizes the conditions under which aliasing does not corrupt the solution. The results are obtained for the hydrological processes in the Zambezi basin. We believe, however, that

Fig. 9 Comparison between aliasing and propagated noise for sub-monthly solutions of different cell sizes, RMS of all 36 months

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significantly corrupting the solution. It was calculated Cell size (km) X Expected accuracy (mm) that a catchment 3300 km wide has an optimal estimation period of 15 days and the one that is 4400 km 4400 11 days 2.2 3300 15 days 2.6 large is most accurate if estimated for not more than 11 2200 Monthly 3 days. All these situations produce an error of 2–3 mm in terms of equivalent water height. The current level of accuracy deliverable by these results are representative for the capabilities of GRACE is such that propagated noise incurs inaccuraGRACE to recover hydrological mass transport procies that only basin-wide recovery of the gravitational cesses in general. field becomes corrupted by temporal aliasing. As technology advances and accuracy increases, it is expected that aliasing becomes increasingly dominant 5 Conclusions for smaller catchments. Thus, under these conditions, temporal aliasing is an issue that should not be Recovering the gravity field fluctuations of hydrologi- overlooked. cal processes, due to their inherent timevarying nature, can be corrupted by aliasing. When solutions are defined as piece-wise constant in time, the optimal es- References timation period is therefore the trade-off between the two conflicting effects of ever-decreasing propagated Data products and access. http://podaac.jpl.nasa.gov/ noise and ever-increasing aliasing. grace/data access.html, April 2007. Different point-mass catchment sizes can be recov- H.C. Winsemius et al. Assessment of gravity recover and cliered with different estimation accuracy. Considering mate experiment (grace) temporal signature over the upper zambezi. Water Resourches Research, 42, W12201, catchments smaller than roughly 2000 km and estimadoi:10.1029/2006WR005192, 2006. tion periods up to a month produces solutions with no noticeable aliasing. As the size of the catchments in- C. Jekeli. The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics creases, the propagated noise decrease, while aliasing and Dynamical Astronomy, 75: 85–101, 1999. remains the same. This invariably leads to the latter Table 1 Space and time trade-off

Image Super-Resolution via Filtered Scales Integral Reconstruction Applied to GOCE Geoid Data Ciro Caramiello, Guido Vigione, Alessandra Tassa, Alessandra Buongiorno, Eric Monjoux and Rune Floberghagen

Abstract In this paper a new approach is presented for enhancing image super-resolution via a filtered scale integral reconstruction. Based on the concept that any sensed quantity (from an instrumental point of view) must be necessarily considered as an averaged value in a suitable time and space continuum, the local reconstruction approach expects to map a space of average values onto another one (filtered scales) with a reduced extension and possibly, in an asymptotic limit, onto the space of points. The only assumption is that measurements are given in the integral average sense i.e., for some reason, they have to be considered as integral average values for given computational domains or cells. The proposed algorithm can be used to correlate averaged values to point values; yet the most reliable results, in a probabilistic sense, can be obtained if the ratio between input and output scales is not too high. The essential concept of this working methodology is the possibility to correlate average measurements to point-wise measurements upon suitably weighting the contribution of the molecules surrounding the inspected one.

Ciro Caramiello CORISTA (Consortium for Research on Advanced Remote Sensing Systems), Napoli, Italy Guido Vigione SERCO SpA c/o ESA/ESRIN, Via G. Galilei 64, Frascati, Italy Alessandra Tassa SERCO SpA c/o ESA/ESRIN, Via G. Galilei 64, Frascati, Italy Alessandra Buongiorno European Space Agency (ESA), http://www.esa.int Eric Monjoux European Space Agency (ESA), http://www.esa.int Rune Floberghagen European Space Agency (ESA), http://www.esa.int

Keywords Geoid · GOCE · Interpolation

1 Introduction The need for a new and reliable n-dimensional interpolation technique has recently gained a paramount importance in several areas of scientific research both from a theoretical and a computational point of view. In this paper a new approach is presented for enhancing super-resolution via filtered scale integral reconstruction. This method, based on an analytical frame, is fairly general and can be applied to any data source. In particular it was applied to GOCE geoid data which have been retrieved with an angular resolution of about 30◦ . By applying the proposed approach, a new interpolated geoid grid can be obtained ensuring the minimization of estimation error in a best fit sense.

2 GOCE Mission Overview The Gravity Field and Steady-state Ocean Circulation Explorer (GOCE) is an ESA mission whose primary objective is to measure the Earth’s Geoid and gravity anomaly field with very high accuracy. The GOCE gravity field models are provided in terms of a set of dimensionless coefficients of a spherical harmonic series up to a maximum degree of the gravity potential. These coefficients are the result of the gravity field determination process. All other quantities delivered together with the products are derived from these coefficients. The coefficients of the spherical harmonic series of the gravitational potential are provided (Rummel et al., 2002) as in the following Eq. (1):

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= rGM

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a r

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(Cnm cos mλ + Snm sin mλ)Pnm (cos θ ) (1) Geoid Heights are defined (Rummel et al., 2004) as in the following Eq. (2):

observations up to so-called Level 2 products. Main Level 2 products will be:

r r r

Precise Science Orbits. Gravity Gradients in different reference frames. Gravity field models as well as quality indicators associated to them.

The GOCE L2 product containing the Geoid Model is called EGM GOC 2 . The gravity field is provided as N (r, θ, λ) = rGM ( Cnm cos mλ γ0 r (2) spherical harmonic series up to a specific degree and n=0 m=0 order. Derived quantities are provided on equi-angular + Snm sin mλ)Pnm (cos θ ) grids. The quality report is provided in PDF format. The product content is listed below: The Spatial Resolution is linked to maximum resolv- r Spherical harmonic series including error estimates. able degree “l” as follows (Koop, 1993): r Grids of geoid heights, gravity anomalies and deflections of the vertical. D = 20000/l (3) r Propagated error estimates in terms of geoid heights. The GOCE Satellite primary payload is composed r Quality report for GOCE gravity field model. by the Electrostatic Gravity Gradiometer (EGG), a three-arm gradiometer composed by highly sensitive accelerometers and by a GPS receiver called the Satellite-to-Satellite Tracking System (SST). GOCE 3 The Integral Reconstruction Algorithm will fly along a very low orbit, at a mean altitude of 250–240 km and inclination of 96.5◦ . Two main uses The integral reconstruction algorithm is based on the can be distinguished (see Beutler et al., 2003): hypothesis that any signal is a mean value relative to n N  ref n   a

r

r

The spatial variations of gravity and Geoid are directly related to density anomalies in lithosphere and upper mantle, respectively, and consequently to interior stresses and ultimately to mass. In this respect GOCE provides important new information to studies of continental and oceanic lithosphere and upper mantle. Secondly, a detailed Geoid surface when combined with satellite altimetry yields the sea surface topography, the quasi-stationary deviation of the ocean surface from its hypothetical surface of rest. Under the assumption of geostrophic balance, the ocean topography can be directly translated into a global map of surface oceanic circulation. Thus, ocean surface circulation becomes directly measurable, globally and uninterruptedly. In conjunction with higher resolution ocean models and ocean measurements, GOCE is expected to improve significantly estimates of global mass and heat transport in the oceans.

a backscattering cell as in Eq. (4). Every cell is part of a macromolecule generally centered in the cell itself which contributes numerically to a local integral bi-dimensional regressive strategy: 1 ϕi = μ(i )

 i

ϕ(P)dω

(4)

where ϕi ∈ Fi−k,i−k+1,...,i,i+ j−1,i+ j is a function definite in the macromolecule relative to i either spreading around it in a concentric fashion (central scheme) or in a directional way (upwind scheme). μ is a norm to be suitably defined. F is the approximant family (polynomial, fourierian, etc . . .) which is defined with n degrees of freedom depending on its order, i.e. the local expansion order. It represents an expansion in an appropriate basis (see Gene et al., 1982): ϕi =

s 

λi (i−k , i−k+1 , . . . , i+ j )ϕ i

(5)

i=1

The European Space Agency has defined a ground where λi depends on all cells in the macromolecule; its system, which will prepare and process the GOCE value is obtained with a least squares approach via a

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singular value decomposition (SVD) which minimizes, 4 The Geoid Reconstruction Algorithm e.g. in the euclideian norm, the reconstruction error: The procedures described above have been applied also Cλ − b = U (AV V λ − b)2 (6) to an analytical function simulating a part of the Earth Geoid surface. The results of application of the integral reconstruction method on the simulated surface, by usAfter that, an integral 2D-regression can be performed: ing different noise degrees, are depicted in the following figures. The following figures show meshes with different r either in a mean value-to-point way: ϕi → ϕ(P) granularities and corresponding reconstructions. yielding a function of a point, in output, from a seOn the left column: two meshes relative to the same ries of mean integral values, in input. inspected area but with different granularity (underr or in a mean value-to-mean value way: ϕi → ϕ j sampled with respect to the surface showed in above i j where i ∈ {} j is a measure decreasing set of do- figures) have been depicted (“theoretical” stands for mains contained in λi defining the so called filtered analytical). On the right column the integral regressive reconstruction algorithm has been applied for each of scale approach. the base lattices, respectively. It is important to highlight that both the μ-interpolated lattices approximate Figure 1 depicts a typical reconstruction result ob- the surface in the inner region whilst, as concerns petained with a set of 1D values; these ones originate e.g. ripheral areas, only the refined μ-lattice is correct. That from a typical DEM survey. One can see the confronta- is not absolutely a problem since the μ-regressive altion of a series of different algorithms based, first of gorithm can be typically and adopted for inner cells. In all, on the two main “philosophical” approaches: sim- other words it is possible to span the entire inspected ple interpolation and integral reconstruction. A multi- areas by resorting to a multiple application of the μple reconstruction families strategy has been adopted regression on several sub-domains. Recall that the “μin order to test several regressive bases. Depending on interpolated-fine” gridding has been obtained directly the particular problem an integral reconstruction family from “μ-interpolated-coarse” considering that in each can yield the best values with respect to the remaining cell the behavior of the shape is completely know since ones. Yet what generally happens is that the reconstruc- the function is analytically locally defined. This extion error, by using a known test signal, is the lowest actly explains what “filtered scales” can achieve: from with respect to classical interpolation. a mean value defined, as an integral value, in a wider cell you can obtain another integral mean value defined in a smaller cell. T

T

5 Conclusions

Fig. 1 Typical reconstruction result obtained with a set of 1D values

These preliminary results concisely demonstrate that Integral Reconstruction Algorithm performance and characteristics are certainly suitable for Reconstructed Geoid Computation. Relative errors obtained on applying proposed algorithm on analytical surface are limited to absolutely acceptable ranges. The same algorithm performances are foreseen to be used with GOCE Geoid Data. The application of the proposed algorithm on GOCE Geoid data is aimed at enhancing the spatial resolution of the Geoid field. Future work will be

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Fig. 2 Typical μ-regressive application for no and low noise level. The left columns (A, C) represents the input data, the right columns (B, D) the interpolated data

devoted to customize the algorithm in order to “pro- Beutler G., Rumme R., Drinkwater M.R., von Steiger R. (eds.) (2003). Earth gravity field from space – from cess” real GOCE Data (non-analytic surface). sensors to earth sciences. Space Sciences Series of Due to the generality of the method, possible Issi 17, ESA GOCE WEB site www.esa.int/ livingweighing (in the norm sense) and/or ad-hoc preproplanet/goce, http://esamultimedia.esa.int/docs/GOCE Level2 cessing techniques are candidate to be adopted for Product User Handbook.pdf. further specialized enhancement of algorithm perfor- Gene H., Van Loan G. (1989). Matrix Computations, 2nd edn. The John Hopkins University Press Baltimore, MD. mances. Yet, probabilistic and/or euristic extensions Han S.C., Jekell C., Shum C.K. (2002). Efficient gravity field could be adopted that could be tuned depending on the recovery using in situ disturbing potential observables from particular ground configuration.

References Balmino G., Perosanz F., Rummel R., Sneeuw N., S¨unkel H., Woodworth P. (1998): European views on dedicated gravity field missions: GRACE and GOCE. ESA Report ID: ESDMAG-REP-CON-001.

CHAMP, Geophys. Res. Letters 29(16) Kluwer. Koop R. (1993). Global gravity field modelling using satellite gravity gradiometry, Publications on Geodesy, New Series 38, Netherlands Geodetic Commission. Marchuk G. (1982). Methods of numerical mathematics. In: Proc. of IAG Symposium, Springer Verlag. Rummel R. (1986). Satellite Gradiometry, Inc. Mathematical and Numerical Techniques in Physical Geodesy, Lecture Notes in Earth Sciences 7, Springer.

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333 Rummel R., Balmino G., Johanessen J., Visser P., Woodworth R. (2002): Dedicated gravity field missions. principles and aims. Journal of Geodynamics 33, 3–20. Rummel R., Gruber Th., Koop R.; (2004) High level processing facility for GOCE: products and processing strategy. Proceedings of the 2nd International GOCE User Workshop, ESA-SP569. Schreiner M., (1994). Tensor Spherical Harmonics and Their Application in Satellite Gradiometry, Universit¨at Kaiserslautern, Fachbereich Mathematik. Schuh, W.-D., (1996): Tailured Numerical Solution Strategies for the Global Determination of the Earth’s Gravity Field, Geod¨atische Institute der T.U. Graz, Graz.

Fig. 3 Color maps of the relative error (B vs. A and D vs. C, Fig. 2) with respect to the max abs value of the current working macromolecule

Rummel R., Sanso F., Gelderen M. v., Brovelli M., Koop R., Miggliaccio F., Schrama E., Sacerdote F. (1993): Spherical harmonic analysis of satellite gradiometry. Netherlands Geodetic Commission, New Series, 39, Delft.

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Fig. 4 Typical μ-regressive application for moderate and severe noise level. The left columns (A, C) represents the input data, the right columns (B, D) the interpolated data

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Fig. 5 Meshes with different granularities (A, C) and corresponding reconstructions (B, D)

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An Error Model for the GOCE Space-Wise Solution by Monte Carlo Methods F. Migliaccio, M. Reguzzoni, F. Sanso` and N. Tselfes

Abstract The aim of the data analysis of the GOCE satellite mission is to estimate spherical harmonic coefficients of the gravitational potential and the corresponding error covariance matrix. However this error estimation is generally very complicated and computationally demanding because of the large number of observations. In particular, this is true for the space-wise approach, which is a collocation solution implemented by several steps. Up to now the error covariance matrix for this approach has been computed under simplifying hypotheses. In order to get a more realistic assessment of the true estimation error a Monte Carlo method can be applied. This requires the generation of several stochastic simulations and the computation of the corresponding solutions. This procedure is made numerically feasible by storing and not recomputing every time the operators that form the space-wise approach. Tests on realistic scenarios are performed and positive results are achieved.

1 Introduction The space-wise approach (Migliaccio et al., 2004, 2006) to the gravity field determination from the GOCE mission (ESA, 1999) is a collocation solution implemented by the following steps (see Fig. 1):

(1) the anomalous potential T along the orbit is estimated by the energy conservation method (Jekeli, 1999; Visser et al., 2003); (2) the coloured noise of the observations (potential and gradiometer) is reduced by a time-wise Wiener “Orbital” Filter (WOF) applied by Fast Fourier Transform (FFT) (Papoulis, 1984); (3) the filtered data are used to predict, by local collocation, a spherical grid of potential T and of second radial derivatives Trr at mean satellite altitude (Tscherning, 2005; Migliaccio et al., 2007a); (4) a vector of spherical harmonic coefficients T, up to a maximum degree L = 250, is computed by harmonic analysis (Migliaccio and Sans`o, 1989; Rummel et al., 1993), i.e. two sets T pot and T grad Keywords GOCE mission · Satellite gradiometry · are computed from T and Trr grids respectively and Space-wise approach · Monte Carlo methods are then combined by a weighted average; (5) the procedure is iterated in order to:

F. Migliaccio DIIAR, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy

M. Reguzzoni Geophysics of the Lithosphere Department, Italian National Institute of Oceanography and Applied Geophysics (OGS), c/o Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy F. Sans`o DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy N. Tselfes DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy

– recover the signal lost in WOF, by exploiting a complementary WOF, – recover the misalignment between instrumental and local orbital reference frame (rotation correction), – converge to an optimal collocation solution (Reguzzoni and Tselfes, submitted). The convergence is reached after three or four iterations. For details about numerical results of these steps, see Migliaccio et al. (2007b).

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Fig. 1 The flow chart of the space-wise approach to GOCE data analysis

Since all operations involved are linear, apart from estimating by the space-wise approach the correspondthe estimation of covariance models which however are ing harmonic coefficients and then computing average not changed through iterations, the space-wise solution and sample error covariances of these solutions. Tˆ can be written as The MC methods (Robert and Casella, 1999) are slowly converging and a large sample would certainly Tˆ = S y 0 , (1) provide us with reliable results; yet the computation of even a single solution S(y 0 ) is a long process (about 10 where S is a linear solver and y 0 is the data vector. days for two months of GOCE data for the space-wise The operator S depends on an a-priori model for two approach). The two most time consuming operations of reasons: this model is used for observation noise es- the space-wise approach are the gridding and the data timation and as a reference field subtracted from the synthesis along the orbit (see Fig. 1). The former is opdata. Once convergence is reached, the obtained solu- timized by computing only once the gridding operator tion Tˆ can be used for improving the noise modelling and storing it, so that it is not recomputed for every MC and for creating a more homogeneous reference field. simulation. The latter is optimized by processing all Therefore the space-wise iterative scheme is repeated MC data sets in parallel. Since the observation points from the beginning, giving rise to a second solver S  for every MC data sets are the same, the computation and the corresponding new solution Tˆ  . Numerical tests of the spherical harmonics required for the synthesis (at (Reguzzoni and Tselfes, submitted) show that an im- every iteration) is made once but used simultaneously for all the MC simulations. Still, taking into account portant improvement is obtained from Tˆ to Tˆ  . Strictly speaking the final operator S, i.e. the combi- the resulting computation burden, only 10 MC realizanation of S and S  , is a non-linear operator, though the tions are used. This however gives rise to a result that second step solver S  provides a good linear approxi- compares well with the true errors, which in our case are known since everything is done by simulation. mation of S in the solution area. The computations presented in the paper are based One essential feature of a sound estimation proceon the End-To-End (E2E) GOCE simulated data that dure is to have a full covariance information on the esinclude two months of realistic observations. The cortimators of the coefficients, namely the full covariance responding signal is generated from EGM96 (Lemoine  matrix of the estimation errors of Tˆ . et al., 1998). Moreover these data include both the erDue to the complexity of the operator S and even of rorless observables and the total errors, which represent  its linear approximation S , it is not possible to derive such an information through a more or less rigorous various effects of random and deterministic nature. The work is divided into two parts. In the first part, covariance propagation. This would be difficult in any the signal is not sampled but kept fixed to the true case due to the large number of GOCE data. So, as it one (EGM96 minus a reference model). 10 samples of has been done for other approaches (see e.g. Alkhatib noise are drawn and, by adding the signal, 10 MC realand Schuh, 2007), it is shown in this paper that a izations are generated. The corresponding 10 sample Monte Carlo (MC) approach can be applied, which basolutions are computed by the space-wise approach. sically consists of sampling the random variables in y 0 ,

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Each solution minus the true model provides a sample of estimation errors and with 10 such samples an error covariance matrix is computed. This is presented in Paragraph 2. In the second part, 10 different samples of signal are drawn. This means that 10 models with the same degree variances as EGM96 minus the reference model are randomly generated and that the signal is then synthesized from these models. Note that, in this case, the computational benefit of synthesizing all the data sets simultaneously is exploited again. A different noise sample is added to each signal and, as before, 10 solutions are computed along with their sample error covariance matrix. This is presented in Paragraph 3. Therefore, the pure noise propagation by MC methods is studied in Paragraph 2, while in Paragraph 3 the stability of the results of Paragraph 2 are evaluated when the signal is not synthesized by the true model but from a randomly generated one with the same stochastic characteristics. In Paragraph 4 we shortly discuss the results obtained.

2 Pure Noise Propagation Noise covariance functions are needed not only for the computation of the space-wise operators S and S  , but also for the simulation of the MC noise samples. Therefore, an estimation procedure is applied (Reguzzoni and Tselfes, submitted). First, an estimate of the harmonic coefficients from degree 2 to 50 from GPS tracking data only is derived. This model is extended with EIGEN-GL04C (F¨orste et al., 2007) from 51 to 360; this initial model is called T i hereafter. A first rough estimate of the noise ν is given by noise ∼ = data − signal(T i )

0.0185

E2

0.018

0.0175 0

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derivatives in the along-track direction Txx is displayed in Fig. 2 for the first 200 s of time lag. It is clear from Fig. 2 that the empirical and true covariance functions fit very well, since their difference in the τ ≤ 200 s window is below the 2% level; outside this window the two plots cannot be distinguished. In Fig. 3 the “true” noise of Txx and two samples drawn from the “estimated” covariance are shown. Their statistical behaviour is quite similar, at least to a pure visual inspection. Starting from the original data set, that can be considered as signal plus noise, y0 = s + ν0,

(4)

the initial space-wise solution Tˆ is computed and the “true” errors δT = Tˆ − T

(5)

are available, as well as their representation in terms of geoid δ N (ϕ, λ) at the level of the ellipsoid. At the 0.4

where signal(T i ) means the specific observational functional applied to the coefficients T i . With this first noise model, by computing the empirical estimator

0.2

t=1

100 s

Fig. 2 Txx true noise covariance (solid black) and its first empirical estimate (dashed grey)

(2)

N 1  Cν (τ ) = ν(t)ν(t + τ ), N

50

E

0 –0.2

(3) –0.4

0

10000

20000

s

30000

the “stationary” covariance of ν is obtained. This is done for covariances and cross-covariances of all the Fig. 3 True errors (black) and simulated errors (grey) for Txx GOCE observables. The case of the second order observable

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same time, 10 sample data sets y k (k = 1, 2, . . . , 10) are generated by adding to the true signal s 10 noise samples ν k drawn from the estimated covariance functions (cfr. Fig. 2), i.e.

Both from Fig. 4 and Table 1 it becomes clear that differences are not large, though significant. In particular the almost double root mean square (rms) of the predicted geoid error in the cap-free area is not acceptable. These effects are attributed to the small difference (6) between the true noise covariance and the one derived yk = s + νk . from the prior global model T i . From each y k a solution, i.e. a vector of coefficients Tˆ k , In order to improve the noise modelling, the initial is computed; the corresponding estimation error vector space-wise solution Tˆ is used, instead of T i , to estimate is given by again the “empirical” covariance functions. This time, the estimated covariances are in even better agreement δ Tˆ k = Tˆ k − T . (7) with the “true” ones, as shown in Fig. 5 for the case of Txx . The idea is to compare the statistical behaviour of δ Tˆ k Repeating now the same scheme presented in the with that of δT in order to see whether reliable error first part of this paragraph, but using these new emestimates from MC method can be derived. To this aim pirical noise covariances, new “predicted” error degree a sample error covariance matrix is computed by variances (cfr. Fig. 6) and new “predicted” geoid errors (cfr. Table 2) are obtained. These results can be 10 1  ˆ ˆT considered reliable and satisfactory, showing that the Cδ Tˆ = δT k δT k . (8) 10 k=1

The corresponding “predicted” error degree variances are compared with the “true” error degree variances (cfr. Fig. 4). “Predicted” and “true” geoid errors, both for the whole ellipsoid and for the ellipsoid without polar caps (|ϕ| ≤ 80◦ ), are compared as well (cfr. Table 1).

0.01830 0.01828 E2 0.01826 0.01824 0

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s 10–14

Fig. 5 Txx true (solid black) and empirical (dashed grey) noise covariance of the final solution. Note the different axis scale compared to Fig. 2

EGM96

10–16

10–18

10–14

10–20

EGM96

10–16 0

50

100 degree

150

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10–18

Fig. 4 Predicted (black) and true (grey) error degree variances of the initial solution Table 1 Predicted and true geoid errors of the initial solution Geoid errors

Predicted (cm)

True (cm)

|ϕ| ≤ 80◦ |ϕ| ≤ 90◦

5.6 20.6

3.2 20.2

10–20 0

50

100 degree

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Fig. 6 Predicted (black) and true (grey) error degree variances of the final solution

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Table 2 Predicted and true geoid errors of the final solution Geoid errors

Predicted (cm)

True (cm)

|ϕ| ≤ 80◦ |ϕ| ≤ 90◦

2.3 17.1

2.4 17.2

MC method, after an improvement of the noise modelling by a previously estimated model, can provide quite sensible estimates of the error statistics, at least when considering the true gravity field and not a sampled one. Note that the true error degree variances between Figs. 4 and 6 and the true geoid errors between Tables 1 and 2 are different. The operator S  of the final spacewise solution performs better because of the already stated improvement of the noise covariance modelling, but also because of the more homogeneous reference model used to compute Tˆ  .

vector T k the corresponding observational signal s k is synthesized. After that, 10 noise vectors ν k (k = 1, 2, . . . , 10) are generated from the initial noise covariance Cν (τ ) (see Eq. (3)), as in Paragraph 2. Thus 10 sets of simulated observations are obtained as yk = sk + νk .

(11)

The space-wise approach is applied to every data set, producing coefficients estimates Tˆ k and the corresponding errors δ Tˆ k = Tˆ k − T k .

(12)

By using all the computed δ Tˆ k the sample error covariance matrix is derived (Eq. (8)). The corresponding error degree variances are shown in Fig. 7 in comparison with the true ones. The errors in terms of geoid on the whole ellipsoid and in the cap-free area only are reported in Table 3. 3 Monte Carlo Method with Simulated As we can see, these results are not much different Signal and Noise from those presented in Paragraph 2 (see Fig. 4 and Table 1); however, since now the gravity field is ranIn collocation theory the estimation error depends also domly generated in such a way that polar caps become on the signal. In a real situation the true signal is not more similar to the rest of the field, the MC solutions known, so signal samples are drawn by assuming that are better in that area. This is reflected in the fact that the gravity field is rotationally invariant. Under this hypothesis spherical harmonic coefficients are considered independent random variables with variances depending on the degree (Heiskanen and Moritz, 1967). 10–14 EGM96 In order to explore the model space in an area not too far from the reality, as it would happen by considering the full degree variances of EGM96, the coeffi- 10–16 cients T are sampled as T sample k = T k = T i + δT k

10–18

(9)

where T i is defined exactly as in Paragraph 2 and δT k are purely random variables from degree variances σn (δT k )2 = σn (T EGM 96 − T i )2 .

(10)

This gives rise to a field which is basically T i plus a spherically invariant part with degree variances of the same order of the difference between EGM96 and the field T i used as reference to compute the initial solution. This difference is assumed as a realistic measure of the actual ignorance of the gravity field. From each

10–20 0

50

100 degree

150

200

Fig. 7 MC estimated (black) and true (grey) error degree variances of the initial solution Table 3 MC estimated and true geoid errors of the initial solution Geoid errors

Predicted (cm)

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|ϕ| ≤ 80◦ |ϕ| ≤ 90◦

5.5 9.8

3.2 20.2

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degree

degree

–20 0 the predicted rms of the geoid error over the whole ellipsoid (see Table 3) is now 9.8 cm against 20.2 cm of the true error rms. –21 50 As in Paragraph 2, the space-wise approach is repeated by using the initial solution Tˆ to produce an improved noise model and an improved reference model. 100 –22 This implies that the MC noise vectors are regenerated according to the new noise model. Also the signal 150 –23 vectors are regenerated because the degree variances (Eq. (10)) change due to the new reference model. 10 sets of spherical harmonic coefficients are computed 200 –24 –200 –100 0 100 200 from the 10 new MC simulated observations, so that a order sample error covariance matrix is predicted for the final space-wise solution Tˆ  . The error degree variances Fig. 9 True error variances of each spherical harmonic coeffiare shown in Fig. 8 and the geiod errors are presented cient of the final solution in Table 4. –20 0 As we can see, while the geoid error statistics in the cap-free area are in quite good agreement, the error rms for the whole ellipsoid is quite different, im–21 50 plying that the error in the polar caps is poorly estimated. This is why the predicted error degree variances –22 100 seem not comparable with the true ones (see Fig. 8). In fact the true error degree variances include a large contribution from the low order coefficients, whose –23 150 representation is concentrated at the poles (Sneeuw and van Gelderen, 1997). This hypothesis is confirmed by inspecting the true and predicted error variances 200 –24 –200

–100

0 order

100

200

Fig. 10 Predicted error variances of each spherical harmonic coefficient of the final solution 10–14 EGM96 10–16

10–18

10–20 0

50

100 degree

150

200

Fig. 8 MC estimated (black) and true (grey) error degree variances of the final solution Table 4 MC estimated and true geoid errors of the final solution Geoid errors

Predicted (cm)

True (cm)

|ϕ| ≤ 80◦ |ϕ| ≤ 90◦

2.3 2.8

2.4 17.2

coefficient by coefficient, displayed in Figs. 9 and 10 respectively. Note that the errors of the low order coefficients are underestimated. Naturally the same effect was not present in Paragraph 2 where the signal was not randomly generated and was always the same for all data sets. The conclusion of this paragraph is that the estimates of the coefficient errors by MC method are reliable not only for the pure case of a true field equal to EGM96, but also for other fields with coefficients “similar” to those of EGM96. As a matter of fact such synthetic fields have more homogeneous characteristics in the polar caps than EGM96 minus the reference model (basically EIGEN-GL04C), with the effect that, for such fields, a better prediction even in the area not covered by data is obtained. Indeed this is not expected to be the case for the true field.

An Error Model for the GOCE

4 Conclusions and Discussion The first interesting conclusion of this paper is that a MC estimation procedure for the space-wise processing of GOCE data is feasible. This requires numerical optimization to cope with the heavy computational burden. For the space-wise approach this is achieved by storing and not recomputing every time the applied operators and by exploiting the parallel synthesis of many data realizations. The second conclusion is that the full covariance matrix of the final solution estimated by the MC method gives a quite accurate picture of the true covariance structure of the estimation errors; in other words, the quality assessment of the solution via the MC method is quite reliable. The third conclusion is that the previous statement holds true for a population of simulated gravity fields to which the true gravity field is likely to belong. Finally a last point is worth underlining. In the space-wise approach the two grids of T and Trr values at satellite altitude give rise to two distinct solutions T pot and T grad which then need to be combined to produce the final model. This was previously done by weighting the two estimates with error degree variances. However, since now the variances of the individual coefficients are available, the weighting could be done in a more precise way. In this respect we should underline that by using methods known in literature as Markov Chain Monte Carlo (MCMC) methods (Gamerman, 1997), in future the full covariance structure of the two vectors T pot and T grad could be utilized to produce the really best linear combination of the two. Acknowledgments This work was performed under ESA contract No. 18308/04/NL/NM (GOCE High-level Processing Facility).

References Alkhatib, H., and W.D. Schuh (2007). Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for large-scale least squares problems, Journal of Geodesy, 81, pp. 53–66. ESA (1999). Gravity Field and Steady-State Ocean Circulation Mission. ESA SP-1233 (1). ESA Publication Division, c/o ESTEC, Noordwijk, The Netherlands.

343 F¨orste, Ch., F. Flechtner, R. Schmidt, R. K¨onig, U. Meyer, R. Stubenvoll, M. Rothacher, F. Barthelmes, H. Neumayer, R. Biancale, S. Bruinsma, J.M. Lemoine and S. Loyer (2007). Global mean gravity field models from combination of satellite mission and altimetry/gravimetry surface data. In: Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Rome, Italy, pp. 163–167. ISBN 92-9092938-3, ISSN 1609-042X. Gamerman, D. (1997). Markov Chain Monte Carlo – Stochastic Simulation for Bayesian Inference. Chapman & Hall, London. Heiskanen, W.A., and H. Moritz (1967). Physical Geodesy. W.H. Freeman and Company, San Francisco. Jekeli, C. (1999). The determination of gravitational potential differences from satellite-to-satellite tracking Celestial Mechanics and Dynamical Astronomy, 75, pp. 85–101. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp and T.R. Olson (1998). The development of the joint NASA GSFC and NIMA geopotential model EGM96. NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, Maryland. Migliaccio, F., and F. Sans`o (1989). Data processing for the Aristoteles mission. In: Proc. of the Italian Workshop on the European Solid-Earth Mission Aristoteles. Trevi, Italy, May 30–31 1989, pp. 91–123. Migliaccio, F., M. Reguzzoni and F. Sans`o (2004). Space-wise approach to satellite gravity field determination in the presence of coloured noise. Journal of Geodesy, 78, pp. 304–313. Migliaccio, F., M. Reguzzoni and N. Tselfes (2006). GOCE a full-gradient simulated solution in the space-wise approach. In: International Association of Geodesy Symposia, ‘Dynamic Planet – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools’, P. Tregoning and C. Rizos (eds), Vol. 130, Springer-Verlag, Berline, pp. 383–390. Migliaccio, F., M. Reguzzoni, F. Sans`o and N. Tselfes (2007a). On the use of gridded data to estimate potential coefficients. In: Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Rome, Italy, pp. 311–318. ISBN 92-9092-938-3, ISSN 1609-042X. Migliaccio, F., M. Reguzzoni, F. Sans`o, N. Tselfes, C.C. Tscherning and M. Veicherts (2007b). The latest test of the spacewise approach for GOCE data analysis. In: Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Rome, Italy, pp. 241–248. ISBN 92-9092-938-3, ISSN 1609-042X. Papoulis, A. (1984). Signal Analysis. McGraw Hill, New York. Reguzzoni, M., and N. Tselfes (submitted). Optimal multistep collocation: application to the space-wise approach for GOCE data analysis. Submitted to Journal of Geodesy. Robert, C.P., and G. Casella (1999). Monte Carlo Statistical Methods. Springer-Verlag, New York. Rummel, R., M. van Gelderen, R. Koop, E. Schrama, F. Sans`o, M. Brovelli, F. Migliaccio and F. Sacerdote (1993). Spherical harmonic analysis of satellite gradiometry. Netherlands Geodetic Commission Publications on Geodesy, New Series, N. 39.

344 Sneeuw, N., and M. van Gelderen (1997). The polar gap, In: Geodetic Boundary Value Problems in View of the One Centimeter Geoid, Lecture Notes in Earth Sciences, F. Sans`o and R. Rummel (eds), Vol. 65, Springer Verlag, Berlin, pp. 559–568. Tscherning, C.C. (2005). Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares

F. Migliaccio et al. Collocation using simulated data. In: International Association of Geodesy Symposia, ‘A Window on the Future of Geodesy’, F. Sans`o (ed), Vol. 128, Springer-Verlag, Berlin, pp. 277–282. Visser, P.N.A.M, N. Sneeuw and C. Gerlach (2003). Energy integral method for gravity field determination from satellite orbit coordinates. Journal of Geodesy, 77, pp. 207–216.

Accuracy Analysis of External Reference Data for GOCE Evaluation in Space and Frequency Domain ¨ K.I. Wolf and J. Muller

Abstract With the upcoming ESA satellite mission GOCE, all components of the gravitational tensor (2nd derivatives of the Earth’s gravitational potential) will be measured globally except of the polar gaps. The highest accuracy level within the measurement band√ width (MBW, 5–100 mHz) is 11 mE/ Hz for the diagonal components of the tensor (1 mE = 10−12 1/s2 ). To meet this accuracy level, the gradiometer will be calibrated and evaluated internally as well as externally. One strategy of an external evaluation includes the use of a global geopotential model in combination with ground gravity data upward continued to satellite altitude. In this study an error estimation for the external reference data is carried out (a) statistically by applying least-squares collocation and (b) empirically in a synthetic environment including (correlated and uncorrelated) noise. The use of synthetic data permits a closed-loop validation in all points. The spectral combination method based on integral formulas with a modified kernel function is applied to compute all components of the tensor. The closed-loop differences are analysed in the space and in the frequency domain. The dependency of the prediction error on the characteristics of the input data (noise level, area size and resolution) is √ shown. An accuracy below the required level of 11 mE/ Hz can be reached combining gravity anomalies with a noise level of 1 mGal and current global geopotential models. K.I. Wolf Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, 30167 Hannover, Germany, e-mail: [email protected] J. M¨uller Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, 30167 Hannover, Germany

Keywords Satellite gradiometry combination · Synthetic Earth model

·

Spectral

1 Motivation The ESA mission GOCE, which is in preparation for launch in 2008, will for the first time apply satellite gradiometry, i.e. the measurement of differential accelerations over short baselines in one satellite. Satellite gradiometry, in combination with GPS-SST (satellite to satellite tracking), is used to determine the Earth’s gravity field, aiming at an accuracy of 1–2 cm for the geoid and 1 mGal for gravity anomalies at 100 km spatial resolution (half wavelength). The recovery of the signal up to these high frequencies (corresponding to a spherical harmonic degree of 200) is the special objective of the gradiometer measurements. The global evaluation of the gradiometer data above a spherical harmonic degree of about 105 is particularly difficult because such a high resolution is not adequately covered by global geopotential models (GPMs) – not even by the ones based on the GRACE mission. Therefore high-quality terrestrial data have to be upward continued for the computation of the reference data for evaluation purposes. The evaluation can only be applied regionally because sufficiently accurate ground data are only available for selected areas. A GPM, representing the long-wavelength part, is introduced in a remove-restore procedure.

2 Computation Methods The computation of reference tensor values from terrestrial data is investigated in several studies (e.g.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

345

¨ K.I. Wolf and J. Muller

346

Bouman and Koop, 2003; Bouman et al., 2005; Denker, 2003; Pail, 2003, 2004). In the present study all tensor components are computed using integral formulas, whereas in (Arabelos and Tscherning, 1998; Arabelos et al., 2007; Wolf, 2007) least-squares collocation is applied. In contrast to (Kern and Haagmans, 2005), here, spectral weights are introduced for the combination of the gravity data with a GPM. In the numerical test scenarios uncorrelated noise as well as correlated noise is considered. At first, the attainable accuracy of the reference values is estimated by leastsquares collocation. Secondly, the remaining noise is evaluated within a synthetic environment. Here, closed-loop differences are analysed in the space but also in the frequency domain. The latter is of special interest for the evaluation of GOCE as explained above. An accuracy level of 1–2 mE in the space domain is aimed for, √ derived from the frequency noise level of 11 mE/ Hz (Drinkwater and Kern, 2006). The symmetric tensor of the 2nd order derivatives of the gravitational potential V reads

V[ij]

⎡ ⎤ Vxx Vxy Vxz ∂2V = = ⎣Vyx Vyy Vyz ⎦ ∂i∂ j Vzx Vzy Vzz

(1)

(2)

N is based on the G RS 80 normal potential. where V[ij] Applying the remove-restore technique residual gravity anomalies

g R = g T − g M

(3)

as differences between the terrestrial gravity anomalies (g T ) and those from a geopotential model (g M ) are used in the computations. The resulting residual tensor R ) have to be restored using again the components (Tˆ[ij] GPM (Tˆ M ): [ij]

M R Tˆ[ij] = Tˆ[ij] + Tˆ[ij] .

The theoretical background of least-squares collocation is described in detail in Moritz (1980). The basic formula for the error prediction part is E Tij Tij = C T R T R − ij

(4)

ij

C T R g R (Cg R g R + E gg )−1 C TT R g R ij

(5)

ij

with the signal covariance matrices C T R T R , C T R g R ij ij ij and Cg R g R and the error covariance matrices E Tij Tij and E gg (i, j = x, y, z). The mathematical background for the derivation of the signal covariances is described in (Tscherning and Rapp, 1974; Tscherning, 1976). All signal covariances are based on the homogeneous and isotropic covariance function CTT of the disturbing potential T CTT =

∞  n=2

with x pointing North, y East and z radially outwards. In the following the tensor is considered with respect to the disturbing potential T as N T[ij] = V[ij] − V[ij]

2.1 Least-Squares Collocation



R 2B σn2 (T ) rPrQ

n+1 Pn (cos(ψPQ)) (6)

with the degree variances σn2 (T ), the radius R B of the Bjerhammer sphere, the radii r P , r Q of the points P and Q, and the Legendre polynomials Pn (cos(ψ)) with respect to the spherical distance ψ. In the remove-restore procedure the covariance function CTT is based on the error degree variances of a reference GPM up to degree n max and above degree n max on the degree variance model of Tscherning and Rapp (1974). The model parameters are modified to adapt the resulting covariance function to an empirical local one which is based on the residual gravity anomalies of the test area, cf. (Knudsen, 1987). Subroutines implemented by Tscherning (1976) are used in an updated version for the computation of the signal covariance values. The error covariance matrix E gg is either a diagonal matrix in the scenarios with uncorrelated noise or a matrix completely filled according to a distance-dependent noise model (see Sect. 3). Observation data in this study are arranged in a geographical grid. Therefore, the resulting matrix Cg R g R + E gg has a block Toeplitz structure, cf. (Wolf, 2006, 2007). This structure is exploited for memory reduction and effective solution of the equation system (e.g., SLICOT-package, cf. Benner et al., 1999). A system with a dimension of 32,400 × 32,400 consisting of mλ = 180 Toeplitz

Accuracy Analysis of External Reference Data

347

blocks, each with mϕ = 180 rows, can be solved in 8 h synthetic environment, which comprises a GPM and on a single Sun Ultra Sparc processor (400 MHz). regional terrestrial gravity data over Central Europe, both including (correlated and uncorrelated) noise. The input data (g T , g M ) and the (error-free) output data (T[ij] ) are based on a synthetic Earth 2.2 Spectral Combination Method model composed of a Digital Topography Model (DTM) and a GPM. Earth’s surface, where the gravity Integral formulas with spectral weighting (Wen- observations are located, is modelled by the DTM, cf. zel, 1982) are used to combine regional terrestrial Fig. 1. The DTM is based on the DTM used by Denker gravity data with a geopotential model for the com- and Torge (1998) for the computation of the European putation of the tensor components Eq. (1). The quasigeoid. The ground-truth GPM (SYNGPM1300S) remove-restore procedure is applied, the residual consists of EIGEN-GRACE02S (n = 2 . . . 104, Reigber tensor components are computed by et al., 2005), EGM96 (n = 105 . . . 360, Lemoine et al., 1998) and GPM98C (n = 361 . . . 1300, R TˆijR = K ij (ψ, α)g R dσ (7) Wenzel, 1999). 4π The test data sets are derived from the σ 0 SYN G PM 1300 S (Fig. 2) by spherical harmonic synthesis where the individual heights of the data with the kernel function ∞  points based onM the DTM are considered. The gravity

n+1 2  anomalies g are based on the SYNGPM1300S R 2n + 1 ∂ sn K ij (ψ, α) = Pn (cos ψ) up to a spherical harmonic degree n max = 360, the ∂i∂ j n−1 rP n=2

(8) 10°

0°

20°

(i, j = x, y, z) based on Stokes’ function, cf. (Wenzel, 1982). The kernel functions depend on the spherical distance ψ and on the azimuth α, except of K zz . The Molodenskii correction terms are neglected in Eq. (7) even though the reference level of the gravity anomalies is the Earth surface. Provided that the highfrequency signal of the gravity data is reduced based 50° on the gravitational effect of a residual terrain model (RTM reduction) the neglect of the individual heights of the observation points causes an effect up to 1 . . . 3 mE in high mountainous regions (Wolf, 2007). The effect decreases with increasing noise of the gravity data due 40° to their loss of weight in the combined solution. The spectral weights sn in Eq. (8) are based on the error degree variance information of the GPM and of 10° 20° 0° the terrestrial gravity data. For the explicit derivatives m of Eq. (8) see (Wolf and Denker, 2005). The integra0 500 1000 1500 2000 tion is evaluated using the 1D-FFT method (Haagmans Fig. 1 Digital topography model of the test area et al., 1993). noise

+

synGpm

+

50°

40°

noise

3 Synthetic Data The quality of the derived tensor components using Eq. (7) is analysed by closed-loop computations in a

Δg M at Dtm heights

T[ij] at Goc ealtitude

Δg T at Dtm heights

Fig. 2 Synthetic data sets based on SYNGPM1300S

¨ K.I. Wolf and J. Muller

348 Table 1 Different noise scenarios and their ID label ID

Level (mGal)

Type

1UC 1C 5C

1 1 5

Uncorrelated Correlated Correlated

deviation of the tensor components is about 0.1 . . . 0.2 E. The radial component Vzz has the largest absolute mean value.

4 Numerical Experiments

◦

2 −4.0·ψ[ ] E g,g = g e .

(9)

The model is derived from a comparison of gravimetric data sets and was already used in different error studies for comparable test areas in Europe, e.g. (Denker, 2003). The test area is located in Central Europe (cf. Fig. 1). The data sets are computed in a 6 geographical grid, limited by 36.5◦ ≤ ϕ ≤ 58.5◦ and −7.0◦ ≤ λ ≤ 27.0◦ . The spectral signal content of the synthetic data is limited to a resolution of about 8 (n max = 1300). The synthetic gravity data represent RTM reduced gravity anomalies. Without this reduction a higher sampling of the input data would be necessary to avoid aliasing errors. The individual heights of the data points are interpolated from the DTM which consists of block mean values with a resolution of 5 . The statistics of the signal content of the tensor components and of the full and residual gravity anomalies are given in Table 2. The signal standard

In the following the attainable accuracy of the derived reference data for the evaluation of the gradiometer data is estimated using different approaches. In Sect. 4.1, a statistical prediction is run applying least-squares collocation. In Sect. 4.2 the results of the closed-loop evaluation are studied in the space domain, Sect. 4.3 deals with their analysis in the frequency domain.

4.1 Statistical Error Estimation The standard deviations of the radial reference tensor component are predicted using least-squares collocation. The absolute accuracy of all other components is better due to their smaller signal content. Least-squares collocation is well suited to study the dependency on the area size, the data noise and the resolution. There is no need to introduce observations, cf. Eq. (5). The predicted standard deviations for different noise scenarios are shown in Fig. 3. The accuracy is estimated for a Resolution [ ] 60

30

15

6

3

[mE]

appropriate noise is generated according to the coefficients’ standard deviations of the SYNGPM1300S. For the terrestrial data g T the complete synthetic model is used and different noise scenarios are investigated, as listed in Table 1. The noise type (correlated and uncorrelated) as well as the the noise level g varies. The correlated noise values are generated using the Cholesky factor of the error covariance matrix, cf. (Wolf, 2006). The noise model is based on (Weber and Wenzel, 1983) and reads

Table 2 Statistic of the signal content of the tensor components and of the gravity anomalies Functional

Mean ± Std [E, mGal]

Vxx Vxy Vxz Vyy Vyz Vzz g T g R

−1374.524±0.201 −0.031±0.085 8.225±0.168 −1372.583±0.236 0.002±0.120 2747.107±0.187 10.07±33.36 −0.08±19.12

Extent of the Area [ ]

Fig. 3 Predicted standard deviations σVLSC of the radial tensor zz component Vzz using least-squares collocation for different scenarios changing extent of the area, resolution and noise (1UC in black, 1C in light grey, 5C in grey)

Accuracy Analysis of External Reference Data

349

4.2 Closed-Loop Differences in the Space Domain The differences between the tensor components directly computed from the SYNGPM1300S and the tensor components derived from the noisy synthetic gravity anomalies (g M , g T ) are analysed in the space domain in all points of the data grid. The differences represent the remaining noise of the solution. In Fig. 4 it can be clearly seen that the noise increases close to the edges of the test area. According to the results from Sect. 4.1 the noise is analysed in the following not for the whole test area but for a smaller inner area (marked in Fig. 4) so that the computation points are not closer than 3.5◦ to the edges of the test area. The standard deviations of the mean of the remaining noise are shown in Fig. 5 for different scenarios. In the case of 1 mGal uncorrelated noise (1UC) the standard deviations are below 1 mE for all tensor

[mE]

0° 10° 20° computation point which is located in the center of an equatorial test area. As expected the accuracy increases when the extent of the area is growing and the data sampling is denser. In a scenario with uncorrelated noise the accuracy of each grid value formally gets better if the input data resolution increases. As a consequence, the estimated accuracy in the least-squares collocation ap50° proach (black bars in Fig. 3) tends close to zero when increasing the sampling density. It is not quite realistic that the correlation between nearby data points is perfectly zero. Therefore, this assumption should be avoided, especially for the estimation of the maximum attainable accuracy. 40° In the scenarios with correlated noise the estimated accuracies do not improve significantly (ca. 2 . . . 3%) when increasing the density of the grid from 6 to 3 . Equally, the improvement is small (ca. 4 . . . 5%) when mE enlarging the area size from 7◦ by 7◦ to 9◦ by 9◦ in 3 6 9 these scenarios. For the closed-loop scenarios the reso- −9 −6 −3 0 lution of the (point) data is therefore chosen with 6 and Fig. 4 Closed-loop differences in Vzz for scenario 1UC. The the computation points whose noise is analysed should analysis of the differences is based on the marked inner area to not be closer than 3.5◦ to the edges of the input data avoid edge effects area. Based on data with a noise level of 1 mGal the accuracy level of 1–2 mE can be reached. It should be mentioned that for block mean values a lower resolution (about 15 ) is sufficient, cf. (Wolf, 2007).

Fig. 5 Standard deviations based on closed-loop differences from different noise scenarios (1UC in black, 1C in light grey, 5C in grey)

components (max. 0.9 mE for Vzz ). The statistically estimated accuracies (Sect. 4.1) are more optimistic (0.6 mE for Vzz , cf. Fig. 3). The higher noise level of the closed-loop results is probably caused by the neglect of the Molodenskii correction terms in Eq. (7) and of the individual heights of the data points, respectively. Correspondingly, the largest differences are located in the alpine region, cf. Fig. 4.

¨ K.I. Wolf and J. Muller

350

In the scenario 1C the standard deviations are also below 1 mE except for the radial component which shows a larger standard deviation of 1.4 mE. This result agrees very well with the statistic estimate of 1.3 mE. Here, the terrestrial gravity anomalies get less weight in comparison with the GPM due to the correlated noise. Therefore, the neglect of the Molodenskii correction terms in Eq. (7) has less effect. Finally the results of the scenario 5C show clearly an insufficient accuracy up to 3.5 mE in Vzz in agreement with the statistic estimate of 3.6 mE.

4.3 Closed-Loop Differences in the Frequency Domain

0°

20°

50°

50°

40°

40°

0°

In order to analyse the remaining noise in the frequency domain the following steps are performed:

10°

10°

20°

Fig. 6 A number of 26 tracks from a 29-days global data set covers the test area

(a) computation of all tensor components in points of a three dimensional grid (spherical surfaces with a 6 data grid; 5 km height difference between each interpolation level), (b) interpolation in points of a simulated GOCE orbit (SCVII, 2000) using a cubic spline approach, (c) rotation of the tensor into the gradiometer reference frame (X G pointing approximately along track, Z G is vertical to it approximately within the orbital plane and Y G completes a right-handed Cartesian system), (d) derivation of closed-loop differences with errorfree tensor values (directly computed from SYN G PM 1300 S ), (e) computation of the spectral densities from the Fig. 7 Spectral densities of simulated noise in Vzz of the syntimes series based on the closed-loop differences. thetic GPM based on the global time series (light grey) and based Due to the limited regional data area only parts of the GOCE tracks can be evaluated. Within the test area a number of 26 tracks from a global 29-days time series covers each at least a time span of 225 s (Fig. 6). In Fig. 7 the spectral density (for Vzz ) of the simulated noise of the synthetic GPM based on the global time series (grey) is compared with the spectral density computed as mean curve from the 26 tracks (black). Within the MBW (dark grey box) the trackwise computed spectral density can be used as approximation of the global one. The white dashed lines in Fig. 7 mark the frequency band which is of special interest for GOCE as

on the 26 tracks (black). The √ solid lines in white mark the accuracy level of 7 and 11 mE/ Hz, respectively. The white dashed lines mark the interesting frequency band for GOCE. The grey box mark (not completely) the MBW of the GOCE gradiometer

explained in Sect. 1. The line at 0.019 Hz corresponds to a spherical harmonic degree 105. The peak formed by the spectral density based on the global times series is caused by the much higher noise of the SYN G PM 1300 S in this frequency band resulting from the EGM96, cf. Sect. 3. A similar peak can also be found in the spectral density curve when the noise simulation is based on the accuracy of the E IGEN GL 04 C, the current combination model from G RACE ,

Accuracy Analysis of External Reference Data

altimetry, terrestrial and other data (GFZ, 2006). The line at 0.036 Hz corresponds to a spherical harmonic degree of 200. Here, the signal of the tensor components approaches zero at GOCE altitude and the noise decreases, too. In Fig. 8 the spectral density of the remaining noise in Vzz based on the combined solution using the GPM and terrestrial data with a noise level of 5 mGal (5C, light grey) and with a noise level of 1 mGal (1C, black) are shown. The curves are clearly closer to zero than the spectral density based on the pure GPM solution. √ In the scenario 1C the required accuracy of 11 mE/ Hz (upper white solid line) can be reached.

Fig. 8 Spectral densities of the remaining noise in Vzz from the combined solution adding terrestrial data to the GPM information (cf. Fig. 7). Scenario 1C in black, scenario 5C in light grey

5 Conclusions The numerical results show that the components of the tensor V[ij] can be computed with an accuracy of 1– 2 mE at GOCE altitude when combining recent global geopotential models with RTM reduced terrestrial gravity data. The resolution of the terrestrial data should be about 6 and their noise level should reach 1 mGal (correlated or uncorrelated). The computation points have to keep a distance of about 3.5◦ from the edges of the regional observation area. Based on the synthetic Earth model it is shown that under these conditions the remaining noise of the computed reference values fall be√ low the accuracy level of 11 mE/ Hz required within the measurement bandwidth of the GOCE gradiometer.

351 Acknowledgments We like to thank C. C. Tscherning for providing his covariance computation subroutines. This is publication no. GEOTECH-303 of the R&D-Programme GEOTECHNOLOGIEN funded by the German Federal Ministry of Education and Research (BMBF) and the German Research Foundation (DFG) under grant 03F0421D.

References Arabelos, D. and Tscherning, C. C. (1998). Calibration of satellite gradiometer data aided by ground gravity data. J. Geod, 72:617–625. Arabelos, D., Tscherning, C. C. and Veicherts, M. (2007). External calibration of GOCE SGG data with terrestrial gravity data: a simulation study. In Proc Joint IAG/IAPSO/IABO Assembly “Dynamics Planet”, Cairns, Australia, Aug. 22– 26, 2005, Vol 130, IAG Symp, pp. 337–344. Springer, Berlin Heidelberg. Benner, P., Mehrmann, V., Sima, V., Van Huffel, S. and Varga, A. (1999). SLICOT – a subroutine library in systems and control theory. NICONET T Rep 97-3. Bouman, J. and Koop, R. (2003). Calibration of GOCE SGG data combining terrestrial gravity data and global gravity field models. In I. N. Tziavos, (Eds.), Proc IAG Int Symp “Gravity and Geoid”, Thessaloniki, Greece, Aug. 26–30, 2002, pp. 275–280. Zeta Publ. Bouman, J., Koop, R., Haagmans, R., M¨uller, J., Sneeuw, N., Tscherning, C. C. and Visser, P. N. A. M. (2005). Calibration and validation of GOCE gravity gradients. In F. Sans`o, (Eds.), A Window on the Future of Geodesy, Proc 36th IAG General Assembly, 23rd IUGG General Assembly, Sapporo, Japan, 2003, Vol 128, IAG Symp, pp. 265–270. Springer, Berlin. Denker, H. (2003). Computation of gravity gradients over Europe for calibration/validation of GOCE data. In I. N. Tziavos, (Eds.), Proc IAG Int Symp “Gravity and Geoid”, Thessaloniki, Greece, Aug. 26–30, 2002, pp. 287–292. Zeta Publ. Denker, H. and Torge, W. (1998). The European gravimetric quasigeoid EGG97 – an IAG supported continental enterprise. In R. Forsberg, M. Feissel, and R. Dietrich, (Eds.), Proc IAG Scientific Assembly “Geodesy on the Move – Gravity Geoid, Geodynamics, and Antartica”, Rio de Janeiro, Brazil, Sept. 3–9, 1997, Vol 119, IAG Symp, pp. 249–254. Springer, Berlin. Drinkwater, M. and Kern, M. (2006). Calibration and Validation Plan for L1b Data Products. Technical Report EOP-SM/1363/MD-md, ESA ESTEC, Noordwijk, The Netherlands, hhtp://www.esa.int/ esaLP/ESAJJL1VMOC LPgoce 0.html. GFZ (2006). Combined Gravity Field Model EIGEN-GL04C, http://www.gfz-potsdam.de/GRACE/results/grav/g005 eigngl04c.html. (25.10.2006). Haagmans, R., de Min, E. and van Gelderen, M. (1993). Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes’ integral. Man Geod, 18:227–241.

352 Kern, M. and Haagmans, R. (2005). Determination of gravity gradients from the terrestrial gravity data for calibration and validation of gradiometric GOCE data. In C. Jekeli, L. Bastos, and J. Fernandes, (Eds.), Proc IAG Int Symp “Gravity, Geoid and Space Missions”, Porto, Portugal, Aug. 30–Sept. 3, 2004, Vol 129, IAG Symp, pp. 95–100. Springer, Berlin Heidelberg. Knudsen, P. (1987). Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data. Bull G´eod, 61(2):145–160. Lemoine, F. G., Kenyon, S. C., Factor, J. K., Trimmer, R. G., Pavlis, N. K., Chinn, D. S., Cox, C. M., Klosko, S. M., Luthcke, S. B., Torrence, M. H., Wang, Y. M., Williamson, R. G., Pavlis, E. C., Rapp, R. H. and Olson, T. R. (1998). The development of the joint NASA GSFC and NIMA geopotential model EGM96, NASA/TP-1998-206861. Technical Report, NASA, Greenbelt, Maryland. Moritz, H. (1980). Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe. Pail, R. (2003). Local gravity field continuation for the purpose of in-orbit calibration of GOCE SGG observations. Advances in Geosciences, 1:11–18. Pail, R. (2004). GOCE Data Archiving and Processing Center (DAPC) Graz, Austrian Space Application Programme, Bridging Phase, Final Report. Technical Report, Technische Universit¨at Graz, Austria, http://www.inas. tugraz.at/forschung/DAPC/projectIb.html. Reigber, C., Schmidt, R., Flechtner, F., K¨onig, R., Meyer, U., Neumayer, K.-H., Schwintzer, P. and Zhu, S. Y. (2005). An earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. Journal of Geodynamics, 39(1):1–10. SCVII (2000). Simulation Scenarios for Current Satellite Missions. IAG Special Commission VII, http://www. geod.uni-bonn.de/apmg/lehrstuhl/simulationsszenarien/sc7/ satellitenmissionen.php (13.07.2006).

¨ K.I. Wolf and J. Muller Tscherning, C. C. (1976). Computation of the second-order derivatives of the normal potential based on the representation by a Legendre series. Man Geod, 1:71–92. Tscherning, C. C. and Rapp, R. H. (1974). Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variance Models. OSU Rep 208. Weber, G. and Wenzel, H.-G. (1983). Error covariance functions of sea gravity data and implications for geoid determination. Marine Geodesy, 75(1–4):199–226. Wenzel, H.-G. (1982). Geoid computation by least-squares spectral combination using integral kernels. In Proc General Meeting of the IAG, Tokyo, May 7–15, 1982, pp. 438–453, Berlin Heidelberg. Springer. Wenzel, H.-G. (1999). Schwerefeldmodellierung durch ultra hochaufl¨osende Kugelfunktionsmodelle. Zeitschrift f¨ur Vermessungswesen, 124(5):144–154. Wolf, K. I. (2006). Considering coloured noise of ground data in an error study for external GOCE calibration/validation. In P. Knudsen, J. Johannessen, T. Gruber, S. Stammer, and T. van Dam, (Eds.), Proc GOCINA Workshop, April, 13–15, 2005, Vol 25, Cahiers du Centre Europ´een de G´eodynamics et de S´eismologie, pp. 85–92. Luxembourg. Wolf, K. I. (2007). Kombination globaler Potentialmodelle mit terrestrischen Schweredaten f¨ur die Berechnung der zweiten Ableitungen des Gravitationspotentials in Satellitenbahnh¨ohe. Ph.D. thesis, Deutsche Geod¨atische Kommission, Reihe C, Nr. 603, www.dgk.badw.de. Wolf, K. I. and Denker, H. (2005). Upward continuation of ground data for GOCE calibration/validation purposes. In C. Jekeli, L. Bastos, and J. Fernandes, (Eds.), Proc IAG Int Symp “Gravity, Geoid and Space Missions”, Porto, Portugal, Aug. 30–Sept. 3, 2004, Vol 129, IAG Symp, pp. 60–65. Springer, Berlin Heidelberg.

Gravity Field Determination at the AIUB – The Celestial Mechanics Approach L. Prange, A. J¨aggi, G. Beutler, R. Dach and L. Mervart

Abstract We present the gravity field model AIUBCHAMP01S, which has been generated using the Celestial Mechanics approach. GPS-derived kinematic positions of low Earth orbiters (LEOs) are used as pseudo-observations to solve for the Earth’s gravity field parameters in a generalized orbit determination problem. Apart from normalized spherical harmonic (SH) coefficients, arc-specific parameters (e.g., accelerometer calibration parameters, dynamical parameters, or pseudo-stochastic parameters) are set up and normal equations are written for all daily LEO arcs. The daily normal equations are combined to weekly, monthly, and annual systems before inversion. The parametrization can be modified on the normal equation level without a new time-consuming set up of the daily normal equations. The results based on one year of CHAMP data demonstrate that the Celestial Mechanics approach is comparable in quality with other approaches.

L. Prange Astronomical Institute (AIUB), University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland, e-mail: [email protected] A. J¨aggi Astronomical Institute (AIUB), University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland G. Beutler Astronomical Institute (AIUB), University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland R. Dach Astronomical Institute (AIUB), University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland L. Mervart Institute of Advanced Geodesy, Czech Technical University, Thakurova 7, 16629 Prague Czech Republic

Keywords CHAMP · Gravity field determination · Pseudo-stochastic parameters · Accelerometer data

1 Introduction The launch of the CHAMP satellite (Reigber et al., 1998) in the year 2000 initiated a new era of gravity field recovery using LEOs. The use of the onboard GPS receiver allowed a precise orbit determination (POD) with unprecedented accuracy, spatial, and temporal coverage. This resulted in the determination of a variety of new global gravity field models using different approaches (Reigber et al., 2003; Wermuth et al., 2004; Mayer-G¨urr et al., 2005). After the launch of the two GRACE satellites in the year 2002, the focus moved to the analysis of the very precise inter-satellite K-band measurements, which allowed a more precise estimation of gravity field models up to higher degrees. The launch of the GOCE satellite with an on-board gradiometer, which is scheduled for the year 2008, shall further advance the gravity field research. While the K-band and gradiometer instruments provide “in-situ” measurements for estimating the high degree coefficients of the Earth’s gravity field, the satellite trajectories are still a major source of information for deriving the low degree coefficients. Moreover, an increasing number of satellite missions not dedicated to gravity field research will be equipped with space-borne GPS receivers, and could therefore contribute to the gravity field research as well. The COSMIC and SWARM missions are examples for such missions. It is therefore our goal to extract the maximum amount of information from GPS-derived kinematic LEO positions using

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a flexible approach, which continues the research performed by J¨aggi (2006) in the field of LEO-POD at the Astronomical Institute of the University of Bern (AIUB).

2 Method The methodology of the Celestial Mechanics approach is based on the monograph (Beutler, 2005). The basic equation is the differential equation for the geocentric motion of the satellite’s center of mass: r¨ = −G

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On the left-hand side of Eq. (1) we find the total acceleration of the satellite. The first term on the righthand side describes the gravitational acceleration of the Earth acting on the satellite (with G: gravity constant; ρ p : density of the Earth at location r p ; dV E : volume element of the Earth; r: satellite position). The second term describes the gravitational attraction of the satellite by other celestial bodies (with m j : mass of the attracting celestial body j; r j : geocentric position vector of the celestial body) treated as point masses. The third term represents the sum of the non-gravitational accelerations (air-drag, radiation pressure from the sun, and the Earth albedo). The gravitational accelerations can be written as gradients of potentials. The potential function of the Earth’s gravity field is conventionally expressed in the 3 Processing Scheme “Earth-fixed” coordinate system by a spherical harmonic expansion (Heiskanen and Moritz, 1967): Gravity field determination using the Celestial Men n max chanics approach is based on kinematic LEO positions

GM aE n  m V (r, λ, φ) = Pn (sin φ) and, optionally, on LEO accelerometer data. The kiner r n=0 m=0 matic orbits (available with a 30 s sampling) were gen(Cnm cos mλ + Snm sin mλ), (2) erated at the AIUB (J¨aggi, 2006) using the Bernese GPS Software (Dach et al., 2007) and the precise point where r, λ, and φ are the spherical coordinates of the positioning (PPP) method. Epochs with less than five satellite in the Earth-fixed coordinate system. GM is the processed GPS satellites are flagged as outliers. A high product of gravity constant and the Earth’s mass; a E level of consistency with the final products (GPS satelis the Earth’s equatorial radius; Pnm are the associated lite clock corrections, Earth rotation parameters, and Legendre functions of degree n and order m; Cnm and GPS satellite ephemerides) computed at the Center for

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Orbit Determination in Europe (CODE) is guaranteed (Hugentobler et al., 2006). The kinematic orbit determination also provides epoch-wise covariance information, which is used for weighting the kinematic data in the gravity field recovery. As kinematic positions are not influenced by information on LEO dynamics, e.g., by an a priori gravity field model, they are well suited for gravity field estimation. In a screening process (reduced-) dynamic orbits are estimated (without improving the SH coefficients) using the kinematic positions as pseudo-observations. Assuming that satellite trajectories are smooth, epochs with striking position differences between the kinematic positions and the (reduced-) dynamic orbits are marked as outliers using a multiple of the RMS error of the orbit determination as threshold. Iteratively the RMS error of the orbit fit and the threshold are lowered by increasing the number of estimated parameters.

The cleaned kinematic positions are used as pseudo-observations in the final orbit and gravity field determination process, where accelerometer data may be optionally introduced. When using accelerometer data, scale and offset parameters are estimated for each arc in addition to the initial conditions, SH coefficients, dynamical parameters and pseudo-stochastic orbit parameters. Following the processing scheme illustrated in Fig. 1, normal equations (NEQs) are computed for each daily arc using a modified version of the orbit determination and parameter estimation program of the Celestial Mechanics software (Beutler, 2005). In subsequent steps all arc-specific parameters are pre-eliminated, and the daily NEQs are combined to weekly, monthly, or annual NEQs. The combined normal equation system is solved by inversion, providing the solution for the unknown parameters as well as the full covariance matrix. For this task we use a modified version of the Bernese GPS Software. The software allows the pre-elimination or deletion of parameters on the NEQ-level without a new, time-consuming assembly of the NEQs.

4 Experiments with Real Data 4.1 Kinematic Orbit Positions

Fig. 1 Processing scheme of the AIUB CHAMP gravity field models

Because the kinematic orbit positions serve as pseudoobservations in the orbit determination and gravity field estimation process, their quality is of crucial importance. The quality of the kinematic data depends on the number and the size of the data gaps, outliers and jumps, and the noise level (depending on the receiver characteristics). The positioning quality varies due to a changing number of tracked GPS satellites or changes in the GPS satellite constellation. Figure 2 from J¨aggi (2006) shows variations of the orbit

Fig. 2 Daily RMS errors of differences between GPS-derived kinematic orbits and reduced-dynamic orbits of the CHAMP satellite in the year 2002 and early 2003

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differences between the CHAMP kinematic orbits and the reduced-dynamic orbit in the year 2002 and early 2003. Because the reduced-dynamic orbit is smooth, most of the variations may be assigned to the kinematic positions. The reasons for the major imprecisenesses of the kinematic positions are too short observation intervals between data gaps, an unfavorable viewing geometry as well as a low number of tracked satellites. The variations in the quality of the kinematic positions contribute to the quality variations of the monthly gravity field solutions shown in Fig. 3, where one of the worst monthly solutions is that of the time interval from DOY (day of the year) 71 to 100 in the year 2002. The corresponding ground coverage (see Fig. 4 (top)) shows the presence of systematic band-shaped data gaps north and south of the equator. The residuals of the reduced-dynamic orbit determination, which are shown for different parametrizations in Fig. 5 are used for the outlier screening of the kinematic positions (see Sect. 3). Different stages (with different thresholds) of the residual screening have been tested to study their influence on the gravity field estimation. Figure 6 shows that the outlier screening affects the quality of the gravity field estimation. Especially the “rough” screening, which removes the largest outliers (threshold > 100 cm for position residuals), is very important, while the rejection of smaller outliers with lower thresholds has a smaller effect on the solutions. A too rigorous screening (thresholds < 10 cm for position residuals) increases the danger of also rejecting information from the observations.

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4.2 Orbit Characteristics and Ground Coverage The orbit characteristics of the satellite has an influence on the quality of the gravity field solution. Features like

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Fig. 7 Gravity field recovery from 20 d of simulated orbits with CHAMP and GRACE orbit characteristics, respectively

repeat orbits, ground coverage, and the orbit height are of special interest. The influence of the orbit height is illustrated in Fig. 7. In a simulation study positions affected by a white noise of 1 cm RMS errors were generated from an orbit with characteristics corresponding to CHAMP respectively GRACE using the EIGEN-2 as introduced gravity field. From both data sets the gravity field was determined with an identical parametrization. We see that the solution estimated from the lower CHAMP-like trajectory is much better for the short wavelenghts of the gravity field, while both results are nearly the same for the long wavelenghts. The change of the orbit height due to orbit decay and maneuvres causes variations in the error of the monthly gravity field solutions.

For CHAMP there are time periods with a strong repeat orbit during the year 2002. In repeat mode the overall gravity field solution quality is reduced. Figure 3 shows the difference degree amplitudes of monthly gravity field solutions. The solutions corresponding to the months, which are illustrated in the ground coverage plots (see Fig. 4) are highlighted in this picture. We see that the solution for DOY 71-100 with systematic data gaps and DOY 131-160 with strong repeat orbits yield the worst results, while the best solution corresponds to DOY 191-220 with a good ground coverage and, in addition, a low RMS error of the kinematic positions. We conclude that the accuracy of monthly CHAMP gravity field solutions significantly differs due to changing orbit

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4.3 Data Weighting The quality of individual kinematic positions differs along the orbit. Therefore, a correct weighting of the pseudo-observations is important for gravity field recovery. The effect of equal, approximative, and correct weighting (using the epoch-wise covariance information) is shown in Fig. 8. In the “equal weight mode” all pseudo-observations have the same weight. In the “approximative mode”, the radial component of the position vector, which is determined worst by GPS, is downweighted by a factor of four w.r.t. the other components. In the “correct mode” the weights are taken from the epoch-wise covariance matrices of the GPS data processing. The latter strategy provides the best results. –5.5 EIGEN–GL04C –6.5 –7.0

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4.4 Sampling Rate of the Kinematic Positions

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For a test period of two months 10 s GPS clocks and kinematic CHAMP positions (J¨aggi, 2006) were used to compute a gravity field solution. The comparison with the solution using 30 s data in Fig. 9 shows that the gravity field recovery considerably profits from the higher number of pseudo-observations. The gain √ does not reach the value of 3, which would be expected for uncorrelated measurements, but especially the higher degrees can be improved by factors up to 1.6 (see Fig. 10). The solution with 5 min pulse interval length also improves for the lowest degrees. This indicates an overparametrization when using 30 s sampling. Therefore, the low degree SH coefficients are much worse than those established with 15 min pulses. When using data with a 10 s sampling, the ratio of

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the number of observations and the number of pulses improves by a factor of 3, which reduces the effect of the overparametrization. The solution with 15 min pulses gains only for the degrees above 20. The limited improvement of the low degrees could indicate that the long wavelength information is already completely contained in the 30 s data, or that there is a limitation in our method. However, the improvement justifies the gravity field recovery from kinematic positions with 10 s sampling in future.

4.5 Influence of Parametrization, Accelerometer Data and Background Models Gravity field recovery tests using different parametrizations revealed that, in order to get a solution of good quality, pseudo-stochastic parameters have to be estimated in addition to the initial conditions and the dynamical parameters. This indicates the presence of effects, which are still not sufficiently considered by the dynamical model and the background models. Figure 11 shows the difference degree amplitudes of annual CHAMP solutions with initial conditions, dynamical parameters, and pulses with different intervals being estimated. The solution with pulse intervals of 5 min provides the best results for the higher degrees, whereas the solution with 15 min pulse intervals provides the best results for the lower degrees. The superposition of both solutions on the

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4.6 Influence of the A Priori Gravity Field It is important to know, how much the estimated gravity field is biased towards the a priori field. Figure 13 shows that the estimated monthly AIUB solutions using different a priori models are almost the same despite using different a priori models. For the Celestial Mechanics approach the influence of the a priori gravity field on the estimated field is negligible, when using kinematic positions as pseudo-observations. Fig. 14 Gravity anomalies [mGal] of the AIUB-CHAMP01S gravity field model up to degree 70

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Fig. 13 Degree amplitudes of different gravity field models and two month AIUB gravity field solutions (using different a priori models), and difference degree amplitudes of monthly AIUB gravity field solutions (using different a priori models) compared to the EIGEN-GL04C model

5 AIUB-CHAMP01S Quality Assessment Using the Celestial Mechanics approach described in Sect. 2, following the processing strategy described in Sect. 3, and taking into account the experience described in Sect. 4, we generated the gravity field model AIUB-CHAMP01S. For this solution a total of 342 days of CHAMP data within the time span March 2002–2003 have been used to create kinematic pseudo-observations with a 30 s sampling. The a priori gravity field model was the EIGEN-2 (Reigber et al., 2003) and the ocean tide model the CSR3.0. Gravity field parameters were estimated up to degree and order 90 (to avoid cutoff effects) without applying any regularization. Apart from the initial conditions dynamical parameters have been

estimated as well as pulses in 5 and 15 min intervals, respectively. The final solution is obtained by the superposition of the solutions with 5 and 15 min pulse intervals on the NEQ level. The AIUB-CHAMP01S model does not make use of accelerometer data. The potential coefficients are available at the webpage http://www.aiub.unibe.ch/content/research/gnss/ gnss research/global gravity field determination/ index eng.html. The time period of the input data is the same as for other one-year CHAMP-only models, which facilitates the comparability with those models. From the computed set of spherical harmonic coefficients, geoid heights and gravity anomalies were computed for a 1◦ × 1◦ grid. Then the differences to the superior reference gravity field model EIGEN-GL04C (F¨orste et al., 2006) have been computed (including a latidute dependent weighting) for different spectral ranges of the SH series (see Table 1). The same has been done for the gravity field models EGM96 (Lemoine et al., 1998) and ITG-CHAMP01S (Mayer-G¨urr et al., 2005). The EGM96 was included, because it is one of the best pre-CHAMP models, the ITG-CHAMP01S, because it is from the generation point of view most similar to our model: It has a comparable approach, uses the same CHAMP data, and is not affected by regularization either. Furthermore, it is probably one of the best CHAMP-only solutions generated so far and therefore was our benchmark. The results in Table 1 reveal that up to degree 50 the two CHAMP-only solutions are comparable in quality, and both of them are better than the pre-CHAMP model. If also the higher degree harmonics (up to degree 70) are taken into account, the AIUB solution gives the best results. Figure 15 shows

Gravity Field Determination at the AIUB

361

Table 1 Comparison of selected gravity field models with EIGEN-GL04C on a latitude-weighted 1◦ × 1◦ grid SH degree and order Compared models

Type of comparison

0–30

0–50

0–70

EGM96-EIGEN-GL04C

Undul. [cm]: RMS Max. Min. Anom. [mGal]: RMS Undul. [cm]: RMS Max. Min. Anom. [mGal]: RMS Undul. [cm]: RMS Max. Min. Anom. [mGal]: RMS

8.2 111.5 −93.3 0.29 1.3 6.7 −6.2 0.04 1.4 7.7 −7.6 0.05

16.7 375.4 −248.5 0.97 5.5 31.2 −31.1 0.37 5.2 30.5 −32.9 0.35

22.4 631.3 −417.2 1.69 26.4 153.9 −161.2 2.57 22.2 137.6 −127.3 2.15

ITG-CHAMP01S-EIGEN-GL04C

AIUB-CHAMP01S-EIGEN-GL04C

We demonstrated that a flexible and robust pseudostochastic parametrization is able to compensate for –6.0 EIGEN–GL04C deficiencies in the background models and facilitates –6.5 good gravity field solutions even in the absence of –7.0 accelerometer data. Thus the Celestial Mechanics –7.5 approach is not only a good complement for sophisti–8.0 cated sensors like K-band or gradiometers, but would –8.5 also be well suited for gravity field recovery from nondedicated satellite missions, which are equipped with –9.0 TUM–2S a good GPS receiver (e.g., SWARM, COSMIC). The ITG–CHAMP01S –9.5 EIGEN–CHAMP03S solutions are virtually not biased towards the a priori AIUB–CHAMP01S –10.0 0 10 20 30 40 50 60 70 80 90 static gravity field. In contrast to J¨aggi et al. (2008), degree of spherical harmonics we concentrate on single satellite studies in this paper. Fig. 15 Difference degree amplitudes of AIUB-CHAMP01S The comparison with other CHAMP-only gravity and other CHAMP gravity field models field solutions shows that our model is comparable in quality with the best alternative models. Experiments show, however, that there is still potential for further the logarithm of difference degree amplitudes of a vaimprovement, e.g., the use of 10 s kinematic positions, riety of CHAMP-only models. Again the good perforwhich will allow us to get the maximum information mance of the AIUB model for the higher degrees is out of the publicly available GPS measurements. visible. On the other hand, there are minor problems related to the lower degree harmonics, especially in the range between degree 15 and 25. Note that TUM-2S Acknowledgments The authors are grateful to the Center for Orbit Determination in Europe (CODE) for using GPS orbit (Wermuth et al., 2004) is a two year solution, whereas solutions and clock corrections. CODE is a joint venture of the EIGEN-CHAMP03S uses 2.5 years of CHAMP the Astronomical Institute of the University of Bern (AIUB, Switzerland), the Federal Office of Topography (swisstopo, data (Reigber et al., 2004). difference degree amplitude (log)

–5.5

Switzerland), and the Federal Agency of Cartography and Geodesy (BKG, Germany).

6 Summary We used the Celestial Mechanics approach for gravity References field estimation based on GPS-derived kinematic positions, to produce the static gravity field model Beutler, G. (2005). Methods of Celestial Mechanics, SpringerAIUB-CHAMP01S from one year of CHAMP data. Verlag.

362 Beutler, G., E. Brockmann, W. Gurtner, U. Hugentobler, L. Mervart, M. Rothacher (1994). Extended orbit modeling techniques at the CODE processing center of the international GPS service for geodynamics (IGS): theory and initial results. In: Manuscripta Geodaetica, 19, 367–386 Dach, R., U. Hugentobler, P. Fridez, M. Meindl (Eds.) (2007). Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern, Switzerland, 2007. Ditmar, P., R. Klees, X. Liu (2007). Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise. Journal of Geodesy, 81(81–96) F¨orste, C., F. Flechtner, R. Schmidt, R. K¨onig, U. Meyer, R. Stubenvoll, M. Rothacher, F. Barthelmes, K.H. Neumayer, R. Biancale, S. Bruinsma, J.M. Lemoine (2006). A mean global gravity field model from the combination of satellite mission and altimetry/gravimetry surface gravity data – EIGENGL04C. In: Geophysical Research Abstracts, Vol. 8, European Geosciences Union. Heiskanen, W.A., H. Moritz (1967). Physical Geodesy. Freeman. Hugentobler, U., S. Schaer, G. Weber et al. (2006). CODE IGS Analysis Center Technical Report 2003/2004. In: International GPS Service 2003–2004 Technical Report, IGS Central Bureau, JPL Pasadena, USA. J¨aggi, A. (2006). Pseudo-Stochastic Orbit Modeling of Low Earth Satellites Using the Global Positioning System. Ph.D. Thesis, Astronomical Institute, University of Bern, Switzerland. J¨aggi, A., G. Beutler, H. Bock, U. Hugentobler (2006). Kinematic and highly reduced-dynamic LEO orbit determination for gravity field estimation. In: Dynamic Planet – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools, edited by C. Rizos et al., pp. 354–361, Springer. J¨aggi, A., G. Beutler, L. Prange, R. Dach (2008). Assessment of GPS observables for Gravity Field Recovery from GRACE. In: This volume.

L. Prange et al. Lemoine F.G., F.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp, T.R. Olsen (1998). The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. In: NASA Technical Paper, NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, USA. Mayer-G¨urr T., K.H. Ilk, A. Eicker, M. Feuchtinger (2005). ITGCHAMP01: a CHAMP gravity field model from short kinematic arcs over a one-year observation period. Journal of Geodesy, 78, 462–480. Reigber C., H. L¨uhr, P. Schwintzer (1998). Status of the CHAMP Mission. In: Towards an Integrated Global Geodetic Observing System (IGGOS), edited by R. Rummel, H. Drewes, W. Bosch, H. Hornik, Springer, Berlin, 63–65, 1998. Reigber C., P. Schwintzer, K.H. Neumayer, F. Barthelmes, R. K¨onig, C. F¨orste, G. Balmino, R. Biancale, J.M. Lemoine, S. Loyer, S. Bruinsma, F. Perosanz, T. Fayard (2003). The CHAMP-only Earth Gravity Field Model EIGEN-2. In: Advances in Space Research, 31(8), 1883–1888. Reigber C., H. Jochmann, J. W¨unsch, S. Petrovic, P. Schwintzer, F. Barthelmes, K.H. Neumayer, R. K¨onig, C. F¨orste, G. Balmino, R. Biancale, J.M. Lemoine, S. Loyer, F. Perosanz (2004). Earth Gravity Field and Seasonal Variability from CHAMP. In: Earth Observation from CHAMP – Results from Three Years in Orbit, edited by C. Reigber H. L¨uhr, P. Schwintzer, J. Wickert, Springer, Berlin, 25–30, 2004. ˇ Wermuth M., D. Svehla, L. F¨old´vary, C. Gerlach, T. Gruber, B. Frommknecht, T. Peters, M. Rothacher, R. Rummel, P. Steigenberger (2004). A gravity field model from two years of CHAMP kinematic orbits using the energy balance approach. In: Geophysical Research Abstracts, Vol. 6, European Geosciences Union.

Robust Trend Estimation from GOCE SGG Satellite Track Cross-Over Differences ¨ F. Jarecki and J. Muller

Abstract Due to their outstanding accuracy and resolution, the innovative satellite gravity gradient (SGG) measurements from the European mission GOCE need dedicated calibration and validation procedures. It has been shown, that comparing the measurements in satellite track cross-overs offer an opportunity to get relative quality information, when applying a straight-forward reduction method. From SGG cross-over differences data sets, calibration parameters can be estimated. Here, the advantage of robust estimation methods over the standard least-squares approach for the analysis of short SGG cross-over data sets is shown. Simulated SGG data with different artificial trends are processed in different subsets. The least-squares approach is feasible to produce trend estimates from the cross-over differences of several days of measurements. Long-term trends can be estimated from cross-overs of a single revolution when applying robust estimation. Anticipating these estimation techniques, crossover validation offers a fast approach to assess independently the quality of space gradiometry with focus on certain time intervals. Keywords Gravity satellite mission · GOCE · Gravity gradiometry · Calibration · Validation · Cross-overs · Robust parameter estimation

1 Introduction The European satellite gravity mission GOCE will realise satellite gravity gradiometry (SGG) by means of an innovative three-axes electrostatic gravity gradiometer in the spring of 2008. This newly introduced measurement technique requires revised calibration and validation approaches. Simulated GOCE gradiometric data sets have been analysed succesfully in satellite track cross-overs, comparing measurements in the same geographical position by applying suitable interpolation and reduction procedures (cf. Jarecki et al. (2006)). Stable parameters of standard error models, like a long-term trend, can be detected from long test data sets (several weeks), as shown in Jarecki and M¨uller (2007). To apply the cross-over method operationally, e.g. in order to monitor the gradiometer’s performance, such parameters should as well be determined from shorter data sets, like single orbit revolutions. Due to single gross errors in the measurements and insufficiencies in the interpolation and reduction approach, standard least-squares estimation is not able to provide optimal results for these shorter data sets. Robust analysis methods, however, turn out to work well in this context.

2 GOCE Satellite Track Cross-Overs F. Jarecki Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany J. M¨uller Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany

Satellite orbit cross-overs (“XOs”) might be interpreted as “the same position in space”. Then, the basic principle of cross-over calibration, as known from Satellite Altimetry and renewed for GOCE, e.g., by Albertella et al. (2000), postulates, that the measurements Vij∗ of

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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364

red Vij,XO = Vij,1GPM − Vij,2GPM ,

(5)

where σred Vij,XO < 2 mE, for XOs with h < 15 km, when using EGM96. After applying this reduction, XO differences Vij,XO = Vij,1 − Vij,2 − red Vij,XO

Fig. 1 Reduction concept for SGG satellite track cross-over differences: altitude and attitude reductions

(6)

for all tensor components Vij can be identified as measurement errors in space domain (cf. Jarecki and M¨uller, 2007). An example data set of reduced XO differences is shown in Fig. 2. In these differences gross errors can be found with thresholds or statistical methods, or calibration parameters can be estimated. 0.2

GOCE’s gravity gradiometer should show no differences in orbit XOs: ∗ ∗ Vij,1 = Vij,2 ,

(1)

where the indices i, j represent the three axes of gradiometer x, y, z, and the indices 1, 2 mean the ascending resp. descending track, forming the actual XO as shown in Fig. 1. Differing from this simple model, XOs of satellite 0 tracks do not mean the same position in space but only E the same (geographical) position ϕ, λ with respect to a well defined projection surface, in this case the (Earth) Fig. 2 Reduced absolute XO differences for simulated tensor component Vzz with simulated errors, 1 day track compared in sphere: the XOs of 30 days

ϕVij,1 = ϕVij,2 ,

λVij,1 = λVij,2 .

(2)

Due to the orbit eccentricity, the satellite altitude h dif3 Linear Trend Estimation fers in those positions: h Vij,1 = h Vij,2 .

(3)

Furthermore, the orientation of the gradiometer differs in the ascending and descending track, causing further differences: Vij,1 = Vij,2 , etc.

(4)

Adopting a standard error model for the gradiometer measurements (according to, e.g., Koop et al., 2002), and neglecting periodic errors, measured gradients VijG might be modelled as VijG (t) = VijR (ϕ(t), λ(t)) + Vij∗ + Vij t.

(7)

Beside a constant bias Vij∗ corrupting the true gradiAll these XO differences, also depicted in Fig. 1 in ent VijR , a linear trend Vij is the basic error feature. It principle, can be reduced with sufficient accuracy using projects linearly into the cross-over differences Eq. (6), modelled differences from a recent geopotential model while the true gradient VijR in the reoccupied position (GPM, e.g. EGM96), as shown in Jarecki et al. (2006): ϕ(t1 ) = ϕ(t2 ), λ(t1 ) = λ(t2 ) and the constant bias Vij∗

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365

disappear due to their relative nature (cf. Jarecki and M¨uller, 2007): VijG = VijG (t1 ) − VijG (t2 ) = Vij · (t1 − t2 ).

(8)

Vij can easily be estimated from XOs along a certain time span using standard least-squares parameter estimation (see, e.g., Koch, 1999, Chap. 3.2), where the differences are identified as observations l of the linear model l + v = Ax.

(9)

x contains the parameters to be estimated, in this case only the trend factor Vij , while the time differences define the design matrix A. The trend parameter is consequently obtained from the least-squares standard solution as estimation xˆ with the estimated standard deviation σˆ . To find the optimal xˆ , the weighted quadratic norm v T Pv of the residuals v, which make the noisy original observations l fit into the linear model, is minimised. The weighting matrix P might be constructed from the difference accuracies and correlations; here the unity matrix is used (P = I ) because of simplicity. The objective function of the minimisation therefore becomes

Fig. 4 Absolute XO differences of the data set from Fig. 3, plotted w.r.t. their position. The geographical distribution of similar differences is rather unstructured

the XO reduction procedure explained in Sect. 2. A geographic plot of the reduced XO differences looks rather unstructured (Fig. 4). In contrast, the linear trend is clearly visible in the data plotted with respect to the time differences (Fig. 5), if the whole data set is considered. To confirm this visual quality assessment, the leastsquares approach described above has been applied to different portions of the data set. As seen from Table 1, the artificial trend of 6.7974 mE d is obtained T

= v v, → min . (10) very well when using XO differences from long time spans, as the redundancy is increasing. In view of the A simulated test data set, containing 30 days of 1 Hz- GOCE calibration, the estimated trend might be subsampled gradients Vzz superposed by an artificial trend tracted from the measurement data to obtain an error of 6.7974 mE d (as shown in Fig. 3), is processed, using

E

2900

2750

2600

0

d

30

Fig. 3 Simulated Vzz time series (30 days) with (grey) and without (black) super-imposed artificial trend (6.7974 mE d ). For the plot, the trend is inflated by 1000

Fig. 5 XO differences of the data set from Fig. 3, plotted w.r.t. the time differences in the XOs. The trend is clearly seen in the time plot

¨ F. Jarecki and J. Muller

366 Table 1 Estimated artificial linear trend (6.7974 mE d ) from XO differences of different combinations of track sections

Track section investigated

XOs from (# XOs)

Estimated trend σˆ

1 arc

30 d (ca. 500) 10 d (ca. 170) 5d (ca. 80) 1d (16) 30 d (ca. 1000) 10 d (ca. 330) 1d (ca. 30)

6.79675 0.03347 6.80794 0.12976 6.79388 0.12742 6.79586 2.15051 6.79785 0.0044 6.7969 0.0847 6.7896 2.2543

1 arc 1 arc 1 arc 1 revolution 1 revolution 1 revolution

 mE  d

Relative error (%) 0.01 0.15 0.05 0.02 < 0.01 < 0.01 0.12

estimated reliably for short measurement intervals, like single revolutions or arcs. The redundancy for a small set of calibration parameters is good: Here, one trend parameter is estimated from 16 XO differences, when investigating one arc with XOs from one day. Nevertheless, the estimation is severely suffering from gross errors in such configurations. Due to the sufficient redundancy, single observations with gross errors might be eliminated or weighted down. Consequently, robust estimation with implicit elimination of gross errors would be an alternative approach for those configurations. Fig. 6 Absolute XO differences of the former erroneous data (Fig. 4), where the estimated trend is removed now. Besides effects of the interpolation algorithm, the differences vanish

4 Robust Trend Estimation

Trend estimation from short time intervals of XO differences suffers from outliers. Therefore, trend estimafree data set (depicted in Fig. 6). Table 1 shows the tion methods, which should be more “robust” against trend parameters, estimated from XO differences from outliers, are investigated in the following. Modelling the data and their assumed systematic the combination of the given time intervals, as well as their relative errors: While the method works for the es- noise characteristics as in Sect. 3, Eq. (7), the sentimation of medium- and long-term trend parameters, sitivity of the parameter estimation against outliers estimated trends based on short parts of the time series is reduced by changing function (Eq. (10)), which is tend to be erroneous and are not statistically signifi- minimised to obtain optimal parameters: Instead of the cant. The latter estimates are especially suffering from quadratic function used in the least-squares approach, interpolation errors in the reduction step (6). When es- “less rapidly increasing functions” (Huber, 1981, timating the trend for a single revolution or arc from p. 162) can be used. Formally, the estimates are the XOs with 1 day of data, the trend estimates might obtained by minimising become erroneous, like stated for the last combination. n  At least, the (formal) accuracy σˆ of the trend estimates Q(v) = ρ(vi ), (11) degrades significantly when using shorter sets of referi=1 ence data. If the gradiometer performance shall be monitored, where ρ is a symmetric function of each observation’s trends (and other calibration parameters) should be residual vi . n is the number of observations. Q replaces

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367

from Eq. (10) in the least-squares approach. To solve the minimising problem, the derivation ψ of the objective function ρ is used: ∂ Q(v) ∂ρ(vi ) ψ(v) = = . ∂vi ∂vi

(12)

Therefore, different robust estimators might be distinguished by their ψ. Here, a “Fair” estimator with 

ψ = arctan(v )

(13)

v

is used, where are rescaled, standardised residuals v. The standardisation step is crucial to ensure scale invariance for the estimates (Huber, 1981). In this step, a tuning constant k = 1.4 is applied, this means, about 5% outliers are assumed to be in the data. According to, e.g., Koch (1999), Chap. 3.8, the robustness of the estimator can be recognised by the characteristics of ψ. If ψ is bounded, the estimation is robust. In the iterative estimation process, the residuals v are weighted by the derivation of ψ, which is consequently called weighting function w. In the case utilised here, 1 w= . 1 + abs(v  )

(14)

Table 2 Results and statistics of the least-squares and robust trend estimates for one revolution from different time spans of XO differences XOs with tracks from

Estimated trend (least-squares) mE d

σˆ trend (1-s) mE d

Estimated trend (robust) mE d

30 days

6.79785

0.0044

6.79750

days 1–10 days 11–20 days 21–30

6.79925 6.80228 6.79789

0.00339 0.04872 0.02970

6.79757 6.79747 6.79748

Mean Std

6.79981 0.00225

days 1–5 days 6–10 days 11–15 days 16–20 days 21–25 days 26–30

6.80295 6.79879 6.79728 6.79688 6.79769 6.79179

Mean Std

6.79756 0.00359

6.79728 0.00050

Single days Mean Std

6.78958 0.02210

6.79478 0.00332

6.79751 0.00006 0.10514 0.12007 0.07215 0.08529 0.05596 0.13191

6.79795 6.79731 6.79715 6.79739 6.79748 6.79642

Table 3 Statistics of trends for all 475 ascending arcs from leastsquares and from robust estimation on the basis of XO differences from different surrounding time spans XOs with tracks from

Mean trend mE d

Std

mE d

Min trend

Max trend

mE

mE

E

Using this weighting function, measurements with d d large residuals are weighted down iteratively and 30 days 0.0014 6.70326 6.94074 are finally practically eliminated. Applying, e.g., 16 Least squares 6.79788 Robust 6.79744 0.0001 6.79667 6.79804 cross-over differences when comparing an arc with 10 days XOs from one day, one or two outliers (approx. fitting Least squares 6.79944 0.0330 6.63393 7.13680 the level of 5% mentioned above) do no harm, as seen Robust 6.79745 0.0001 6.79706 6.79779 from the results of the test computations: Tables 2 1 day 0.0168 6.77417 6.91497 and 3 show the benefits of the robust method for short Least squares 6.79916 6.79746 0.0001 6.79716 6.79773 term trend estimation. In Table 2, the estimated trend Robust parameters from the XO differences of one revolution of the simulated GOCE orbit with different parts of the orbit are shown. The estimation was carried out using the least-squares technique as well as with the robust “Fair” estimator. The different parts of the 2780 time series are highlighted in Fig. 7, while the table contains the mean trend and its standard deviation 2745 (std) for each set of combinations. For the longer data sets, each single estimate is shown with its formal 2710 accuracy σˆ as well. Both, the formal accuracy of the 0 30 d least-squares estimates and the consistency of each set of comparable estimates significantly decreases, Fig. 7 30-days time series of tensor component Vzz , with 5 days when the time span (and number) of XOs used as intervals indicated, as well as revolution No. 2 and, as example, reference decreases. Generally spoken, the robust the ascending arc No. 233

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trend estimates fit about one magnitude better with the real trend than the least squares figures. Furthermore, the single robust estimates spread much less. Consequently, their set scatter (std) comes out roughly one magnitude better for each data set. The time difference between the measurements under investigation and the crossing arcs does not play an important role: Comparing the second arc from the first day of the data set with days 20 to 25 gives the best result, whereas – astonishingly – the comparison with the closest data is worst. Table 3 presents the statistics of the trend parameters estimated for each ascending orbit track arc (475 ascending track arcs in the 30 days data series) from the combination with different time spans of crossing descending arcs. These descending arcs are taken from the temporal proximity of the arc under investigation: The time span given, e.g. ten days, means descending arcs from the directly preceding and following five days with respect to the ascending arc under investigation. As an example for monitoring the behaviour of the instrument, Fig. 8 shows the actual parameter estimation for the last data set in Table 3: Here, the trend is obtained from the XOs of each ascending arc with the descending arcs from the closest 24 h of data. When trying to use least-squares estimation for single day data sets, the mean trend comes out much to large. Single estimates even give relative errors of 1.7% (and in the case of ten days of reference data, even up to 5%, due to a larger number of outliers actually being used in the estimation). In contrast, the robust trend estimates

are precise in every single case, and maximum relative errors of the smooth data sets drop to 0.005%.

5 Conclusions Reduced cross-over differences of SGG data are very well suited for the relative quality assessment of the original measurements. Calibration parameters, such as linear trends and periodic perturbations, can be estimated from XOs with several weeks of reference data with standard least-squares estimation. Robust estimation extends the use of this method to the estimation of linear trends from shorter sets of reference data. Thus, it provides an excellent method for GOCE gradiometer monitoring, especially if long-term trends are removed in a previous step. Robust estimation might significantly improve the estimation of periodic calibration parameters from short-term XO differences, which is suffering from outliers even more than linear trend estimation. This application of the method is currently under investigation. Acknowledgments The German Ministry of Education and Research (BMBF) supports the GOCE-GRAND project, in whose context this study was carried out, within the GEOTECHNOLOGIEN geoscientific programme under grant 03F0421D. This is publication no. GEOTECH-305 of that programme.

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E

6.9

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Fig. 8 Trends for each ascending arc, from least-squares (grey) and from robust estimation (black) on the basis of XO differences from one day of temporally nearby data

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Reference Frame Consistency in CHAMP and GRACE Earth Gravity Models Christopher Kotsakis

reference frame that ensues from implementing an optimal inversion procedure to various data sets which are used for the model’s development. In the case of processing satellite tracking data, for example, such EGM-related global reference frames can be realized through an appropriate set of constraints that is applied to a number of satellite tracking stations in order to remove the rank deficiency of the input data that lead to the estimation of the low-degree EGM coefficients; see Pavlis (1998). This occurs if in the processing of the satellite tracking data one solves simultaneously for the tracking station positions (and possibly velocities) along with the geopotential harmonic coefficients. Alternatively, one may decide to adopt some existing ITRF-type frame to fix a-priori the EGM’s reference system, and then process the tracking data (and other types of geodetic observables coming from dedicated satellite gravity missions such as CHAMP or GRACE) exclusively for geopotential recovery. In any case, if we want to maintain a coherent framework for the comparison of current and future EGMs, and to evaluate their prediction accuracy for gravity field functionals, it is important to investigate the consistency of their inherent reference frames. Compared to the standard direct approach followed Keywords GRACE · CHAMP · Reference frame · by the IERS for the realization of the International Spherical harmonics · Helmert transformation Terrestrial Reference Frame (ITRF) in terms of recurring estimates for the 3D geocentric Cartesian 1 Introduction coordinates at a number of control stations (Altamimi et al. 2002), an EGM solution carries through its All gravity field functionals obtained from a global harmonic coefficients (including the conventionally Earth Gravity Model (EGM) depend on the terrestrial fixed or estimated zero- and first-degree coefficients, and its associated scaling factors GM and a) all the required information to constrain the origin, orientaChristopher Kotsakis tion and scale of a spatial reference system. Different Department of Geodesy and Surveying School of Engineering Aristotle University of Thessaloniki Univ. Box 440, EGMs provide alternative ‘indirect realizations’ of a Abstract After the launch of the CHAMP and GRACE satellite missions, an increasing number of spherical harmonic models have become available for the long-, medium- and short-wavelength mapping of the Earth’s gravitational field. In view of the need for a coherent comparison between such Earth Gravity Models (EGMs) and a detailed evaluation of their accuracy for various gravity field functionals, it is important to investigate the consistency of their inherent reference frames, especially when their use is intended for high precision studies. Following the methodology described in an earlier paper by Kleusberg (Manuse Geod, 5: 241–256) the Helmert transformation parameters among the inherent reference frames for several CHAMP and GRACE models are estimated in this paper. In particular, the differences between the corresponding spherical harmonic coefficients for a given pair of EGMs are parameterized through a 3D similarity transformation model, whose weighted least-squares adjustment yields valuable information for the origin stability, the orientation consistency and the spatial scale variation between their underlying reference frames.

Thessaloniki 54124, Greece, e-mail: [email protected] M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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3D Cartesian coordinate system, whose transformation parameters are estimable from the differences of their corresponding spherical harmonic coefficients (Kleusberg 1980). The consistency of the global reference frames inherent in different EGMs is a critical issue that affects several tasks, including the determination of combined estimates for gravity field signals by merging information from different geopotential models, the comparison and accuracy evaluation of different EGMs, and the calibration of their error models using terrestrial and/or satellite data. Moreover, the origin, orientation and scale stability among the EGMs’ reference frames is crucial for estimating and properly monitoring Earth change parameters (e.g. mean sea level changes, Earth rotation variations, mass re-distribution, etc.) through global gravity field mapping, as well as for integrating geometric and ‘gravimetric’ reference systems and data sets that are linked to them. There are several investigations in the geodetic literature which have focused on reference frame consistency aspects in global geopotential models (e.g. Anderle 1974, Rapp and Rummel 1976, Grappo 1980, Lachapelle and Kouba 1981, Kirby and Featherstone 1997). Despite the wide-ranging interpretations of their results, most of these studies have relied on a common methodology, notably the comparison of geometrically derived (i.e. through satellite positioning techniques) and EGM-based geoid undulations over a global, continental or regional network of leveling benchmarks. The main drawback of such evaluation techniques is that the reference frame transformation parameters (usually three translation components and one scale change factor are only considered), which are obtained from the comparison of geometric and EGM-based geoid heights, are inevitably distorted due to the high correlation caused by the non-uniform distribution and limited spatial coverage of the control points. Even in cases with a truly global and uniform distribution of data points (Schaab and Groten 1979), the conclusions drawn from such an evaluation scheme are likely to be obscured by the coupling of long-wavelength biases from the EGM-geoid heights into the estimates of the reference frame transformation parameters; see Weigel (1993). Furthermore, the geoid height is a totally insensitive variable to the rotation of the underlying reference system about the symmetry axis of the adopted refer-

C. Kotsakis

ence ellipsoid. Hence, it is impossible to estimate this particular datum parameter solely from the differences N GPS − N EGM . The latter are able to recover only variations in the direction of the mean Earth rotation axis that is implied in each geoid type, and not orientation differences in the zero-meridian planes of their associated reference frames. The objective of the present paper is to implement an alternative approach for testing the consistency of the reference frames inherent in different EGMs. The formulation given in Kleusberg (1980) for the Helmert spatial transformation of a spherical harmonic series is employed to estimate the origin, orientation and scale variations among the global reference frames implied in recent CHAMP and GRACE geopotential models. The benefit of such an approach is that it does not involve the intermediate computation and comparison of other gravity field functionals (e.g. geoid heights), but it relies entirely on the original harmonic coefficients of the Earth’s gravitational potential that are provided by the specific EGMs under comparison.

2 Methodology Our study is based on the mathematical model given by Kleusberg (1980) for the linear approximation of the Helmert (or 3D Euclidean similarity) transformation for the Earth’s gravitational potential harmonic coefficients, under translations, rotations and scale variation in the underlying Cartesian coordinate system. Although the linearized transformation formulae derived in that study are theoretically less rigorous than other versions that appeared in previous references (e.g. Goldstein 1984), they are fairly precise for practical use in global geodetic studies where small (differential) reference frame perturbations are typically involved (Pavlis 1998). Due to the limited extent of the paper, only a brief overview of the Helmert transformation formulae for spherical harmonic coefficients will be given, without presenting any additional mathematical details or derivations; for a complete description, see the original paper by Kleusberg (1980). Let us consider the usual expansion of the Earth’s gravitational potential in terms of a spherical harmonic series (Heiskanen and Moritz 1967)

Reference Frame Consistency in CHAMP

373

⎧ ⎫ C nm cos mλ⎬ ⎨

 n GM a + V (r, λ, ϕ) = P nm (sin ϕ) ⎭ r n,m r ⎩ S nm sin mλ (1) The set {C nm , S nm } contains unitless, fullynormalized spherical harmonic coefficients that are obtained from a global EGM up to a maximum degree of expansion n max . The quantities GM and a correspond to the geocentric gravitational constant and mean equatorial radius, which are the basic scaling factors that are conventionally associated with an EGM solution, while P nm (·) represent the fully-normalized associated Legendre functions of degree n and order m. In most EGMs the zero-degree normalized coefficient C 0,0 is usually treated as an errorless quantity, with its value fixed a-priori to 1. Based on Eq. (1), such a constraint gives an initial noise-free approximation for the Earth’s gravitational potential V (r, λ, ϕ) ≈

GM GM C 0,0 = r r

where xcm , ycm , z cm are the Cartesian coordinates of the geocenter with respect to the reference frame associated with the series expansion in Eq. (1). The first-degree coefficients in most EGMs are conventionally fixed to zero, thus enforcing a geocentricity constraint for the global reference system that should be used for the computation of the Earth’s gravitational potential (and its functionals) from a spherical harmonic expansion. In several recent CHAMP and GRACE models non-zero estimates and formal error variances are provided for their first-degree coefficients, which are handled as additional unknown parameters within the EGM development process. The resulting Cartesian coordinates for the Earth’s geocenter and their accuracy level, with respect to the inherent reference frames for these geopotential models, are given in Table 1. In general, the expansion of the Earth’s gravitational potential V (·) in a different reference frame than the one implied in Eq. (1) takes the form

(2) V (r  , λ , ϕ  ) =

which corresponds to a point-mass (or homogeneous sphere) Earth model, with the origin of the EGM’s reference frame located at the Earth’s center of mass (geocenter). Note that in several geopotential models (e.g. GRIM5, GFZ96, TUM1S, TUM2S) the zero-degree coefficient is not conventionally fixed to 1, and its actual value is associated with a formal statistical error that corresponds to the uncertainty of the geocentric gravitational ‘constant’ GM which is estimated anew within the EGM development process. In such cases, the following formula relates the accuracy of the zero-degree coefficient with the uncertainty of the associated value for the geocentric gravitational constant σC 0,0 =

σGM GM

GM  a n r  n,m r  ⎧  ⎫ ⎪ ⎨C nm cos mλ ⎪ ⎬ × + P nm (sin ϕ  ) (5) ⎪ ⎩ S  sin mλ ⎪ ⎭ nm

where r  , λ , ϕ  are the coordinates of the evaluation point with respect to a new Cartesian terrestrial coordinate system. Note that the scaling factors GM and a are assumed to retain their conventional values in both reference systems. Let us assume that the two reference frames differ according to the standard linearized Helmert transformation model (e.g. Soler 1998)

(3)

The first-degree normalized coefficients C 1,0 , C 1,1 , S 1,1 are physically related to the Cartesian shifts of Table 1 Geocenter’s Cartesian coordinates (and their formal accuracy level) with respect to the inherent reference frames in rethe origin of the EGM’s reference frame (in which the cent CHAMP/GRACE models (in mm) geodetic coordinates r, λ, ϕ should refer to) with re- Model xcm ycm z cm spect to the geocenter. In fact, we have the well-known EIGEN-CG03C −5.5 ± 4.8 −3.4 ± 4.8 −1.5 ± 4.4 relationships (e.g. Kirby and Featherstone 1997) EIGEN-CG01C −3.8 ± 6.1 1.2 ± 6.1 −1.2 ± 5.5 C 1,0

z cm xcm ycm = √ , C 1,1 = √ , S 1,1 = √ a 3 a 3 a 3

(4)

EIGEN-CHAMP03S TUM2S TUM1S

−2.7 ± 3.3 0.2 ± 0.2 −1.1 ± 0.2

6.1 ± 3.3 0.7 ± 0.2 −2.1 ± 0.2

−9.2 ± 3.4 0.2 ± 0.2 −15.1 ± 0.2

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C. Kotsakis

⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ tx x x − x δs εz −ε y ⎣ y  − y ⎦ = ⎣t y ⎦ + ⎣−εz δs εx ⎦ ⎣ y ⎦ z − z tz ε y −εx δs z ⎡

Table 2 The conventional values for the geocentric gravitational

(6) constant (GM) and the mean equatorial radius (a) associated with the EGMs that are used for the numerical tests in this paper Model

GM (in m3 s−2 )

a (in m)

n max

EIGEN-CG03C EIGEN-CG01C EIGEN-GL04C EIGEN-CHAMP03S EIGEN-GRACE02S EIGEN-GRACE01S TUM2S TUM1S EIGEN2 EIGEN1S

398600.4415 × 109 398600.4415 × 109 398600.4415 × 109 398600.4415 × 109 398600.4415 × 109 398600.4415 × 109 398600.4418 × 109 398600.4360 × 109 398600.4415 × 109 398600.4415 × 109

6378136.46 6378136.46 6378136.46 6378136.46 6378136.46 6378136.46 6378137.00 6378137.00 6378136.46 6378136.46

360 360 360 140 150 140 60 60 140 119

where tx , t y , tz are the translation parameters between the two frames, εx , ε y , εz are the rotation angles about the axes of the first frame (anticlockwise rotations assumed positive), and δs corresponds to the unitless factor of their differential spatial scale change. Based on the Cartesian coordinate transformation in Eq. (6) and the fact that the gravitational potential V (·) is independent of the reference coordinate system in which we choose to perform its evaluation through a spherical harmonic series, a set of linearized transUsing as reference the GM and a values associated formation formulae can be derived between the coef  ficients {C nm , S nm } and {C nm , S nm }. These formulae with the first model {C nm , S nm }, the required re-scaling were originally published by Kleusberg (1980), and for the coefficients of the second model has the form they can be expressed in terms of the general equations    

   GM  a  n C nm (orig) C nm = (9)    GM a S nm S nm (orig) C nm − C nm = δC nm (tx ) + δC nm (t y ) + δC nm (tz ) + δC nm (εx ) + δC nm (ε y )

(7)

where GM  and a  are the scaling factors associated with the original harmonic coefficients of the second model.   and The coefficients {C nm , S nm } computed from Eq. (9) are compatible with the values GM and a associated  S nm − S nm = δS nm (tx ) + δS nm (t y ) with {C nm , S nm } and they should be used, instead of (8) {C nm (orig), S nm (orig)}, in the formulation of Kleus+ δS nm (tz ) + δS nm (εx ) + δS nm (ε y ) berg’s transformation model (which actually requires + δS nm (εz ) + δS nm (δs) that both models refer to the same conventional scaling The analytical expressions of the above equations factors). are rather cumbersome, requiring more than one page for their detailed writing, and they will not be given here. In general, for each harmonic coefficient C nm or 3 Least-squares Estimation Using   S nm being transformed, the righthand side of the above Kleusberg’s {Cnm , Snm } → {Cnm , Snm } equations depends on the values of the ‘nearby’ coeffiTransformation Model cients C n−1,m+1 , C n,m−1 , C n−1,m−1 , C n,m+1 , C n−1,m , S n−1,m+1 , S n,m−1 , S n−1,m−1 , S n,m+1 and S n−1,m ; for more details, see Kleusberg (1980). For an arbitrary pair of EGMs, a linear system of equaDifferent EGMs are not always compatible with the tions can be formed based on Kleusberg’s (1980) transsame numerical values for their associated scaling fac- formation model tors GM and a (see Table 2). In order to evaluate the consistency of the reference frames inherent in differy=Ax+v (10) ent EGMs through a least-squares (LS) inversion of Kleusberg’s transformation model, a re-scaling of their where y is a vector that contains the differences   original spherical harmonic coefficients may be needed C nm − C nm and S nm − S nm of their corresponding so that they refer to the same conventional values for harmonic coefficients, x is an unknown vector with the the geocentric gravitational constant and mean equato- Helmert transformation parameters of their underlying rial radius. reference frames, A is a full-rank design matrix + δC nm (εz ) + δC nm (δs)

Reference Frame Consistency in CHAMP

375

whose elements depend on the harmonic coefficients from which we can infer the uncertainty in the EGM’s {C nm , S nm } of the first ‘reference’ EGM, and v is a zero-degree coefficient as residual vector due to non-datum-related errors in the coefficients of both EGMs. σC 0,0 = σGM /GM (15) In our study, the vector x of the Helmert transformation parameters is estimated through a weighted LS Note that the above conventional scheme for assignadjustment of Eq. (10), using the following guidelines: ing error variances to the zero- and first-degree har– the differences according to Eqs. (7) and (8) are monic coefficients is employed only for EGMs that formed for the entire harmonic spectral bandwidth use the a-priori assumptions C 0,0 = 1 and/or C 1,0 = n, m = 0 − 70. If the zero- and first-degree coeffi- C 1,1 = S 1,1 = 0. For EGMs which are accompanied cients of both EGMs under comparison are a-priori by non-zero error variances for their zero- and/or firstfixed to 1 and 0, respectively, then the first four ele- degree harmonics, the weight determination according to Eq. (11) employs the formal error variances of these ments of the observation vector y will be zero; – a diagonal weight matrix P is used for the LS adjust- coefficients, as given in their official release. 

ment. The weight for each ‘observation’ C nm − C nm  and S nm − S nm is taken as pi =

σ 2

C nm

1 + σC2

4 Numerical Results (11a) nm

Ten different CHAMP/GRACE EGMs have been used for our numerical tests, which are listed and in Table 2 along with their associated scaling 1 pi = 2 (11b) factors GM and a. Their spherical harmonic coσ  + σ2 efficients and the corresponding error variances S nm S nm have been obtained from the website of the Interrespectively. In order to assign realistic error vari- national Centre of Global Earth Models (ICGEM) ances to the zero- and first-degree harmonic coeffi- at the Geo-ForschungsZentrum (GFZ), Potsdam cients (in cases of EGMs where these coefficients are (http://icgem.gfz-otsdam.de/ICGEM/ICGEM.html). a priori fixed to the error-free values 1 and 0, respec- In the case where both formal and calibrated error tively), the following conventions have been adopted. variances are provided for a particular EGM (e.g. Conventional (A-Priori) EGM ‘Geocentricity’ Accu- EIGEN-GRACE02S), the latter have been adopted for the statistical weight determination according to racy Level (11). It should be pointed out that all tested EGMs are σxcm = σ ycm = σz cm = 0.01 m (12) expressed in the tide-free system. EIGEN-CG03C is adopted as the ‘reference’ model from which we can infer the uncertainty in the EGM’s linked to the {C nm , S nm } coefficients that appear first-degree coefficients as in Kleusberg’s transformation model; see Eqs. (7) √ σC 1,0 = σz cm /(a 3) (13a) and (8). The estimated Helmert parameters will thus be consistent with the transformation scheme GRF1 → GRF2, where GRF1 is the inherent reference √ σC 1,1 = σxcm /(a 3) (13b) frame of EIGEN-CG03C, and GRF2 is the inherent reference frame of every other EGM that is tested. Hence, the ingoing values {C nm , S nm } for comput√ σ S 1,1 = σ ycm /(a 3) (13c) ing the observation vector y in (10) will always correspond to the original harmonic coefficients of EIGEN  CG03C, whereas the ingoing values {C nm , S nm } are obConventional (A-Priori) ‘GM’ Uncertainty Level tained by applying the rescaling formula (9) to the orig(IERS Conventions, McCarthy and Petit 2004) inal harmonic coefficients of every other EGM to be σGM = 0.8 × 106 m 3 s −2 (14) tested. Based on our particular selection of EGMs (see

376

C. Kotsakis

Table 2) such a re-scaling is necessary only for TUM2S comparisons between global terrestrial reference and TUM1S. frames (Soler and van Gelder 1987, Boucher and An example of the correlation matrix for the Altamimi 2001, Heflin et al. 2002); see also Schaab estimated transformation parameters from the leastand Groten (1979). squares adjustment of Kleusberg’s model is given in – in terms of orientation stability for the mean Earth Table 3, for the comparison between EIGEN-CG03C rotation axis (i.e. rotation angles εx and ε y ), the asand EIGEN-CG01C. It is seen that the correlations sociated reference frames of the tested EGMs exbetween all possible combinations of the Helmert hibit variations in the order of 10−2 –10−3 arcsec. transformation parameters is essentially negligible, On the other hand, the rotation angle about the mean thus suggesting a well-conditioned adjustment model. Earth rotation axis (εz ) appears to have much larger Note that similar patterns of statistical correlation have values (by 1–3 orders of magnitude, up to a few arcbeen obtained in all the adjustment tests that were sec!), a fact that suggests the existence of possible performed for our study. systematic differences in the definition and realizaThe results for the estimated transformation paramtion of the zero-meridian plane within the EGMs’ eters, along with their formal statistical accuracy (1σ reference frames; uncertainty level), are given in Table 4. Based on the – the references frames associated with most of the values given in this table, the following comments can tested EGMs show a scale stability at the ppb be made: level, or better. Notable exceptions are the models TUM1S and EIGEN2, where the estimated scale – the inherent reference frames of the tested EGMs factor reaches the values of −16.21 and 8.34 ppb, are consistent, in terms of their origin position, at respectively. Note that the harmonic coefficients of the level of 1–2 cm; each tested model have been properly re-scaled be– an interesting result is the evident bias in the tz fore the adjustment so that they refer to the GM and values, compared to the equatorial translation a values of the reference model EIGEN-CG03C. components tx and t y . A similar ‘z-shift’ effect Thus, the estimated parameter δs given in Table 4 is has been also reported in other studies on direct Table 3 Correlation matrix for the estimated transformation parameters between the inherent reference frames in the EIGEN-CG03C and EIGEN-CG01C models, as obtained from the least-squares adjustment using Kleusberg’s (1980) model tx ty ty εx εy εz δs

tx

ty

1.0000000

0.0000082 1.0000000

tz 0.0000013 −0.0000037 1.0000000

εx 0.0000312 0.0004224 0.0001308 1.0000000

εy −0.0000625 −0.0000699 0.0011952 −0.0012470 1.0000000

εz −0.0000929 −0.0044572 0.0007413 −0.0020342 0.0028387 1.0000000

δs 0.0013781 −0.0002247 0.0008539 −0.0001379 −0.0000411 −0.0000483 1.0000000

Table 4 Helmert transformation parameters of the inherent reference frames in CHAMP/GRACE EGMs (with respect to EIGENCG03). The differences of spherical harmonic coefficients with their error variances up to n max = 70 have been used, including the zero- and first-degree harmonics from all models Model

tx (mm)

t y (mm)

tz (mm)

εx (mas)

ε y (mas)

εz (mas)

δs (ppb)

EIGEN-GL04C 6.3 ± 8.5 −0.3 ± 8.5 14.1 ± 8.3 5.19 ± 3.75 −0.93 ± 3.73 −71.07 ± 53.34 0.59 ± 2.18 EIGEN-CG01C 2.1 ± 6.0 2.2 ± 6.0 4.2 ± 5.5 −0.67 ± 4.11 −2.82 ± 4.09 95.21 ± 61.40 0.42 ± 2.21 EIGEN-CHAMP03S 2.7 ± 9.4 1.0 ± 9.3 5.6 ± 8.8 15.98 ± 22.18 20.52 ± 22.02 −582.88 ± 356.97 0.53 ± 4.53 5.7 ± 14.5 0.7 ± 14.5 15.3 ± 13.1 7.91 ± 15.45 −4.07 ± 15.29 −5380.54 ± 1239.92 1.55 ± 5.97 TUM2S(∗) TUM1S(∗) 4.3 ± 12.8 −1.1 ± 12.8 −0.9 ± 11.5 109.53 ± 15.84 49.15 ± 15.74 −4099.31 ± 1306.46 −16.21 ± 5.28 EIGEN-GRACE02S 6.6 ± 11.0 2.6 ± 11.0 7.6 ± 10.6 −4.04 ± 6.66 −7.18 ± 6.59 −15.60 ± 98.64 0.22 ± 2.85 EIGEN-GRACE01S 5.6 ± 8.1 0.6 ± 8.1 15.7 ± 8.0 5.02 ± 27.52 −5.63 ± 27.56 −69.55 ± 217.89 0.01 ± 2.08 EIGEN2 0.3 ± 14.1 2.7 ± 14.1 −11.4 ± 13.9 10.76 ± 5.16 27.92 ± 5.14 1681.47 ± 253.13 8.34 ± 3.59 EIGEN1S 5.4 ± 20.8 0.3 ± 20.8 15.1 ± 20.4 15.94 ± 77.52 22.30 ± 85.41 1766.71 ± 2444.75 0.04 ± 5.31 (∗) The spherical harmonic coefficients with their error variances up to n max = 60 have been used for testing these models.

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not influenced by any artificial scaling differences geoid heights). It is hoped that the results obtained that may exist among the harmonic coefficients of from this study will be of interest for many gravity field specialists and EGM development teams for the the tested EGMs. purpose of identifying possible discrepancies in the The square roots of the a-posteriori variance factors physical/mathematical models and/or data sets used in obtained from the previous adjustment tests are shown the compilation of modern global geopotential models. in Table 5. In most cases their values are close to 1, indicating that the choice of the diagonal weight maAcknowledgments The author would like to thank Nikos Pavlis trix P which was formed according to Eqs. (7) and (8) for his useful remarks, and also to acknowledge the two anonyis fairly realistic. The most notable exceptions occur mous reviewers for their comments. for the adjustment tests performed with the TUM1S and TUM2S models, the results of which suggest that the original error variances for their harmonic coefficients are rather optimistic. On the other hand, the test References results obtained with EIGEN-GL04C, EIGEN-CG01C and EIGEN-GRACE01S show that their harmonic co- Altamimi Z, Sillard P, Boucher C (2002) ITRF2000: A new release of the International Terrestrial Reference Frame efficients were probably assigned a lower statistical for earth science applications. J Geoph Res, 107(B10): weight than they deserve, within the least-squares ad2214–2232. justment of Kleusberg’s transformation model. Anderle RJ (1974) Transformation of terrestrial survey data to Table 5 Square root of the a-posteriori variance factor for the least-squares adjustments corresponding to the cases shown in Table 4 Model

σˆ o

EIGEN-GL04C EIGEN-CG01C EIGEN-CHAMP03S TUM2S TUM1S EIGEN-GRACE02S EIGEN-GRACE01S EIGEN2 EIGEN1S

0.77 0.78 1.60 2.99 2.64 1.01 0.73 1.27 1.87

5 Conclusions Using Kleusberg’s (1980) linearized model for the Helmert transformation of the spherical harmonic coefficients of the Earth’s gravitational potential, a number of CHAMP/GRACE global geopotential models have been tested in terms of their reference frame consistency. Compared to other techniques that have been used for similar purposes, our methodology relies on the direct adjustment of the differences   {C nm − C nm } and {S nm − S nm } according to a similarity-type transformation model that relates the underlying EGMs’ reference frames, without computing other intermediate gravity field functionals (e.g.

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C. Kotsakis Soler T (1998) A compendium of transformation formulas useful in GPS work. J Geod, 72(7/8): 482–490. Soler T, van Gelder BHW (1987) On differential scale changes and the satellite Doppler system z-shift. Geophys J R Astr Soc, 91(3): 639–656. Weigel G (1993) Geoid undulation computations at Doppler satellite tracking stations. Manusc Geod, 18(1): 10–25.

The Fast Analysis of the GOCE Gravity Field Xu Xinyu, Li Jiancheng, Jiang Weiping, Zou Xiancai and Chu Yonghai

Abstract The semi-analytical (SA) approach is proposed for the fast determination of gravity field from the GOCE (Gravity Field and steady-state Ocean Circulation Explorer) observations. In this paper, we discuss the principle and the characteristics of the SA approach in detail, and point out that based on the application of FFT and the introduction of the nominal orbit (circular, exact repeat orbit, uninterrupted measurement time series, and constant inclination), the SA approach can effectively recover the potential coefficients from large observations. This method can give a fast diagnosis of the GOCE system performance in parallel to the running of the GOCE mission by comparing the estimated noise characteristics of the SGG (Satellite Gravity Gradiometry) time series with the gradiometer error PSD (Power Spectrum Density) (prior). The performance of this method is evaluated using the simulated observations generated in different cases (ideal and practical). The results in this paper show that this method can be applied to the practical cases of non-circular, non-repeat orbits and non-constant inclination in the time series of Xu Xinyu School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Li Jiancheng School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Jiang Weiping School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Zou Xiancai School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Chu Yonghai School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

observations by the iteration algorithm which can eliminate or weaken the approximate errors in the observation model. The test examples show that to recover the high degree gravity field model, the optimal regularization parameter should be selected in the Kaula regularization technique. Keywords GOCE · FFT · Semi-analytical approach · Gravity field

1 Introduction The GOCE satellite is to be launched in May 2008, with its main objectives being to recover the geoid with 1-cm accuracy and 100 km spatial resolution corresponding to about degree and order 200. The major observations for the gravity field recovery are the SST-hl (Satellite-to-Satellite Tracking in the high-low mode) orbit data and the gradiometry data (ESA 1999). In the 20 months period, the GOCE mission will provide hundreds of millions of observations, based on which the established gravity field model with degree and order 200 contains about 40,000 potential coefficients. Strictly solving the involved equations is time-consuming and requires a high-capacity computer. Therefore it is significant to develop a fast method to determine the coefficients. In the past few years, many authors have made their efforts to seek effective methods for the solution of the gravity field parameters from extra-high degree equations (Rummel et al. 1993; Koop 1993; Schuh 1996; Sneeuw 2000; Pail and Plank 2002). The SA method is one of some fast effective methods. In this method, the gravitational potential is expressed as a function

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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of the orbit’s Kepler elements a, e, I, , ω, M (the major axis, eccentricity, inclination, right ascension of the ascending node, argument of perigee, mean anomaly) which can be considered as a function of time, and is called time-wise approach. Wagner and Klosko (1977) introduced this method to satellite geodesy. Kaula (1983) analyzed the satellite data using the method, and Schrama (1990, 1991) applied it to the analysis of SGG data. Sneeuw (1991, 2000) derived the formula of the gravitational potential expressed by the orbit’s elements based on group theory and quantum mechanics. The formula is consistent to the one given by Kaula (1966). In this paper, the GOCE operational orbit is simulated by the integration method. The gradiometry data and the disturbing potential data along the orbit are computed using the spherical synthesis method. And the white noises for the SST data and the colored noises for SGG data are also simulated. Based on the simulated data, we estimate the potential coefficients using the SA method. The results show that the corresponding iteration procedure can eliminate or weaken the approximate errors in the observation model, and that the SA approach is a fast and valid method for the GOCE mission. The potential coefficients are also estimated using the different regularization parameters (small, proper, big). The comparison with the ‘true’ model EGM96 suggests that the optimal regularization parameter should be selected in the Kaula regularization technique. The combined solutions of the SGG observations and the SST data show that the long-wavelength error of gravity field solution can be effectively reduced by the SST data.

GM R r, θ, λ

geocentric constant mean equatorial radius of the Earth geocentric spherical coordinates (radius, co-latitude, longitude) l,m degree, order C lm , S lm fully normalized l degree and m order potential coefficients P lm (cos θ ) fully normalized associated Legendre functions.

The gravitational potential can also be expressed as a function of orbit’s Kepler elements a, e, I, , ω, M. Provided that e ≈ 0, the expression of the gravitational potential is: V (r, I, u, ) = 

l+1 l ∞ l GM    R F lmk (I ) R r l=0 m=0 k=−l[2]

C lm −S lm

l−m:even l−m:odd



S cos ψmk + lm C lm



l−m:even sin ψmk l−m:odd

(2) where F lmk (I ),

the normalized inclination function ψmk = ku + m u = ω + M, the argument of latitude  = − θG , the longitude of the ascending node, θG , the Greenwich sidereal time. “[2]” denotes the step length of the variant k The differences between the above formula and the one given by Kaula (1966) are that k = l − 2 p substitutes p and the inclination function is a normalized one. Here, l and k have the same parity. Equation (2) can also be expressed as the time series:

2 Theoretical Background There are many representations of the gravity field model. The most popular one is in terms of a spherical harmonic series expansion, which is a function of the spherical coordinates: ∞ l  GM   R l+1 V (r, θ, λ) = R r l=0 m=0

(C lm cos mλ + S lm sin mλ)P lm (cos θ ) (1) where

Vij (t) =

lmax  lmax 

ij

[ Amk cos ψmk (t)

m=0 k=−lmax ij + Bmk sin ψmk (t)]

(3)

where Vij is the second order derivatives of the gravitational potential in the local north-oriented reference frame (x-axis points to the north, y-axis is directed to the west, z-axis has the geocentric radial direction, and x, y, z form a right-handed system). lmax is the maximum resolvable harmonic degree. And ψmk (t) = ku(t) + m(t)

(4)

The Fast Analysis of the GOCE Gravity Field ij

Amk ij Bmk



GM = R 

αlm =

 lmax l+1  R α ij λlk F lmk (I ) lm βlm r

l=lmin[2] l−m:even C lm

−S lm

381

l−m:odd



, βlm

S = lm C lm

l−m:even

3 Simulation and Discussion (5)

l−m:odd

ij

where λlk is the eigenvalues of the second order differential operators. For the components Vxx , Vyy , Vzz , the eigenvalues are −(l+1+k 2 )/(r 2 ), −((l+1)2 −k 2 )/(r 2 ) and (l + 1)(l + 2)/(r 2 ) respectively. According to the above formulation, the SA method can be performed by two steps. In the first step, the ij ij lumped coefficients (LC) Amk , Bmk are evaluated by the FFT technique. In the second step, the potential coefficients are estimated based on the linear relationship between the potential coefficients and LC. We can recover the potential coefficients order by order from the LC. Here we assume that the normal equation has a block-diagonal structure. In principle, the orbit and the observations in the SA approach should satisfy the following conditions (Sneeuw 2000):

r r r

The nominal orbit: the orbit is circular (eccentricity e = 0), inclination I of the orbit is constant, and the orbit is exact repeated. In the one repeat period, the nodal days Ne and the orbit revolutions No do not have a common divisor. The maximum degree of the potential model holds lmax < 1/2No . The observations are continuous, and the sampling rate is constant.

Pail and Wermuth (2003) and Preimesberger and Pail (2003) proved that the deviations from these requirements can be dealt with an iterative method. The gradiometry observations of the GOCE mission contain the colored noises, and have high accuracy only in the bandwidth 0.005–0.1 Hz (ESA 1999). Therefore, a filter strategy has to be used in the frame of a least square adjustment procedure. The SA approach can simplify such filter strategy, because it transforms the observations into the frequency domain where the convolution filter procedure is a simple multiplication. The SA approach is the fastest method at present with the application of the FFT technique, the blockdiagonal normal equation and the fast filter strategy. However the solution by the SA is not the final because it introduces the approximation in the observation model.

In order to test the approach, the simulated data sets (orbit, SGG, SST observations and noises) are generated as follows:

r

r

r r

Orbit: according to the requirements of the GOCE mission, non circular, sun-synchronous 29 days/467 revolutions near repeat orbit is simulated based on the Earth gravity model EGM96 (Lemoine et al. 1998), truncated up to the degree and order 70. The inclination I is 96.7◦ and the mean altitude is 245 km. SGG observations: based on the model EGM96 (truncated up to degree and order of 80 and 200 respectively) and the normal ellipsoid GRS80, only the component Tzz of the disturbing gravity gradient tensor is computed along the orbit with the sampling interval of 5 s. SST observations: based on the EGM96 truncated to degree and order 200 and the normal ellipsoid GRS80, the disturbing potential is computed along the orbit with the sampling interval of 5 s. Noises: first, the white noises with variance 0.5 m2 /s2 are simulated; second, according to the error PSD (Power Spectrum Density) function of the gradiometer (Ditmar et al. 2003), the colored noises are simulated based on AR (auto-regressive) model. The analytical PSD function corresponding to the Tzz component is:

S( f ) =

S0 1−e

−

f f0

(6)

where √ S0 = 3.2 mE/ Hz f 0 = 27 cpr/Tr ≈ 0.005 Hz ‘cpr’ stands for cycles per revolution, and Tr is the average revolution period. The square root of the PSD of the simulated noise and the analytical function PSD are shown in Fig. 1. First, we apply the SA method to the noise free gradiometry data to recover the Earth gravity model with degree and order 80. Figure 2 shows the convergence of the SA method in terms of the deviations of the estimated coefficients from the initial ‘true’ EGM96 model represented by a degree RMS defined in Eq. (7). The dashed line, the solid line and the bold solid line

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Fig. 1 Square root of the PSD of the simulated noise and analytical function (the green is analytical PSD and the blue is from the simulated error time series)

and valid. And from the plot, the iteration procedure can completely eliminate the errors caused by the non-circular, non-constant inclination and non-exact repeat orbit. This conclusion is the basis for the following numerical tests. A simulated colored noise time series is added to the simulated Tzz observations. Figure 3 displays the degree rms of the iteration solutions estimated by Tzz observations in terms of the colored noises, and the bold solid line represents the final solution. From the plot, we can see that the accuracy of the long-wavelength solution is not good, which reflects the decreased measurement accuracy below the measurement bandwidth (cf. Fig. 1). The geoid differences between the final solution and the ‘true’ model EGM96 are presented in Fig. 4. The plot shows that the accuracies at the polar areas are poor. Tables 1 and 2 also show the property that the statistical accuracy in terms of geoid in the ‘±90◦ ’ area is better than the one in the ‘±80◦ ’ area. The expressions ‘±90◦ ’ and ‘±80◦ ’ represent the areas from −90◦ to 90◦ and from −80◦ to 80◦ in latitude respectively. When the maximum degree and order of the gravity field model is up to 200, the regularization technique should be applied in the procedure of least-squares (Kusche and Klees 2002). Because of the properties of the GOCE mission, the downward continuation, polar gaps and colored noises, will cause the normal equation matrix to be ill-conditioned. In this paper, the Kaula regularization technique is used. From Figs. 3 and 4, we can see that although the regularization technique is applied in solving the normal

Fig. 2 Degree RMS of the iteration solutions in terms of noise free gradiometry data

represent the EGM96, the iteration solutions and the final solution respectively.    σl = 

l 1  (EST) (EGM96) 2 (EST) (EGM96) 2 [(C lm − C ) + (S lm − S lm ) ] 2l + 1 m=0

(7)

Here ‘EST’ denotes the estimated coefficients and ‘EGM96’ refer to ‘true’ model EGM96. Figure 2 illustrates that the final solution can converge to the ‘true’ model EGM96 perfectly, which suggests that the Fig. 3 Degree RMS of the iteration solutions by Tzz observations concept and the realization of the method are correct in terms of the colored noises

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Fig. 4 Geoid Height Deviations from the ‘true’ Reference Model EMG96 (unit: m)

Table 1 Statistical deviation from the reference model EGM96 in terms of the geoid height according to the different observations Observations Max(m) Min(m) RMS±90◦ (cm) RMS±80◦ (cm) SGG 6.56 SGG + SST 3.31

−6.82 −3.65

75.49 26.45

13.88 4.50

Table 2 Statistical deviation from the reference model EGM96 in terms of the geoid height using the different regularization parameters α

Max(m)

Min(m)

RMS±90◦ (cm)

RMS±80◦ (cm)

109.5 1012.5 1014.5

30.13 5.96 4.72

−16.52 −6.77 −5.84

231.15 77.17 87.56

41.95 13.84 35.36 Fig. 5 Degree RMS of the iteration solutions by Tzz observations in terms of the colored noises

equation, the low degree errors of the gravity field solution can not be controlled effectively. So, the combined solutions from the simulated SST observations with the white noises and the gradiometry observations with the colored noises are estimated. The iteration solutions in terms of the degree RMS are displayed in Fig. 5, and the final solution (dotted line) from the Tzz observations are also provided in the plot. The bold solid line represents the final solution. The statistical properties of the deviation from the reference model EGM96 in terms of the geoid height of the final solution are summarized in Table 1. From Fig. 5 and Table 1, we can see that the accuracy of the long-wavelength solution is improved obviously. The accuracy of the combined final solution in terms of geoid is up to 26.45 and 4.50 cm in the ‘±90◦ ’ and ‘±80◦ ’ areas respectively. In order to test the impact of the regularization parameter on the solutions, the potential coefficients are

estimated based on the different regularization parameters. The error spectra of the coefficients (deviations from the ‘true’ model EGM96) are plotted in Fig. 6, which correspond to the regularization parameters a 109.5 , 1012.5 and 1014.5 respectively. Table 2 summarizes the statistical properties of the deviation from the reference model EGM96 in terms of the geoid height using the different regularization parameters. From Fig. 6 and Table 2 the optimal regularization parameter α is 1012.5 , and the RMS of the difference between the computed geoid and that of the ‘true’ model EGM96 is 77.17 cm and 13.84 cm in the ‘±90◦ ’ and ‘±80◦ ’ areas respectively. Figure 6 also shows that if the regularization parameter is too large, it will lead to an over-smoothed solution, especially to the low order coefficients.

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Fig. 6 Coefficient error spectra of the solutions based on the different regularization parameters

4 Conclusions The SA approach is proposed to estimate the potential coefficients from the SGG and SST observations of the GOCE mission. The test results show that it is a fast and valid method for the GOCE gravity field recovery. The iteration technique can effectively reduce the model errors coming from the non-circular, non-constant inclination, non-exact repeat orbit and the colored noises. The optimal regularization parameter should be selected for the optimal solution in the Kaula regularization scheme. Acknowledgments This research was supported by the National Natural Science Foundation of China (No. 40637034). The authors thank Dr. R. Pail for numerous valuable discussions and suggestions. They are also grateful to Prof. C.C. Tscherning.

References Ditmar P, Kusche J, R Klees (2003). Computation of Spherical Harmonic Coefficients from Gravity Gradimetry Data to Be Acquired by the GOCE Satellite: Regularization Issues. Journal of Geodesy, Vol. 77, pp. 465–477. ESA (1999). Gravity Field and Steady-State Ocean Circulation Mission. Reports for mission selection, The Four Candidate Earth Explorer Core Missions, SP-1233(1), European Space Agency, Noordwijk. Kaula WM (1966). Theory of Satellite Geodesy. Blaisdell Publishing Company. London. Kaula WM (1983). Inference of Variations in the Gravity Field from Satellite-to-satellite Range-rate. Journal of Geophysical Research, 88, pp. 8345–8350.

Koop R (1993). Global Gravity Field Modelling Using Satellite Gravity Gradiometry. Netherlands Geodetic Commission. Publications on Geodesy. No. 38, Delft, The Netherlands. Kusche J, R Klees (2002). Regularization of Gravity Field Estimation from Satellite Gravity Gradients. Journal of Geodesy, Vol 76, pp. 359–368. Lemoine, FG, SC Kenyon, JK Factor, RG Trimmer, NK Pavlis, DS Chinn, CM Cox, SM Klosko, SB Luthcke, MH Torrence, YM Wang, RG Williamson, EC Pavlis, RH Rapp and TR Olson (1998). The Development of the Joint NASA GSFC and the NIMA Geopotential Model EGM96, NASA TP/-1998206861, NASA Goddard Space Flight Center, 575 pp. Pail R, G Plank (2002). Assessment of Three Numerical Solution Strategies for Gravity Field Recovery from GOCE. Satellite Gravity Gradiometry Implemented on a Parallel Platform, Journal of Geodesy, Vol. 76, pp. 462–474. Pail, R., M Wermuth (2003). GOCE SGG and SST Quick-look Gravity Field Analysis. Advances in Geosciences, Vol. 1, pp. 5–9. Preimesberger, T., R Pail (2003). GOCE Quick-look Gravity Solution: Application of the Semi-analytic approach in the Case of Data Gaps and Non-repeat Orbits. Studia Geophysica et Geodaetica, Vol. 47, pp. 435–453. Rummel R, van Gelderen M, Koop R, Schrama E, Sans´o F, Brovelli M, Miggliaccio F, F Sacerdote (1993). Spherical Harmonic Analysis of Satellite gradiometry. Neth. Geod. Comm., Publications on Geodesy, 39, Delft, The Netherlands. Schuh W.-D (1996). Tailored Numerical Solution Strategies for the Global Determination of the Earth’s Gravity Field. Mitteilungen Geod. Inst. TU Graz, 81, Graz University of Technology, Graz. Schrama E (1990). Gravity Field Error Analysis: Applications of GPS Receivers and Gradiometers on Low Orbiting Platforms, NASA TM-100769, GSFC Greenbelt Md. 20771. Schrama E (1991). Gravity Field Error Analysis: Applications of Global Positioning System Receivers and Gradiometers on Low Orbiting Platforms, Journal of Geophysical Research, Vol. 96 No. B12, pp. 20,041–20,051.

The Fast Analysis of the GOCE Gravity Field Sneeuw N (1991). Inclination Functions, TU delft, Fac. of Geodesy, FMR Internal Report 91.2 Sneeuw N (2000). A Semi-analytical Approach to Gravity Field Analysis from Satellite Observations, Deutsche Geod¨atische Kommission, Reihe C, Nr.527. Wagner, CA., SM Klosko (1977). Gravitational Harmonics from Shallow Resonant Orbits, Cel. Mech. Vol. 16, pp. 143–163.

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GRACE Application to the Receding Lake Victoria Water Level and Australian Drought J.L. Awange, M.A. Sharifi, W. Keller and M. Kuhn

Abstract Lake Victoria in Africa, the world’s second largest freshwater lake, has been experiencing receding water levels since 2001. As it recedes, more than 30 million people who depend on it for livelihood are facing a disaster. Similarly, Australia is facing its worst drought on record with the livelihoods of a few million people at stake. G ravity Recovery A nd Climate Experiment (GRACE) data for 45 months (i.e., April 2002– April 2006) are employed to analyze these emerging challenges by measuring variations in the stored water. The results indicate a general decline in the lake Victoria basins water level at a rate of 1.83 km3 /month.

transport, domestic and livestock uses, see Fig. 1. It supports a livelihood of more than 30 million people living around the lake (Riebeek, 2006). Though the lake has continued to attract worldwide attention due to its importance, its recent decline has caused grate alarm as to whether the lake is actually drying up! According to Nicholson (1998, 1999) the lake level has experienced significant fluctuation since 1960s. However, the lake level has drastically been receded in the last five years (Awange and Onganga, 2006). The lake levels have dropped by more than 1.1 m below the 10 year average (Kull, 2006). Lake’s surface water, together with ground water, soil moisture and snow constitute stored water. Keywords Lake Victoria · Australian drought · Changes in the lake’s water level are directly related to the variation of the water stored in its basin which GRACE mission · Water level contributes 20% in form of river discharge. Since the lake is supplied both by direct rainfall (Nicholson, 1998) and the river discharges, a decrease in 1 Introduction stored basin water will contribute to a drop in the lake level (Awange et al., 2007). Lake Victoria, the second largest freshwater lake in Changes in basin’s stored water is directly linked the world by area, is a source of water for irrigation, into the local gravity variations. The launch of Gravity Recovery A nd Climate Experiment (GRACE) mission in 2002 has opened a new era in environmental J.L. Awange Department Spatial Sciences, Western Australia Sciences geodesy. It is the first gravity dedicated mission Center and The Institute for Geoscience Research, Curtin in low-low mode which supplies gravity monthly University of Technology, Bentley, Australia solutions up to medium wavelengths (Keller and M.A. Sharifi Sharifi, 2005). GRACE satellites had been recognized Surveying and Geomatics Engineering Department, Faculty of by Hildebrand (2005) as having potential to provide Engineering, University of Tehran, Iran the first space based estimate of terrestrial stored W. Keller ground water. Institute of Geodesy, Universit¨at Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany Considering the GRACE outstanding capabilities in M. Kuhn monitoring stored water changes (Tapley et al., 2004; Department of Spatial Sciences, Western Australia Sciences Ramillien et al., 2005) motivates us to analyze the Center and The Institute for Geoscience Research, Curtin south-west Australian drought using the local gravity University of Technology, Bentley, Australia M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Fig. 1 Lake Victoria basin (from Kayombo and Jorgensen (2006))

variation. As Fig. 2 shows, western Australia has been experiencing the lowest rainfall on record. In the map label three phrases are used for expressing rainfall deficiencies. For a small part of the region (lowest on record) the rainfall was the lowest since at least 1900. The area with severe deficiency label has the lowest 5% of historical totals. The territory experiencing rainfall deficiency between 5 and 10% is labelled as serious deficiency. We could expect to capture some significant changes in the gravitational acceleration of the area during the drought period. Using the Stokes coefficients of monthly Earth’s gravity solutions the gravity changes over Western

Australia are analyzed during 2002–2006. Reducing the stored water is clearly depicted in the local surface mass density. For mapping more regional details, analysis of regional variations using the localized base functions is recommended.

2 Methodology In order to obtain time varying gravity field due to mass redistribution over the study areas, the larger effect of gravity variation due to the mass of the Earth, which

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Fig. 2 Western Australian rainfall deficiencies 2004–2007 (source: http://www.bom.gov.au/cgi-bin/silo/)

which occur over a timescale shorter than one month, are then removed using atmospheric and ocean models (e.g. Wahr et al., 1998). The resulting difference in Eq. (1), which is called gravity anomaly or geoid variation is due to water storage changes. If we δG(t) = G(t) − G 0 , (1) consider δClm (t) and δSlm (t) to be normalized Stokes coefficients expressed in terms of millimeters of to give the monthly time-variable geoid δG(t). geoid height, with l and m being degree and order Changes mostly related to the atmosphere and ocean

is always constant G 0 corresponding to nearly 99% of the total field is computed from the long-term temporal mean and removed by subtracting it from the monthly geoid (G(t)) measured by GRACE at a time t, i.e.,

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respectively, the time-variable geoid in Eq. (1) is expanded in terms of spherical harmonic coefficients as δG(θ, λ, t) =

l L  

(δ C¯ lm (t) cos mλ

l=0 m=0

+δ S¯ lm (t) sin mλ) P¯ lm (cos θ )

(2)

conservation in the Earth system, δ Cˇ 00 is equal to zero for every j. 4. Calculate the variation in Equivalent Water Thickness (EWT), EWT(φ, λ) j =

σ (φ, λ) j ρw

(6)

where L is the maximum degree of decomposition, in meters or equivalently. θ is the co-latitude, λ the longitude and P¯ lm fully normalized Legendre polynomial (Heiskanen and (7) EWT(φ, λ) j = σ (φ, λ) j Moritz, 1967). The first step in the analysis of GRACE data of states the variations in millimeters. the lake basin as well as the western Australia re- 5. Spatially average over the specified area using the gion would provide an estimate of total water storage weighted mean formula: change that includes all ground water, soil moisture, snow, ice, and surface waters. Stokes coefficients are n  1 converted to those of the equivalent water changes acEWT j = n σ (φk , λk ) j  k (8) cording to the following computational scheme: k=1  k k=1

1. Compute residual Stokes coefficients of the jth monthly solution by removing the long-term where  k is the area of the kth cell within the study area. temporal mean of C¯ lm and S¯ lm This computational scheme is employed and the

¯ 

¯ 

 achieved numerical results are shown in the following N Clm 1  C¯ lm δ Clm section. = ¯ − (3) Slm j S¯ lm i δ S¯ lm j N i=1

where N is the total number of monthly solutions. 2. Convert residual Stokes coefficients into residual 3 Numerical Analysis surface density coefficients δ Cˇ lm and δ Sˇ lm Among the CSR, GFZ and JPL publicly available

ˇ 

 ρavg 2l + 1 δ C¯ lm δ Clm monthly solutions, we utilized the CSR monthly = (4)  ¯ ˇ models. Using Eq. (5), monthly surface mass density 3ρw 1 + kl δ Slm j δ Slm j variation from GRACE data for the period of April where kl is the load Love number of degree l 2002 up to April 2006 were then computed. The results (Wahr et al., 1998). ρavg = 5517 kg/m3 and are presented in Figs. 3, 4 and 5. A comparison of the ρw = 1000 kg/m3 are the average density of the monthly EWT variations from 2002 to 2005 gives a clear picture of the variation in the stored water of the Earth and water respectively. 3. Compute the spatial variations of the surface den- Lake Victoria basin. In 2003 (Fig. 3), January, May, and June recorded a sity rise in EWT by 400–500 mm. The rest of the months l L   indicate values between 100 and 200 mm. Moving on  δσ (φ, λ) j = Rρw (δ Cˇ 00 ) j + to 2004 (Fig. 4), all the months except April indicated l=1 m=0 negative EWT values, thus loss of water. In May, only  (δ Cˇ lm cos mλ + δ Sˇ lm sin mλ) j P¯ lm (sin φ) (5) the North–West part of the basin shows a positive EWT value of about 100 mm. This is the part of the basin where R = 6, 378, 137 m is the radius of the which normally receives high amount of rainfall. July Earth and δσ stands for the surface mass den- and August also show some positive EWT values of sity variations in kg/m2 . Due to the total mass about 100–200 mm in the Eastern part of the basin.

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phenomena occurred in the lake basin. For instance, March, April and May signal the heavy rainfall periods of the basin. Inter-annual comparison of the EWT variation from 2002 to 2006 (Fig. 6) shows a dramatic decline in the lake Victoria basin level. The blue color

2004−1

In 2005 (Fig. 5), a fall in the geoidal height occurs in all months except July, signifying a further drop in water level within the basin. Interannual comparison of EWT variation in few successive years leads to a far better understanding of

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(positive EWT) has gradually disappeared and from 2005 the whole area is covered by red color (negative EWT). The GRACE satellite observations are almost measured in the along-track direction. Due to the nearly polar orientation of the satellites’ orbit, one could expect the south-north error stripes (Sneeuw et al., 2006). Implementation of spatial averaging filters could partially improve the solution (Jekeli, 1981). EWT variations over the lake basin are spatially averaged over the whole area using Eq. (8) A fiveyear time-series of EWT variations are depicted in Fig. 7. One see catastrophically declining stored water

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Fig. 7 Lake Victoria basin accumulated equivalent water thickness variation between 2002 and 2006

level of the basin. The basin equivalent water height dramatically decreases at a rate of 6.20 mm/month. The changes are expressed in equivalent water volume in Fig. 8. The results indicate a general decline in the lake basin’s water level at the rate of 1.83 km3 /month. Similarly, the variation of the monthly geopotential models over the western part of Australia is portrayed in terms of EWT from April 2002 up to April 2006. For instance, the variations from 2003 through 2005 are depicted in Figs. 9, 10 and 11. Interannual comparison of EWT variations over the study area indicates a tendency towards negative values. In 2003, for instance, the equivalent thickness

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varies at a range of [−100 200] mm, while it reaches – From an analysis of the GRACE gravity field variations over the lake basin, a fall of the geoid −400 mm in 2005. However, the results show that level at a rate of 1.6 mm/year was observed. This the hydrological signal over Australia is unrealistisignifies an annual loss of the basins stored water cally variable, which is due to inappropriateness of during the period of 2002–2006. Since the basin the GRACE data processing and filtering methods catchment discharge contributes about 20% of for regional applications, coupled with the likely the Lakes hydrology, the reduction of the basins small hydrological signal in Australia. A much more stored water contributed to the decline of the 20% Australia-focussed reprocessing of the GRACE data is input to the lake and thus reduction in the Lake’s therefore needed for any useful hydrological signals to waters. be extracted. – The increased withdrawal of the water from the lake basin could also be attributed to the expansion of the 4 Conclusion Owen Falls (now consisting of the original Naluabaale Dam and the new Kiira Dam extension) as pointed out in EAC (2006). The following facts have been established following the satellite analysis of Lake Victoria basin and western – For this region with sparse terrestrial data, the results present the first independent non-terrestrial Australia:

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diagnosis of the fall of Lake Victoria water level (i.e., the fall is directly related to the decline in the basin’s waters) – For the case of Australia, the results show that the hydrological signal over Australia is unrealistically variable, which is due to global support of spherical harmonics and filtering methods, coupled with the likely small hydrological signal in Australia. GRACE analysis of Lake Victoria’s basin has thus helped to reveal the trend in the decline of the basins stored water. This however could have contributed to a reduction of the 20% discharge into the lake. Furthermore, a much more Australia-focussed reprocessing of the GRACE data is needed for extracting useful hydrological signals.

References Awange J.L., O. Onganga (2006). Lake Victoria: Ecology, Resources and Environment. Springer, Heidelberg Awange, J.L., M.A. Sharifi, G. Ogonda, J. Wickert, E.W. Grafarend, M. Omulo (2007). The falling Lake Victoria water level: GRACE, TRMM and CHAMP satellite analysis of the lake basin. Water Resour Manage, DOI 10.1007/s11269-0079191-y EAC (East African Community) (2006). Lake Victoria basin commission. Special report on the decline of water levels of Lake Victoria. EAC Secretariat, Arusha, Tanzania Hildebrand P.H. (2005). Toward an improved understanding of the global fresh water budget. A symposium on living with a limited water supply 85th AMS Annual Meeting, 9–13 January, San Diego Heiskanen W.A., H. Moritz (1967). Physical Geodesy. Freeman and Company, London

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Fig. 11 Annual variation in South-West Australia 2005 Jekeli, C. (1981). Alternative methods to smooth the Earths gravity field. Department of Geodetic Science and Surveying, Ohio State University, Columbus Kayombo S., S.E. Jorgensen (2006). Lake Victoria: experience and lessons learned brief. International Lake Environment Committee, Lake Basin Management Initiative, Kusatsu, Japan Keller, W., M.A. Sharifi (2005). Gradiometry using a satellite pair. J Geodesy 78:544–557 Kull D. (2006). Connections between recent water level drops in Lake Victoria, dam operations and drought. Available at: http://www.irn.org/programs/nile/pdf/060208vic.pdf. Cited 18th May 2006 Nicholson S.E. (1998). Historical fluctuations of Lake Victoria and other lakes in the Northern Rift Valley of East Africa. In: Lehman JT (ed) Environmental change and response in East African lakes. Kluwer, Dordrecht, pp 7–35 Nicholson S.E. (1999). Historical and modern fluctuations of lakes Tanganyika and Rukwa and their relationship to rainfall variability. Clim Change 41: 53–71

Riebeek H. (2006). Lake Victorias falling waters. http://earthobservatory.nasa.gov/Study/Victoria/printall.php Ramillien G., F. Frappart, A. Cazenave, A. G¨untner (2005) The variation of land water storage from an inversion of 2 years GRACE geoids. Earth Planet Sci Lett 235:283301s Sneeuw, N., M.A. Sharifi W. Keller (2006). Gravity recovery from formation flight missions. In: Proc of IAG Symposia Hotine-Marussi Symposium of Theoretical and Computational Geodesy: Challenge and Pole of Modern Geodesy, Wuhan, 29 May–2 June 2006 Tapley B.D., S. Bettadpur, J.C. Ries, P.F. Thompson, M.M. Watkins (2004). GRACE measurements of mass variability in the Earth system. Science 305: 503–505 Wahr J., M. Molennar, F. Bryan (1998). Time variability of the Earths gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103(B12): 30205–30229

Updated OCTAS Geoid in the Northern North Atlantic – OCTAS07 O.C.D. Omang, A. Hunegnaw, D. Solheim, D.I. Lysaker, K. Ghazavi and H. Nahavandchi

Abstract A new gravimetric geoid (OCTAS07v2) is generated using Stokes’ formula with gravity data as input. As local gravity data, a combination of land gravity data, new and old airborne gravity data, and adjusted marine gravity data has been used. All marine gravity data has been error screened and quality assured by removing dubious data and adjusting the data when necessary. Voids in the gravity data distribution were patched with gravity data from satellite altimetry. The OCTAS07v2 geoid was estimated using the remove-compute-restore technique. The longwavelength signal of the local gravity data was reduced using a Wong-Gore modified Stokes’ function. The long-wavelength part was represented by a global gravity field model based on GRACE data. The OCTAS07v2 geoid model was combined with the OCTAS07 MSS model to create a synthetic Mean Dynamic Topography (MDT) model. In comparison with the OCCAM MDT, our new synthetic MDT O.C.D. Omang Geodetic Division, NMA, Kartverksveien 21, N-3511 Hønefoss, Norway, e-mail: [email protected] A. Hunegnaw Geodetic Division, NMA, Kartverksveien 21, N-3511 Hønefoss, Norway D. Solheim Geodetic Division, NMA, Kartverksveien 21, N-3511 Hønefoss, Norway D.I. Lysaker Department of Mathematical Sciences and Technology, UMB, POBox 5003, N-1432 A◦ s, Norway K. Ghazavi Department of Geomatics, NTNU, Høgskoleringen 7G, N-7491 Trondheim, Norway H. Nahavandchi Department of Geomatics, NTNU, Høgskoleringen 7G, N-7491 Trondheim, Norway

model gave a std. dev. of the residuals of 11 cm. A comparison to the main northern North Atlantic currents show many similar features. Keywords OCTAS · Geoid · MDT · MSSH · Adjusted marine gravity data · New airborne gravity data

1 Introduction The Norwegian Ocean Circulation and Transport between the north Atlantic and the arctic Sea (OCTAS) Project, running from 2003 to 2008, focuses on the ocean circulation in the Fram Strait and adjacent sea with the main objective to improve the sea surface topography determination and to study the impact on ocean modelling, see Fig. 1. A central quantity for studying and understanding the ocean circulation is the MDT, which is the difference between the mean sea surface heights (MSSH) and the geoid. The MDT provides the absolute reference surface for the ocean circulation and is, in particular, expected to improve the determination of the mean ocean circulation. This, in turn, will advance the understanding of the role of the ocean mass and heat transport in climate change. Up to the expected launch of GOCE, the gravimetric geoid is not known with sufficient accuracy to allow full use of the massive sea surface height information, which several satellite altimetry missions have regularly provided since the early nineties, in global ocean circulation analysis. However, in a few marine regions in the world, sufficient in-situ information about the Earth’s gravity field exists to compute a more accurate geoid. The region covering the Norwegian and Greenland Sea between Greenland, Iceland, Norway and the

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Fig. 1 OCTAS study area

UK, including the Fram Strait is one of these regions. One goal of the OCTAS project is therefore to determine an accurate geoid in the Fram Strait and the adjacent seas. New airborne gravity measurements have been carried out. The marine gravity data have been adjusted and error screened based on the new airborne gravity data. Both, new airborne gravity data and adjusted marine gravity data will improve the quality of the gravimetric geoid. The new geoid is used together with an accurate MSSH to determine the MDT.

2 Airborne Gravity Survey In a joint cooperation between Geoid and Ocean Circulation in the North Atlantic (GOCINA) and Ocean

Circulation and Transport Between North Atlantic and the Arctic Sea (OCTAS), new airborne gravity data was collected during the summer 2003 in the Northern North Atlantic. In Fig. 2 the OCTAS part of the airborne measurement campaign is visible. For information about the airborne gravity data collected by GOCINA see e.g. Forsberg et al. (2004). The airborne survey was carried out with an aircraft equipped with GPS receivers, laser altimetry, Inertial Navigation Systems (INS), and a modern La-Coste & Romberg marine gravimeter. The measurements were done around Greenland, Svalbard, Jan Mayen and along the Norwegian coast. This additional survey (including the GOCINA measurements from Greenland over Iceland to Norway) was particularly important since it ties in as many of the different marine surveys as possible, allowing a data check and improvement by

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Fig. 2 New airborne gravity data measured by OCTAS during 2003

crossover adjustment as well as filling of some major data voids. A total of 9222 measurements divided into 35 profiles of airborne gravity tracks have been processed. An internal cross-over computation (airborne gravity data only) shows a RMS of 1.6 mGal, while the comparison with the original marine data gives a RMS of 4.51 mGal, see Table 1. Table 1 Statistics of the cross-over computations for the airborne data (internal: airborne data only; marine: cross-overs with marine data). Values in mGal Internal Marine

N

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Min

Max

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3 Marine Gravity Data The main marine gravity data set used in this study has been acquired from BGI, NGDC, Norwegian Mapping Agency (NMA), and from international and national oil companies. The data set was recently improved with a major airborne gravity survey campaign performed in 2003 under the scope of the GOCINA and OCTAS project, see Sect. 2. Marine gravity measurements are, in principle, very precise; military tests with the BGM-3 sea gravimeter, now commercially available, achieved RMS cross-over

errors of only ±0.38 mGal (Bell and Watts, 1989). For a variety of reasons, the accuracy of the available marine gravity anomalies does not match this precision. Some relate to measuring gravity on an imperfectly stabilised platform, while others are due to systematic instrument errors, loosing reference to an absolute gravity datum and uncertainties in the navigation system in terms of course, speed, and position errors, affecting the E¨otv¨os correction. Wessel and Watts (1988) review these problems in depth. Our strategy involves pre-processing the raw gravity data followed by a network adjustment. Pre-processing aims at reducing the dynamical errors associated with courses changes, smoothing out high-frequency noise, and removing spikes and blunders. Network adjustment aims at removing the systematic effects of datum offsets, different gravity reference systems and drifts in the gravity meter zero. The basic component of our pre-processing algorithm is the line-segment. A line-segment is a component of a survey where the ship’s course is adequately straight. Point-to-point vectors are compared with chosen criteria for breaking surveys into linesegments: a break can be triggered by a large change in course azimuth or an excessive gap between points. For each line-segment, we represent the a long-track free-air anomaly as well as the eastings and northings, defining the ship’s position, by a continuous function. Chebyshev polynomials represent our best estimate for the true shape of the gravity anomaly profile, smoothing out point-to-point noise. Statistics derived from the residuals between the fitted curve and the point data are used to estimate the stationary random component of the data errors. The subsequent network adjustment is to suppress the remaining systematic errors. The network adjustment model fits an independent datum shift parameter to each survey or survey leg. For any survey with a sufficient number of crossing points, a drift is included as an additional parameter. The adjustment estimates these model parameters by minimising the cross-over errors in the least squares sense, weighting the observed free air anomaly at the crossing according to the standard deviation of the polynomial curve fit for that line-segment. For the approximately 45,000 cross-over points in the northern Atlantic Ocean, the network adjustment reduces the standard deviation of the cross-over errors from 4.13 to 1.64 mGal. Similarly, the difference between KMS02 altimetry anomalies and shipborne and

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airborne data improved, with the adjustment reducing the standard deviation of the differences from 8.15 to 6.07 mGal. The differences between marine free-air anomalies before and after adjustment are shown in Fig. 3. The figure shows that the network adjustment has identified datum shifts for several surveys, probably resulting from bad harbour ties.

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from the Nordic Geodetic Commission (NKG) gravity data base, while over Greenland, Iceland and Svalbard data were selected from the Arctic Gravity Project (ArcGP). The marine areas were covered by a combination of adjusted marine gravity data (see Sect. 3), new and old airborne measurements (see Sect. 2), and gravity data from satellite altimetry [KMS-02 (Andersen and Knudsen, 1998)]. Voids in the marine and airborne gravity data were patched with KMS02 data. The combined gravity data set was build up of complete Bouguer anomalies on land and free-air anomalies at sea. We are using the remove-restore technique in combination with the residual terrain model (RTM)/Helmert method (Forsberg, 1984; Omang and Forsberg, 2002). The heights are averaged into a smooth reference surface href with a resolution of approx. 50 km. A Bouguer plate approximation is used to obtain the reduced anomalies by gred = g + 2π Gρh ref − gggm ,

Fig. 3 Illustrated is the change in marine gravity data due to error screening and adjustment of the data

4 Global Geopotential Model

(1)

where gggm is the global geopotential model. As GGM we used the EIGEN-GL04S1 as described in Sect. 4. The reduced gravity data gred is gridded and Faye anomalies, gfaye , are obtained after restoring the RTM terrain effect 2π Gρ(h −h ref ). The residual quasigeoid is estimated using the multi-band spherical 2DFFT (Forsberg and Sideris, 1993) approach

In the computation of the updated geoid (OCTAS07v2) ζres = F −1 [F(gfaye )F(S τ (ψ))], (2) we have used the global geopotential model (GGM) EIGEN-GL04S1 from GFZ. It is a satellite-only where F and F −1 are the Fourier and the inverse gravity field model based on GRACE data from Fourier transform, respectively. Februar 2003 to July 2005, excluding Jan 2004, and S τ (ψ) is the Wong-Gore modified Stokes’ function LAGEOS data from Februar 2003 to Februar 2005. (Wong and Gore, 1969) The tide system is tide-free. The model is complete to degree and order 150. For more information see ∞  2n + 1 GRACE science results at http://www.gfz-potsdam.de/ (3) Pn (cos ψ), S τ (ψ) = n−1 pb1/op/grace/results/grav/g006 eigen-g104s1.html. n=τ where τ is the truncation degree, and Pn is Legendre polynomials. As Eq. (3) indicates, the Wong-Gore 5 Geoid Determination modification gives a kernel function, summing up only terms from degree τ to infinity. The long wavelength To obtain a gravity coverage without large gaps, a compart of the signal is thereby not taken into account, and bination of several different data sets has been used. the changing of τ compares to some degree to the seCovering the Nordic and Baltic countries, and parts lection of different capsizes. of the European continent, gravity data were obtained

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Restoring the GGM gives the quasigeoid, ζ = ζres + ζggm

(4)

Over sea, or where the height equals zero, the quasigeoid ζ equals the geoid, N. The new geoid model is referred to as OCTAS07v2. In Fig. 7 the effect of using adjusted marine gravity data instead of unadjusted data in the geoid computation is illustrated, compare to Fig. 3. As shown there are differences up to more than 50 cm, and several surveys give large changes in the geoid, e.g. a survey along latitude 75◦ N. Also changes do occur due to the fact that several points have been deleted.

from the interpolation of the data using least-squares collocation. The OCTAS07 MSS model was also further validated using available global and regional models (KMS01, KMS03, KMS04, CLS01, CLS04, GSFC00, OCTAS06 MSS and OSU95). The mean and standard deviation of differences between the OCTAS07 MSS and KMS04 MSS models are 0.5 and 10.4 cm, respectively. These values are 0.8 and 7.2 cm, respectively, when the OCTAS07 MSS and the OCTAS06 MSS models are compared. The model OCTAS07 MSS is illustrated in Fig. 4.

6 OCTAS07 MSSH Multiple high-latitude observing satellite radar altimetry data, including ENVISAT (cycles 10–52), ERS2 (cycles 1–85), ERS-1 ERM (phases C and G) and GFO (cycles 37–168) data, are used to determine the MSS model, called OCTAS07 MSS. These data have been cross-validated using the multiple altimetry data base (the so-called stack files), generated at the Ohio State University. In this study, several experimental models have been derived and their consistency and accuracy have been derived and their consistency and accuracy have been evaluated, respectively. The OCTAS07 MSS model is developed with the mean tracks of TOPEX/POSEIDON as a reference. In this model, the annual, semi-annual as well as sea surface trends were removed. The resolution of the OCTAS07 MSS model is 3 min in latitude and 6 min in longitudes. The OCTAS04v1 geoid model is used later on for the comparisons and MDT computations and it is not used directly in the MSS computations. Note that OCTAS04v1 is a gravimetric geoid computed in the region in 2004. The long-wavelength portion of this geoid model is determined from the global geopotential model GGM01C up to degree and order 200. The OCTAS04v1 is calculated using the remove-restore technique and the Wong-Gore modified Stokes’s function with truncation degree 80. The OCTAS07 MSS model ranges between 15 and 70 m in the study region. The internal consistency or the quality estimate for the OCTAS07 MSS model ranges from 2 to 5 cm over the study region. This quality estimate is output

Fig. 4 The OCTAS07 MSS model

7 Synthetic MDT A synthetic MDT model was computed from a combination of the OCTAS07v2 geoid, the OCTAS07 MSS and the OCCAM MDT (Webb et al., 1998), mathematically given as, MDT = OCTAS07 MSS − OCTAS07v2.

(5)

The synthetic MDT was smoothed, to remove short scale features, using a Gaussian shaped filter with a resolution of 1 degree, i.e. a low-pass filter. In Fig. 5 our synthetic MDT, derived from OCTAS07v2 geoid and the OCTAS07 MSS model, is illustrated. The synthetic MDT is labeled synthetic since it is not based on any oceanographic data.

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= OCTAS07 MSS – Geoid – OCCAM MDT) are listed. All values in meter

Fig. 5 A synthetic MDT derived from OCTAS07 MSS and OCTAS07v2 geoid model. MDT is low-pass filtered

8 Discussion A new geoid model, OCTAS07v2, was calculated using adjusted marine gravity data, airborne data, ArcGP gravity data and EIGEN-GL04S1 as GGM, see Fig. 6. In Table 2 the new geoid model and all older OCTAS geoids are used in the computation of the residuals

= MSSH − Geoid − OCCAMMDT,

(6)

where MSSH is OCTAS07 MSS. The subscript sea and ice indicates that the land areas are removed in the first case, while in the latter also areas north of Svalbard and along the east Greenland coast (where sea-ice is located) has been removed from the OCTAS07 MSS model.

Fig. 6 The OCTAS07v2 geoid model

Geoid

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0.297 0.299 0.298 0.295 0.292 0.281 0.281

0.129 0.114 0.122 0.113 0.117 0.112 0.111

As illustrated in Table 2 the residuals of Eq. (6) has decreased since the first OCTAS geoid model, to a level of approximately 28 cm or 11 cm if all sea area is included or if areas north of Svalbard and along the northeast coast of Greenland is excluded, respectively. In theory, however, the residuals, , of Eq. (6) should equal zero. A synthetic MDT was generated using OCTAS07v2 geoid and OCTAS07 MSS model. From the synthetic MDT an velocity field was generated by taking the derivative with respect to ψ and λ. The estimated velocity field of the synthetic MDT, illustrated in Fig. 8, show similar oceanographic features of the North Atlantic current as Fig. 1. Especially, along the coast of Norway, the splitting west of Trondheim (65◦ N, 5◦ E) and west of Lofoten (70◦ N, 10◦ E), and the current north in direction of Svalbard. The current from Artic basin and southwards along the Greenland coast is also clearly visible. Main problem areas are especially north of Svalbard and close to the coast (e.g. northeast

Fig. 7 Difference between geoid with adjusted marine data and one without

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velocity field of the synthetic MDT give a good representation of the major oceanographic features/-currents in the Northern North Atlantic. Acknowledgments The OCTAS project is supported by the Norwegian Research Council. We would also like to thank two anonymous reviewers for their comments and careful corrections of the paper.

References

Fig. 8 An MDT derived from OCTAS07 MSS and a geoid model based on adjusted marine gravity data. MDT is low-pass filtered. Velocity is added

cost of Greenland). This is due to sea ice in the north and MSSH models having problems determining the correct sea height along the coastlines.

9 Conclusions The marine gravity data in the Northern North Atlantic has been error screened and adjusted using new airborne gravity data. The standard deviation of the crossover errors for the marine gravity data was reduced from 4.1 to 1.6 mGal. A new geoid was computed based on adjusted marine gravity data and EIGEN-GL04S1. The new geoid model OCTAS07v2 show a small improvement compared to the older versions of the OCTAS geoids. A new synthetic MDT was computed by combining OCTAS07v2 geoid and OCTAS07 MSS model. The

Andersen, O. B. and Knudsen, P. (1998). Global marine gravity field from the ERS-1 and GEOSAT geodetic mission altimetry. J. Geophys. Res., 103(C4), 8129. Bell, E. and Watts, A. (1989). Evaluation of BGM-3 sea gravity meter system onboard R/V Conrad. J. Geophys., 51(7), 1480–1493. Forsberg, R. (1984). A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Technical Report 355, Dept of Geodetic Science and Surveying, Ohio State University, Columbus. Forsberg, R. and Sideris, M. (1993). Geoid computations by the multi-band spherical FFT approach. Manuscripta Geodaetica, 18, pp 82–90. Forsberg, R., Olesen, A., Vest, A., Solheim, D., Hipkin, R., Omang, O., and Knudsen, P. (2004). Gravity field improvements in the north atlantic region. In 2nd International GOCE User Workshop. ESA. Omang, O. and Forsberg, R. (2002). The northern european geoid: A case study on long-wavelength geoid errors. J. Geod., 76(6–7), 369–380. Webb, D., de Cuevas, B., and Coward, A. (1998). The first main run of the occam global ocean model. Technical Report 34, Southampton Oceanography Centre. Wessel, P. and Watts, A. (1988). On the accuracy of marine gravity measurements. J. Geophys. Res., 93(B1), 393–413. Wong, L. and Gore, R. (1969). Accuracy of geoid heights from modified Stokes kernels. Geophys. J. R. Astr. Soc., 18, 81–91.

New Combined Geoid Solution HGTUB2007 for Hungary ´ Gy. Toth

consistent height system, where the normal heights are compatible with the heights of the GPS and the quasigeoid model. Although due to moderate topography, the maximum difference between quasigeoid and geoid ´ am, 1999), it cannot be neis 28 mm for Hungary (Ad´ glected for high-precision geoid determination. Several gravimetric quasigeoid solutions have recently been produced for Hungary using Stokes integral and FFT methods (HGEO2000: Kenyeres et al. (2000) and HGTUB2000 T´oth and R´ozsa (2000)). Several of these solutions have been fitted at GPS/leveling points to the Hungarian National GPS Network (OGPSH) to facilitate the use of GPS to produce heights which are consistent with the national height system of Hungary (HGGG2000&2004: Kenyeres et al. (2000)). Besides of gravimetric data, however, various other data sets are available in Hungary, e.g. astrogeodetic deflections and surface gravity gradients, and it would be desirable to use these together with gravimetric data to obtain a refined solution. Another benefit would be the direct utilization of GPS/leveling data during the · combination process. By assigning large weights to such a dataset, the solution can effectively be “fitted” to the height system defined by GPS and national normal heights (Marti, 2006). This is more logical than a more-or-less “ad hoc” fitting of a gravimetric solution to the existing GPS/leveling data.

Abstract A new quasigeoid solution HGTUB2007 was computed for Hungary using least-squares collocation technique for the first time by combining different gravity datasets. More than 300 000 point gravity data were interpolated onto a 1.5 × 1 geographical grid consisting of 26, 478 values in the IGSN71 gravity system. The selected subset of these gravimetric data were combined with 138 astrogeodetic deflections and gravity gradients available at more than 25, 000 points in the least-squares collocation procedure. Topographic information was provided by SRTM3 data at 3 × 3 resolution. We have used the GPM98CR model and a GRACE GGM02-based combined model as a global geopotential reference to our new solution. Several solutions were produced and compared by combining different datasets. The final solution was chosen to fit to the national GPS/leveling network of Hungary with a very high weighting. As a quick evaluation of the solution with GPS/Leveling data shows, the obtained accuracy is about 2–4 cm in terms of standard deviation of geoid height residuals. Keywords Least-squares Combination solution

collocation

·

Geoid

1 Introduction

One of the main goals of the computation of a regional high precision quasigeoid model is to set up a Gy. T´oth Department of Geodesy and Surveying and Physical, Budapest University of Technology and Economics Geodesy and Geodynamics Research Group of BUTE-HAS, H-1521 Budapest, Hungary, M¨uegyetem rkp. 3.

2 Data Sets and Data Reduction Procedure The following data sets were available for our computations:

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´ Gy. Toth

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Fig. 1 Data sets used for the combined geoid solution

r

mean free-air gravity anomalies in 1 × 1.5 blocks based on 300 000+ point gravity data, 138 astrogeodetic vertical deflections (both ξ and η, North and East component) given in HD72 and transformed into GRS80, surface gravity gradients (torsion balance stations) at 27 005 points, GPS/leveling data of the Hungarian National GPS Network (OGPSH) at 340 points.

For reduction of the observations two different sets of geopotential coefficients were utilized. The first was r the GPM98CR geopotential model by Wenzel (which is identical with EGM96 up to harmonic degree 180, used to degree and order 720). Moreover, as an option, r the GRACE GGM02C model up to degree and order 200 that has been combined with GPM98CR model r with a linear “transition band” between degrees 180– 200 (GGM02CB model; maximum degree and order 720) was used. This model has the same maximum deWe have selected only a subset of several of the gree and order as the GPM98CR, and it is used for testabove data sets to make the combination solution ing how the new GRACE low-frequency geopotential numerically tractable, i.e. to allow the in-core solution information affects the solution. of collocation equations within the available comAs a further model we introduced the fixed mass puter memory. (Another option would be to consider model of SRTM3 heights over the whole area and comthe paging of normal equations to disk during the puted RTM corrections on all the data taking into accomputation.) count the maximum wavelength included in the geopoThe final datasets contained the following measuretential models. ments: Our previous tests (T´oth and R´ozsa, 2006) showed r 6678 mean free-air gravity anomalies in 2 × 3 that GRACE-based models fit better to the gravity anomalies in Hungary (smaller bias and better agreeblocks r 276 vertical deflection components (ξ and η) ment between the two fields in terms of standard r 7452 gravity gradients deviation). GRACE models, however, produced worst r 94 GPS/leveling points (recently leveled) fit to the OGPSH compared to EGM96 based models. In the data reduction process the GRACE-based Additionally, we introduced 267 European Gravi- geopotential model also produced slightly larger stanmetric Quasigeoid (EGG97) heights (Denker and dard deviation and bias of residuals except on gravity Torge, 1998) outside of Hungary just to avoid drifting data (see Table 1). Maybe this situation will change away of the solution in these areas. Table 1 Standard deviation / bias of residuals

Dataset

Gravity [mGal]

ξ [arcsec]

η [arcsec]

GPS/leveling [m]

Original data GPM98CR + RTM GGM02CB + RTM

±17.92/14.90 ±7.77/ − 4.41 ±7.69/ − 1.06

±2.47/0.55 ±1.22/0.02 ±1.20/ − 0.24

±2.17/2.85 ±1.06/ − 0.06 ±1.09/ − 0.11

±0.12/0.23 ±0.12/0.23 ±0.14/0.61

New Combined Geoid Solution HGTUB2007

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with the forthcoming next generation high resolution Table 2 Comparison between model and data variances with 2 EGM model. The standard deviation of gravity gradi- model parameters C0 = 57.7 mGal , D = 5 km, T = 9 km. The variance of gravity data equals to C0 ents (not shown), remained the same (about ± 15 E) after the reduction process. For efficient reduction of Gravity field quantity Data variance Model variance such data, however, a high degree geopotential model Height anomaly ±0.120 ±0.083 should have been used. ±1.13 Vertical ±1.1–1.4 deflection Horizontal gravity gradient

3 Calculation of the Quasigeoid

±14–15 E

±15.2 E

The residual gravity field of all observations was interpolated by least squares collocation with the self-consistent planar logarithmic covariance model by Forsberg (1987). The model features simple, closed formulas for autocovariances (ACF) and cross covariances (CCF) of geoid undulations, gravity anomalies, deflections of the vertical, and second-order gradients, both at the reference plane and aloft. Planar approximation is justified by the use of a high degree reference field (up to n max = 720) and the relatively small computation area (3.5◦ × 7◦ ) (Forsberg, 1984). The gravity disturbance (anomaly) model ACF C(g h 1 , g h 2 ) = −C0

4  i=1

+

αi ln(Di + h 1 + h 2

! s 2 + (Di + h 1 + h 2 )2

Fig. 2 RTM gravity anomaly ACF – empirical and model

(1)

depends both on the heights of the points h 1 , h 2 and the horizontal distance s. It is non-singular for all gravity field functionals considered and through the constants αi and Di it depends on the following three free parameters only: C0 , D and T. The first parameter C0 acts as a scale factor of the gravity disturbance ACF, the shallow depth parameter D produces a high-frequency attenuation and the parameter T a low-frequency attenuation. For complete formal ACF and CCF terms of the anomalous potential and its first and second derivatives see the Appendix of Forsberg (1987). The above three ACF parameters have been determined by fitting the model (1) to residual gravity data empirical covariances. A reasonable agreement has been found between model and actual gravity data variances according to Table 2. with the following model parameters: C0 = 57.7 mGal2 , D = 5 km, T = 9 km. Figures 2 and 3 show the best models for gravity resp. quasigeoid (GPS/leveling) ACFs. As a remark,

Fig. 3 GPS/leveling ACF – empirical and model

the national height system of Hungary uses normal heights instead of orthometric heights, therefore it is proper to estimate quasigeoid heights. In the following test computations these gravity ACF parameters were used. It was also found that the two most sensitive data sets with respect to the variation of the covariance

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model parameters are the deflections of the vertical and gravity gradients. Combination of gravity, vertical deflection and GPS/leveling data was then straightforward using the selected self-consistent covariance model. We tried different weights of the observables and examined the residuals at GPS/leveling points. Gravity gradients, however has been treated in a two-step process. First, horizontal gravity gradients at selected 3726 sites were used with error standard deviation of ±5 E in combination with astrogeodetic vertical deflections to predict deflection components at the gravity gradient sites. This error figure comes from our experiments of combining gravity and gradient data. The astrogeodetic deflections were used with their proper weights (±0.3 ) in the combination. Then these computed vertical deflections with their prediction errors were used in a second step in combination with gravity, vertical deflection and GPS/leveling data to optimally estimate residual quasigeoid heights on the final grid consisting of 14,700 points. The argument behind this two-step process was the numerical instability of collocation equations that resulted while including gravity gradients in the solution, and probably this was due to the inhomogeneous coverage of gravity gradients (see Fig. 1). Because the CCF of gravity gradients and vertical deflections has a much shorter correlation length than gradient-geoid CCF (3.5 km instead of 17 km, cf. Fig. 4), the numerical instability of equations disappeared in the two-step process.

Fig. 4 Gradient-geoid and gradient-deflection CCFs

4 Comparison of Solutions In the sequel we discuss differences between the following combined quasigeoid solutions:

r r r

astrogravimetric (AG) astrogravimetric and GPS/leveling (AGG) astrogravimetric, GPS/leveling and gradiometric (AGGG).

Some of the solution have been fitted to GPS/leveling quasigeoid heights by assigning very large weights to these data. Of course, this procedure certainly hides some problems of the GPS/leveling dataset, but it is desirable for users of GPS to compute heights in the existing national height system of Hungary. In the fitted AGGG solution HGTUB2007 GPS/leveling data have been assigned very large weights to constrain the solution at 94 recently leveled OGPSH sites (Fig. 5). The prediction errors are thus practically found to be zero near the OGPSH points as well as about 1 cm inside and 5 cm outside of Hungary. The comparison of AG and AGG quasigeoid solutions (Fig. 6) shows a mean bias of 32 cm and standard deviation of 17 cm. A NW-SE tilt can also be seen. Comparison of OGPSH and EGG97 by Kenyeres and Vir´ag (1998) also showed a significant, 23 cm bias and a NW-SE tilt. At several GPS/leveling sites of the national GPS network (OGPSH) there are large discrepanices, which can be attributed to problems of the GPS/leveling dataset (Kenyeres and Vir´ag, 1998). Comparison of two fitted AGG quasigeoid solutions based on the two different geopotential models shows a zero mean and standard deviation of residuals 10.7 cm (Fig. 7). Larger residuals can only be seen outside of Hungary. It is interesting to see the quasigeoid height differences due to the inclusion of gravity gradients in the combination solution. These AGG and AGGG height anomaly differences, as it can be seen in Fig. 8, may reach up to 10 cm in certain parts of the country. Finally, we compared our recent solution with our former gravimetric HGTUB2000 quasigeoid model (T´oth and R´ozsa, 2000) computed by the spectral combination technique. For this model the GPM98CR model was used with gravity data consisting of 58,800 1 × 1.5 mean Faye anomaly blocks for the whole computation area.

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8 46 45 .

.2

41

40 .4

41 41.8.6

43 .4

44

45

44.6

44 .8

42 .4 42 .6

.8 43 44

45 .6

46 .2

44 . 4

44.8

4 42 2 .4 43 .2 .6 43

42 .2

42 .8

20

21

.6 44

19

43 43 .4.2 43 .8 .2 44

43

18

43

43 . 2

45 45 .2

.2

17

.6

16

43

43

4 5 .4 45 46.8

45.6

44

43

43

44 .4

43.2

.8 43

43.4

44

4 4 .4

44 .8 44 .6

44.2

46 45 .8

45

42

43 .843.6 43 .4

.2 43

45 .2

39 39 .6 40 4 0 .8 .2 40 .4 40 .6 40 .8 41 41 .4 41 .6 42 42.2 4 1 .8 42 .8

.2 41 .4 41 .6 41

45 .4

.8 41

43

8 42 .

45.4

40 .6

43

.6

43

43 .2

42 .2

43

43

42.6

43 .4 43

44 .8 44.6 44 .2 44.4

.8 38 39 .2 39

42

.6 42

.4 45

.6 43

46.6

46 .4

.4 388.6 3

.8

40 .8

42

42 .2 42 .4

46 .2

44 .2

455 .2 4

46.5

43

44

.4 44 6 . 44 .8 44

45 .6

47

46

44

.2 43

43.6 43.8

42 .6

43 .4

2

.6.8 4466 47

47.5

.4

38 38 .2

9 .4 33399.6.8 0 4 0.2 4

43

42.4 42 .6

8 43 . 44

.6 43

. 45 45 45 .4

48

.2 4433.4

4 2 .8

42

43.2 .2 43

.2 4411 .4

43.2

48.5

43

42 .8

43

45.4

49

22

Fig. 5 HGTUB2007 fitted astrogravimetric, gradiometric, GPS/leveling (AGGG) quasigeoid solution (m). GRACE-based GGM02CB geopotential model was used and SRTM3 heights

Fig. 6 Astrogravimetric minus fitted astrogravimetric GPS/leveling solution differences (m). Black circles denote GPS/leveling sites and squares outside of Hungary denote places where EGG97 quasigeoid undulations have been introduced into the combined solution

Problems with GPS/leveling

410

´ Gy. Toth

Fig. 7 Differences of AGG solutions based on GM98CR and GGM02C geopotential models (m)

Fig. 8 Quasigeoid height differences (m) due to inclusion of gravity gradients into the fitted AGG solution. Small dots indicate locations of included horizontal gravity gradients

It can be seen on Fig. 9 that the differences may reach up to 90 cm North of Hungary (high mountains), but inside Hungary the discrepancies are much smaller, mainly below 10 cm. These large differences are probably due to the fact that in the HGTUB2000 model gravity data outside Hungary have also been included. The residuals of the combined solution at OGPSH sites may reach ±6 cm with standard deviation of

±2.1 cm (Fig. 10). From the EUVN GPS/leveling sites (Ihde et al. 2000), 7 stations fall into the computation area and these has also been used for checking of the fitted AGGG solution. In this case the bias is −0.136 m whereas the standard deviation of residuals is ±4.2 cm (Fig. 11). We note, that this bias is in agreement with the one found between heights of the EUVN and the Hungarian leveling network, which is about 14 cm.

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Fig. 9 HGTUB2007 – HGTUB2000 quasigeoid differences (m)

Fit of HGTUB2007 at 94 OGPSH points std : ±0.021 m

0

OGPSH points

Fig. 10 Quasigeoid height differences between HGTUB2007 and GPS/leveling at 94 OGPSH sites (m)

BAKSIP

–0.3

–0.06

no fit

std : ±0.098 m

bias/tilt fit

std : ±0.042 m

UZHGO

–0.2

–0.04

KAMEN

–0.136mm -0.136

TIMISO

–0.02

–0.1

PENC

0

NADAP

0.02

CSANAD

difference (m)

geoid height difference (m)

0.04

HGTUB2007 differences

0.1

0.06

EUVN GPS leveling sites

5 Conclusions and Future Work

Fig. 11 Quasieoid height differences between HGTUB2007 and GPS/leveling at 7 EUVN sites (m)

A new quasigeoid solution has been achieved for Hungary for the first time by combining different gravimetric, astrogeodetic, GPS/leveling and gravity gradient data sets. As it was mentioned, there are some issues with the used GPS/leveling data which should be fixed. The GPS data were collected by rapid measurement technology (fast-static), processed with non-scientific software provided by vendors of receiver hardware and OGPSH was established mainly for horizontal control.

The (third-order) levelings of these sites are also sometimes problematic. Therefore, the vertical component may be weak. New precise GPS/leveling measurements are desirable to achieve greater consistency between the different kinds of gravity field data. Although the number of available gravity data is satisfactory, there is still the issue that several areas in Hungary are to be densified with future gravity measurements: “white spots” do exist.

412

It was mentioned in Sect. 2 that the fit of GRACEbased combined geopotential models to the OGPSH is not satisfactory and this issue is still to be investigated. Thus, our solution would benefit from the forthcoming new and consistent high resolution EGM models based on recent gravity field missions CHAMP and GRACE, so we plan to use these models as soon as they will be available. The use of synthetic mass models, especially the litospheric model of the Pannonian Basin (Papp and Kalm´ar, 1996) would be beneficial not only for data reduction, but also for checking the computation methodology/software. The use data provided by the forthcoming GOCE mission is also planned in proper combination with our regional data. Due to the above mentioned problems, our new quasigeoid solution should be regarded as a preliminary test version and it will be updated to the final solution when these issues are fixed. Acknowledgments This research was sponsored by the Hungarian Scientific Research Fund (OTKA T046718 and K60657). Gravity data provided by the Hungarian Institutions ¨ FOMI and ELGI are acknowledged as well as the use of GRAVSOFT software by C.C.Tscherning and R. Forsberg. Comments and suggestions of two anonymous reviewers helped to improve the paper and thereby are also gratefully acknowledged.

References ´ am, J. (1999). Difference Between Geoid Undulation and Ad´ Quasigeoid Height in Hungary. Boll. Geof. Teor. Appl, Vol. 40, No. 3–4, pp. 571–575. Denker, H., Torge, W. (1998). The European Gravimetric Quasigeoid EGG97 – An IAG Supported Continental Enterprise.

´ Gy. Toth In: R. Forsberg, M. Feissel, R. Dietrich (eds.): Geodesy on the Move – Gravity, Geoid, Geodynamics and Antarctica. IAG Symp. Vol. 119, 249–254. Springer Verlag, Berlin, Heidelberg, New York. Forsberg, R. (1984). Local Covariance Functions and Density Distributions. Reports of the Department of Geodetic Science and Surveying, Report No. 356, The Ohio State University, Columbus, Ohio, June. Forsberg, R. (1987). A New Covariance Model for Inertial Gravimetry and Gradiometry. Geophys Res, Vol. 92, No. B2, pp 1305–1310. ´ am, J. Gurtner, W. Harsson, B.G. Sacher, M. Schl¨uter, Ihde, J. Ad´ W. W¨oppelmann (2000). The EUVN Height Solution – Report of the EUVN Working Group – EUREF Symposium 2000, Tromsø, 22–24 June 2000, Norway. Kenyeres, A. Vir´ag, G. (1998). Testing of Recent Geoid Models with GPS/Leveling in Hungary. Proceedings of the Second Continental Workshop on the Geoid in Europe. Vermeer M. ´ am J. (Eds.), Reports of the FGI, 98:4, pp 217–223. and Ad´ Kenyeres, A. R´ozsa, Sz. Papp, G. (2000). HGEO2000: A Step put Forward to the cm-accuracy Gravimetric Geoid in Hungary. Poster presented at the GGG2000 Conference, Banff, Canada, August 2000. Marti, U. (2006). Comparison of High Precision Geoid Models in Switzerland. Proceedings of the IAG Scientific Assembly, Cairns. Proceedings of the IAG, Vol. 130, pp 377–382, Springer Verlag, Hamburg. Papp, G., Kalm´ar, J. (1996). Toward the Physical Interpretation of the Geoid in the Pannonian Basin using 3-D Model of the Lithosphere. IGeS Bulletin, N. 5., pp. 63–87. T´oth, Gy. R´ozsa, Sz. (2000). New Datasets and Techniques – An Improvement in the Hungarian Geoid Solution. Paper presented at Gravity, Geoid and Geodynamics Conference, Banff, Alberta, Canada July 31–Aug 4. T´oth, Gy. R´ozsa, Sz. (2006). Comparison of CHAMP and GRACE Geopotential Models with Terrestrial Gravity field Data in Hungary. Acta Geod. Geoph. Hung., Vol. 41, No. 2, pp 171–180.

Regional Astrogeodetic Validation of GPS/Levelling Data and Quasigeoid Models Christian Voigt, Heiner Denker and Christian Hirt

Abstract In the context of a GOCE regional validation and combination experiment in Germany, a work package within the framework of the GOCE-GRAND II project, gravity observations, vertical deflections and GPS/levelling data are collected as independent data sets. The observation of absolute gravity values is carried out by the Bundesamt f¨ur Kartographie und Geod¨asie (BKG), while the vertical deflections are observed by the Institut f¨ur Erdmessung (IfE) using the Hannover digital transportable zenith camera system TZK2-D. The vertical deflections have an accuracy of approx. 0.1 arc seconds and are arranged along a North–South and East–West profile. The two profiles have a length of about 500 km each with a spacing of 2.5–5 km between adjacent stations. Furthermore, a national GPS and levelling data set of about 900 stations with an accuracy of approx. 1 cm is available for Germany. The analysis of the vertical deflections is carried out by the astronomical levelling method, resulting in two (quasi)geoid profiles. The accuracy of the profiles is expected to be at the cm level. A cross-validation of both the vertical deflection and GPS/levelling data is realised by traversing the profiles through all nearby GPS/levelling stations (approx. 40 in total). In addition, comparisons are performed with the German Christian Voigt Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany, e-mail: [email protected] Heiner Denker Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, D-30167 Hannover, Germany Christian Hirt Department Geomatik, HafenCity Universit¨at Hamburg, Hebebrandstraße 1, 22297 Hamburg, Germany

Combined QuasiGeoid 2005 (GCG05) and the purely gravimetric solution EGG07 (European Gravimetric Quasigeoid 2007). Keywords Digital zenith camera system · Deflections of the vertical · Astrogeodetic quasigeoid profile · GPS/levelling data · Quasigeoid models

1 Introduction Within the next few years, improved high-resolution global gravity field models are anticipated from the GOCE mission. The expected accuracies are about 1–2 cm in terms of geoid undulations and 1 mgal for gravity, both at a resolution of about 100 km (see, for example, ESA 1999). Then, from a combination of the GOCE based global gravity field models (expected to be available up to spherical harmonic degree and order 250) with regional terrestrial data sets, an accuracy of about 1 cm is expected for the complete geoid spectrum. In this context, accurate and independent data sets are essential for the combination process as well as for the validation of the results. Therefore, a regional validation and combination experiment is carried out in Germany as a work package within the framework of the GOCE-GRAND II project. In the first stage, new absolute gravity observations, GPS/levelling data, as well as astronomic vertical deflections are collected. While the validation of the terrestrial gravity data base is carried out by the Bundesamt f¨ur Kartographie (BKG) using absolute gravity observations at field stations as spot-checks, a new regional data set of astronomically determined vertical deflections is collected

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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by the Institut f¨ur Erdmessung (IfE) as a completely independent observation type. The vertical deflections are also employed for a cross-validation of the existing GPS/levelling data. During recent years, the economic and precise determination of vertical deflections with an accuracy of approx. 0.1 was realised by the development of digital transportable zenith camera systems such as the TZK2-D (IfE; Hirt 2004) and DIADEM (Geodesy and Geodynamics Laboratory GGL, ETH Zurich; B¨urki et al. 2004). Both zenith camera systems were extensively employed for local and regional gravity field determinations in Germany (Hirt and Flury 2007), Switzerland (M¨uller et al. 2004, B¨urki et al. 2005) and Greece (M¨uller et al. 2006). In this study, the TZK2-D is employed for the determination of vertical deflections along a North–South and East–West profile with a spacing of 2.5–5 km between adjacent stations. For the cross-validation with the GPS/levelling data, each profile traverses through approx. 20 nearby GPS/levelling control points. At present, a few vertical deflection observations are still missing in the East–West profile, while the North– South profile is completely finished. Hence, this publication discusses mainly the results from the North– South profile. Finally, the astrogeodetic North–South profile is compared with the existing German Combined QuasiGeoid 2005 (GCG05; Liebsch et al. 2006) as well as the purely gravimetric quasigeoid EGG07 (European Gravimetric Quasigeoid 2007; Denker et al. 2007). In addition, the gravimetric quasigeoid models (GCG05 and EGG07) are evaluated by GPS/levelling control points along the profile; corresponding evaluations of gravimetric quasigeoid models by GPS/levelling data can be found, for example, in Lysaker et al. (2007) and Krynski and Lyszkowicz (2006). In this contribution, the three basic data sets, namely astronomically determined vertical deflections, GPS/levelling data and gravimetric quasigeoid models, are described in Sect. 2. The astrogeodetic quasigeoid computation is described in Sect. 3, while Sect. 4 and 5 deal with the comparison of the North–South astrogeodetic quasigeoid profile with corresponding values from GPS/levelling and gravimetric quasigeoid models. Emphasis is put on the analysis of the differences between the relevant data sets and their possible origins.

C. Voigt et al.

2 Data Sets In this section, the main features and accuracies of all relevant data sets are described.

2.1 Vertical Deflections Since September 2006, vertical deflections have been determined astronomically along two profiles across Germany using the Hannover digital transportable zenith camera system TZK2-D. The determination of the vertical deflections is based on the automatic determination of the direction of the plumb line in combination with the determination of ellipsoidal coordinates by GPS measurements. For further details on the TZK2-D system see Hirt (2004). The vertical deflection stations are arranged along a North–South and East–West profile (see Fig. 1). The north-south profile has a length of 540 km and runs

Fig. 1 Astrogeodetic profile lines and GPS/levelling points

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Table 1 Characteristics of the astrogeodetic profiles North–South East–West Length (km) Number of stations Average spacing (km) Number of GPS/levelling points

540 137 3.9

518 61/133 (observed/planned) 3.9

20

23

Table 2 Statistics of 23 double observations in different nights on stations along the North–South profile [ ] ξ η

Mean

RMS

Min

Max

−0.021 0.025

0.097 0.121

−0.210 −0.268

0.172 0.231

25 20

ξ ["]

15 10 5 0 –5

0

100

200 300 Distance [km]

400

500

0

100

200 300 Distance [km]

400

500

8 6 4 η ["]

from the Harz Mountains in the north to the Bavarian Alps in the South. In total, 137 vertical deflection stations are observed, having an average spacing of 3.9 km (from 2.5 to 5 km). The station spacing is reduced to approx. 2.5 km in regions where rapid gravity field changes occur, while in homogeneous areas, a spacing of nearly 5 km is selected. The vertical deflection observations along the north-south profile have been completely finished. Along the East–West profile with a length of 518 km, running from Lusatia in the east to the M¨unsterland region in the west, approx. 80% of the 133 vertical deflection stations have been determined up to now. Therefore, the cross-validation with the GPS/levelling data and the gravimetric quasigeoid models is only carried out with the vertical deflections along the north-south profile. For a statistics of the profiles see Table 1. Because of the high degree of automation of the zenith camera system, it was possible to determine vertical deflections on up to 12 stations in a single night. The accuracy of these vertical deflections is stated to be approx. 0.1 (see, for example, Hirt and Flury 2007). This figure is in close agreement with the statistics of the differences between 23 double observations on identical points in different nights along the northsouth profile (see Table 2). For further details and accuracy analyses of the digital zenith camera system TZK2-D see, for example, Hirt (2004) and Hirt and Flury (2007). Figure 2 shows the vertical deflection components ξ (North–South) and η (East–West) along the North– South profile.

2 0 –2 –4

Fig. 2 Observed vertical deflections on stations along the North– South profile (from North to South)

2.2 GPS/Levelling Data Figure 1 shows that the North–South profile was set up to traverse through 20 nearby GPS/levelling points, which has an effect on the azimuths of the corresponding profile sections. The GPS/levelling points no. 1, 2 and 4 are located in Thuringia, while point no. 3 and points 5–20 are located in Hesse and Bavaria, respectively (Thuringia, Hesse and Bavaria are federal states in Germany). All GPS/levelling points are extracted from a national data set of approx. 900 points with a spacing of 25–30 km. The national data set was collected by the BKG. Two versions of this data set exist, one from 2003 and one from 2005. The major difference between the two data sets is a re-adjustment of the SAPOS GPS reference stations, which was introduced at the beginning of 2004. However, the federal states of Germany used individual procedures to implement the new GPS reference frame, ranging from re-observation and re-adjustment to different transformation procedures, which may introduce some inhomogeneities in the GPS coordinates. It should be noted that the 2005 GPS/levelling data set was also

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C. Voigt et al. 2

Δζ [cm]

Δζ [cm]

data from the above mentioned 2005 national data set. This model was calculated by BKG and IfE using 1 2 3 5 6 8 9 10 11 12 13 14 15 16 1718 1920 two different approaches, the final model being derived 0 1 4 7 simply by averaging both individual solutions. The ac–1 curacy of the GCG05 model is stated to be 1–2 cm, for –2 further details see Liebsch et al. (2005). As a second quasigeoid model, the purely gravi–3 Δζ GPS/levelling points 2005-2003 metric solution (without GPS/levelling data) EGG07 –4 0 100 200 300 400 500 (European Gravimetric Quasigeoid 2007) is employed. Distance [km] This solution was computed at IfE using the spectral Fig. 3 Differences between GPS/levelling data sets 2005 combination method with integral formulas within the and 2003 framework of the European Gravity and Geoid Project (EGGP), see, for example, Denker (2006) and Denker used for the computation of the German Combined et al. (2007). The differences between the 2005 GPS/levelling QuasiGeoid 2005 (GCG2005). For further details on data set and the GCG05 and EGG07 quasigeoid the GPS/levelling data sets see, for example, Liebsch models are depicted in Fig. 4. The figure shows that et al. (2005). the GCG05 model, including the GPS/levelling data, The differences between the height anomalies of closely resembles the GPS/levelling control points the 2005 and 2003 GPS/levelling points along the (introduced in GCG05 with a standard deviation of north-south profile are shown in Fig. 3, ranging from about −3 to +2 cm. While the differences between the about 1 cm). The EGG07 differences exhibit mainly two data sets represent mainly a bias in Bavaria (pts. long wavelength structures, except at pt. no. 4 where a no. 5–20, except point no. 7), a more irregular pattern peak of about 3 cm appears to be present. is found at the beginning of the profile (pts. no. 1–4). 5 The exact reason for this behaviour is presently unknown. 3 The accuracy of the quasigeoid heights derived from 2 5 6 7 12 13 14 15 16 1 GPS/levelling data depends on the one hand on the accuracy of the ellipsoidal heights determined with GPS –1 1 3 4 8 9 10 11 17 18 19 20 (1–2 cm) and on the other hand on the accuracy of the Δζ GPS/levelling points-GCG05 –3 precise geometric levelling, where an accuracy of apΔζ √ GPS/levelling points-EGG07 prox. 1 mm / km yields 23 mm over the whole profile –5 0 100 200 300 400 500 length of 540 km. Distance [km] The accuracy issue is especially important in the context of the cross-validation of the astronomic and Fig. 4 Comparison of GPS/levelling data with the quasigeoid models GCG05 and EGG07 GPS/levelling data as well as the detection of possibly existing systematic errors in geometric levelling and GPS ellipsoidal heights.

3 Astrogeodetic Quasigeoid Profile 2.3 Gravimetric Quasigeoid Models

The method of astronomical levelling is documented very well (see, for example, Heiskanen and The gravimetric quasigeoid models used in this work Moritz 1967, Torge 2001). Hence, only a summary of are all based on the combination of global gravity the most important equations is given in this section. In this study, vertical deflections ε at the surface of field models with terrestrial gravity anomalies and terthe Earth according to the definition of Helmert are utirain data. In addition to this, the German Combined Quasi- lized to compute quasigeoid height differences. The baGeoid 2005 (GCG05) includes also GPS and levelling sic equation is

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2 ξ1,2 = −

N εds − E 1,2 ,

(1)

1

where ζ is the height anomaly difference between points 1 and 2, ds is the line element, and E N is the normal height reduction. The negative sign results from the definition of the vertical deflections. The vertical deflection ε is the deflection component in path direction, specified by the azimuth α. It can be computed from the observed north-south and east-west component ξ and η: ε = ξ cos α + η sin α.

(2)

Furthermore, the normal height reduction E N can be computed by 2 N E 1,2

= 1

g − γ045 γ045

dn+

γ 1 − γ045 γ045

H1N −

γ 2 − γ045 γ045

H2N . (3)

In the above equation, g is the surface gravity which is available with an accuracy of about 1 mgal, γ 1 , γ 2 are the mean normal gravity values at stations 1 and 2, H1N , H2N are the corresponding normal heights, ξ

observed

ξ

topographic

ξ

15

15

10

10

5

0

–5

–5

15

100

η

200 300 Distance [km]

observed

η

topographic

400

η

0

500 15

100

η

top removed

ξ

interpolated

ξ

top restored

200 300 Distance [km]

top removed

η

interpolated

400

η

500

top restored

10 η ["]

10 η ["]

top removed

5

0

0

ξ

20

top removed

ξ ["]

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Furthermore, a gross error in the vertical deflection data has a bias-like impact on the height anomalies for the remaining part of the profile. The magnitude of this bias naturally depends on the magnitude of the gross error in the vertical deflections and on the station separation. For example, a gross error of 1 can already cause a bias of up to 2 cm over a 4 km long path.

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Since the method of astronomical levelling only provides differences of height anomalies, an absolute level 2 has to be defined. In the comparison between the as1.5 trogeodetic quasigeoid profile (shown in Fig. 7) with the GPS-levelling data and the gravimetric quasigeoid 1 models, the absolute level is taken from the GCG05 0.5 model at the first station. The height anomalies are 0 shown in Fig. 8, while the corresponding differences 0 100 200 300 400 500 are illustrated in Fig. 9. Distance [km] Figure 9 shows that long wavelength differences Fig. 7 Astrogeodetic quasigeoid profile with a range of up to 14 cm occur between the astrogeodetic quasigeoid profile and the purely gravimetric quasigeoid heights are shown in Fig. 7. Incidentally, solution EGG07. For the combined GCG05 model, the the astrogeodetic solution provides only differences of differences decrease to a range of about 12 cm. Regarding the differences between the astrogeodetic profile height anomalies without any absolute datum. Before comparing the astrogeodetic results with the and the GPS/levelling control points (2005 data set), GPS/levelling data sets and the gravimetric quasigeoid two station groups can be distinguished, namely points models, the effect of different error sources in the ver- no. 1–4 and 5–20. When looking only at the Bavarian GPS/levelling points (pts. no. 5–20), a small longtical deflection data is discussed. If only random errors with a magnitude of 0.1 ex- wavelength difference exists, while the differences in ist in the vertical deflection data, an accuracy of 2.2 cm the northern section (pts. no. 1–4) appear to be quite can be expected for the quasigeoid profile over the full random. Between the northern and Bavarian section, a length of 540 km; this figure is resulting from simple jump of almost 5–6 cm shows up (between pts. no. 4 error propagation with uncorrelated observation errors, and 5). It should be pointed out, that this is exactly at neglecting any signal omission errors and systematic error components. Furthermore, from Eq. (2) it is obvi48 ous that in our case (North–South profile with azimuths 10 47.5 20 11 12 8 9 α near 180◦ ) the north-south vertical deflection compo7 5 6 47 13 nent ξ is more critical than the east-west component η. 4 19 46.5 3 14 In order to detect systematic and gross errors, dou16 18 15 2 46 ble observations have been carried out on 23 stations 17 1 in different nights. From Table 2 it can be seen that 45.5 ζQuasigeoid profile ζ GPS/levelling points ζGCG05 ζ EGG07 the mean values of the differences are not significantly 45 0 100 200 300 400 500 different from zero, and secondly, the maximum differDistance [km] ence of 0.27 is lying within the 3σ tolerance. Hence, no systematic difference could be detected between ob- Fig. 8 Astrogeodetic quasigeoid profile, GPS/levelling data set 2005, quasigeoid models GCG05 and EGG07 servations on different days.

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the border between the two federal states Thuringia and Bavaria, and this is also the region where the 2003 and 2005 GPS/levelling data show larger irregular discrepancies. When comparing with the 2003 GPS/levelling data, the jump between the northern and southern section reduces to about 3 cm (see also Sect 2.2 and particularly Fig. 3). Hence, in this region, some inconsistencies may exist in the GPS/levelling data sets. In order to prove that the jump in the differences between the 4th and 5th point is not caused by signal omission errors or systematic errors in the astrogeodetic solution, a densification of the astrogeodetic stations has been carried out in this region lately, but with no significant change in the results. Finally, the differences between the astrogeodetic profile and the GPS/levelling points are within ±1– 2 cm for each of the two station groups. This reveals a quite good consistency, considering the predicted accuracy for the astrogeodetic quasigeoid profile (2.2 cm), the selected station spacing of 3.9 km on average, and the stated accuracy of the GPS/levelling points (1–2 cm).

5 Summary and Conclusions In this contribution, a regional astrogeodetic quasigeoid profile has been computed by applying the method of astronomical levelling, using high-precision vertical deflections determined with the Hannover digital zenith camera system TZK2-D. With an accuracy of the vertical deflections of 0.1 , and an average station spacing of 3.9 km, a more than 500 km long profile has been attained with an accuracy at the cm level.

In comparison with a GPS/levelling data set from 2005, a jump of nearly 6 cm was found between Thuringia (4th point) and Bavaria (5th point), and a small long-wavelength difference exists for the Bavarian GPS/levelling stations (points no. 5–20). Hence, at the border between Thuringia and Bavaria some inconsistencies may exist in the GPS/levelling data sets which have to be further investigated. Acknowledgments This is publication no. GEOTECH – 304 of the R&D-Programme GEOTECHNOLOGIEN funded by the German Ministry of Education and Research (BMBF) and German Research Foundation (DFG), Grant 03F0422A.

References B¨urki B., A. M¨uller, H.-G. Kahle (2004). DIADEM: the new digital astronomical deflection measuring system for highprecision measurements of deflections of the vertical at ETH Zurich. IAG GGSM 2004, Porto, Portugal, CD-ROM Proceedings. B¨urki, B., M. Ganz, C. Hirt, U. Marti, A. M¨uller, P.V. Radogna, A. Schlatter, A. Wiget (2005). Astrogeod¨atische und gravimetrische Zusatzmessungen f¨ur den GotthardBasistunnel. Report 05-34, Bundesamt f¨ur Landestopographie (swisstopo), Wabern, Schweiz. Denker, H. (2006). Das Europ¨aische Schwere- und Geoidprojekt (EGGP) der Internationalen Assoziation f¨ur Geod¨asie. ZfV 131, pp. 335–344. Denker, H., J.-P. Barriot, R. Barzaghi, D. Fairhead, R. Forsberg, J. Ihde, A. Kenyeres, U. Marti, M. Sarrailh, I.N. Tziavos (2007). The Development of the European Gravimetric Geoid Model EGG07. IUGG XXIV General Assembly “Earth: Our Changing Planet”, Perugia, Italy, July 2–13, 2007, this volume. ESA (1999). From E¨otv¨os to milligal. Final report ESA/ESTEC, Contract No. 13392/NL/GD. Heiskanen, W.A., and H. Moritz (1967). Physical Geodesy. W.H. Freeman and Company, San Francisco.

420 Hirt, C. (2004). Entwicklung und Erprobung eines digitalen Zenitkamerasystems f¨ur die hochpr¨azise Lotabweichungsbestimmung. Wissenschaftliche Arbeiten der Fachrichtung Vermessungswesen der Universit¨at Hannover Nr. 253. Hirt, C., and J. Flury (2007). Astronomical-topographic levelling using high-precision astrogeodetic vertical deflections and digital terrain model data. J Geod Online First. DOI: 10.1007/s00190-007-0173-x. Krynski, J., and A. Lyszkowicz (2006). Centimetre quasigeoid modelling in Poland using heterogeneous data. Proceedings of the 1st International Symposium of the International Gravity Field Service, “Gravity Field of the Earth”, Harita Dergisi, Special Issue 18, pp. 37–42, Ankara, Turkey. Liebsch, G., U. Schirmer, J. Ihde, H. Denker, J. M¨uller (2006). Quasigeoidbestimmung f¨ur Deutschland. DVWSchriftenreihe, No. 49, pp. 127–146. Lysaker, D.I., O.C.D. Omang, B.R. Pettersen, D. Solheim (2007). Quasigeoid evaluation with improved levelled height data for Norway. J Geod 81(9): pp. 617–627 DOI: 10.1007/s00190-006-0129-6.

C. Voigt et al. M¨uller, A., B. B¨urki, H.-G. Kahle, C. Hirt, U. Marti (2004). First results from new high-precision measurements of deflections of the vertical in Switzerland. IAG Symposia 129, pp. 143–148, Springer Verlag. M¨uller, A., B. B¨urki, P. Limpach, H.-G. Kahle, V.N. Grigoriadis, G.S. Vergos, I.N. Tziavos (2006). Validation of marine geoid models in the North Aegean Sea using satellite altimetry, marine GPS data and astrogeodetic measurements. Proceedings of the 1st International Symposium of the International Gravity Field Service, “Gravity Field of the Earth”, Harita Dergisi, Special Issue 18, pp. 90–95, Ankara, Turkey. Torge, W. (2001). Geodesy. Third Edition. W. de Gruyter, Berlin, New York. Voigt, C., and H. Denker (2006). A study of high frequency terrain effects in gravity field modelling. Proceedings of the 1st International Symposium of the International Gravity Field Service, “Gravity Field of the Earth”, Harita Dergisi, Special Issue 18, pp. 342–347, Ankara, Turkey.

Insights into the Mexican Gravimetric Geoid (GGM05) D. Avalos, M.C. Santos, P. Van´ıcek and A. Hern´andez

Abstract The precise geoid determination technique from the University of New Brunswick (UNB) was adopted in Mexico to compute the national geoid model GGM05. To generate it, the input data was treated carefully and the theoretical background is well established. However, the final assessment gives unsatisfactory biases at level of metres. Some preliminary results of the corresponding research expose the possible sources for those large errors. The reference geoidal heights are the main suspect. Hence, in order to obtain a reliable assessment for GGM05, the issue of building better references has to be addressed. New developments like recent vertical movements modeling, rigorous orthometric heights estimation and precise positioning shall be combined to help removing uncertainties from the reference data, resulting in a better understanding of the Mexican gravimetric geoid.

1 Introduction

The Mexican Gravimetric Geoid (GGM) is an ongoing effort from INEGI, the federal institution in Mexico responsible for the national geodetic control. After many years depending on foreign institutions and technicians to obtain the national geoid model, the Mexicans opted to learn the UNB Stokes-Helmert technique for precise geoid computation (Van´ıcˇ ek et al., 1999) to make it themselves. The mathematics of this technique is well established, leading the computation of geoidal heights with uncertainty of 1 cm. Currently, the estimated accuracy of GGM05’s geoidal heights is 36 cm rms (INEGI, 2007). This value results form a comparison against 1377 point values of geoidal height derived from GPS/BM references using the well known formula N = h − H , where N, h and H correspond to geoidal, geodetic and orthometric heights. Regional and systematic Keywords Geoid assessment · Orthometric height · biases between GPS and BM observations have been Geoidal height detected, some reaching 2 m at the central part of the country, as indicated in Fig. 2. These differences are well beyond the reasonably expected since all height D. Avalos estimations were assumed to have centimetre accuracy. Department of Geodesy and Geomatics Engineering, University But some questions raised around the reference data of New Brunswick, 15 Dineen Drive, Fredericton, because it is already aged. The levelling observations New Brunswick, Canada, E3B5A3 used were made, in average, 30 years before the GPS M.C. Santos Department of Geodesy and Geomatics Engineering, University data were collected. This leads to a mandatory analof New Brunswick, 15 Dineen Drive, Fredericton, ysis of the computational process and the reference New Brunswick, Canada, E3B5A3 data used. P. Van´ıcek Preliminary results of the analysis are showing that Department of Geodesy and Geomatics Engineering, University time variation in heights is one of the main reasons for of New Brunswick, 15 Dineen Drive, Fredericton, discrepancies between GGM05 and the reference data. New Brunswick, Canada, E3B5A3 Since heights are continuously changing due to geoA. Hern´andez INEGI, Direcci´on General Adjunta de Geom´atica, Av. Nacozari dynamical processes and the surveying data cannot be Sur 2301, Aguascalientes, Mexico, 20270

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gathered in short periods of time, it is necessary to have geodetic tools to deal with time varying heights. Until now, the studies of recent vertical crustal movement on the Mexican territory have been devoted to small regions like city areas. These are not enough to explain long wavelength biases. About the input data, there has been an effort to guarantee the use of a reliable dataset of gravity values and to assemble the first continuous digital elevation model with high resolution for national coverage.

Fig. 1 Geoid Model GGM05 (courtesy INEGI, 2007)

2 The GGM05 Model was not available at that time, and one of the main enhancements is in the program to compute downward continuation of Helmert gravity anomalies. Now GGM05 is offered as official national geoid model, but its characteristics are not fully investigated yet. The GPS/BM assessment of GGM05 brings the map of differences between geoidal heights at 1377 points on the Mexican area is presented in Fig. 2. Surprisingly the differences grow up to 2 m at some regions when it was expected to have few decimetres in the worse case. Research is now taking place to find out the causes for such a high difference between the two sources in order to answer the next questions: where are the errors coming from?, is that difference a real error in GGM05 model?, if not, then what is the best feasible way to perform a reliable evaluation in the Mexican scenario?. The first step to accomplish this research is gathering evidence which help discriminating error sources and its possible magnitude. Some preliminary results are exposed in Sect. 3 and they bring the issues to be faced in further stages. The second step is to design

Digital elevation models of free air gravity anomaly, lateral density of topography and orthometric height are the main input in the Stokes-Helmert technique for geoidal height computation. From the beginning it was a challenge to obtain these models in the Mexican case, since the available data did not qualify in coverage and specifications. A strong campaign to accelerate gravity data collecting and building the homogeneous models of lateral density and orthometric heights with nationwide coverage was organized. Then the final input data set for GGM05 was released after discriminating outliers and suspicious values which could not be validated with the resources inside INEGI. The coverage, distribution, and general quality of gravity observations was greatly enhanced, but still not the ideal: there are at least one observation in every cell 2.5 min in latitude and longitude. Other data sources used to complete the input was the low frequencies of EIGEN2 geopotential model for reference spheroid, TOPEX-POSEIDON altimetry to derive free air gravity anomaly at the sea, and GRIM4 global topographic elevation model for computing for zone contributions to the gravity field. GGM05 model has estimated geoidal heights ranging between −48 and +6 m. The resolution is 2.5 min in latitude and longitude and coverage from 86 to 119◦ west in longitude and 14 to 33◦ north in latitude, as illustrated in Fig. 1. The computations to obtain GGM05 follow the UNB technique, Helmertizing the mean free air gravity anomalies to perform downward continuation to the cogeoid and evaluate the Stokes formula. Data processing was performed using SHGEO package version 2001 from the geodesy research group at UNB (Janak, 2001). The newest version of the software Fig. 2 Difference in geoidal height, GPS/BM – GGM05

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an appropriate set of geodetic tools to assess geoidal heights in the present situation. We believe that it is feasible to apply recent developments and concepts to disadvantageous situations to obtain reliable and more accurate assessments.

3 On the Assessment of Geoidal Heights The effort to investigate error sources began looking for geoidal height estimations independent from the GPS/BM. Previous geoid models with similar resolution and coverage, like MEX97, GEOID03 and EGM96 (all made by institutions in the United States of America), were compared to GGM05. In this case the results suggest that systematic errors may be affecting the GGM05 especially along the central part of the country, which coincides with the GPS/BM reference. The accuracy of these previous models is highly correlated since they use basically a common gravity data base. However, the quality of gravimetric data used for the Mexican territory was highly deteriorated (Lemoine et al., 1998), and the final accuracy of their geoidal heights over Mexico is difficult to ascertain (Smith, 2005). We support the idea that GPS observations on benchmarks (GPS/BM) can be the best way to estimate geoidal heights, but there are many issues to consider in levelling and GPS techniques so that they become reliable reference at centimetre level. Observing the data used for GPS/BM it can be noticed that a significant difference in epoch for data collecting exists. The GPS measurements of geodetic heights were made between years 2001 and 2004, while levelling comes from the NAVD88 network, which is a Helmert approximation to orthometric height, observed between 1950 and 1980.

Fig. 3 Subsidence for Aguascalientes city. Units: cm/year (Courtesy R. Esquivel)

model of vertical velocities (h) show estimated geodetic heights decreasing from 2 to 18 cm per year (Esquivel et al., 2004). Figure 3 shows the contour lines of subsidence over Aguascalientes. For an insight about the velocities in long wavelength we took advantage of a long relevelled segment across the central region in Mexico and passing over Aguascalientes. This line was originally observed during the 50’s and the newest observation was made from 2001 to 2005. In order to compare height differences, the Helmert approximate orthometric height from NAVD88 was propagated from one benchmark taken arbitrarily. At the end of the line, benchmark QT136 was chosen as reference for its position is far from areas with known high dynamicity. Hence the height differences obtained are interpreted as mainly vertical movements relative to QT136. These differences are presented in Fig. 4, ranging from +45 to −130 cm. Figure 4 illustrates the changes in shape of the profile sketched by Fig. 5. Some systematic trends can be noticed, which suggest that NAVD88 network is no longer compatible with recent observations. But there is a sudden change of sign in the height differences 4 Time Variation of Height over the area of Aguascalientes. On the side of geodetic height estimations from For Mexicans it is a well known fact that height GPS, the methodology implemented L1 and L2 frechanges in time since it can be noticed visually at quencies observations over periods of 3 h. Differenmany places. As an example of the expected magni- tial post processing, associated to reference stations in tude of recent vertical crustal movements, we quote Mexico was performed then. However, no velocities the study about subsidence devoted to the urban area on the reference stations were considered. This arise of Aguascalientes. Combining relevelled segments and the question of probable systematic biases added to the GPS observations over years 2003–2004, the derived geodetic heights.

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modeling in Mexico. Its theoretical background supports centimetre accuracy but the existing tools for the final assessment cannot guarantee similar level of uncertainty. It is important to know all of the characteristics of geoid models like GGM05, but the lack of precise orthometric height estimations prevents us from performing the GPS/BM combination to obtain the precise geodetic heights needed for the assessment. Mexico seems to be a nice playground for the study of vertical movements due to its tectonic dynamicity. The new results give some evidence that vertical crustal movements should be regarded as one of the main factors of bias for old levelling. Hence, analysis is enforced to model and learn about the magnitude of recent vertical crustal movements. This implies the creation of a model of vertical crustal movements as detailed as possible, making use of relevelled lines, sea surface topography estimations, tide gauge and geological information. We believe that a reliable assessment of GGM05 at few centimetres level is possible after coupling the adequate geodetic tools. Besides regarding the vertical dynamicity it can be implemented an approximation to rigorous orthometric heights (as proposed by Kingdon et al., 2005), taking advantage of by-products created during the Fig. 4 Differences in the Helmert orthometric height, relative to GGM05 computations. On the side of GNSS processBMQT136 ing, since the realistic accuracy of geodetic heights obtained might be easily biased by more than one centimetre, we shall address revision to the methodology to guarantee that error sources are treated as best as possible. An alternative means of estimating absolute vertical crustal movements is GNSS data processing in Precise Point Positioning mode for reference stations. A time series over the last 10 years may provide useful information which could be incorporated as part of the model of vertical crustal movements. It should be kept in mind that an error of about 10% is introduced when geodetic height velocities are used to model orthometric heights (Van´ıcˇ ek, 1986). In order to determine the difference between geodetic and orthometric height velocities, knowledge on the geoidal height velocity would be also required. Fig. 5 Sketch of the relevelled segment

5 Conclusions and Future Work The GGM05 model is a national vertical reference made from the best quality data ever used for geoid

Acknowledgments Support for this research is provided by the GEOIDE Centre of Excellence, the Mexican National Council of Science and Technology (CONACyT), and by the main data provider National Institute of Statistics, Geography and Informatics (INEGI), Mexico.

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References Esquivel, R., Hern´andez A., Zerme˜no M.E. (2004). An´alisis geod´esico de la subsidencia en la Ciudad de Aguascalientes. In: Revista Cartogr´afica del IPGH No. 78–79, Mexico. INEGI, Direcci´on General de Geograf´ıa (2007). Modelo geoidal GGM05. Metodolog´ıas y Sistemas de Consulta, http://www.inegi.gob.mx. Janak, J. (2001). Stokes-Helmert’s GEOid software Reference Manual. A technical report of the Department of Geodesy and Geomatics Engineering, University of New Brunswick, Canada. Kingdon R., Van´ıcˇ ek P., Santos M. C., Ellmann A., Tenzer R. (2005). Toward an Improved Orthometric Height System for Canada. Geomatica Vol. 59, No. 3. Lemoine F.G., Kenyon S.C., Factor J.K., Trimmer R.G., Pavlis N.K., Chinn D.S., Cox C.M., Klosko S.M., Luthcke S.B., Torrence M.H., Wang Y.M., Williamson R.G., Pavlis E.C.,

425 Rapp R.H., Olson T.R. (1998). The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. NASA/TP1998-206861, July 1998. Santos M. C., Van´ıcˇ ek P., Featherstone W., Kingdon R., Ellmann A., Marin B-A., Kuhn M., Tenzer R. (2005). The Relation Between Rigorous and Helmert’s Definitions of Orthometric Heights, Journal of Geodesy 80:691–704, on line DOI 10.1007/s00190-006-0086-0. Springer-Verlag 2006. Smith D. A. (2005). Mexico geoid heights (MEXICO97), NOAA/National Geodetic Survey of the United States of America, http://www.ngdc.noaa.gov. Van´ıcˇ ek, P., Krakiwsky E.J. (1986). Geodesy: The Concepts. Elsevier 2nd rev. ed., North-Holland, Amsterdam. Van´ıcˇ ek, P., Huang J., Nov´ak P., Pagiatakis S., V´eronneau M., Martinec Z., Featherstone W.E. (1999). Determination of the Boundary Values for the Stokes-Helmert Problem. Journal of Geodesy, Springer Berlin, Vol. 73, No. 4, pp. 180–192.

An Improved Geoid in North Eastern Italy P. Sterzai, F. Coren, N. Creati, I. Marson and M. Maso

Abstract An improved geoid, called ADBVE2006 (Autorita di Bacino di Venezia 2006), has been calculated for the north-east of Italy using new land and marine gravity data together with new high resolution multibeam bathymetric data. A standard processing procedure has been applied to gravity data in order to compute the Free Air and the Bouguer anomalies. The gravity reference datum is IGSN71. The long wavelenghth part of the gravity field has been modelled by EGM96. The computation has been carried out by the “remove-restore” spectral technique (Stokes’ approach) using the software GRAVSOFT. The goal of the calculated model is to improve the geoid estimation in the coastal areas, where the calculated geoids usually suffer from the lack of gravimetric data as well as a good bathymetry.

Keywords Gravity · Geoid · North eastern Italy

1 Gravity Dataset The gravity data on land have been acquired with a LaCoste-Romberg model D-018 gravimeter with ZLS (Zero Length Spring) feedback during a terrestrial survey. The land gravity data consist of 641 new points and the OGS (Istituto Nazionale di Oceanografia e di Geofisica Sperimentale, www.inogs.it) Italian gravity database. So far, gravity data were missing for the Adriatic Sea, especially along the coast. In 2005, during an OGS sea survey exploration, a large part of the Adriatic Sea data gap has been filled with new gravimetric data. Also bathymetric high resolution multibeam data have been acquired, see Figure 1. The final dataset with of more than 28 500 points has an approximate distribution of one point every 500 m2 . A geographic grid with a spacing of 0.02 over an area ranging from 44.75 to 46.75◦ in latitude and from 10.70 to 14.00◦ in longitude has been created. The Free-Air and complete Bouguer gravity anomalies have been calculated using the standard density value of 2670 kg/m3 .

P. Sterzai Istituto Nazionale di Oceanografia e Geofisica Sperimentale, Borgo Grotta Gigante 32/c, 34010, Sgonico (TS), Italy F. Coren Istituto Nazionale di Oceanografia e Geofisica Sperimentale, Borgo Grotta Gigante 32/c, 34010, Sgonico (TS), Italy N. Creati Istituto Nazionale di Oceanografia e Geofisica Sperimentale, Borgo Grotta Gigante 32/c, 34010, Sgonico (TS), Italy I. Marson Istituto Nazionale di Oceanografia e Geofisica Sperimentale, Borgo Grotta Gigante 32/c, 34010, Sgonico (TS), Italy and Universit`a di Trieste M. Maso Autorit`a di Bacino dei Fiumi dell’Alto Adriatico, Dorsoduro 3593, 30123, Venezia, Italy

2 The ADBVE 2006 Geoid The calculation has been done using the software package GRAVSOFT, a program developed by the Department of Geophysics of Copenhagen (Tscherning et al., 1992). ADBVE2006 has been calculated using the well known “remove-restore” procedure (Schwarz et al., 1990; Groten, 1980). The procedure can be summarized as follows:

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Fig. 1 Gravimetric measurements used for the ADBVE2006 calculation

removed and substituted with data from a checked GTOPO30 digital elevation model. A Gaussian smoothing filter has been applied to merge the DEMs. The final gravimetric geoid has been fitted in order to minimize the difference between the gravimetric geoid and the GPS/leveling geoid called GEOTRAV accepted as ground truth (Morelli, 2001). This fitting has been done using an iterative scheme. First the geoid ondulation is calculated trough a bilinear interpolation at known GPS/leveling points. A difference matrix is created subtracting the GPS/leveling geoid ondulation to the gravimetric geoid ondulation. This difference matrix is interpolated by a least-square weighted collocation method on a regular grid and added to the initial geoid gravity model. The difference between the new hybrid gravimetric geoid model and the GPS/leveling The global geopotential model EGM96 (Lemoine geoid ondulation is calculated again. If there is still an anomalous difference the whole procedure is repeated et al., 1998) has been used in the computation. The height model used for the compu- iteratively till a good solution is found. Figure 3 shows the difference between the ADtation is the global elevation model SRTM BVE2006 geoid and the corresponding benchmarks (http://www2.jpl.nasa.gov/srtm/). Since the SRTM data show some errors or gaps especially in areas with of the IGM (Istituto Geografico Militare). The differhigh slopes (Alps) the original SRTM data have been ences ranges from −0.25 to 0.26 m with a standard 1 – Remove a gravity anomaly field (global spherical harmonic model evaluated at the geoid) from Helmert gravity anomalies based on local gravity measurements and digital elevation data. This leaves “residual gravity anomalies” (Rapp, 1993). Free-air anomalies are used to approximate Helmert gravity anomalies. 2 – Compute “residual co-geoid undulations” by a spherical Fourier representation of the Stokes integral (Sjoberg, 1986). 3 – Restore a geoid undulation field (spherical harmonic model evaluated at the geoid) to the “residual co-geoid undulations” and add the indirect effect (computed from digital elevation data) to form the geoid undulations.

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Fig. 2 The ADBVE2006 geoid

deviation of 0.11 m. This new high resolution geoid allows to create accurate hydrological models and to transform ellipsoidal to orthometric heights with a high accuracy (better than 0.13 m). The standard deviation of the computed ADBVE2006 geoid is 11 cm if compared with GEOTRAV (Morelli, 2001), measurements (Fig. 3). Looking at Fig. 3, it is possible to spot that the error distribution is not homogeneous since the higher values are located along the Italian national borders and in mountain areas. Anyway, the geoid misses some input data from the eastern area (towards Slovenia because the gravimetric database is not yet available) and from the north where the rugged terrain that is a characteristic of the alpine area prevents a large scale data acquisition due to logistics issues. Obviously this implies a lower accuracy of the calculated geoid for those areas. Nevertheless the ADBVE2006 geoid can be considered of good quality, especially in central flat areas as well as along the coastal regions, where the impact on anthropic activity is higher.

The geoid model has also been implemented into specific coordinate conversion software that has been developed by OGS specifically for the ADBVE. The software named Geoid Corrected Coordinate Converter has embedded the geoid ADBVE2006 matrix at the resolution of 0.02 and allows to transform planar and altimetric coordinates between Roma 40 datum to WSG84 and vice versa. This program, written in Python uses the Proj.4 libraries; it adopts a new computed transformation matrix between the WGS84 and Roma40 system, avoiding the need of roto-translation parameters.

3 Conclusions A new improved local geoid in North-Eastern Italy has been calculated using an integration of gravimetric land measurements from the OGS Italian gravity database that has been extended with other 641

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Fig. 3 The standard deviation of the ADBVE2006 geoid compared with GEOTRAV ground control points, units are in meters

new acquired points on land and a new marine dataset collected by the OGS’s research vessel R/V OGS Explora by means of a Bodenseewerke marine gravimeter; this dataset is also comprehensive of multibeam profiles that have been used to enhance the bathymetric model. This wide integration of dataset allows to compute a high resolution geoid model called ADBVE2006 that is almost continuous in the transition between land and marine domains and grants a high accuracy along coastal areas. The new geoid has been tested and verified by Prof. R. Barzaghi from Politecnico di Milano, whose cooperation is thankfully acknowledged.

References Groten, E. (1980). Geodesy and Earth’s Gravity Field, Duemmler and Bonn. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, R.H. Rapp, and T.R. Olson (1998). The development of the

joint NASA/GSFC and the National Imagery and Mapping Agency (NIMA) geopotential models, EGM96, NASA. TP1998-206861, 575 pp. Morelli, M. (2001). Il modello altimetrico GEOTRAV98 un’utile soluzione operative (The altimetric model GEOTRAV98: a useful operational tool). Bollettino di Geodesia e Scienze Affini (Boll. Geod. Sci. Affini) ISSN 0006-6710 CODEN BGSAA9, 2001, Vol. 60, No 1, pp. 67–76 (6 ref.) Rapp, R.H. and Y.M. Wang (1993). Geoid undulation differences between geopotential models. Surveys in Geophysics, Vol. 14, No 4–5, pp. 373–380. Schwarz, K.P., M.G. Sideris, and R. Forsberg (1990). The use of FFT techniques in physical geodesy. Geophysical Journal International, 100, pp. 485–514. Sjoberg, L.E. and H. Fan (1986). A comparison of the modified Stokes formula and Hotine’s formula in Physical Geodesy. Report n.4, the Royal Institute of Technology, Stockolm. Tscherning, C.C., R. Forsberg and P. Knudsen (1992). The GRAVSOFT package for geoid determination. Proc. 1. Continental Workshop on the Geoid in Europe, Prague, May 1992, pp. 327–334.

Towards a New Global Digital Elevation Model P.A.M. Berry, R.G. Smith, J.A. Freeman and J. Benveniste

Abstract The SRTM dataset provides the first nearglobal elevation model of the Earth’s continental land surface at 3 resolution. This unique dataset is currently used for a range of applications. However, there are a number of known quirks and omissions in these data, such as voids over inland water and erroneous heights from forest canopy; in addition a recently completed global analysis of these data utilising a dataset derived from multi-mission satellite radar altimetry has revealed spatially correlated error signatures possibly caused by incorrect heights in ground control points used to seed the solution (Berry et al., Remote Sensing of Environment, 106(1): 17–27). This paper presents the initial results from work currently underway towards an enhanced global elevation dataset, ACE2, a follow-on to the highly successful ACE GDEM. This development utilises over 67 million height measurements derived from multimission satellite radar altimetry using an expert system approach to validate and augment the SRTM dataset to derive a full global GDEM. One of the key results is that in many regions the SRTM dataset is more accurate than its stated vertical precision of +/ − 16 m.

P.A.M. Berry EAPRS Laboratory, De Montfort University, Leicester, UK, LE9 1BH R.G. Smith EAPRS Laboratory, De Montfort University, Leicester, UK, LE9 1BH J.A. Freeman EAPRS Laboratory, De Montfort University, Leicester, UK, LE9 1BH J. Benveniste European Space Agency, Earth Observation Applications Department, ESRIN, Via Galileo Galilei, I-00044, Frascati, RM, Italy

1 Introduction The SRTM mission collected a unique dataset which was utilized to produce a near-global Digital Elevation Model at 3 arcsec resolution (USGS, 2005). This unique dataset provided the first high resolution elevation data over much of the earth’s land surface. However, the data suffered from a number of artifacts, resulting from a combination of instrument and processing constraints (e.g. Smith & Sandwell, 2003: Berry et al., 2007). This, together with the voids in the DEM where terrain interactions with the active radar illumination precluded the production of height information (such as inland water) have limited the utility of the SRTM dataset for some applications. Results from a number of independent regional validations against ground truth (e.g. Denker, 2004; Kellndorfer et al., 2004; Kiamehr & Sj¨oberg, 2005; Koch & Heipke, 2001) confirm both the good accuracy of much of this dataset and the presence of spatially correlated anomalies. Accordingly, it was decided to investigate whether the SRTM dataset could be effectively enhanced by merging with a unique database of altimeter generated heights with near-global distribution, derived from the ERS-1 Geodetic Mission (Capp, 2001). This dataset, derived by reprocessing the Altimeter echoes from the Geodetic Mission using an expert system approach (Berry et al., 1997; Hilton et al., 2003), produced over 67 million height measurements with an average across-track spacing of 4 km and an along-track sampling of 350 m (Capp, 2001). Following an initial analysis of the SRTM DEM (Berry et al., 2007) it was decided to work towards fusing these datasets, supplemented by data from ERS-2, EnviSat, Topex and Jason-1, to create a new

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enhanced GDEM, ACE2, at the full 3 resolution. The EGM 96 geoid is used throughout to transform the altimeter ellipsoidal heights into estimates of orthometric heights, to be consistent with the SRTM model (Lemoine et al., 1998).

2 Methodology The first step in developing a fused GDEM is the detailed analysis of the two datasets; the outcomes from this analysis are the subject of this paper. After retracking the entire ERS-1 Geodetic Mission dataset, and subsetting those points lying within the SRTM latitude limits, a first comparison was performed on every pixel of the SRTM dataset overflown by the altimeter to give an overview of the agreement between these two independent measurements: detail on the processing methodology and initial global results have previously been published (Berry et al., 2007). Following on from this, a more detailed spatial analysis was undertaken. In order to assess where altimeter derived heights could improve the SRTM dataset, the percentage of altimeter echoes successfully acquired by the ERS-1 Geodetic Mission was calculated globally. All of the areas with less than 50% of the theoretical maximum number of waveforms acquired within a 30 × 30 arcsec area were excluded. This constrained the analysis to those regions where the density of altimeter data was such that a very comprehensive analysis of the SRTM data was possible (55,212,947 altimeter points covering 71% of the continental land mass within the SRTM latitude limit). Every SRTM pixel overflown by the altimeter was then compared with the corresponding Altimeter height measurements. The results are shown in Table 1, banded by height difference. Table 1 Global height difference statistics (SRTM – Altimeter)

Height difference range (SRTM – Altimeter) (m) −56 to −40 −40 to −24 −24 to −8 −8 to +8 8 to 24 24 to 40 40 to 56 Total

To examine the spatial distribution of these differences, the difference dataset was binned into 7.5 arcmin bins, and the average difference calculated. A global plot of these results is shown in Fig. 1. It is clear that over a large proportion of the world’s land surface, there is a very good agreement between the two instruments with a significant proportion of these two datasets showing agreement to better than ±8 m. One clear area of disagreement occurs over the rainforests, where the SRTM retrieves heights from within the upper canopy whilst the altimeter penetrates to the underlying ground (Berry et al., 2007). Note that the apparent discrepancy over the Amazon river network in Fig. 1 is in fact a consequence of the binning up of the data for plotting: small parts of the surrounding rain forest have been included in the ‘river’ pixels, which has caused an apparent height bias. A detailed examination of the Amazon river network heights with the SRTM heights however confirms that the representation of these heights within the SRTM are better than shown in Fig. 1. However, the general representation of inland water is variable within the SRTM dataset; this is the subject of a separate detailed investigation currently underway. It is currently planned that the majority of the heights over inland water will be replaced by altimeter derived annual mean height values in ACE2. Most of Australia shows extremely good agreement, and an almost complete assessment is possible due to the relatively flat nature of the majority of the terrain and therefore the extremely good altimeter coverage. Elsewhere coherent spatial patterns of agreement are seen in the flatter regions where the altimeter coverage is good, with generally good agreement over much of the assessed surface. The results in Table 1 and Fig. 1 show that the difference between the two datasets is less than the SRTM stated accuracy (±16 m) for 77% of those areas where the altimeter coverage is better than 70%

Number of points 611,340 1,331,598 4,899,870 33,485,657 9,176,845 3,890,974 1,816,663 55,212,947

Percentage of total in range (−56 to +56 m) 1.1 2.4 8.9 60.6 16.6 7.0 3.3

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Fig. 1 A comparison of the SRTM dataset with ERS-1 Geodetic Mission heights globally, for areas with near complete Altimeter height coverage, binned to a resolution of 7.5 . Excluded areas are shown in white Table 2 Global height difference Statistics fine-scale (SRTM – Altimeter)

Height difference range (SRTM – Altimeter) in metres −16 to −10 −10 to −6 −6 to −2 −2 to 2 2 to 6 6 to 10 10 to 16 Total

of the theoretical maximum. Statistics were therefore generated (Table 2) to examine these areas in more detail; this revealed that over a significant proportion of the assessed SRTM values the agreement between the SRTM heights and the altimeter heights is extremely good. Accordingly, a spatial plot was created for all pixels where the difference was less than ±16 m (Fig. 2). Surprisingly this revealed a high spatial correlation in these ‘good’ regions, with many areas where the SRTM heights are accurate to within ±6 m. Intricate spatial patterns of differences are evident, the causes of which are currently under investigation. Of particular interest are the very distinct patterns emerging over the Sahara desert in Northern Africa and the Kalahari Desert in Southern Africa; these are in part due to differences in the way that the two instruments perceive dune fields. This assessment was then used to build a quality matrix by grading the differences between the altimeter and the SRTM heights more finely

Number of points 2,192,632 3,083,826 8,122,657 12,434,156 8,582,975 4,365,307 3,876,280 42,657,833

Percentage of total in range (−16 to +16 m) 5.1 7.2 19.0 29.1 20.1 10.2 9.1

for the full range of differences including those within the SRTM stated accuracy.

3 Next Steps In order to create the new fused ACE2 DEM for areas outside of the SRTM latitude limits, a fusion of ERS1/2, EnviSat, Jason-1 and TOPEX data will be used to create the best possible height model. Assessment of this primarily flat topography has already shown that very good altimeter coverage is obtained over these regions, which facilitates this part of the work. Where voids occur for inland water, the heights will be determined by using the database of altimeter derived River and Lake heights (Berry et al., 2005). Over those regions where the rough topography precludes satisfactory coverage from the ERS-1 Geodetic Mission,

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Fig. 2 A comparison of the SRTM dataset with ERS-1 Geodetic Mission heights globally, for areas within the SRTM stated accuracy of ±16 m. Excluded areas are shown in white

a profile analysis will be performed using the EnviSat RA-2 data (Benveniste et al., 2002); this instrument dynamically switches between three modes of operation in order to maintain lock on the underlying terrain, and therefore successfully captures data in all but the most severe terrain. The results of these comparisons will be used to populate the quality matrix, and may also be used to correct apparent anomalies in the vertical and/or horizontal referencing of the SRTM DEM on a case by case basis.

both vertical and horizontal misplacement. The projected release date for the new Global Model, ACE2, is November 2008. Acknowledgments The authors would like to thank ESA for providing the ERS-1/2 and EnviSat data and JPL for providing the SRTM dataset and TOPEX waveform data. Jason-1 data were kindly supplied by AVISO.

References 4 Discussion This initial analysis has shown that over many areas where the ERS-1-GM has good coverage the SRTM is performing well within its stated accuracy (for 77% of these areas) and in some areas shows vertical precision of better than 6 m. The SRTM values for these areas will be passed through unchanged to ACE2 and detailed information on the differences will be put into an associated quality matrix. The next step is to investigate areas where significant differences occur, and altimeters are known to have good coverage, such as over the rainforests; here the SRTM data may be replaced with a DEM created from altimeter heights. For many areas it is expected that the SRTM data will be fused with control arcs of altimeter data to assist with

Benveniste et al., 2002; J. Benveniste, ENVISAT RA-2/MWR product handbook, Issue 1.2, PO-TN-ESR-RA-0050, European Space Agency, Frascati, Italy. Berry P.A.M., Jasper A., Bracke H., 1997; Retracking ERS-1 altimeter waveforms over land for topographic height determination: an expert system approach, ESA Pub. SP-414, Vol. 1, 403–408. Berry P.A.M., Garlick J.D., Freeman J.A. and Mathers E.L., 2005; Global inland water monitoring from multi-mission altimetry, Geophysical Research Letters, 32 (16), L16401, DOI: 10.1029/2005GL022814. Berry P.A.M., Garlick J.D., Smith R.G., 2007; Near-global validation of the SRTM DEM using satellite radar altimetry, Remote Sensing of Environment, 106(1) Pages 17–27, DOI: 10.1016/j.rse.2006.07.011 Capp P., 2001; Altimeter Waveform Product ALT.WAP compact user guide, Issue 4.0, PF-UG-NRL AL-0001, Infoterra Ltd, UK. Denker H., 2004; Evaluation of SRTM3 and GTOPO30 terrain data in Germany, International Association of Geodesy

Towards a New Global Digital Elevation Model Symposia, Gravity, Geoid and Space Missions, V129, ed C. Jekeli et al., pp 218–223, Springer Verlag, Berlin. Hilton R. D., Featherstone W. E., Berry P. A. M., Johnson C. P. D., Kirby J. F., 2003; Comparison of digital elevation models over Australia and external validation using ERS1 satellite radar altimetry, Australian Journal of Earth Sciences, 50(2), April, pp. 157–168(12), DOI:10.1046/j.1440 0952.2003.00982.x Kellndorfer J.M. et al, 2004; Vegetation height estimation from Shuttle Radar Topography Mission and National Elevation Datasets, Remote Sensing of Environment, 93; 339–358. Kiamehr, R., Sj¨oberg, L.E., 2005; Effect of the SRTM Global DEM on the determination of a high resolution geoid model; a case study in Iran, Journal of Geodesy, 79(9), p 532, December. Koch A., Heipke C., 2001; “Quality assessment of digital surface models derived from the Shuttle Radar Topography

435 Mission (SRTM), Geoscience and Remote Sensing Symposium, 2001, IGARSS ’01, IEEE 2001 International, vol. 6, no.pp. 2863–2865. Lemoine F.G., Kenyon S.C., Factor J.K., et al., 1998; The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP 1998–206861. Smith B. & Sandwell D., 2003; Accuracy and resolution of Shuttle Radar Topography Mission Data, Geophysical Research Letters 30(9), 1467. USGS, 2005 USGS., Shuttle Radar Topography Mission (SRTM)—“Finished” products, U.S. Geological Survey. URL: http://edc.usgs.gov/products/elevation/srtmbil.html, (last date accessed: 30 June 2005).

Symposium GS003

Earth Rotation and Geodynamics Conveners: V. Dehant, C.L. Huang

Contribution of Non-Tidal Oceanic Mass Variations to Polar Motion Determined from Space Geodesy and Ocean Data ¨ and F. Seitz F. Gottl

Abstract In order to assess the contribution of nontidal oceanic mass changes to polar motion, equatorial oceanic excitation functions are determined from combinations of space geodetic techniques and ocean data. Satellite altimetry provides accurate information on sea level anomalies (SLA) which are caused by mass and volume changes of seawater. Since Earth rotation is solely affected by mass variations the volume (steric) effect has to be reduced from the observations in order to infer oceanic contributions to Earth rotation. Oceanic polar motion excitations from reduced SLA are compared with respective results from ocean models. Contributions of ocean currents, atmospheric and hydrological effects are added in order to validate the oceanic excitations from altimetry with independent geodetic observations which reflect the integral effect of a multitude of geophysical processes in the Earth system. This requires an investigation of accuracy and consistency of the combined data sets. The study reveals that model-only combinations of atmospheric, oceanic and hydrological excitations agree better with geodetic observations than combinations which include excitations from altimetry. Possible reasons could be errors in the steric reduction of SLA and/or the compensation of erroneous patterns in atmosphere data by numerical ocean models. Keywords Oceanic mass variations · Polar motion · Satellite altimetry · Steric effect F. G¨ottl Deutsches Geod¨atisches Forschungsinstitut (DGFI), Alfons-Goppel-Strasse 11, D-80539 Munich, Germany F. Seitz Earth Oriented Space Science and Technology (ESPACE), Technische Universit¨at Munchen, Arcisstr. 21, D-80290 Munich, Germany

1 Introduction Observation techniques of the motion of the Earth rotation axis with respect to the Earth’s surface have continuously improved over the last decades. While optical astrometry allowed for observations of polar motion with an accuracy of about one meter, modern space geodetic techniques such as Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR) and Global Navigation Satellite Systems (GNSS) provide results at a millimetre level. However the geophysical interpretation of the time series in terms of contributions from particular geodynamic processes is a big challenge since the observations reflect the integral effect of redistribution and motion of masses within and between the individual subsystems of the Earth. Large effort has been made in the assessment of mass transports in atmosphere, hydrosphere and solid Earth in interdisciplinary research from geophysical modelling as well as from terrestrial and space borne observations. Strictly speaking, geodetic observations of polar motion reflect the motion of the Earth rotation axis with respect to the Terrestrial Reference Frame (TRF). Consequently systematic errors in TRF computations might show up as artificial signals in polar motion time series. Therefore an improved geophysical understanding of dynamic processes and their interactions is important in order to obtain independent estimates of Earth rotation variations. In this article we discuss results of oceanic polar motion excitation functions which have been derived from satellite altimetry observations. Different data sets for the reduction of steric sea level variations (volume changes due to changes of temperature and salinity) are applied in order to separate the effect of

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SLA are provided with a temporal resolution of 10 days. They are averaged to monthly mean fields in order to be consistent with other data sets as explained below. As mentioned above, Earth rotation is affected by mass variations but not by volume changes of seawater. Therefore the steric effect has to be reduced from observed SLA. It is estimated from physical ocean parameters such as temperature and salinity changes in different depth layers of the ocean. Fofonoff and Millard (1983) give a detailed explanation about transient density anomalies using the equation of state of seawater. Vertical integration of density anomalies within a water column yield steric sea level variations; for details see Chambers (2006). Here we apply numerical corrections of the steric effect as provided from Ishii et al. (2006). Alternatively we use monthly three 2 Data Sources dimensional temperature and salinity fields which are based on objective analyses. They are offered from the In this study all analyses are performed on the basis of National Oceanographic Data Centre (NODC) as papolar motion excitation functions. Excitation functions rameters of the World Ocean Atlas 2005 (WOA05) are the mathematical description of geophysical effects (Antonov et al. (2006); Locarnini et al. (2006)). After the steric corrections have been applied, on Earth rotation. They comprehend angular momentum variations due to changes of the Earth’s inertia ten- the reduced SLA are interpreted as oceanic mass sor (mass term) and due to motions of masses with re- signals which are expressed in equivalent water spect to the reference frame (so-called relative angular heights (EWH). Those are subsequently converted momenta; motion term). The advantage of an evalua- into polar motion excitations (mass term) χ1O and χ2O tion in the excitation domain is the absence of free os- (units: rad = 180◦ /π · 3600 · 1000 mas) by cillations of the Earth such as the Free Core Nutation  2π π ewh 1 and the Chandler oscillation. C nm ewh Q P nm (cos θ Q ). ewh = 4π 0 S nm 0  " cos mλ Q (1) sin θ Q dθ Q dλ Q , sin mλ Q 2.1 Oceanic Polar Motion Excitations ocean mass redistributions from sea level anomalies as observed by the TOPEX/Poseidon Extended Mission. The oceanic mass excitations are subsequently compared with respective time series from the ocean models OMCT (Ocean Model for Circulation and Tides) and ECCO (Estimating the Circulation and Climate of the Ocean). Further we compare the total effect of atmospheric, oceanic and hydrological excitations with the geodetic observations of polar motion provided by the International Earth Rotation and Reference Systems Service (IERS) in its C04 series. The comparisons will be performed for model-only combinations as well as for combinations in which the oceanic mass effect is derived from altimetry.

Oceanic polar motion excitations are determined from combinations of satellite altimetry and ocean data from hydrographical observations and models. Satellite altimetry provides accurate information on SLA with respect to a mean sea surface which are caused by mass and volume changes of seawater. At Deutsches Geod¨atisches Forschungsinstitut (DGFI) SLA are estimated from observations of the TOPEX/Poseidon Extended Mission which have been reduced by environmental and geophysical effects such as troposphere, ionosphere, sea state bias and tides. Additionally the classical inverse barometer (IB) correction has been applied. The latter accounts for the hydrostatic response of the ocean surface to atmospheric pressure variations.

 " ewh −0.684 3ρw C nm C nm · = ewh S nm 2n + 1 Rρ e S nm

(2)

and χ1O χ2O

"

1.608R 2 Me · = C−A ewh

#

 5 C 21 . · S 21 3 ewh

(3)

In these equations C nm and S nm denote normalized coefficients of the spherical harmonic expansion of the EWH (units: m). C nm and S nm are dimensionless normalized Stokes coefficients; λ Q and θ Q stand for the spherical geocentric coordinates of the

Contribution of Non-Tidal Oceanic Mass Variations to Polar Motion

 1.608 OAM 1 , · = 0.684 ·

(C − A) OAM 2  " 1.608 χ1o oam1 · = χ2o

(C − A) oam2

χ1O χ2O

"

(4) (5)

and "  O χ1ocean χ + χ1o . = 1O ocean χ2 χ2 + χ2o

(6)

20 0

1

χO [mas]

Figure 1 displays equatorial oceanic mass excitation functions from reduced SLA and the ocean models. Especially in χ1 significant discrepancies are obvious. Correlation coefficients between all displayed curves range from 0.18 to 0.77 for χ1 and from 0.45 to 0.87 for χ2 , where the highest values occur between the two solutions from reduced SLA. The corresponding RMS differences range from 4.1 to 8.5 mas for χ1 and from 4.7 to 8.0 mas for χ2 . While the range of the values of the curves, i.e. approximately ±20 mas in both components, represents the absolute contribution of the oceanic mass term to polar motion excitation, the RMS values are related to the uncertainty of this geophysical effect. The results of the oceanic mass excitation functions from altimetry shall be compared to precise geodetic observations of polar motion. However there is no

−20 2003

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2005

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20 0

O

χ2 [mas]

integration point Q; P nm (cos θ Q ) denote the fully normalized Legendre functions of degree n and order m; ρw = 1000 kg m−3 is the density of fresh water, ρe = 5517 kg m−3 is the average density of the Earth and R = 6, 378, 136.46 m its mean radius. In Eq. (2)  explains the effect of the factor 0.684 = 1 + k2 + kan the yielding of the Earth’s body to surface loads where k2 = −0.305 is the solid Earth load Love number of  = −0.011 accounts for the effects degree 2 and kan of mantle inelasticity. The factor 1.608 = [ (C − A)]/[(C − A+ Am + C AC )σ0 ] in Eq. (3) comprehends the effects of core-mantle decoupling and rotational deformation (Gross (2007); Chen et al. (2000)). Numerical values are: = 7.292115 · 10−5 rad s−1 (mean angular velocity of the Earth), C = 8.0365 · 1037 kg m2 and A = 8.0101 · 1037 kg m2 (axial and equatorial moments of inertia), Am = 7.0999 · 1037 kg m2 (equatorial moment of inertia of the mantle), AC = 9.1168 · 1036 kg m2 (equatorial moment of inertia of the core), C = 2.546 · 10−3 (ellipticity of the core) and σ0 = 2π/(434.2 · 86164) rad s−1 (observed Chandler frequency). Resulting oceanic excitation functions χ1O and χ2O are compared with respective time series from the ocean models ECCO and OMCT. The Special Bureau for the Oceans (SBO) of the Global Geophysical Fluids Centre (GGFC) provides six-hourly series of OAM (oceanic angular momenta; mass term) and oam (motion term) computed from the ECCO run kf049f (Gross et al. (2005)). The model is forced by wind stress, heat and freshwater fluxes from the atmospheric reanalyses of NCEP/NCAR (National Centres for Environmental Prediction/National Centre for Environmental Research) (Kalnay et al. (1996)); an exact inverse barometric response of the ocean surface to atmospheric pressure variations is assumed. ECCO assimilates altimetry and Expendable Bathythermograph (XBT) data. Besides we use six-hourly OAMand oam-series computed from OMCT (Thomas et al. (2001)). This model is forced by wind stress, 2m-temperature and freshwater fluxes from the operational analysis of the ECMWF (European Centre for Medium-Range Weather Forecasts). Like in ECCO no atmospheric pressure forcing is taken into account in the applied OMCT version. To be consistent with the results from altimetry the modelled OAM- and oam-series are averaged into monthly mean values. Oceanic polar motion excitations χ1ocean and χ2ocean (units: rad) are computed from

441

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Fig. 1 Solutions for equatorial oceanic mass excitations from reduced SLA (steric reductions from Ishii (grey fat line) and from WOA05 (black fat line)) and from the models ECCO (grey thin line) and OMCT (black thin line). A mean value for the period between 2003 and 2006 is subtracted from each of the time series

¨ and F. Seitz F. Gottl

atmo 1

χ

50 0 −50 2003

2004

2005

2004

2005

50 0

2

χatmo [mas]

direct measure for the oceanic contribution since the observations reflect the integral effect of mass and motion terms of a multitude of geophysical processes. The largest effects are due to mass transports in the atmosphere, the oceans and the continental hydrosphere. Consequently atmospheric and hydrological effects as well as the oceanic motion term must be either removed from the geodetic observations or – vice versa – added to the oceanic mass effect in order to achieve comparable time series. In the following we will deal with the latter possibility.

[mas]

442

2.2 Atmospheric Polar Motion Excitations Atmospheric angular momentum time series AAM (mass term) and aam (motion term) due to pressure changes and winds are provided from the NCEP reanalyses by the Special Bureau for the Atmosphere (SBA) of the GGFC. The time series are available with a temporal resolution of 6 h. Since the effect of atmospheric pressure forcing is excluded in all oceanic data sets an IB correction has been adopted, i.e. atmospheric pressure have been replaced with an average value over the oceans (Zhou et al., 2006). In the framework of the DFG (Deutsche Forschungsgemeinschaft) research unit FOR584 “Earth Rotation and Global Dynamic Processes” AAM and aam-series from six-hourly operational ECMWF analyses are available. Like in the case of NCEP an IB correction has been adopted. The AAM and aam-series are averaged into monthly mean values and converted into atmospheric polar motion excitations χ1atmo and χ2atmo (units: rad) by  " 1.608 χ1A AAM 1 , · = 0.684 · χ2A

(C − A) AAM 2  " 1.608 χ1a aam1 · = χ2a

(C − A) aam2

−50 2003

Fig. 2 Atmospheric excitation functions of polar motion (mass and motion term) from NCEP (black) and ECMWF (grey). A mean value for the period between 2003 and 2006 is subtracted from both time series

feature high correlation coefficients that amount to 0.70 for χ1 and 0.98 for χ2 . Comparatively small RMS values for χ1 (5.8 mas) and χ2 (4.1 mas) indicate a low uncertainty of the equatorial atmospheric excitation functions, i.e. the atmospheric effect is known much better than the oceanic one. The curves of χ2 are characterised by significantly larger amplitude variations (values between −59 and 35 mas) than the curves of χ1 (values between ±18 mas). This can be explained by the geographical dependency of the excitation functions. Regions in which mass redistributions have a large influence on χ2 are located along the ±90◦ -meridians which cross the major landmasses of Eurasia and North and South America. In contrast χ1 is mainly influenced by mass variations around the 0◦ and 180◦ -meridians which cross the Atlantic (7) and the Pacific ocean. Since atmospheric pressure changes are very small over the oceans due to the IB (8) correction, the contributions are much smaller in this component.

and "  A χ + χ1a χ1atmo = 1A atmo χ2 χ2 + χ2a

(9)

2.3 Hydrological Polar Motion Excitations

Figure 2 displays the combined effect of mass and motion terms of the atmospheric polar motion excitation functions. Both signals are very similar and

Hydrological excitation functions are computed from monthly fields of global water storage variations. Respective data sets from the model LDAS (Land Data

Contribution of Non-Tidal Oceanic Mass Variations to Polar Motion

Assimilation System) developed at National Oceanic and Atmospheric Administration (NOAA) Climate Prediction Centre (CPC) are provided in monthly resolution by the Special Bureau for Hydrology (SBH) of the GGFC. Furthermore we derived monthly fields of EWH changes from surface water, groundwater and snow from the LaD (Land Dynamics) model (Milly and Shmakin, 2002). Modelled water storage variations are transformed into hydrological polar motion excitations χ1H and χ2H (units: rad) following Eqs. (1), (2) and (3). Currently no hydrological relative angular momentum series are offered by the SBH. However the motion effect is very small because of the comparatively slow transport processes of continental water masses. Figure 3 compares the hydrological mass excitation functions of polar motion from both models. The curves feature very large discrepancies which are also reflected by the correlation coefficients which amount to 0.57 for χ1 and −0.22 for χ2 . Especially the curves of χ2 show poor agreement and they are even out of phase. The RMS differences between the curves are 4.2 mas for χ1 and 10.9 mas for χ2 . Consequently the hydrological mass effect causes a similar uncertainty in the total equatorial excitations as the oceanic mass effect.

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3 Validation of the Integral Effect with Geodetic Observations We compare combinations of the above data sets with the integral effect from independent geodetic observations: [1] atmospheric effect from ECMWF combined with the oceanic mass effect from altimetry (steric reduction from WOA05) and the oceanic motion effect from OMCT; [2] atmospheric effect from ECMWF combined with the oceanic mass effect from altimetry (steric reduction from Ishii et al. (2006)) and the oceanic motion effect from OMCT; [3] atmospheric effect from NCEP combined with the oceanic effect from ECCO; [4] atmospheric effect ECMWF combined with the oceanic effect from OMCT. Results of the combinations [1], [3] and [4] are shown in the upper panels of Fig. 4 for χ1 and of Fig. 5 for χ2 . geo The signals are compared with excitation series χ1 geo and χ2 (units: as = 1000 mas) which are computed from observed polar motion (C04 series of the IERS).

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Fig. 3 Excitation functions of polar motion due to mass redistributions within the continental hydrosphere (LDAS: black; LaD: grey). A mean value for the period between 2003 and 2006 is subtracted from both time series

Fig. 4 Upper panel: combined atmospheric and oceanic χ1 components of the excitation functions (combination [1] in lightgrey (fat), [3] in dark-grey (fat) and [4] in light-grey (thin)) are compared with the geodetic χ1 -component of the excitation function (black). Lower panel: combined atmospheric, oceanic and hydrological χ1 -components of the excitation functions (same combinations like upper panel plus hydrological effect from LDAS)

¨ and F. Seitz F. Gottl

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Fig. 5 Upper panel: combined atmospheric and oceanic χ2 components of the excitation functions (combination [1] in lightgrey (fat), [3] in dark-grey (fat) and [4] in light-grey (thin)) are compared with the geodetic χ2 -component of the excitation function (black). Lower panel: combined atmospheric, oceanic and hydrological χ2 -components of the excitation functions (same combinations like upper panel plus hydrological effect from LDAS)

The relation between the excitation functions and the polar motion coordinates (x, y) (units: as) is given by geo

χ1

geo χ2

=x+

2Q 2π 2 Tc (1 + 4Q )

= −y +

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(2Q x˙ − y˙ )

(10)

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

12 9 6 3 0

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Fig. 6 Upper panel: correlation coefficients of the combined geophysical χ1 -components of the excitation functions with respect to the geodetic observations. Lower panel: corresponding RMS differences (combinations [1], [2], [3] and [4] in light grey without hydrological effect, in dark grey with hydrological effect from LDAS and in black with hydrological effect from LaD)

correlation wrt. C04

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(Gross (1992)). Thereby TC = 433.1 d is the Chandler period and Q = 170 is the so-called quality factor which accounts for the damping of the Chandler oscillation due to friction. The correlation coefficients and RMS differences (see Figs. 6 and 7) show that the model-only combinations [3] and [4] correspond better with geodetic observations than combinations in which satellite altimetry data are applied. Although the steric corrections are certainly not perfect, the main reason for the better agreement of [3] and [4] is, that the model-only combinations are more consistent. The phase relations between these data sets match since the ocean models are forced by the respective atmospheric data. Since the mass excitations from altimetry, the motion excitations from OMCT and the atmospheric excitations

rms wrt. C04 [mas]

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Fig. 7 Upper panel: correlation coefficients of the combined geophysical χ2 -components of the excitation functions with respect to the geodetic observations. Lower panel: corresponding RMS differences (combinations [1], [2], [3] and [4] in light grey without hydrological effect, in dark grey with hydrological effect from LDAS and in black with hydrological effect from LaD)

from ECMWF are not in phase, artificial signals due to the non-compensation of counteracting signals are likely. Here additional investigations are necessary.

Contribution of Non-Tidal Oceanic Mass Variations to Polar Motion

When the hydrological effect is additionally considered (lower panels of Figs. 4, 5, 6 and 7) the agreement is slightly improved in the case of LDAS. When LaD is applied, the agreement gets worse. Although the influence of continental hydrology is very small, it seems that LDAS reproduces continental hydrological processes better than LaD. Nevertheless the knowledge of continental hydrology is limited due to the lack of observations. Improvements can be expected from the observations of the Gravity Recovery and Climate Experiment (GRACE).

4 Conclusions The validation of oceanic mass excitation functions of polar motion by means of independent precise geodetic observations requires the use of geophysical data sets and models. When different data sets are combined consistency is a crucial factor. Above results showed that geophysical model-only combinations of atmospheric, oceanic and hydrological excitations agree better with geodetic observations than combinations in which satellite altimetry and physical ocean data are considered. Erroneous patterns of the atmospheric excitations are compensated by the corresponding ocean model which uses the atmospheric data as input variables. This is, of course, not the case when excitations from reduced SLA are combined with the atmospheric excitations. In order to improve the excitation functions from altimetry further investigations are necessary: On the one hand improved data sets for SLA will be derived from multi mission solutions considering the dynamic atmospheric correction (DAC) which is based on the barotropic model MOG2D (Mod`ele aux Oudes de Gravit´e – 2 dimensions). On the other hand alternative data sets for the steric correction are expected from the future Soil Moisture and Ocean Salinity satellite mission (SMOS). Acknowledgments This paper was developed within the subproject “Integration of Earth rotation, gravity field and geometry using space geodetic observations” of the DFG research unit FOR584 (Earth Rotation and Global Dynamic Processes). The authors thank R. Gross for his helpful review of the manuscript.

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We are grateful to M. Thomas for offering the OMCT data. NCEP reanalysis data are provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA. ECCO data are a contribution of the ECCO Consortium for Estimating the Circulation and Climate of the Ocean funded by the National Oceanographic Partnership Program (NOPP).

References Antonov, J.I., R.A. Locarnini, T.P. Boyer et al. (2006). World Ocean Atlas 2005. Volume 2: Salinity. S. Levitus, Ed. NOAA Atlas NESDIS 62, U.S. Government Printing Office, Washington, D.C., pp. 182. Chambers, D.P. (2006). Observing seasonal steric sea level variations with GRACE and satellite altimetry. J. Geophys. Res., 111, 10.1029/2005JC002914. Chen, J.L., C.R. Wilson, R.J. Eanes et al. (2000). A new assessment of long-wavelength gravitational variations. J. Geophys. Res., 105, 0148-0227/00/2000JB900115. Fofonoff, N.P. and R.C. Millard (1983). Algorithms for computation of fundamental properties of seawater. Unesco Technical Papers in Marine Science, 44. Gross, R.S. (1992). Correspondence between theory and observations of polar motion. Geophys. J. Int., 109, pp. 162–170. Gross, R.S. (2007). Earth rotation variations – long period. in: Treatise on Geophysics, Volume 3 – Geodesy, Elsevier, in press. Gross, R.S., I. Fukumori, and D. Menemenlis (2005). Atmospheric and oceanic excitation of decadal-scale Earth orientation variations. J. Geophys. Res., 110, B09405. Ishii, M., M. Kimoto, K. Sakamoto et al. (2006). Steric sea level changes estimated from historical ocean subsurface temperature and salinity analyses. J. Oceanography, 62, pp. 155–170. Kalnay, E., M. Kanamitsu, R. Kistler et al. (1996). The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Meteor. Soc., 77, pp. 437–471. Locarnini, R.A., A.V. Mishonov, J.I. Antonov et al. (2006). World Ocean Atlas 2005. Volume 1: Temperature. S. Levitus, Ed. NOAA Atlas NESDIS 61, U.S. Government Printing Office, Washington, D.C., pp. 182. Milly, P.C.D and A.B. Shmakin (2002). Global modeling of land water and energy balances Part I: the land dynamics (LaD) model. J. Hydrometeorology, 3, pp. 283–299. Thomas, M., J. S¨undermann and E. Maier-Reimer (2001). Consideration of ocean tides in an OGCM and impacts on subseasonal to decadal polar motion excitation. Geophys. Res. Lett., 28(12), pp. 2457–2460. Zhou, Y.H., D.A. Salstein and J.L. Chen (2006). Revised atmospheric excitation function series related to Earth’s variable rotation under consideration of surface topography. J. Geophys. Res., 111, 10.1029/2005JD006608.

Simulation of Historic and Future Atmospheric Angular Momentum Effects on Length-of-day Variations with GCMs Timo Winkelnkemper, Florian Seitz, Seung-Ki Min and Andreas Hense

Abstract This paper focuses on atmospheric winddriven effects on changes in length-of-day (LOD). A 20th century simulation has been carried out using the ECHAM5 standalone atmosphere general circulation model (GCM). The spectrum of the resulting time series for LOD shows typical structure patterns which resemble geodetic observations. Furthermore a future scenario run for the period 2000–2100 driven by SRES A1B forcing scenario shows a strong increase in the axial atmospheric angular momentum (AAM) which implies a lengthening of the LOD. For the scenario runs the coupled atmosphere ocean GCM ECHO-G has been used. The extent of the simulated changes in axial AAM exceeds results from former studies. By 2100 the model shows an increase in axial AAM of about 10 percent compared to present day conditions. The strongest trends in zonal windspeed are detected in the Southern Hemisphere for mid and higher latitudes in the upper troposphere. The reason for this trend can be found in the thermal wind equation. The westerly winds in high levels are directly related to the magnitude of the horizontal, north-south, gradient in temperature averaged from the Timo Winkelnkemper Meteorological Institute University of Bonn, Auf dem H¨ugel 20, 53121 Bonn, Germany, e-mail: [email protected] Florian Seitz Earth Oriented Space Science and Technology (ESPACE), Technische Universit¨at M¨unchen, Arcisstr. 21, 80333 Munich, Germany, e-mail: [email protected] Seung-Ki Min Climate Research Division, Environment Canada, 4905 Dufferin Street, Toronto, Ontario M3H 5T4, Canada, e-mail: [email protected] Andreas Hense Meteorological Institute University of Bonn, Auf dem H¨ugel 20, 53121 Bonn, Germany, e-mail: [email protected]

Earth’s surface to the height of the level. The future scenario runs show significant strengthening in this gradient at higher levels. Keywords Atmospheric angular momentum (AAM) · Earth rotation · Length-of-day (LOD) · Atmospheric excitation · Climate change · GCM · ECHAM5 · ECHO-G

1 Introduction General circulation models are able to simulate mass movements and mass concentrations on a global scale in a realistic way. Due to enormous mass displacements and motions relative to the rotating Earth the atmosphere and oceanic hydrosphere have an important impact on Earth rotation parameters (ERPs). Besides well predictable tidal effects those two components of the Earth system explain the largest part of the variance of ERPs on subdaily to interannual scales (Lambeck, 1980). Variations in LOD are closely linked to those in the axial AAM. The derivation of the relation between LOD and axial AAM is straightforward from the law of angular momentum conservation (Gross et al., 2004, see Sect. 3). In this paper we focus on the atmospheric motion term effect on LOD. Axial AAM changes are highly correlated with short-term changes in LOD. Hence, on seasonal to interannual time scales, the AAM becomes a dominant excitation mechanism of LOD. On decadal time scales, significant angular momentum transfer between the Earth’s liquid core and solid mantle can be detected (Hide, 1969) and exceeds the axial AAM influence on LOD.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Our research concentrates on the ability of the ECHAM5 GCM (Roeckner et al., 2003) to reproduce axial AAM variations associated with changes in LOD. The simulation covers the period from 1880 to 2003. From the early 1960s space geodetic techniques allowed for the observation of the Earth’s rotation with increasing accuracy. A 45 year time series of LOD from 1962 to present is provided by the International Earth Rotation and Reference Systems Service (IERS) in its well-known C04 series. Possible long term trends under a climate change scenario have been studied in previous investigations (de Viron et al., 2002). As far as only atmospheric effects on LOD are considered, those earlier studies predict an increase in global relative axial AAM over the 21st century which is associated with a slowing of the angular velocity of the Earth. The main reason for this was pointed out by Lorenz and DeWeaver (2007) who detected significantly increased and stronger westerly winds in mid and high latitudes particularly in the Southern Hemisphere under a climate change scenario.

2 Atmospheric Simulations Atmospheric GCM simulations have been carried out from 1880 to 2003 with ECHAM5.3.02. An ensemble of three runs has been created by disturbing the initial conditions. The latter were extracted from a short preindustrial control run with five-year intervals. To obtain realistic states of the atmosphere a broad set of forcing factors was used. It includes greenhouse gas concentrations, an aerosol climatology, volcanic aerosols, solar variability (Froehlich and Lean, 1998) and sea surface temperature (SST) data as well as sea ice concentration (SIC) data from the Hadley Centre’s reconstruction (Rayner et al., 2006). To avoid a systematic underestimation of variance when linearly interpolating from monthly means to daily values – as the model expects as input – a filter has been applied to the SST and SIC data (Taylor et al., 2000). The resolution of the model is T63 in the horizontal and 31 layers in the vertical with the 10 hPa level defining the top of the model atmosphere. This represents a global grid consisting of 192 times 96 gridpoints. The distance of two neighbouring grid points is 1.875◦ . This resolution allows to simulate small scale

T. Winkelnkemper et al.

troughs which could have an impact on the globally integrated AAM. The A1B scenario run for a time span of 200 years (2000–2200) was performed by the free coupled atmosphere-ocean sea ice model ECHO-G (Legutke and Voss, 1999; Min et al., 2005, 2006). SST and SIC are internal model variables and not prescribed as boundary conditions. The model consists of the ECHAM4 (Roeckner et al., 1996) atmosphere GCM and the HOPE-G (Wolff et al., 1997) ocean model. These model runs were part of the 4th assessment report (AR) of the Intergovernmental Panel on Climate Change (IPCC, 2007). The A1B scenario describes a future world of rapid economic growth and global population that peaks in mid-century and slowly declines thereafter while new efficient technologies will then be introduced. CO2 , CH 4 , N2 O, NOx , CO and SO2 concentrations are derived from this economic and demographic scenario. Figure 1 illustrates ECHO-G temperature anomaly projections for different scenarios. These projections show an increase of global temperature by nearly 3◦ K for the A1B scenario compared to present day conditions. The atmospheric resolution of ECHO-G is T30 in the horizontal (corresponding to a grid spacing of 3.75◦ ) and 19 layers in the vertical. The oceanic resolution is T42 (2.8◦ ) with a meridional refinement up to 0.5◦ in the tropics and 20 vertical layers.

3 Variations of LOD Angular momentum fluctuations in the various components of the Earth (so-called subsystems) are reflected by temporal variations of Earth rotation. The solid Earth’s reaction on the redistributions and motions of masses follows from the law of angular momentum conservation within a closed system: L = Iω + h.

(1)

In this equation I denotes the combined tensor of inertia of the solid Earth and its fluid subsystem, h stands for angular momenta with respect to the reference frame (so-called relative angular momenta) and the vector L on the left hand side denotes the total angular momentum of the whole system. The Earth rotation vector is denoted by ω. If external torques

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temperature anomaly (in K)

Fig. 1 ECHO-G temperature anomalies for scenario A1B, B1 and PIcntrl (pre-industrial control run) with observations (reference 1880–1920)

year

due to the gravitational influence of Sun and Moon are neglected, the Earth can be viewed as a closed system. Equation (1) is a simplification of the well-known Liouville differential equation (Lambeck, 1980). It is valid on condition of two assumptions: First, the Earth rotation vector and the angular momentum vector h are considered to be parallel. Second, the tensor of inertia is assumed to be rotationally symmetric with respect to the rotation axis of the Earth. Since both assumptions are not perfectly fulfilled, the orientation of the Earth rotation vector changes with respect to an Earth-fixed reference frame (polar motion) which is not subject of this paper. For studies on LOD these effects are minor (H¨opfner, 1996). Variations of Earth rotation are computed with respect to a uniformly rotating geocentric reference frame with its z-axis, about which it rotates, pointing approximately to the direction of the maximum moment of inertia. The angular velocity of the uniform rotation is one revolution per sidereal day:

= 2π/86, 164 s−1 . Due to geophysical processes and gravitational torques this uniform rotation is

slightly disturbed and the Earth rotation vector can be written as ⎛ ⎞ m 1 (t) ω(t) = · ⎝ m 2 (t) ⎠ , m i 1. (2) 1 + m 3 (t) The dimensionless quantities m i denote small disturbances of the uniform rotation (Munk and MacDonald, 1960). Fluctuations of the angular velocity of the Earth are equivalent to changes of length-of-day (LOD). They follow from temporal variations of the absolute value of the Earth rotation vector |ω(t)|, which are computed by ! |ω(t)| = m 1 (t)2 + m 2 (t)2 + (1 + m 3 (t))2 ≈ (1 + m 3 (t))

(3)

The error of LOD due to this approximation is in the order of 10−16 s and is negligible. The relation between the variations |ω(t)| and length-of-day changes follows from the definition of LOD, as the duration of one revolution of the Earth reduced by 86,400 s:

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2π − 86400 s. |ω(t)|

surface pressure ps are obtained from the model’s output. η is the standardized vertical coordinate of the atmospheric GCM. Boundaries of η are 0 for the top When higher order terms, external torques and temof the atmosphere and 1 for the Earth’s surface. poral changes in the tensor of inertia are neglected, the insertion of Eqs. (2) into (1) yields for the third component (Gross et al., 2004) LOD(t) =

(4)

4 Results

m 3 (t) = −h z (t)/ Cm

(5)

From the ECHAM5 simulations a 2 m-temperature anomaly time series has been derived. Since the simulation is driven by observed or reconstructed SSTs and SICs, temperature values over the ocean were excluded because they are highly affected by the SST. Therefore a high correlation with an observed temperature time series would not necessarily demonstrate any model skill. In Fig. 2 the three ensemble members are represented by the dashed lines. The correlation with the observed temperatures is very high even though data points over the ocean have been excluded. The dashed lines show a fairly small spread which indicates a small Vatm model uncertainty. The solid bold line (observations) 3 is right in between the dashed lines during many time r h AAM = ps ucos2 ϕdϕdλdη (7) z intervals suggesting that the simulations are quite g η λ ϕ realistic. From the atmospheric simulations the global axEquation (7) shows the global mass integration over the atmosphere expressed in spherical coordinates. ial AAM is calculated by the integration shown in In the horizontal plane the integration is done over Eq. (7). The model is discretising zonal windspeed ϕ (latitude) and λ (longitude). Zonal windspeed u and on 31 model levels (η-levels). To avoid interpolation

where Cm is the polar moment of inertia of the Earth’s crust and mantle. Polar motion is related to the temporal variation of m 1 (t) and m 2 (t). Equations (3) and (4) show the dependance of LOD(t) on m 3 (t). In the following we will focus on atmospheric effects on length-of-day changes. As mentioned above, it is well known that the largest part of observed variations of LOD on time scales from months to a few years is caused by changes in axial relative atmospheric angular momentum h AAM due to zonal wind variations z (Gross et al., 2004): AAM hz = ρr 2 (r × v)z dV (6)

global 2m-temperature anomaly over land ensemble member

0 –0.5

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through mountains the vertical integration is done over these hybrid η-levels. In contrast to pressure levels the η-levels follow the terrain and are by definition above surface. Integrations over pressure levels tend to overestimate global axial AAM. The simulated AAM time series are plotted together with observed LOD variations from the IERS C04 series in Fig. 3. Decadal changes due to core-mantleinteraction have been removed from the observations Fig. 3 Time series of simulated axial AAM and LOD (7 yr running mean removed)

by substracting a seven year running mean. The effect of long-periodic tides has been removed by filtering. Short periodic tide effects are purged by only considering monthly mean values. The remaining signal is highly correlated with the axial relative AAM time series. Not only the annual and semi-annual cycle but also El Ni˜no peaks (e.g., 1983) are well reconstructed. A fingerprint of this typical oscillation can be found in the wavelet spectrum of the simulated AAM (Fig. 4)

0 –1.0e-03

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8.0e+25 9.0e+25 1.0e+26 1.1e+26 1.2e+26 1.3e+26 1.4e+26 1.5e+26

compared sim. axial AAM and lod time series

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Fig. 5 Time series of global axial AAM (5 yr running mean) ECHO-G A1B scenario

where a lot of energy is apparent in a broad band with periods from three to six years. The ECHO-G scenario runs show a strong trend in the axial AAM (Fig. 5). Under this A1B climate change scenario the atmosphere induces an increased global AAM. By the year 2100 the increase amounts to 10 percent compared to the 2001–2010 mean. In Fig. 6 trends of zonal windspeeds at 300 hPa are displayed. Southern Hemisphere jet stream gets strengthened. In the Northern Hemisphere the region of strongest trends is the Northeast Atlantic. Generally

Fig. 6 Zonal windspeed trends at 300 hPa for 21st century from ECHO-G A1B runs (ensemble mean), shaded areas have no significant trend)

trends over land are weaker. This change pattern is in good consistency with the results of former studies (e.g., Lorenz and DeWeaver, 2007). In the year 2100 the increased axial AAM would cause a lengthening of a solar day by about 0.65 ms. An integration of the computed LOD values over the whole 21st century leads to the prediction that the Earth will be about 11.86 s behind schedule by 2100. However it should be kept in mind that this result comprehends solely the effect of the atmospheric motion term which is superimposed by other effects. For instance

Simulation of Historic and Future AAM effects

the slowing of Earth rotation due to the loss of angular momentum to the moon will lead to an additional delay of 40 s until the year 2100.

5 Discussion The ECHAM5 simulations of atmospheric wind-driven effects on LOD appear to be very realistic since the 2m-temperature profiles of the ensemble members show little departure from each other and agree well with observations. The well reproduced global temperature anomaly time series is also pointing at realistic circulation patterns. Since axial AAM explains more than 80 percent of the variance of LOD on interseasonal and interannual time scales (Gross et al., 2004), the derived LOD series looks very reasonable, too. Not only characteristics in the frequency domain but also amplitudes of the simulated axial AAM are very well captured and exceed results of previous low resolution GCM studies in quality. Improvements in the models have been achieved in the representation of the orography due to higher horizontal resolution which should produce more realistic winds in mountainous regions. Furthermore simulated winds at the tropopause may have large absolute errors due to large magnitudes and due to a rudimentary treatment of the stratosphere in the previous models. Obviously higher model resolution tends to produce more realistic results. This is not trivial since we are dealing with axial AAM, i.e. a globally integrated variable (Stuck, 2002). ECHO-G A1B scenario runs show a very strong century-long increase in AAM. The dynamic reason of the increase cannot be retraced easily. Strongly increased temperatures in the upper tropics enhance the meridional temperature gradient. In the inner tropics the tropopause level is rising. Therefore the tropopause slope towards the poles is getting intensified. Stratospheric cooling and tropospheric warming amplify the meridional temperature gradient. Due to thermal wind balance westerlies become stronger. The increase in kinetic energy at the tropopause level is consistent with the rise in tropopause height because synoptic waves are trapped in the troposphere (Lorenz and DeWeaver, 2007). This does not contradict the popular impression that the Arctic is especially warming, as the Arctic warming is primarily confined to the very near surface levels.

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For validation purpose one should compare results to other state-of-the-art models. Abarca del Rio (1999) found trends in reanalysis data which explain an increase in LOD of about 0.5 ms/century during the last 50–60 years. The amplitude of zonal windspeed trends for the 21st century seems to be very strong in the ECHO-G runs compared with other model results. Consequently the predicted change in LOD of about 0.65 ms/century is significantly higher than the value of 0.181 ms/century which was derived from an CMIP2 (Meehl et al., 2000) model ensemble in the analysis of de Viron et al. (2002) even though the increase in CO2 (1% per year) is higher compared to the A1B-scenario (∼0.6% per year). In the CMIP2 study the HadCM2 model reached 0.53 ms/century. This shows that the uncertainty of models is a big issue. On interdecadal time scales other effects become more dominant and cannot be neglected (de Viron et al., 2002). Among those are the angular momentum loss to the sun and moon due to tidal friction (2 ms/century), sea level rise (0.05 ms/century), core motion 0.1–0.2 ms/century and the pressure term of the atmosphere (−0.075 ms/century). Nevertheless axial AAM changes will play a key role in future LOD variations. Acknowledgments This paper was developed within a project supported by DFG grants HE 1916/9-1 and DR 143/12. The authors thank the Deutsches Klimarechenzentrum (DKRZ), Hamburg, Germany, for providing the CPU time for the standalone runs, the Hadley Centre for providing the SST/SIC data and the IERS for providing the C04 time series.

References Abarca del Rio, R.: The influence of global warming in Earth rotation speed. Ann. Geophys., 17, 806–811, 1999. de Viron, O., V. Dehant, H. Goosse, and M. Crucifix: effect of global warming on the length-of-day. Geophy. Res. Lett., 29(7), 10.1029/2001GL013672, 2002. Fr¨ohlich, C. and J. Lean: The Sun’s total irradiance: cycles, trends, and related climate change uncertainties since 1976, Geophys. Res. Lett. 25(23), 4377, 1998. Gross, R.S., I. Fukumori, D. Menemenlis, and P. Gegout: Atmospheric and oceanic excitation of length-of-day variations during 1980–2000. J. Geophys. Res., 109, B01406, doi:10.1029/2003JB002432, 2004. Hide, R.: Interaction between the Earth’s liquid core and solid mantle. Nature 222, 1055–1056; doi:10.1038/2221055a0, 1969.

454 H¨opfner, J.: Seasonal oscillations in length-of-day. Scientific Technical Report, No.: STR96/03, 1996. IPCC: Climate Change 2007: The Physical Science Basis. Intergovernmental Panel on Climate Change, http://www. ipcc.ch/ipccreports/ar4 – wg1.htm, 2007. Lambeck, K.: The Earth’s Variable Rotation: Geophysical Causes and Consequences. Cambridge University Press, New York, 1980. Legutke, S. and Voss, R.: The Hamburg Atmosphere-Ocean Coupled Circulation Model ECHO-G. Technical Report No. 18, German Clim. Comput. Cent., Hamburg, Germany, 1999. Lorenz, D. J., and E. T. DeWeaver: Tropopause height and zonal wind response to global warming in the IPCC scenario integrations, J. Geophys. Res., 112, D10119, doi:10.1029/2006JD008087. 2007. Meehl, G. A., G. J. Boer, C. Covey, M. Latif, and R. J. Stouffer: The Coupled Model Intercomparison Project (CMIP), Bull. Amer. Meteorol. Soc., 81, 313 318, 2000. Min, S.-K., S. Legutke, A. Hense, and W.-T. Kwon: Internal variability in a 1000-year control simulation with the coupled climate model ECHO-G – I. Near-surface temperature, precipitation and mean sea level pressure. Tellus, 57A, 605–621. 2005. Min, S.-K., S. Legutke, A. Hense, U. Cubasch, W.-T. Kwon, J.H. Oh, and U. Schlese: East Asian climate change in the 21st century as simulated by the coupled climate model ECHOG under IPCC SRES scenarios. J. Meteor. Soc. Japan., 84, 1–26. 2006. Munk, W. H. and MacDonald, G. J. F.: The Rotation of the Earth: A Geophysical Discussion. Cambridge University Press, New York, 1960.

T. Winkelnkemper et al. Rayner, N. A., P. Brohan, D. E. Parker, C. F. Folland, J. J. Kennedy, M. Vanicek, T. Ansell and S.F.B. Tett: Improved analyses of changes and uncertainties in sea surface temperature measured in situ since the mid-nineteenth century: the HadSST2 data set. J. Clim., 19(3) pp. 446–469. 2006. Roeckner, E., K. Arpe, L. Bengtsson, M. Christoph, M. Claussen, L. Dmenil, M. Esch, M. Giorgetta, U. Schlese and U. Schulzweida: The atmospheric general circulation model ECHAM-4: Model description and simulation of present-day climate, Rep. 218, Max-Planck-Institute (MPI) for Meteorology, Hamburg, Germany, 1996. Roeckner, E., G. Buml, L. Bonaventura, R. Brokopf, M. Esch, M. Giorgetta, S. Hagemann, I. Kirchner, L. Kornblueh, E. Manzini, A. Rhodin, U. Schlese, U. Schultzweida und A. Tompkins: The atmospheric general circulation model ECHAM 5. PART I: Model description, Rep. 349, MPI for Meteorology, Hamburg, Germany, 2003. Stuck, J.: Die simulierte axiale atmosph¨arische Drehimpulsbilanz des ECHAM3-T21 GCM, PhD thesis, Bonner Meteorologische Abhandlungen, 56, Asgard, Sankt Augustin, 2002. Taylor, K. E., D. Williamson and F. Zwiers: The sea surface temperature and sea-ice concentration boundary conditions of AMIP II simulations. PCMDI report No. 60, 20 pp, 2000. Wolff, J. O., E. Meier-Reimer and S. Legutke: The Hamburg ocean primitive equation model, DKRZ 13, 98 pp., German Clim. Comput. Cent., Hamburg, Germany, 1997.

Contributions of Tidal Poisson Terms in the Theory of the Nutation of a Nonrigid Earth V. Dehant, M. Folgueira, N. Rambaux and S.B. Lambert

Abstract The tidal potential generated by bodies in the solar system contains Poisson terms, i.e., periodic terms with linearly time-dependent amplitudes. The influence of these terms in the Earth’s rotation, although expected to be small, could be of interest in the present context of high accuracy modelling. We have studied their contribution in the rotation of a non rigid Earth with elastic mantle and liquid core. Starting from the Liouville equations, we computed analytically the contribution in the wobble and showed that the presently-used transfer function must be supplemented by additional terms to be used in convolution with the amplitude of the Poisson terms of the potential and inversely proportional to (σ − σn )2 where σ is the forcing frequency and σn are the eigenfrequencies associated with the retrograde free core nutation and the Chandler wobble. These results have been detailed in a paper that we published in Astron. Astrophys. in

V. Dehant Royal Observatory of Belgium, 1180 Brussels, Belgium, e-mail: [email protected] M. Folgueira Instituto de Astronom´ıa y Geodesia (UCM-CSIC), Facultad de Ciencias Matem´aticas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain Observatoire de Paris, SYRTE/UMR 8630-CNRS, 61, Avenue de l’Observatoire, 75014 Paris, France N. Rambaux Royal Observatory of Belgium, 1180 Brussels, Belgium Department of Mathematics, Facult´es Univ. N.D. de la Paix, 5000 Namur, Belgium, e-mail: [email protected] S.B. Lambert Royal Observatory of Belgium, 1180 Brussels, Belgium Observatoire de Paris, SYRTE/UMR 8630-CNRS, 61, Avenue de l’Observatoire, 75014 Paris, France, e-mail: [email protected]

2007. In the present paper, we further examine the contribution from the core on the wobble and the nutation. In particular, we examine the contribution on extreme cases such as for wobble frequencies near the Free Core Nutation Frequency (FCN) or for long period nutations. In addition to the analytical computation, we used a time-domain approach through a numerical model to examine the core and mantle motions and discuss the analytical results. Keywords Precession · Nutation · Poisson terms

1 Scientific context In view of the great improvement in the measurement of the Earth Orientation Parameters, it is necessary to propose further analytical developments of nutation and to consider all the phenomena providing contributions at the microarcsecond level (see e.g., Dehant et al. 2003). In this paper, we show that several purely periodical terms of nutations with periods of 1, 18.6, and 10,467.6 years must be re-estimated at this level of precision, when taking into account the tidal potential generated by bodies in the solar system containing Poisson terms, that is to say, periodic terms with linearly time-dependent amplitudes. As shown in Folgueira et al. (2007), the consideration of a linear time dependence in the amplitude of the forcing potential produces a change in the amplitudes of some of the purely periodic terms. These terms must be added to the adopted model MHB2000 of Mathews et al. (2002, see also McCarthy and Petit 2004). The results have

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

455

456

V. Dehant et al.

  " Aα k i dV21 been detailed in the Astron. Astrophys. paper that we = −3i 2 V21 − + V21 , (1) − published (Folgueira et al., 2007). These results show kfluid

dt a that the most important contributions are stemming from nutations of high amplitude near the Free Core Nutation Frequency (FCN) or from very long period   q0 * dωf q0 * dω nutations. In the present paper, we further examine + 1 + h 1f + i (1 + αf )ωf 1 + hf 2 dt 2 dt the contribution from the core on these extreme cases 3 q0 dV21 by computing analytically these limits. In order to be = 2 hf . (2) dt a

2 closer to the numerical integration that we have performed, we have considered here the simplification of where, a nondeformable Earth. Our paper is organized as follows. After describing the differential equations of the rotation of an elastic V is the tesseral part of the potential, 21 Earth with a liquid core (Sects. 2 and 3), we explain A, A , and A and C, C , and C are the principal mof m f m two different approaches, a semi-analytical developments of inertia of the whole Earth, of the liquid core, ment (Sect. 4.1) and a numerical integration (Sect. 4.2), and of the mantle, for the equatorial and z-axis (printo obtain the coefficients of the nutation due to the tidal cipal) mass repartition, Poisson terms. This work is also contributing to the ob- is the mean angular of the Earth, jectives of the DESCARTES Sub-project entitled: “Geo- α = C−A and α = Cf −Af are the dynamical flattenings f A Af physical effects of adopting the new solutions for the of the whole Earth and of the liquid core, Earth’s rotation in the framework of the new parame- k is the Love number expressing (without dimension) ters adopted by the IAU 2000 Resolutions”. the mass redistribution potential at the surface of the Earth induced by a forcing expressed in terms of a gradient of an external potential and acting on the 2 The Differential Equations whole Earth, k 1 is the Love number expressing (without dimension) This section attempts to show firstly the differential the mass redistribution potential at the surface of the equations describing the rotation of an elastic Earth Earth (and evaluated there) induced by a pressure at with a liquid core of which the solution will be used the core-mantle boundary (CMB), to determine the influence of the tidal Poisson terms on h is the Love number expressing (without dimension) f the Earth’s rotation. The formulation and the methodthe deformation of the CMB induced by a volumic ology used for the derivation of the solution are briefly potential evaluated at the surface, given at the end of this section. For details of the math- h is the Love number expressing (without dimension) 1f ematical process for obtaining the solution, we refer the deformation of the CMB caused by a forcing exthe reader to the paper of Folgueira et al. (2007, see pressed in terms of a gradiant of an external potential also Dehant et al. 1993 and Greff-Lefftz et al. 2002). evaluated at the CMB, The basic equations are the angular momentum bal- h is the Love number expressing (without dimension) 1f ance equations for the whole Earth and the liquid core the deformation of the CMB caused by an inertial relating the angular velocity vectors of the whole Earth, pressure on the CMB, ω, and of the liquid core, ω f , to the tidal external poten- k fluid is the secular Love number

also called the fluid tial. In an Earth-fixed frame of reference, the equations αAG Love number kfluid = 3 2 a 5 , and for the equatorial component ω = ω1 + iω2 are (see, 3

2 aeq e.g., Sasao et al. 1980; Hinderer et al. 1987): q0 is defined as where ME is the Earth’s mass. It 



(

)

k1 dω dωf + Af + Aα kfluid dt kfluid dt ( )   k k1 − i Aα 1 − ω + i Af + Aα ωf kfluid kfluid A 1+α

k

GM E

must be noted that we have used here the notations introduced in Dehant et al. (1993). These notations are perfectly equivalent to those used in MHB2000. Mathews et al. (2002) used the compliances instead of the Love numbers, grouping the Love numbers with q20 .

Contributions of Tidal Poisson Terms

457

We consider that the external potential contains contributions due to the so-called Poisson terms, that is to say, terms of the form t × eiσ t . The sets of fundamental arguments employed in tide and nutation theories of the rigid Earth are not linear in time and therefore they do include small nonlinear terms. This causes slow variations of the fundamental frequencies and hence also of the frequencies of the spectral terms. This is particularly true for the arguments arising from planetary perturbations. The Poisson terms stem from the approximation for short time scales of the long term variations arising from planetary perturbations. Usually one indeed prefers to work with constant frequencies; therefore, one splits the argument into a part linear with time and a remaining part involving the second and higher powers of time. This second part being very small, the sine or cosine of the argument can be written as a purely sinusoidal part of which the amplitude has quadratic and higher degree dependences on the time variable. Within a secular theory of the orbital motion of the Earth and of the other planets of the Solar System, linear as well as non-linear changes in the obliquity appears. Terms in the spectral representation which have amplitudes depending polynomially on time are usually referred to as Poisson terms. The external tide generating potential can therefore be written: V21 = V21,0 + V21,1 t + V21,2 t 2  + (V21,n,0 + V21,n,1 t)eiσn t ,

(3)

potential, into Eqs. (1) and (2), provides a set of ten algebraic equations, which allows us to directly obtain the expressions for ω0 , ω1 , ω2 , ωn0 , ωn1 , ωf0 , ωf1 , ωf2 , ωfn0 and ωfn1 in terms of the coefficients of the external potential: V21,0 , V21,1 , V21,2 , V21,n,0 and V21,n,1 . The part related to V21,n,0 and V21,n,1 reduces to:

=

3 +

T0 (σ )V21,n,0 + T1 (σ )iV 21,n,1 , +T0 (σ )V21,n,1 t eiσn t ,

a2

ω = ω0 + ω1 t + ω2 t 2  + (ωn,0 + ωn,1 t)eiσn t ,

(4)

n

ωf = ωf,0 + ωf,1 t + ωf,2 t 2  + (ωf,n,0 + ωf,n,1 t)eiσn t .

(5)

n

(6)

and the reduced solution for the core wobble ωf is:

n

where the amplitudes V21,n,0 and V21,n,1 are both related to the spectral component of frequency σn . Moreover, we assume that the solutions of Eqs. (1) and (2) can be written as follows:

(

αAk − q20 h f Af kfluid σCW − kfluid Am σ − σCW )  (α − q20 h f )Af σFCN + V21,n,0 Am (σ − σFCN )  σCW + − (σ − σCW )2 )  (α − q20 h f )Af σFCN + iV 21,n,1 Am (σ − σFCN )2 ( αAk − q20 h f Af kfluid σCW + − kfluid Am σ − σCW  ) q0  σFCN (α − 2 h f )Af + V21,n,1 t eiσn t , Am (σ − σFCN )

3 ω= 2 a

(

(αk − kfluid q20 h f ) A kfluid Am ) 2 (α − q20 h f ) 2 A σCW + + V21,n,0 σ − σCW Am (σ − σFCN ) ( ) 2 (α − q20 h f ) 2 A σCW + + iV 21,n,1 (σ − σCW )2 Am (σ − σFCN )2 ( 2 (αk − kfluid q20 h f A σCW + − + kfluid Am σ − σCW  ) (α − q20 h f ) 2 A + (7) V21,n,1 t eiσn t Am (σ − σFCN )

3 ωf = 2 2 a

−

3 Analytical Solutions The substitution of the above expressions (4) and (5) for ω and ωf , as well as Eq. (3) for the external

=

3 +

T f 0 (σ )V21,n,0 + T f 1 (σ )iV 21,n,1 , +T f 0 (σ )V21,n,1 t eiσn t .

a2

(8)

458

V. Dehant et al.

In the above expressions,

Tnondef 1 (σ ) = −



σFCN

 A q0 = − 1 + αf − h 1f Am 2

is the frequency of the nearly diurnal free wobble (NDFW) associated with the free core nutation (FCN), and expressed in the terrestrial frame,  σFCN = σFCN + is its counterpart in the celestial reference frame, and σCW

A α = Am

1−

k

+

(9)

Am (σ − σFCNnondef )2

(14)

Tnondef 0 and Tnondef 1 are called transfer functions for the non-deformable Earth (nondef); with,   A σFCNnondef = − 1 + αf Am



kfluid

σCWnondef (σ − σCWnondef )2  σFCN α Af nondef

(15)

(10) σCWnondef =

A α Am

(16) is the Chandler wobble (CW) frequency in the terrestrial frame. If we consider that the Earth is completely rigid (rig; If we consider that the Earth is not deformable (non- σCW rig = σEuler ), the core contribution disappears def stands for non-deformable; this is the case consid- so that: ered in the section on Numerical Integration), all Love   numbers become zero so that: 3 σEuler ωrig = 2 V21,n,0 − σ − σEuler a    3 σCWnondef σEuler ωnondef = 2 − + − iV 21,n,1 σ − σCWnondef a (σ − σEuler )2 )   "  σFCNnondef α Af σEuler + V21,n,0 + − V21,n,1 t eiσn t , Am (σ − σFCNnondef ) σ − σEuler  3 + σCWnondef Trig0 (σ )V21,n,0 = 2 + − a (σ − σCWnondef )2 ) +Trig1 (σ )iV 21,n,1  σFCN α Af , nondef + iV 21,n,1 +Trig0 (σ )V21,n,1 t eiσn t , (17) Am (σ − σFCNnondef )2  σCWnondef with, + − σ − σCWnondef  )  (18) σEuler = α σFCN α Af nondef + V21,n,1 t eiσn t Am (σ − σFCNnondef ) 3 + = 2 Tnondef 0 (σ )V21,n,0 a +Tnondef 1 (σ )iV 21,n,1 , +Tnondef 0 (σ )V21,n,1 t eiσn t

(11)

σCWnondef =

A σEuler Am

(19)

Trig0 (σ ) = −

σEuler σ − σEuler

(20)

where (12)

where Tnondef 0 (σ ) = − +

σCWnondef σ − σCWnondef  σFCN α Af nondef Am (σ − σFCNnondef )

Trig1 (σ ) = −

Trig0 (σ ) σEuler = 2 σ − σEuler (σ − σEuler )

(21)

(13) Trig0 and Trig1 are called transfer functions for the rigid Earth (rig).

Contributions of Tidal Poisson Terms

459

In the next paragraph, we compute the difference between the non-deformable Earth and the rigid Earth response for the wobble (ωnondef −ωrig ). The analytical expression is thus: ωnondef − ωrig =

r

 σFCN 3α Af nondef  ) a 2 Am (σ  − σFCN nondef   iV 21,n,1 + V21,n,1 t (25) V21,n,0 +   (σ − σFCN ) nondef

ωnondef − ωrig =

3 a2

 −

σCWnondef σEuler + σ − σCWnondef σ − σEuler )  σFCN α Af nondef V21,n,0 + Am (σ − σFCNnondef ) ( σCWnondef σEuler + − + 2 (σ − σCWnondef ) (σ − σEuler )2 )  σFCN α Af nondef iV 21,n,1 + Am (σ − σFCNnondef )2  σCWnondef σEuler + − + σ − σCWnondef σ − σEuler )   σFCN α Af nondef V21,n,1 t eiσn t + Am (σ − σFCNnondef )

r

(22)

which can also be written: 3

ωnondef − ωrig =

a2

{ (Tnondef 0 (σ ) − Trig0 (σ ))(V21,n,0 + V21,n,1 t) , +(Tnondef 1 (σ ) − Trig1 (σ ))iV 21,n,1 eiσn t (23) For the retrograde diurnal wobble corresponding to nutation, Eq. (22) can be written using the fact that σ = − + σ  : ωnondef − ωrig =

3 a2 (



 σFCN nondef

)

α Af V21,n,0 Am ) (  σFCNnondef α Af iV 21,n,1 + 1+   2 Am (σ − σFCN ) nondef ( )  σFCN α Af V21,n,1 t nondef + 1+   (σ − σFCN ) Am nondef " eiσn t (24) 1+

 (σ  − σFCN ) nondef

 First, for σ  close to σFCN , Eq. (24) can be writnondef ten:

r

where we see that, for a frequency close to the FCN resonance, the term in V21,n,1 may largely be enhanced. For a very long nutation period (σ  close to zero), Eq. (24) can be written:   3α Af iV 21,n,1

ωnondef − ωrig = (26)  σFCN a 2 2 Am nondef which shows that for the precession or the very long nutation period, we may have a contribution from the core due to the Poisson terms in the forcing. It must be noted that Ferr´andiz et al. (2004) have also considered the effects of a liquid core on the Poisson terms and have evaluated an additional core contribution to precession related this. They have found an effect at the level of 20 milliarcsecond (mas) per century. Starting from Eq. (24), we can also evaluate the effects for the classical large nutations such as at 13.66 days, 0.5 and 1 year, 9.3 and 18.6 years. We have noted that the transfer function for the Poisson part of the potential (the second term of Eq. (24)) is larger than the transfer function for the periodic part of the potential (the first term of Eq. (24)) by a factor of about 200 for both the prograde annual nutation and the retrograde semi-annual nutation, if the frequencies are expressed in cycle per day. The effective contributions to the Earth nutations depend however on the ratio between V21,n,1 and V21,n,0 . For the two highest terms mentioned here and far from long period as well as far from the FNC period, i.e. the semi-annual retrograde nutation and the prograde annual nutation, the final contribution is at the level of 10−6 , which is very small.

The only contributions to purely periodic nutation amplitude from the Poisson part of the forcing would then This equation may be considered in different be either for very long periods or for nutation close frequency ranges within the nearly diurnal fre- to the FCN. They come from the existence of a core within the Earth. quency band:

460

V. Dehant et al.

ε Poisson = 0.2 cos(−l  ) − 0.2 cos

4 Contributions to the Nutation: Two Approaches

−2.3 cos(2l  − 2F + 2D − 2 ) +0.3 cos(− )

(27)

4.1 Semi-Analytical Development

where l  is the mean anomaly of the Sun, F is the mean As the ratio of the non-rigid Earth circular nutation angular distance of the Moon from the ascending node, to the rigid Earth circular nutation (the so-called B- D is the difference between the mean longitudes of the ratio of Wahr, 1981) is identical to the ratio of the Sun and the Moon, and is the mean longitude of the non-rigid Earth wobble to the rigid Earth wobble, the ascending node of the lunar orbit. These terms correabove expressions obtained for the wobbles can be spond to the retrograde annual nutation, the prograde used for determining the nutation B-ratios. By using 18.6 year nutation, the nutation at the very long period a semi-analytical development of the tide generating of 10,467.6 years appearing in Roosbeek’s tidal potenpotential of Roosbeek (1998, see also Roosbeek and tial used here, and the retrograde 18.6 year nutations. Dehant 1998), we then evaluate numerically the am- As explained in Sect. 3, they correspond to very long plitude of each rigid Earth nutation. This allows us period nutation or to a nutation close to the FCN. Figures 1(a) and 2(a) show the time evolution of to compute the non-rigid Earth nutations for the each each of four terms found to be above 0.1 μas for the nunutation frequency, either in the prograde part of the tation in longitude and obliquity. Figures 1(b) and 2(b) spectrum or in the retrograde part of the spectrum. The combination (sum and differences potentially di- represent the global contribution coming from the sum vided by the sine of the mean obliquity) of prograde of these four terms. and retrograde nutations allows to compute the nutation in longitude ψ and in obliquity ε. Identifying 4.2 Numerical Integration the purely-periodic terms and the Poisson terms in the tide generating potential it is then possible to compute the contribution to the nutation due to the tidal Poisson In parallel to the analytical approach developed in Folgueira et al. (2007) as well as in the previous terms. The contributions for the nutation in longitude ψ sections, we have calculated the nutations of the and in obliquity ε that are above the level of 0.1 μ as Earth numerically using a numerical integration approach. For obtaining such rotational motions of (microarcseconds), are written as follows: the Earth, we have used the SONYR model (Bois, Journet, Vokrouhlivk`y, and Rambaux) allowing to Poisson  ψ = −0.4 sin(−l ) + 0.5 sin compute rotational and orbital motions of terrestrial +5.9 sin(2l  − 2F + 2D − 2 ) planets. SONYR is the acronym of Spin-Orbit N-bodY −0.7 sin(− ) Relativistic model, a dynamical model of the solar System that includes the coupled spin-orbit motions of Fig. 1 (a) On the left, time evolution of the each contributing term of the purely periodic nutation in longitude coming from the tidal Poisson terms of the forcing. The time is measured in Julian years and the amplitudes in microarcseconds. (b) On the right, time evolution of the total contribution to periodic nutation in longitude due to the tidal Poisson terms of the forcing

(a)

(b)

Nutation in longitude

2.5

P = – 3823298.12 d

6

2

4

1.5

2

1

P = 6798.38 d

0.5 0 –0.5

0 –2

P = – 365.26 d

–4 5

10

15

20

25 30 P = – 6798.38 d

Nutation in longitude

–6

10000 20000 30000 40000 50000 60000 years

Contributions of Tidal Poisson Terms Fig. 2 (a) On the left, time evolution of the each contributing term of the purely periodic nutation in obliquity coming from the tidal Poisson terms of the forcing. The time is measured in Julian years and the amplitudes in microarcseconds. (b) On the right, time evolution of the total contribution to periodic nutation in obliquity due to the tidal Poisson terms of the forcing

461

(a)

(b) Nutation in obliquity

Nutation in obliquity P=–3823298.12 d

2

2 1.5

1

1

0

0.5

–1

P=–365.26 d 0

10000 20000 30000 40000 50000 60000 years

5

10

years

P=6798.38 d

20 15 P=–6798.38 d

the terrestrial planets and of the Moon. It is described in a series of papers (see a review in Bois (2000) and reference therein; Bois and Vokrouhlick`y (1995); Rambaux et al. (2007)). In the frame of the present paper and in order to compare with the analytical results developed in the previous sections (see Eq. (22) for instance), we have in particular calculated the difference between the nutation of a rigid Earth and the nutation of a two-layered Earth model. As seen from Eq. (22), there are three contributions from the existence of a core, among which one contribution related to V21,n,0 and two new contributions related to V21,n,1 , the Poisson part of the forcing potential. As the numerical approach is based on a forced model of the celestial bodies motions and not on potential-form forcing, it is impossible to separate the three contributions. It is however interesting to compare the results at the global level as, in principal, if the time-dependent amplitudes of the nutations can be separated from their constant amplitudes, it is possible to isolate the term in V21,n,1 t from the two other terms in V21,n,0 and V21,n,1 . However, the terms coming from the coupling between the core and the mantle (i.e. the non-rigid part of the transfer function in the frequency domain) and from the periodic forcing (the terms in V21,n,0 in Eq. (22)) are mixed with the contributions of the purely-periodic terms coming from the Poisson terms in the rotation of the mantle and the core (the terms in V21,n,1 in Eq. (22)). To that aim, we have taken the following approach. First, we neglect in a first approximation the purely relativistic terms in the post-newtonian development. We then compute the dynamical behavior of the two rotational motions of the mantle and the core, for an Earth

25

–2

consisting of a completely homogeneous body and for an Earth consisting of two layers. In order to be able to consider core-mantle coupling, we have included the Poincar´e model for the core in the SONYR model (see, e.g., Poincar´e 1910; Moritz 1980). The coupling mechanism between the mantle and core in this model is called inertial coupling, and is due to the pressure of the fluid on the core-mantle boundary. It is possible at this level to compare the results from the two Earth models. It must be noted that in our numerical approach we have ignored the deformation contributions and did only consider the core existence for the geophysical part of the equations. Table 1 provides the ratios between the non-rigid contributions and the rigid contributions for the main prograde and retrograde nutations of the Earth. In this case the terms coming from the transfer function resulting from the existence of a core and the terms coming from the Poisson part of the forcing are mixed. As our analytical developments have shown in Sects. 2 and 3, the main contribution is not coming from the Poisson terms in the forcing, but rather from the core FCN resonance. We find that the contribution of the prograde 18.6 years nutation is smaller than the unity whereas the ratio for all other nutations are greater than 1 as in the case of Wahr’s B-ratio (Wahr 1981). Table 1 Ratio non-deformable nutation amplitude over the rigid nutation amplitude Nutation period 18.6 years Semi-annual Annual Monthly

Prograde nondef/rig

Retrograde nondef/rig

0.9914806597 1.220587363 1.851327564 1.130579228

1.007487569 1.088899310 1.068475799 1.122725790

462

For the next step, we want to determine the amplitudes of the purely Poisson terms (the terms in V21,n,1 t) in the Earth orientation and the amplitudes of the purely periodic terms (the two other terms in V21,n,0 and V21,n,1 ). This determination involves the Fourier analysis of the Earth orientation obtained in a N-body integration in terms of constant and time-dependent amplitudes for each nutation. This formal decomposition of long-term motions in Fourier series and Poisson terms is however very tricky as the length of the time series used for this decomposition is of importance in this approach. For most of the terms, we can only estimate the sum of the two contributions, and we can only estimate the ratios and the difference between the total non-rigid contribution and the rigid contribution as done above for the construction of the table. For the terms of which the period is smaller than the time span of the time series, it is in principal possible to separate the Poisson terms from the periodic contribution to nutations by fitting an amplitude and a term proportional to the time for each frequency. However, as seen from the previous sections, the time dependent parts are very small. Additionally, many terms interfere with one another and the identification of purely-periodic and Poisson amplitudes becomes impossible. One possibility to obtain a clear separation, and thus to obtain the impact on the purely-periodic nutations of the Poisson terms in the forcing, would be to compute two orbital motions of the Earth reconstructed using a tidal potential for the Earth with and without Poisson terms and to compare the results from the numerical integration approach for both cases. This will allow obtaining their respective contributions to the Earth rotation. Additionally, for the nutation close to the FCN, it is impossible to perfectly clear the he nutation time series from the free core nutation excitation if the initial conditions are not properly chosen or if no dissipation is taken into account. A method for getting independence from the initial conditions has been developed for Mercury by Bois and Rambaux (2007) and could possibly be applied for the Earth. Acknowledgments We are really thankful to Nicole Capitaine and Chengli Huang for their fruitful reviews of our paper. The research was carried out in the Section ‘Time, Earth Rotation, and Pace Geodesy’ of the Royal Observatory of Belgium and received financial support from the Descartes Prize Allowance (MF, NR), for which we express our sincere appreciation.

V. Dehant et al.

References Bois, E., 2000. Connaissance de la libration lunaire a` l’`ere de la t´el´em´etrie laser-Lune, C. R. Acad. Sci. Paris, t. 1, S´erie IV, 809–823. Bois, E., and D. Vokrouhlick`y, 1995. Relativistic spin effects in the Earth-Moon system, A & A 300, 559. Bois, E., and N. Rambaux, 2007. On the oscillations in Mercury’s obliquity, Icarus, in press. Dehant, V., J. Hinderer, H. Legros, and M. Lefftz, 1993. Analytical approach to the computation of the Earth, the outer core and the inner core rotational motions, Phys. Earth Planet. Inter. 76, 259–282. Dehant, V., M. Feissel-Vernier, O. de Viron, C. Ma, M. Yseboodt, and C. Bizouard, 2003. Remaining error sources in the nutation at the submilliarcsecond level, J. Geophys. Res. 108(B5), 2275, DOI: 10.1029/2002JB001763. Ferr´andiz, J.M., J.F. Navarro, A. Escapa, and J. Getino, 2004. Precession of the nonrigid Earth: effect of the fluid outer core, Astron. J. 128, 1407–1411. Folgueira, M., V. Dehant, S.B. Lambert, and N. Rambaux, 2007. Impact of tidal Poisson terms to non-rigid Earth rotation, A & A 469(3), 1197–1202, DOI: 10.1051/0004-6361:20066822. Greff-Lefftz, M., H. Legros, and V. Dehant, 2002. Effect of inner core viscosity on gravity changes and spatial nutations induced by luni-solar tides. Phys. Earth Planet. Inter. 129(1–2), 31–41. Hinderer, J., H. Legros, and M. Amalvict, 1987. Tidal motions within the earth’s fluid core: resonance process and possible variations, Phys. Earth Planet. Inter. 49(3–4), 213–221. Mathews, P. M., T. A. Herring, and B. A. Buffett, 2002. Modeling of nutation and precession: new nutation series for nonrigid Earth and insights into the Earth’s interior, J. Geophys. Res. 107(B4), DOI: 10.1029/2001JB000390. McCarthy, D.D., and G. Petit (Eds.), 2004. Conventions 2003. IERS Technical Note 32, Publ. Frankfurt am Main: Verlag des Bundesamts f¨ur Kartographie und Geod¨asie. Moritz, H. and I.I. Mueller, 1987. Earth Rotation: Theory and Observation. The Ungar Publishing Company, New York. Poincar´e H., 1910. Sur la pr´ecession des corps d´eformables. Bulletin Astronomique, Serie I, 27, 321–356. Rambaux, N., T. Van Hoolst, V. Dehant, and E. Bois, 2007. Inertial core-mantle coupling and libration of Mercury, A & A 468(2), 711–719. Roosbeek, F., 1998. Analytical developments of rigid Mars nutation and tide generating potential series, Celest. Mech. Dynamical Astron. 75, 287–300. Roosbeek, F., and V. Dehant, 1998. RDAN97: An analytical development of rigid Earth nutations series using the torque approach, Celest. Mech. Dynamical Astron. 70, 215–253. Sasao, T., S. Okubo, and M. Saito, 1980. A simple theory on dynamical effects of stratified fluid core upon nutational motion of the Earth, Proc. IAU Symposium 78, ‘Nutation and the Earth’s Rotation’, Dordrecht, Holland, Boston, D. Reidel Pub. Co., 165–183. Wahr, J.M., 1981. The forced nutations of an elliptical, rotating, elastic and oceanless earth, Geophys. J. R. Astron. Soc. 64, 705–727.

Consistency of Earth Rotation, Gravity, and Shape Measurements ´ Geoffrey Blewitt and Peter J. Clarke Richard S. Gross, David A. Lavallee,

Abstract Degree-2 spherical harmonic coefficients of the surface mass density determined from independent GRACE, SLR, GPS, and Earth rotation measurements are compared to each other and to a model of the surface mass density obtained by summing the contributions of atmospheric pressure, ocean-bottom pressure, land hydrology, and a mass-conserving term. In general, the independent measurements are found to be quite consistent with each other and with the model, with correlations being as high as 0.87 and with the model explaining as much as 88% of the observed variance. But no measurement technique is found to be best overall. For the different degree-2 coefficients, measurements from different techniques are found to agree best with the model. Thus, each measurement technique contributes to understanding the degree-2 surface mass density of the Earth.

1 Introduction

The rearrangement of mass within the surficial fluid layers of the Earth, including the atmosphere, oceans, and water, snow and ice stored on land, causes the Earth’s gravitational field to change, causes the Earth’s rotation to change by changing the Earth’s inertia tensor, and causes the Earth’s shape to change by changing the load acting on the solid, but not rigid, Earth. Large-scale changes in the Earth’s gravitational field have been measured for more than two decades by satellite tracking and more recently by the CHAMP and GRACE satellite missions. Changes in the Earth’s rotation have been measured since the 1970s by the space-geodetic techniques of satellite and lunar laser ranging (SLR and LLR) and very long baseline interferometry (VLBI) and more recently by the global positioning system (GPS). GPS can also be used to Keywords Earth rotation · Earth shape · Gravity · measure large-scale changes in the Earth’s shape by Surface mass density precisely positioning the sites of a global network of ground-based GPS receivers. On time scales of months to a decade, loading of the solid Earth by surface fluRichard S. Gross ids dominates the observed variations in each of these Jet Propulsion Laboratory, California Institute of Technology, three fundamental geodetic quantities (gravity, rotaMail Stop 238-600, 4800 Oak Grove Drive, Pasadena, CA tion, and shape). Since these three quantities all change 91109, USA in response to the same changes in surface mass load, David A. Lavall´ee observations of these quantities must be consistent with School of Civil Engineering and Geosciences, Newcastle each other. University, Cassie Building, Newcastle upon Tyne NE1 7RU, UK Here we investigate temporal variations in the Geoffrey Blewitt degree-2 spherical harmonic coefficients of the surface Nevada Bureau of Mines and Geology and Seismological mass density, which corresponds to temporal variations Laboratory, University of Nevada, Mail Stop 178, Reno, NV in (1) the Earth’s ellipsoidal shape, (2) the strongest 89557, USA harmonic coefficient of the Earth’s gravitational field, Peter J. Clarke and (3) the only coefficient that relates to the Earth’s School of Civil Engineering and Geosciences, Newcastle University, Cassie Building, Newcastle upon Tyne NE1 7RU, moment of inertia and hence to its rotation in space. UK Thus we choose to investigate degree-2 coefficients M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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because they present the best opportunity to test the consistency between three fundamentally different measurement techniques and with the largest-scale models of the Earth.

ΔClm + iΔSlm =

a (1 + kl )K lm

2



M

Ωo

Ylm (θ, φ)Δσ (θ, φ)dΩ (4)

2 Stokes Coefficients and Surface Mass Density

where the integral extends over the surface Ωo of the Earth. When the change in surface mass density is expressed as a change in the pressure Δp(θ, φ) that is acting on the surface of the solid Earth by Δσ (θ, φ) = Δp(θ, φ)/g, then Eq. (4) becomes:

The gravitational potential U (ro ) of the Earth can be expanded in spherical harmonics in the usual geodetic manner as (e.g., Kaula 1966):

ΔClm + iΔSlm

U (ro ) =

∞ l  GM   a l (Clm cos mφo ro ro

(1)

where ro (ro , θo , φo ) is some external field point, G is the gravitational constant, M is the mass of the Earth, a is some reference radius that is less than ro and which is taken here to be the radius of the Earth, and the P˜ lm are the 4π -normalized associated Legendre functions of degree l and order m. The dimensionless expansion coefficients Clm and Slm are known as Stokes coefficients and are the normalized multipole moments of the Earth’s density field ρ(r) (e.g., Chao and Gross 1987): Clm + i Slm K lm = (1 + kl ) l Ma

r l Ylm (θ, φ)ρ(r) dV (2) Vo

2 ! (2 − δm0 )π 2l + 1

Ωo

Ylm (θ, φ)Δp(θ, φ)dΩ

where g is the gravitational acceleration that has been assumed here to be constant over the surface of the Earth. By expanding the surface mass density in the unnormalized spherical harmonics used by Blewitt and Clarke (2003), Eq. (4) can be used to relate changes in the spherical harmonic coefficients of the surface mass density Δσlm to changes in the Stokes coefficients as: Δσlm =

1 M Nlm (ΔClm + iΔSlm ) 1 + kl a 2

(6)

where the Nlm are normalization factors: Nlm



where the integral extends over the volume Vo of the Earth, the Ylm are the fully normalized surface spherical harmonic functions, the kl are the load Love numbers of degree l, and the K lm are normalization factors: K lm = (−1)m

a2 Mg

(5)

l=0 m=0

+Slm sin mφo ) P˜ lm (cos θo )

= (1 + kl )K lm

  2l + 1 (2 − δm0 )(2l + 1)(l − m)! 1/2 = (7) 4π (l + m)!

The above equations will be used below to infer changes in the degree-2 spherical harmonic coefficients of the surface mass density from measured changes in the degree-2 Stokes coefficients. They will also be used to determine the degree-2 coefficients of the surface mass density from models of atmospheric surface pressure, ocean-bottom pressure, and continental water storage.

(3)

From Eq. (2), changes in the Stokes coefficients caused by changes in density that are concentrated in an infinitesimally thin layer at the surface of the Earth can be written in terms of changes in the surface mass density σ (θ, φ) as:

3 Models of Surface Mass Density Models of the mass density of the surface geophysical fluids are used to help assess the consistency of the measurements. The particular models used are

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described in this section. The sum of these models will and was constrained by in situ and satellite data including expendable bathyther-mograph data and altimetric be compared to measurements in Sect. 4. measurements of sea surface height. The ECCO/JPL series designator of this model run is kf049f. Atmospheric surface pressure was not used to force 3.1 Atmospheric Surface Pressure the ocean model. Instead, the response of the oceans to changes in surface pressure has been taken into acThe atmospheric surface pressure series used here count here when modeling the effects of atmospheric is from the National Centers for Environmental Pre- pressure variations (see Sect. 3.1). Since the ocean model is formulated under the diction / National Center for Atmospheric Research Boussinesq approximation it conserves volume rather (NCEP/NCAR) reanalysis project (Kalnay et al. 1996). Changes in the degree-2 spherical harmonic coeffi- than mass. Artificial mass variations caused by this cients of the gravitational potential caused by changes approximation have been removed here by computing in surface pressure were computed using Eq. (5) the effect on the bottom pressure of a spatially uniform by assuming an inverted barometer response of the layer added to the sea surface that has just the right oceans to the imposed surface pressure variations. fluctuating thickness to a posteriori impose mass The inverted barometer response of the oceans was conservation (Greatbatch 1994). Like the atmospheric surface pressure series, computed by replacing the atmospheric pressure at changes in the degree-2 spherical harmonic coeffieach oceanic grid point with the mean atmospheric cients of the gravitational potential caused by changes pressure calculated over the domain of the global in ocean-bottom pressure were computed using oceans. Degree-2 coefficients of the surface mass density Eq. (5). Degree-2 coefficients of the surface mass were computed from the degree-2 coefficients of the density were computed from the degree-2 coefficients gravitational potential using Eq. (6). In order to match of the gravitational potential using Eq. (6). In order to the temporal resolution and sampling interval of the match the temporal resolution and sampling interval GRACE measurements, 30-day averages of the 6-h val- of the GRACE measurements, 30-day averages of the ues were first formed and then the 30-day values were 12-hour values were first formed and then the 30-day linearly interpolated to the epochs of the GRACE mea- values were linearly interpolated to the epochs of surements. The final values for the modeled degree- the GRACE measurements. The final values for the 2 variations in surface mass density caused by atmo- modeled degree-2 variations in surface mass density spheric surface pressure variations were obtained by caused by ocean-bottom pressure variations were detrending the interpolated values by removing just the obtained by detrending the interpolated values. mean and the trend that was determined from a simultaneous least-squares fit for a mean, a trend, and periodic terms having frequencies of 1 cycle per year (cpy), 2 3.3 Land Hydrology cpy, and 3 cpy.

3.2 Ocean-Bottom Pressure The ocean-bottom pressure series used here is from a data assimilating oceanic general circulation model that was run at the Jet Propulsion Laboratory (JPL) as part of their participation in the Estimating the Circulation and Climate of the Ocean (ECCO) consortium (Stammer et al. 2002). The model was forced with twice daily wind stress and daily surface buoyancy fluxes from the NCEP/NCAR reanalysis project

The land hydrology model used here is the Euphrates version of the Land Dynamics (LaD) model of Milly and Shmakin (2002). This model consists of gridded values of the surface mass density of snow, root-zone soil water, and groundwater given at monthly intervals from January 1980 to May 2005. The sum of the individual snow, soil water, and groundwater mass densities is used here. Changes in the degree-2 spherical harmonic coefficients of the gravitational potential caused by the total continental water storage was computed using Eq. (4) and degree-2 coefficients of the surface mass density

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were computed from the coefficients of the gravitational potential using Eq. (6). The resulting monthly degree-2 surface mass density values of the continental water were linearly interpolated to the epochs of the GRACE measurements and then detrended.

3.4 Global Surface Mass Conservation Since mass can neither be created nor destroyed, the total mass of the atmosphere, oceans, and continental water must be a constant. Of course, the mass of an individual component of the surface geophysical fluids like the atmosphere can change as water in its various phases cycles through it. But the total mass of all components must be a constant. The atmosphere and land hydrology models used here capture the mass changes in these surface fluids, but the ocean model does not. The mass of the ocean model used here does not change because it is not forced by any mechanism that would cause its mass to change. So global mass conservation has been imposed here by adding a layer of water to the surface of the ocean model of just the right time varying thickness to make the total mass of the atmosphere, oceans, and water stored on land a constant. Since the surface mass density of the land hydrology model used here was given at monthly intervals, the mass-conserving ocean layer was computed at monthly intervals. Changes in the degree-2 spherical harmonic coefficients of the gravitational potential caused by the mass-conserving layer were computed using Eq. (4) and degree-2 coefficients of the surface mass density were computed from the coefficients of the gravitational potential using Eq. (6). The resulting monthly degree-2 surface mass density values of the mass-conserving layer were linearly interpolated to the epochs of the GRACE measurements and then detrended.

4 Consistency of Measurements 4.1 GRACE Temporal variations in the Earth’s global gravitational field have been measured from space by the Gravity

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Recovery and Climate Experiment (GRACE) since the spacecraft were launched in March 2002 (Tapley et al. 2004). Spherical harmonic coefficients of the gravitational potential are being determined from the GRACE observations by each of the three centers of the GRACE Science Data System located at the University of Texas at Austin Center for Space Research (UTCSR), the GeoForschungsZentrum Potsdam, and the Jet Propulsion Laboratory. The release-1 (RL01) and release-4 (RL04) coefficients determined by UTCSR have been used here. The atmosphere/ocean de-aliasing product has been added back to the degree-2 coefficients in order to recover the full gravitational signal. While UTCSR removed the effects of the ocean pole tide from the RL04 coefficients (Bettadpur 2007), they did not remove them from the RL01 coefficients. So they were removed here using the self-consistent equilibrium ocean pole tide model of Desai (2002). Since the UTCSR RL04 coefficients were sometimes given at slightly different epochs than the RL01 coefficients, the RL04 coefficients were linearly interpolated to the epochs of the RL01 coefficients. Degree2 coefficients of the surface mass density were then computed from the degree-2 gravitational field coefficients using Eq. (6). In order to match the duration of the GPS-derived surface mass density series, the GRACE series were truncated at May 2005. The 34 monthly coefficients spanning April 2002–May 2005 were then detrended. Figure 1 displays the resulting degree-2 coefficients of the surface mass density as measured by GRACE, with the UTCSR RL01 coefficients being given by the open circles, and the RL04 coefficients being given by the filled circles. Also shown as a thick solid line is the sum of the models of the degree-2 mass coefficients due to atmospheric surface pressure, oceanbottom pressure, land hydrology, and the global massconserving ocean layer. Figure 2 gives the correlations between the observed and modeled series, as well as the percentage of the observed variance explained by the modeled series. With a correlation of 0.83, the (2,0) cosine coefficients of the two GRACE solutions are significantly correlated with each other. And while the (2,1) and (2,2) sine coefficients are quite similar to each other, having correlations of 0.81 and 0.95, respectively, there is less agreement between the cosine coefficients, with the correlations being 0.40 and 0.55 between the (2,1)

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Degree-2 Coefficients of Surface Mass Density

Fig. 1 Observed and modeled degree-2 spherical harmonic coefficients of the surface mass density. The observed coefficients are from the GRACE UTCSR RL01 solution (open circles), the GRACE UTCSR RL04 solution (filled circles), SLR (stars), GPS (crosses), and Earth rotation (diamonds). The modeled coeffi-

cients (solid line) are the sum of those due to atmospheric surface pressure, ocean-bottom pressure, land hydrology, and the global mass-conserving ocean layer. A mean and a trend have been removed from each of the displayed series

and (2,2) cosine coefficients, respectively. The (2,1) and (2,2) sine coefficients also agree much better with the models than do the cosine coefficients. For example, the models explain 58.9% of the observed variance of the RL04 (2,1) sine coefficient and have a

correlation of 0.78 with it, but have a correlation of only 0.26 with the RL04 (2,1) cosine coefficient and the variance actually increases when the models are removed. The RL04 (2,0) coefficient agrees only slightly better with the models than the RL01 coefficient.

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Fig. 2 Zero-lag correlation between the various observed degree-2 surface mass density coefficients and with the modeled coefficients. For each coefficient, the greatest correlation between the observed and modeled series is given in bold, while that between the independently observed series is given in bold

italics. Shown in parentheses is the percentage of the observed variance explained by the models, where for each coefficient the greatest variance explained is given in bold. A negative value for the variance explained is obtained when the variance increases when the modeled series is removed

4.2 Satellite Laser Ranging

replacement values are determined from ranges to five geodetic satellites (Lageos-1, Lageos-2, Starlette, Stella, and Ajisai) using procedures and background models consistent with those used to determine the UTCSR RL01 coefficients and are therefore meant to replace the zonal degree-2 coefficient of just this solution. The degree-2 gravitational field coefficients from SLR used here are these replacement values, with the non-zonal coefficients provided to us by Cheng (personal communication, 2007) and coming from the same SLR solution as the zonal coefficient. Since the SLR series used here is meant to replace the degree-2 coefficients of the UTCSR RL01 GRACE solution, it was pre-processed in the same manner as the UTCSR RL01 series. The atmosphere/ocean de-aliasing product was added back to the degree-2

Temporal variations in the low-degree spherical harmonic coefficients of the Earth’s gravitational field have been determined from laser ranging measurements to the Lageos-1 satellite since it was launched in May 1976 (e.g., Cheng and Tapley 2004). Because the zonal degree-2 gravitational field coefficient determined in early GRACE solutions was less accurate than that determined by satellite laser ranging (SLR), and since many users of the early GRACE solutions either removed the zonal degree-2 coefficient or replaced it with an SLR-based coefficient, the GRACE project started providing replacement values of the zonal degree-2 coefficient based upon SLR measurements (Cheng and Ries 2007). The

Consistency of Earth Rotation, Gravity, and Shape Measurements

coefficients in order to recover the full gravitational signal and the effects of the ocean pole tide were removed using the self-consistent equilibrium ocean pole tide model of Desai (2002). Since the SLR coefficients were sometimes given at slightly different epochs than the RL01 coefficients, the SLR coefficients were linearly interpolated to the epochs of the RL01 coefficients, and they were truncated to span the same duration as the RL01 coefficients. Degree-2 coefficients of the surface mass density were computed from the SLR degree-2 gravitational field coefficients using Eq. (6), and were then detrended. The SLR degree-2 surface mass density coefficients are given by the stars in Fig. 1. The SLR coefficients agree similarly with the two GRACE solutions except for the (2,1) cosine coefficient where it agrees much better with the RL01 solution. The SLR (2,0) cosine coefficient agrees much better with the models than does either GRACE solution, with the models explaining 88.3% of the SLR variance and having a correlation with the SLR of 0.94. But the non-zonal coefficients of the GRACE solutions generally agree better with the models than do the non-zonal SLR coefficients.

4.3 Global Positioning System Following the method given by Blewitt and Clarke (2003) and using the basis functions of Clarke et al. (2007), data from the global network of GPS receivers as reanalyzed at the Scripps Institution of Oceanography (SIO) have been used here to determine changes in the low-degree and order spherical harmonic coefficients of the shape of the Earth’s surface, and hence of the surface mass density that is causing the shape to change. The resulting degree-2 coefficients of the surface mass density span January 1996–May 2005 at fortnightly intervals. In order to match the temporal resolution and sampling interval of the GRACE measurements, monthly averages of the fortnightly GPS values were first formed and then the monthly values were linearly interpolated to the epochs of the GRACE measurements. The GPS degree-2 surface mass density coefficients were then detrended. The GPS degree-2 surface mass density coefficients are given by the crosses in Fig. 1. The (2,2) sine and cosine coefficients from GPS have greater variability than

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the GRACE or SLR coefficients, but the other degree2 coefficients have similar variability. The (2,1) cosine coefficient from GPS agrees nearly as well with the models as does the GRACE RL01 coefficient.

4.4 Earth Rotation The observed Earth orientation series used here is the COMB2005 combined series of Gross (2006). Prior to using Earth orientation measurements to study surface mass loads, other sources of excitation should be removed from the measurements. Solid Earth and ocean tidal effects have thus been removed from the COMB2005 excess length-of-day values using the models of Yoder et al. (1981) and Kantha et al. (1998), respectively. The effects of the termensual (9.12- and 9.13-day), fortnightly (13.63- and 13.66-day), and monthly (27.55-day) ocean tides have been removed from the COMB2005 polar motion excitation series using the model of Gross et al. (1997). The effects of atmospheric winds have been removed using the wind angular momentum series (Zhou et al. 2006) obtained from the International Earth Rotation and Reference Systems Service (IERS) Special Bureau for the Atmosphere (Salstein 2003) that was computed from the wind fields of the NCEP/NCAR reanalysis project. The effects of oceanic currents have been removed using the current angular momentum series (Gross et al. 2005) obtained from the IERS Special Bureau for the Oceans (Gross 2003) that was computed from the currents of the ECCO/JPL data assimilating ocean model kf049f. And decadal-scale variations in length-of-day, caused predominantly by core-mantle interactions, have been removed by applying a highpass filter with a cutoff period of 4 years. The same highpass filter was applied to the polar motion excitation values in order to remove their decadal variations. In order to match the temporal resolution and sampling interval of the GRACE measurements, 30-day averages of the daily Earth orientation values were first formed and then the 30-day values were linearly interpolated to the epochs of the GRACE measurements. Degree-2 zonal and (2,1) sine and cosine coefficients of the surface mass density were computed from the length-of-day and polar motion excitation series using

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Eq. (11) and (12) of Gross et al. (2004). The surface mass density coefficients were then detrended. The degree-2 surface mass density coefficients from Earth rotation measurements are given by the diamonds in Fig. 1. They are in good agreement with the other measurements and agree better with the modeled (2,1) sine coefficient than the other measurements.

5 Discussion and Summary Degree-2 coefficients of the surface mass density determined from GRACE, SLR, GPS, and Earth rotation measurements have been compared to each other in order to assess their consistency. Depending on the particular degree-2 coefficient, the correlations between independent measurements range from a low of 0.03 [between the SLR and UTCSR RL04 measurements of the (2,1) cosine coefficient] to a high of 0.87 [between the SLR and GPS measurements of the degree-2 zonal coefficient]. While the correlation between the UTCSR RL01 and RL04 measurements of the (2,2) sine coefficient is higher at 0.95, these measurements are both based on GRACE observations and are therefore not independent of each other. The worst agreement between the measurements is found for the (2,1) cosine and (2,2) cosine coefficients where the correlations between the measurements are usually less than the 95% significance level of 0.51. The measurements were also compared to the sum of the models of the degree-2 mass density coefficients of atmospheric surface pressure, ocean-bottom pressure, land hydrology, and a global mass-conserving ocean layer. The SLR measurements were found to agree best with the modeled degree-2 zonal coefficient, the Earth rotation measurements with the modeled (2,1) sine coefficient, and the GRACE measurements with the other modeled degree-2 coefficients although the GPS measurements agree nearly as well as the GRACE measurements with the modeled (2,1) cosine coefficient. Except for the (2,1) cosine coefficient, the GRACE UTCSR RL04 solution was found to agree as well or better with the models than the RL01 solution. In conclusion, we find that three fundamentally different measurement techniques (based on gravity, geometry, and rotation) give mutually consistent results that agree well with large-scale models of

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mass redistribution near the Earth’s surface. We therefore expect all three techniques to contribute to future understanding of global-scale redistribution of surface mass. Acknowledgments We thank the IGS analysis center at Scripps Institution of Oceanography for freely providing GPS SINEX files for use by the geodetic community. The work of one of the authors (RSG) described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Support for the work of RSG was provided by the Earth Surface and Interior Focus Area of NASA. Support for GB was provided by NASA grants NNG04G099G and NAG513683. PJC acknowledges grant NER/A/S/2001/01166 from the UK Natural Environment Research Council, and DAL was supported by a Royal Society University Research Fellowship.

References Bettadpur, S. (2007). UTCSR level-2 processing standards document for level-2 product release 0004. GRACE 327-742, CSR-GR-03-03, 17 pp., Center for Space Research, Univ. Texas, Austin. Blewitt, G., and P. Clarke (2003). Inversion of Earth’s changing shape to weigh sea level in static equilibrium with surface mass redistribution. J. Geophys. Res., 108(B6), 2311, doi:10.1029/2002JB002290. Chao, B.F., and R.S. Gross (1987). Changes in the Earth’s rotation and low-degree gravitational field induced by earthquakes. Geophys. J. R. Astr. Soc., 91, pp. 569–596. Cheng, M.K., and B.D. Tapley (2004). Variations in the Earth’s oblateness during the past 28 years. J. Geophys. Res., 109, B09402, doi:10.1029/2004JB003028. Cheng, M.K., and J. Ries (2007). Monthly estimates of C20 from 5 SLR satellites. GRACE Technical Note 05, 2 pp., Center for Space Research, Univ. Texas, Austin. Clarke, P.J., D.A. Lavall´ee, G. Blewitt, and T. van Dam (2007). Basis functions for the consistent and accurate representation of surface mass loading. Geophys. J. Int., in press, doi:10.1111/j.1365-246X.2007.03493.x. Desai, S.D. (2002). Observing the pole tide with satellite altimetry. J. Geophys. Res., 107(C11), 3186, doi:10.1029/2001JC001224. Greatbatch, R.J. (1994). A note on the representation of steric sea level in models that conserve volume rather than mass. J. Geophys. Res., 99, pp. 12767–12771. Gross, R.S. (2003). The GGFC Special Bureau for the Oceans: Past progress and future plans. In: Proceedings of the IERS Workshop on Combination Research and Global Geophysical Fluids. Richter, B., W. Schwegmann, and W.R. Dick (eds). pp. 131–138, IERS Technical Note No. 30, Bundesamts f¨ur Kartographie und Geod¨asie, Frankfurt, Germany. Gross, R.S. (2006). Combinations of Earth orientation measurements: SPACE2005, COMB2005, and POLE2005. Jet Propulsion Laboratory Publ. 06-3, 26 pp., Pasadena, Calif.

Consistency of Earth Rotation, Gravity, and Shape Measurements Gross, R.S., B.F. Chao, and S. Desai (1997). Effect of longperiod ocean tides on the Earth’s polar motion. Prog. Oceanogr., 40, pp. 385–397. Gross, R.S., G. Blewitt, P. Clarke, and D. Lavall´ee (2004). Degree-2 harmonics of the Earth’s mass load estimated from GPS and Earth rotation data. Geophys. Res. Lett., 31, L07601, doi:10.1029/2004GL019589. Gross, R.S., I. Fukumori, and D. Menemenlis (2005). Atmospheric and oceanic excitation of decadal-scale Earth orientation variations. J. Geophys. Res., 110, B09405, doi:10.1029/2004JB003565. Kalnay, E., M. Kanamitsu, R. Kistler, W. Collins, D. Deaven, L. Gandin, M. Iredell, S. Saha, G. White, J. Woollen, Y. Zhu, M. Chelliah, W. Ebisuzaki, W. Higgins, J. Janowiak, K.C. Mo, C. Ropelewski, J. Wang, A. Leetmaa, R. Reynolds, R. Jenne, and D. Joseph (1996). The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Met. Soc. 77, pp. 437–471. Kantha, L.H., J.S. Stewart, and S.D. Desai (1998). Long-period lunar fortnightly and monthly ocean tides. J. Geophys. Res., 103, pp. 12639–12647. Kaula, W.M. (1966). Theor. Satellite Geod. Blaisdell, Waltham, Mass. Milly, P.C.D., and A.B. Shmakin (2002). Global modeling of land water and energy balances. Part I: the Land Dynamics (LaD) model. J. Hydrometeor., 3(3), pp. 283–299.

471 Salstein, D.A. (2003). The GGFC Special Bureau for the Atmosphere of the International Earth Rotation and Reference Systems Service. In: Proceedings of the IERS Workshop on Combination Research and Global Geophysical Fluids. Richter, B., W. Schwegmann, and W.R. Dick (eds). pp. 131–138, IERS Technical Note No. 30, Bundesamts f¨ur Kartographie und Geod¨asie, Frankfurt, Germany. Stammer, D., C. Wunsch, I. Fukumori, and J. Marshall (2002). State estimation improves prospects for ocean research. Eos Trans. Amer. Geophys. Union, 83(27), pp. 289–295. Tapley, B.D., S. Bettadpur, M. Watkins, and C. Reigber (2004). The gravity recovery and climate experiment: Mission overview and early results. Geophys. Res. Lett., 31, L09607, doi:10.1029/2004GL019920. Yoder, C.F., J.G. Williams, and M.E. Parke (1981). Tidal variations of Earth rotation. J. Geophys. Res., 86, pp. 881–891. Zhou Y.H., D.A. Salstein, and J.L. Chen (2006). Revised atmospheric excitation function series related to Earth variable rotation under consideration of surface topography. J. Geophys. Res., 111, D12108, doi:10.1029/2005JD006608.

Current Estimation of the Earth’s Mechanical and Geometrical Parameters A.N. Marchenko

Abstract The Earth’s mechanical and geometrical parameters were estimated from the simultaneous adjustment of the 2nd-degree harmonic coefficients of six gravity field models and seven values of the dynamical ellipticity HD all reduced to the common MHB2000 precession constant at epoch J2000. The transformation of 2nd-degree harmonic coefficients in the case of a finite commutative rotation was developed to avoid uncertainty in the deviatoric part of inertia tensor. This transformation and the exact solution of eigenvalueeigenvector problem are applied to determine (a) the static components and accuracy of the Earth’s tensor of inertia at epoch 2000 and (b) the time-dependent constituents of the inertia tensor. Special attention was given to the computation of temporally varying components of the Earth’s inertia tensor, which are based on the time series of 2nd-degree coefficients through GRACE observations. A remarkable stability in time of the position of the axis A of inertia as the parameter of the Earth’s triaxiality is discussed.

adjustment at the epoch 1997 of four sets of 2nd degree gravitational harmonic coefficients C2m , S2m of the Earth’s gravity models and six values of dynamical ellipticity HD . Most recent gravity field models give more accurate determinations of the time-dependent coefficients C2m (t), S2m (t) from GRACE observations. In addition, latest determinations of the dynamical ellipticity HD derived from the precession constant through VLBI are based on the non-rigid Earth’s rotation theory. This paper will focus on (a) the estimation of the principal moments and products of inertia from the adjustment of new sets of C2m , S2m , and HD at the epoch 2000; (b) the direct computation of time-dependent components of the inertia tensor based on the GRACE time series of C2m (t), S2m (t). The fully normalized coefficients C2m , S2m are selected from the six gravity field models: EGM96, GGM01S, GGM02C, EIGEN-CHAMP03S, EIGENGRACE02S, EIGEN-GL04S1. The time variable coefficients in these solutions are referred to different Keywords Principal axes and moments of inertia · epochs with a spacing of 18 years in between. The secular change in the 2nd-degree zonal coefficients Dynamical ellipticity · GRACE · Precession constant C20 = 1.1628 · 10−11 yr−1 is adopted for these models together with the simple linear model for C21 , S21 represented by the mean pole’s drift with the reference 1 Introduction mean pole coordinates xp (t0 ) = 0.054 , yp (t0 ) = 0.357 at the epoch t0 = 2000 in accordance with The preceding solution for the Earth’s principal IERS Conventions 2003 (McCarthy and Petit, 2004). axes (A, B, C), principal moments of inertia (A, B, To be consistent the following transformations were C), and other fundamental constants was created applied beforehand: (1) prediction of C2m (t), S2m (t) to in (Marchenko and Schwintzer, 2003) from the the common epoch 2000; (2) reduction of C20 to the adopted zero-frequency-tide system; and (3) scaling of these coefficients to common values GM and a. A.N. Marchenko Thus, a consistent set of the Earth’s fundamental National University “Lviv Polytechnic”, Institute of Geodesy, parameters was estimated from the simultaneous Bandera St. 12, 79013 Lviv, Ukraine M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

473

474

A.N. Marchenko

adjustment of the new set of C2m , S2m of six Earth gravity field models and seven values HD of the dynamical ellipticity all transformed to the common value pA = 50.2879225 /yr of the MHB2000 precession constant at epoch J2000 (Mathews et al., 2002). The solution for static components consists of the adjusted five 2nd-degree harmonic coefficients related to the reference IERS pole at the epoch 2000 (with the mean pole coordinates xp (t0 ) = 0.054 , yp (t0 ) = 0.357 ), the principal moments of inertia (A, B, C) of the Earth, HD , the orientation of principal axes, the coefficients in the Eulerian dynamical equations, and the geometrical polar and equatorial flattenings. Temporally varying components of the Earth’s tensor of inertia were found (additionally to secular changes) from the GRACE time series of C2m (t), S2m (t) and adjusted value of the dynamical ellipticity HD , by involving the secular variation HD of the dynamical ellipticity HD .

2 Transformation of 2nd Degree Harmonic Coefficients Through Lambeck’s Formulae Let us express the vector g = [C20 ; C21 ; S21 ; C22 ; S22 ]T of the harmonic coefficients, adopted in the frame XYZ, via the vector gZ = [A20 ; A21 ; B21 ; A22 ; B22 ]T given in the coordinate system X Y Z with a small angle between the axes Z and Z  , expressed by the mean pole coordinates. Simultaneous adjustment of appropriate sets of C2m , S2m to the adopted reference pole based on the Lambeck’s formulae (Lambeck, 1971) was considered in Marchenko and Schwintzer (2003) by means of the equation:

solution of associated spherical triangles. It is easy to verify that after some simple algebra the following relationships are valid tan xp = tan θp cos λp ,

tan yp = − tan θp sin λp , (3)

which give exact expressions for the polar coordinates θp , λp considered at the unit sphere. Hence, Eq. (3) will get a special importance for similar to Eq. (1) transformation, where the non-orthogonal matrix Pxy will replace by some orthogonal matrix Rθλ , which is depended on θp , λp derived now from Eq. (3): θp is the polar distance of the axis Z and λp is the longitude of this axis defined by Eq. (3). Thus, to avoid uncertainty in the deviatoric part of inertia tensor in the case of different sequences of finite rotations we will use in contrast to (Marchenko and Schwintzer, 2003) a commutative rotation about the nodes line of XYZ and X Y Z systems with the following transformation of the coordinate vector r = Q · r from XYZ to X Y Z frame, where according to Madelund (1957) Q = R3 (−λP )R2 (θP )R3 (λP ) is the rotation matrix depended on the polar coordinates (3) of the axis Z in the system XYZ. The corresponding relationship between harmonic coefficients (in XYZ and X Y Z systems) via the (5 × 5)-orthogonal matrix Rθλ reads ⎛

g = Rθλ · gZ

r11 ⎜−r12 ⎜ =⎜ ⎜−r13 ⎝ r14 r15

r12 r22 r23 −r24 −r25

r13 r23 r33 −r34 −r35

r14 r24 r34 r44 r45

⎞ r15 r25 ⎟ ⎟ r35 ⎟ ⎟ · gZ , (4) r45 ⎠ r55

where g = Pxy · gZ ,

(1)

where the matrix Pxy depends only on the mean pole coordinates xp , yp at chosen epoch (including also order 2 terms) with the following relationships for the so-called amplitude θp and azimuth λp : xp = θp cos λp ,

yP = −θp sin λp ,

(2)

introduced by Lambeck (1971) in the planar approximation. To avoid the approximation (Eq. (2)) and the corresponding non-orthogonal matrix Pxy we will determine the angles θp and λp by means of the pole coordinates xp , yp , considered at the unit sphere, from the

r11 = (3 cos2 θP − 1)/2, √ r12 = − 3 cos λP sin θP cos θP , √ r13 = − 3 sin λP sin θP cos θP , √ r14 = ( 3 sin2 θP u1 )/2, √ r15 = 3 cos λP sin λP sin2 θP ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = cos λP sin θP (u2 − cos θP ), ⎪ ⎪ ⎪ ⎪ ⎪ = − cos λP sin θP u4 , ⎪ ⎪ ⎪ ⎪ 2 ⎪ = − sin λP sin θP (2 cos λP u3 + 1),⎪ ⎪ ⎪ ⎪ ⎭ 2 2 = cos λP cos θP + sin λP u2 ,

r22 = cos2 λP u2 + cos θP sin2 λP , r23 r24 r25 r33

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (5)

Current Estimation of the Earth’s Mechanical and Geometrical Parameters

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ = − cos λP sin θP u5 , ⎪ ⎪ ⎬ 2 2 2 2 = −(4 cos λP sin λP u3 − cos θP − 1)/2, ⎪ ⎪ ⎪ ⎪ = cos λP sin λP u23 u1 , ⎪ ⎪ ⎪ ⎭ 2 2 2 = 2 cos λP sin λP u3 + cos θP , (5)

r34 = sin λP sin θP (cos θP − 2 cos2 λP u3 ), r35 r44 r45 r55

475

u1 = 2 cos2 λP − 1, u2 = 2 cos2 θP − 1, u3 = cos θP − 1, u4 = cos θP u1 + 2 sin2 λP , u5 = 2 cos2 λP + 2 cos θP sin2 λP − 1.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Then, the inverse transformation admits the sentation with the orthogonal matrix RTθλ :

g˜ = [A20 , 0, 0, A22 , 0]T of the two nonzero harmonic coefficients A20 , A22 in the coordinate system of the Earth’s principal axes of inertia A B C. Assuming our initial information consisting of the vector g of 2nd-degree coefficients together with their variance-covariance matrix, we will use for such a transformation the exact closed solution of the eigenvalue problem given in Marchenko and Schwintzer (2003) together with accuracy estimation by rigorous error propagation. If the vector g of C2m and S2m is given, the solu(6) tion of the eigenvalue problem provides the computation of the corresponding A20 , A22 in the principal axes system via the eigenvalues of the quadratic form representing the degree 2 potential. By involving the dynamical ellipticity HD we can determine the normalrepre- ized by the factor 1/Ma2 principal moments of inertia A, B, and C:

gZ = RTθλ · g.

(7)

The last relationship together with Eqs. (5) and (6) will be considered as basic equation for the adjustment to the adopted IERS reference pole of different 2nd degree coefficients gZ chosen as observations according to various gravity field models. It has to be pointed out, that in contrast to the Lambeck’s formulae adopted by the IERS Conventions 2003 in the planar approximation, the transformation of the degree 2 potential from XYZ to X Y Z system applying the matrix Q allows to keep the first invariant I1 of the deviatoric part H of inertia tensor. In this case all elements of the 3 × 3 matrix H (XYZ frame), expressed through C2m , S2m , or all elements of the 3 × 3 matrix H (X Y Z system), expressed via A2m , B2m , are connected by the commutative orthogonal rotation that leads to I1 = Trace(H) = Trace(QHQT ) = Trace(H ) = 0. If Eqs. (1 and 2) are used instead of Eqs. (3, 4, 5, 6 and 7), we get the non-zero first invariant I1 = Trace(H ) = 0 ≈ 10−15 .

3 Earth’s Mechanical and Geometrical Parameters Now we consider the transformation of the coefficients C2m , S2m , defined in the Earth’s-fixed geocentric coordinate system XYZ, to the vector

√ HD = (2C − A − B)/2C ⇔ C = − 5A20 /HD (8a) A=

√ √ 5A20 (1 − 1/HD ) − 15A22 /3

B=

√ √ 5A20 (1 − 1/HD ) + 15A22 /3

(8b) (8c)

Then the orientation of the principal axes in the XYZ frame is based on the exact solution of eigenvector problem, using C2m , S2m only without the dynamical ellipticity HD . Table 1 summarizes some current estimates of HD and the precession constant pA . The first five values of HD were discussed in Dehant et al. (1999) as ‘the best values to be used in the nutation theory’ in the year 1999. Note that HD = 0.0032737875 determined by Mathews (2000) was omitted in this study, in view of its preliminary character before the final HD adopted in the MHB2000 non-rigid Earth’s nutation theory (Mathews et al., 2002). With HD , A20 , A22 , the computation √ of C, B, A, and the trace Tr(I) = A + B + C = 5A20 (2 − 3/HD ) = 3Im of the Earth’s inertia tensor according to Eqs. (8a), (8b) and (8c) are straightforward. By this we get a direct dependence of A, B, C, Tr(I), and the mean moment Im of inertia on the adopted gravity field model and on the treatment of the permanent tide in the C20 ≈ A√20 harmonic coefficient. The difference B − A = 2 15A22 /3 is also slightly dependent of a permanent tide system because C20 enters into the computation of the coefficient A22 . It is assumed that

476

A.N. Marchenko

Table 1 Determinations of the dynamical ellipticity HD . Transformed HD -values to the common MHB2000 precession constant pA = 50.2879225 /yr are given in brackets

Precession const. pA [ /yr], J2000

Reference Williams, (1994) Souchay and Kinoshita, (1996) Hartmann et al., (1999) Bretagnon et al., (1998) Roosbeek and Dehant, (1998) Mathews et al., (2002) (MHB2000) Fukushima, (2003)

the HD values are related to the zero-frequency tide system (Bursa, 1995; Groten, 2000). From the initial values of the dynamical ellipticity HD given in Table 1 (assumed to refer to J2000) six values differ in the adopted according to IERS Conventions 2003 (IAU2000 Precession-Nutation model) MHB2000 precession constant. To transform the associated quantities from different pA to the common value pA = 50.2879225 /yr the differential relationship of Souchay and Kinoshita (1996) δHD = 6.4947 · 10−7 δpA was applied to compute the values HD given in brackets in Table 1, which were used in the following adjustment. Thus, the principal moments A, B, C are determined from a least-squares adjustment of the astronomical and geodetic parameters, all referred to a common permanent tide system. As “observations” are taken (a) the seven values for HD (Table 1) and (b) six sets of A20 , A22 , taking into account their variance-covariance matrices. Using Eqs. (8a), (8b) and (8c) we get the overdetermined system of non-linear observation equations (j) (i) with respect to A, B, and C. HD (i = 1, 2, . . . k), A20 ,

HD 0.0032737634 (0.0032737779) 0.0032737548 (0.0032737693) 0.003273792489 (0.0032737745) 0.003273766818 (0.0032737813) 0.0032737674 (0.0032737819) 0.0032737949 0.0032737804 (0.0032737783)

50.287700 (50.2879225) 50.287700 (50.2879225) 50.288200 (50.2879225) 50.287700 (50.2879225) 50.287700 (50.2879225) 50.2879225 50.287955 (50.2879225)

where A0 , B0 , C0 are some approximate values (j) (j) 0 of the principal moments; A20 = A20 − A20 , (j)

(j)

0

(j)

(j)

A22 = A22 − A20 , HD = HD − H0D ; and X = [A; B; C]T is the vector of corrections provided by iterations based on the normal equation system following from Eq. (9). A number of iterations depends on the initial values A0 , B0 , C0 in Eq. (9), which were found from Eqs. (8) using some set of (q) 0 (q) 0 (p) HD = H0D , A20 = A20 , A22 = A22 adopted in zero approximation. Usually it is sufficient to make 2–3 iterations. For each of the 7 values HD an identical standard deviation σ7H = ±0.737 · 10−8 derived from the scattering about the mean value was assumed for the weighting in the subsequent adjustment. Another set of the six HD gave a smaller value of the standard deviation σ6H = ±0.456 · 10−8 (Marchenko and Schwintzer, 2003). The final solution is given here at the epoch 2000 derived from the 7 values of H(i) D from Table 1 plus six sets of 2nd degree harmonic coefficients of the following gravity field models EGM96, GGM01S, GGM02C, EIGEN-CHAMP03S, EIGEN(j) and A22 (j = 1, 2, . . . l) are treated as observations GRACE02S, and EIGEN-GL04S1 (7 + 12 = (i) with ε being an error component. For k values of HD 19 linear equations) taken from the International (j) (j) and for 1 sets of A20 , A22 of 1 gravity field mod- Center for Global Earth’s Models (http://icgem.gfzels we form after linearization (see, Marchenko and potsdam.de/ICGEM/ICGEM.html). Applying the (j) (j) Schwintzer 2003) the system of (k + 2l) observation covariance matrices for every set of A20 , A22 ) we get adjusted principal moments given in Table 2. equations Let us consider the adjustment of different C2m , S2m according to various gravity field models to the adopted ⎛ ⎞ ⎛ 0 (i) ⎞ IERS reference pole. If the coefficients C2m and S2m − 2C1 0 − 2C1 0 A0 +B HD 2C20 ⎟ ⎜ are given, Eq. (7) allow us to compute all coefficients ⎟ ⎜ ⎜ √ (j) 1 1 1 ⎟ + ε, (9) A and B related to X Y Z . Applying Eq. (7) A ⎜ 2 5 2√5 − √5 ⎟ · X = ⎜ A20 ⎟ ⎠ ⎝ 2m 2m 21 ⎝ ⎠ (j) 3 3 and B21 then read √ − √ 0 A22 2 5

2 5

Current Estimation of the Earth’s Mechanical and Geometrical Parameters Table 2 Results of the adjustment of the astronomical HD and geodetic A20 , A22 parameters (zero-frequency-tide system; GM = 398, 600.4415 km3 /s2 ; a = 6, 378, 136.49 m; epoch = 2000) Parameter Solved A B C Derived Im HD (C − A) · 10−6 (C − B) · 10−6 (B − A) · 10−6 α = (C − B)/A β = (C − A)/B γ = (B − A)/C A20 · 10−6 A22 · 10−6 1/f 1/fe

7 HD and 6 gravity field models 0.329612745 ± 0.00000050 0.329620007 ± 0.00000050 0.330699011 ± 0.00000050 0.329977254 ± 0.00000050 0.0032737797 ± 0.0000000050 1086.266876 ± 0.000080 1079.004543 ± 0.000080 7.262334 ± 0.000054 (3273.5523 ± 0.0050) · 10−6 (3295.5126 ± 0.0051) · 10−6 (21.9606 ± 0.0002) · 10−6 −484.1694083 ± 0.000032 2.8126870 ± 0.000034 298.256474 ± 0.000012 91, 435.4 ± 1.1

477

system following from Eq. (7) with the two additional conditions. Taking for all (1 = 6) gravity models (EGM96, GGM01S, GGM02C, EIGEN-CHAMP03S, EIGENGRACE02S, and EIGEN-GL04S1) the harmonic (j) (j) coefficients A2m and B2m in the X Y Z frame as observations, we get in this way the adjusted C2m , S2m – coefficients (at the epoch 2000), given in Table 3. These coefficients restore exactly the adopted mean pole coordinates xp = 0.054 , yp = 0.357 , if inserted into the following expressions xp =

√ ( 3 C20 + C22 )C21 + S22 S21

yp = −

2

2

2

3C20 − C22 − S22

,

√ ( 3 C20 − C22 )S21 + S22 C21 2

2

2

3C20 − C22 − S22

(11a)

,

(11b)

following from Eqs. (3), (4), (5), (6) and (7). Applying the exact equations, the orientation of the principal axes A, B, and C are computed for the adjusted set of B21 = r13 C20 + r23 C21 + r33 S21 − r34 C22 − r35 S22 , degree 2 coefficients (Table 4). (10b) It should be pointed out, that such ‘high’ accuracy of C21 , S21 in Table 3 are result from the application of where the harmonic coefficients A21 and B21 must be the mentioned conditions A21 = B21 = 0 [Eqs. (10a) zero by definition, if the axis Z and the figure axis C and (10b)]. Accuracy of the latitudes of the principal are coinciding at the epoch t0 . Eq. (10a) and (10b) give axes A, B, and C and the longitude of the axis C in a tool to test whether gravity field models are referred to a common axis C. Computation of A21 and B21 for Table 3 Results of the adjustment of the geodetic adopted xp = 0.054 and yp = 0.357 at the epoch C2m , S2m parameters (zero-frequency-tide system; t0 = 2000 (IERS Conventions 2003) leads to the con- GM = 398, 600.4415 km3 /s2 ; a = 6, 378, 136.49 m;   clusion that the reference systems of considered grav- xp = 0.054 , yp = 0.357 ; epoch = 2000) 6 Gravity field models ity field models do not exactly match (differences up Parameter −484.16940829 ± 0.000034 to one order greater than standard deviations given for C20 · 106 6 C · 10 −0.00022261 ± 4.0 · 10−11 21 C21 , S21 ). 6 0.00144760 ± 1.1 · 10−10 To avoid these differences we determine only one S21 · 10 6 C22 · 10 2.43934361 ± 0.000021 adjusted set of C2m and S2m to the adopted IERS refS22 · 106 −1.40029495 ± 0.000021 erence pole at the epoch 2000. For 1 chosen gravity models we initially compute the harmonic coefficients Table 4 Spherical coordinates of the principal axes and their ac(j) (j) A2m , B2m (j = 1, 2, . . . l) and then use Eq. (7) as the curacy according to adjusted C2m , S2m (epoch 2000) linear observation equation system with respect to the Axis Latitude [degree] Longitude [degree] 5 unknowns C2m , S2m of the vector g for each j. The A −0.000040 345.0711 orthogonal matrix RT0λ of this system depends only on ±0.0002 ±0.3 · 10−9 the polar coordinates (3) selected at the epoch 2000. B 0.000092 75.0711 ±0.0002 ±0.2 · 10−9 Two additional conditions are added to Eq. (7) accord89.999900 278.6014 ing to Eqs. (10a) and (10b) with zero left-hand sides. C ±0.2 · 10−5 ±0.1 · 10−10 The vector g results from the solution of the normal A21 = r12 C20 + r22 C21 + r23 S21 − r24 C22 − r25 S22 , (10a)

478

Table 4 for the adjusted C2m , S2m again reflect an influence of the additional conditions A21 = B21 = 0 which were initially introduced via adjustment to the adopted IERS reference pole fixed at the epoch 2000 by the mean pole coordinates in Eq. (3). After transformation of adjusted C2m , S2m to the principal axes system we get the same numerical values of the coefficients A20 , A22 from Table 2. Hence, their combination with the adjusted HD = 0.0032737797 ± 0.000000005 gives the same values for other parameters from Table 2. Therefore, Tables 2, 3, and 4 illustrate one consistent set of the Earth’s fundamental parameters at the epoch 2000 with the adopted treatment of the permanent tide. In comparison with previous results (Marchenko and Schwintzer, 2003) based on Eq. (1) we get generally a similar accordance between the simultaneous adjustment of astronomical and geodetic constants and the separate adjustment of the degree two harmonic coefficients only to the IERS reference pole. But differences between adjusted C2m , S2m based on Eqs. (1), (2), (3), (4), (5), (6) and (7) have values about 10−15 that corresponds to the non-zero I1 in the case of Eqs. (1) and (2).

4 Earth’s Time-Dependent Parameters from GRACE

A.N. Marchenko

Tables 2 and 4 were determined now as time-dependent for each related moment of time t on the total period from 2002 to 2007 years. Because of a great number of various parameters computed for every t we give below only their evolution for the axes C and A of inertia. For other illustrations it is sufficient to show in this study only mean values of some time-dependent quantities obtained by averaging their instant values from 2002 to 2007. Figures 1 and 2 show temporal changes of the longitudes of the axes C and A. Table 5 demonstrate the mean longitudes of these axes and mean values of the angle γ˜ between two axes of the gravitational quadrupole, located in the plane of the axes A and C (Marchenko, 1998): cos γ˜ = (3A22 +

√

3A20 )/(A22 −

√ 3A20 ),

(C − B)/(C − A) = (1 − cos γ)/2. ˜

(12a)

(12b)

A comparison of each C2m (t), S2m (t) taken from various centers of analysis leads to the conclusion about systematic differences presented in these series. However some derived parameters illustrated by Table 5 and Fig. 2 are stable. In contrast to the evident temporal change of the figure axis C (Fig. 1) we get a remarkable stability in time of the position of

The time-dependent 2nd degree harmonic coefficients - CNES-GRGS - CSR Release01 C2m (t), S2m (t) were taken from the International - GFZ Release03 Center for Global Earth’s Models (http://icgem.gfz- JPL Release02 potsdam.de/ICGEM/ICGEM.html) and extracted for the following GRACE time series: (a) CNES-GRGS; (b) CSR Release 01; (c) GFZ Release 03; (d) JPL Release 02. These C2m , S2m with a step size from 10 days (CNES-GRGS) to one month (other solutions) were applied to the direct computation of temporally varying components of the inertia tensor and other associated parameters. To be consistent the following transformations were used to values C2m (t), S2m (t) as in the case of time-independent constituent: (1) reduction of C20 to a common zero-frequency tide system and (2) scaling of these coefficients to common values of GM = 398, 600.4415 km3 /s2 and a = 6, 378, 136.49 m. Fig. 1 Longitude of the axis C of inertia from CNES-GRGS, Taking into account the adjusted dynamical ellip- CSR, GFZ, and JPL time series for the period from 2002 to 2007 ticity HD = 0.0032737797, all parameters listed in years

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of the Earth’s tensor of inertia requires the dynamical ellipticity HD according to Eqs. (8a), (8b) and (8c). To compute the values A, B, and C from the adjusted HD = 0.0032737797 (related to J2000) and the GRACE C2m (t), S2m (t) at each given time t, which is different from the standard epoch 2000, an additional correction δH to HD should be applied. Special study of the GRACE C2m (t), S2m (t) led to a non-stable determination of the secular variation of A22 adopted finally as A22 = 0. As a result, we assume the non-tidal variation δC in the moment of inertia C as a function of C20 only (Yoder et al., 1983) ‘as zonal forces do not change the revolution shape of the body’ (Melchior, 1978) and come to δA = δB = −δC/2 for the trace Tr(I) = const (Rochester and Smylie, 1974). By this ˙ D = −7.8453 · 10−11 yr−1 we get the secular change H ˙ D (t − t0 ) for given in Table 6. This amounts to δH = H the reduction of HD from the year t0 = 2000 to each Fig. 2 Longitude of the axis A of inertia from CNES-GRGS, moment t related to the C (t), S (t) GRACE time 2m 2m CSR, GFZ, and JPL time series for the period from 2002 to 2007 −10 for the reduction series. (Note that δH = −5.5 · 10 years of HD from the year 2000 to 2007). Then, applying for parameters connected with Table 5 Mean longitudes of the principal axes A, C and mean ˙ values of the angle γ˜ between two quadrupole axes for the period C20 = 1.1628 · 10−11 yr−1 the following linear from 2002 to 2007 ˙ − t0 ), where F˙ = dF(t)/dt and dependence δF(t) = F(t Solutions Longit. A Longit. C Angle γ˜ t0 is chosen reference epoch, we give in Table 6 the CNES-GRGS 345.0715◦ 280.860◦ 170.61988◦ corresponding estimates of some secular changes in CSR Release 01 345.0711◦ 279.336◦ 170.61990◦ the frame of the linear model δF(t) = F(t ˙ − t0 ). ◦ ◦ GFZ Release 03 345.0712 280.229 170.61986◦ Among parameters from Table 6 all secular changes JPL Release 02 345.0711◦ 279.469◦ 170.61993◦ ˙ excluding γ˙ and have the same order as variation A 20 p˙ A . According to Williams (1994) the variation p˙ A the inertia axis A derived from GRACE (Fig. 2). was called by the J˙2 precession rate with the estiProcessing of the CHAMP quarterly solutions (Reig- mated range (−11.6 to − 16.8) × 10−3 [ /cy2 ], which ber et al., 2003) for C2m (t), S2m (t) (period from 2000.9 to 2003.4 years) produces the same conclusion Table 6 Secular changes in some astronomical and geodetic about stability of the axis A with the mean longitude parameters corresponding to the secular drift in the coefficient λA = 345◦ .0706E. It has to be pointed out, that the A20 ≈ C20 (t0 = 2000) direction of the principal axis A is considered in the Parameter Secular variation F˙ = dF/dt precession-nutation theory (Bretagnon et al., 1998; ˙ = 1.1628 · 10−11 yr−1 A20 ≈ C20 A 20 √ ˙ Trace(I) Roosbeek and Dehant, 1998) like the parameter of the −11 yr−1 ˙ D = − 5A H HD 20 3C2 = −7.8453 · 10 Earth’s triaxiality or the longitude λA of the major δHD  2 ˙ pA p˙ A = HD / δpA = −0.0121 /cy √ ˙ axis of the equatorial ellipse. Thus, the adjusted to ˙ = 5A20 = 0.8667 · 10−11 yr−1 A A the IERS reference pole at the epoch 2000 numerical √ 3˙ ˙ = 5A20 = 0.8667 · 10−11 yr−1 ◦ ◦ B B value λA = 345 .0711E ± 0 .0002 (Table 4) in terms 3√ ˙ ˙ = − 2 5A20 = −1.7334 · 10−11 yr−1 C of accuracy estimation agrees perfectly with those C √ 3˙ from Table 5 and may be recommended for the Earth’s α = C−B α˙ = − 5A20 (C−B+3A) = −7.8970 · 10−11 yr−1 2 A √ ˙ 3A ◦ ◦ rotation theory: λA = 14 .9289 W ± 0 .0002. β = C−A = −7.8968 · 10−11 yr−1 β˙ = − 5A20 (C−A+3B) B 3B2 √ ˙ If orientation of the principal axes of inertia, the (B−A) γ = B−A γ˙ = 5A20 = 5.7552 · 10−16 yr−1 2 C √ 3C angle γ, ˜ and some other parameters depend only on ˙ 3 5 A 20 ˙f = − = −3.9001 · 10−11 yr−1 2 C2m (t), S2m (t), the determination of temporal changes f - CNES-GRGS - CSR Release01 - GFZ Release03 - JPL Release02

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√ is depended on the adopted J˙2 = − 5C˙ 20 having the value ‘about 0.7% classical acceleration induced by ecliptic motion and two orders of magnitude larger than tidally induced accelerations’. Williams’ J˙2 precession rate p˙ A = −0.014 [ /cy2 ] given in 1994 agrees well with those from Table 6. Because p˙ A − 0.0121[ /cy2 ] was based on the secular variation C˙ 20 = 1.1628 · 10−11 yr−1 adopted in all selected gravity field models this parameter also may be recommended for the Earth’s rotation theory. Table 7 lists mean values of the principal moments of inertia A, B, and C derived from the C2m (t), S2m (t) GRACE time series, the adjusted ˙ D = −7.8453 · 10−11 yr−1 HD = 0.0032737797, and H (Table 6) by averaging their instant values on timeperiod from 2002 to 2007. The comparison of these mean values of GRACE-only principal moments of inertia from Table 7 with the adjusted quantities A, B, and C given in Table 2 leads to a good accordance in terms of accuracy estimation.

of the MHB2000 precession constant) were used in the general adjustment with respect to the Earth’s principal moments of inertia A, B, and C normalized by the factor 1/Ma2 . Results of these adjustments of geodetic and astronomical “constants” represent one set of consistent parameters related to chosen epoch J2000 and given in Tables 2 and 3 as the time-independent constituent of the Earth’s tensor of inertia and other related parameters. Time-dependent components of the Earth’s inertia tensor were found from the GRACE time series of the time-dependent C2m (t), S2m (t) coefficients taken from four solutions. On the one hand, a remarkable stability in time of the position of the inertia axis A was detected from GRACE and CHAMP solutions. On the other hand, the secular change of all static parameters including the dynamical ellipticity and precession constant was estimated. (Computed periodic components at seasonal and shorter time scale require additional study and we omit here the corresponding discussion.) Because the longitude λA of the principal Table 7 Mean values of the principal moments of inertia A, B, axis A is considered in the nutation theory like the and C from the GRACE series of C2m , S2m , adjusted HD = parameter of the Earth’s triaxiality, the obtained value ˙ D = −7.8453 · 10−11 yr−1 for the time- λ = 14◦ .9289W ± 0◦ .0002 can be recommended for 0.0032737797, and H A period from 2002 to 2007 the Earth’s rotation theory together with the J˙2 precesSolutions A B C sion rate p˙ A = −0.0121[ /cy2 ] of the precession conCNES-GRGS 0.32961281 0.32962007 0.33069908 stant p . A CSR R01 GFZ R03 JPL R02

0.32961283 0.32961263 0.32961269

0.32962009 0.32961990 0.32961995

0.33069909 0.33069890 0.33069895

5 Conclusions The transformation of the second-degree harmonic coefficients C2m , S2m for the case of a finite commutative rotation was developed via Lambeck’s formulae for polar coordinates (considered at the sphere) to avoid uncertainty (about 10−15 ) in the deviatoric part of inertia tensor. This transformation was used in the individual adjustment of the geodetic only parameters C2m , S2m (adopted in the zero frequency tide system) of six gravity field models to the IERS reference pole given by the conventional mean pole coordinates xp = 0.054 , yp = 0.357 at the epoch 2000 (IERS Conventions 2003). The same sets of C2m , S2m coefficients together with seven values of the dynamical ellipticity HD (all transformed to the common value PA = 50.2879225 /yr

Acknowledgments I would like to thank T. Herring and anonymous reviewer for their insightful comments, which led to improvements in the discussion of these results. I wish to address my special sincere thanks to V. Dehant and P. Holota for the attention and support of these researches in their presentation at the XXIV General Assembly of IUGG2007 in Perugia.

References Bretagnon, P., G. Francou, P. Rocher, and J.L. Simon (1998) SMART97: A new solution for the rotation of the rigid Earth. Astronomy and Astrophysics, 329, pp. 329–338. Bursa, M. (1995) Primary and derived parameters of common relevance of astronomy, geodesy, and geodynamics. Earth, Moon and Planets, 69, pp. 51–63. Dehant, V. et al. (1999) Considerations concerning the non-rigid Earth nutation theory. Celestial Mechanics and Dynamical Astronomy, 72, pp. 245–309. Groten E. (2000) Parameters of common relevance of astronomy, geodesy, and geodynamics. Journal of Geodesy, 74, pp. 134–140.

Current Estimation of the Earth’s Mechanical and Geometrical Parameters Hartmann, T., M. Soffel, and C. Ron (1999) The geophysical approach towards the nutation of a rigid Earth, Supplement Series, Astronomy & Astrophysics, 134, pp. 271–286. Lambeck K. (1971) Determination of the Earth’s pole of rotation from laser range observations to satellites. Bulletin G´eod´esique, 101, pp. 263–281. Madelund E. (1957) Die Mathematischen Hilfsmittel des Physikers, Berlin, Gottingen, Heidelberg, Springer-Verlag, Marchenko A.N. (1998) Parameterization of the Earth’s Gravity Field. Point and Line Singularities. Lviv Astronomical and Geodetic Society, Lviv. Marchenko A., Schwintzer P. (2003) Estimation of the Earth’s tensor of inertia from recent global gravity field solutions. Journal of Geodesy, Vol. 76, p. 495–509. Mathews, P.M. (2000) Improved models for precession and nutation. In: Proceedings of IAU Colloquium 180 “Towards Models and Constants for Sub-Microarcsecond Astrometry”, Naval Observatory, Washington, pp. 212–222. Mathews, P.M., Herring T.A., and Buffet, B.A. (2002) Modeling of nutation-precession: new nutation series for nonrigid Earth, and insights into the Earth’s interior, Journal of Geophysical Research, Vol. 107, No. B4, 10.1029/2001JB000390. McCarthy D. and Petit G. (2004) IERS Conventions (2003), IERS Technical Note, No. 32, Verlag des Bundesamts fur Kartographie und Geodasie, Frankfurt am Main.

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Melchior P. (1978) The Tides of the Planet Earth. Pergamon, Oxford. Reigber Ch., Jochmann H., W¨unsch J., Neumayer K.-H., Schwintzer. P (2003) First insight into temporal gravity variability from CHAMP. In: Reigber Ch., L¨uhr, H., Schwintzer, P. (eds.), In: “First CHAMP Mission Results for Gravity, Magnetic and Atmospheric Studies”. Springer-Verlag BerlinHeidelberg, pp. 128–133. Rochester, M.G., and D.E. Smylie (1974) On changes in the trace of the Earth’s inertial tensor. Journal of Geophysical Research, 79 (32), pp. 4948–4951. Roosbeek, F., and V. Dehant (1998) RDAN97: An analytical development of rigid Earth nutation series using the torque approach. Celestial Mechanics, 70, pp. 215–253. Souchay, J., and H. Kinoshita (1996) Corrections and new developments in rigid Earth nutation theory: I. Luni-solar influence including indirect planetary effects, Astronomy and Astrophysics, 312, pp. 1017–1030. Williams, J.G. (1994) Contributions to the Earth’s obliquity rate, precession and nutation, Astronomical Journal, 108, pp. 711–724. Yoder, C.F., J.G. Williams, J.O. Dickey, B.E. Schutz, R.J. Eanes, and B.D. Tapley (1983) Secular variation of earth’s gravitational harmonic J2 coefficient from Lageos and nontidal acceleration of earth rotation. Nature, 303, pp. 757–762.

The Earth’s Global Density Distribution and Gravitational Potential Energy A.N. Marchenko

Abstract The Earth’s global density model given by the restricted solution of the 3D Cartesian moments problem inside the ellipsoid of revolution was adopted to preserve in this way the external gravitational potential up to second degree/order, the dynamical ellipticity, the geometrical flattening, and six basic radial jumps of density as sampled for the PREM model. Comparison of lateral density anomalies with estimated accuracy of density leads to the same order values in uncertainties and density heterogeneities. Hence, four radial density models were chosen for the computation of the Earth’s gravitational potential energy E: LegendreLaplace, Roche, Bullard, and Gauss models. The estimation of E according to these continuous density models leads to the inequality with two limits. The upper limit EH agrees with the homogeneous distribution. The minimum amount EGauss corresponds to the Gauss’ radial density. All E-estimates give a perfect agreement between EGauss , the value E derived from the piecewise Roche’s density with 7 basic shells, and the values E based on the four simplest models separated additionally into core and mantle only. Keywords Earth’s density distribution · propagation · Gravitational potential energy

Error

1 Introduction Determination of the Earth’s volume density distribution δ(ρ, ϑ, λ) from external potential data requires A.N. Marchenko National University “Lviv Polytechnic”, Institute of Geodesy, Bandera St. 12, 79013 Lviv, Ukraine

a solution of the known inverse problem of the Newtonian potential. If the planet’s gravitational potential energy E and density at the surface are accepted as additional information, this problem transforms from an improperly posed to a properly posed problem with its possible solution for the 3D density δ(ρ, ϑ, λ) through the three-dimensional Cartesian moments (Mescheryakov, 1977). According to Gauss (1840) the search of the stationary value E can be treated as one of central subjects of the potential theory. A remarkable summary of the Gauss’ problem reads: “minimum and maximum potential energy correspond to physically (for the Earth) meaningless cases: a surface distribution and a mass point. The ‘true’ Earth lies somewhere in between” (Moritz, 1990). It is obvious that the potential energy E can be estimated from the density and internal gravitational potential. However only few E-values for the homogeneous Earth (Mescheryakov, 1973; Rubincam, 1979; Moritz, 1990) and the planet differentiated into homogeneous mantle and homogeneous core (Rubincam, 1979) are found in the literature. Thus, the question remains: how can we evaluate better this ‘true’ Earth and the corresponding potential energy E. This study focuses on (a) the determination of the Earth’s global density distribution and (b) the estimation of the gravitational potential energy E using continuous and piecewise density models. The Earth’s mass and principal moments of inertia represent initial information for the unique solution of the restricted Cartesian moments problem providing in this way the density δ(ρ, ϑ, λ) and the potential energy E. The principal moments of inertia given in Marchenko (2007) were used for the computation of the 3D global density δ(ρ, ϑ, λ). It should be pointed out, that accuracy of the global density and potential

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

483

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⎫ ⎪ ⎪ ⎪ ⎪ = (B + C − A)/2,⎬ = (A − B + C)/2,⎪ ⎪ ⎪ ⎪ = (A + B − C)/2,⎭

I000 = 1,

energy was derived especially to restrict the possible solution domain in such a way that a reasonable solution may be selected either from 3D-spatial or radial density inside the ellipsoidal or spherical planet.

I200 I020 I002

2 The Earth’s Global Density Distribution Let us consider the mathematical model of the 3D global density distribution δ(ρ, ϑ, λ) derived by (Mescheryakov et al., 1977) inside the Earth having a shape of the ellipsoid of revolution with the flattening f and the semimajor axis a. According to Mescheryakov (1991) the exact (restricted by the order 2) solution of the three-dimension Cartesian moments problem for δ(ρ, ϑ, λ) reads δ(ρ, ϑ, λ) = δ(ρ)R + δ(ρ, ϑ, λ),

(1)

δ(ρ, ϑ, λ) = K + ρ2 (K1 sin2 ϑ cos2 λ

⎫ √ 5A20 (1 − 1/HD ) − 15A22 /3,⎪ ⎪ ⎬ √ √ B = 5A20 (1 − 1/HD ) + 15A22 /3, ⎪ √ ⎪ ⎭ C = − 5A20 /HD .

A=

√

I000 = I000 − IR 000 ,

I200 = I200 − IR 200 ,

I020 = I020 − IR 020 ,

I002 = I002 − IR 002 .

 (4)

In the relationships above χ = 1 − f, the dimensionless Cartesian moments I000 , I200 , I020 , and I002 (I100 = I010 = I001 = 0) of the density of a gravitating body (Grafarend et al., 2000) can be computed through the Earth’s mass M and dimensionless principal moments of inertia A, B, and C normalized by the factor 1/Ma2 :

(6)

The reference model δ(ρ)R includes individual information about density jumps, the mean density δR m, R and the mean moment of inertia Im , which have been selected preliminary for the construction of the radial profile δ(ρ)R . In contrast to Mescheryakov (1991) [Eqs. (1), (2), (3) and (4)] the Cartesian moments IR 000 , R , and IR of the reference density δ(ρ) were IR , I R 200 020 002 derived here for one common set of the conventional constants δm and Im of the model (1) and density jumps entering into δ(ρ)R :

+ K2 sin2 ϑ sin2 +K3 cos2 ϑ), (2) where δ(ρ)R is the piecewise reference radial density model with radial density jumps such as PREM (Dziewonski and Anderson, 1981), δ(ρ, ϑ, λ) is some anomalous density with the following components ⎫ 5 I002 ⎪ ⎪ K = δm [5I000 − 7(I200 + I020 + )], ⎪ ⎪ 4 χ2 ⎪ ⎪ ⎪ ⎪ 35 I002 ⎪ ⎪ ⎪ K1 = − I ), δm (3I200 + I020 + 000 ⎬ 2 4 χ 35 I002 ⎪ ⎪ K2 = − I000 ), ⎪ δm (I200 + 3I020 + ⎪ ⎪ 4 χ2 ⎪ ⎪ ⎪ ⎪ 35 I002 ⎪ ⎭ K3 = δm (I200 + I020 + 3 2 − I000 ), ⎪ 4 χ (3)

(5)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

δR m , δm

IR 000 =

R 3IR m δm , 2δm (χ2 + 2) ⎪ ⎪ ⎪ ⎪ ⎪ R ⎪ δ 3 · χ2 IR ⎪ m m ⎭ = , 2 · δm (χ2 + 2)

R IR 200 = I020 =

IR 002

1 δR m =3

δ(ρ)R ρ2 dρ, 0

2(χ2 + 2) IR = m 3δR m

1

(7)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ 4 ⎪ δ(ρ)R ρ dρ.⎪ ⎪ ⎭

(8)

0

Thus, in these formulae ρ (0 ≤ ρ ≤ 1) is the relative distance from the origin of a coordinate system to an internal current point; ϑ and λ are the polar distance and longitude of this point; δm is the convenient mean density; HD is the dynamical ellipticity; A20 , A22 are the fully normalized (non-zero) harmonic coefficients adopted here as Stokes constants in the Earth’s principal axes system OABC. Therefore, this 3D global density (Eq. (1)) is given in the geocentric coordinate system of the principal axes of inertia and agreed with the

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485

Earth’s mass and the principal moments of inertia to Table 1 Initial parameters and their accuracy preserve in this way the external gravitational potential Reference Adopted parameters from zero to second degree/order, HD , the geometrical Groten, (2004) G = (6.67259 ± 0.0003) · 10−11 m3 kg−1 s−2 flattening f, and density jumps. Groten, (2004) GM = (398600.4415 ± 0.0008) · 109 m3 s−2 Marchenko, A = 0.32961274 ± 0.0000005 The radial density δ(ρ)R is also treated within the (2007) ellipsoid of revolution if we use the formula re = Marchenko, B = 0.32962001 ± 0.0000005 R(1 − 2f · P2 (cos ϑ)/3) for the radius vector re by (2007) neglecting f2 (Moritz, 1990), where P2 (cos ϑ) is the Marchenko, C = 0.33069901 ± 0.0000005 2nd-degree Legendre polynomial. This formula results (2007) 1/f = 298.25650 ± 0.00001 from the average of re over the unite sphere that gives Marchenko and the mean radius R = 6371 km. According to Mesch- Schwintzer, 2003 eryakov (1991) Eqs. (1) and (2) are valid for a homothetic stratification when f = const inside the ellipsoidal Earth. Hence, if the set of the internal ellipsoidal surfaces ˜re is labeled by the associated mean radius r of a sphere we have and the gravitational constant G = (6.673 ± 0.010) · −11 m3 kg−1 s−2 suggested by the IERS Conventions 10   2 r ˜re 2003 (McCarthy and Petit, 2004) are selected, we get ˜re = r 1 − f · P2 (cos ϑ) ⇒ ρ = = . (9) σδm = 0.08 g/cm3 . According to the IAG recommen3 R re dations for G and GM (see Table 1) another mean den3 By averaging δ(ρ, ϑ, λ) over ellipsoidal surfaces we sity δm = (5.5145 ± 0.0026) g/cm was estimated. Thus, the global density distribution and accuracy define the piecewise radial density δ(ρ) as at different depths were based on the value δm = ⎫ (5.5145±0.0026) g/cm3 , the flattening f, and the prin⎪ δ(ρ) = δ(ρ)R + [K + ρ2 D], ⎬ 

  cipal moments of inertia A, B, and C from Table 1. The I002 35 ⎪ δm 5 I200 + I020 + − 3I , D = ⎭ 000 principal moments of inertia (given here in the zero 12 χ2 (10) frequency tide system) are results from the adjustment (Marchenko, 2007) of the 2nd-degree harmonic coefwith the treatment of the reference density δ(ρ)R ficients of 6 gravity field models and 7 values HD of within the ellipsoidal Earth. Since the radius ρ is the dynamical ellipticity all transformed to the comconstant for each ˜re , the radial densities δ(ρ)R and mon value of precession constant at epoch J2000. The δ(ρ) are also constant by Eq. (10) at the ellipsoidal reference radial density profile δ(ρ)R in Eq. (1) was selected in the form of the simple piecewise Roche’s law surface Eq. (9). Accuracy estimation of the 3D continuous global separated into seven basic shells (Marchenko, 2000), density was derived from error propagation, keeping which is slightly different from the PREM-density. Therefore, with δ(ρ)R known as exact constituent, in mind that information about accuracy of the mean R , and den- the accuracy estimation σ density δR , the mean moment of inertia I δ(ρ,ϑ,λ) of the 3D continum m ˜ sity jumps in different piecewise radial models δ(ρ)R ous global density distribution δ(ρ, ϑ, λ) (based only (such as PREM) are not found in literature or were on the Earth’s mechanical parameters) and lateral dennot easily accessible to the author. For this reason we sity heterogeneities (Eq. (2)) are straightforward. Comwill consider the reference density δ(ρ)R as some exact parison of these lateral density anomalies δ(ρ, ϑ, λ) constituent or “normal density”. Hence, the variance- (Fig. 1) with the accuracy σδ(ρ,ϑ,λ) of the continuous covariance matrix of the principal moments of iner- constituent at the same depths (Fig. 2) leads generally tia, accuracy of the mean density σδm , and accuracy at least to values of the same order in uncertainties and of the flattening σf were chosen as initial information density heterogeneities taken for various depths. Because discussed uncertainties are increasing when ra(Table 1). The accuracy σδm of the mean density δm requires dius ρ is decreasing to zero (origin) we will use below additional remarks because this value represents a scale only radial density models for further determination of factor of the considered theory. If δm = 5.514 g/cm3 the Earth’s gravitational potential energy.

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Fig. 1 Density anomalies [g/cm3 ] δ(ρ, ϑ, λ) (Eq. (2)) at the mantle/crust boundary (r = 6346.6 km)

g/cm3

Fig. 2 Accuracy σδ(ρ,ϑ,λ) of the continuous constituent of 3D density distribution [g/cm3 ] at the mantle/crust boundary (r = 6346.6 km)

g/cm3

3 Estimation of the Gravitational Potential Energy

For the determination of the potential energy E we will examine additionally to the homogeneous Earth the following radial-only continuous density profiles: Legendre-Laplace law, Roche’s law (as solutions of As well-known the computation of the Earth’s gravithe Clairaut’s equation), Bullard’s model, and Gaustational potential energy is based on the following exsian (normal) distribution (see Fig. 3). Therefore, in orpression (Moritz, 1990): der to determine the gravitational potential energy E we use density laws from Table 2 initially for the spherical 1 E=− δ · Vi · dτ, (11) Earth. 2 τ The parameters of the simplest density models (Fig. 3) listed in Table 2 were derived in the closed where δ is the Earth’s density, Vi is the internal graviform (Marchenko, 2000) from the solution of the tational potential, and τ is the planet’s volume.

The Earth’s Global Density Distribution and Gravitational Potential Energy (r) Gauss’ model

Legandre-Laplace model

PREM model Bullard’s model

487

internal potentials we find final expressions given in Table 3 for the estimation of the potential energy E of the spherical Earth. With adopted δm and the dimensionless mean moment of inertia Im = 0.3299773±0.0000005 also taken from Table 1 numerically we get estimations of the energy E given in Table 4, which includes E-estimates given by Mescheryakov (1973) and Rubincam (1979) for further comparisons. Thus, there are two limits for all computed E: EGauss ≤ EEarth ≤ EHomogeneous .

Roche’s model

r [km]

Fig. 3 Legendre-Laplace, Roche, Bullard, and Gauss continuous densities compared with the PREM-density model δ(ρ) [g/cm3 ]

Table 2 Expressions for different radial density models Model

Mathematical expression

Homogeneous planet Legendre-Laplace law Roche’s law Bullard’s model Gauss’ model (Marchenko, 2000)

δ(ρ) = δm = const δ(ρ) = δ0 sin(γρ)/(γρ) δ(ρ) = a + bρ2 δ(ρ) = a + bρ2 + cρ4 δ(ρ) = δ0 exp(−β2 ρ2 )

inverse problem based on the well-known conditions to keep δm , Im , and the density δs at the Earth’ surface (Moritz, 1990). The parameters δ0 and a (Tables 2 and 3) represent the density at the origin. Initially we derive relationships for the internal potential Vi corresponding to these density laws. Then, applying Eq. (11) to the density models from Table 2 and

(12)

The upper limit EH agrees with the homogeneous Earth. The minimum amount EGauss corresponds to the Gauss’ model. Such assertion has an evident mathematical explanation. The first term of the Taylor series expansion of EGauss from Table 3 represents the gravitational potential energy of the homogeneous Earth EH . Globally speaking every model from Table 3 includes the main term equal to EH . But the sum of other terms with EH gives a smaller E than the value EH . It has to be pointed out, that the energy E derived by Mescheryakov (1973) as 2E = −Vm M was based Table 4 Estimations of the gravitational potential energy E (spherical Earth) Model

E, ergs

Mescheryakov, 1973 Rubincam, 1979 Homogeneous planet Legendre-Laplace law Roche’s law Bullard’s model Gauss’ model

−2.34 × 1039 −2.45 × 1039 −2.2419 × 1039 −2.4595 × 1039 −2.4802 × 1039 −2.4716 × 1039 −2.5009 × 1039

Table 3 The gravitational potential energy E of the spherical Earth for different radial density models Model

Mathematical expression

Homogeneous planet

E = −16π2 Gδ2m R5 /15 

4π2 Gδ20 R5 2 γ − 3 sin γ cos γ + 1 2 cos E=− γ γ4 16π2 GR5 2 2 (21a + 24ab + 7b ) E=− 315 5  2  16π GR 286a(36b + 25c) + 9009a2 + 3(1001b2 + 1404bc + 495c2 ) E=− 315135   √ √ √ π2 Gδ20 R5 2 π exp(−β2 )erf(β) 2πerf( 2β) 2) − − 2 exp(−2β E=− β 2β β4

Legendre-Laplace law Roche’s law Bullard’s model Gauss’ model

488

on the known Earth’s mass M and the mean-value theorem after preliminary computation of the mean value Vm of the internal potential Vi in Eq. (11). The estimation of E given by Rubincam (1979) was found for the spherical Earth differentiated into homogeneous core and homogeneous mantle with one jump at the coremantle boundary. We will apply a similar approach to the above-discussed profiles using the direct approximation of the PREM density by these four simplest piecewise models separated into two shells with the same basic jump at the core-mantle boundary. Table 5 illustrates results of such approximation in the form of r.m.s. deviations from the PREM density based in every case on the additional conditions to keep δm , Im , and δs . Despite the best value of r.m.s. for the Bullard’s model we prefer to use below a simpler Roche’s law due to a smaller number of the parameters aj and bj (j = 1, 2, . . . k) introduced for each shell. The comparison of E-values from Tables 4 and 5 gives better-quality agreement between all values of E when the basic jump of density at the core/mantle boundary is taking into consideration. E-estimates given in Table 5 fulfill again to the inequality Eq. (12) with the two limits EGauss and EH from Table 4. All values of E in the case of these piecewise models from Table 5 are very close to the minimum amount EGauss . For this reason the accuracy σEGauss of EGauss was derived under the assumption that σEGauss depends only on accuracy of δm and Im given above. Numerically we get EGauss = (−2.5009 ± 0.0025) × 1039 ergs. Hence, if a spherical Earth differentiates into present-day core and mantle we get in view of the estimated accuracy σEGauss = ±0.0025 × 1039 ergs a perfect accordance between E-values corresponded to the layered Legendre-Laplace, Roche, Bullard, and Gauss models with 2 shells. This quantity σEGauss

A.N. Marchenko

is certainly larger than E-estimates contained in the 2nd-degree harmonics (Rubincam, 1979) and for this reason we will use again radial-only piecewise model for the determination of the potential energy E of the ellipsoidal Earth. The internal potential Vi inside the ellipsoid of revolution with the radial density δ(r = ρ · R) was adopted according to Moritz (1990, p. 41). For the homothetic stratification f = const we get sphera

Vi = Vi

4πG = r

+ Vell i

r

R δ(r)r dr + 4πG

δ(r)rdr + Vell i ,

2

r

0

(13) Vell i

8πG · f = P2 (cos ϑ) 3r

r δ(r)r2 dr 0

−

8πG · f P2 (cos ϑ) 3r

r δ(r)r4 dr,

(14)

0

the internal potential of the ellipsoidal Earth (Eq. (13)) in the form of the internal potential of the spherical sphere planet Vi reduced to Vi by the ellipsoidal reducell tion Vi (Eq. (14)). Equations (13) and (14) allow the direct computation of the potential energy E in the following way E = ESphere + Eell ,

(15)

if inserted into Eq. (11). Then, taking into account the flattening f we will determine the ellipsoidal reduction Eell beforehand. Since the values E of the piecewise radial models with one jump (Table 5) are very close to the lower limit EGauss in Eq. (12) it Table 5 Estimations of E for the spherically symmetric Earth is enough to estimate Eell by applying the Gauss’ with basic jump of density at the core/mantle boundary (2 shells) continuous model inside the ellipsoid with the homoR.m.s. thetic stratification. Numerically we get EGauss ≈ ell deviation from 0.000045 × 1039 ergs two orders smaller value than PREM accuracy σEGauss = ±0.0025 × 1039 ergs. Hence, it is (core-mantle sufficient to adopt the reduction Vell Model E, ergs area), g/cm3 i = 0 in Eq. (15) for the internal potential V . 39 i Rubincam, 1979 −2.45 × 10 – If the expression for E is known, the piecewise 0.430 Legendre-Laplace law −2.4944 × 1039 0.409 Roche’s law −2.4938 × 1039 PREM profile is one of a most suitable densities for 0.322 Bullard’s model −2.4907 × 1039 the estimation of the potential energy E, although 0.437 Gauss’ model −2.4940 × 1039 this problem is not discussed in the literature. Due

The Earth’s Global Density Distribution and Gravitational Potential Energy

489

the value 0.06 g/cm3 for the most important in our case core-mantle area and increases only to 0.24 g/cm3 for the total Earth (core-mantle-crust). With Vell i = 0, r0 = 0, and a current point lied within the j shell at the distance r, the substitution of Eq. (16) into Eq. (13) provides the expression for the internal potential

(r)

Piecewise PREM model with 11 shells

4πG  Vi (r) = r j

η

4πG δ1 (r)r dr + r

r δj (r)r2 dr

2

l=1 η−1

rj

rj+1

η+1 k  δj+1 (r)rdr + 4πG δ1 (r)rdr.

+ 4πG

Piecewise Roche’s model with 7 shells

1=j+1 η

r

r [km]

(17) Fig. 4 Piecewise Roche-density model with 7 shells compared with PREM-density model δ(ρ) [g/cm3 ]

Then we substitute Eq. (16) into Eq. (17), obtaining

to polynomials of different powers adopted for each shell there are significant difficulties in the derivation of such relationship for E in the case of the PREM model. Therefore we will apply another appropriate model represented by polynomials of identical even powers within every shell. Because the PREMprofile agrees well with the piecewise Roche model (Marchenko, 2000) consisting from 7 shells (Fig. 4), we use this Roche density as initial information in the following form

Vi (r) =

δj (r) = aj + bj

r 2 R

  GMj 3bj 5 4πG 5 + aj (r3 − r3j ) + (r − r ) j r 3r 5R2   bj 4 4πG j aj (r2j+1 − r2 ) + (r − r4 ) + Cint , + 2 2R2 j+1 (18) ⎫  j  ⎪ ⎪ 4π  3b1 5 ⎪ 3 3 5 ⎪ Mj = (r1 − rl−1 ) , a1 (r1 − rl−1 ) + ⎪ ⎪ 2 3 ⎬ 5R l=1

j

Cint =

 ⎪ k  ⎪ 4πG  b1 4 ⎪ 4 ⎪ (r − r ) a1 (r21+1 − r21 ) + 1+1 1 .⎪ ⎪ 2 ⎭ 2 2R 1=j+1

, a0 = b0 = 0,

(19)

(16)

Thus, according to Eqs. (17) and (18) in the case of the piecewise Roche’s density (Eq. (16)) the internal potential Vi (r) at the arbitrary current point P can be formed from the four parts: first two terms and last two terms in Eq. (18) represent the potentials of the Earth’s layers lied below and above P, respectively. By this, after some algebraic manipulation with Eq. (18) inserted into Eq. (11) we get a simple possibility of the deterTable 6 Piecewise density Roche’s model separated into 7 basic mination of the gravitational potential energy E. The shells as sampled for the PREM (Marchenko, 2000) result is

where j = 0, 1, 2, . . . k, k is the number of shells (k = 7), aj and bj are the known coefficients of the model (16) given for each shell separately (Table 6) with the artificial zero shell a0 = b0 = 0 involved here for the generalization of basic formulae. Note also that r.m.s. deviation between these models (Fig. 4) has

Shell 1 (Inner core) 2 (Outer core) 3 (Lower mantle) 4 (Upper mantle) 5 (Upper mantle) 6 (Upper mantle) 7 (Crust)

aj , g/cm3

bj , g/cm3

rj , km

13.061 12.483 6.370 6.058 5.784 6.057 6.622

−8.891 −8.343 −2.574 −2.577 −2.524 −2.903 −3.952

1221.5 3480.0 5701.0 5971.0 6151.0 6346.6

k 

rj

E = −2π

δj (r)Vi (r)r2 dr =

j=1 rj−1

Ej = −2π

8  m=1

cmj ,

k  j=1

⎫ ⎪ ⎪ ⎪ Ej ,⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(20)

490

A.N. Marchenko

where Ej expresses the contribution of each j-shell in spherically symmetric Earth differentiated into core the total value E by and mantle only. We may assume that the quantity E = −2.4910 × 1039 ergs will be close to E-value of ⎫ 2 2 the PREM model, taking into account a minimum conc1j = aj Mj (rj − rj−1 )/2, ⎪ ⎪ ⎪ ⎪ tribution E7 of the crust (Table 7) into the total E and ⎪ ⎪ c2j = 4πGaj (aj Aj + bj Bj ), ⎪ ⎪ r.m.s. deviation between PREM and piecewise Roche’s ⎪ ⎪ ⎪ c3j = 4πGaj (aj Aj + bj Cj ), ⎪ models. ⎪ ⎪ ⎪ j ⎪ 3 3 ⎬ c4j = aj Cint (rj − rj−1 )/3, (21) c5j = bj Mj (r4j − r4j−1 )/(4R2 ), ⎪ ⎪ ⎪ ⎪ ⎪ 4 Conclusions ⎪ c6j = 4πGbj (aj Cj + bj Dj /R2 ),⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c7j = 4πGbj (aj Bj + bj Dj /R2 ),⎪ ⎪ ⎪ ⎪ The global density model inside the ellipsoidal Earth j 5 5 2 ⎭ c8j = bj Cint (rj − rj−1 )/(5R ), was adopted as exact solution of the restricted threedimensional Cartesian moments problem for δ(ρ, ϑ, λ) under the conditions to conserve the Earth’s mass, the ⎫ geometrical flattening, and all principal moments of Aj = (3r5j−1 − 5r3j−1 r2j + 2r5j )/30, ⎪ ⎪ ⎪ ⎪ inertia. This model includes the reference radial den⎪ 7 5 2 7 2 Bj = (5rj−1 − 7rj−1 rj + 2rj )/(70R ), ⎬ (22) sity profile δ(ρ)R selected in the form of the piecewise Cj = (3r7j−1 − 7r3j−1 r4j + 4r7j )/(84R2 ), ⎪ ⎪ Roche’s models with 7 basic shells, taking into account ⎪ ⎪ ⎪ 9 5 4 9 2 ⎭ density jumps as sampled for PREM. With δ(ρ)R choDj = (5rj−1 − 9rj−1 rj + 4rj )/(180R ). sen as exact constituent, the accuracy σδ(ρ,ϑ,λ) of continuous global density was derived from the consistent With adopted piecewise Roche’s density (Fig. 4, set of the Earth’s mechanical parameters. Comparison Table 6) we get the estimation of the potential energy E of the lateral density anomalies δ(ρ, ϑ, λ) with the (Table 7) by means of the contributions Ej of each shell accuracy σδ(ρ,ϑ,λ) at the same depths leads generally 39 (Eq. (20)). The obtained quantity E = −2.4910 × 10 at least to values of the same order in uncertainties and ergs agrees well with E-estimates from Table 5 based density heterogeneities. on the radial profiles with one jump at the core-mantle As a result, only radial density models were adopted boundary and fulfills to the inequality Eq. (12) at the for the determination of the gravitational potential envicinity EGauss = (−2.5009 ± 0.0025) × 1039 ergs of ergy E. Relationships for E were derived in the followthe minimum amount. In view of the accuracy σEGauss = ing cases: (1) continuous Legendre-Laplace, Roche, ±0.0025 × 1039 ergs we get a remarkable accordance Bullard, and Gauss radial density laws; (2) the same rabetween E = −2.4910 × 1039 ergs derived from the dial models with one added jump of density at the corepiecewise Roche’s density with 7 basic shells as sammantle boundary (2 shells); (3) the piecewise Roche’s pled for PREM and the values E given by the simprofile separated into 7 shells. The estimation of E acplest piecewise Legendre-Laplace, Roche, Bullard, and cording to various continuous density laws gives the Gauss models with 2 shells all corresponded to the following result: there are two limits for all computed E. First one agrees with the homogeneous distribution. Table 7 Estimations of the gravitational potential energy E de- Second one corresponds to the Gauss’ radial density. rived from the piecewise Roche’s model (Table 6) Finally all determinations of the potential energy E Shell Contribution Ej of each shell, ergs were made for the spherical Earth since the ellipsoidal 1 (Inner core) −0.0541 × 1039 reduction Eell gives two orders smaller quantity than 2 (Outer core) −0.9159 × 1039 the estimated accuracy σEGauss = ±0.0025 × 1039 ergs. 3 (Lower mantle) −1.1625 × 1039 Taking into account σEGauss we get a perfect agreement 4 (Upper mantle) −0.1527 × 1039 between EGauss = (−2.5009 ± 0.0025) × 1039 ergs, 5 (Upper mantle) −0.0954 × 1039 the potential energy E = −2.4910 × 1039 ergs derived 6 (Upper mantle) −0.0998 × 1039 39 7 (Crust) −0.0104 × 10 from the piecewise Roche’s density with 7 basic shells, Total gravitational potential energy: −2.4910 × 1039 ergs and the values E given by the four simplest piecewise

The Earth’s Global Density Distribution and Gravitational Potential Energy

Legendre-Laplace, Roche, Bullard, and Gauss models with 2 shells (core and mantle only). Among continuous densities for the Earth’s interior given in Fig. 3 the Gaussian distribution (based on the Earth’s fundamental parameters δm , Im , δs only) allows a better-quality representation of the general trend of the planet’s piecewise density. By this, the Gauss’ model leads to the reliable estimation of the lower limit EGauss of the potential energy E, answering in this manner on the question above about the gravitational potential energy E of the ‘true’ Earth (Moritz, 1990): all piecewise density models give E-values at the vicinity EGauss = −2.5009 × 1039 ergs of the lower limit of Eq. (12). Acknowledgments I would like to thank J. Davis for significant comments, which led to improvements in the discussion of these results. I wish to address my special sincere thanks to V. Dehant and P. Holota for the attention and support of my investigations in their presentation at the XXIV General Assembly of IUGG2007 in Perugia.

References Dziewonski A.M. and Anderson D.L. (1981) Preliminary reference Earth model. Physics of the Earth and Planetary Interiors, Vol. 25, pp. 297–356. Grafarend E., Engels J., and Varga P. (2000) The temporal variation of the spherical and Cartesian multipoles of the gravity field: the generalized MacCullagh representation. Journal of Geodesy, Vol. 74, pp. 519–530. Groten E. (2004) Fundamental parameters and current (2004) best estimates of the parameters of common relevance to

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astronomy, geodesy and geodynamics. Journal of Geodesy, Vol. 77, pp. 724–731. Marchenko A.N. (2000) Earth’s radial density profiles based on Gauss’ and Roche’s distributions. Bolletino di Geodesia e Scienze Affini, Anno LIX, No. 3, pp. 201–220. Marchenko A.N. (2007) Current estimation of the Earth’s mechanical and geometrical parameters. Paper presented at the IUGG General Assembly 2007, IAG Session GS003 “Earth Rotation and Geodynamics”, Perugia, Italy, 2–13 July, 2007. Marchenko A.N. and Schwintzer P. (2003) Estimation of the Earth’s tensor of inertia from recent global gravity field solutions. Journal of Geodesy, Vol. 76, p. 495–509. McCarthy D. and Petit G. (2004) IERS Conventions (2003), IERS Technical Note, No. 32, Verlag des Bundesamts fur Kartographie und Geodasie, Frankfurt am Main, 2004. Mescheryakov G.A. (1973) On the estimation of some values characterizing the internal gravity field of the Earth. Geodesy, Cartography and Aerophotosurveying, Lvov, No. 17, pp. 34–40 (in Russian). Mescheryakov G.A. (1977) On the unique solution of the inverse problem of the potential theory. Reports of the Ukrainian Academy of Sciences, Kiev, Series A, No. 6, pp. 492–495 (in Ukrainian). Mescheryakov G.A. (1991) Problems of the Potential Theory and Generalized Earth. “Nauka”, Moscow, 1991. 203 p. (in Russian). Mescheryakov G.A., Shopjak I.N., and Dejneka Yu.P. (1977) Function’s representation inside the Earth’s ellipsoid by means of the partial sum of a generalized Fourier series. Geodesy, cartography and aerophotosurveying, No 21, pp. 55–62, Lvov (in Russian). Moritz, H. (1990) The Figure of the Earth. Theoretical Geodesy and Earth’s Interior. Wichmann, Karlsruhe. Rubincam D.P. (1979) Gravitational potential energy of the Earth: a spherical harmonic approach. Journal of Geophysical Research, Vol. 84, No. B11, pp. 6219–6225.

Analysis of Geophysical Variations of the C 20 Coefficient of the Geopotential L.I. Fern´andez

Abstract The temporal variations of the C20 spherical harmonic coefficient of the geopotential are estimated from the length of day (LOD) and compared with the C20 variations due to geophysical contributions. In particular, we analyzed the agreement of the hydrological C20 changes as estimated by the Land Dynamics Model (LaD) model. The computation spans between January 1980 and May 2004 for the hydrological model. The contribution of atmospheric mass redistributions, along with the oceanic mass terms and solid Earth tides were removed from the geodetic C20 time series for computing residuals. Afterward the hydrological influence was investigated. After eliminating seasonal variations, the hydrological excitation seems to be not adequate to explain the inter-annual variations found in the C20 residuals. The luni-solar precession and nutation of the Earth depend on the dynamical flattening (H); which is related to the principal moments of inertia of the whole planet. H is linked to the C20 coefficient of the Earth’s potential, which is regularly determined by space geodetic techniques. In this work, we also estimate the seasonal variations on H due to geophysical causes. These results should be useful to investigate the geophysical considerations in the computation of the IAU 2000 precesionnutation model.

L.I. Fern´andez Facultad de Cs. Astron´omicas y Geofisicas de la Universidad Nacional de La Plata, Paseo del Bosque s/n, B1900FWA, Buenos Aires, Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina

Keywords Length-of-day · Hydrology · Geopotencial · Gravity · Earth rotation

1 Introduction The excitation functions to the Earth’s rotational variations considered in the Liouville equations can be mathematically expressed as the sum of two terms: the mass and the motion terms of the angular momentum. The mass term is related to the masses redistributions inside the Earth, i.e. surface mass load, variations due to changes of atmospheric surface pressure, ocean bottom pressure and continental water storage, including snow and ice on land. This mass redistribution related to the dynamics of the internal processes; also change the Earth’s gravitational field by changing the Earth’s tensor of inertia. The motion terms, which do not have direct connection with gravity field, are associated with angular momentum exchange between the solid and fluid parts of the Earth’s system, i.e. atmospheric winds and ocean current variations. Angular momentum exchange related to flow in rivers and streams, for example, is likely to contribute negligibly (Chen and Wilson, 2003). Under the conservation of the angular momentum the Liouville equations provide both polar motion and the Earth’s rotation rate or equivalently the length of day (LOD), which are components of the Earth’s rotational variations. They are an important data source for validating and constraining geophysical models dealing with the mass transport into the Earth’s system. Consequently, neglecting the motions of mass elements with respect to the reference frame, the Liouville equations can be used to study the mass term

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

493

494

L.I. Fern´andez

variations of the excitation functions from the different geophysical sources. The normalized time variations in the degree-two zonal term of the Earth’s gravity field are related to the LOD excitation function (χ3 ) by (Chen et al., 2005; Bourda 2007), C20

  −(1 + k2 ) 3 Cm = √ χ3 + TR 2 5 MR2 0.753

(1)

where Cm is the principal inertia moment of the Earth’s mantle (7.1236 × 1037 kg m2 ) (Eubanks, 1993), M and R are the Earth’s mass and mean radius respectively, k2 (−0.301) is the degree 2 load Love number accounting for the elastic yielding effects and TR is the change in the trace of the Earth’s inertia tensor. Following Chen et al., 2005; C20 variations caused by TR changes are too small (10−13 , less than 0.1% of the variation measured) and they can be neglected. The relationship between the variations of the Stokes coefficients and the respective variations of the Earth’s dynamical flattening is (Bourda and Capitaine, 2004) √ MR2 H = − 5 C20 C

(2)

where C is the principal moment of inertia of the Earth. The coefficient (C/MR2 ) will be evaluated using: (i) C, M and R values from International Association of Geodesy (IAG), (Groten, 1999), (ii) values computed by (Bourda and Capitaine, 2004) by using the Clairaut theory and assuming hydrostatic equilibrium. Thus H will be determined from the geodetic Earth rotation time series as well as the different geophysical sources that contribute to the global mass redistributions into the Earth’s system. In particular, the fluctuation in the groundwater storage and the polar ice sheets, through the seasonal ice and snow loading and melting as modelled by the Land

Dynamics Model (LaD) (Milly and Shmakin, 2002) is compared with other sources of seasonal variations in the Earth’s inertia tensor. The hydrological data time series was computed using a simulation of global continental water (LaDWorld) based on the climatologically-driven hydrological model LaD. In particular the calculation was performed on the version LaD Danube. It was released in September 2004 and it runs from January 1980 through April 2004. Table 2 Values for the coefficient (MR2 /C) used for the computation of H IAG 99 (Groten, 1999)

(Bourda and Capitaine, 2004)

3.023879

3.017774

2 Data and Models 2.1 Earth Rotation Daily excitation functions χ3 are computed from the Length-of-Day (LOD) values of the IERS multitechnique combined C04 solution. The time spans between January 1980 and May 2004. Decadal variations of the Earth rotation are generally attributed to interactions between Earth’s core and the mantle. In particular, they are linked to the torques in the fluid core aligned with the rotation axis. These long period terms (i.e. periods of 10 years or more) are not supposed to appear at C20 variations due to contributions from atmosphere, oceans and hydrology. This is the reason why such variations have to be filtered out from the C20 series derived from LOD (and denoted as C20 geodetic hereafter), before the comparison. The low-frequency components are eliminated from the data series by means of a high-pass Vondr´ak filter. The Vondr´ak filter (Vondr´ak, 1977) is a smoothing method based on the Whitaker-Robinson method.

Table 1 Amplitude and phase of annual and semi-annual C20 changes estimated from the geodetic time series (IERS EOP C04), Atmosphere + Ocean (AO) and the LaD hydrological model. Time series spans from January 1980 to December 2002 Annual C20

amplitude

Semi-annual (×10−10 )

phase [degrees]

amplitude (×10−10 )

Geod (C04) 0.740 222 0.110 Atm + Ocn 0.716 220 0.120 Hydro (LaD) 0.650 43 0.100 The phase (φ) is defined as sin(2πft + φ), and the epoch t is referred to January 1st, 0h TU

phase [degrees] 173 342 150

Analysis of Geophysical Variations

495

hereafter named as LaD. The groundwater density distribution data has been generated within the Continental Water, Climate, and Earth-System Dynamics Project and is provided on a global 1◦ × 1◦ grid. Data outputs are available at monthly intervals for the period between January 1980 and May 2004. The numerical values refer to the first day of each month. Data quantities represent the mass of water per grid cell given in units of [kg/m2 ]. Frozen cells are assumed to be permanently glaciated and global ice melting and sublimation 2.2 Oceanic Model is considered. The water storage within a non-glaciated cell is Oceanic angular momenta (OAM) were obtained from composed of a snow pack store ws , a root zone store the Special Bureau for the Oceans (SBO) of the IERS. wr , and a groundwater store wg . Glaciated cells conThe time series applied in this study were computed sist of the snow pack store and a glacier-ice store wl . from an unconstrained version of the global ocean cir- In general form, the total water storage W of a cell is culation model ECCO (Estimating the Circulation and given by the Climate of the Ocean) (Gross et al., 2004). The ECCO data is available on an irregular grid between W = ws + wr + wg + wl (3) 72.5◦ S and 72.5◦ N. The latitudinal spacing of the grid points is 1/3◦ between ±10◦ around the equator and (Milly and Shmakin, 2002). After reading the ground1◦ towards the higher latitudes; the longitudinal spac- water values from the LaD model, we computed the ing is 1◦ . Time frame and temporal resolution of the time series of Stokes coefficients C20 due to the ECCO OAM are consistent with the atmospheric an- continental water storage redistributions. According to gular momenta series of National Centers for Environ- Wahr et al. (1998), mental Prediction (NCEP) (Kalnay et al., 1996). (1 + k2 )R 2 C20 = 5M 2.3 Atmospheric Model (4) W (φ  , λ)P 20 (cos φ  ) sin φ  dφ  dλ Atmospheric data are based on 2.5◦ ×2.5◦ surface press sure and wind fields of the reanalyses of the NCEP. As atmospheric pressure forcing is not taken into ac- where P 20 (cos φ) is the normalised associated Legencount by the global ocean circulation model ECCO, an dre function and W (φ , λ) represents the variations inverse barometric (IB) correction was applied to the of the water storage in units of [kg/m2 ], and φ is the pressure fields (i.e. the pressure was set to zero over the co-latitude (90◦ − φ), with φ the latitude. This integral was evaluated numerically. For each oceans) prior to the computation of atmospheric angular momenta (AAM) (Salstein et al., 1993). The data computation point we considered a surrounding 4◦ × set which spans the time interval between January 1st 4◦ sub-block of the grid data and performed a two1980 and December 31st 2002 at daily intervals was dimensional interpolation by using a polynomial of deprovided by the Special Bureau for the Atmosphere gree 3 in both latitude and longitude. Thus from Eq. (4), monthly hydrological contribu(SBA) within the Global Geophysical Fluids Center tions to C20 were estimated for the period from Jan(GGFC) of the IERS. uary 1980 to May 2004. Its main objective is to remove noise from observations and it is routinely applied at the International Earth Rotation and Reference Systems Service (IERS) to smooth Earth Orientation Parameters (EOP) time series (http://hpiers.obspm.fr/eop-pc). In this case, we applied a Vondr´ak filter with a halfwith period of 1300 days in order to remove periods longer than 7 years but keeping the seasonal terms.

2.4 LOD Excitation Function from the LaD Model

3 Comparison of Seasonal Variability

Variations of the degree 2 and order 1 Stokes coeffi- Firstly, in order to get the same time interval for all cients due to continental hydrology are computed from the data time series, a Savitzky-Golay smoothing filter the Land Dynamics Model (Milly and Shmakin, 2002), of degree 2 with 30 points was applied. This filter is

496

L.I. Fern´andez

nothing but an unweighted linear least-square fit using a polynomial of the specified degree to derive the filter coefficients. The Savitzky-Golay filter is normally used for smoothing noise fluctuations from observational data. The best performance of this filter will occur when the adjusted polynomial accurately describes the analytical function (Savitzky and Golay, 1964). In the frequency domain, the method shows to be effective at preserving the high-frequency components of the signal (Press et al., 1992). The filter was adopted to smooth the time series of atmospheric and oceanic excitations as well as the geodetic excitation at 30-day intervals. Because the influence of the mass global redistributions from Atmosphere and Ocean on Earth’s rotation rate at a broad band of frequencies is very well known (Chen et al. 2000, 2004, 2005; Gross et al., 2004), we add both time series in C20 Atm + Ocn. Figure 1 shows the expected good agreement between the C20 geodetic variations and the C20 variations due to Atmosphere + Oceans. The hydrological estimation of C20 shows a different behaviour and it is nearly out of phase. This effect should obey to a cancellation between the water content of land and ocean + water vapour, in driving the Earth rotational rate change. A similar effect was observed for the hydrological model CDAS-1 by Chen et al., 2000. They compared the remaining variations of LOD not accounted by the atmosphere (winds + pressure) with respect to the hydrological and oceanic excitations. They found that,

on seasonal scale, hydrological and oceanic excitations are out of phase. For this work, in order to explore how the agreement is between C20 geodetic and the sum of all the geophysical model predictions, the LaD hydrological contribution is added to the C20 variations due to Atmosphere + Oceans. Afterwards, we plot the C20 geodetic variations along with the variations due to Atmosphere + Oceans + Hydrology as estimated by the LaD model (not shown). The plot shows relatively larger discrepancies at interannual periods and this fact is perhaps a symptom of a non effective runoff of the hydrological model. The amplitude and phase of the annual and semiannual zonal C20 variations from the different geophysical causes are estimated by a least squares adjustment and listed in Table 1. The amplitude of the hydrological annual contribution is smaller than the respective ones EOP-derived and atmospheric+oceanic C20 variations. The amplitude of the annual contribution is comparable for all the data time series. The semi-annual hydrological contribution is one order of magnitude smaller than the atmospheric + oceanic global mass transports in the system Earth. Moreover, the amplitudes of both terms, annual and semi-annual, are close to each other between (atmosphere + ocean) and hydrology (LaD). These results were compared against amplitudes and phases of annual and semi-annual variations estimated by least squares for C20 changes by Chen and Wilson (2003). They computed their results for C20

3.00E –10 C20 geod

C20 atm + ocn

C20 Lad

2.00E –10 1.00E –10 0.00E +00 –1.00E –10 –2.00E –10 –3.00E –10 1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Fig. 1 The LOD-derived C20 variations (geodetic), the C20 variations due to Atmosphere + Oceans (atm + ocn) and ΔC20 estimated by the LaD hydrological model

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variations estimated from Earth’s rotation parameters and the sum of atmosphere, ocean and water effects using the Land Data Assimilations System data (LDAS) model. For the C20 geodetic variations, there is a good agreement for the amplitude of the annual term but the value of the amplitude of the semi-annual term is one order of magnitude bigger for Chen and Wilson (2003). The estimation of all phases differs by about π. For the geophysical excitations to the C20 changes, it is not possible to perform a direct comparison between the LaD and LDAS hydrological models from (Chen and Wilson, 2003); results. Nevertheless, the annual and semi-annual amplitudes computed in this work for (atmosphere + ocean) and hydrology (LaD) are in fairly good agreement with respect to those estimated by Chen and Wilson, 2003 for the C20 variations due to atmosphere + oceans + LDAS model. In order to qualify the hydrological seasonal and inter-annual contribution as estimated by LOD to the C20 variations, the hydrological C20 time series was compared with the C20 estimated from EOP and atmospheric and oceanic excitation functions. Seasonal and shorter term signals were removed by smoothing with a low-pass Vondr´ak filter (Vondr´ak, 1977). Although the Vondr´ak filter is usually applied as a smoothing procedure, it can also be used as an effective frequency filter. In that case, one must take into account that the narrower is the filter, the

more the amplitude of the signal decreases. The same holds for amplitudes near both edges (Vondr´ak, 1977). The applied filter has a half-width period of 500 days and a cut-off period of 3 years, removing thus the 95% of the signal at periods lower than 1 year. The atmospheric and oceanic influence was removed from the C20 geodetic time series by subtracting the C20 Atm + Ocn variations (hereafter AO). The resulting variations after filtering both, geodetic–AO and hydrological C20 , are shown in Figure 2. This figure shows that the geodetic derived C20 variations corrected by atmosphere and oceans are dominated by a strong inter-annual oscillation with a period near 5 years as estimated through a spectral analysis (Figure 3). This feature does not appear in C20 hydrological variations. This finding agrees with (Chen et al., 2005; Bourda et al. 2005) results when comparing the C20 variations estimated from Earth Orientation Parameters (EOP) with the respective Earth’s gravity field variations computed from Satellite Laser Ranging (SLR) results. From Figure 3 it is also noticeable a common peak near 2 years. The cross correlation between C20 G-AO (geodetic EOP-estimated without atmospheric and oceanic contributions) and C20 LaD (hydrological model LaD prediction) was also computed. The correlation coefficient shows a peak at 0.4 but not at zero phase lag.

1.5E –10 1E –10

C20 LaD seasonal detrended

C20 geod-(atm + ocn) seasonal detrended

5E –11 0 –5E –11 –1E –10 –1.5E –10 1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

years

Fig. 2 The low-pass Vondr´ak filtered EOP-derived (geodetic) C20 after Atmosphere+Ocean (atm+ocn) contributions subtracted. In gray line, the low-pass Vondr´ak filtered hydrological C20 time series (LaD)

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5 Discussion

4 C20 G-AO

3.5

C20 LaD Periodogram values

3 2.5 2 1.5 1 0.5 0 0

10 20 30 40 50 60 70 80 90 100 110 120 Period (months)

Fig. 3 Periodogram values of EOP-derived C20 after atmospheric + oceanic contributions were removed (G-AO) (black line) and C20 as estimated from the hydrological LaD model (gray line) Seasonal and shorter terms are filtered before spectral analysis

4 Link Between C 20 and the Earth’s Dynamical Flattening Classically, the astronomical determinations of the Earth’s dynamical ellipticity (H) are based on observations and they can differ by about 10−7 depending on the authors (Mathews et al., 2002; Bourda and Capitaine, 2004). Assuming that H can be expressed as the sum of a constant and a variable part (H + ΔH ) and using Eq. (2); the seasonal variations on ΔH caused by different geophysical causes are estimated. These results can be useful to estimate the influence of the variable geopotential on precession-nutation model. Tables 3 and 4 shows the least-squares adjustment of the seasonal terms contributing to the geophysical ΔH . Values were computed applying IAG (Groten, 1999) and (Bourda and Capitaine, 2004) values for the constant (MR2 /C) as shown in Table 2.

The changes of the mass redistribution in the Earth’s system can be computed from geophysical models as variations in the standard geopotential coefficient C20 . Besides, gravity field data (i.e. the Stokes coefficients) are routinely obtained from orbitography of geodetic satellites (Biancale et al., 2000) or from dedicated space gravimetric missions like CHAMP (CHAllenging Minisatellite Payload; Reigber et al., 2002) or GRACE (Gravity Recovery And Climate Experiment; Reigber et al., 2005). Thus, the analysis and comparison of C20 from individual phenomena allows testing and improving different global geophysical models. These results can also be applied to estimate H variations in order to investigate the geophysical considerations in the computation of the IAU 2000 precesionnutation model (Mathews et al., 2002). We compared different C20 time series at interannual periods in order to investigate the relationship between the continental water global mass redistributions as estimated by the LaD model and the C20 geodetic variations after removing atmospheric and oceanic influence. From this study, it was found that the hydrological variations are not adequate to explain the inter-annual variations found in the C20 residuals. The strong seasonal variability in the C20 is due to the well-known large scales zonal patterns into the atmospheric, oceanic and hydrological global circulation systems (Chen et al., 2000; Gross et al. 2004; Milly and Schmakin, 2004). Chen et al. (2005) affirmed that the inter-annual variation near 5 years in geodetic derived C20 might be caused by errors in atmospheric winds as modelled by NCEP reanalysis. If the inter-annual feature found in 5 years has atmospheric origins and it is linked with the model used by the NCEP reanalysis, the elimination of the

Table 3 Least-squares adjustment of seasonal terms in the H contributions. Values were computed by using IAG 99 (Groten, 1999) values for (MR2 /C) Annual

Semi-annual

H

Amplitude (×10−10 )

Phase [degrees]

Amplitude (×10−10 )

Phase [degrees]

Geod (C04) Atm + Ocn Hydro (LaD)

5.001 4.843 4.988

44.69 40.05 210.78

0.797 0.819 0.093

263.20 251.97 24.04

The phase (φ) is defined as sin(2π ft + φ), and the epoch is referred to January 1st, 0h TU

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Table 4 Least-squares adjustment of seasonal terms in the H contributions. Values were computed by using (Bourda and Capitaine, 2004) values for (MR2 /C) Annual H

Amplitude

Semi-annual (×10−10 )

Phase [degrees]

Amplitude (×10−10 )

Geod (C04) 4.991 44.69 0.795 Atm + Ocn 4.833 40.04 0.820 Hydro (LaD) 4.978 210.77 0.093 The phase (φ) is defined as sin(2π ft + φ), and the epoch is referred to January 1st, 0h TU

Phase [degrees] 263.19 251.97 24.04

atmospheric excitements in the EOP-derived C20 , Bourda G. 2007 Length-of-day and space-geodetic determination of the Earth’s variable gravity field. J. Geod. Doi: will contaminate the data time series. 10.1007/s00190-007-0180-y. The spectral common peak near 2 years seems to be Bourda G., Capitaine N. 2004 Precession, nutation, and space a feature of filtered applied to remove seasonal terms geodetic determination of the Earth’s variable gravity field. A that a feature associated with the global redistributions & A 428, Pp. 691–702. of mass from continental waters + ice and snow. Bourda G., Fern´andez L. Schuh H. 2005 Earth gravity field and LOD variations related to hydrological excitations PreMoreover, Clarke et al. (2005) asseverated that the sentation at IAG Meeting Dynamic Planet 2005. Cairns. effect of surface loading obtained by adding mass disAustralia. tributions from different individual phenomena is not Chen J.L., Wilson C.R. 2003 Low degree gravitational changes self-consistent. They found that the sum of forcing confrom the Earth rotation and geophysical models. Geophys. straints of gravitational consistency is necessary to asRes. Lett. 30, 24, 2257, doi: 10.1029/2003GL018688. sure that the total load mass is conserved. They also af- Chen J.L., Wilson C.R., Chao B.F., Shum C.K., Tapley B.D. 2000. Hydrological and oceanic excitations to polar motion firmed that the effect is significant and non negligible. and length-of-day variation. Geophys. J. Int. 141 149–156. In fact, they used ECCO model for the circulatory Chen J.L., Wilson C.R., Tapley B.D., Ries J.C. 2004. Low ocean redistribution, NCEP reanalysis data for the atdegree gravitational changes from GRACE: validation and mospheric surface pressure and LaD model for the coninterpretation. Geophys. Res. Lett. 31, L22607, doi: 10 1029/2004GL021670. tinental hydrology, estimating an amplification of 16% in the annual variation of the Earth’s dynamical ellip- Chen J.L., Wilson C.R., Tapley B.D. 2005 Interannual variability of low-degree gravitational change, 1980–2002. J. Geod. 78, ticity due to mass loading. Pp. 535–543. For the reasons explained above, it is suggested Clarke P.J., Lavallee D.A., Blewitt G., van Dam T.M., Wahr J.M. to repeat the calculations and the comparison from 2005 Effects of gravitational consistency and mass conservaindependent determinations of the zonal C20 (SLR, tion on seasonal surface mass loading models. Geophys. Res. Lett. 32, L08306, doi: 10.1029/2005GL022441. GRACE) by using other models for the atmospheric excitation and to take into account the effect of sea- Eubanks T.M. 1993. Variations in the orientation of the Earth Contributions of Space Geodesy to Geodynamics. In: Smith, sonal surface mass loadings in the new results. D., Turcotte, D. (Eds), Contributions of Space Geodesy to Acknowledgments The author thanks Nicole Capitaine and an anonymous reviewer for the critical reading and the valuable suggestions that improve the presentation of this manuscript. The author also acknowledges the Advisory Board of the Descartes Nutation Prize for the six-month postdoctoral fellowship at the Institute of Geodesy and Geophysics (Technical University of Vienna) in 2005 to work on this topic. Finally, I am grateful to the Institute of Geodesy and Geophysics for the financial support that allows the presentation of this work at the XXIV IUGG General Assembly.

References Biancale R., Balmino G., Lemoine J.-M. and coauthors 2000. A new global Earth’s gravity field model from satellite orbit perturbations: GRIM5-S1. Geophys. Res. Lett. 27, 22. 3611–3614. Doi: 10.1029/2000GL011721.

Geodynamics: Earth Dynamics. Geodynamics Series A.G.U. (24), 1–54. Gross R.S., Fukumori I, Menemenlis D., Gegout P. 2004. Atmospheric and Oceanic Excitation of length of day variations during 1980–2000. J. Geophys. Res., 109 (B01406), doi: 10.1029/2003JB002432. Groten E. 1999. Report of Special Commission 3 of IAG. (http://www.gfy.ku.dk/∼iag/HB2000/part4/groten.htm). Kalnay E., Kanamitsu M., Kistler R & coathors. 1996. The NMC/NCAR 40-year Reanalysis Proyect. Bull. Amer. Meteor. Soc., 77, 437–471. Mathews P.M., Herring T.A., Buffett B.A. Modeling of nutation and precession: new nutation series for nonrigid Earth and insights into the Earth’s interior. J. Geophys. Res., 107, B4, 10.1029/2001JB000390. Milly P.C.D., Shmakin A.B. 2002. Global modeling of land water and energy balances. Part I: the Land dynamics (LaD) model. J. Hydrometeorology, 3, 283–299.

500 Press W.H., Vetterling W.T., Teukolsky S.A., Flannery B.P. 1992. Numerical Recipes in FORTRAN: The Art of Scientific Computing. Cambridge University Press, New York. Reigber C., Balmino G., Schintzer P, Biancale R. and coauthors. 2002. A high quality global gravity field model from CHAMP GPS tracking data and accelerometry (EIGEN-1S). Geophys. Res. Lett. 29, 14, doi: 10.1029/2002GL015064. Reigber C., Schmidt R. Flechtner F, K¨onig R. and coauthors. 2005. An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. J. Geodyn., 39, 1, 1–10. doi: 10.1016/j.jog.2004.07.001. Salstein D.A., Kann D.M., Miller J.A., Rosen R.D. 1993. The sub-bureau for atmospheric angular momentum of the

L.I. Fern´andez international earth rotation service: a meteorological data center with geodetic applications. Bull. Am. Meteorol. Soc., 74, 67–80. Savitzky A., Golay M.J.E. 1964. Smoothing and differentiation of data. Anal. Chem., 36, 1627–1639. Vondr´ak J. 1977. Problem of smoothing observational data II. Bull. Astron. Inst Czech. 28, 84–89. Wahr J.M., Molenaar M., Bryan F. 1998. Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J. Geophys. Res., 103 (B12), 98JB02844.

Analysis of Mass Variations in Northern Glacial Rebound Areas from GRACE Data ¨ ¨ Holger Steffen, Jurgen Muller and Heiner Denker

Abstract Since 2002 the Gravity Recovery and Climate Experiment (GRACE) satellite mission is mapping the Earth’s gravity field, showing variations due to the integral effect of mass variations in the atmosphere, hydrosphere and geosphere. After reduction of oceanic and atmospheric contributions as well as tidal effects during the GRACE standard processing, monthly solutions of the gravity field are provided by several institutions. The solutions of the analysis centres differ slightly, which is due the application of different reduction models and centre-specific processing schemes. In addition, residual signals from insufficient pre-processing of the transmitted satellite data may be present. We present our investigation of mass variations in the areas of glacial isostatic adjustment (GIA) in North America and Northern Europe from GRACE data, especially from the latest release of the GFZ Potsdam. One key issue is the separation of GIA parts and the reduction of the observed quantities by applying dedicated filters and models of hydrological variations. In a further step, we analyse the results of both regions regarding their reliability, and finally a comparison to results from geodynamical modelling is presented. Our results clearly show that the quality of the GRACEderived gravity change signal benefits from improved Holger Steffen Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, 30167 Hannover, Germany, e-mail: [email protected] J¨urgen M¨uller Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, 30167 Hannover, Germany Heiner Denker Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, 30167 Hannover, Germany

reduction models and dedicated analysis techniques. Nevertheless, the comparison to results of geodynamic models still reveals differences, and thus further studies are in progress. Keywords GRACE · Mass variation · Glacial isostatic adjustment · Global hydrology models

1 Introduction The ice sheets, which were built up during the Last Glacial, depressed due to their enormous weight the Earth’s surface by several hundreds of metres. The crust was caused to sink into the fluid mantle, and mantle material was forced to flow away from underneath the loaded area. Since the Last Glacial Maximum at 22,000 years BP, when the ice sheets began to melt, the surface rebounds due to the buoyancy of the displaced material relative to the mantle. This process, termed glacial isostatic adjustment (GIA), is still observable today. On the Northern Hemisphere the regions of Fennoscandia and North America are characterised by present-day uplift caused by GIA. The uplift of the crust reaches values up to around 1 cm/yr in the centre of both regions (Ekman, 1996; Johansson et al., 2002; Sella et al., 2007). This effect can be observed geometrically, e.g. by GPS or spirit levelling (e.g. Ekman, 1996; Ekman and M¨akinen, 1996; Milne et al., 2001; Johansson et al., 2002; Scherneck et al., 2003; Kuo et al., 2004), and gravitationally by absolute gravimetry (e.g. M¨uller et al., 2005; Wilmes et al., 2005; Lambert et al., 2006; Timmen et al., 2006) or by satellite missions like the Gravity Recovery and Climate Experiment (GRACE) (e.g.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Wahr and Velicogna, 2002; Peltier, 2004; Velicogna and Wahr, 2004; M¨uller et al., 2006b; Tamisiea et al., 2007; van der Wal et al., 2007). In this paper, we use the monthly GRACE solutions to determine the GIA-induced gravity changes. During the mission duration (up to now more than five years), a temporal gravity change of at least 100 nm/s2 (∼3 mm geoid change) is expected in the centre of the Fennoscandian land uplift area (the Bothnian Bay, M¨uller et al., 2006a), and the configuration of GRACE is suitable to determine this magnitude of variation (Wahr et al., 1998). Gravity field variations of the Earth result from the integral effect of oceanic, atmospheric and hydrological mass movements and those caused by dynamics in the Earth’s interior. Unfortunately, in addition to these mass variations the monthly solutions may also contain residual signals from insufficient preprocessing. Hence, GIA-induced mass variations in the rebound areas of Fennoscandia and North America have to be extracted from the whole signal with dedicated filters and analysis techniques (see e.g. Wahr and Velicogna, 2002; Velicogna and Wahr, 2004). Furthermore, the GRACE gravity fields require smoothing to reduce the effects of errors present in shortwavelength components. As the smoothing radius decreases, these errors manifest themselves in maps of surface mass variability as long, linear features generally oriented north to south (so-called stripes, see Swenson and Wahr, 2006, for more information). Several filter techniques have been published, especially in the last years (Han et al., 2005b; Sasgen et al., 2006; Swenson and Wahr, 2006; Kusche, 2007), and a thorough summary can be found in Kusche (2007). Commonly the Gaussian filter is used, which is based upon methods of isotropic Gaussian smoothing outlined in Jekeli (1981), and it was introduced in Wahr et al. (1998) for GRACE monthly gravity fields. This filter depends on the spherical harmonic degree l and is a normalised spatial average to compensate for poorly known, short-wavelength spherical harmonic coefficients. Han et al. (2005b) developed a non-isotropic filter with coefficients depending on both degree l and order m. It is constructed like the Gaussian filter, but with variation of the average radius with the harmonic order, which leads to a compression of Gaussian signals in the NS direction as compared to the EW direction. Swenson and Wahr (2006) presented a non-isotropic filter for

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the so-called “destriping”, the decorrelation of the GRACE coefficients. They examined the spectral signature of the correlated errors, and removed the correlated signals with a method using polynomials. Their filter clearly reduces the presence of stripes in the GRACE gravity fields, but unfortunately, it may remove real signals as well. The above mentioned filter were applied in this study. Using adequate filters, temporal gravity variations can be recovered. A quite large number of investigations used GRACE data to determine variations caused by hydrological, cryospheric or oceanic effects (see e.g. Tapley et al., 2004; Velicogna and Wahr, 2004; Wahr et al., 2004; Famiglietti et al., 2005; Frappart et al., 2005; Han et al., 2005a; Chen et al., 2006; Schmidt et al., 2006; Seo et al., 2006; Swenson and Milly, 2006; Yeh et al., 2006; Munekane, 2007). Up to now, the number of investigations to GIA using GRACE is rather small, which is mainly due to the fact that hitherto the mission duration was too short. With the new Release 4 (RL04) of the three main analysis centres University of Texas at Austin, Center for Space Research (CSR), Jet Propulsion Laboratory (JPL), and the GeoForschungsZentrum Potsdam (GFZ), as well as solutions provided by the University of Bonn and the Centre National d’Etudes Spatiales the number of investigations will probably increase. In this study, we determine secular and periodic gravity changes in both Fennoscandia and North America from the GRACE monthly solutions RL04 as provided by CSR, JPL and GFZ. We compare the resulting variations with hydrological and geodynamic models.

2 Comparison of Monthly Solutions All analysis centres have reduced the atmospheric and oceanic contributions as well as the tidal effects during the standard GRACE processing by applying corresponding global models. The number of the monthly solutions provided by the three main analysis centres differs due to various reasons. GFZ has so far released 52 monthly gravity field solutions, from August 2002 to April 2007, with gaps in September 2002, December 2002 to January 2003, June 2003 and January 2004. CSR has provided 57 solutions, from April 2002 until May 2007 with gaps in May 2002, June 2002 and June 2003. The JPL solution envelopes the period from January 2003 to November 2006; there are 46

Analysis of Mass Variations in Northern Glacial Rebound Areas from GRACE Data

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i = 1 indicates the semi-annual and i = 2 the annual period. The accuracy of the determined parameters is about 13 mm for the amplitudes and about 7 mm/year for the secular trend, when applying a Gaussian filter with 400 km radius. In this study we focus on the secular trend B. Figure 1a shows the secular trend in Fennoscandia as determined with a Gaussian filter from the GFZ solutions. A clear signal in the order of more than 30 mm/year is visible for the central area around the Bothnian Bay, which can be related to GIAinduced gravity changes. This value is at least 15% of the largest annual variation of around 200 mm in Fennoscandia in the GRACE data (observed in 2006) and thus a clear signal over the considered time period. In comparison to previous studies (M¨uller et al., 2006b), our results confirm, that the centre of the secular signal from GRACE is located in the expected uplift centre. The CSR solution (Fig. 1b) also shows a clear signal, but more shifted to the west of the Scandinavian Peninsula with a smaller magnitude of at most 30 mm/year. Furthermore, the ellipsoidal shape of the uplift area is smaller, and it is directed SW to i=1 (1) NE compared to the NS extension of the GFZ solution. +Di sin(ωi t). The smallest signal in extension and magnitude is In the above equation t is the time difference rel- found for the JPL solution (Fig. 1c). The magnitude ative to the month of the first monthly solution. Index of 20 mm/year is smaller by one third compared

monthly solutions available with a gap in June 2003. Each monthly GRACE gravity field solution consists of a set of Stokes coefficients Clm and Slm up to degree and order 120 with corresponding “best guess” calibrated errors. Due to the higher errors at shorter wavelengths, the spherical coefficients are only considered up to degree and order 50 (which in principle corresponds itself to a rectangular box filtering), followed by Gaussian filtering with a radius of 400 km. Considering the present accuracy of the spherical harmonic models and the longer time span of the monthly solutions, 400 km are meanwhile appropriate for continental areas. Finally, in this analysis the GRACE gravity values are converted into columns of equivalent water thickness using the equations given in Wahr et al. (1998) to facilitate the comparison with hydrology models. The gravity values dg(ϕ, λ, t) were computed on a 2◦ × 2◦ grid for each monthly solution. Then secular (B) and periodic (amplitudes Ci and Di of typical periods ωi ) gravity variations are determined over the corresponding time span t for every grid point: i=2 dg(ϕ, λ, t) = A + Bt + Ci cos(ωi t)

Fig. 1 Secular variation after Gaussian filtering with 400 km radius in Fennoscandia (a–c) and North America (d–f) determined from GRACE monthly solutions as provided by GFZ (a and d), CSR (b and e) and JPL (c and f), units: mm/year

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to GFZ and CSR, while the uplift area only covers Central Sweden. All solutions indicate a positive trend between 20◦ to 40◦ E and 50◦ to 55◦ N, which might be caused by hydrological processes. Also here the largest values are detected for the GFZ solution, while the smallest are found for the JPL solution. The differences between the three GRACE centres are caused by the different processing techniques, the different time spans of the input data, changed processing and filtering strategies as well as by the use of different reduction models during the standard processing. Figure 1d shows the secular gravity changes in North America obtained from the GFZ monthly solutions. A double peak of maximum positive gravity change is found west and east of the southern Hudson Bay, which indicates the former ice sheet maximum. The signal value is about 50 mm/year. In contrast to the results for Fennoscandia, the CSR solution (Fig. 1e) shows nearly the same behaviour in location and value as the GFZ solution. Nevertheless, the peak positions are slightly shifted to the west. The JPL solution (Fig. 1f) shows again the smallest signal with a magnitude of around 35 mm/year, which is smaller by one third compared to GFZ and CSR. All solutions

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indicate large negative gravity changes in Greenland, the southern Great Plains and south-eastern Alaska, which are related to hydrological processes (Great Plains) and to ice mass losses (Greenland, Alaska). In Fig. 2 we compare different filtering techniques, which were shortly summarised in Sect. 1. The top row shows the results for Fennoscandia, the bottom row the results for North America. Obviously, the non-isotropic filters after Han et al. (2005b) (Fig. 2b) and Swenson and Wahr (2006) (Fig. 2c) do a stronger smoothing compared to the Gaussian filter (Fig. 2a), especially in EW direction. Furthermore, the non-isotropic filters decrease the amplitudes of the GRACE-derived signal– by around 6% with “destriping” filter and by around 30% using the filter from Han et al. (2005b). Looking at a specific filter, the trend signal is still a clear signal of at least 15% of the maximum annual variation observed by GRACE during the last 5 years. Thus, the secular trend in eastern Europe is also decreased, but additionally the structure with two maximums areas is melted into a single region. Unfortunately, such a result is also observed for North America, and the double peak of the uplift signal almost disappears (Fig. 2e and f). As this structure is known from gravity measurements (Pagiatakis and Salib, 2003) and confirms

Fig. 2 Comparison of different filters in Fennoscandia (a–c) and North America (d–f) determined from GRACE monthly solutions provided by GFZ. (a and d) Gaussian (isotropic) filter with 400 km. (b and e) Non-isotropic filter after Han et al. (2005b). (c and f) Destriping filter after Swenson and Wahr (2006) and Gaussian smoothing with 400 km. Units mm/year

Analysis of Mass Variations in Northern Glacial Rebound Areas from GRACE Data

modelling results (Peltier, 2004), these non-isotropic filter are not well-suited for rebound investigations. Therefore, the Gaussian filter is used for further analysis in this study.

3 Comparison to Hydrology and Geodynamic Models Besides the secular gravity change, significant periodic variations are present in the GRACE data, which are mainly caused by hydrological processes. Numerous studies have proved that GRACE is able to detect continental water storage changes (e.g. Seo et al., 2006; Swenson and Milly, 2006; Yeh et al., 2006). In this investigation, we compare the results from the GFZ monthly solutions to the hydrological models WaterGAP Global Hydrology Model (WGHM, D¨oll et al., 2003) and Land Dynamics World (LaDWorld, Milly et al., 2002). WGHM is based on global hydrological and meteorological data sets. For each 0.5◦ grid cell, WGHM computes time series of monthly runoff and river discharge (D¨oll et al., 2003). The WGHM is provided us by the GFZ (courtesy of Andreas G¨untner and Roland

Fig. 3 Comparison of global hydrology models WGHM (b) and LaDWorld (d) as well as the result of the geodynamical 1D modelling (c) to the secular variation of the GFZ GRACE solution (a) in Fennoscandia. Units mm/year. Here, the considered time span for the hydrology models and the GRACE solution covers 46 months only

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Schmidt) in form of monthly solutions of Stokes coefficients up to degree and order 100 from February 2003 to December 2006, using the GRACE gravity field format. These values are also converted into columns of equivalent water thickness using the equations given in Wahr et al. (1998). We do not use here the December 2006 to better compare to LaDWorld, which ends in November 2006. LaDWorld is a series of retrospective simulations of global continental water and energy balances, created by forcing the Land Dynamics (LaD) model (Milly et al., 2002). In this study, we use the sum of simulated variables for snow water equivalent, soil water and shallow ground water, representing the main quantities of interest in this investigation. The data are provided in monthly solutions from January 1980 until November 2006 in a 1 × 1◦ grid in units of water column. We use the time span of February 2003 until November 2006. In order to compare the hydrological data with the GFZ GRACE data, both hydrological model are resampled to a 2 × 2◦ grid, and smoothed by Gaussian filtering. In both hydrology models the continental icesheets are not included. In Fig. 3a the GRACE trend in Fennoscandia is shown for the corresponding time span with its maximum of more than 30 mm/year. Figures 3b and

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d illustrate the secular variation of the hydrological models WGHM and LaDWorld, respectively. A comparison shows discrepancies at least between the two hydrological models. WGHM highlights a positive trend of around 15 mm/year in Fennoscandia and a negative trend in the East European Plain. In contrast, LaDWorld shows no significant trends or peaks. Western Europe is characterised by an increase of at most 10 mm/year, while northeast Europe experiences a decrease of 10 mm/year. These values are quite small small compared to largest annual variations in the hydrology models of at least 200 mm/year. Compared to the GRACE result, the rate of hydrology is much smaller than the detected uplift signal. Hence, the major gravity changes in Fennoscandia seem to be not significantly affected by hydrological processes, when considering WGHM and LaDWorld. But one has to consider that these hydrology models are not developed for long-term investigations, as hydrologists

Fig. 4 As Fig. 3, but for North America

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are more interested in short-term (seasonal) variations (G¨untner and Petrovic, pers. comm., 2007). In Canada (Figs. 4b and d), the secular variation of the hydrological models WGHM and LaDWorld is again smaller than the uplift signal (Fig. 4a). Both models show positive trends in southeast Alaska and over Baffin Island, with the largest values associated with LaDWorld. For WGHM a large contribution is also found from Alberta to Manitoba, which encompasses quite well the drainage basin of the Nelson River. In contrast, LaDWorld has the maximum at the Great Lakes. It can be clearly seen from the models that the hydrological influence in the uplift region around the Hudson Bay is smaller than the uplift values. Nevertheless, one has to be careful with any further interpretation. As mentioned above, the hydrology models cannot trace the trends expected from the GRACE solutions. In addition, ice mass changes have to be considered. For example, both hydrology models show a

Analysis of Mass Variations in Northern Glacial Rebound Areas from GRACE Data

positive trend in the source region of the Yukon River, but GRACE observes a decrease, which is due to retreat of glaciers in the Coast Mountains. In a last step, we compare the GRACE solutions with results of a geodynamical modelling using the software package ICEAGE (Kaufmann, 2004). An iterative procedure in the spectral domain up to degree 192 following the pseudo-spectral approach outlined in Mitrovica et al. (1994) and Mitrovica and Milne (1998) is used to determine spherically symmetric (1D), compressible, Maxwell-viscoelastic Earth models for Fennoscandia and North America with the help of the global ice model RSES (Research School of Earth Sciences, Canberra), and relative sea level data. The calculation method has been extensively described in Steffen and Kaufmann (2005). The best-fit 3-layer (lithosphere, upper mantle, lower mantle; inviscid core as assumed lower boundary condition) Earth model found for Fennoscandia has a lithosphere thickness Hl = 120 km, an upper mantle viscosity ηUM = 4 × 1020 Pa s and a lower mantle viscosity ηLM = 1023 Pa s. The best-fit Earth model for Canada is characterised by a lithosphere thickness Hl = 150 km, and upper and lower mantle viscosities ηUM = 7 × 1020 and ηLM = 2 × 1022 Pa s, respectively. The modelled secular trends for each area (Figs. 3c and 4c) are compared to the GRACE solution (Figs. 3a and 4a), respectively. For Fennoscandia the location of the uplift maximum and its value fit quite well (Fig. 3), but the extension of the uplift area from the modelling is directed SW to NE, which is basically due to the geometry of the ice-sheet model. The difference between the two is mainly due to the isotropic filtering and the simple Earth model. Comparing these results shows that the trend detected by GRACE in Eastern Europe is not GIA-induced, and therefore it has a strong hydrological part, which cannot be explained by the hydrology models (Figs. 3b and d) at the moment. This result strongly supports the revision of global hydrology models for this kind of investigation. Looking at Canada (Fig. 4a and c), slight differences stand out in the uplift region. The model area is smaller than observed with GRACE and the distance between the two peaks decreases. Nevertheless, the maximum values are comparable. In addition, it becomes clear that the negative trends over Greenland and the Coast Mountains as seen by GRACE are not related to GIA induced by the last ice age.

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4 Summary and Outlook In this study we investigated temporal gravity variations on the Northern Hemisphere based upon monthly GRACE gravity field solutions provided by the three main analysis centres GFZ, CSR and JPL. We focused on the secular trend and analysed the uplift signal with different filtering techniques and hydrological and geodynamic models. The GRACE data clearly show temporal gravity variations in Fennoscandia and North America, and the secular variations are in good agreement with other studies (e.g. Lambeck et al., 1998; M¨uller et al., 2006b; van der Wal et al., 2007). The non-isotropic filter techniques have to be improved for GIA investigations, as they tamper with the expected rebound signal. In addition, our results show that recent global hydrology models have to be revised for the use in geodynamical investigations. Furthermore, we used a 1D earth model to compare with the detected land uplift effect. The location and orientation of the modelling results agree well with the GRACE uplift signal. In the future, more refined earth models (e.g. 3D models) shall be developed and compared to the GRACE solutions, which may help to improve the estimation of the land uplift signal from GRACE data and to discriminate between the different assumptions and parameters used in the geophysical uplift models. Acknowledgments We would like to thank the GRACE science team for overall support, GFZ, CSR, and JPL for providing the monthly GRACE solutions, and Petra D¨oll and her group for making the WGHM data available. Many thanks also to Kurt Lambeck for providing the RSES ice model and Georg Kaufmann for providing the modelling software ICEAGE (Kaufmann, 2004). We would also like to thank Roland Schmidt (GFZ), Andreas G¨untner (GFZ), Svetozar Petrovic (GFZ), Sean Swenson (University of Colorado) and Wouter van der Wal (University of Calgary) for helpful discussions. We are grateful for numerous comments and suggestions by Olivier de Viron and John Wahr. This research was funded by the DFG (research grant MU1141/8-1).

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Analysis of Mass Variations in Northern Glacial Rebound Areas from GRACE Data Tamisiea, M. E., Mitrovica, J. X., Davis, J. L. (2007). GRACE gravity data constrain ancient ice geometries and continental dynamics over Laurentia. Science 316, 881, doi:10.1126/science.1137157. Tapley, B., Bettadpur, S., Ries, J., Thompson, P., Watkins, M. (2004). GRACE measurements of mass variability in the Earth System. Science 305, 503–505. Timmen, L., Gitlein, O., M¨uller, J., Denker, H., M¨akinen, J., Bilker, M., Pettersen, B. R., Lysaker, D. I., Omang, O. C. D., Svendsen, J. G. G., wilmes, H., Falk, R., Reinhold, A., Hoppe, W., Scherneck, H.-G., Engen, B., Harsson, B. G., Engfeldt, A., Lilje, M., Strykowski, G., Forsberg, R. (2006). Observing Fennoscandian gravity change by absolute gravimetry. In: F. Sans, A. J. Gil (Eds.) “Geodetic Deformation Monitoring: From Geophysical to Engineering Roles”, Springer, IAG Symp. 131, 193–199. van der Wal, W., Wu, P., Sideris, M. G. (2007). The use of GRACE data for postglacial rebound studies in NorthAmerica. J. Geodyn., in prep. Velicogna, I., Wahr, J. (2004). Ice mass balance in Greenland from GRACE. Geophys. Res. Lett. 32/18, L18505, doi:10.1029/2005GL023955.

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Report of GGP Activities to Commission 3, Completing 10 Years for the Worldwide Network of Superconducting Gravimeters David Crossley and Jacques Hinderer

Abstract The Global Geodynamics Project was renewed for the foreseeable future at the IUGG in Perugia. The second cycle (called GGP2) of operations started in 2003, and followed the initial years of GGP1 (1997–2003). Thus GGP has now completed 10 years of collecting data from all the currently operating superconducting gravimeters (SGs). During the last cycle, GGP operations have gone smoothly for most stations, but with the inevitable instrumental problems. We have lost (at least temporarily) stations Boulder and Bandung, but gained an instrument in MunGyung (S. Korea) and two instruments in Hsinchu (Taiwan). New installations were recently done in Pecny (Czech Republic) and Dehradun (India), and several other locations in the US and Asia are being contemplated in the next cycle of GGP (2007–2011). Over the past two years, we have worked to prepare raw GGP data (at sampling times of 1–5 s) for inclusion into the IRIS data set for the seismologists to use in normal mode studies of the Earth. A successful GGP Workshop was held in Jena, Germany in March 2006, and the first official Asian SG Workshop took place in March 2007 in Taiwan, hosted by our colleagues in Hsinchu. Of continuing interest within GGP is the issue of combining measurements from absolute gravimeters at the SG stations for a variety of long-term studies of the gravity field such as tectonic uplift, subduction zone slip, and determination of the Earth’s centre of mass with respect to the terrestrial reference field. GGP has David Crossley Department of Earth and Atmospheric Sciences, Saint Louis University, 3642 Lindell Blvd. St. Louis, MO 63108, USA, e-mail: [email protected] Jacques Hinderer EOST/Institute de Physique du Globe, 5 Rene Descartes, 67084 Strasbourg Cedex, France

now become involved with the development of GGOS, a project that intends to coordinate the use of many different geodetic data sets for future ease of access. Keywords Superconducting · gravimeter · GGP · gravity field · time-variable gravity.

1 Introduction and Station Review We show in the Fig. 1 the location of most of the stations in the GGP network for the decade 1997–2007. The cluster of stations in Europe and N.E. Asia are shown separately in Figs. 2 and 3 respectively. Two stations in the network stopped recording during the current reporting period (2003–2007). These were Bandung, Indonesia, and Boulder, Colorado. The situation in Bandung was instrument failure due to a severe storm that flooded the facility and lifted the instrument off its supporting pillar. The internal components in the dewar were OK, but the damage was too severe to be easily or quickly repaired. There has been talk of reconstituting a station in Indonesia, but in another location. The station and instrument are run by Kyoto University, Japan. The situation in Boulder is different. Due to the retirement of key personnel in Boulder, repair of a data acquisition failure in May 2004 was not done, and no further data has been available to GGP, although some data has been recorded. NOAA has ownership of the instrument, and it is still functioning; for the moment we are informed it will stay at TMGO (Table Mountain Gravity Observatory) until further decisions are made. Suggestions about transferring it to Alaska, or to another site in the US have been discussed within GGP and passed on to NOAA.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Fig. 1 Distribution of SG stations: light circles are operating, dark circles are stopped, diamonds are new installations, and squares are stations projected for the next 2–3 years

Fig. 3 As previous figures, but for N.E. Asia. There are two instruments at Hsinchu

Fig. 2 GGP stations in Europe, excepting Ny Alesund (Fig. 1). The north German station on the Baltic Sea is in the planning stage

Two other stations had troubles that have resulted in data interruptions. One is the instrument in Kyoto that has had tilt problems for a long time that has been difficult to fix (Iwano and Fukuda, 2004). Also the instrument in Syowa, Antarctica has had drift problems since 2006, and we have not seen data for some time.

Report of GGP Activities to Commission 3

Thankfully all other stations in the network are functioning well and sending data to the GGP database at ICET. Most of these stations have now brought themselves up to date at least into early 2007. Only one station so far has not sent data to GGP. New data is now available from the instrument in Kamioka (Japan), at an underground installation in Mozumi Mine that is more famous for the location of the neutrino detection facility (Super-Kamiokande). Several strainmeters are also installed in the tunnels. Two new stations have been recording since 2005: MunGyung in S. Korea, and Kamioka in Japan. The former is run by J.-W. Kim of Sejong University, Korea, also associated with the geodesy group at Ohio State University. We are still awaiting data from MunGyung. Two new instruments were installed in Hsinchu, Taiwan, in 2006 by C. Hwang and colleagues of the National Chiao Tung University. One had to be returned to GWR for repairs but is expected back imminently. The other is recording well in an underground facility in Hsinchu and we have seen preliminary data. Finally, two very recent installations were completed early this year (2007). One is at the historic Geodetic Observatory in Pecny, Czech Republic. Another was installed at the Wadia Institute of Himalayan Geology in Dehradun, India to study earthquake precursory signals in conjunction with other geophysical methods. Both installations were successful and we look forward to receiving their data. A preliminary look at data from Dehradun is encouraging. As for future stations, we are anticipating the installation of an SG in Manaus, Amazon Basin, as part of the effort of GFZ (Potsdam) to characterize the very large annual hydrology signal over the basin seen by the GRACE satellites. The anticipated date is Fall 2007. We also know of several other planned locations identified on the maps. There will be an SG in Texas to monitor hydrology (C. Wilson, U. Texas, Austin), and a second instrument in Wuhan to be installed by the Chinese Bureau of Seismology. The installation of an SG in Tahiti is in the works, with an instrument being considered in the southern US at Sunspot in New Mexico, the site of the lunar laser ranging facility. The German group is planning to locate an SG station on the Baltic Sea coast (Fig. 2). Overall the list of stations will soon reach 30, of which at least 25 should be functioning within the next year or two.

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2 GGOS Along with many other IAG organizations, GGP will contribute data to the Global Geodetic Observing System (GGOS) as part of the extensive range of goals identified in the GGOS Reference Document (GGOS2020). The combined use of SGs and AGs to monitor the long term changes in the surface gravity field has now become commonplace and we look forward to the opportunity to see SGs installed at more of the fiducial (or core) stations of the proposed GGOS ground-based network. One of the projects involving ground-based SG and AG data will be to constrain the position of the Earth’s Center of Mass as a fundamental variable in the determination of the gravity field for space and satellite missions. Of the current GGP stations, only a few are installed at space-geodetic stations (e.g. Wettzell, TigoConcepcion, Ny-Alesund, Sutherland). The majority were installed for specific tasks such as hydrology or tectonics-based research, but even though SGs might be installed for one purpose, the data cam be used in a wide variety of projects. A fuller report on the relationship between GGP and GGOS can be found in Crossley and Hinderer (2007).

3 ICET The future location and organization of the International Center for Earth Tides was discussed at the IUGG in Perugia, 2007. At a meeting of the ICET Directing Board, a recommendation was made by in favor of the proposal by University of French Polynesia (UFP), Tahiti, under the directorship of Jean-Pierre Barriot. This rather exotic physical location was favored over a competing proposal from central Europe primarily on the basis of the infrastructure available for the task at UFP. Also, the Directing Board hoped they would be invited to frequent meetings at UFP. In particular, UFP has a strong computing effort, and there will be a major overhaul of the ICET Website and new arrangements to be made with GFZ concerning the operation of the GGP database. In other respects, ICET has played a strong supporting role, under the Directorship of Bernard Ducarme, e.g. for the processing and analysis of GGP data. For these and other services re-

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lated to GGP, we have been assured that the necessary effort will be forthcoming from UFP.

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eas.slu.edu/GGP/ggpnews.html. We summarize them by year. 2004 (a) Earth Tides Symposium, Ottawa, Canada. Because the ETC overlaps almost completely with the GGP community, the ET meetings are equivalent to stand-alone GGP meetings in terms of scientific importance. The papers from this symposium are available as a special publication of the Journal of Geodynamics (Volume 41 (1–3), 2006) http://www.sciencedirect.com/science/ journal/02643707 (b) In October 2004 there was an International Symposium on Remote Sensing, held on Jeju Island, Korea and organized by the Geohazard Information Laboratory at Sejong University. Among the papers presented, several were from GGP (e.g. Kim et al., 2005; Crossley and Hinderer, 2005; Neumeyer, 2005) perspectives and the conference served to introduce SG technology to Korean geodesists. (c) A special issue of Journal of Geodynamics (Volume 38, 2004) was published to summarize the achievements of the first 6-year phase of GGP. It can also be accessed from the Science Direct website listed above.

We have had a project now for almost 2 years, to get GGP raw data (recorded at 1, 2, or 5 sec sampling) into the global seismic database IRIS (Incorporated Research Institutions for Seismology). There have been some difficulties in specifying the complete SG instrument response in the very precise format required by seismology. To a geodesist, the issue is very simple: what is the calibration (or scale) factor? It is the DC response of the instrument to convert volt to acceleration, a topic with a long history in gravimetry, and it is known that SGs can be calibrated to better than 0.1% in amplitude (approaching 0.02% in some experiments). Such a specification is incomplete for seismology, because we also need an accurate phase calibration; we note this is also important for ocean tide loading (e.g. Bos and Baker, 2005). The phase response at high frequencies can only be accurately measured using special equipment not normally available at SG stations. Information on the technique is available at http://www.eas.slu.edu/GGP/phasecal.html. We have struggled somewhat with the issue and this has delayed the project. But now we have the means to finalize this issue with the help of seismologists. This will enable us to 2006 follow the example of station Membach, where through the efforts of M. Van Camp, data from the MB SG (spe- (a) GGP Workshop, Jena. A workshop on “Analysis of Data from Superconducting Gravimeters and cial seismic output) has been sent to IRIS for more than Deformation Observations regarding Geodynamic a year. We are aided in this venture by GFZ Potsdam, Signals and Environmental Influences” was held in and we promise some real results from other GGP staJena in March. This brought together GGP memtions soon. bers and those working on the environmental influences on gravity. The proceedings and papers from 5 Workshops and Special Publications this workshop are available online through ICET (Bulletin d’Informations des Marees Terrestre) at http://www.astro.oma.be/ICET/bim/141.html GGP normally meets once a year in connection with a major meeting or as a stand-alone workshop. We here review the various occasions, following 2007 the IUGG in Sapporo, 2003. As always, the shorter meetings are for business only, while the specialized (a) GGP Workshop, Taiwan. The Taiwanese hosted the “First Asia workshop on superconducting workshops (in 2004, 2006) are the major opportunity gravimetry” in Hsinchu, Taiwan, March 12–15. to exchange scientific information. All events are It had the distinction of being the only workshop reported in GGP Newsletters that are distributed (so far) to be held within a tunnel complex only a to the GGP mailing list, available at http://www.

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few meters from the SG room itself (the activities and Taiwan. We minimize overlap with the companof the conference of course were recorded in the ion paper on GGOS (Crossley and Hinderer, 2007), as data). The purpose of the workshop was to empha- there we treat some of the connections between SGs size the increasing number of SGs in Asia, and to and AGs, hydrology and geodesy. focus attention on scientific problems in this part of the world. Of special interest is the very high 6.1 Instrumental uplift rate of Taiwan itself, which is currently to be monitored both by AG and SG measurements. Scientific papers from the workshop can be Some recent SG installations appear in publications found under the Scientific Program on the website that are helpful references on the operation of the varihttp://space.cv.nctu.edu.tw/SG/Asia workshop.html. ous stations. An example is Tigo-Concepcion in Chile, (b) Of interest to GGP may be the upcoming series of where the initial operation of GWR R038 – a new type volumes on the Treatise on Geophysics, Geodesy of remotely controlled SG – is introduced by Wilmes Volume, edited by T. Herring and G. Schubert. It et al. (2006). We show one of their plots (Fig. 4) that includes an extensive treatment of SG issues, as has some interesting aspects. The estimated drift apwell as articles on AGs and other topics of interest pears very small (0.12 μGal / yr), though this has not to GGP. The volumes will be released at the Fall been verified with the typical AG measurements. The most intriguing information is a strong annual signal AGU in San Francisco (Hinderer et al., 2007a). (c) IUGG, Perugia. A GGP Meeting was held on of amplitude about 6 μGal that remains, even after the Thursday 5 July 5:30–7:30 pm, following the effect of rainfall and groundwater changes have been business meeting of the Earth Tides Commission removed from the data. The co-located VLBI measure(ETC). We are happy to say that Gerhard Jentzsch ments are also affected by the height variations of about (U. Jena, Germany) was unanimously approved as 10 mm, with the anticipated polarity (i.e. height reducthe continuing President of the ETC for a further tion with gravity high), one of the first indications of 4-year term. His leadership both in tides and the usefulness of gravity measurements at fiducial stations (see discussion in Crossley and Hinderer, 2007). gravity is a valued asset to GGP. The MunGyung installation (Kim et al., 2007) shows very promising data and a correlation of 2008 groundwater with gravity with a diurnal periodicity. (a) ETC Symposium, 1–5 September, Jena, Ger- A report on early data from the Hsinchu SGs was many. The 16th International Symposium on given by Kao et al. (2007), highlighting the difference Earth Tides will be held in Jena next year, between the instruments SG048, that is working well, following the tradition of several GGP meeting and SG049. The latter had a larger-than-expected drift being held there in the past (see above). This and was sent back to the manufacturer for correction. SGs have become relatively well known, and much symposium is distinguished by combining the publicized, instruments; it is thus rare to see new topic of Earth Tides with Crustal Deformation, uses appear in research. An interesting suggestion Geophysical Fluids, as well as GGP. In this way, was made by Shiomi (2007) on the testing of the the ETC is hoping to broaden its base of interest universality of the free fall principle using SGs (note from tides, and tide-related topics, to a wider the paper title does not imply “testing the free fall of coverage of gravimetry-related topics and crustal geodynamics. The meeting involves several SGs”!). The physics involves detecting the differential motion of the inner core of the Earth as it orbits the sub-commissions of IAG. Sun, supposing the former can move preferentially with respect to the mantle.

6 Some Results from the GGP Workshops 6.2 Tides, Ocean Loading, and Sea Level We have space for just a few of the many interesting results from the last 4 years, here taken from the ET Tides are still an active research topic using SGs, symposium in Ottawa and the GGP Workshops in Jena particularly the aspect of ocean tide loading. Boy

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Fig. 4 Residual gravity at TC in Chile. Tides, pressure, polar motion, and groundwater-induced gravity variations have been removed (Wilmes et al. 2006)

et al. (2006a) analyzed gravity at 6 SG stations for the long-period tides, and then corrected the amplitude and phase of each monthly (Mm) and fortnightly (Mf) tidal constituent using an anelastic solid earth model. The residual amplitude and phase is then the observed ocean tide, compared to several models. The results shown in Fig. 5 indicate large observed errors that span most of the range of ocean tide models. This is in contrast with similar analyses at diurnal periods where the SG data are able to discriminate some differences in ocean tide loading (Boy et al. 2003, Bos and Baker, 2005). Long period tides are influenced by many factors such as instrument handling, site location, data quality, and hydrology. Shallow water tides detected by SGs was covered in a paper by Boy et al. (2004). SGs have contributed significantly to tides in two other areas. One is in more precise measurements of the free core nutation (FCN) period by Sato et al. (2004) and Ducarme et al. (2007). These developments have particularly resolved the issue of the Q of the FCN and brought the results more in line with geodetic techniques such as VLBI. At long periods, SG measurements of polar motion have revealed new insights into the long period admittance of the Earth through determination of the gravimetric factors (Ducarme et al., 2006; Harnisch and Harnisch, 2006a). SGs have been used successfully to investigate atmospheric tides (Boy et al., 2006b). Sato et al. (2001) used SG records from Esashi, Canberra, and Syowa to

demonstrate the importance of the steric correction to sea surface height (SSH) change. The steric part does not involve any additional mass change and hence does not alter gravity by loading; thus inland SG measurements are a unique tool to distinguish between steric and non-steric SSH components.

6.3 Earthquakes and Normal Modes Hinderer et al. (2007b) generated a solution to the static deformation of the Earth following the Sumatra 2004 event, using the summation of normal modes with a time – dependent distributed source event. This study did not use GGP data because unfortunately station Bandung was not operational, and there was no other SG close enough to measure the static displacement (∼100 μGal near the fault). The event was picked up by the GRACE satellites and has been widely reported. Nevertheless, the normal modes excited from the 2004 Andamen-Sumatra earthquake were very well recorded over the GGP network, and this has generated some publications. Roult et al. (2006) measured the frequencies and Q’s of several of the lower frequency modes that are well recorded by SGs and Rosat et al. (2007) analyzed 0 S0 amplitudes at SG stations and found a geographic pattern that is similar to that expected from the coupling of 0 S0 to the mode 0 S5 due to lateral heterogeneity in the upper mantle. A somewhat

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Fig. 5 Fortnightly and monthly tides as observed by 6 SGs compared to ocean tide models (Boy et al. 2006a)

similar study by Crossley and Xu (2007) showed that across the GGP array (Fig. 6). The SG-determined amthe Q and initial amplitude of 0 S0 was very consis- plitude had significantly less scatter than amplitudes tently measured as 5550 and 0.156 μGal respectively measured from a much larger number of GSN (Global 0.180

6500

0.175 6000

Q

0.165 5500 0.160

0.155

5000

Fig. 6 Q (left axis, open circles) and initial amplitude (black squares) for mode 0 S0 from the Sumatra earthquake at all 18 GGP SG sites

0.150

4500

cb es h1 h2 m1 ma mb mc me ny s1 s2 st Station

tc

vi w1 w2 wu

0.145

initial amplitude (µGal)

0.170

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Seismograph Network) Seismometers because of the much better calibration of the SGs.

groundwater-and gravity, leading to a well-determined admittance of 2.5 μGal m−1 .

6.4 Hydrology

6.5 Tectonics

Hydrology effects have become the prime target for long period studies with SGs because they are so easily measured. An example is Imanishi et al. (2006) who showed the effectiveness of a simple bucket (or tank) model to explain the gravity changes with rainfall (as done by several previous studies). In Fig. 7 it is seen that the gravity decreases sharply within about 1 h, and then recovers after about 1 month. These time scales are similar to the results of Wilmes et al. (2006) for Tigo-Concepcion and Harnisch and Harnisch (2006b) for Bad Homburg. Another study of note is that of Virtanen et al. (2006) who clearly separated the effects of local, regional, and global hydrology for station Metsahovi in Finland. This station is above ground, and demonstrates a very high correlation (0.83) between

As mentioned above, one of the Taiwan SGs is being repaired and when it returns, it will probably be situated somewhere along the ATGO (Absolute Gravity of the Taiwanese Orogen) profile (Masson et al., 2007), shown in Fig. 8. The easternmost station (AG1) is on an island and the last point (AG9) is on the West coast. Taiwan has one of the highest uplift rates in the world (up to 5 cm / yr). The ATGO project will require frequent AG measurements at the 9 stations shown, together with GPS, and the SG will be at one of the sites. With a gravity / height ratio of −0.2 μGal / mm this should yield a strong signal of > 4 μGal per year (larger than the drift of most SGs). The SG will be able to verify the hydrological part of this signal, together with soil moisture and groundwater observations. Other tectonically-related SG projects have been discussed in connection with GGOS goals (Crossley and Hinderer, 2007), but essentially many of them remain as future prospects to be undertaken with colocated AG measurements.

6.6 GGP and GRACE The 6-8 SG stations in Europe have been used as potential ground truth for the GRACE mission since 2002. In a recent paper, Boy and Hinderer (2006) summarized the seasonal changes in hydrology as detected by GRACE and at the ground-based GGP stations, compared to the changes predicted by hydrology models. Figure 9 shows comparisons for two stations: Wettzell, above ground, and Moxa, below. Neumeyer et al. (2006) also did this type of comparison with similar results, but also included some precise modeling of the 3-D atmospheric attraction effect and a consideration of the additional vertical displacement effect on the gravimeters that is not seen by GRACE. Despite the apparent ‘success’ of such comparisons, Fig. 7 From top to bottom: (a) rainfall, (b) modeled gravity, these results are problematic because the satellite (c) SG observed, and (d) SG corrected, for station Matsushiro data cannot give accurate results over wavelengths (Imanishi et al., 2006) <600–1000 km. A different method using all the

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Fig. 8 The 9 AG stations across a transect of Taiwan to measure uplift (Masson et al., 2007)

SG data together to find a smoothed field has been pursued by Crossley et al. (2005). The ground and satellite fields are compared by analysis of the EOF eigenfunctions and principal components, using all the spatial and temporal data together. This approach

gives a more satisfying averaging of the ground fields. Figure 10 shows a recent result for the CNES/GRGS 10-day GRACE solution. Good agreement is evident in the time evoluation of the 3 first principal components of the spatial data sets.

Fig. 9 GRACE-GGP comparisons for Wettzell where groundwater correlates with gravity. At Moxa the SG series is uncorrelated with other signals (Boy and Hinderer, 2006)

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Fig. 10 Principal components at 10-day sampling from GRACE CNES/GRGS solution (grgs), 7 European GGP ground stations 10 day (ggp10d02) and GLDAS hydrology (hyd10)

Acknowledgments GGP appreciates the cooperation of the many dedicated individuals who maintain their instruments and willingly share their data. We are indebted to Bernard Ducarme and his assistants at ICET for their unfailing support of GGP. We also acknowledge the retirement of Tadahiro Sato who has pioneered many new ideas and projects in gravimetry, particularly with SGs. We also thought Jurgen Neumeyer had retired, but apparently not; he has been instrumental (no pun intended) in the installation of new SGs.

References Bos, M.S. and T.F. Baker, 2005, An estimate of the errors in gravity ocean tide loading computations, J. Geod. 79, 50–63. Boy, J.-P. and J. Hinderer, 2006, Study of the seasonal gravity signal in superconducting gravimeter data, J. Geodyn., 41, 227–233. Boy, J.-P., M. Llubes, J. Hinderer, and N. Florsch, 2003, A comparison of tidal ocean loading models using superconducting gravimeter data, J. Geophys. Res., 108, (B4), 2193, doi:10.1029/2002JB002050. Boy, J.-P., M. Llubes, R. Ray, J. Hinderer, N. Florsch, S. Rosat, F. Lyard, T. Letellier, 2004. Non-linear oceanic tides observed by superconducting gravimeters in Europe, J. Geodyn., 38, (3–5), 391–405. Boy, J.-P., M. Llubes, R. Ray, J. Hinderer, and N. Florsch, 2006a, Validation of long-period oceanic tidal models with superconducting gravimeters, J. Geodyn., 41, 112–118. Boy, J.-P., R. Ray, and J. Hinderer, 2006b, Diurnal atmospheric tide and induced gravity variations, J. Geodyn. 41, 253–258. Crossley, D. and J. Hinderer, 2005, Using SG arrays in hydrology in comparison with GRACE satellite data, with extension to seismic and volcanic hazards, Kor. J. Rem. Sens., 21 (1), 31–49.

Crossley, D. and J. Hinderer, 2007, The contribution of GGP superconducting gravimeters to GGOS, submitted to IAG Proceedings of the IUGG, Perugia 2007. Crossley, D. and Y. Xu, 2007, Amplitude and Q of 0 S0 from the Sumatra earthquake as recorded on superconducting gravimeters and seismometers, paper presented in Session JSS011, IUGG, Perugia. Crossley, D., J. Hinderer, and J.-P. Boy, 2005. Time variation of the European gravity field from superconducting gravimeters, Geophys. J. Int., 161 (2), 257–264. Ducarme, B., A.P. Venedikov, J. Arnoso, X.D. Chen, H.-P. Sun, R. Vieira, 2006. Global analysis of the GGP superconducting gravimeter network for the estimation of the pole tide gravimetric factor, Proc. 15th Int. Symp. Earth Tides, J. Geodyn., 41, 334–344. Ducarme, B., H.-P. Sun, J.Q. Xu, 2007. Determination of the free core nutation period from tidal observations of the GGP superconducting gravimeters network, J. Geod., 81, 179–187, (doi:10.1007/s00190-006-0098-9). Harnisch, G., and M. Harnisch, 2006a, Study of the long term gravity variations based on data of the GGP cooperation, Proc. 15th Int. Symp. Earth Tides, J. Geodyn., 41, 318–325. Harnisch, G., and M. Harnisch, 2006b, Hydrological influences in long gravimetric data series, J. Geodyn., 41, 276–287. Hinderer, J., D. Crossley, and R. Warburton, 2007a. Superconducting gravimetry, in: Treatise on Geophysics, Vol 3., eds. T. Herring and G. Schubert, Elsevier, in press. Hinderer, J., C. de Linage, L. Rivera, J.-P. Boy, S. Lambotte, and R. Biancale, 2007b, The gravity signature of earthquakes, paper presented at First Asian Workshop on SGs, Taiwan, March 2007. Imanishi, Y., K. Kokubo and H. Tatehata, 2006, Effect of underground water on gravity observations at Matsushiro, Japan, J. Geodyn. 41, 221–226.

Report of GGP Activities to Commission 3 Iwano, S. and Y. Fukuda, 2004. Superconducting gravimeter observations without a tilt compensation system, Phys. Earth Planet. Int., 147, 343–351. Kao, R., C.C. Cheng, C.W. Lee, and C. Hwang, 2007. Analysis of SG T048 observations at Hsinchu: signal and noise, paper presented at First Asian Workshop on SGs, Taiwan, March 2007. Kim, J.-W., J.-S. Jeon, and Y.-S. Lee, 2005. Geohazard monitoring with space and geophysical technology, Korean J. Rem. Sens., 21 (1), 51–27. Kim, J.W., T.H. Kim, H.J. Park, J. Neumeyer, I. Woo, and J.S. Jeon, 2007. Analysis of SG measurement at MunGyung Observatory, Korea, paper presented at First Asian Workshop on SGs, Taiwan, March 2007. Masson, F., J. Hinderer, C. Hwang, C.C. Chen, C.W. Lee, and N. Le Moigne, 2007. Absolute gravity of the Taiwanese Orogen: the ATGO project, presented at First Asian Workshop on SGs, Taiwan, March 2007. Neumeyer, J., 2005. Efficiency of superconducting gravimeter observations and future prospects, Kor. J. Rem. Sens., 21 (1), 15–29. Neumeyer, J., F. Barthelmes, O. Dierks, F. Flechtner, M. Harnisch, G. Harnisch, J. Hinderer, Y. Imanishi, C. Kroner, B. Meurers, S. Petrovic, Ch. Reigber, R. Schmidt, P. Schwintzer, H.-P. Sun, and H. Virtanen, 2006, Combination of temporal gravity variations resulting from superconducting gravimeter (SG) recordings, GRACE satellite observations and global hydrology models, J. Geod, 79, 573–585.

521 Rosat, S., S. Watada, and T. Sato, 2007, Geographical variations of the 0S0 normal mode amplitude: predictions and observations after the Sumatra-Andaman earthquake, Earth Planets Space, 59, 307–311. Roult, G., S. Rosat, E. Clevede, R. Millot-Langet, and J. Hinderer, 2006, New determinations of Q quality factors and eigenfrequencies for the whole set of singlets of the Earth’s normal modes 0S0, 0S2, 0S3 and 2S1 using superconducting gravimeter data from the GGP network, J. Geodyn. 41 (1–3), 345–357. Sato, T., Fukuda, Y., Aoyama, Y., McQueen, H., Shibuya, K., Tamura, Y., Asari, K., and Ooe, M., 2001, On the observed annual gravity variation and the effect of sea surface height variations, Phys. Earth Planet. Int., 123, 45–63. Sato, T., Tamura, Y., Matsumoto K., Imanishi, Y., MacQueen, H., 2004. Parameters of the fluid core resonance inferred from superconducting gravimeter data, J. Geodyn., 38, 375–389. Shiomi, S., 2007, Testing the universality of free fall with superconducting gravimeters, paper presented at First Asian Workshop on SGs, Taiwan, March 2007. Virtanen, H., M. Tervo, and M. Bilker-Koivula, 2006, Comparison of superconducting gravimeter observations with hydrological models of various spatial extents, Bull. d’Inf. Marees Terr., 142, 11361–11365. Wilmes, H., A. Boer, B. Richter, M. Harnisch, G. Harnisch, H. Hase, and G. Engelhard, 2006, A new data series observed with the remote superconducting gravimeter GWR R038 at the geodetic fundamental station TIGO in Concepci’on (Chile), J. Geodyn. 41, 5–13.

European Tidal Gravity Observations: Comparison with Earth Tide Models and Estimation of the Free Core Nutation (FCN) Parameters B. Ducarme, S. Rosat, L. Vandercoilden, Xu Jian-Qiao and Sun Heping

Abstract The data of sixteen West European tidal gravity stations have been reprocessed. The tidal gravity factors have been corrected for ocean loading effect using the mean of 9 different ocean tides models with different grid size (0.5◦ , 0.25◦ and 0.125◦ ). For the principal tidal waves O1 and M2 the standard deviation of the corrected amplitude factors δc is lower than 0.1%. For O1 the value δc = 1.15340 ± 0.00023 lies between the Dehant, Defraigne and Wahr, 1999 (DDW99) elastic model and the Mathews, 2001 (MAT01) inelastic model. For M2 the value δc = 1.16211 ± 0.00020 fits very well the DDW99 and the MAT01 inelastic models. For K1 the mean result δc = 1.13525 ± 0.00032 (7 Global Geodynamics Project stations and Pecny) fits the MAT01 model to better than 0.05%. We determine the FCN parameters using either generalized least-squares or a Bayesian approach and find values between 428 and 432 sidereal days for the eigenperiod and quality factor Q values around 15,000.

B. Ducarme Research Associate NFSR, Royal Observatory of Belgium, Av. Circulaire 3, B-1180, Bruxelles, Belgium S. Rosat Royal Observatory of Belgium, Av. Circulaire 3, B-1180, Bruxelles, Belgium L. Vandercoilden Royal Observatory of Belgium, Av. Circulaire 3, B-1180, Bruxelles, Belgium Xu Jian-Qiao Institute of Geodesy and Geophysics, Chinese Academy of Science, 54 Xu Dong Road, Wuhan 430077, China Sun Heping Institute of Geodesy and Geophysics, Chinese Academy of Science, 54 Xu Dong Road, Wuhan 430077, China

Keywords Tidal gravity observations · Global Geodynamics Project (GGP) · Tidal gravity models · Free Core Nutation (FCN)

1 Introduction A cluster of 7 Global Geodynamics Project (GGP, Crossley et al., 1999) stations equipped with recent “compact tidal” (CT) or “double sphere” (CD) Superconducting Gravimeters (SG) is operating in Europe, in the vicinity of 9 independently calibrated tidal gravity stations. All the tidal gravity observations have been reprocessed using the tidal analysis software ETERNA (Wenzel, 1996) to determine, for a given tidal wave, the observed amplitude A and phase difference α with respect to the astronomical tide, i.e. the vector A = (A, α) in polar coordinates (Fig. 1). The amplitude factor δ is defined as the ratio A/Ath (Melchior, 1983) with respect to the astronomical tide of amplitude Ath . Several models of the Earth response to the tidal forces have been developed in the last ten years (Dehant et al., 1999; Mathews, 2001). The tidal gravity observation vector A(δAth , α) can be compared with the body tide models R(Ath · δth , 0), if we subtract the tidal loading effects L(L, λ) to get the so called “corrected” tidal parameters: amplitude factor δc and phase difference αc . Ac (δc Ath , αc ) = A − L

(1)

The tidal loading vector L, which takes into account the direct attraction of the water masses, the flexion of the ground and the associated change of potential, is generally evaluated by performing a convolution integral

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between 48 and 50◦ of latitude. At these latitudes the reference waves O1 and M2 have roughly the same amplitude. The interest of Western Europe is that the ocean tide loading is very weak in the diurnal band (less than 0.15 μGal i.e. 0.5% of O1 amplitude only). An accuracy of 10% on the evaluation of the loading will insure an accuracy of 0.05% on the corrected amplitude factor δc (O1). For M2, the situation is quite different as the tidal loading is 10 times larger, close to 1.5 μGal. The loading computation accuracy required for 0.1% accuracy on the corrected tidal factors is 2%, which could be more difficult to achieve.

2 Calibration of the Instruments Fig. 1 Phasor plot of a given tidal frequency showing the relationship between the observed tidal amplitude vector A(A, α), the Earth model R(R,0) and the computed ocean tides load vector L(L, λ)

We can clearly divide the stations into two groups: the GGP stations equipped with modern CT and CD superconducting gravimeters (SG) and the other ones, where between the ocean tide models and the load Green’s older model T SG’s or spring gravimeters were used. The GGP SG’s have been calibrated using parallel function computed by Farrell (1972). The error on the corrected tidal gravimetric factors tidal recording with absolute FG5 gravimeters, as dedepends thus on the error of the tidal gravity observa- scribed in Francis (1997). The instruments used in the additional stations, tions on one side and of the ocean tides loading comexcept in Brussels and Pecny, have been calibrated putation on the other. against the Hannover vertical calibration line (KangThe tidal gravity stations in Western Europe form ieser and Torge, 1981; Kangieser et al., 1983), either ◦ an East–West profile between 4 and 16 of longitude directly or indirectly. This is the case of T018 in (Table 1). All stations except Medicina are located Potsdam, which has been adjusted on the LaCosteRomberg (LCR) D02, calibrated against the Hannover Table 1 Selected tidal gravity stations in Europe. CD, CT, T: Su- standard (Neumeyer, personal communication), and perconducting Gravimeters, ASK: modified Askania, L: LaCoste LCR ET19 in Schiltach adjusted on the mean of LCR & Romberg ET: Tidal LaCoste D014 and G709 (Wenzel et al., 1991). The LCR ET13 Station Latitude◦ Longit.◦ Instr. has been used in several stations around the world by Brussels B 50.799 4.358 T003 the team of the Proudman Oceanographic Laboratory, Membach B 50.609 6.007 CT21 and can be considered as a subset directly connected Walferdange L 49.665 6.153 A233 to the Hannover system (Baker et al., 1991). Strasbourg F 48.622 7.684 CT26 In Brussels the T003 has been adjusted on the tidal Schiltach D 48.333 8.333 ET19, D014, L709 Karlsruhe D 49.012 8.413 L299 factors of a SCINTREX CG3M gravimeter record Z¨urich CH 47.367 8.550 ET13 (Ducarme and Somerhausen, 1997). In Walferdange, Bad Homburg D 50.229 8.611 CD30 the ASK233 has been adjusted on the same gravimeter. Chur C 46.850 9.533 ET13 In Pecny the two spring gravimeters were simultaneHannover D 52.387 9.712 L079, L299 Moxa D 50.645 11.616 CD34 ously calibrated by a parallel recording with the FG5 Medicina I 44.522 11.645 CT23 215 (Palinkas, 2006). Concerning the instrumental Wettzell D 49.146 12.879 CD29 error we can separate the calibration of the instrument Potsdam D 52.381 13.068 T018 acting on the amplitude factor δ from the instrumental Pecny CZ 49.920 14.790 A228, L137 Vienna A 48.248 16.361 CT25 delay and timing error affecting only the phase α. It is

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indeed possible to get rid of the calibration error by considering the ratio M2/O1. An evaluation of the precision of the calibrations based on absolute FG5 gravimeters is obtained by comparison of different instruments calibrated simultaneously i.e. the two components of the dual sphere instruments at Bad Homburg, Moxa and Wettzell (Kroner et al., 2005) and the two spring gravimeters at Pecny. The amplitude factors agree within 0.05% and the phase differences within the associated RMS errors (less than 0.05◦ ). A comparison of LCR instruments calibrated on the Hannover baseline is possible in Schiltach (D014, L709) and in Hannover (L079, L299) (Timmen and Wenzel, 1994). These instruments are less coherent with discrepancies higher than 0.1%. Comparison between different calibration methods is possible in Membach, where the gravimeter SCINTREX CG3M recorded in parallel with CT021. The agreement is better than 0.1%. Further evaluation of the accuracy of the calibration will be given in Sect. 4 by comparison of the corrected tidal amplitude factors in different stations.

3 Tidal Loading Evaluation We used 9 ocean tide models with different grid resolution 0.5◦ × 0.5◦ (CSR3, CSR4, FES95, GOT00, NAO99, ORI96), 0.25 × 0.25◦ (FES02, TPX06) and 0.125 × 0.125◦ (FES04). Details and references can be found in Boy et al., 2003 and Ducarme et al., 2007b. For the first generation of models (CSR3, FES95 and ORI96), the effect of the imperfect mass conservation is corrected on the basis of the code developed by Moens (Melchior et al., 1980). The dispersion of the amplitude and the phase of the load vector L for the 9 models, expressed by the standard deviation, is close to 0.03 μGal (12◦ ) for O1. For M2 it decreases from 0.07 μGal (3◦ ) at Brussels to 0.03 μGal (1.5◦ ) at the East of the profile. It is close to 0.1% of the tidal waves amplitude everywhere except at Brussels and Membach, which are closer to the coast. The fact that the dispersion is the same for the large M2 loading as for the small O1 loading means probably that we reach a limit of precision concerning the computation of the load vector. Using the mean of the 9 models, the associated RMS error on the corrected tidal fac-

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tor will be close to 0.03% for O1 and 0.04% for M2. However, it does not take into account eventual systematic errors in the tidal loading evaluation, which can be larger than the error estimates of SG measurements (Boy et al., 2003).

4 Corrected Tidal Factors We can now study the dispersion of the tidal factors after correction by the mean of the 9 ocean tide models (Table 2). In our EW profile we can consider the stations equipped with the new generation of SG’s i.e. CT and CD instruments, called here. “GGP profile”, on one side and instruments of different type: old SG in Brussels and Potsdam (not calibrated using AG) and spring instruments on the other. If we compare the corrected amplitude factors δc along the GGP profile (upper part of Table 2), we can derive an associated standard deviation of 0.08%. It includes calibration and differential tidal loading errors. The calibration error is certainly the main limiting factor, as the standard deviation on the ratio M2/O1 is only 0.02%. As a matter of fact, the corrected phase differences αc are slightly positive nearly everywhere. It could be due either to an over-correction of the instrumental time lag or to a systematic error in the ocean tidal loading. The first possibility is unlikely as different types of instrument have been used. If we consider that the phase misfit is only due to an incorrect tidal loading correction and that the loading error is the same for both in phase and out of phase components, it puts an upper limit to the error on the corrected amplitude factor at the level of 0.05% for M2, 0.025% for O1 and 0.06% for the ratio δc (M2)/δc (O1). The results of Brussels and Membach are offset by more than 2.5σ from the mean, for the two parameters αc (M2) and δc (M2)/δc (O1). These stations are indeed closest to the ocean. It was thus decided to eliminate them from the mean for M2 and for the ratio δc (M2)/δc (O1). There is no systematic difference, concerning the mean values, between the GGP profile and the additional stations at the level of 0.02%. Moreover the standard deviations of both groups are the same. It seems thus reasonable to adopt the global mean of the

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Table 2 Corrected tidal parameters (9 ocean models) STD: standard deviation O1

M2

Corrected factors

M2/O1

Corrected factors

Wave

δc

αc

GGP profile Membach Strasbourg Bad Homburg Moxa Medicina Wettzell Vienna Mean

1.15359 1.15366 1.15450 1.15315 1.15334 1.15237 1.15224 1.15326

0.018 0.010 0.021 0.029 0.010 0.032 0.031 0.022

STD Additional stations Brussels Walferdange Potsdam Pecny Schiltach Karlsruhe Hannover Z¨urich Chur Mean STD

0.00078

0.009

1.15405 1.15506 1.15413 1.15443 1.15293 1.15339 1.15350 1.15226 1.15173 1.15350 0.00106

–0.028 –0.013 0.018 0.021 0.011 0.046 0.053 –0.006 0.002 0.012 0.026

Global Mean STD RMS

1.15340 0.00092 0.00023

0.016 0.021 0.0005

δc

αc

δc (M2)/δc (O1)

r1.16142

∗ 0.119

∗ 1.00679

1.16232 1.16287 1.16159 1.16237 1.16116 1.16092 1.16187

0.045 0.074 0.063 0.011 0.048 0.029 0.046

1.00751 1.00725 1.00732 1.00783 1.00763 1.00753 1.00747

0.00072

0.023

0.00021

r1.16077

∗ 0.093

1.16294 1.16324 1.16342 1.16180 1.16198 1.16240 1.16147 1.16110 1.16229 0.00085

–0.037 0.034 0.026 0.043 0.048 0.040 –0.001 0.006 0.020 0.029

1.00682 1.00789 1.00779 1.00769 1.00745 1.00775 1.00799 1.00814 1.00762 0.00041

1.16211 0.00081 0.00020

0.031 0.029 0.007

1.00755 0.00034 0.00008

∗ 1.0058

∗ 2σ

offset, not taken into account in the mean • not taken into account in the mean

15 series as the most probable value of the corrected tidal parameters for O1 and to exclude Brussels and Membach for M2. Using the ocean tide models GOT99, NAO99 and FES99, Baker and Bos (2003) give for Europe mean GGP O1 corrected amplitude factors 1.1533, 1.1532, 1.1536 respectively. For spring gravimeters they quote values 1.1522, 1.1521, 1.1525. For M2 the results are 1.1614, 1.1616, 1.1620 for GGP and 1.1605, 1.1605, 1.1613 for spring instruments. Our set of spring and SG gravimeters is more coherent, even if some anomalies are still existing. The subset observed with ET13 at stations Chur and Z¨urich provides amplitude factors 0.1% lower than the mean of the stations. Inversely the stations Brussels and Walferdange, based on SCINTREX calibration, seem 0.1% higher than the global mean as well as Pecny. It is surprising, as this station has been very carefully calibrated with FG5 215 (Palinkas, 2006). In Vienna, a parallel recording with the same FG5 provided an increase of the calibra-

tion by a factor 1.0025 (Meurers, personal communication). If this correction is applied we get in Vienna δc (O1) = 1.1551, closer to the Pecny value.

5 Stability of the Corrected Factors To get a better idea of the load computation errors we subdivided the 9 ocean tide models in two subgroups of 6 models each, either the models with a coarse 0.5×0.5◦ grid (CSR3, CSR4, FES95, GOT00, NAO99, ORI96) or the most recent ones (CSR4, FES02, FES04, GOT00, NAO99, TPX06). The models CSR4, GOT00 and NAO99 are common to the two groups. For the most recent models we used two different computing approaches: with mass correction (MCOR) or without mass correction (NOMC) (Ducarme et al., 2007b).

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Table 3 Mean corrected tidal parameters using different combinations of ocean tide models Wave

O1

Ocean tide model

Corrected factors

0.5 × 0.5◦ Recent NOMC Recent MCOR Mean STD Global (Table 2)

M2∗ Corrected factors

δc

αc

δc

αc

δc (M2)/δc (O1)

1.1534 1.1534 1.1535 1.1534 0.0001 1.1534

0.02 0.01 0.04 0.02 0.01 0.02

1.1623 1.1616 1.1622 1.1620 0.0004 1.1621

0.02 0.03 0.02 0.02 0.00 0.03

1.0078 1.0073 1.0075 1.0075 0.0002 1.0076

∗

Brussels and Membach excluded STD: standard deviation

Table 3 provides a comparison of the mean corrected tidal amplitude factors and phase differences, for three individual solutions. The stations Brussels and Membach, which are the closest to the ocean, are more sensitive to the choice of the ocean tide models. It is the reason why we discarded them for M2. For O1 the maximum discrepancy between the different combinations is at the level of 0.02%, while it reaches 0.05% for M2. The standard deviation is close to 0.015% for O1, so that the global mean should have a better precision, considering only the loading error. For M2 the standard deviation is larger (0.04%). It probably means that we cannot determine the mean corrected tidal factors of M2 with a much better precision. It is close to the RMS error of 0.03% (O1) and 0.04% (M2) estimated in Sect. 3.

6 Comparison with Body Tide Models The most recent body tide models are the DDW99 (Dehant et al., 1999), hydrostatic (H) or nonhydrostatic/inelastic (NH) models, and the MAT01 (Mathews, 2001) NH model. The hydrostatic version of MAT01 is practically identical to DDW99/H. The theoretical tidal amplitude factors δth , given in Table 4 should be compared with the corrected ones δc given in Table 3. The RMS error on the mean tidal factors can be computed as the combination of the dispersion of the series for O1 and M2 (0.02% in Table 2) on one hand, and the oceanic loading computation error discussed in Sect. 3 and 5 (0.025% for O1 and 0.04% for M2) on the other hand. A conservative evaluation leads to

Table 4 Theoretical amplitude factors δth O1 K1 M2 δth (O1)/δth (K1) δth (M2)/δth (O1)

DDW/H

MAT01/NH

DDW/NH

1.1528 1.1324 1.1605 1.0180 1.0067

1.1540 1.1349 1.1616 1.0168 1.0066

1.1543 1.1345 1.1620 1.0174 1.0066

an RMS error of 0.03% on the mean corrected tidal factors for O1 and 0.045% for M2. As the discrepancies between models are of the order of 0.1%, we can draw some conclusions concerning the best fitting models. For O1 the mean ratios between the observed and theoretical amplitude factors are: DDW/H 1.00051, MAT01/NH 0.99948, DDW/NH 0.99923. The observations fall between DDW99/H and MAT01/NH models (Fig. 2). For M2 the mean ratios are: DDW/H 1.00137, MAT01/NH 1.00040, DDW/NH 1.00013. All the individual solutions shown in Fig. 3 and Table 3 fit to better than 0.02% to the DDW/NH model and MAT01/NH agrees within the 2σ error range. There is still one important parameter to consider, the ratio M2/O1. For the ratio M2/O1 the observation error is lower (0.008% in Table 2) while the loading computation error could reach 0.06% (Sect. 5). The final error should be lower than 0.06%. The theoretical value 1.0067 is practically the same for all the considered body tide models (Table 4), but disagrees with the observations (Table 3) by 0.09%, i.e. close to twice the associated RMS error. The situation is slightly improved if we consider only the 6 recent ocean tide models, NOMC option.

528

B. Ducarme et al.

Fig. 2 Ratio to theoretical models for O1: From left to right 0.5 × 0.5◦ , NOMC, MCOR and all Light-gray: DDW/H, dark-gray MAT01/NH, white DDW/NH

O1 1.001

1.0005

ratio

1 DDW/H MAT01/NH DDW/NH

0.9995

0.999

0.9985

0.998 0.5 × 0.5

Fig. 3 Ratio to theoretical models for M2: From left to right 0.5 × 0.5◦ , NOMC, MCOR and all Light-gray: DDW/H, dark-gray MAT01/NH, white DDW/NH

NMCOR MCOR ocean models

all

M2 1.002 1.0015 1.001 DDW/H MAT01/NH DDW/NH

ratio

1.0005 1 0.9995 0.999 0.9985 0.5 × 0.5

7 The Liquid Core Resonance

NMCOR MCOR ocean models

all

(Membach, Strasbourg, Medicina, Vienna, Wettzell, Bad Homburg, Potsdam, Moxa) and Pecny. It should be As the ocean tidal loading seems to be very well con- pointed out also that, for most ocean tides models, the strained in the diurnal band, we are in a very good po- loading corrections of these constituents have to be exsition to study the Free Core Nutation (FCN). The standard deviation on K1 (Table 5) is lower than 0.1%. It is clear that the observed ratio δc (O1)/δc (K1) does not Table 5 Corrected tidal parameters (9 ocean models) O1 K1 δc (O1)/δc (K1) agree with any one of the theoretical models of Table 4. Wave This fact is well known (Ducarme et al. 2007a) and STATION δc δc is due to the choice of the FCN model. The discrep- Membach 1.15359 1.13568 1.01577 1.15366 1.13547 1.01602 ancy can be removed by computing the FCN parame- Strasbourg 1.15450 1.13600 1.01629 ters (eigenperiod T and quality factor Q) directly from Bad Homburg Moxa 1.15315 1.13507 1.01593 the observations. Medicina 1.15334 1.13480 1.01634 The determination of the FCN parameters requires Wettzell 1.15237 1.13424 1.01598 1.15224 1.13404 1.01605 a very precise determination of the small diurnal con- Vienna 1.15443 1.13672 1.01558 stituents ψ1 and ϕ1 . We keep thus only the stations Pecny Mean 1.15341 1.13525 1.01600 with the best signal to noise ratio i.e. the GGP profile STD 0.00083 0.00090 0.00025

European Tidal Gravity Observations

529

trapolated from the values obtained for the main diurnal constituents (Q1, O1, P1 and K1). The load vectors are normalized first, dividing them by the corresponding theoretical amplitude in the tidal potential. It is necessary to take into account that the core resonance does also affect the ocean tides (Wahr and Sasao, 1981). Before interpolation we have thus to correct this resonance effect for the main diurnal waves, especially K1 , then interpolate or extrapolate the weaker components and apply again the resonance on the results. This procedure, described in Xu et al. (2004), has been applied on the real and imaginary parts of the ocean load vectors. As a matter of fact we need only to consider 5 tidal frequencies O1, P1, K1, ψ1 and ϕ1 . Additional constituents such as Q1, NO1, J1 and OO1 do not improve the solution. We computed the resonance parameters (Table 6), using a generalized least-squares approach and imposing non-negative Q values (Florsch and Hinderer, 2000). Q=

σR = 10x 2σ I

the most complete information on the FCN resonance parameters. Besides, this method enables to respect the non-Gaussian distribution of the Q-parameter. The results of the Bayesian inversion are the probability laws for each parameter (Tarantola and Valette, 1982). By integration of the other two parameters (real a R and imaginary a I parts of the resonance strength), we can compute the joint probability density function (pdf) for T and x. We have for example:  pa R ,a I (x, T ) =

(4)

where p is a probability density function. Such pdf is plotted in Fig. 4 for the dataset corresponding to the oceanic model based on 6 recent tidal maps with mass correction (MCOR). Similar pdfs are obtained for the other datasets. A further integration leads to the marginal pdf for x and T (formula 5 for × and Fig. 4 (b)).

(2) 

or

pT,a R ,a I (x) = x = log10 (Q)

p(x, T, a R , a I )da R da I

p(x, T, a R , a I )da R da I dT (5)

(3)

From these laws of probability, we can estimate mean values for Q and for T (Table 7), which can be compared to the least-squares results in Table 6. Notice that in our case, the law of probability for Q is nearly Gaussian. There is a good agreementbetween both inversion methods, within the 95% confidence interval of the least-squares adjustment. Notice that in Table 7 we give the mean values of the parameters and not the values for the maximum of probability (Tarantola and Valette, 1982) as in Fig. 4. Notice that in Table 7 we give the mean values of Table 6 Free core nutation parameters computed using a conthe parameters and not the values for the maximum of strained generalized least-squares approach with 95% confidence probability (Tarantola and Valette, 1982) as in Fig. 4. interval We find values ranging from 428 to 432 sidereal Tfcn (sid. day) Q δth (O1)/δch (K1) days, with associated Q values around 15,000. The so9 Maps 430.9 17637 1.0164 lution obtained with the 0.5 × 0.5◦ maps alone is less (426.4,435.6) (9456,130695) ◦ ◦ 32601 1.0161 0.5 × 0.5 429.7 stable. It is a well known fact that the inversion of FCN (391.2,476.7) (8827, ∞) parameters from tidal gravity observations alone is unRecent 432.2 14605 1.0164 derestimating Q (Florsch and Hinderer, 2000). A rea(NOMC) (427.2,437.5) (8628,47524) son is that tidal loading corrections are not so well conRecent 429.9 12369 1.0162 (MCOR) (425.2,434.7) (7563,33933) strained, especially for ψ1 , which is the wave closest to the resonance (Sato et al., 2004).

where σR and σI are the real and imaginary parts of the nearly diurnal resonance frequency. These results of Table 6 are in good agreement with previous determinations (Sato et al., 2004; Ducarme et al., 2007a). As expected, the ratio δth (O1)/δth (K1) is now very close to the experimental value 1.016. Florsch and Hinderer (2000) have proposed a Bayesian method to invert for the resonance parameters, since such a probabilistic approach allows

530 Fig. 4 (a) Joint probability density function for x = log10 (Q) and T. (b) Marginal probability laws for x and T. The maximum of probability is reached for T = 428 days and Q = 11,220

B. Ducarme et al. (a)

(b)

calibrated additional stations. This network forms and East-West profile extending from Brussels (B) to Tfcn (sid. day) Q Vienna (A). 9 maps 428.7 15,436 To eliminate the tidal loading effect from the obser427.5 276,345 vations we used 9 ocean tides models CSR3, CSR4, 0.5 × 0.5◦ Recent (NOMC) 430.1 14,906 Recent (MCOR) 428.4 12,753 FES95, FES02, FES04, GOT00, NAO99, ORI96 and TPX06. For the principal waves O1 and M2 the standard deviation of the mean corrected amplitude factor δc is lower than 0.1% for both data sets and the mean values agree to better than 0.05%. 8 Conclusions We compared the corrected tidal amplitude factors δc with theoretical tidal amplitude factors δth derived In this study we used two data sets: seven GGP tidal from the DDW99 H and NH models as well as from gravity stations, equipped with the most recent CT the MAT01/NH model. The differences between these or CD superconducting gravimeters, and nine well Table 7 Free core nutation parameters derived from the Bayesian inversion method

European Tidal Gravity Observations

different models are slightly larger than 0.1%. We can expect a global error budget close to 0.03% for O1. For M2 the error budget is 0.045%. We can thus draw some conclusions concerning the best fitting models. For O1 the solutions obtained with the different subsets of ocean tide models agree closely. The corrected mean tidal factor δc (O1) = 1.1533 ± 0.0002 lies between the DDW99/H and MAT01/NH models. The MAT01/NH value 1.1540 is closer to the observations than the DDW99/NH one. For M2 the global solution δc = 1.16211 ± 0.00020 fits the DDW99/NH value 1.1620, but MAT01/NH is still within the associated 2σ error range. For K1 the mean result δc = 1.13525 ± 0.00032 (7 GGP stations and Pecny) fits the MAT01 model to better than 0.05%. The best solution for the ratio M2/O1 (1.00727) is given by the combination of the 6 recent ocean tide models. This value is within 0.05% of the theoretical models value 1.0067. The ratio O1/K1 (1.0160) is closest to the MAT01 model. The discrepancy is due to the choice of the FCN model and can be removed by computing the FCN parameters directly from the observations. We determined the FCN parameters using two different inversion methods: generalized least-squares and Bayesian. Both methods agree for any one of the combination of ocean tide models but discrepancies are still large between models. We find values between 428 and 432 sidereal days for the eigenperiod and quality factor Q values around 15,000. The only way to improve the estimation of the FCN using tidal gravity observations is to improve the ocean tide models. The adoption of the MAT01/NH model as a standard for computing modeled tidal parameters seems to be the best compromise as it fits the observations with discrepancies close to 0.05%. Acknowledgments This paper wishes to pay tribute to all the distinguished geodesists and geophysicists who performed very accurate tidal gravity observations across Europe during the last 50 years, with a special mention of late Professors P. Melchior and H.-G. Wenzel. The SG data are taken from the GGP data base and the other data sets from the data base of the International Center for Earth Tides (ICET), where data owners deposited their data. We wish to thank several colleagues who helped a lot to trace back the calibration of different stations: T. F. Baker, M. & G. Harnisch, C. Kroner, B. Meurers, N. d’Oreye, J. Neumeyer, V. Palinkas, H. Wilmes and W. Z¨urn. The authors wish also to thank N. Florsch for his helpful commitment in the initial computation of the Bayesian inversion method.

531

References Baker, T. F., Bos, M. S. (2003) Validating Earth and ocean models using tidal gravity measurements. Geophys J. Int., 152, 468–485. Baker, T. F. Edge R. J., Jeffries G. (1991). Tidal gravity and ocean tide in Europe. Geoph. J. Int., 107, 1–11. Boy, J.-P., Llubes, M., Hinderer, J., Florsch, N. (2003). A comparison of tidal loading models using superconducting gravimeter data. Geoph. J. Int., 108, 2193, doi:10.1029/2002JB002050. Crossley, D., Hinderer, J., Casula, G., Francis, O., Hsu, H. T., Imanishi, Y., Jentzsch, G., K¨aa¨ ria¨ınen, J., Merriam, J., Meurers, B., Neumeyer, J., Richter, B., Shibuya, K., Sato, T., Van Dam, T. (1999). Network of superconducting gravimeters benefits a number of disciplines. EOS, 80, 11, 121/125–126. Dehant V., Defraigne P., Wahr J. (1999). Tides for a convective Earth. J. Geophys. Res., 104, B1, 1035–1058. Ducarme, B., Somerhausen, A. (1997). Tidal gravity recording at Brussels with a SCINTREX CG3-M gravimeter. Bull. Inf. Mar´ees Terrestres, 126, 9611–9634. Ducarme, B., Sun, H.-P., Xu, J.-Q. (2007a). Determination of the free core nutation period from tidal gravity observations of the GGP superconducting gravimeter network; J. Geodesy, 81, 179–187, doi:10.1007/ s00190-006-0098-9. Ducarme, B., Timofeev, V. Y., Everaerts, M., Gornov, P., Parovishnii, V., van Ruymbeke, M. (2007b). A trans-siberian tidal gravity profile (TSP) for the validation of ocean tides loading corrections. J. Geodynamics, doi:10.1016/j.jog.2007.07. Farrell, W. E. (1972). Deformation of the Earth by surface load. Rev. Geoph., 10, 761–779. Florsch, N., Hinderer, J. (2000) Bayesian estimation of the Free Core Nutation parameters from the analysis of precise gravity data. Phys. Earth Planet. Int., 117, 21–35. Francis, O. (1997). Calibration of the C021 superconducting gravimeter in Membach (Belgium) using 47 days of absolute gravity measurements. International Association of Geodesy Symp., Gravity, Geoid and Marine Geodesy, Segawa et al. (eds.), 117, 212–219. Kangieser, E., Torge, W. (1981). Calibration of LaCosteRomberg gravity meters Model G and D. Bull. Inf. Bur. Grav. Int., 49, 50–63. Kangieser, E., Kummer K., Torge, W., Wenzel H.-G. (1983). Das Gravimeter-Eichsystem Hanover. Wissenschaftliche Arbeiten der Fachrichtung Vermessungswesen der Universit¨at Hannover, 1203 Kroner C., Dierks O., Neumeyer J., Wilmes H. (2005). Analysis and observations with dual sensor superconducting gravimeters. Phys. Earth Planet. Int., 153(4), 210–219. Mathews, P. M. (2001). Love numbers and gravimetric factor for diurnal tides. Proc. 14th Int. Symp. Earth Tides. J. Geod. Soc. Jpn., 47(1), 231–236. Melchior P. (1983). The tides of the planet Earth, 2nd Edition, Pergamon Press, Oxford, 641pp. Melchior, P., Moens, M., Ducarme, B. (1980). Computations of tidal gravity loading and attraction effects. Royal Observatory of Belgium, Bull. Obs. Mar´ees Terrestres, 4, 5, 95–133.

532 Palinkas V. (2006). Precise Tidal measurements by spring gravimeters at the station Pecny. J. Geodynamics, 41, 14–22. Sato, T., Tamura, Y., Matsumoto, K., Imanishi, Y., Mac Queen H. (2004). Parameters of the fluid core resonance inferred from superconducting gravimeter data. J. Geodynamics, 38, 375–389. Tarantola, A., Valette, B. (1982). Inverse problems-quest for information. J. Geophys., 50, 159–170. Timmen, L., Wenzel H.-G. (1994). Improved gravimetric Earth tide parameters for station Hannover. Bull. Inf. Mar´ees Terrestres, 119, 8834–8846.

B. Ducarme et al. Wahr, J. M., Sasao, T. (1981) A diurnal resonance in the ocean tide and in the Earth’s load response due to the resonant free core nutation. Geoph. J. R. Astr. Soc., 64, 747–765. Wenzel, H. G. (1996). The nanogal software: earth tide data preprocessing package. Bull. Inf. Mar´ees Terrestres, 124, 9425–9439. Wenzel, H. G., Z¨urn, W., Baker, T. F., (1991). In situ calibration of LaCoste-Romberg Earth Tide Gravity meter ET19 at BFO Schiltach. Bull. Inf. Mar´ees Terrestres, 109, 7849–7883. Xu, J. Q., Sun, H. P., Ducarme, B. (2004) A global experimental model for gravity tides of the Earth. J. Geod., 38, 293–306.

Physical Modelling to Remove Hydrological Effects at Local and Regional Scale: Application to the 100-m Hydrostatic Inclinometer in Sainte-Croix-aux-Mines (France) L. Longuevergne, L. Oudin, N. Florsch, F. Boudin and J.P. Boy

Abstract New inclinometers devoted to hydrological studies were set up in the Vosges Mountains (France). Two orthogonal 100-m base hydrostatic inclinometers were installed in December 2004 as well as a hydrometeorological monitoring system for the 100-km2 hydrological unit around the inclinometer. As inclinometers are very sensitive to environmental influences, this observatory is a test site to confront hydrological modelling and geodetic observations. Physical modelling to remove hydrological effects without calibrating on geodetic data is tested on these instruments. Specifically, two deformation processes are most important: fluid pressure variations in nearby hydraulically active fractures and surface loading at regional scale.

1 Introduction Hydrological phenomena mix with geodynamical processes on a broad band of frequencies, ranging from long-term and annual variations, to short period disturbances of 1 h (e.g. Evans and Wyatt 1984; Crossley et al., 1999; Zadro and Braitenberg, 1999). Some authors (e.g. Dal Moro and Zadro, 1998) concluded that hydrological effects should be removed before studying other deformation or dynamic signals. L. Longuevergne UMR Sisyphe, Universit´e Pierre et Marie Curie, France L. Oudin UMR Sisyphe, Universit´e Pierre et Marie Curie, France N. Florsch UMR Sisyphe, Universit´e Pierre et Marie Curie, France F. Boudin G´eosciences Montpellier, Universit´e Montpellier II, France J.P. Boy EOST, Universit´e Louis Pasteur, Strasbourg, France

Two methodologies have arisen to investigate hydrological effects. Both of them lead to relatively good results. They are however very different in terms of processes being modelled and spatial “extent” of the investigation: (i) The first one focuses on local deformation (Newtonian attraction for gravimetry). It is based on correlations between geodetic observations and local hydrological measurements – rainfall, piezometric, and soil moisture measurements (Bower and Courtier, 1998; Crossley et al., 1998; Kroner et al. 2001; Van Camp et al. 2006; Meurers et al. 2007). Both linear and nonlinear models have also been used (Wolfe et al., 1981; Yamauchi, 1987; Latynina et al. 1993). In this case, fitting on (integrative) geodetic measurements somehow plays the role of “distribution” of the measured hydrological variable of local significance. Temporally speaking, this correction is a-priori limited by the temporal sampling of the environmental monitoring. (ii) A somewhat different deterministic approach considers the physical modelling of the contribution using global hydrological models (e.g. Lad World (Milly et al., 2002a), or GLDAS (Rodell et al. 2004) considering global surface loading and Newtonian attraction. No calibration was made on geodetic data (Dong et al., 2002; Boy and Hinderer, 2006). This methodology is limited by the spatial smoothing of global models (the tightest description is based on a 0.25◦ grid), and their temporal sampling. Note that Virtanen et al. (2006) has used hydrological models of several spatial extents in order to improve this approach. This work is based on the fact that all environmental signals are correlated (especially on annual time scale), so calibration using geodynamic data could lead to incomplete correction of environmental noise (Tervo

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

533

534

et al., 2006). As a consequence physical modelling is needed, i.e. every deformational process mixed in geodynamic measurements should be investigated separately. From a hydrological point of view both the amount and the position of water masses have to be modelled. This decoupling between geodetic deformation and hydrological modelling is the next step to accurate corrections of geodynamic time series (see Hasan et al., 2006; Longuevergne et al. 2006; Naujok et al., 2007). The possibility to use geodetic data as a tool for hydrology is at stake. In order to investigate hydrological contribution to geodetic observations, a new observatory devoted to hydrological studies has been set up in Sainte-Croixaux-Mines, in the Vosges Mountains (eastern France). We chose to install the geodetic instruments that are considered to be the most sensitive to environmental influences: inclinometers. Two processes of deformation induced by water generate significant geodynamical effects on recorded data: local hydraulic pressure fluctuations in nearby conductive fractures and surface loading at regional scale.

2 Spatial Scales Associated with Hydrological Effects

L. Longuevergne et al.

particular because of the indirect effects of oceans on the determination of tidal parameters (e.g. Bos and Baker, 2005). Each geodetic instrument has a different spatial sensitivity when dealing with loading process. Agnew (2001) investigated how instruments “see” a uniform surface loading. He has plotted “loading maps” using “deformational distance” for gravity, radial displacement, tilt and strain observations. This work has been followed by Llubes et al. (2004). They proposed 3 scales of interest which must be considered when dealing with environmental loading (see Fig. 1): global scale, i.e. mass redistribution on the entire earth, regional scale, i.e. distances between 1 and 10 km, and finally local scale, 1 km around the instrument. We keep the same scales as reference for all deformation processes. Orders of magnitude of observed deformation determined from previously cited works are indicated in Table 1.

Table 1 Spatial scales associated with different geodetic measurements and associated orders of magnitude (from previously cited works and this work) GPS Gravimeters Inclinometers

Local

Regional

Global

negligible 50 nm.s−2 500 nrad

negligible negligible 50 nrad

10 mm 50 nm.s−2 Negligible

The earth is an elastic body that deforms under mass redistribution of different fluid layers on its surface (ocean, atmosphere, continental water). Surface loading has been studied for a long time in

Fig. 1 Situation of inclinometers and illustration of spatial scales. On the left, local scale and installed instruments in their environment. The N37◦ E inclinometer is installed orthogonal the metallic vein (dark blue line), the N120◦ E inclinometer is installed inside the exploited metallic vein. In the centre, regional

scale illustrated with colored topography. Contours are yearly rainfall [mm], thick line is the limit of the studied catchment, the triangle at the outlet is the water flow measurement. On the right, global scale

Physical Modelling to Remove Hydrological Effects

535

3 Inclinometers Devoted to Hydrological year, and an important horizontal rainfall gradient is also observed (80 mm.km−1 ). To take into account this Studies spatial variability, 15 daily-sampling raingauges are used in this study. A single raingauge near the incliThe choice of the installation site is a first consideranometer would have underestimated the total rainfall tion to record good data, and requires special care (e.g. of 10% (see Fig. 1 for annual rainfall amount). Daily Melchior, 1973). An old mine was chosen in Saintestream flow at the outlet of the catchment is measured Croix-aux-Mines (Vosges Mountains, east of France). by the French direction for environment (DIREN). Figure 1 shows the two orthogonal branches of the mine. The mine excavated into gneissic rocks and the 160-m height rock covering, ensuring the environment to be very stable (temperature variation is around 4 Local Scale Deformation 10−1◦ C over the year). We set up two orthogonal 100-m base hydrostatic inclinometers, based on the The comparison between water drainage out of the communicating vessels’ principle (see Agnew, 1986), mine and daily decimated tilt time series (see Fig. 3) using a design recently developed by Boudin (2004). shows that hydraulic pressure variations in nearby The N37◦ E instrument has been monitoring crustal conductive fractures generate the major part of the tilting since December 2004 with a temporal sam- recorded deformation. Evans and Wyatt (1984) atpling of 30 s and a resolution better than 10−10 rad tributed this deformation to the change in the aperture (0.02 mas). of a hydraulically active fracture. Indeed, 20 m from It is notable that a perfect coupling between gneissic the north vessel the exploited metallic sulphide vein rock and instrument has been achieved. This is impor- plays the role of a preferential path for water. Because tant since the use of inserted material is considered of the limited thickness of the fractures, an influx of as responsible of important drifts (see Fig. 2 for raw water induces a quick variation of water height, and data). The data has proved to be of good quality for thus a quick variation of pressure. both long-term and short-term monitoring (see Boudin The signature of this deformation is the 250 et al., 2007 for normal modes generated by 2004 nrad.month-1 quick variation during 2 months on the Sumatra-Andaman earthquake). Unfortunately, several N37◦ E instrument, orthogonal to the fracture, and lightning and associated electrical problems have to be very few effect on the N120◦ E instrument, installed reported, which caused the gaps in time series. quasi-parallel to the fracture (see Fig. 2, in April 2006, In the same time, the hydrometeorological moni- October 2006 and February 2007). toring of the 108 km2 hydrological unit (catchment) The quantification of this deformation can be seen, around the instrument has been set up in order to mon- as a first step, with first order models. Mechanically itor mass balance as precisely as possible. A meteoro- speaking, deformation is linked to water height varilogical station has been installed. The studied catch- ations in the fracture. Hydrologically speaking, wament receives a total rainfall amount of 1300 mm a ter flow is linked to the exponential of water height Hourly N37 inclinometer data Daily N37 inclinometer data Hourly N120 inclinometer data Daily N120 inclinometer data

Fig. 2 Raw data of both orthogonal inclinometers (N37 instrument below, N120 instrument above). Note the 100-nrad amplitude tides, and the repeated 250 nrad.month−1 -“drift” during 2 monthes on N37 instrument only

Inclinometric deformation [nrad] and Effective rainfalls [mm]

400 300

250 nrad.month–1

200 100 0 –100 –200 –300

Jan05

Jan06

Jan07

536 300

150

200 100 100 0 –100 50 –200 –300

Jan05

Apr05

Jul05

variations, i.e. water height variations is linked to the logarithm of water flow, the hydrological units having an integrative behaviour. These considerations are plotted on Figure 3. The water supply of these fractures is driven by the hydrological unit on top of the hill. Geophysical studies show 10- to 20-deep altered gneiss (i.e. ≈ sand) covering a fractured media. Soils have the capacity to absorb water before letting it infiltrate. As a consequence, they play the role of a low-pass filter concerning water infiltration. This explains the long term contribution of this local hydrological unit. The quantification of this deformation without calibration on geodetic data is difficult, for both hydrological and mechanical reasons. Firstly, the mine drainage gathers water that does not come from the deforming fracture. Secondly, is it necessary to correctly model this hydromecanical deformation and the joint behaviour. In the following work, this deformation has been “corrected” thanks to the calibration of the logarithm of mine drainage on geodetic data. As a consequence, long term corrected data is affected by this approximate correction. An accurate calculation of the fracture deformation will need further studies, on both mechanical and hydrological considerations.

Oct05

Jan06

Apr06

Jul06

Oct06

Jan07

Water flow out of the mine [m 3.j–1 ]

Inclinometric deformation [nrad]

Fig. 3 Daily decimated N37◦ E tilt signal (dotted red curve) vs. logarithm of water flow out of the mine (continuous black curve). Blue bars are estimated monthly time derivative of stored water variations (i.e. Rainfall minus potential evapotranspiration minus runoff)

L. Longuevergne et al.

Apr07

We use the Green function formalism to calculate tilt deformation induced by stored water variations within the catchment (Farrell, 1972; Pagiatakis 1990; Guo et al., 2004). This calculation needs both the amount (i.e. the stored water variations) and the location of water masses.

5.1 Amount of Water

A robust calculation of the mass balance within the catchment relies on the capacity to assess realistically each part of the hydrological cycle at catchment scale: dS dS dt = P − AE − (Q + Q sub ), where dt is the time derivative of stored water on the earth crust, P is precipitation rate, AE actual evapotranspiration rate, and finally Q + Q sub the runoff amount, respectively surface runoff and subsurface runoff (e.g. Lettemaier, 2005). Evapotranspiration is a nonlinear process that is difficult to measure directly (e.g. Strangeways, 2003). Subsurface runoff may also be a substantial part of surface runoff (e.g. Le Moine et al., 2007). Thus we chose to assess water losses by calibrating a rainfall-runoff 5 Regional Scale Deformation hydrological model. We use the 4-parameter lumped model GR4J (Perrin et al., 2003). The model is caliInclinometers are sensitive to surface loading at re- brated using the daily streamflow measured at the outgional scale. As a consequence, we study the unit of let of the catchment. The Nash and Sutcliffe (1970) crihydrological significance of the same size, i.e. the 108- teria is up to 0.8 in calibration (1 indicating a perfect km2 catchment around the instrument (see Fig. 1 – re- description of the reality). More details may be found in (Longuevergne et al., 2006). gional scale).

Physical Modelling to Remove Hydrological Effects

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5.2 Position of Water Masses

As a consequence, inclinometers may be seen as a tool for the validation of more complicated distributed The second part of our methodology consists in dis- hydrological models, since they are sensitive to the potributing the determined stored water layer over the sition of water masses inside the hydrological unit. catchment surface. This question is important because a tiltmeter is an oriented instrument. In the following work, we have assumed stored water variations are located in the bed of the valleys.

6 Discussion

5.3 Comparison Between Model and Observations

The modelling of hydrological contribution to geodetic observations is difficult since all environmental signals are correlated and mixed differently in geodetic The discrepancies between modelled and observed measurements. As a consequence, the use of simple deformation help to understand the difficulties asso- physical models (Longuevergne et al., 2006; Meurciated with hydrological surface loading (see Fig. 4). ers et al., 2007), as well as hydrological experiments Indeed, the inclinometer is a “differential balance” to focus on single process (Kroner and Jahr, 2006) between what is north and what is south to the instru- are very helpful to estimate good orders of magnitude ment, that is why it can detect when the hydrological and understand “what is happening”. When looking at behaviour of the catchment cannot be considered as monthly time derivative of stored water variations plotuniform (snow accumulation periods, storms observed ted on Figure 3 (i.e. measured rainfall minus measured only in the northern part of the catchment in May). On potential evapotranspiration minus measured runoff), it the contrary, the predicted regional deformation is in seems that the agreement with tilt data to be correct afgood agreement when the catchment behaviour can be ter time integration. Unfortunately, surface loading is considered as “homogeneous” (i.e. stable precipitation of the order of 50 nrad, and cannot explain alone the 500 nrad recorded deformation. field).

Fig. 4 Predicted modelled regional surface loading in light red vs. recorded N37◦ E tilt signal (dark blue curve). For more clarity, the zero of precipitations is taken to be −40. Global scale hydrological loading has also been calculated, its long-period contribution is negligible (light green curve)

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Note that global hydrological loading has also been evaluated using GLDAS (Rodell et al., 2004) global hydrological model. This deformation is only 4 nrad (see Fig. 4).

7 Conclusions We have proposed a methodology to physically model the hydrological contribution to geodetic signals without calibrating on geodetic data. It is based on a 3-step approach: 1. Determine deformation process and control the mechanical (or dynamical) behaviour at the spatial scale of the deformation. 2. Calculate the amount of water in the associated hydrological unit using hydrological models adapted at the spatial scale of interest. 3. Evaluate the position of water masses Concerning surface loading at regional scale, results show that modelled short-term fluctuations of stored water are consistent with inclinometer measurements, when the spatial distribution of water within the catchment can be considered as “constant” (i.e. stable precipitation field in time). Conversely, large discrepancies exist during storms and snow accumulation periods because the inclinometer is also sensitive to the position of water stored within the catchment. As a consequence, we believe that geodesy could bring interesting additional information on both the quantification and the distribution of water within the catchment. Geodesy could therefore become an interesting tool for hydrological studies, i.e. provide a better understanding of hydrological processes and a more accurate calibration of hydrological models. Acknowledgments The first author is particularly grateful to V´eronique Dehant. Thanks to her recommendation, IAG awarded the first author a grant to present this work at the IUGG meeting in Perugia. We also would like to strongly thank the volunteers whose work benefits to the distribution of rainfall over the catchment: Roger and Gitta Zenner, Michel Gangloff, and more especially Michel Kammenthaler, Thierry and Franc¸oise Vincent for their incredible contribution since the beginning of this project. The authors also thank DIREN for the availability of data. This study has been carried out within the framework of the ECCO/PNRH Project “Hydrology and geodesy” coordinated by Nicolas Florsch and financed by the French research agency under the number ANR-05-009.

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References Agnew, D. C. (1986). Strainmeters and tiltmeters, Rev. Geophys., 24, 579–624 Agnew, D. C. (2001). Map projections to show the possible effects of surface loading, J. Geod. Soc. Japan, 47, 255–260 Bos, M.S., Baker, T.F. (2005). An estimate of the errors in gravity ocean tide loading computations. Geodesy 79(1): 50–63, doi: 10.1007/s00190-005-0442-5 Boudin, F. (2004) D´eveloppement d’un inclinom`etre longue base pour des mesures tectoniques, th`ese, universit´e Paris-7, Paris. Boudin, F., Bernard, P., Longuevergne, L., Florsch, N., Bour, O., Esnoult, M., Courteille, C., Caudal, J. (2007). A silica long base tiltmeter with high stability and resolution, submitted in Rev. Sci. Instrum. Bower, D., Courtier, N. (1998) Precipitation effects on gravity measurements at the Canadian absolute gravity site. Phys Earth Planet Int 106(3–4): 353–369, doi:10.1016/S00319201(97)00101-5 Boy, J.P., Hinderer, J. (2006) Study of the seasonal gravity signal in superconducting gravimeter data. J. Geodyn. 41(1–3): 227–233, doi:10.1016/j.jog.2005.08.035 Crossley, D., Xu, S., van Dam, T. (1998). Comprehensive analysis of 2 years of SG data from table mountain, Colorado. In: Ducarme B., Paquet P. (eds) Proc. 13th Int. Symp. Earth Tides, Observatoire Royal de Belgique, Brussels, Schweizerbart’sche Verlagsbuchhandlung, pp 659–668 Dal Moro, G., Zadro, M. (1998). Subsurface deformations induced by rainfall and atmospheric pressure: tilt/strain measurements in the NE-Italy seismic area, Earth Planet. Sci. Lett. 164, 193–203 Dong, D., Fang, P., Bock, Y., Cheng, M. K., and Miyazaki, S. (2002). Anatomy of apparent seasonal variations from GPSderived site position time series, J. Geophys. Res, 107(B4), doi:10.1029/2001JB000573 Evans, K. and F. Wyatt (1984). “Water table effects on the measurement of earth strain,” Tectonophysics, vol. 108, pp. 323–337 Farrell, W.E. (1972). Deformation of the earth by surface loads, Rev. Geophys. Space Phys., 10, 761–97 Guo, J. Y., Li, Y. B., Huang, Y., Deng, H. T., Xu, S. Q. & Ning, J. S. (2004). Green’s function of the deformation of the Earth as a result of atmospheric loading. Geophys. J. Int. 159(1), 53–68. doi: 10.1111/j.1365-246X.2004.02410.x Hasan, S. Troch, P., Boll, J., Kroner, C. (2006). Modeling of the hydrological effect on local gravity at Moxa, Germany. J Hydrometeor 7(3):346-354, doi:10.1175/JHM488.1 Kroner C., Jahr, T. (2006). Hydrological experiments around the superconducting gravimeter at Moxa Observatory. J Geodyn 41(1–3):268–275, doi:10.1016/j.jog.2005.08.012 Latynina, L.A., Abashizde, V.G., Alexandrov, Kapanadze, A.A., Karmaleeva, P.M., (1993). Deformation observations in epicentral areas. Fizika Zemli 3, 78–84, in Russian Le Moine, N., Andr´eassian, V., Perrin, C. and Michel C. (2007). How can rainfall-runoff models handle intercatchment groundwater flows? Theoretical study based on 1040 French catchments, Water Resour. Res., 43, W06428, doi:10.1029/2006WR005608

Physical Modelling to Remove Hydrological Effects Lettemaier, D.P. (2005). Observations of the global water cycle – global monitoring networks in Encyclopedia of Hydrologic Sciences, vol. 5, edited by M.G. Anderson and J.J. McDonnel, pp. 2719–2732, John Wiley, Hoboken, N.J. Llubes, M., N. Florsch, J. Hinderer, L. Longuevergne, and M. Amalvict (2004). Local hydrology, the Global Geodynamics Project and CHAMP/GRACE perspective: Some cases studies, J. Geodyn., 38, 355–374 Longuevergne, L., Florsch, N., Boudin, F., Vincent, T., Kammenthaler, M. (2006). Hydrology and geodesy : an hydrological perspective, BIM 142, 11387–11398 Melchior (1973). Some problems of strain and tilt measurements, Phil. Trans. R. Soc. Lond. A. 203–208 Meurers, B., Van Camp, M., Petermans, T. (2007). Correcting superconducting gravity time-series using rainfall modelling at the Vienna and Membach station and application to Earth tide analysis. J Geodesy, in press, doi:10.1007/s00190-0070137-1 Milly, P. C. D., and A. B. Shmakin (2002). Global modeling of land water and energy balances. Part I: the land dynamics (LaD) model, J. Hydrometeorol., 3(3), 283–299 Naujok, M., Kroner, C., Jahr, T., Krause, P., Weise, A. (2007). Gravimetric 3D modelling and observation of timedependant gravity variations to improve small-scale hydrological modelling, IUGG meeting, Perugia, 2–13 July 2007 Nash, J.E., et Sutcliffe, J.V. (1970). River flow forecasting through conceptual models, 1, A discussion of principles, J. Hydrol., 10, 282–290 Pagiatakis, S.D. (1990). The response of a realistic Earth to ocean tide loadings, Geophys. J. Int. 103 (1990) 541–560 Perrin, C., Michel, C., Andrassian, V. (2003). Improvement of a parsimonious model for stream flow simulation, J. Hydro. 279(1–4), 275–289

539 Rerolle, T., Florsch, N., Llubes, M., Boudin, F. Longuevergne, L. (2006). Inclinometry, a new tool for the monitoring of aquifers? C. R. Geoscience 338 (2006) Rodell, M., P. R. Houser, U. Jambor, J. Gottschalck, K. Mitchell, C.-J. Meng, K. Arsenault, B. Cosgrove, J. Radakovich, M. Bosilovich, J. K. Entin, J. P. Walker, D. Lohmann, and D. Toll, (2004). The global land data assimilation system. Bull. Amer. Meteor. Soc. 85, 381–394 Strangeways, I. (2003). Measuring the Natural Environment Cambridge University Press, Cambridge. doi:10.2277/0521529522 Tervo, M., Virtanen, H., Bilker-Koivula, M., 2006, Environmental loading effects on GPS time series, Bull. d’Inf. Mar´ees Terr. 142: 11407–11416 Van Camp M., Vanclooster M., Crommen O., Petermans T., Verbeeck K., Meurers B., van Dam T., Dassargues A. (2006). Hydrogeological investigations at the Membach station, Belgium, and application to correct long periodic gravity variations. J Geophys Res 111(B10403), doi:10.1029/2006JB004405 Virtanen H., M. Tervo, M. Bilker-Koivula (2006). Comparison of superconducting gravimeter observations with hydrological models of various spatial extents. Bull. d’Inf. Mar´ees Terr. 142: 11361–11368. Wolfe, J.E., Berg, E., Sutton, G.H. (1981). The change in strain comes mainly from the rain: Kippa, Oahu. Bull. Seismol. Soc. Am. 71, 1625–1635. Yamauchi, T. (1987). Anomalous strain response to rainfall in relation to earthquake occurrence in the Tokai area, Japan J. Phys. Earth 35, 19–36. Zadro M., Braitenberg C. (1999). Measurements and interpretations of tilt-strain gauges in seismically active areas, Earth Sci. Rev. 47, 151–187.

Monitoring Crustal Movement in the Qinghai-Tibetan Plateau Using GPS Measurements from 1993 to 2002 Weiping Jiang, Xiaohui Zhou, Caijun Xu and Jingnan Liu

Abstract The Tibetan Plateau is the center-piece of the broad India-Eurasia collision zone, and a natural laboratory to study large-scale continental deformation. We analyze Tibet campaign data from 1993 to 2002 and some continuous Global Positioning System (GPS) stations around Tibet using GAMIT/GLOBK software. The data analysis involves two major procedures. The first procedure uses the GAMIT software to estimate parameters such as station position and orbital trajectory on a daily basis from the union of two data sets: (a) the campaign stations, and (b) long-running continuous GPS stations around Tibet. In the second procedure, we combine the daily solutions with global GPS sub-networks (IGS1, IGS2, IGS3), which are provided by Scripps Institute of Oceanography (SIO), using the GLOBK software in a “regional stabilization” approach in order to estimate the positions and velocities. Then we discuss present-day crustal movement of Qinghai-Tibetan Plateau, and determine the strain rate. The conclusion is that the Himalayan block is mainly under compressing strain, and the maximum compressing rate is −98.5 ± 4.2 nstrain/yr and with the extension rate of 26.7 ± 2.8 nstrain/yr in the direction of N127.1 ± 0.7◦ E; the middle part of the Tibet block Weiping Jiang GNSS Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Xiaohui Zhou School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Caijun Xu School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Jingnan Liu GNSS Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

is mainly under compressing strain, and the maximum compressing rate is −20.3 ± 1.2 nstrain/yr in the direction of N39.0 ± 2.0◦ E with the extension rate of 10.8 ± 1.6 nstrain/yr in the direction of N129 ± 2.0◦ E. Both principal compressing strain rate and extension rate trails off from the Himalayan block to the middle part of the Tibet block. Besides, GPS result shows that Tibet Plateau uplifts with 3–5 mm per year. Keywords GPS · Tibet · Deformation · strain rate

1 Introduction The Tibetan plateau lies between the Indian and Eurasian plates, and is a complex region with many earthquakes and active faults. Now Tibet is moving significantly relative to Eurasia and undergoing significant internal deformation (Chen, 2002). The high elevation of the Himalayas and Tibet is resulted from the continuing collision between India plate and Asia plate, which started more than 45–50 million years ago. The problem of crustal movement in the Qinghai-Tibetan Plateau is a hot topic of the present day international geodetic world. Since 1993, GPS was used to monitor the movement in Southern Tibet Plateau. Liu et al. (2000) has conducted a research on current crustal movement and strain rates using GPS data between 1993 and 1995 in Tibetan Plateau. It displays the characteristic of vertical movement of part of Tibetan Plateau. Wei et al. (2001) has processed the Climbing Project data from 1992 to 1999 using GAMIT/GLOBK software and obtained the velocity field in China mainland relative to “average” Eurasian

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plate. Wang et al. (1998) processed GPS data using GIPSY software and obtained GPS velocity field of north-eastern Tibet block. Jiang et al. (2002) obtained GPS crustal movement velocity field relative to regional gravity datum and found that there was a bulk sinistral twisting between Gangqing block and Alashan block. Zhang et al. (2002) simulated and studied the block demarcation, movement and deformation in the northeast margin of Qinghai-Tibet using DDA model. Liu et al. (2000) studied and analyzed GPS data of Tibetan Plateau GPS network from 1993 to 2000 (totally 3 campaigns) and they got a good result (Xu et al. 2000). The method they used is that Lhasa (LHAS) station is used as fiducial site. They translated the coordinate and velocity of LHAS station in the International Terrestrial Reference Frame (ITRF) 97 to transient epoch at first, determined position of GPS stations of every campaign respectively, and then solved the velocity field according to its displacement and time interval. The method they used neither solved the coordinates of GPS stations in a global frame, nor determined the velocities by united adjustment. To obtain more accurate GPS results, we adopt unified data processing to mitigate the effect of changes in the network geometry as a function of time. We have analyzed all campaigns and continuous GPS data collected in southern Tibetan Plateau from 1993 to 2002, using the GAMIT/GLOBK software (King and Bock, 2003; Herring, 1999). This approach determines the station position and velocity vectors in a single, self-consistent reference frame (Arnadottir et al. 2006; Jiang et al. 2001), whereas previous studies used different frames or different reference sites for different surveys. Firstly, we combine the observed data with selected IGS data to solve the baseline, obtain daily loose solution, and then combine global IGS network data to solve GPS velocity field and strain field, and finally make tectonic interpretation.

2 GPS Data Processing 2.1 GPS Observations A total of 15 GPS sites in southern Tibet lie within roughly N110◦ E transect about 1400 km long and 66 km wide, extending from Geolmud south to Nyalam. Data were collected between 1993 and 2002

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with a total of five campaigns. 12 GPS sites were measured in the first campaign, 14 GPS sites were measured in the second campaign, and 15 GPS sites were measured since 2000. The 15 sites are Geolmud, Budongquan, Erdaogou, Yanshiping, Wenquan, Anduo, Naque, SuoCounty, Lhasa, Gyangze, Lhaze, Nyalam, Saga, Xigaze, and Yadong. The first GPS campaign took place from July 18 to August 24, 1993; the second one took place from June 5 to July 5, 1995; the third one took place from June 5 to June 25, 1997; the fourth one was carried out from May 24 to August 13, 2000; and the fifth one was carried out from July 12 to July 22, 2002. There were two observation sessions at every site in each campaign from 1993 to 1997 campaign. One was in daytime and the other was at night, and every observation session lasted about 9–12 h. Since 2000, every site of each campaign was measured continuously 2–3 days. The GPS monitoring sites are mainly deployed in the Tibet block, Himalayan block, and Qaidam block. The GPS-chain network crosses the Karakorum, Altun, Qilian Mountain, Longmen Mountain, Xianshuihe, Anninghe, Zemuhe, Xiaojiang and Honghe fault et al. large-scale strike slip boundary faults, and it covers many rift systems. So it can be used to monitor blocks movement and their boundary activities to forecast earthquake and to conduct research on block relative movement (Liu et al. 2000; Xu, 2002).

2.2 GPS Data Analysis The data analysis involves two major procedures as described elsewhere (Feigl et al. 1993; McClusky et al. 2000). The first procedure uses the GAMIT software to estimate parameters such as station position and orbital trajectory on a daily basis. The second procedure uses the GLOBK software to obtain time series and velocity field. In the first procedure, raw campaign data were analyzed in 24-h daily solutions along some global permanent sites using GAMIT. The strategy of daily solution is described as the following (1) Daily solution is calculated using an ionsphericfree combination (LC) double-difference, phase solution. (2) 13 global permanent sites (ALGO, BRMU, GRAZ, KOKB, LHAS, MADR, METS, WUHN,

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POL2, SHAO, TAIW, TROM and USUD) are constrained as reference stations by their uncertainties. Precise satellite orbits and Earth rotation parameters from the International GNSS Service (IGS) are used and tightly constrained according the related accuracy. As for the campaign data collected prior to 1994, SIO satellite orbits are used. GPS satellite orbit parameters are estimated together with all station coordinates. The sample interview is 30 s, and the cut-off angle is set to 15 degree. Tropospheric delay corrections are applied using Saastamoinen model. Dry Niell mapping function with 13 zenith delay estimates for each station. Antenna phase center corrections for satellites and receivers, and the models use IGS phase center calibrations. Ocean tide loading correction is applied using the model provided by GAMIT. Model corrections for satellite clock offsets (the clock parameters are taken from Broad-cast Ephemeris); Model corrections for receivers clock offsets (the clock offsets are computed from pseudoranges).

the second step of the second procedure, we estimate the positions and velocities of the other stations with respect to the frame defined by the regional stabilization. For the continuously observing stations, we place a lower bound on their uncertainties by adding 2 and 5 mm of white noise to the horizontal and vertical components respectively, to account for the correlation between successive estimates for the station positions. We estimate a velocity for a station if it has at least two occupations separated by at least 0.9 years.

(3)

(4) (5) (6)

(7)

(8) (9)

In the second procedure, there are totally 50 daily loose solutions in our network from 1993 to 2002. We use GLOBK software to obtain time series and velocities of stations in Tibet Plateau. In the first step of second procedure, we combine the daily loose solutions with 3 global IGS subnetworks (IGS1, IGS2 and IGS3) which are from SIO in a regional stabilization approach (McClusky et al. (2000)). This stabilization procedure defines a reference frame by minimizing, in the least squares sense, the departure from a priori values based on the International Terrestrial Reference Frame (ITRF) 2000 (Altamimi et al. 2002), of the positions and velocities for a set of 27 well-determined stations in and around Tibet Plateau (VILL, STJO, KOUR, BRMU, THU1, ALGO, YELL, USUD, ZWEN, METS, TROM, GRAZ, POTS, KOSG et al.). These stations thus move in a “no net rotation” (NNR) reference frame approximately aligned with the ITRF2000. To decrease the sensitivity of the frame to any errors in the vertical velocities (e.g., satellite antenna phase centers or ionospheric effects), we assign lower weights (larger a priori uncertainties) to the vertical velocity estimates by a factor of 10. In

3 GPS Velocity Field As described in data analysis, we determine velocity fields in ITRF2000 (See Figs. 1 and 2). In the figure, the ellipse at tip of arrow represents error of velocity of GPS stations, and bold lines represent main active segments. Table 1 describes velocities of GPS stations in ITRF2000. Considering the data adopted in this paper from GPS observation, we use the plate movement model determined by ITRF2000 rather than geological model NUVEL-1A. Besides, as for the vertical velocity of station 0003, we use data from 1993, 1995 and 1997, because of discrepancy of monument position. The monument position of 2000 and 2002 adopts upper monument while the ones of 1993, 1995 and 1997 adopts lower monument. From Table 1, we could divide these sites into two profiles: south of Himalaya (0012, 0011, 0010, 0013 and 0009) and Tibet block (0008, 0006, 0005, 0007, 0004, 0003, 0014, 0002 and 0001). These sites in south of Himalaya presents an increase from west to east of the EW velocity vector, which illustrates the east part of Tibet plate is mainly extrusion because Indian plate subducts towards Tibetan Plateau; the south-north velocity vector of these sites decreases from south to north, which embodies present shortening in the North–South direction of Himalayan block. The velocities of these sites in Tibet block show the east-west velocity vector decreases from west to east, whose variation represents mainly extrusion, since these stations were deployed in the east fringe of Tibet block; the north-south velocity vector of these sites decreases integrally, and the velocity difference between the southmost station and the northmost station is 5.3 ± 0.3 mm/yr, which represents present shortening in north-south direction of Tibet block. From the velocity field angle of whole

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Fig. 1 GPS horizontal velocities 1993–2002 for sites in southern Tibet in the ITRF2000 Frame

Fig. 2 GPS vertical velocities 1993–2002 for sites in southern Tibet in the ITRF2000 Frame

survey area, the law of velocity changes from west to east in east-west direction increases firstly and decreases again, where the maximal velocity sites in east-west direction focus on the central part of Tibetan Plateau, and thus the extension of the Tibetan Plateau in the east-west direction is mainly in the middle part of the Tibet block. The average extension east-west velocity of the eight sites 0002, 0003, 0004,

0005, 0006, 0007, 0008, 0014 relative to Geolmud (0001) in the Tibet block is 9.2 ± 0.4 mm/yr, which is consistent with the result of 8.7 ± 6.4 mm/yr (Liu et al. 2000), however accuracy of result improves; the extension east-west velocity in the Himalayan block is about 11.2 ± 0.2 mm/yr, which is in accordance with 5–10 mm/yr estimated from seismic materials (Baranowski et al. (1984)), and (10 ± 5) mm/yr

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Table 1 GPS velocities relative to Eurasia plate in the ITRF2000 Frame Site

Number

Lon (◦ )

Lat (◦ )

Vew (mm/yr)

Vns (mm/yr)

Vup (mm/yr)

Geolmud Budongquan Erdaogou Yanshiping Wenquan Amdo Naque Lhasa Gyangze Lhaze Nyalam Saga Xigaze SuoCounty Yadong

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 A015

94.9 93.9 92.8 92.1 91.9 91.7 92.0 91.1 89.6 87.6 86.0 85.2 88.9 93.8 88.9

36.4 35.5 34.6 33.7 33.1 32.3 31.5 29.7 28.9 29.1 28.3 29.4 29.2 31.9 27.4

11.0 ± 0.4 15.6 ± 0.6 18.3 ± 0.5 20.4 ± 0.3 25.5 ± 0.4 21.3 ± 0.3 20.7 ± 0.2 17.0 ± 0.2 12.3 ± 0.3 13.3 ± 0.3 8.1 ± 0.6 8.8 ± 0.3 13.7 ± 0.7 22.6 ± 0.4 11.9 ± 0.3

5.8 ± 0.3 4.0 ± 0.6 6.6 ± 0.4 7.9 ± 0.2 13.8 ± 0.3 8.6 ± 0.2 11.1 ± 0.2 17.1 ± 0.2 20.5 ± 0.2 23.1 ± 0.2 26.1 ± 0.6 23.3 ± 0.2 22.6 ± 0.7 6.1 ± 0.3 27.3 ± 0.3

3.4 ± 1.0 1.2 ± 2.2 3.5 ± 2.2 5.2 ± 0.8 2.3 ± 1.3 4.8 ± 0.8 1.6 ± 0.6 0.8 ± 0.6 4.7 ± 0.9 3.2 ± 0.7 5.0 ± 2.6 4.0 ± 0.9 1.4 ± 2.9 2.5 ± 1.0 0.3 ± 1.2

Latitude (Lat) and Longitude (Lon) are given in degress north and east, respectively. Station velocities in east–west direction and north-south direction and their uncertainties are given in mm/yr. North and East directions are supposed to be positive.

obtained from satellite image materials (Armijo et al., (1986)), and a little less than the seismological rate of (18 ± 9) mm/yr (Monlar and Deng 1984)). As to the result of Wang et al. (2002), they pointed out that the Lhasa block moves at a rate between 27 and 30 mm/yr in the direction of NE30◦ –47◦ , Qiangtang block moves at rate of (28 ± 5) mm/yr in the direction of N60◦ E, and the rates of Qaidam block are 12–14 mm/yr in the similar direction of movement to the Qiangtang block. According to the division of Wang et al. (2002), our results in the corresponding blocks show a good agreement that the velocity of 0001 in Qaidam block is about 12.5 mm/yr in the direction of N62◦ E, the average velocity of Qiangtang block is about 23.3 mm/yr in the direction of N67◦ E, and the average velocity of Lhasa block is 25.9 mm/yr in the direction of N28◦ E, and with a better accuracy and reliability. Five repeated GPS campaigns materials show that Tibetan Plateau is uplifting vertically. If average uplifting velocity of the five sites 0003, 0004, 0006, 0007 and 0008 by five observations in the Tibet block is used to represent the uplifting velocity of the Tibet block, the uplifting velocity of the Tibet block is about (3.2 ± 0.5) mm/yr, which is in accordance with the result (3.6±4.0 mm/yr) of Liu et al. (2000), and with better accuracy and reliability. If average uplifting velocity of the four sites 0009, 0010, 0011 and 0012 by five campaigns in the Himalayan block is used to represent the uplifting velocity of the Himalayan block (excluding the observation of site 0013 since its observation is

limited), the uplifting velocity of the Himalayan block is about (4.2 ± 0.7) mm/yr, and it is different from the result ((6.6 ± 6.6) ∼ (9.1 ± 5.1) mm/yr) from Liu et al. (2000). The possible reason is that the time of first two campaigns observations was relative short and the model (such as tropospheric delay) they used was not perfect. We can speculate it from the large error in the vertical direction from the first three campaign results. Besides, our result is in accordance with 5.8 mm/yr, the value by Zhang et al. (1991) using State Class One levelling data. If the Indian block is moving toward the north at the rate of 50 mm/yr, about 40–50% of it will be absorbed by the Himalayan block, and about 10 ∼ 20% by the Tibet block. The extending velocity ratio of the Tibet block in the east-west direction to that in the North– South direction is nearly 2.3:1, and the extending velocity ratio of the Himalayan block in the East–West direction to the uplifting velocity in the vertical direction is nearly 2.7:1, and the extending velocity ratio of the Tibet block in the east-west direction to the uplifting velocity in the vertical direction is nearly 7.3:1.

4 Interpretation of Crustal Movement in Tibetan Plateau As to the internal structure of Tibetan Plateau, YadongGulu rift, Karakorum-Jiali fault (Armijo et al. (1986, 1989)) and Kunlun fault are regarded as internal main

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active faults of Tibet block. Since these faults generally have the active rates of about 10 mm/yr, we can divide Qinghai-Tibetan Plateau into three subsections: east of Yadong-Gulu rift, west of Yadong-Gulu rift, between the north of Karakorum-Jiali fault and south of Kunlun fault and north of Kunlun fault, and each block has their own velocity. We divide the Plateau into these blocks due to realistic consideration. Since our materials are very limited, we can not further divide the blocks into subsections from the GPS observed velocity field angle (Wang, (2004); Chen, (2002)). In order to understand the space strain characteristics of Tibetan Plateau, we divide these 14 stations into two groups. According to classification of Wang (2004), these 14 stations were divided into two kind of subareas. One is bounded by sets of faults without main active rifts in its internal subarea, and the other is the area with main active rifts. According to this principle, the survey area can be divided into four subregions. These subregions are segmented by Karakorum-Jiali fault, Kunlun fault and Qilian Mountain. There are six stations in the south of Karakorum-Jiali fault, and there are seven stations between the north of Karakorum-Jiali fault and south of Kunlun fault, and station LHAS lies in the east of Yadong-Gulu rift. Since there is only one site in the east of Yadong-Gulu rift, it is not convenient to solve the strain parameters, and thus it is excluded from computation. Thus survey area is divided into two parts: block between south of Kunlun fault and north of KarakorumJiali fault and block within south of Karakorum-Jiali fault. The average strain rate can be solved independently correspond to each block, using QOCA (Dong et al. 1998) software to determine the maximal principal strain rate, azimuth of maximal principal strain rate and maximal shear strain rate (Drew and Snay (1989); Feigl et al. (1990); Chen (2002)) shown in Table 2 and Fig. 3. The face-to-face arrow represents compression strain rate and inverse arrow represents extension strain rate in Fig. 3.

In the Table 2, Ia, Ib, IIa, IIb represent: Ia: subregion south of Karakorum-Jiali fault (from Wang (2004)); Ib: subregion south of Karakorum-Jiali fault (this paper); IIa: subregion between Karakorum-Jiali fault and Kunlun fault (from Wang (2004)); IIb: subregion between Karakorum-Jiali fault and Kunlun fault (this paper). Table 2 made comparison of maximal principal strain rate, azimuth of maximal principal strain rate and maximal shear strain rate of block south of Karakorum-Jiali fault and block between south of Kunlun fault and north of Karakorum-Jiali fault of our results and results of Wang (2004). Eight sites were used by Ia in Table 2, whose distribution is almost the same as the 6 sites (0007, 0009, 0010, 0011, 0012 and 0013) used by Ib, because there are three sites in Lhasa and nearby in Ia. There are 7 sites (0001, 0002, 0003, 0004, 0005, 0006 and 0014) in IIb. The ratio of maximum contraction rate to the maximum extension rate of two source materials is consistent. The ratio of area south of Karakorum-Jiali fault is about 4, and the ratio of area between Karakorum-Jiali fault and Kunlun fault is about 2. From the angle of quantity, strain parameters of area between Karakorum-Jiali fault and Kunlun fault are consistent. However, as to the area south of Karakorum-Jiali, though the difference of azimuth of maximal principal strain rate is very small, both the value of maximal principal strain rate and the value of tensile strain are two times larger than the result of Wang (2004). The mean square error of parameters in Ia is big, and Chen (2002) explained that “The misfit in the the south of the Karakorum-Jiali Fault Zone (“S.KJFZ”) is significantly larger than those in other networks, implying that the site velocities in S.KJFZ are more heterogeneous than in the rest of the plateau, and cannot be interpreted by the uniform strain plus block motion model.” Our result is relative ideal since the error of parameters is small. Thus further research need to make on the strain parameters of this region. The solution results show:

Table 2 Strain rate parameters in the Tibetan Plateau Sub-regions

Number of sites

ε˙ 1 nstrain/yr

ε˙ 2 nstrain/yr

Azimuth (◦ )

(˙ε1 − ε˙ 2 )/(2) nstrain/yr

(˙ε1 + ε˙ 2 ) nstrain/yr

Ia Ib IIa IIb

8 6 10 7

−41.3 ± 11.6 −98.5 ± 4.2 −20.5 ± 2.2 −20.3 ± 1.2

11.9 ± 5.4 26.7 ± 2.8 7.8 ± 2.5 10.8 ± 1.6

37.6 ± 5.2 37.1 ± 0.7 32.0 ± 3.3 39.0 ± 2.0

26.6 ± 7.3 62.6 ± 2.6 14.1 ± 1.8 15.6 ± 1.0

−29.3 ± 10.7 −78.1 ± 5.1 −6.4 ± 2.4 −9.5 ± 2.2

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Fig. 3 Strain rate parameters in Tibetan Plateau

extensive in the east-west direction, and uplifting in the vertical direction. The maximal shear strain rate of Himalayan block is 62.6 ± 2.6 nstrain/yr, and the maximal areal dilatation is −78.1 ± 5.1 nstrain/yr. The maximal shear strain rate of Tibet block is 15.6 nstrain/yr, and the maximal areal dilatation is −9.5 ± 2.2 nstrain/yr. The uplifting velocity of Tibet block is about (3.2 ± 0.5) mm/yr and the uplifting velocity of Himalayan block is about (4.2 ± 0.7) mm/yr. These results reflect present crustal deformation and strain situation of Tibetan Plateau, and it is consistent with the present tectonic deformation characteristics by geologic and seismic data. This result also shows that the three-dimensional crustal movement of the Tibetan Plateau is probably constrained by many factors. GPS observation describes the present threeThe maximal principal strain rate and extension strain dimensional crustal kinematics well, and further rate decrease from the Himalayan block to the Tibet GPS observations and research will play a key role block. Besides, Table 1 shows both the Himalayan in solving the kinematic mechanism of the present block and the Tibet block is mainly uplifting. movement in the Tibetan Plateau. 1. The Himalayan block is mainly under compressive strain. The strain rate achieves −98.5 ± 4.2 nstrain/yr in N37.1 ± 0.7◦ E direction, and extension strain rate achieves 26.7 ± 2.8 nstrain/yr in N127.1 ± 0.7◦ E direction. Maximal shear strain rate achieves 62.6 ± 2.6 nstrain/yr, and areal dilatation achieves −78.1 ±5.1 nstrain/yr. 2. The Tibet block is mainly under compressive strain. The strain rate achieves −20.3 ± 1.2 nstrain/yr in N39.0 ± 2.0◦ E direction, and extension strain rate achieves 10.8 ± 1.6 nstrain/yr in N129.0 ± 2.0◦ E direction. Maximal shear strain rate achieves 15.6 ± 1.0 nstrain/yr, and areal dilatation achieves −9.5 ±2.2 nstrain/yr.

5 Conclusions The result of five GPS precise repeated surveying indicates that the crustal movement in the Tibetan Plateau is still compressive in the north-south direction,

Acknowledgments We are grateful to the individuals who contributed the acquisition of the GPS data that form the basis for this paper. The maps in this paper were generated using the public domain Generic Mapping Tools (GMT) software (Wessel and Smith (1998)). This research was supported by National Natural Science Foundation of China, Project No. 49974001 and 49234070 and National Basic Research Programs of China, Project No. 2006CB701300.

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References Altamimi, Z., P. Sillard, and C. Boucher (2002). ITRF2000: A new release of the International Terrestrial Reference Frame for earth science applications. J. Geophys. Res., 107, B10, 2214, doi: 10.1029/2001JB00561. Armijo, R., P. Tapponnier, J.L. Mercier, and T.L. Han (1986). Quaternary extension in southern Tibet: field observations and tectonic implications. J. Geophys. Res., 91, B14, pp. 13803–13872. Armijo, R., P. Tapponnier, and T.L. Han (1989). Late Cenozoic right-lateral strike-slip faulting in southern Tibet. J. Geophys. Res., 94, B3, pp. 2787–2838. Arnadottir, T., W.P. Jiang, K.L. Feigl, H. Geirsson, and E. Sturkell (2006). Kinematic model of plate boundary deformation in southwest Iceland derived from GPS observations. J. Geophys. Res., 111, B07, doi: 10.1029/2005JB003907. Baranowski, J., J. Armbruster, L. Seeber, and P. Molnar (1984). Focal depths and fault plane solutions of earthquakes and active tectonics of the Himalaya. J. Geophys. Res., 89, pp. 6918–6928. Bock, Y., S. Wdowinski, P. Fang, J. Zhang, S. Williams, H. Johnson, J. Behr, J. Genrich, J. Dean, M. van Domselaar, D. Agnew, F. Wyatt, K. Stark, B. Oral, K. Hudnut, R. King, T. Herring, S. Dinardo, W. Young, D. Jackson, and W. Gurtner (1997). Southern California Permanent GPS Geodetic Array: continuous measurements of regional crustal deformation between the 1992 Landers and 1994 Northridge earthquakes. J. Geophys. Res., 102, pp. 18,013–18,033. Chen, Q. (2002). Crustal deformation along the San Andreas fault and within the Tibetan plateau measured using GPS. Ph.D thesis, University of Alaska Fairbanks. Dong, D., T.A. Herring, and R.W. King (1998). Estimating regional deformation from a combination of space and terrestrial geodetic data. J. Geodesy, 72, pp. 200–214. Drew, A.R., and R.A. Snay (1989). DYNAP: software for estimating crustal deformation for geodetic data. Tectonophysics, 162, pp. 331–343. Feigl, K.L., D.C. Agnew, Y. Bock, D. Dong, A. Donnellan, B.H. Hager, T.A. Herring, D.D. Jackson, T.H. Jordan, R.W. King, S. Larsen, K.M. Larson, M.H. Murray, Z. Shen, and F.H. Webb (1993). Space geodetic measurement of crustal deformation in central and southern California, 1984–1992. J. Geophys. Res., 98, pp. 21,677–21,712. Herring, T.A. (1999). GLOBK: Global Kalman filter VLBI and GPS analysis program. Massachusetts Institute of Technology, Massachusette. Jiang, W., J. Liu, and S. Ye (2001). The systematical error analysis of baseline processing in GPS network. Geomatics and Information Science of Wuhan University, 26(3), pp. 196–199. (in Chinese) Jiang, Z., X. Zhang, and D. Cui (2002). Recent horizontal movement and deformation in the northeast margin of QinghaiTibet block. Chin. J. Geophys., 44(5), pp. 637–644. (in Chinese)

W. Jiang et al. King, R.W., and Y. Bock (2003). Documentation for the GAMIT Analysis Software. Mass. Inst. Technol., Cambridge. Liu, J., C. Xu, C. Song, C. Shi, W. Jiang, and L. Dong (2000). Using repeated precise GPS campaigns for research of presentday crustal movement and strain in the Qinghai-Tibetan Plateau. Chin. Sci. Bull., 45(24), pp. 2658–2663. (in Chinese) McClusky, S., S. Balassanian, A. Barka, C. Demir, S. Ergintav, I. Georgiev, O. Gurkan, M. Hamburger, K. Hurst, H. Kahle, K. Kastens, G. Kakelidze, R. King, V. Kotzev, O. Lenk, S. Mahmoud, A. Mishin, M. Nadariya, A. Ouzounis, D. Paradissis, Y. Peter, M. Prilepin, R. Reilinger, I. Sanli, H. Seeger, A. Tealeb, M.N. Toksoz, and G. Veis (2000). Global positioning system constraints on plate kinematics and dynamics in the eastern Mediterranean and Caucasus. J. Geophys. Res., 105, B3, pp. 5695–5720. Monlar, P., and Q. Deng (1984). Faulting associated with large earthquakes and the average rate of deformation in central and eastern Asia. J. Geophys. Res., 89, pp. 6203–6228. Wang, Q. (2004). GPS determination of present-day crustal deformation in China: velocity field and tectonic implication. Ph.D thesis, Wuhan University. (in Chinese) Wang, Q., X. You, W. Wang, and Z. Yang (1998). GPS measurement and current curstal movement across the Himalaya, Crustal Deformation and Earthquake, 18(3), pp. 43–50. (in Chinese) Wang, Q., P. Zhang, Z. Niu, J.T. Freymueller, X. Lai, Y. Li, W. Zhu, J. Liu, R. Biham, and K.M. Larson (2002). Presentday crustal movement and tectonic deformation in China continent. Science in China (Series D), 45, pp. 865–874. Wei, Z., W. Duan, and X. Wu. (2001). Large-scale contemporary crustal movement in mainland of China. In: Symposium on Processing and Explaining of High-Accuracy GPS Measurement, pp. 53–63. (in Chinese) Wessel, P. and W. H. F. Smith (1998). New improved version of the Generic Mapping Tools Released, EOS Trans. AGU, 79, pp. 579. Xu, C. (2002). The Kinematic Models of Crustal Movement and Tectonic Stress Field in the Tibetan Plateau, Beijing: SinoMaps Press. (in Chinese) Xu, C., J. Liu, C. Song, W. Jiang, and C. Shi (2000). GPS measurement of present-day up-lift in the Southern Tibet. Earth Planets Space, 52, pp. 735–739. Zhang, Q., Y. Zhou, X. Lu, Q. Xu (1991). Uplift velocity of modern Qinghai-Tibetan Plateau. Chin. Sci. Bull., 36(7), pp. 529–531. (in Chinese) Zhang, X., Z. Jiang, B. Chen, H. Li (2002). Preliminary research on present block deformation, movement and deformation in northeast Margin of Qinghai-Tibet. J. Geodesy Geodynamics, 22(1), pp. 63–67. (in Chinese)

Impact of Local GNSS Permanent Networks in the Study of Geodynamics in Central Italy G. Fastellini, F. Radicioni and A. Stoppini

Abstract The DICA (Department of Civil and Environmental Engineering) of Perugia University currently operates two local GNSS permanent stations networks in central Italy. These networks supply various types of positioning services, which are currently being used mostly by surveyors and mapping agencies. This kind of networks also have a strong potential for the monitoring of crustal deformations in local areas. In the specific case, such applications are particularly interesting because the covered region (Umbria and central Italy) is characterized by a relevant seismicity. In the present paper, both networks are described, as for stations location, data availability, and performances in terms of positioning services at different accuracy levels. The possibilities for applications on local geodynamics with different approaches are then discussed, showing the results of a test performed with displacements simulated by means of a mechanical device.

1 Introduction

Regional and local networks of GNSS permanent stations are spreading all around the world in most technically advanced countries, covering the territory with a growing density of stations and offering a wide series of services, many of which were unpredictable in the early times of satellite positioning. The high concentration of stations in some small areas and the high accuracy of the GNSS positioning are elements of a great interest for the study of local geodynamics phenomena, e.g. regional crustal deformations, seismic activity (analysis before and after events, forecasts) and so on. The DICA Department at Perugia University is currently operating two local GNSS permanent stations networks. The station distribution is quite dense, with an average distance of only about 40 km from each other. Until now, these networks have been mostly used by surveyors, companies and mapping agencies, taking advantage of the different positioning services Keywords GNSS · Permanent stations · Deformation in applications related to mapping, engineering and cadastre. Still, we are convinced of the great potential control · Real time positioning of such dense networks for the analysis of local crustal deformations, to be likely analysed together with the monitoring of the seismic activity in Umbria and surrounding areas. It is therefore our plan to improve both networks G. Fastellini in function of such kind of applications, implementing Department of Civil and Environmental Engineering, University appropriate analysis procedures and software compoof Perugia, Via G. Duranti 93, 06125 Perugia, Italy nents in that direction. F. Radicioni A first step consists in refining the analysis of coDepartment of Civil and Environmental Engineering, University ordinate time series, on a daily or weekly basis. Anof Perugia, Via G. Duranti 93, 06125 Perugia, Italy other approach is to define methodologies for real time A. Stoppini Department of Civil and Environmental Engineering, University analyses. On the latter purpose, some tests based on of Perugia, Via G. Duranti 93, 06125 Perugia, Italy

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displacements impressed by a mechanical device have been performed: the first results are presented in this paper.

2 GNSS Networks Operated by DICA As referred above, DICA is operating two GNSS permanent stations networks:

r

GPSUMBRIA, the GNSS official network of the Umbria Region, offering services both in post-processing and real time. It comprises 10 stations, covering the whole territory of Umbria (Fig. 1). Information about this network and its data (publicly available at the moment) can be collected on the web site http://www.gpsumbria.it/ r LABTOPO, a local network set up for research purposes, putting together a bundle of GNSS permanent stations of different Universities, schools, public administrations and private companies. It presently comprises 21 stations (including all GPSUMBRIA sites), covering a wide area in central Italy from Rimini to Rome (Fig. 2). At Fig. 1 GPSUMBRIA network the present time, this network only offers postprocessing services. Info and data (mostly free) and processing methods the post-processing users can are available through the web site http://labtopo. reach a positioning accuracy of about one centimeter. ing.unipg.it/labtopo/index.php. Post-processing data are currently distributed for free in order to promote their use. The monumentation of the station sites of the above Real time positioning services (network code cornetworks is very stable: all stations of GPSUMBRIA rections – NDGPS and network carrier phase correcmatch the EUREF-IGS specifications, and most sta- tions – NRTK) are supplied by the GPSUMBRIA nettions of LABTOPO are set up with equivalent char- work to registered users. The registration is currently acteristics. Two GPSUMBRIA stations (UNPG – Pe- free because the real time services are still in a “prorugia and UNTR – Terni) are part of the EUREF EPN motional” phase, but this distribution policy will likely network and the EUREF-IP real-time project. change soon, with the transition to a fully operational All stations are equipped with Topcon GPS- phase. GLONASS geodetic receivers. In most cases the The real time functionality of GPSUMBRIA netantennas are of the choke-ring type. Most antennas work is presently achieved by means of the software R of GPSUMBRIA network have been individually GNSMART by Geo++ (W¨ ubbena et al., 2001). calibrated (absolute calibration) with the Automated Phase corrections are distributed to the users in VRS R PCV Field Calibration by Geo++/GeoService (Virtual Reference Station) or FKP (Fl¨achen Korrektur Parameter) mode, using the RTCM 2.3 format (cor(references: http://www.geopp.de). All data of both networks are submitted every day to rection types 18, 19 or 20, 21). The users can get the an automatic quality control and distributed to the sci- corrections through a direct connection to a stack of entific and technical community through the network GSM modems set up at the network control center or web sites, in form of daily or hourly RINEX files at (preferred approach) through the network own NTRIP sampling rates of 30, 5 and 1 s, to be used for post- caster, using the NTRIP protocol described by Weber processing applications. With appropriate observation et al. (2005).

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Fig. 2 LABTOPO network

The real time software performs (by means of GNNET – a module of GNSMART – Fig. 3) a continuous computation of the network, which beside its primary function (ambiguities and biases determination) constitutes a powerful instrument for the network analysis and control.

3 Analysis of Coordinate Time Series A first “classic” possibility for the use of GNSS permanent network data in Geodynamics is to compute postprocessed network solutions from the RINEX files at a regular interval, e.g. on a monthly, weekly or daily basis, obtaining time series of three-dimensional coordinates to be then analyzed and filtered with appropriate algorithms. For this purpose, an automated procedure for computing network daily solutions has been set up for GPSUMBRIA network, using BPE (Bernese Processing Engine module, Bernese Software v. 5.0). The same procedure will also be implemented for the

LABTOPO network. The analyses are performed in ITRS and ETRS geodetic systems. The graphs in Fig. 4 show as example the results of one of the first series analysed, comprising 24 not consecutive ETRS89 daily solutions (June-August 2005) for the station UNPG (Perugia). The solutions have been obtained using GPS and GLONASS data, and BKG precise ephemerides. In the analysis has been adopted the Saastamoinen tropospheric model, with the Niell mapping functions. Other relevant Bernese options selected for this elaboration are: OBS-MAX criterion for baselines forming, and QIF strategy for ambiguity resolution. Considering that our research aims to a local deformation analysis, the connection to ETRS89 has been obtained linking the network to a number of local points of the national geodetic network IGM95 (Italian realization of the ETRS89, see Surace, 1997). Planimetric coordinate variations are limited into a range of ±2 mm, while the height component shows bigger oscillations (±8 mm). Of course, most of such apparent coordinate variations is constituted by noise, which has to be removed through an improved

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Fig. 3 GNNET (a GNSMART module) real-time network graph for GPSUMBRIA

elaboration including appropriate filtering and smoothing algorithms, presently under developing and testing. The time series analysis methods adopted by EUREF for EPN network in the ETRS89 datum are of a particular interest. More information on EUREF time series processing strategy and products can be found in Kenyeres and Bruyninx (2004) and Kenyeres (2006). An implementation of EUREF-like procedures and a comparison of our results with the EUREF time series (for the stations that are part of EPN) will be included in the future GPSUMBRIA data processing flow.

4 Use of Network Real Time Software: A Test with Simulated Displacements An alternative approach which appears possible for an automatic site monitoring is to use data derived from

the software of network real time positioning services (like GNSMART, GNNET, see par. 2). This software performs a continuous network computation, so it is possible to record solutions at an assumed sampling rate (e.g. 1 min) on a specific log file. The network real time software, in order to work correctly (its function is essentially to compute ambiguities and other biases for all stations and satellites), needs strongly constrained coordinates for most stations. Therefore, to explore the possibilities of such software for monitoring purposes, a test has been made including in the network a control point subject to a weaker constrain, in order to show the possibility to measure deformations of the order of a centimeter. An experiment has been set up within the GPSUMBRIA network by de-constraining the coordinates of an auxiliary station (UPG2), temporarily included in the network, as shown in Fig. 5. The figure contains an

Impact of Local GNSS Permanent Networks Fig. 4 ETRS89 coordinate variation series (mm vs. number of daily solution) obtained for UNPG station by means of Bernese 5.0 BPE

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8 6 4 2 0 –2 –4 –6 –8 –10

8 6 4 2 0 –2 –4 –6 –8 –10

8 6 4 2 0 –2 –4 –6 –8 –10

excerpt from the option file of GNNET module; values after each station name indicate constrain values for N, E, H components in meters. In the auxiliary station a choke-ring antenna connected to a GPS/GLONASS receiver of the same type of the other network stations was installed. The antenna has been put in site by means of a bidirectional “translation sled”. This is a mechanical device (Figs. 6 and 7) capable to impose micrometric displacements in two orthogonal directions (which have been put in coincidence with northing and easting), measuring them at a 0.1 mm accuracy by means of two vernier scales. Starting from an initial position (zero values on N and E scales), progressive displacements have been imposed in steps of 1 cm separated

Fig. 5 Adding to GPSUMBRIA network a de-constrained auxiliary station (UPG2, evidenced)

by pauses of 1.5 h (3 h for the last movement), in the following order: 1 cm N; 1 cm N; 1 cm W; 1 cm W.

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320000

315000

310000

305000

300000

295000

290000

285000

280000

–5

Fig. 8 Displacement in N direction (mm) vs. time (GPS weekly seconds)

5 0 –5 –10

Fig. 6 Choke-ring antenna mounted on the “translation sled” –15 –20

320000

315000

310000

305000

300000

295000

290000

285000

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–25

Fig. 9 Displacement in E direction (mm) vs. time (GPS weekly seconds)

Fig. 7 Detail of the “translation sled”

The graphs in Figs. 8 and 9 show the comparison between imposed (recognizable by the steps) and computed coordinates, expressed in millimeters, referred to time in GPS weekly seconds. It is possible to see that the displacements have been revealed by the real time software, but with an approximation of a few millimeters and a consistent time delay. In other words, the network solution shows a sort of “inertia”, which is likely due to the Kalman approach used by the GNNET software. Still, the software has substantially been able to catch the imposed movements. It is our plan to perform further series of tests with different values of the constrain for the control points, to investigate the influence of the constrain size.

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5 Further Possibilities and Final Remarks promising of them on our networks, in order to conThere are of course many other possibilities, described in scientific literature and technical papers, for the continuous monitoring of control points, which can be part of the local GNSS network or not (e.g. supplementary points equipped with a continuously operating GNSS receiver). Deformation control software products like R or 3D Tracker R offer sub-centimeter motion Hydra detection in real time. Some tests on this kind of software have been carried out with success at our department by Radicioni (2000). To operate in the frame of a permanent network it is obviously necessary to build up an appropriate interface between the station data flow and the software. A promising approach, still to be experimented on our networks, is to use RTCM data stream derived from the network (in VRS or FKP mode) for obtaining high accuracy NRTK solutions in a control point, by means of appropriate sequential algorithms taking into account that the point is “fixed” but can have small displacements, deploying in time progressively or in a seldom way (e.g. in the case of a seismic event). The relevance of the data acquired and recorded from a local permanent GNSS network for the study of local geodynamics is more than evident. There are many possible approaches for the deformation analysis, and it is our goal to test the most interesting and

tribute to the establishment of a local deformation control service for Umbria and central Italy.

References Kenyeres, A. (2006), New products at the EPN time series special project: status report. In: Proc. of Riga EUREF Meeting, Latvia, June 14–17 2006, EUREF Publication No. 16. Kenyeres, A., Bruyninx, C. (2004), EPN coordinate time series monitoring for reference frame maintenance. GPS Solutions, Vol. 8 pp. 200–209, 2004. Radicioni, F. (2000), Testing a real-time GPS system for quasistatic deformation monitoring of ground surface and large structures. In: Proc. of Millennium Meeting Poland-Italy, Krakow, Poland, June 28–July 2, 2000. Special Issue of Reports On Geodesy, Warsaw University of Technology, Sledzinski J, Ed. Surace, L. (1997): La nuova rete geodetica nazionale IGM95 : risultati e prospettive di utilizzazione. In: Bollettino di geodesia e scienze affini, 1997, Vol. 56, No. 3, pp. 357–378. Weber, G., Gebard, H., Dettmering, D. (2005), Networked Transport of RTCM via Internet Protocol (NTRIP). In: “A Window on the Future of Geodesy”, Proceedings of the International Association of Geodesy. IAG General Assembly, Sapporo, Japan June 30–July 11, 2003. Series: International Association of Geodesy Symposia, Vol. 128, Sanso, F. (Ed.). W¨ubbena, G., Bagge, A., Schmitz, M. (2001), RTK networks R GNSMART: Concepts, implementation, based on Geo++ results. In: Proc. of ION GPS 2001, Salt Lake City, Utah, September.

New Results Based on Reprocessing of 13 years Continuous GPS Observations of the Fennoscandia GIA Process from BIFROST M. Lidberg, J.M. Johansson, H.-G. Scherneck, G.A. Milne and J.L. Davis

Abstract We present new and improved 3D velocity fields of the Fennoscandian Glacial Isostatic Adjustment (GIA) process derived from more than 4800 days (13 years) of data at more than 80 permanent GPS sites. We use the GAMIT/GLOBK software package for the GPS analysis. The solution has an internal accuracy at the level of 0.2 mm/yr (1 sigma) for horizontal velocities at the best sites. We present our results both in the ITRF2000 and in the new ITRF2005 reference frames, and discuss the difference in vertical rates associated with the choice of reference frame. Our vertical velocities agree with results derived from classic geodetic methods (tide-gauge, repeated levelling, and repeated gravity observations) at the 0.5 mm/yr level (1 sigma). We also compare the observations to predictions derived from a GIA model tuned to fit the new data and get agreement on the sub-millimeter level.

M. Lidberg Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, Lantm¨ateriet, SE-801 82, G¨avle, Sweden J.M. Johansson Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, SP Technical Research Institute of Sweden, Box 857, SE-501 15, Bor˚as, Sweden H.-G. Scherneck Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, Sweden G.A. Milne Department of Earth Sciences, Durham University, Science Labs, Durham DH1 3LE, UK J.L. Davis Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS 42, Cambridge, MA 02138, USA

Keywords Fennoscandia land uplift · Glacial isostatic adjustment · Continuous GPS · Reference frames

1 Introduction The BIFROST (Baseline Inferences for Fennoscandian Rebound Observations Sea Level and Tectonicas) project was started in 1993. The first primary goal was to establish a new and useful three-dimensional measurement of the movements in the earth crust, able to constrain models of the GIA (glacial isostatic adjustment) process in Fennoscandia. Station velocities based on analysis of observations at permanent GPS stations in Sweden and Finland from the period August 1993 to May 2000 was presented in Johansson et al. (2002) and Scherneck et al. (2002) together with a through description of the BIFROST network. These velocities have then been used to constrain a viscoelastic selfgraviting model of the Fennoscandian GIA process (Milne et al. 2001). Updated station velocities were presented in Lidberg et al. (2007), based on data from the period 1996 to mid 2004. Since the early phase of the BIFROST effort, especially up to mid 1996, comprise a period of intensive development and hardware changes at the GPS sites, producing shifts in the position time series, the solution in Lidberg et al. shows smaller uncertainties in station velocity compared to Johansson et al. (2002). Some additional sites in Norway, Denmark and northern Europe were included in the solution, but no update of the GIA model was presented. In this study we have re-analyzed all BIFROST data up to November 2006. The extended period allows us to include also more recent GPS sites in northern

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Europe, resulting in a denser sampling of the GIA process. This is most evident for Norway, but partly also for continental Europe. In addition to ITRF2000, we have aligned our solution to the new ITRF2005 reference frame. The solutions are also compared to an updated GIA model.

2 The BIFROST GPS Network The sites included in the analysis presented here are shown in Fig. 1. A description of the BIFROST GPS

Fig. 1 The extended BIFROST network. Filled dots mark sites available in the public domain through EPN or IGS. Diamonds mark sites in national densifications

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network, originally composed of the permanent GPS network of Sweden and Finland, and here improved by also including more sites in Norway, Denmark, and sites in the EUREF Permanent Network (EPN), may be found in Lidberg et al. (2007). By including stations also outside of the former ice sheet we include also the subsiding forebulge, and may eventually determine the area where the effects of the GIA process must be taken into account. The additional stations are also needed for reference frame realization when the regional BIFROST analysis is combined with networks from global analysis (see Sect. 3 below).

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alization in one step. We have however not used this facility in this study. The measure to solve for a scale factor in the 3.1 Analysis of GPS Data GLOBK step may be put in question. Our principal argument for doing so is that we do not trust the long We have used the GAMIT/GLOBK software pack- term stability in the scale achieved from GPS analysis, age (Herring et al. 2006a-c), developed at the Mas- given the applied processing strategy. See further the sachusetts Institute of Technology (MIT), Scripps In- discussion in Lidberg et al. (2007). stitution of Oceanography, and Harvard University for analysis of the GPS data. Main characteristics of the GAMIT/GLOBK soft- 3.2 Reference Frame Realization ware are that dual frequency GPS observations from each day are analyzed using GAMIT applying the dou- The purpose of our study is to derive a 3D velocity field ble differencing approach. The results are computed of the deformation of the crust in Fennoscandia domiloosely constrained Cartesian coordinates for stations, nated by the ongoing GIA process. In order to resolve satellite orbit parameters, as well as their mutual de- the slow and small-scale deformation of the region, a pendencies. Results from analyzed sub-networks are terrestrial reference frame (TRF) consistent over the then combined using GLOBK, which is also used for period of analysis is needed. We also would like to reference frame realization. avoid pertirbations from individual stations as far as For the GAMIT part of the analysis we have applied possible. The natural choice was therefore to adapt to an identical strategy as in Lidberg et al. (2007). That is the International Terrestrial Reference Frame (ITRF) a 10◦ elevation cut off angle, atmospheric zenith de- as global constraints. We have thus constrained our solays where estimated every second hour (piece-wise- lutions to booth the ITRF2000 (Altamimi et al. 2002), linear model) using the Niell mapping functions (Niell as well as to the new ITRF2005 (Altamimi et al. 2007) et al. 1996) together with daily estimated gradient pa- reference frames. rameters. Elevation dependent weighting based on a Our a priori choice has been to follow the stratpreliminary analysis was applied, and GPS phase am- egy applied in Lidberg et al. (2007), where the rebiguities were estimated to integers as far as possible. gional BIFROST analysis was combined with global Station motion associated with ocean loading and solid networks analyzed by SOPAC, comprising major part Earth tides were modeled, and a priori orbits from the of the IGS sites. The sites used for reference frame Scripps Orbit and Permanent Array Center (SOPAC), realization of this combined regional and global anal“g-files”, were used. In the analysis corrections for an- ysis are shown in Fig. 2. For the ITRF2000 realizatenna phase centre variations (PCV) have been applied tion we needed to use sites which are well determined according to the models relative to the AOAD/M T ref- in ITRF2000 and whose position time series are staerence antenna models. ble (low noise, no non-linear behaviour etc) for the In the second step of the processing, GLOBK was analyzed time span (see further discussion in Lidberg used to combine our regional sub networks with an- et al. 2007). The ITRF 2005 is computed from time sealyzed global networks into single day unconstrained ries of position estimates. Therefore shifts have been solutions. Finally, constraints that represent the refer- introduced in the stations position and velocity estience frame realization was applied by using a set of mates of the ITRF2005 solution, when considered apglobally distributed fiducial stations and solving for propriate. Thus it has been possible to find 78 canditranslations, rotations and a scale factor, as well as dates for reference frame realization that fulfils the staa slight adjustment of the satellite orbit parameters. bility criterion, compared to 43 for ITRF2000. The result comprises stabilized daily station positions, satellite orbit parameters, and earth orientation parameters (EOPs) (Nikolaidis et al. 2002). Using GLOBK 3.3 Evaluation of TRF Realization it is also possible to combine several days (and years) of GAMIT results and estimate initial site positions and Figure 3 shows the number of sites actually used and velocities, and apply constraints for reference frame re- post fit residuals in the realization of the ITRF2000 and

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a

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Fig. 2 Sites used to as candidates for constraining the combined BIROST + SOPAC global analysis to ITRF. A: the 43 sites used for ITRF2000. B: the 78 sites used for ITRF2005. See text

ITRF2005 reference frames for the combined regional BIFROST and global SOPAC analysis. For the ITRF2000 stabilization in Fig. 3a and b we see an improvement in the results up to about 1998. But then we also see a slight decrease in the results after about 2000 (the RMS decrease first, then increase after 2000 – opposite for used sites). For the ITRF2005 stabilization in Fig. 3c and d we see also here an improvement up to about 1998, but then the results are

stable at a high accuracy level from 1999. Our interpretation of this is that the new ITRF2005 have a better internal accuracy compared to the older ITRF2000. Figure 4 shows de-trended time series plot of daily position estimates in ITRF2005 from Vilhelmina (VIL0) (64◦ N) for the complete period of analysis (Aug 1993–Oct 2006). We see the effect of some disadvantageous antenna radom models used at most Swedish sites (except Onsala) between 1993 and 1996

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cause velocity differences depending on the choice of tide model for the GPS analysis. Data from a set of global IGS sites for the 5 year period 2000.0–2005.0 was analyzed using GAMIT/GLOBK applying the IERS 2003 and IERS 1996 (denoted IERS 1992 in the paper) tide models respectively. The results showed latitude dependent differences in vertical velocity of ∼ −0.35 mm/yr at high latitudes increasing to ∼ +0.2 mm/yr at equatorial latitudes (symmetric about the equator). In our regional analysis of the BIFROST network we have consistently used the IERS 1996 tide model. The global sub-network from SOPAC have also been analyze using the IERS 1996 tide model up to beginning of 2006, when SOPAC changed to IERS 2003 in their processing. Possible limitations in the old IERS 1996 tide model may therefore contribute to the bent vertical time series. Based on the discussion above we have performed our own analysis of a small global network with global coverage including 35 selected sites, applying the IERS 2003 tide model. Then we have combined the BIFROST sub-networks with this network in order to give our results a global connection, without using the products from SOPAC. The global network is supposed to:

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cover the globe sufficiently well connect the regional and the global sub-networks (8 common sites – cp. Figs. 1 and 5) include a sufficient number of “good” sites for reference frame realization (23 sites).

Our 35 sites global network is shown in Fig. 5 (black squares) together with the 23 sites used as candidates for reference frame realization (light circles). (Johansson et al. 2002). Because of this, the period In Fig. 6 we show an example from Vilhelmina of before mid-1996 has been excluded from further a de-trended time series plot of daily vertical posiinvestigation in this analysis. We also see non-linear tion estimates. The vertical “banana-shape” is maybe behavior in the vertical position. This “banana”-shape, not eliminated, but at least heavily reduced. It may be (or possibly a change of vertical rate by mid 2003 noted that in this combined solution the IERS 2003 tide or early 2004) is visible in most of our high latitude model was used for the global 35 site network, while sites (possibly above 55◦ N) and seems to be more the regional BIFROST analysis was performed using pronounced towards north. We also stress that long the old IERS 1996 model. uninterrupted time series are needed to see this phenomenon. The cause of the bent vertical time series is not yet understood but we can think of a number of 3.4 Two Step Reference Frame Approach candidate causes (see discussion in Sect. 7). One candidate may be the modeling of tides. In our analysis we have “stable” sites with +10 year Watson et al. (2006) show that aliasing effects may observations, and “new” sites with <5 year. If the

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Fig. 4 Detrended time series plots of daily position estimates (n,e,u) from Vilhelmina (VIL0) (64◦ N) for the complete period of analysis (Aug 1993–Oct 2006)

vertical time series in our network still show a bend- 1. First we determine position and velocity for 27 “good” sites (no breaks in time series etc during ing shape, then a new site with data from e.g. 2002 will the complete period 1996–2006), based on the 7 get a biased high vertical rate compared to sites with year period 1998.0–2005.0 (grey marker in Fig. 6). 10 years data. We also would like to take advantage of These regional reference frame sites (italic site id in the old stable site we have in our BIFROST network to Fig. 1) are mainly from Sweden, some Finnish sites, improve velocity estimates at the more recent sites. and some IGS sites from continental Europe. Therefore we have applied a two step approach for reference frame realization for our final results:

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Ny-Alesund . kir0 reyk joen vil0 . mets onsa .

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Fig. 5 Network of 35 sites (black squares) and the 23 sites (light circles) used for reference frame realization Fig. 6 De-trended time series plot of daily vertical position estimates for Vilhelmina. The regional analyses have been combined with the global network in Fig. 5

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2. In the second step we apply a 6-parameter transfor- all the sites in the BIFROST network. The procedure mation (no scale) of all daily solutions to the re- also has the effect of reducing the noise in the time series, where the common mode noise, evident in global gional frame defined in 1. GPS networks, has a considerable contribution. An exThe procedure above has been carried out booth usample of time series from Kivetty (Finland) is given ing the ITRF2000 and the ITRF2005 reference frame. in Fig. 7. Note the outliers in the vertical component, For outlier editing in stage 1, and for the selection of and occasionally also in horizontal components. This “good” sites, we have used the “tsview” tool (Herphenomenon has been attributed to snow accumulation ring 2003). While computing the station position and on the GPS antenna and is most pronounced at northern velocity, we have simultaneously estimated parameters inland station (cf. Johansson et al. 2002, and discussion for annual and semi-annual seasonal variations. in Lidberg et al. 2007). The purpose of outlier editing is to remove erro3.5 Time Series Analysis and Data Editing neous samples from disturbing the velocity estimates, and to retrieve a “clean” data set that belong to one The outcome of the procedure described above is daily stochastic distribution, where the residuals can be used station positions in a well defined reference frame for for estimating the precision of the derived parameters.

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M. Lidberg et al. KIVE North Offset 2632277.21972 m _ 0.03 nrms = 0.32 wrms = 1.3 mm # 2645 rate (mm/yr) = 12.65 +

Fig. 7 Time series plots (neu) of daily position estimates for Kivetty (KIVE) after applying the 2 step reference frame approach. The north and east components has been de-trended (mm)

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In this study we have (again) used the tsview tool to retrieve station velocities in north, east and up components, estimating seasonal parameters, outlier editing, and introducing shifts in the time series when appropriate. Reliable accuracy estimates of derived station velocities presuppose that the character of the noise of the position time series is known a priori or can be estimated. A common method is to determine the spectral

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index and amplitude on the noise using maximum likelihood estimation (MLE) (e.g. Williams 2003, Williams et al. 2004). In this work we have however used the “realistic sigma” function of tsview, where formal uncertainties in derived parameters (assuming white noise) are scaled by a factor derived from the residuals assuming a Gauss-Markov process (Lidberg et al. 2007). The noise scaling factor is usually in the range of 3–5.

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4 Results

5 New GIA Model

The results from the process above are two 3D velocity fields constrained to the ITRF2000 and ITRF2005 reference frames respectively. The purpose of this work is however to study crustal deformations within the area influenced by the GIA-process. We therefore would like to present our results in relation to the stable part of the Eurasian tectonic plate. To do so we have removed the plate tectonic motion from our ITRF2000 based solution using the ITRF2000 No-Net Rotation (NNR) Absolute Rotation Pole for Eurasia. The velocity field constrained to ITRF2005 is then rotated to best fit the rotated ITRF2000 field. RMS of residuals in this fit is at the 0.1 mm/yr level. The results are shown in Figs. 8 and 9.

In this section we compare our GPS derived station velocities with predictions based on a geophysical GIA model. We use an updated version of the BIFROST model presented in Milne et al. (2001). The revised model provides an optimum fit to the new GPS solution in Lidberg et al. (2007). The model comprises the ice model of Lambeck et al. (1998) and an Earth viscosity model defined by a 120 km thick lithosphere, an upper mantle viscosity of 5 × 1020 Pas and a lower mantle viscosity of 5×1021 Pas. For comparison, the optimum values obtained for the older GPS solution (Johansson et al. 2002) were, respectively, 120 km, 8 × 1020 Pas and 1022 Pas.

Fig. 8 Horizontal station velocities from the ITRF2000 and ITRF2005 solutions (dark grey and light grey arrows with 95% probability ellipses respectively), together with the GIA model (black arrows) rotated to fit the GPS derived velocities

566 Fig. 9 Vertical station velocities from the ITRF2005 and ITRF2000 solutions (two leftmost bars), the updated GIA model (black) and values according to Ekman (rightmost bar when available)

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In order to check our results we look at the horizontal velocities at sites with the longest site records, located at stable part of Eurasia where we initially would assume the effect from the GIA process to be small. The RMS value of the north and east velocities for the seven sites BOR1, BRUS, JOZE, LAMA, POTS, RIGA, and WTZR is at the 0.5 mm/yr. This indicates a fairly successful reference frame realization, compared to the formal errors of the ITRF2000 Euler pole for Eurasia (0.02 mas/yr or 0.6 mm/yr).

Looking at the horizontal velocities in Fig. 8, we may notice that these velocities, which should be relative to “stable Eurasia”, may have a bias of several 0.1 mm/yr. The GPS-derived velocities at the best sites in Germany and Poland may not be zero and random, and we would expect zero horizontal velocity close to land uplift maximum, UME0 or SKE0, rather than somewhere between MAR6 and SUN0. The agreement with the GIA-model is good with RMS of residuals of 0.4 and 0.3 mm/yr for the north and east components for the 71 common points.

New Results Based on Reprocessing of 13 years

6.2 Vertical Velocities In Fig. 9 and Table 1 we compare vertical rates from our ITRF2000 and ITRF2005 based solutions to the updated GIA-model and to values presented by Ekman (1998). The latter are based on apparent land uplift of the crust relative to local sea level observed at tide gauges during the 100-year period 1892–1991, repeated geodetic leveling in the inland, an eustatic sea level rise of 1.2 mm/yr, and rise of geoid following Ekman and M¨akinen (1996). The standard errors of these rates (here denoted “Ekman”) are estimated to between 0.3 and 0.5 mm/yr (larger values for inland stations). Table 1 Statistics from comparison of vertical rates from the solutions presented here, the updated GIA-model, and values from Ekman (1998). See text Difference from ITRF2000 based solution (mm/yr)

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First we note the differences in vertical velocities (mean difference 0.4 mm/yr) between GPS solutions constrained to ITRF2000 and ITRF2005. Such a systematic difference has been reported from the work with ITRF2005 (e.g. Altamimi 2006), but our difference is somewhat smaller than expected. We also note that booth our GPS solutions give larger vertical rates compared to booth the new GIA-model and the tidegauge rates presented by Ekman.

7 Discussion We have shown agreement between GPS-derived station velocities and predictions from an updated GIA model at the 0.5 mm/yr level in horizontal and vertical components. The crustal motions we observe in northern Europe can, on average, thus be explained to this level of accuracy! Some individual sites, usually with short observation span, may have larger discrepancies. There are also large differences in the northern-most area. We think this is due to remaining effects from the “banana-shaped” vertical time series. We don’t understand the causes of this effect yet. However candidates

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for explanation may be effects from succession of GPS satellite block type (Ge et al. 2005), and contribution from higher order ionospheric terms (Kedar et al. 2003) which have not been corrected for here. The tide models may still contribute to the bent time series since the global sub-network (Fig. 5) is computed using the IERS 2003 model, but the regional BIFROST subnetworks are still processed using the old IERS 1996 tide model. In order to achieve a considerable increase in accuracy of station velocity from GNSS analysis, a number of measures should be considered. A consistent use of the best tide model is obvious. Higher order ionospheric effects should be taken into account. Atmospheric loading may be included at the observation level (e.g. Tregoning and van Dam 2005). New mapping functions are now available (e.g. Tesmer et al. 2007). IGS has now changed from relative to absolute calibrated antenna phase center variations (PCVs) in analysis and products. To take advantage of these improvements, re-processing of a considerable amount of the IGS-sites would in principle be needed. Especially if the approach of global connection for reference frame realization described here is applied. These efforts have fortunately started (e.g. Steigenberger et al. 2006). It may also be noted that some of the possible improvements may be difficult to implement in practice to get full advantage of them. E.g. the PCV determined in absolute antenna calibration may be valid for an isolated antenna, but these properties may change due to electromagnetic coupling and scattering effects when the antenna is attached to its foundation (Granstr¨om 2006). The stability in the reference frame is of crucial importance in order to determine un-biased absolute vertical displacements relative to the earth centre of mass. We note the difference between the latest and former versions of ITRF. Even though ITRF2005 have better internal accuracy (Fig. 3), the ITRF2000 derived values are in better agreement with the GIA model and values obtained from classic geodetic methods. An attempt to derive a GIA-model based on the ITRF2005 GPS velocities gave unreasonable results. A closely related point is the availability of stations within the IGS that have not undergone an antenna installation change for > 10 years. The number of such stations is limited. We consider our results to be very good from the perspective of applications like geodetic reference frame management for GIS applications and

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“geo-referencing”. However, for sea level work (using booth GPS and tide gauge observations), aiming at the 0.1 mm/yr level grate care must be taken. In order to reach this goal, it is crucial to continue research into long term stability in GNSS analyses as well as reference frame realization. Acknowledgments The authors thanks Nocolas Bergeot for reviewing the paper and giving helpful suggestions and comments.

References Altamimi, Z. (2006). Station positioning and the ITRF. ILRS Workshop, Canberra October 16–20, 2006. www. ilrscanberraworkshop2006.com.au/workshop/day2/Monday1700.pdf. Sited 19 August 2007. Altamimi, Z., P. Sillard, C. Boucher (2002). ITRF 2000: a new release of the International Terrestrial Reference Frame for earth science applications. J. Geophys. Res., 107(B10),2214, doi:10.1029/2001JB000561. Altamimi, Z., X. Collilieux, J. Legrand, B. Garayt and C. Boucher, (2007). ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and earth orientation parameters. J. Geophys. Res., doi: 10.1029/2007JB004949. In press. Ekman, M. (1998). Postglacial uplift rates for reducing vertical positions in geodetic reference systems. In: Proceedings of the General Assembly of the Nordic Geodetic Commission, May 25–29, 1998, Edited by B. Jonsson, ISSN 0280-5731 LMV-rapport 1999.12. Ge, M., G. Gendt, G. Dick, F.P. Zhang, C. Reigber (2005) Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys. Res. Lett. 32 L06310 doi:10.1029/2004GL022224. Granstr¨om, C. (2006). Site-dependent effects in high-accuracy applications of GNSS. Licentiate Thesis, Technical Report No 13L, Department of Radio and Space Science, Chalmers University of Technology, Sweden. Herring, T.A. (2003). MATLAB tools for viewing GPS velocities and time series GPS Solutions 7:194–199. Herring, T.A., R.W. King, S.C. McClusky (2006a). Introduction to GAMIT/GLOBK, release 10.3. Department of Earth, Atmospheric, and Planetary Sciences, MIT, http://wwwgpsg.mit.edu/∼simon/gtgk/docs.htm. Cited August 2007. Herring, T.A., R.W. King, S.C. McClusky (2006b). GAMIT reference manual, release 10.3. Department of Earth, Atmospheric, and Planetary Sciences, MIT, http://wwwgpsg.mit.edu/∼simon/gtgk/docs.htm. Cited August 2007.

M. Lidberg et al. Herring, T.A., R.W. King, S.C. McClusky (2006c). GLOBK reference manual, release 10.3. Department of Earth, Atmospheric, and Planetary Sciences, MIT, http://wwwgpsg.mit.edu/∼simon/gtgk/docs.htm. Cited August 2007. Johansson, J.M., J.L. Davis, H.-G. Scherneck, G.A. Milne, M. Vermeer, J.X. Mitrovica, R.A. Bennett, B. Jonsson, G. Elgered, P. El´osegui, H. Koivula, M. Poutanen, B. O. R¨onn¨ang, I. I. Shapiro (2002). Continuous GPS measurements of postglacial adjustment in Fennoscandia 1. Geodetic result. J. Geophys. Res., 107(B8), 2157, doi:10.1029/2001JB000400 Kedar, S, G.A. Hajj, B.D. Wilson, M.B. Heflin (2003). The effect of the second order GPS ionospheric correction on receiver positions, Geophys. Res. Lett., 30(16) 1829 doi:10.1029/2003GL017639. Lambeck, K, C. Smither, P. Johnston (1998). Sea-level change, glacial rebound and mantle viscosity for northern Europe. Geophys. J. Int., 134, 102–144. Lidberg, M, J.M. Johansson, H.-G. Scherneck, J.L. Davis (2007). An improved and extended GPS derived velocity field for the glacial isostatic adjustment in Fennoscandia. Journal of Geodesy, 81 (3), pp 213–230. doi:10.1007/s00190-0060102-4. Milne, G.A., J.L. Davis, J.X. Mitrovica, H.-G. Scherneck, J.M. Johansson, M. Vermeer, H. Koivula (2001). Space-Geodetic Constraints on Glacial Isostatic Adjustments in Fennoscandia. Science 291, 2381–2385. Steigenberger, P., M. Rothacher, R. Dietrich; M. Fritsche, A. R¨ulke, and S. Vey (2006), Reprocessing of a global GPS network, J. Geophys. Res., 111, B05402, doi:10.1029/2005JB003747. Tesmer, V., J. Boehm, R. Heinkelmann, H. Schuh (2007). Effect of different tropospheric mapping functions on the TRF, CRF, and position time-series estimated from VLBI. J. Geodesy, 81 (6–8), pp. 409–421. DOI: 10.1007/s00190-0060126-9 Tregoning, P., T. van Dam (2005). Effects of atmospheric pressure loading and seven-parameter transformations on estimates of geocenter motion and station heights from space geodetic observations, J. geophys Res, 110, B03408, doi:10.1029/2004JB003334. Watson, C., P. Tregoning, R. Coleman (2006). Impact of solid Earth tide models on GPS coordinate and tropospheric time series. Geophys. Res. Lett. 33 L08306 doi:10.1029/2005GL025538. Williams, S.P.D., Y. Bock, P. Fang, P. Jamason, R.M. Nikolaidis, L. Prawirodirdjo, M. Miller, and D.J. Johnson (2004). Error analysis of continuous GPS position time series, J. Geophys. Res., 109, B03412, doi:10.1029/2003JB002741.

Analysis of GPS Data from An Antarctic Ice Stream R. Dach, G. Beutler and G.H. Gudmundsson

Abstract Temporal variations in the flow of an active ice stream are analyzed using GPS data collected over a period of two months at six different locations. The diameter of the network is about 60 km. The ice stream moves with a velocity of about one meter per day. The kinematic data are processed using three different strategies: zero-difference network solution, Precise Point Positioning, and double-difference network solution with resolved carrier phase ambiguities. The solutions are compared with regard to the quality of the resulting coordinate time series. Special attention is paid to the positional accuracy as a function of temporal frequency for these different analysis methods as the overall aim of the measurements is to estimate temporal variability in ice flow. Keywords GPS · GNSS · Kinematic positioning · Ambiguity-fixing · Precise point positioning

1 Introduction To process GNSS data we usual distinguish zeroand double–difference processing schemes. Both are mathematically equivalent. The zero-difference method allows to solve for the receiver and satellite R. Dach Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland e-mail: [email protected]

clock corrections, in the double-difference approach we have access to the integer nature of the carrier phase ambiguities. Integer carrier phase ambiguities may also be introduced as conditions of sum (or by adequate technologies) to a zero-difference solution. Resolving the integer phase ambiguity parameters is essential for obtaining optimal results (e.g., in the repeatability of station coordinate estimation). Recent studies at the AIUB (J¨aggi et al., 2008) concerning the gravity field recovery have shown that not all GNSS applications benefit from the introduced integer ambiguities. In that study, the orbits of the two GRACE satellites were generated once with real valued ambiguities and once with resolved integer ambiguities. On one hand, the relative validation of the orbits using K– band measurements between the two satellites clearly demonstrated the benefit of introducing the integer ambiguities. On the other hand, the gravity fields obtained from both orbits were nearly identical. For a more detailed description of this experiment we refer to J¨aggi et al., (2008). This result may be surprising at first sight, but can be explained by the fact that the epoch–to–epoch coordinate differences are dominated by the noise of the carrier phase measurements, which is identical for both solutions. Hence, only the long-term characteristics become more stable as the carrier phase ambiguities are fixed. Here we focus on the impact that resolving the carrier phase ambiguities has on the accuracy as a function of temporal frequency.

G. Beutler Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland G.H. Gudmundsson British Antarctic Survey High Cross, Madingley Road, Cambridge, CB3 0ET, UK

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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2 Description of the Dataset

Yet another receiver (FLET) was placed at the top of a nearby ice dome (Fletcher Promontory) where Six GPS stations were installed on the Rutford Ice little ice movement is to be expected. The nearest Stream, West Antarctica, from end of December 2003 tracking station of the International GNSS Service (IGS) with coordinates available in the IGS05 referuntil February 2004 (see Fig. 1): ence frame — the IGS realization of the ITRF2005 – One of the receivers, C − 20, was located on the considering the corrections between relative and (freely floating) ice shelf some 20 km down-stream absolute antenna phase center variation modeling (Schmid and Rothacher (2003); Ferland (2006)) – is from the grounding line. – A second one, station C + 00, was at the grounding O’Higgins (station ID’s OHI2 and OHI3). Most of the sites were equipped with LEICA SR530 line marking the division between the grounded ice receivers and LEIAT502 antennas. The station C + 30 steam and the floating ice shelf. – Four more stations C + 10, C + 20, C + 30, and on the ice stream was occupied with a TRIMBLE C + 40 were located at increasing distances of 10, 4000SSE receiver and a TRM22020.00+GP antenna. 20, 30, and 40 km, respectively, upstream from the None of the antennas were covered by a radome. The two IGS stations in O’Higgins (OHI2/OHI3) grounding line.

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Fig. 1 Location of the GPS sites on the Rutford ice stream

Analysis of GPS Data from An Antarctic Ice Stream Fig. 2 Number of GPS satellites contained in the RINEX observation files for the 9 stations in the campaign A – character indicates incomplete observations in the bin, a ∗ character is printed in a bin where more than 9 satellites are observed. In the first line (0000) the number of maximum observable GPS satellites above an elevation angle of 14◦ are listed

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0000 |787767767789889998988******99997878656887677988*99999899*989*988 -----+---------------+---------------+---------------+---------------+ C-20 |-665577677898898988789****988776767556887677878*99898889*989887C+00 |6665577--7898898978789****988776767555887677978*99898889*9898777 C+10 |6665577677898898978789****988766767555887677--8*99898889*9898777 C+20 |6765577677898898978889****988766766555887677978*99898889*989--77 C+30 |6665376677888899978578**9*988766766556887577778999898878*9898788 C+40 |-765577677898898978889****988766767555887677978*99898889*989877FLET |6765577677888898978789***99887767675558876--978*99898889*9798777 OHI2 |97766889**889****8**999***7*9998798868899868******99899**9998998 OHI3 |8644434686466656644445536755446775324547554556756774346567655456 -----+---------------+---------------+---------------+---------------+ 0 12 24 Hours for day 2004-01-20

are equipped with an AOA SNR-8000 ACT receiver with an AOAD/M T antenna (radome DOME) respective with an ASHTECH Z18 receiver with an ASH701941.B antenna (radome SNOW). In Fig. 2 the number of satellites observed by the nine GPS stations is plotted for one day. The measurements to GLONASS satellites at stations OHI3 are not considered. This day is typical for all other days of observation. Note that there are intervals with only 5 GPS satellites in view, which limits the quality of a kinematic solution (where we have to solve for 4 parameters per epoch). Unfortunately, the elevation mask was set to 14◦ during the measurement campaign. Figure 2 contains in the first line an artificial station with all GPS satellites in view considering the same elevation cut-off. Comparing the number of observed satellites for this artificial station with the content of the RINEX files there are many intervals with one or two missing satellites (PRN 23 was unhealthy for this day from midnight until 23: 00UT). To increase the number of available observations significantly it is necessary to decrease the elevation mask considerably during the measurement campaign. With an elevation mask of 3◦ , for example, at least 8 satellites would have been visible over the entire day.

2.1 Glaciological Background The experimental setup was designed to study tidally induced temporal variations in the flow of Rutford Icestream. Figure 3 shows linearly detrended stake positions in north direction as functions of time. The most

surprising aspect of the results is the temporal variation in horizontal movement with a period of about two weeks Gudmundsson (2006). This is clearly seen in the top part of Fig. 3. Superimposed on this longperiod component are smaller dirunal and semidiurnal components (Fig. 3 (bottom)). The biggest diurnal and semi-diurnal variation in horizontal position results for the station on the grounding line. The size of the amplitude is decreasing rapidly for the stations located inland (becoming effectively zero at a distance of 30 km from the grounding line). Further details of the variation of different tidal constituents with distance are given in Gudmundsson (2006).

3 GPS Data Processing The analysis was performed with the Bernese GPS Software, Version 5.0, which is available to the user community (Dach et al., 2007).

3.1 Connection to the ITRF In a first step the 2000 km baseline from O’Higgins to the station on the Fletcher Promontory was processed assuming that both stations are static within each daily solution. These daily solutions were then combined to a common solution considering the full covariance information and by introducing the coordinates and velocities from the IGS05 realization of the ITRF2005 for OHI2 as known, and a linear velocity for station FLET within this interval.

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From these two months of data we obtained a velocity for the Fletcher Promontory station of −0.1721 m/year for the North and 0.2579 m/year for the East component. This corresponds to 0.8 mm horizontal movement per day. As expected given the fact that this station was located at the top of an ice dome with a snow accumulation of a about 0.5 m water equivalent, the vertical velocity component was the largest one, or −0.8957 m/year (−2.5 mm/day). Figure 4 shows the residuals of the daily solutions after the linear station movements for FLET have been removed. In the North and East component the RMS are 2.2 mm and 2.3 mm, respectively, whereas for the vertical component the RMS is 7.6 mm. These values are achieved after the resolution of about 80% of the carrier phase ambiguities.

3.2 Local Network on the Ice Stream In a second step the coordinates for the station on the Fletcher Promontory were fixed for each day using the coordinates and velocities obtained in the previous step. One independent set of coordinates was estimated for each epoch (sampling 30 s) for each of the six moving receivers on the ice stream. The troposphere for these six sites was modeled by estimating corresponding parameters every two hours for each of the stations except for FLET because of the small extension of the network. Based on these assumptions the following solutions were computed using only the carrier phase measurements:

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1. a zero-difference network solution1 where satellite and receiver clock corrections are independently computed as additional parameters for each epoch, 2. a PPP (precise point positioning) solution1 for each of the six stations on the ice stream introducing the satellite clock corrections obtained in step 1., 3. a double-difference network solution from six baselines between the FLET station and the receivers on the ice stream with real valued carrier phase ambiguities, and 4. a double-difference network solution using the same baselines as in 3., but introducing the integer values for about 80% of the carrier phase ambiguities. All solutions are based on the same set of observations. The preprocessing of the data was carried out only once. The baseline solutions use only observations that were also used to obtain to the zero-difference solution. In this way, the results from the four solutions for the six kinematic sites on the ice stream are comparable.

1 Because no pseudorange data are used one carrier phase ambiguity per station and day has to be constrained to prevent the singularity because of the correlation to the clock parameters. This is the only modification withe respect to the officially distributed version 5.0 of the Bernese GPS Software.

4 Intercomparision of Kinematic Solutions 4.1 Expectation from Theory The solutions 1. and 2. are expected to be nearly identical because a perfect consistency of the introduced satellite clock corrections with the satellite orbits and the software models are guaranteed. In addition, the data used for the PPP contributed to the computation of the satellite clocks. The second aspect is usually not met in the case of a PPP. Mathematically, the solutions 1. and 3. are expected to give the same results, as well, because the double-difference solution can be derived from the zero-difference solution by pre-eliminating the receiver and satellite clock parameters. For practical reasons, in the Bernese GPS Software only an independent set of baselines is considered for the double-difference solution. Due to this simplification a few observations may be lost in the double-difference with respect to the zero-difference solution (two independent baselines between the stations A and B and A and C, respectively, contain common observations between A and B and A and C, but not necessarily all observations between B and C). Another aspect artificially degrading the doubledifference solution is that the ambiguities are simply merged from two zero-difference observation files when forming the baseline observation files. They are

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not updated according to the new observation scenario in the new double-difference solution. As a consequence, ambiguity parameters with a small number of observations are estimated, even if they weaken the parameter estimation. Such ambiguity parameters are eliminated by excluding the corresponding observations from the processing, when the list of ambiguities is updated for the baselines.

This direct comparison by forming differences of the resulting time series only shows that the kinematic positions obtained from all three solutions with realvalued ambiguities (1. zero-difference network, 2. PPP, and 3. double-difference network) are closer to each other than the fourth solution based on resolved integer ambiguities. The comparison does not tell which of the solutions is the best.

4.3 Day Boundary Discontinuities 4.2 Differences in the Results

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Each day’s observations were processed independently. The actual movement of the icestream over each 30 s interval is small or less than about 0.5 mm. The size of this discontinuity with respect to the actual expected movement, and with respect to the individual epoch within each daily solution, gives some indication of the accuracy of the results. Table 2 contains the RMS of these epoch differences between two consecutive daily solutions for two of the six stations of the ice stream. It can be seen that for three of the solutions (1. zero-difference network, 2. PPP, and 3. double-difference network) the RMS values are similar, whereas for the double-difference solution with introduced integer ambiguities (4.) the RMS—at least for the horizontal component—is significantly smaller. This is in accordance with the expectation that resolving the ambiguity parameters improves the GNSS derived results. For comparison Table 3 contains the RMS value of the epoch-to-epoch difference within each daily solution. These values are of course obtained from many more data points (nearly 3000 epoch differences more) and have in this way a better statistical certainty. The RMS values from different solutions show no significant difference between the four time series of kinematic positions.

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The differences between the three different kinematic solutions (PPP and double-difference with and without ambiguity resolution) relative to the zero-difference network solution (with estimated receiver and satellite clock corrections) were computed and the RMS values were computed for each station and component (see Table 1). These results confirm the expectations. Especially for the vertical component (U) the mean differences are larger for the ambiguity-fixed solution w.r.t. the other solutions. The same holds for the two horizontal components – but in some cases to a lesser extent. Table 1 RMS of the differences between the kinematic solutions 1. zero-difference network solution 2. PPP using the satellite clocks from 1. 3. double-difference solution, no ambiguity resolution 4. double-difference solution with ambiguity resolution RMS of difference of the solutions Station

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This result indicates that the noise of the kinematic solutions, as it is reflected in Table 3, is dominated by the noise of the phase measurements (including multipath effects). It is remarkable that—at least for the horizontal components—the RMS of the epoch–to–epoch differences between consecutive days in Table 2 for solution 4. (with introduced integer ambiguities) reaches almost the same values. For the vertical component the RMS of the epoch-to-epoch difference between two consecutive days is by a factor of about two larger than the general noise level of the solution as judged from the RMS of the epoch-to-epoch differences within each daily solution.

4.4 Allan Deviation In the timing community the Allan deviation (σ (τ ), Allan (1987)) is a widely used statistical tool to characterize the stability of atomic clocks for different time intervals τ (starting from a few minutes/hours for short-term stability up to several days for long-term stability) from a time series of clock values (x1 . . . x N ). For a perfect clock a constant frequency is expected — meaning that a drift of the obtained time values is permitted. The Allan deviation considers the values referring to three epochs separated by an interval length of τ :    σ (τ ) = 

N −2  1 (xi+2 − 2xi+1 + xi )2 2(N − 2)τ 2 i=1

We assume parts of the time series separated by more than a five minute data gap to be independed, and, in contrast to the practice in the timing community, do not use a triplet of values to compute the Allan deviation where such a break is found. In particular for long intervals with low number of observed satellites (see

Fig. 2) this condition reduces the number of triplets of values contributing to the Allan deviation. Nevertheless this assumption is reasonable because some of these independent parts show offsets comparable to the discontinuities between the daily solutions if they are not connected by ambiguity parameters. Figure 5 shows the Allan deviations for one of the stations on the ice stream (C + 30) computed from the time series of kinematic positions obtained with the different processing methods. The linear movement of the ice stream is considered by the Allan deviation (analogue to the permitted linear drift of time values from a clock). For the three processing strategies with real valued ambiguities (1. ZD–FLOAT: zero–difference network, 2. PPP(ZD): PPP with satellite clock corrections from 1., and 3. DD–FLOAT: double–difference network) a similar behavior in the solutions for the horizontal components is observed (see Fig. 5). With increasing length of the time interval (τ ) the slopes of the curves are at first equal to −1. This corresponds to the phase noise of the carrier phase measurements itself. For 1 < τ < 6 hours the curves are almost flat indicating that biases existing in the solutions that are constant over these intervals — the carrier phase ambiguity parameters. For longer intervals up to about 12– 15 h the slope is again −1 but with a higher noise level than for the intervals shorter than 1 h. Note that because of small number of contributing triplets the Allan deviation for intervals longer that 15 hours should be not interpreted. The fourth solution with resolved ambiguities (4. DD–FIXED) starts for short intervals with the same noise behavior as the other solutions. Up to an interval length of two hours the slope is also −1. For longer intervals it is only 0.9 which can be seen as a consequence of the discontinuities between the independent daily solutions acting as long-term biases. For the vertical component a slope of about −1 over the entire interval can be observed for all solutions. The noise level in the kinematic solutions for the vertical component is significantly higher than for the horizontal components, what is to be expected from the general experience in the GNSS data processing (e.g., due to the geometrical coverage of satellites, correlation with other parameters like troposphere or receiver clock corrections)—in particular if no observations to satellites with low elevations are available (Rothacher and Beutler (1998); Dach et al., (2003)).

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Fig. 5 Allan deviation for station C + 30 on the ice stream computed from the time series of kinematic positions obtained from different GNSS processing strategies: ZD–FLOAT zero–difference network solution PPP(ZD) PPP using the satellite clocks from 1. DD–FLOAT double–difference network solution with estimated real valued ambiguities DD–FIXED double–difference network solution with introduced integer ambiguities PPP(IGS) PPP using the satellite clocks from IGS

For comparison an alternative PPP solution was added to Fig. 5. Instead of using satellite corrections derived from the zero-difference network solution as was done in the PPP (PPP(ZD)) solution described above, the combined satellite clock corrections provided by the IGS were introduced (PPP(IGS)). At first; the IGS provided at that time satellite clock corrections only every 5 min with the consequence that the sampling for this solutions has to be adapted accordingly. Secondary, the consistency between all introduced products (satellite orbits and clock corrections as well as Earth orientation parameters) for the combined IGS product cannot be as good as for an internal zero-difference network solution: For the IGS product different analysis centers with different software packages containing different analysis models and individual analysis strategies are mixed with a sophisticated weighting scheme, which cannot be reproduced with one of the processing software packages. The influence of this circumstance is clearly indicated by the Allan deviation for intervals up to 1 h by a slope of about −0.5 indicating a colored noise. The impact of the unresolved phase ambiguities is observed as well in this solution. Figure 6 shows Allan deviations for two doubledifference solutions (with and without resolved integer ambiguities). As the Allan deviations of the zerodifference network solution and the PPP solution are both similar to that of the double-difference solution with real–valued carrier phase ambiguities these are not shown. The impact of the non-linear movement of the ice stream due to the sub-daily ocean tides—as they are already indicated in Fig. 3 — is clearly seen in the Allan deviations of Fig. 6 for the horizontal components. The stations closely located to the grounding line (respective to the shelf ice) show a larger peak in the Allan deviation for interval around 6 hours. Because the noise level of the solution with introduced integer ambiguities is in any case lower than the noise level indicated by the Allan deviation for the solution with real-valued ambiguities, the resolution of the ambiguities to their integer value helps to improve the solution and brings out in more detail the sub-daily variations in the movement of the ice stream.

4.5 Investigating the Vertical Component Figures 5 and 6 show no improvement in the Allan deviation for the time series of the kinematic solution

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with resolved integer ambiguities compared to the solution with real-valued ambiguities for the vertical component. As mentioned above, one possible reason may be the correlation of the vertical component with the troposphere model parameters, which also have to be estimated. For very short baselines with a length of a few meters, no troposphere modeling is needed. A simple analysis of the observations of the phase measurements of the first frequency, where only phase ambiguities and station coordinates have to be solved for, is sufficient. In such a case no correlations can occur. At the O’Higgins site two receivers, separated by only 3 m submit data to the IGS. For this short baseline a kinematic solution was calculated for one of the stations, with the coordinates of the other fixed. Two solutions, one with real-valued, and another one with integer ambiguities, were generated. From these two kinematic time series the Allan deviations were computed and displayed in Fig. 7. Interestingly, these solutions show the same Allan deviations for both solution types for the vertical component (as those in the Figs. 5 and 6). Hence, the correlation of the vertical component with other parameters that had to be solved for cannot explain why the ambiguity resolution does not improve the Allan deviations for the vertical component. The only difference to the local network solutions is a lower noise level of the kinematic solutions, explained by the lower noise level of the L1 phase measurements compared to the ionosphere-free linear combination which was used for the processing of the local network on the ice stream.

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We confirmed by comparing the kinematic coordinate time series computed from GPS data on the Rutford Ice Stream with different processing strategies that the accuracy of a Precise Point Position (PPP) solution is improved by calculating the satellite clock corrections in a initial zero-difference processing step as opposed to using the IGS satellite clock corrections. Doubledifference network solution with introduced integer ambiguities improves the accuracy further. The Allan deviation is well suited to analyze the noise characteristics of kinematic GNSS solutions if the rover receivers follow a linear trajectory. The horizontal components of a kinematic solution with resolved ambiguities reveal noise characteristics of the phase measurements in the Allan deviation. For the vertical component we see the same noise behavior in the Allan deviation for solutions with real-valued and resolved ambiguities. Nevertheless, when studying, e.g., the discontinuities at the day boundaries we see the benefit of resolving the ambiguities to their integer values. Only very short-term characteristics of a kinematic solution (shorter than 1 h for terrestrial sites) do not benefit from the ambiguity resolution. The solution is here dominated by the phase noise of the GPS observations. The analysis of the kinematic solutions for GPS data from Rutford Ice Stream confirms the previously proposed explanation (J¨aggi et al., 2008) for the apparent conflict between the improvement of the orbits of the two GRACE satellites due to resolving the ambiguities

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and the similarity of the derived gravity field from the Dach, R., U. Hugentobler, P. Fridez, and M. Meindl, Eds. (2007), Bernese GPS Software, Version 5.0, User Manual, AstroGRACE orbits without and with ambiguity resolution. nomical Institute, University of Bern. The main difference between the Allan deviation for Ferland, R. (2006), IGS05 Fine Tuning, IGS-Mail, 5455, the LEO compared to the terrestrial stations is that the available at http://igscb.jpl.nasa.gov/mail/igsmail/...2006/ benefit from the ambiguity resolution for the kinematic msg00178.html. positions can be seen already for intervals of about Gudmundsson, G.H. (2006), Fortnightly variation in the flow velocity of rutford ice stream, West Antarctica, In: Nature, 5 min for GRACE because of the much shorter mean 444(doi:10.1038), pp. 1063–1064. validity interval for the ambiguities. Acknowledgments The Astronomical Institute at the University of Bern (AIUB) hosts the Center for Orbit Determination in Europe (CODE), which is a joint venture of the AIUB, the Federal Office of Topography (Swisstopo, Switzerland), and the Federal Agency for Cartography and Geodesy (BKG, Germany). The results presented in this paper benefit from the work of the IGS analysis center CODE, (Hugentobler et al. (2006)).

References Allan, D.W. (1987), Time and Frequency (Time-Domain) Characterization, estimation, and prediction of precision clocks and oscillators. In: IEEE Trans. on Ultrason., Ferroelect., Freq. Contr., 34, pp. 647–654. Dach, R., G. Beutler, U. Hugentobler, S. Schaer, T. Schildknecht, T. Springer, G. Dudle, and L. Prost (2003), Time transfer using GPS carrier phase: error propagation and results, In: J. Geod., 77(1–2), pp. 1–14.

Hugentobler, U., M. Meindl, G. Beutler, H. Bock, R. Dach, A. J¨aggi, C. Urschl, L. Mervart, M. Rothacher, S. Schaer, E. Brockmann, D. Ineichen, A. Wiget, U. Wild, G. Weber, H. Habrich, and C. Boucher (2006), CODE IGS Analysis Center Technical Report 2003/2004, in IGS 2004 Technical Reports, edited by Ken Gowey et al., IGS Central Bureau, Jet Propulsion Laboratory, Pasadena, California, USA, in press. J¨aggi A., G. Beutler, L. Prange, L. Mervart, and R. Dach (2008), Assessment of GPS observables for gravity field recovery from GRACE. This Volume. Rothacher, M., and G. Beutler (1998), The role of GPS in the study of global change, In: Phys. Chem. Earth, 23(9–10), pp. 1029–1040. Schmid, R., and M. Rothacher (2003), Estimation of elevationdependent satellite antenna phase center variations of GPS satellites, In: J. Geod., 77(7–8), pp. 440–446.

Dynamic Analysis of Crustal Movements Along the Dead-Sea Rift L. Shahar and G. Even-Tzur

Abstract The geodetic-geodynamic network of Israel, named G1, was constructed in the 1990s, with the aim of understanding recent tectonics and deformation in Israel. The network contains approximately 150 points which were built according to high-standard technical specifications to ensure their high geotechnical stability. In this research we use measurements of 60 points, from the northern part of the G1 network together with five permanent GPS stations to monitor current crustal movements across the Dead Sea Rift (DSR) in the north of Israel. The DSR is the northern part of the Syrian-African Fault. It extends from southern Turkey to the Aqaba Gulf and separates the Arabian plate from the Sinai sub-plate. Recent researches show that the DSR behaves as a locked strike-slip fault. The research makes use of the Two-Step analysis of dynamical networks. This method is based on a first step of independent adjustment of every measurement campaign to a set of network points coordinates, followed by a second step in which we use the coordinates as pseudo-observations for adjusting a set of kinematical or dynamic parameters. The least squares estimation of the second step assumes an ideal mathematical model of the tectonics. While the correlation between the mathematical model and the physical phenomenon is not absolute, the solution is biased due to the correlation. That is the main reason for using prior information for matching of the model to the phenomenon. We use the locked fault model to describe the dynamic behavior of points in the north of Israel. It relies L. Shahar Faculty of Civil and Environmental Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel G. Even-Tzur Faculty of Civil and Environmental Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel

on an arctangent function and depends on four independent parameters. The locked fault model suffers from an instability relationship between its parameters since it deals with extremely different scales of its parameters. By using prior knowledge we evaluate the model parameters by enforcing constraints on the adjusted parameters. A second solution of the model parameters is presented by extended free dynamic network. In this method we adjust the kinematical parameters simultaneously with the dynamical softened parameters. A third solution is achieved by regulation of the normal matrix in certain ways. The paper presents results which based on two GPS measurement campaigns carried out in 1996 and 2002. Applying different solutions strengthens the possibility of correct use of the mathematical model to achieve results in close proximity to the actual physical behavior and enables reaching a reliable solution. The solution indicates a left lateral motion of the DSR in a rate of 2–7 mm/yr. In addition, the system is characterized by a high level of model noise which indicates an unsuitability of the mathematical model to the geophysical reality. Keywords Crustal movement · Deformation · Two-step analysis · G1 network · Northern Israel · Dead-Sea Rift · Carmel Tirza Fault Belt

1 Introduction The Dead-Sea Rift (DSR) is the northernmost part of the Syrian-African Fault. This fault separates between the African plate which is located on its western side

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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and the Arabian plate which is located eastern to it. The Sinai sub-plate is located on the DSR’s western side, as an integral part of the African plate. Relative to the Eurasia plate, both of the plates move to the north but with different velocities: the Arabian plate moves faster than the Sinai sub-plate. Due to this difference, the rift is defined by a left horizontal movement. The tectonic activity surroundings the rift created some arms of secondary fault systems on its two sides. The most important secondary system is the Carmel-Tirza Fault Belt (CTFB) which is located on its western side (Fig. 1). Schattner et al. (2006) presents the division of the Sinai sub-plate into three morphotectonic units: the southern is the Negev unit, the central is the JudeaSamaria unit and the northern is the Galilee-Lebanon unit (Fig. 2). Actually, the CTFB separates the JudeaSamaria and the Galilee-Lebanon units. The GalileeLebanon unit was formed by the youngest tectonic activity and we can see some independent sources that support this claim. For example, Fig. 3 presents the sharp continental shelf margins of the Galilee-Lebanon unit compared to the dull margins of the Judea-Samaria Fig. 2 Sinai sub-plate’s morpho-tectonic units unit. This sharpness is a geomorphologic evidence for the high level of the tectonic activity in this unit.

Fig. 3 A bathymetric shaded relief map of Galilee-Lebanon and Judea-Samaria units and its continental shelf margins (Schattner et al., 2006)

Fig. 1 Carmel-Tirza fault belt (CTFB), a secondary system which located western to the main rift, the Dead-Sea Rift (DSR) (Hofstetter et al., 1996)

During the ninetieths the G1 network was established by the Survey of Israel and the Geological Survey of Israel for monitoring the primary and the secondary faults in Israel. The network serves as a geodetic-geodynamic network in Israel and contains about 150 points that are homogeneously scattered inside the country. About a third of the network points are located in the northern part of Israel. The location

Dynamic Analysis of Crustal Movements

of the points was planned mainly according to geological considerations, in order to enable an optimal monitoring of the deformations in the primary and the secondary known faults over Israel. The network points were built according to high technical specifications to ensure their high geotechnical stability (Karcz, 1994). The network was planned to be measured twice a decade and until now it has been measured in 1996 and in 2002. In both measurements campaigns, four GPS dual frequency receivers were used. In 1996 the network was measured with Trimble SSE and SSI receivers and in 2002 with Trimble 5700 receivers with Zephyr Geodetic antennas. The 1996 campaign included two measurements of each point in two independent sessions of about 24 h (Ostrovsky, 2001). In 2002 each point was measured independently at least in three sessions of about 8 h (Even-Tzur et al., 2004). In addition, permanent GPS sites, which located in northern Israel, were in operation and were used as a part of this research; one of them operated in 1996 and five in 2002, in total 60 points took part in this research (Fig. 4).

Fig. 4 The network points, including 55 G1 points and 5 permanent GPS sites, which located in northern Israel

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2 The Analysis Method There are some Least-Squares estimation methods for deformation analysis; most of them are simultaneous methods which use all of the measurements as a base for the adjustment of a movement model for the network points according to the general linear model (Koch, 1999): l + ν = Ax

(1)

There are n observations and u unknowns parameters, therefore the observations vector l and the residuals vector ν are [n, 1] vectors, the coefficients matrix A is [n, u], and the unknown vector x is [u, 1]. The adjustment system’s variance factor of unit weight is a function of the sum of squares of residuals (SSR), the degrees of freedom r , and the weights matrix P: 2

σ 0 = ν T Pν/r

(2)

The evaluation of the unknowns’ vector is: x = Q A T Pl

(3)

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campaign and to evaluate the deformation distinction ability of the network. 2. The Two-Step method is defined with modularity Q = N −1 = (A T P A)−1 (4) that contributes to future monitoring. While the simultaneous adjustment requires the measurements We can calculate the variance-covariance matrix data for each campaign, the Two-Step method re(VCM) as: quires only some calculated parameters of the first step: the adjusted coordinates set and its VCM, the 2 2 2   x = σ 0 (A T P A)−1 = σ 0 N −1 = σ 0 Q (5) SSR and the degree of freedom of each campaign. 3. The first step intermediate results can be an efficient Papo and Perelmuter (1993) presented the Two-Step platform for the detection of gross errors, problemanalysis method for deformation analysis of dynamic atic campaigns and anomaly network points. networks. The first step includes an independent ad4. The analysis of advantages and disadvantages of the justment of each measurements campaign for a static Least-Squares estimation, simultaneous or not, and set of Cartesian coordinates; therefore, this step does relations to robust methods, shows that the main not include any treatment of the network’s geometriLeast-Squares estimation disadvantage is the abcal changes. In the second step, the first step’s adjusted sence of gross error detection inside the adjustment parameters are used as pseudo observations for adprocessing. Chen et al. (1987) developed an effijustment of the approximated movement model which cient technique for a priori measurements’ gross ertreats the geometrical changes of the network. The secror detection. This technique neutralizes the main ond step credibility depends on the correlation between disadvantage of the Least-Squares estimation and the chosen model and the geophysical reality. enables the use of the Two-Step method and the utiTherefore, the first step solutions is: lization of its advantages in deformation analysis. where the cofactor matrix Q is the inverse of the normal matrix:

l I + ν I = AI x I

(6)

The result of the first step represents the network points’ estimation for any of the k measurement campaigns and serves as observations for the second step, i.e. l II = x I . Therefore the second step solution is: x I + ν II = AII x II

3 The Mathematical Model 3.1 Movement Modeling

Control networks’ modeling approaches are based on three branches of mechanics: statics, kinematics and while the indexes I and II represent the first and the dynamics. The classical geodetic control network is a second steps, accordingly. static network that assumes stability of the network’s The unknowns’ VCM can be calculated with the geometry and actually does not take the time dimenuse of Eq. (5), where the second step SSR is (Even- sion into consideration. Default of the stability condiTzur, 2004): tion distorts the network because of the absence of the time dimension. This cancels out any reference to the

k    T  T II II II I I I SS R = ν P ν + Pi νi νi (8) time-dependent effects on the control network. Kinematic modeling includes the time as fourth dii=1 mension and enables the calculation of movement paAdjustment of the measurements without gross er- rameters. These types of models are usually reprerors reveals the same result and precision estimation sented by a Taylor series, in both concepts, simultaneous or not. However, some l advantages in the use of the Two-Step method are  ( ti )n f (x) = f (n) (t0 ) (9) noticed: n! n=1 1. The first step results include a set of adjusted coordinates and VCM of each measurement campaign. Where ti is the difference of the reference time These sets enable to control the quality of each t0 and the measurement time ti , t = ti − t0 . The (7)

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f (x) vector is the difference of the location xi at the measurement time and the reference location x0 at the time t0 : f (x) = xi − x0 . Usually, the kinematic model is a first or second order function that includes the reference location and the movement parameters. In addition to the reference location, a first order polynom presents the velocity as a linear model of velocity scalar field: xi − x0 = x˙ ti

parameters as barometric pressure, atmospheric and gravitational data etc. (Dong et al., 1998; Papo and Perelmuter, 1991; Welsch, 1996). In a case where many of the physical parameters are not measurable or calculable, it is possible to evaluate them and to check the model’s compatibility. The model’s noise represents the model’s incompatibility to the geophysical reality; therefore, the final model matching is possible by testing a few similar models (10) and choosing the model with the lowest noise level.

where x˙ is the network points’ velocities vector. In addition to the velocity field, a second order term presents 3.2 Dynamic Model Regularization the acceleration scalar field as a quadratic model:

Problem

xi − x0 = x˙ ti +

x¨ ti 2 2

(11)

The mathematical modeling of physical phenomena is subjected to some typical problems. While where x¨ is the network’s points’ accelerations vector. modeling of a physical phenomenon with discrete In this case, the velocity field behavior is not neces- mathematical tools a regularization problem may sarily constant, and the velocity changes are expressed occur. Hadamard (1902) claims that every physical in the acceleration field. Just as the linear model phenomenon defined by a matchable unique ideal assumes a constant velocity, the quadratic model mathematical model which can be supported by assumes a constant acceleration. Simple calculation the phenomenon’s data continuously. A function without degree of freedom requires two measurement which describes a physical phenomenon is usually campaigns for the linear model and three for the consists physical parameters. Using measurements quadratic model. Additional measurements increase date for physical parameters calculation is an inverse the system’s degree of freedom and precision, enables problem. Sometimes, the physical parameters are gross error detection and requires an adjustment defined with a high rate of sensitivity for physical process (Papo and Perelmuter, 1982; Welsch and phenomena changes which are measured in a way Heunecke, 2001). In spite of the kinematical concept which makes their calculation through conventional in the geophysical parameters estimation, it is regarded mathematical tools difficult. Numeric solution of a as acceptable way for deformation monitoring in continuous physical phenomenon usually requires the phenomenon’s discretization. A discrete function certain cases. The small number of measurement campaigns, the which describes a continuous physical phenomenon varied tectonic movements occur inside the network is usually characterized with an ill-posed problem, area and the absence of physical data bring to the kine- which denies the numeric stability of the calculation matical modeling of monitoring networks (Papo and of its physical parameters. Inverse problem functions are usually characterized with ill-posed problems. A Perelmuter, 1994). Static or kinematical models are private cases functional condition which is not characterized with of the dynamical model. Numerously measurement an ill-posed problem is defined as well-posed. The campaigns of the network which is located inside an meaning of such a case is a potential existence of a area with known or measurable physical parameters stable solution. However this is guaranteed only in the enable the effective use of the dynamic model. In most case of a well-conditioned situation. A situation which cases, the dynamic model is a second order differential is not described as well-conditioned is defined as an equation which includes kinematical parameters that ill-conditioned problem whose quantifying is done depend on measured, adjusted or a priori known using a condition number, which is equal or greater physical parameters. Therefore, sometimes the model than 1. The ill-condition number problem is stricter is determined also according to measured physical as the higher the condition number is. Condition

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number of a matrix C can be calculated (Golub and Van Loan, 1996) as:     cond(C) = C −1  C

(12)

In a case of normal matrix N , we can calculate the condition number as: λmax (N ) cond(N ) = (13) λmin (N ) where λmax (N ) and λmin (N ) are the maximum and the minimum normal matrix’ eigenvalues, accordingly.

4 The Network Solution 4.1 The Network’s Kinematical Analysis A priori knowledge can be used for the determination of the characteristics of the tectonic structure. The quality of this knowledge has a direct influence on the suitability of the mathematical model to the geophysical reality. Hence, it is desired to improve this knowledge by intermediate concluding during the tectonic monitoring. The geological, the geophysical and the geodetic research provide valuable apriori knowledge of the physical characteristic of the Dead-Sea Rift (DSR), for example knowledge about its velocity and its depth of locking. The geological research indicates on a sinistral horizontal shifting of the rift of about 105 km roughly since the beginning of the Miocene, in the last 20–23 million years. Therefore the fault’s average velocity estimation is about 4–5 mm/yr. This estimation refers to the DSR’s southern part, south to the Sea of Galilee. North to this region, the shifting is smaller and reaches values of about 80 km. In the complicated geophysical reality of northern Israel, it is difficult to identify the contributions of the separate tectonic motions which comprise the general motion of the region. For example, a point which is located in the Galilee is apparently influenced by the DSR, the CTFB and other active faults. G1 network is defined as a geodynamic network which is able to monitor first and second order faults in Israel. Its density and the additional permanent GPS sites increase the network’s ability to identify the contributions of the main influence factors, i.e. the DSR and the CTFB.

Actually, the forces which move the DSR are the ones which indirectly move its secondary systems; hence, there is a correlation between the DSR and the secondary systems. However, most of the research is focused on the DSR’s tectonics, hence, minimization of the influence of secondary systems’ tectonics (mainly the CTFB’s tectonics) is necessary. The network areas which are located western to the DSR are suspected as influenced from both systems, DSR and CTFB. For DSR’s tectonics monitoring, points which are with CTFB significant influence should be filtered. Application of kinematical analysis before dynamic analysis is an efficient tool for spotting areas whose major tectonic source is not the DSR activity. Because of the small number of measurement campaigns of the G1 network, which is located in a complex fault area, a kinematic analysis which includes the application of the Two-Step method was done. In the first step, the measurements were processed using semi-scientific software of GPS vectors processing. Furthermore, gross errors were filtered. After filtering the gross error, the database consisted of 69 independent vectors from the 1996 campaign and 74 from the 2002 campaign. The measurements of each campaign were adjusted into a free static network. The free network precision indicates the network potential in horizontal deformation monitoring. The second step includes a kinematical analysis of 34 points, which were measured in both campaigns. The analysis started with a general congruency test (Cooper, 1987) with 5% significance level. The result strengthens the apriori hypothesis which claims the existence of a horizontal deformation inside the network. Partial congruency tests, which examine interior stability of areas, were applied on different geographical areas. These tests indicate three areas characterized with interior stability: the first is the Golan area, which is located on the Arabian plate, eastern to the DSR and includes the Golan-Heights and Mt. Hermon (Fig. 5). The second one is the Eastern Galilee area, and the third one is the Carmel area which includes the Carmel Mountain and its coastal plane. The Carmel area is located south to the CTFB; therefore the analysis with the fixed Carmel datum has the potential to isolate the areas which are not influenced significantly by the CTFB. This analysis indicates a significant motion of the Golan area in the direction of NNW at a rate of about 4–7 mm/yr and a motion of the Eastern Galilee in the same direction at a rate of about 3–4 mm/yr (Fig. 6). The fact that

Dynamic Analysis of Crustal Movements Fig. 5 A shaded relief map of the geographical parts in northern Israel

Fig. 6 Kinematical result of the horizontal network relative to the Carmel datum. The Carmel datum area is marked as a shaded relief map. An error ellipse represents a precision of a point velocity with 5% significance level. A line inside an ellipse represents a point velocity in such a way that a deviation the line from its borders represents a significant velocity of a point in the direction of the line, compared with the datum

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Fig. 7 Kinematical result of the horizontal network relative to the Golan datum. The Golan datum area is marked as a shaded relief map. An error ellipse represents a precision of a point velocity with 5% significance level. A line inside an ellipse represents a point velocity in such a way that a deviation the line from its borders represents a significant velocity of a point in the direction of the line, compared with the datum

two interior-stable areas in the both DSR’s sides move in the same direction indicates the absence of significant horizontal influence of the CTFB on the Eastern Galilee. Additionally, analysis of the motion of the Eastern Galilee relative to the Golan datum indicates significant sinistral motion of 2–3 mm/yr in the direction of SSE (Fig. 7). This analysis supports the knowledge about the DSR as a left transform.

4.2 Dynamic Model Fitting to the Dead-Sea Rift A dynamic analysis includes a physical interpretation of the kinematic parameters, in this case the velocities of the network points. A common interpretation of the DSR’s tectonics is given by the tangential velocity distribution model (Weertman and Weertman, 1964):

Fig. 8 Dynamic model of infinite locked strike-slip fault. A point velocity νx is a function of its orthogonal distance y from a fault with velocity of V = 5 mm/yr, which is locked up to depth of D = 10 km. These values pronounce an apriori estimation of the Dead-Sea Rift (Even-Tzur, 2004)

Dynamic Analysis of Crustal Movements

νx = π −1 V tan−1

589

y D

(14) the ill-posed problem which is defined by the internal scaling. Ostrovsky (2005) analyzed the G1 network’s measurements and presented a slip rate of 4.5 mm/yr where νx represents the velocity with the fault direc- with standard deviation of 0.8 mm/yr, but with relation of a point at an orthogonal distance y from an tive to a single point which located near to the DSR. infinite locked fault. The parameter D represents the Wdowinsky et al. (2004) analyzed the DSR’s tectondepth of locking and V is the free transform velocity. ics in the North of Israel with the use of 5 permaFigure 8 demonstrates this interpretation graphically. nent GPS sites and also the IGS site in Damascus, The G1 Network design and the measurements cam- UDMC. They adjusted measurements conducted durpaign were conducted according to the a priori knowl- ing the years 1996–2002 using the dynamic model of edge; therefore measurements of points, which are lo- infinite locked strike-slip fault. The adjustment system cated at a large distance from the rift, were empha- was characterized as well-conditioned by the use of sized, by means of multiple long measured vectors and constraints of some of the physical parameters. The the addition of new points in the Golan-Heights eastern result of this research suggests slip of 2.4–3.3 mm/yr part and in the Carmel area, in 2002. The model does with standard deviation of 0.1–0.6 mm/yr which refers not describe the geophysical reality ideally, because of to fixed depths of locking between 6 and 18 km, acthe complex tectonic texture of the area, but it serves cordingly. This range of values represents the solution as a satisfactory starting point. based on two points which are located south to the CTFB, therefore they are not influenced by the DSR’s tectonics only, but also, allegedly, by the tectonics of the CTFB, involving some faults which are character4.3 The Dynamic Model Solution ized with a left motion. The current dynamic analysis includes a constrained The simple solution of minimum constraints which solution with a fixed location of the rift and with a solves the datum defect (Koch, 1999) is character- fixed depth of locking. The analysis includes comparized with ill-posed problem for this nonlinear model. ison of three combinations of network’s points which Papo (1986) proposed a method which extends the have been analyzed separately: the entire network (34 network solution and solves the physical parameters points), a sub-network (in total 26 points) that omits 8 which are defined in the model, in addition to the con- points located south to the CTFB and a sub-network ventional network’s parameters. This solution is char- that includes the Golan and the Eastern Galilee points acterized with ill-conditioning. Actually, analysis of (20 points). Table 1 presents the results based on the the model’s physical parameters reveals the extreme three combinations, including the adjusted slip meadifference in the proportion of the physical parame- sures and their precision as a function of a discrete ters: while the depth of locking is evaluated to be a few depth value of up to 20 km (which is an evaluation of kilometers, the rift velocity is evaluated as a few mil- the crustal’s thickness in northern Israel). limeters per year only. Therefore, it is required to solve Table 1 Faults velocity and the posterior variance factor (PVF, σˆ 02 ) for the three sub-networks as a function of the locking depth. The lowest posterior variance factor’s rates are prominent Areas:

The entire network (34 points)

Galilee Golan sub-network (26 points)

Locking D. [km]

Velocity [mm/yr]

Std. [mm/yr]

PVF σˆ 02 Velocity [mm/yr]

Std. [mm/yr]

PVF σˆ 02 Velocity [mm/yr]

Std. [mm/yr]

PVF σˆ 02

0 2 4 8 12 16 20

1.9 2.5 2.9 3.7 4.5 5.1 5.8

0.6 0.8 0.9 1.2 1.5 1.8 2.1

2.436 2.380 2.385 2.406 2.432 2.458 2.479

0.6 0.8 0.9 1.2 1.5 1.8 2.2

2.626 2.520 2.514 2.535 2.576 2.614 2.645

0.6 0.7 0.8 1.1 1.4 1.7 2.1

2.144 2.053 2.043 2.030 2.044 2.070 2.115

1.9 2.6 3.0 3.9 4.7 5.3 6.0

Eastern Galilee Golan sub-network (20 points)

2.1 2.7 3.2 4.3 5.5 6.4 7.4

590

The sum of squares of residuals (SSR) and the posterior variance factor (PVF) σˆ 02 depends on two factors: the raw data’s precision and the suitability of the mathematical model to the geophysical reality. In this research, we used the SSR as an effective index for examination of the suitability of the mathematical model to the geophysical reality. Table 1 presents the PVF. Since the apriori variance factor is 1, this term represents the model’s noise for the three combinations of points. The table demonstrates the following facts: 1. In the three cases the results indicate on DSR’s sinistral slip at a rate of 1.9–7.4 mm/yr with changeable precision. The accepted values depend on the constrained value of the depth of locking: as the locking is deeper, the slip velocity increases while its precision decreases. 2. For an increasing assumption of fixed locking depth, the result for the rift motion velocity increases and the precision decreases. 3. The velocity obtained with the smallest subnetwork (which includes the Golan and the Eastern Galilee only) is the fastest. The velocity accepted using the sub-network which includes the Western Galilee points is the slowest. 4. In the three sub-networks, the minimal model noise is achieves with 2–8 km depth of locking. 5. The lowest model noise is obtained using the smallest sub-network.

5 Summary and Conclusions Since an efficient procedure for gross error filtering was used, the Two-Step method which is based on Least-Squared Estimation, was found to be an optimal method for dynamic network analysis. The dynamic solution indicates a sinistral slip of the DSR in a rate of 2–7 mm/yr with standard deviation of 0.5–2 mm/yr, accoddingly, but in addition, the system is characterized by a high level of model noise which indicates an unsuitability of the mathematical model to the geophysical reality. For the comparison, the kinematic analysis indicates a 2–3 mm/yr. The comparison of the results obtained using the different sub-networks does not enable to asses the CTFB’s motion characteristics since the CTFB’s

L. Shahar and G. Even-Tzur

movement direction is not parallel to the DSR’s movement directions. The results support the hypothesis that the DSR is a locked fault. This conclusion is relevant to the DSR’s part which passes through the network area only. We restrict this conclusion with the notion that the results are based on two measurement campaigns only.

References Chen Y. Q., M. Kavouuras and A. Chrzanowski (1987). A strategy for detection of outlying observations in measurements of high precision. The Canadian Surveyor. 41 (4), pp. 529–540. Cooper M. A. R. (1987). Control Surveys in Civil Engineering. Collins, London. Dong D., T. A. Herring and R. W. King (1998). Estimating regional deformation from a combination of terrestrial geodetic data. Journal of Geodesy. 72, pp. 200–214. Even-Tzur G. (2004) Variance factor estimation for two-step analysis of deformation networks. Journal of Surveying Engineering. 130(3), pp. 113–118. Golub G. H. and C. F. Van Loan (1996). Matrix Computations. 3rd ed. The Johns Hopkins University Press, Baltimore. Hadamard J. (1902). Sur les probl`emes aux d´eriv´ees partielles et leur signification physique. Princeton University Bulletin. 13, pp. 49–52. Hofstetter A., T. van Eak and A. Shapira (1996). Seismic activity along fault branches of the Dead Sea-Jordan transform system: The Carmel-Tirtza fault system. Tectonophysics. 267 (1–4). pp. 317–330. Karcz I. (1994). Geological considerations in design of the seminal Dead Sea Rift Network. Perelmuter Workshop on Dynamic Deformation Models, Haifa. Workshop Proceedings. pp. 39–55. Koch K. R. (1999). Parameter Estimation and Hypothesis Testing in Linear Models. Springer-Verlag, New-York. Ostrovsky H., (2005). The G1 Geodetic-Geodynamic Network: Results of the G1 GPS Surveying Campaigns in 1996/1997 and 2001/2002. Technical Project Report, Survey of Israel, Tel-Aviv. Papo H. B. (1986). Extended free net adjustment constraints. NOAA Technical Report NOS 119 NGS 37, Rockville. Papo H. B. and A. Perelmuter (1982). Deformation as reflected in the kinematics of a network of points in Jo´o I. and A. Detrek¨oi (Eds.), Doformation Measurements. Budapest, pp. 439–451. Papo H. B. and A. Perelmuter (1991). Dynamical modeling in deformation analysis. Manuscripta Geodetica. 18(5), pp. 295–300. Papo H. B. and A. Perelmuter (1993). Two step analysis of dynamical networks. Manuscripta Geodetica. 18(6), pp. 422–430.

Dynamic Analysis of Crustal Movements Papo H. B. and A. Perelmuter (1994). Episodic (static) and kinematic models as special cases of the general dynamic deformation models. Perelmuter Workshop on Dynamic Deformation Models, Haifa, Israel. Proceedings. pp. 341–348. Schattner U., Z. Ben-Avraham, M. Lazar and C. Huebscher (2006) Tectonic isolation of the Levant basin offshore Galilee-Lebanon – Effects of the Dead Sea fault plate boundary on the Levant continental margin, eastern Mediterranean. Journal of Structural Geology. 28, pp. 2049–2066. Wdowinski S., Y. Bock, G. Bear, L. Prawirodirdjo, N. Bechor S. Naman R. Knafo Y. Forrai and Y. Melzer (2004). GPS measurements of current crustal movements along the Dead Sea fault. Journal of Geophysical Research. 109, pp. 1–16.

591 Weertman J. and J. R. Weertman (1964). Elementary dislocation theory. Macmillan, New-York. Welsch W. M. (1996). Geodetic analysis of dynamic processes: classification and terminology. The 8th FIG International Symposium on Deformation Measurements, Hong-Kong. Proceedings. pp. 147–156. Welsch W. M. and O. Heunecke (2001). Models and terminology for the analysis of geodetic monitoring observations. The 10th FIG International Symposium on Deformation Measurements, Orange. Proceedings. pp. 390–412.

Symposium GS004

Positioning and Applications Conveners: C. Rizos, S. Verhagen

GPS-Based Monitoring of Surface Displacements in the Mud Volcano Area, Sidoarjo, East Java H.Z. Abidin, M.A. Kusuma, H. Andreas, M. Gamal and P. Sumintadireja

Abstract Since 29 May 2006, gas and hot mud has been gushing from the ground in Sidoarjo, East Java, Indonesia. As of late September 2006 scientists assume that the eruption may be a mud volcano forming, and may be impossible to stop. Surface displacements of the area, both in the vertical and horizontal directions, are expected due to this massive mud extrusion. GPS observations, both in campaign and continuous modes, were conducted to study this surface displacement phenomenon. GPS surveys were performed on up to about 20 stations using dual-frequency geodetic-type receivers with observation session lengths of about 5–7 h. Nine GPS campaigns have been conducted between June 2006 and June 2007. GPS continuous subsidence monitoring was conducted on five to ten stations, and started on 22 September 2006. A field survey to check the surface representation of the subsidence phenomenon was also conducted.

H.Z. Abidin Geodesy Research Division, Institute of Technology Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia, e-mail: [email protected] M.A. Kusuma Geodesy Research Division, Institute of Technology Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia H. Andreas Geodesy Research Division, Institute of Technology Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia M. Gamal Geodesy Research Division, Institute of Technology Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia P. Sumintadireja Applied Geology Research Division, Institute of Technology Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia

Based on the GPS-derived results it was determined that the area about 3–4 km around the extrusion centre experiences surface displacements. Rates of horizontal displacement are about 0.5–2 cm/day, while vertical displacements are about 1–4 cm/day, with rate increasing towards the extrusion centre. These observed surface displacements might consist of ground relaxation due to mudflow, loading due to the weight of mud causing the area to compact, land settlement and/or reactivation of the geological structures. Keywords Mud volcano · Sidoarjo · Deformation · Monitoring · Indonesia · GPS survey

1 Introduction The Sidoarjo mudflow, also informally abbreviated as LUSI from Lumpur Sidoarjo (lumpur is the Indonesian term for mud), is an ongoing eruption of gas and mud from the earth in the sub-district of Porong, Sidoarjo in East Java, Indonesia (20 km south of Surabaya). Since its extrusion on 29 May 2006, the mud has caused significant livelihood, environmental and infrastructure damage (see Fig. 1). The volumes of erupted mud increased from the initial 5000 m3 /day in the early stage to 120, 000 m3 /day in August 2006. Peaks of 160,000 and 170, 000 m3 /day of erupted material follow earthquakes swarms during September 2006; in December 2006 the flux reached the record-high level of 180, 000 m3 /day; and in June 2007 the mud volcano was still expelling more than 110, 000 m3 /day (Mazzini et al., 2007). On 26 September barriers built to hold back the mud failed,

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Fig. 1 Mud volcano of Sidoarjo

resulting in the flooding of more villages. On 24 November a gas pipeline exploded near the mud extrusion centre, killing several people. As of late September 2006, scientists assume that the eruption may be a mud volcano forming, and may be impossible to stop (van Noorden, 2006). So far, all efforts to stop the flow have failed. Considering the effects of mud loading, ground relaxation due to mud outflow, land settlement caused by surface works (e.g. construction of dykes) and/or possible reactivation of the geological structures in the area, it can be expected that the surface in the mud volcano area of Sidoarjo will be deformed (displaced), both horizontally and vertically. Information about ground displacements in this mud volcano area will be useful for several activities, such as determination of dangerous areas, land use planning, relocation and development of infrastructure, management of mud storage and flow on the surface, and planning and the development of a hazard mitigation system (see Fig. 2). Considering the importance of this ground displacement information, it should be monitored and studied. In this case, the characteristics of the expected ground displacement phenomenon are studied using GPS observations, both in campaign and continuous modes.

Landuse planning

Determination of dangerous area

Relocation & development of infrastructure

CHARACTERISTICS OF GROUND DISPLACEMENTS

Planning and development of hazard mitigation system

Management of mud storage and flow on the surface

Estimation of subsurface geological structure

Fig. 2 Importance of ground displacement information

2 Mud Volcano of Sidoarjo Mud volcano systems are common on Java Island, particularly in the East Java province, as shown in Fig. 3. Beneath the island of Java is a half-graben lying in the east-west direction, filled with over-pressured marine carbonates and marine muds (Stewart and Davies, 2006). It forms an inverted extensional basin that has been geologically active since the Paleogene epoch (Matthews and Bransden, 1995). The basin started to become over-pressured during the

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Fig. 3 Location of mud volcanoes in East Java, Indonesia

Oligo-Miocene period. Some of the over-pressured mud escapes to the surface forming mud volcanoes, which have been observed at Sangiran Dome and near Purwodadi City. On 28 May 2006, PT Lapindo Brantas targeted gas in the Kujung Formation carbonates in the Brantas PSC area by drilling a borehole named the Banjar-Panji 1 exploration well. The drill string went into a thick clay seam (500–1,300 m deep), and then sands, shells, volcanic debris and into permeable carbonate rocks (van Noorden, 2006). At 5:00 a.m. local time (UTC+8), the drill string went deeper to about 2,834 m (9,298 ft), after which water, steam and a small amount of gas erupted at a location about 200 m southwest of the well. Two further eruptions occurred on the second and the third of June about 800–1,000 m northwest of the well, but these stopped on 5 June 2006 (Davies et al., 2007). During these eruptions, hydrogen sulphide gas was released and local villages observed mud at a high temperature, around 60◦ C or 140◦ F (Normile, 2006). According to a model developed by a geologist (Davies et al., 2007) the drilling pipe penetrated the over-pressured limestone, causing entrainment of mud by water. The influx of water to the well bore caused a hydro-fracture, but the steam and water did not go through the borehole; they penetrated the surrounding over(burden)-pressured strata. The pressure formed some fractures around the borehole to propagate to the surface 200 m away from the well. The most likely cause of these hydraulic fractures in the shallowest strata is the unprotected drill string with a steel casing (Davies et al., 2007). Borehole protection by steel

casing has been a common procedure in oil or gas exploration. However, Mazzini et al. (2007) suggest a new conceptual model where the eruption is triggered by fracturing followed by vertical migration of overpressured mud. Although circumstantial, based on the observations collected, they indicate that the 27 May 2006 earthquake might have triggered the eruption of Sidoarjo mud volcano. This earthquake of magnitude 6.3 occurred at 5:54 a.m. local time on 27 May 2006, with an epicenter 280 km west-southwest of the eruption, near Yogyakarta (U.S. Geological Survey, 2006). Up to 11 May 2007, the mud had covered an area of about 7 km2 , as shown in Table 1. Moreover, by the end of May 2007, the mud had flooded and destroyed 10,426 houses, 33 schools, 4 offices, 31 factories, 65 mosques, 2 religious schools, 1 orphanage, and 28 other buildings (Kompas, 2007). Besides this damage, 14 people were killed, about 10,000 people had to be evacuated, and 2,441 lost their jobs in the affected factories. 5,397 students, 405 teachers, and 46 staff of the destroyed schools have also been affected.

Table 1 Increased area of mud coverage Date

Area of mud coverage (km2 )

29 August 2006 17 September 2006 20 October 2006 22 April 2007 11 May 2007

3.58 4.35 4.59 6.71 6.92

598

3 GPS-Derived Ground Displacements Nine (9) GPS campaigns have been conducted to study the ground displacements around the mud volcano area of Sidoarjo. The surveys were conducted on 29 June–2 July 2006, 23–25 July 2006, 26–29 Aug 2006, 17–20 Sep 2006, 12–15 Oct 2006, 4–11 Feb 2007, 10–12 March 2007, 5–11 May 2007 and 9–13 June 2007. GPS surveys were conducted using dual-frequency geodetic-type receivers, with observation session lengths of about 5–7 h. Due to the change in mud coverage area, the number of observed GPS stations were different from survey to survey, and the observed stations could not always be the same. The locations of GPS stations were also restricted by the mud coverage and its progression. Examples of some GPS stations are shown in Fig. 4. The characteristics of observed GPS stations are shown in Table 2. Data processing of the GPS survey data was conducted using the scientific GPS processing software Bernese 4.2 (Beutler et al., 2001). In general, standard deviations of the estimated coordinates are of the order

Fig. 4 Some GPS stations around Sidoarjo mud volcano

H.Z. Abidin et al. Table 2 Configuration of GPS monitoring networks Survey

Number of points

Coverage radius from the main vent (km)

1 2 3 4 5 6 7 8 9

14 12 19 17 9 9 8 23 29

0.5–1 0.5–1 3–4 3–4 3–4 3–4 3–4 3–4 3–4

of several mm in both horizontal and vertical components, as shown in Fig. 5. The GPS derived displacements from the first three GPS campaigns conducted in June, July and August 2006 are shown in Fig. 6. These GPS results show that the surface displacements in the mud volcano area of Sidoarjo have both horizontal and vertical components. In these first three months of mud extrusion it can be seen that the rates of displacements are increasing with time. In this period, the rate of horizontal and

GPS-Based Monitoring of Surface Displacements

Standard Deviation (mm)

8

6

4

Northing

2 Easting

Ellipsoidal Height

0 Components of Coordinates

Fig. 5 Box plots of the standard deviation of coordinates

vertical displacements were up to 2 and 4 cm/day, respectively; and vertical displacements are dominated by subsidence. Based on GPS results and also on visual field observation, the affected area of displacements up to end of August 2006 is still contained to about 1 km around the extrusion centre. Starting from the third campaign, more GPS stations were observed. But at the same time, several stations close to the main vent had to be abandoned, since they were already covered by mud. The reference station, which firstly located about 4 km south of the extrusion centre, was also moved about 12 km away to the north of mud volcano area. The results obtained from 9 GPS campaigns, as summarised in Table 3, show that ground displacements around the mud volcano area have both temporal and spatial domains, either in their rates or directions. However in general, the vertical displacements are in the form of subsidence and the horizontal displacements are toward the area about 300–400 m northnorthwest of the main vent of mud extrusion. The results in Table 3 also show that since mid-October 2006 the rates of displacements outside of a 1 km radius from the main vent were slowing down. In addition to episodic GPS observations, from 22 September 2006, GPS continuous monitoring was also conducted on five to ten stations with various operational lengths. The two most important GPS continuous stations were RW01 and RW02, both located about 500 m from the main vent of mud extrusion, as shown in Fig. 7. RW01 station operated between 22 September 2006 and 23 January 2007, while RW02 station operated between 6 November and 9 December 2006.

599

Operation of the two stations had to be terminated due to unsafe working environment caused by the continuing mud extrusion. The mud now covers the two locations. The subsidence observed by these two GPS continuous stations is shown in Fig. 8. The observed subsidence rate of RW02 is much higher than that of RW01, namely about 3.8 cm/day compared to 1.8 cm/day (see Fig. 8). The horizontal displacements of the two stations are about 0.6 cm/day (RW01) and 1.0 cm/day (RW02), with the directions toward the deepest cone of subsidence indicated in Fig. 7. The results also show that about 7–8 months after the first mud extrusion, the subsidence around the main vent area exhibits a linear trend without showing any sign of slowing down.

4 Surface Representation of Displacements The horizontal and vertical ground displacements as detected by GPS observations should theoretically have a surface representation. Therefore a field survey in and around the mud volcano area was also conducted in this study, in order to check the surface representation of the ground displacement phenomenon. Several months after the first mud extrusion, the ground displacements had caused cracking in the walls and floors of houses located even about 1–2 km from the main vent. The cracking mainly occurred on the northern and eastern sides of the main vent of mud extrusion. The ground displacements also affected a concrete toll bridge in the area. Because of cracking and deformation, the bridge has had to be dismantled for safety reasons. The explosion of the underground gas pipeline on 24 November 2006 was partly caused by the on-going ground subsidence in the area, in combination with ground settlement caused by dyke construction above it. The explosion location is inside the deepest cone of subsidence (see Fig. 7). Subsidence in and around the mud extrusion area is, in combination with the increase in the mud extrusion rate, also responsible for increasing the mud coverage area as previously shown in Table 1.

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Fig. 6 GPS derived displacements, July–August 2006

PBRK Vertical and Horizontal Displacements (cm) June - July 2006

5.0 TOLL

14.3 5.1

V

7.5

H 5 Cm

7.2 RIG1

2.6

SIRN

6.3

6.7 PSKO

–29.2 JTRJ

2.9

500 m

–3.7

–5.7

PBRK Vertical and Horizontal Displacements (cm) July - August 2006

27.6 TOLL

60.8

14.3

–44.7

V

H 10 Cm

10.5 –44.3

SIRN

RIG3

500 m 15.2 JTRJ

–91.3 –105.3

–19.7

On 27 September 2006, the ground displacements had also caused a dextral (right lateral) movement of a railroad located on the western side of the mud extrusion, as shown in Fig. 9. Considering this dextral movement of the railroad and the existence of subsidence and uplift of some GPS points (see Table 3), especially in the western side of mud extrusion, it can be hypothesised that the mud volcano of Sidoarjo has reactivated the geological structure in the area.

5 Concluding Remarks GPS results show that the surface displacements in the mud volcano area of Sidoarjo have both horizontal and vertical components in the time period studied, with the following characteristics:

r

Surface displacements have both spatial and temporal variations.

GPS-Based Monitoring of Surface Displacements Table 3 Summary of GPS-derived displacements

601

Period

Vertical displacement

Horizontal displacement

June–July 2006

Rate < 1.1 cm/day; subsidence (all stations) Rate < 3.6 cm/day; subsidence (all stations) Rate < 2.1 cm/day; subsidence and uplift Rate < 2.4 cm/day; mostly subsidence Rate < 0.3 cm/day; subsidence (all stations) Rate < 0.3 cm/day; mostly subsidence Rate < 0.2 cm/day; subsidence and uplift Rate < 0.2 cm/day; subsidence and uplift

Rate < 0.3 cm/day; directions: all to north-west of the main vent Rate < 0.9 cm/day; directions: all to north-west of the main vent Rate < 0.7 cm/day; directions: mostly to north-west of the main vent Rate < 0.9 cm/day; directions: mostly to north-west of the main vent Rate < 0.3 cm/day; directions: all to north-west of the main vent Rate < 0.3 cm/day; directions: some to north-west of the main vent Rate < 0.1 cm/day; directions: some to north-west of the main vent Rate < 0.1 cm/day; directions: no specific pattern of direction

July–Aug 2006 Aug–Sep 2006 Sep–Oct 2006 Oct 2006–Feb 2007 Feb–March 2007 March–May 2007 May–June 2007

Fig. 7 Location of RW01 and RW02 stations

225.8cm cm 225.8

GPS Continuous Result RW01 Station 22 Sep 2006 - 23 Jan 2007

Fig. 9 Dextral movement of railroad at km+39.2 on 27 September 2006

Subsidence rate : 1.8 cm/day 122.4 days days 122.4

124 cm

GPS Continuous Result RW02 Station 06 Nov - 09 Dec. 2006

r r

Subsidence rate : 3.8 cm/day 33 33 days days

Fig. 8 Subsidence rates of RW01 and RW02 stations

r

Rates of horizontal displacements are about 0.5– 2 cm/day, while vertical displacements are about 1– 4 cm/day, with the rate increasing towards the extrusion centre. Vertical displacements are dominated by subsidence, although some uplift has occurred. The affected area of displacement tends to increase with time, and a year after the mud extrusion, the affected area has expanded to about 3–4 km around the extrusion centre.

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Although GPS surveys are able to provide precise and accurate ground displacement vectors, in order to have better and more detailed information on ground displacement characteristics around the mud volcano area of Sidoarjo, the GPS survey method should be integrated with other monitoring techniques such as levelling, tiltmeter measurements and InSAR (Interferometric Synthetic Aperture Radar) (Massonnet and Feigl, 1998). With more available data and information, more reliable deformation and mechanism modelling can be performed. Fortunately, ERSDAC (Earth Remote Sensing Data Analysis Center) of Japan conducted a ground deformation study of the Sidoarjo mud volcano area using the InSAR technique based on ALOS (Advanced Land Observing Satellite) PALSAR data (JAXA, 2007). The maximum subsidence rates obtained by this InSAR study (see Table 4) are in agreement with those obtained from our GPS observations, namely around 1–4 cm/day. The InSAR results, however, show that the deformed area caused by the mud volcano of Sidoarjo up to February 2007 is larger than that covered by the GPS network, as shown in Figure 10. Besides the subsidence area around the mud volcano; the InSAR results also show that there are also other subsidence and uplift regions in the northeast and northwest of the mud volcano area. It should also be noted from Fig. 10 that InSAR is quite weak in estimating the deformation of the area close to the main vent, since most of the area is covered by mud. In this case, GPS stations that can still be located on certain dry ground inside the mud-covered area, can provide the deformation information. The present coverage of the GPS network extends to only about a 4 km radius from the main vent, which is represented by the middle ellipse of subsidence in Fig. 10. In the near future more GPS points will be added in two other deformed areas indicated by this InSAR result.

Table 4 InSAR-derived subsidence (Deguchi et al., 2007)

Period

Max. Interval Maximum subsidence rate (day) subsidence (m) (cm/day)

19 May–4 Oct 06 138 4 Oct–19 Nov 06 46 19 Nov 06–4 Jan 07 46

1.26 1.58 0.96

0.9 3.4 2.0

Besides trying to obtain a full deformation picture related to this mud volcano phenomenon by integrating GPS and InSAR results, further research will try to answer a few important questions related to displacements caused by the mud volcano of Sidoarjo, namely: 1. What is the real spectrum of horizontal and vertical displacements in both the spatial and temporal domain? 2. What is the best (spatial and temporal) model for these displacements, and how to integrate geodetic and geological data in modelling? 3. What are the future impacts of these ground displacements on infrastructure around the area?

References Beutler, G., H. Bock, E. Brockmann, R. Dach, P. Fridez, W. Gurtner, U. Hugentobler, D. Ineichen, J. Johnson, M. Meindl, L. Mervant, M. Rothacher, S. Schaer, T. Springer and R. Weber (2001). Bernese GPS Software Version 4.2. U. Hugentobler, S. Schaer, P. Fridez (Eds.), Astronomical Institute, University of Berne, 515 pp. Davies, R.J, R.E. Swarbrick, R.J. Evans and M. Huuse (2007). Birth of a mud volcano: East Java, 29 May 2006. GSA Today, February 17 (2). Deguchi, T. (2007). Monitoring of Land Deformation by Interferometric SAR Using ALOS/PALSAR Data: Application for Mud Volcano in Sidoarjo, East Java, Powerpoint presentation file, 31 July, ERSDAC (Earth Remote Sensing Data Analysis Center), Japan. Deguchi, T., Y. Maruyama, C. Kobayashi (2007). Monitoring of Deformation Caused by the Development of Oil and Gas Field Using PALSAR and ASTER Data, Powerpoint presentation file, 14 January, ERSDAC (Earth Remote Sensing Data Analysis Center), Japan. JAXA (2007). Website of Japan Aerospace Exploration Agency (JAXA), page on Advance Land Observing Satellite(ALOS), Address http://www.eorc.jaxa.jp/ALOS/, Accessed date: 4 June 2007. Kompas (2007), Indonesian National Newspaper, Saturday, 26 May 2007, pp. 33–40. Massonnet, D. and K.L. Feigl (1998). Radar interferometry and its application to changes in the Earth’s surface. Reviews of Geophysics, Vol. 36, No. 4, November, pp. 441–500. Matthews, S.J. and P.J.E. Bransden (1995). Late cretaceous and cenozoic tectono-stratigraphic development of the East Java Sea Basin, Indonesia, Marine and Petroleum Geology, 12 (5): 499–510. Mazzini, A., H. Svensen, G.G. Akhmanov, G. Aloisi, S. Planke, A. Malthe-Sørenssen, B. Istadi (2007). Triggering and dynamic evolution of the LUSI mud volcano, Indonesia. Earth and Planetary Science Letters, 261 (2007), pp. 375–388.

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Fig. 10 InSAR-derived displacements, between 19 May 2006 and 19 February 2007; after Deguchi (2007)

Normile, D. (2006). GEOLOGY: Mud eruption threatens villagers in Java, Science, 313 (5795): 1865, 29 September. Stewart, S.A. and R.J. Davies (2006). Structure and emplacement of mud volcano systems in the South Caspian Basin, AAPG Bulletin, 90 (5): 771–786. May, DOI:10.1306/11220505045.

U.S. Geological Survey (2006). http://earthquake.usgs.gov/ eqcenter/eqinthenews/2006/usneb6/. van Noorden, R. (2006). Mud Volcano Floods Java, available at http://www.nature.com/news/2006/060828/full/news0608281.html, Retrieved on 2006-10-18.

Differential and Precise Point Positioning in the South American Region with lonosphere Maps S.M. Alves Costa, Alberto Luis da Silva, Newton Jose´ de Moura Jr, Mauricio Alfredo Gende and Claudio Antonio Brunini

Abstract This paper presents two strategies for improving GPS coordinate solutions using regional ionosphere maps in South America, a region susceptible to strong ionospheric activity. Since 2005, the La Plata Ionospheric Model (LPIM) has been used to routinely generate Vertical Total Electron Content (VTEC) maps at the La Plata Processing Center (CPLAT) using about 50 GPS stations in South America. VTEC is modelled using a linear combination of undifferenced GPS carrier phase observations in modified dip latitude or modip: tan μ = √

I cos ϕ

Where μ is moddip latitude, I the magnetic dip and ϕ the geographic latitude. A particular approach is used to solve the hardware related biases. S.M. Alves Costa Coordination of Geodesy, Brazilian Institute of Geography and Statistics – IBGE, Av. Brasil 15671, CEP: 21241-051, Rio de Janeiro, RJ, Brazil, e-mail: [email protected] Alberto Luis da Silva Coordination of Geodesy, Brazilian Institute of Geography and Statistics – IBGE, Av. Brasil 15671, CEP: 21241-051, Rio de Janeiro, RJ, Brazil, e-mail: [email protected] Newton Jos´e de Moura Jr Coordination of Geodesy, Brazilian Institute of Geography and Statistics – IBGE, Av. Brasil 15671, CEP: 21241-051, Rio de Janeiro, RJ, Brazil, e-mail: [email protected] Mauricio Alfredo Gende Facultad de Ciencias Astron´omicas y Geof´ısicas Universidad Nacional de La Plata, Paseo Del Bosque S/N 1900, La Plata, Argentina, e-mail: [email protected] Claudio Antonio Brunini Facultad de Ciencias Astron´omicas y Geof´ısicas Universidad Nacional de La Plata, Paseo Del Bosque S/N 1900, La Plata, Argentina, e-mail: [email protected]

Since 2003, the Instituto Brasileiro de Geografia e Estatistica (IBGE) has collaborated in order to maintain the SIRGAS Reference Frame in South America, delivering weekly coordinate solutions using the same set of permanent stations that the CPLAT uses for computing ionosphere maps. This work examines two possible benefits of VTEC maps in GPS processing, namely, solving ambiguities in differential solutions and mitigating ionospheric biases in undifferenced single-frequency solutions. In differential processing, ionosphere maps are applied in the ambiguity resolution step when using the Bernese GPS processing software. For undifference processing, ionosphere maps are input to mitigate ionospheric biases when using the Precise Point Positioning (PPP) software developed by the Geodetic Survey Division (GSD), Natural Resources Canada (NRCan). In order to validate the authors’ proposal, data from four different periods in 2006 from a network of GPS stations near the ionospheric anomaly region in the South Atlantic and equatorial regions were tested with and without VTEC regional ionosphere maps. Keywords Regional network · Ionosphere maps · PPP

1 Introduction For more than a decade the SIRGAS project (Geodetic Reference System for the Americas) has been working to generate an optimal reference frame based on satellite observations and to establish a geodetic reference network. Presently more than 70% of South American countries have adopted SIRGAS as their official reference

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frame and are establishing permanent GPS stations in their territories. In the latest meeting of the SIRGAS Working Group I – WG I (Reference Frame), it was decided to establish five pilot processing centres: Instituto Nacional de Estad´ıstica Geograf´ıa e Inform´atica – INEGI, IBGE, Instituto Geogr´afico Agust´ın Codazzi – IGAC, Universidad Nacional de La Plata – UNLP and Instituto Geogr´afico Militar de Argentina – IGMA (SIRGAS 2007). At the same time a “call for participation” for a pilot experiment devoted to establishing a regular service to compute and deliver South American Regional Ionosphere Maps (SA-RIMs) of the Vertical Total Electron Content (VTEC) based on (but not exclusively) SIRGAS observations, was issued. This call encouraged research to improve the performance of the currently available GPS-based VTEC models in the South American region, where the presence of the Equatorial and South Atlantic anomalies result in an extremely complex VTEC pattern. Another initiative of the SIRGAS project has been to provide an online service for GPS data processing using the Precise Point Positioning (PPP) approach. The Geodetic Survey Division of Natural Resources of Canada has kindly provided IBGE their PPP software (CSRS-PPP) in the scope of the National Geospatial Framework Project. In this paper the authors explore the possibility of using SA-RIMs to improve the accuracy of PPP coordinate solutions. This paper is organised as follows. Section 2 describes the data set used in this study and gives a characterization of the ionosphere maps used, and of the La Plata Ionospheric Model (LPIM). A brief overview of the PPP software follows. Section 3 presents the results obtained by PPP and differential processing. Section 4 gives an outlook to future work.

Table 1 Receiver/antenna equipment used in the experiment

2 Materials and Methods The five pilot processing centres of SIRGAS have the task to generate weekly GPS solutions since week 1400. The individual results of these centres are compared and combined. After this experimental period, the idea is to collaborate with IGS Regional Network Associate Analysis Center (IGS RNAAC SIR) in order to generate the final solutions for the South and Central American permanent network. Data from the Regional GPS network on the South American continent were used in the experiments. Daily/weekly solutions from this network are produced by La Plata University and IBGE, as the pilot Processing Centre of SIRGAS. Data from four days (9, 120, 177 and 258) of 2006 were selected to generate South American Regional Ionosphere Maps – SA-RIM (Brunini et al. 2007). 50 stations in South America were selected to produce SA-RIM, while for the IGS maps 13 South American stations were used. For the analysis of results considering these two ionosphere maps (SA-RIM and IGS-GIM), 9 stations with a good geographical distribution in South America and with different receiver/antenna equipment (see Table 1) were used.

2.1 IGS Global Ionosphere Maps Since the creation of the IGS Ionosphere Working Group, the results from five analysis centres have been combined to provide the community with twodimensional world-wide grids of VTEC (Feltens and Schaer 1998). These final IGS Global Ionosphere Maps (IGS-GIMs) result from the weighted average of the individual GIMs computed by each analysis centre. The main characteristics of the IGS GIMs are:

Station

Receiver

Antenna

BRFT CONZ

LEICA GRX1200PRO TPS E GGD (177 e 258) JPS LEGACY (09 e 120) TRIMBLE 4000SSI AOA BENCHMARK ACT ASHTECH Z-FX TRIMBLE NETRS TRIMBLE 4000SSI TRIMBLE 4000SSI ASHTECH Z-FX

LEIAT504 TPSCR3 GGD

NONE CONE

TRM29659.00 AOAD/M T ASH700700.B TRM41249.00 TRM29659.00 TRM29659.00 ASH700700.B

NONE NONE NONE NONE NONE NONE NONE

CUIB LPGS MCLA NAUS PARA SMAR VARG

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Fig. 1 Permanent stations in South America

– use of geomagnetic latitude for the interpolation process, – the distribution of the VTEC over a single layer shell at a height of approximately 450 km is represented by a spherical harmonic expansion up to a maximum degree and order of 15, and every day is split into 12 periods of 2 h and a full set of expansion coefficients is adjusted for every period. In order to achieve a homogenous world-wide coverage, data from about 300 stations are used.

2.2 La Plata Ionospheric Model The SA-RIMs are computed using the La Plata Ionospheric Model (LPIM) (Brunini et al. 2004). The main characteristics of LPIM are the slant TEC (STEC)

calibration procedure and the use of the modip latitude for VTEC interpolation (Rawer 1984). LPIM implements a de-biasing procedure that improves data calibration in relation to other procedures commonly used for such purposes. After calibration, slant TEC measurements are converted to vertical TEC and mapped using local time and modip latitude. The use of modip latitude smoothes the spatial variability of VTEC, especially in the South American low latitude region and, hence, allows for a better VTEC interpolation. LPIM implements the Regional Ionosphere Maps (RIMs) computation in three stages: (i) pre-processing; (ii) calibration; and (iii) interpolation. The main task performed by the pre-processing step is the computation of the ionospheric observable (Iob ) from dual-frequency carrier-phase GPS observations. This step relies on the fact that the range-delay in GPS measurements induced by the ionosphere is

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proportional to the STEC along the satellite-receiver line-of-sight and inversely proportional to the square of the signal frequency. Hence, subtraction of simultaneous observations at different frequencies leads to an observable in which all frequency-independent effects disappear, while the ionospheric and any other frequency-dependent effects remain present: L 1,ar c = sT EC + B R + B S + Car c + ε L ,

the network. To accomplish this task the equivalent VTEC is approximated by a bilinear expansion dependent on the geographic longitude, λ I , and the modip latitude, μ I , of the pierce point: νTEC ≡ a(t) + b(t).(λ1 − λ0 ). cos(μ1 ) + c(t).(μ1 − μ0 )

(4)

(1)

where t is the Universal Time of the observation and λ0 and μ0 are the geographic longitude and the modip where L 1,ar c is the carrier-phase Iob –the sub-index ar c latitude of the observing receiver respectively. The derefers to every continuous arc of carrier-phase obser- pendence on time of the expansion coefficients is apvations, which is defined as a group of consecutive proximated with ladder functions: observations along which carrier-phase ambiguities do not change-; B R and B S are the so-called satellite and α(t) = αi , for ti ≤ t < ti + δt, receiver inter-frequency biases (IFB) for carrier-phase i = 1, 2, . . . , (5) observations; Car c is the bias produced by carrierphase ambiguities in the Iob ; and ε L is the effect of where α represents any one of the coefficients a, b or noise and multipath. All terms of Eq. (1) are expressed c; and δt is the interval of validity of every planar apin Total Electron Content Units (TECU), 1 TECU be- proximation (5m in the case of LPIM). ing equivalent to 1016 m−2 . Finally, the observations are calibrated and the Dual-frequency P-code observations lead to an anal- equivalent VTEC are estimated from eq. (3): ogous Iob : PI = sTEC + b R + b S + ε P ,

(2)

where the meaning of the terms is similar to Eq. (1) with the following distinctions: different satellite and receiver IFBs (b R and b S instead of B R and B S ); there is no ambiguity term; and the effect of noise and multipath, ε P , is about 100 times greater. The calibration module of LPIM relies on the thinshell and mapping function approximations (Davies and Hartmann 1997). Based on these approximations, Eq. (1) is written in the following way: L I,ar c = sec(z ) · νT EC + γar c + ε L ,

(3)

where sec(z ) is the mapping function, z being the zenith distance of the satellite as seen from the point where the signal intersects the thin-shell (the so-called pierce point) that, in the case of LPIM, is located 450 km above the Earth’s surface; νT EC ≡ cos(z ) · sT EC, is the equivalent VTEC at the pierce point; and γar c ≡ B R + B S + Car c , is a calibration constant that encompasses, all together, receiver and satellite IFBs and the ambiguity term. The calibration, i.e., the estimation of γar c for every continuous arc, is performed independently for every observing receiver in

ˆ ≡ νTEC + νT EC

εL L I,ar c − γˆar c = . sec(z ) sec(z )

(6)

In order to ensure calibration of complete arcs, LPIM processes, in any run, 36h of data from −6h to ˆ values for the interval 30h UT, but retains the νT EC from 0h to 24h UT. The calibration procedure relies on the estimation of an independent calibration constant for every observed arc and every observing receiver (Eq. (3)) that encompasses altogether the ambiguity term and the receiver and satellite DCBs. This procedure avoids the leveling errors due to P-code multipath that depends on the receiver/antenna configuration. The interpolation problem arises from the need to create a regular data grid from scattered observations in order to map the VTEC in the desired region. It starts by adjusting a set of coefficients of a spherical harmonic expansion up to degree and order 15, every 1h UT to the equivalent VTEC computed from the observations of all receivers of the network. A well-known rule for interpolation states that the smoother the function to be interpolated, the better the accuracy of the interpolated values. This simple rule helps to explain why the use of the modip latitude improves the accuracy of the VTEC interpolation in

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Fig. 2 Snapshots of La Plata ionospheric maps for four epochs in the year 2006

the South American region compared to the results obtained with the more commonly used geomagnetic latitude (Azpilicueta et al. 2005). After the coefficients are estimated they are used to compute a regular grid of VTEC every 1 h. Due to the presence of the South Atlantic Geomagnetic Anomaly, the largest deviation between modip and geomagnetic isolines occurs at low-latitudes in the South American region and the surrounding oceanic areas and VTEC interpolation should perform better if the modip latitude (instead of the geomagnetic latitude) is used. Since July 2005, the GESA laboratory belonging to the Facultad de Ciencias Astron´omicas y Geof´ısicas of the Universidad Nacional de La Plata (http://cplat.fcaglp.unlp.edu.ar) computes hourly SARIMs (VTEC maps) in the framework of the SIRGAS pilot experiment. Figure 2 shows snapshots of RIM from 19 h UT of days 9, 120, 177 and 258 of 2006. It represents the VTEC behavior for the region in study.

orbits, clocks, and ionosphere maps (only in code processing), and is equivalent to phase-differential GPS processing. The data used in the PPP software were only L1 (code-only) observations. The software requires an input ionosphere map to correct for the first-order ionospheric delays. Data from each epoch of 2006 was used to generate solutions with RIM from La Plata and GIM from IGS.

3 Results The SA-RIM and IGS-GIM were used in different executions of two software packages: the CSRS-PPP and the Bernese GPS Software version 5.0, developed by the University of Bern, Switzerland (Dach et al. 2007).

3.1 Results with CSRS PPP Software

PPP processing using single-frequency receivers requires mandatory external ionospheric information. 2.3 The PPP Software IGS-GIMs or SA-RIM are useful for this processing strategy and have shown similar performances (Fig. 3). The CSRS-Precise Point Positioning (CSRS-PPP) In the space domain (direct comparison of the grids) software used in this work was developed by Natural VTEC mean differences between these two models Resources Canada. It can process GPS observations are lower than 6 TECUs for latitudes lower than −30◦ from single- or dual-frequency receivers, operating in but increases in the anomaly region, particularly near static or kinematic mode (T´etreault et al. 2005). It can the Amazon where fewer data are available, up to 15 achieve accurate positioning results with data from TECUs (Gende et al. 2007). Three types of analyses only one station. The precision of results obtainable were carried out to compare the results obtained with by PPP is based on the precise IGS products for SA-RIM and IGS-GIM in the position domain.

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Fig. 3 Plots of results at station BRFT day 009 with model LPIM and GIM for a period of 24 h

(a) As shown in Fig. 3, the analysis of the coordinate accuracy using 24 h of data does not reveal significant differences between IGS-GIM and SA RIM derived ionosphere corrections. In most cases, the standard deviation of the horizontal component reached 20 cm after the first 2 h decreasing to 5 cm after 24 h. The standard deviation of the vertical component reached about 50 cm after 2 h decreasing to 15 cm after 24 h. (b) Figure 4 shows the analysis of the coordinate accuracy in sections of 2 h demonstrating again the equivalence between the IGS GIM and the SARIMs. No obvious time dependent signals were detected over a period of 24 h. The standard deviation of the horizontal component was between 20 and 30 cm and the standard deviation of the vertical component, was between 50 and 60 cm for all 2 h solutions. Two stations showed atypical behavior in that the standard deviation of latitude was about 10% higher than the standard deviations for longitude (La Plata and Concepci´on). The standard deviation for the vertical component reaches 1 m for station La Plata on day 177. (c) As shown in Fig. 5, the analysis of the differences between IGS weekly solutions and the authors’

PPP solutions obtained using SA-RIM or IGSGIM seems to indicate a better agreement with the IGS weekly solution for the PPP solutions using IGS GIMS instead of SA-RIMs. In some cases, height differences to the IGS week solution between both kinds of PPP solutions remain at ±1 m after 24 h. Furthermore, the height difference compared to the IGS weekly solution has a different sign when using IGS GIM or SA-RIM. In general, after 24 h of observations, the differences remain at the 20 cm level for latitude and longitude.

3.2 Results with the Bernese Software The interest in using the Bernese Software is to compare the results of the ambiguity resolution step using ionosphere maps. The complete South American network was used and two solutions were produced: (a) Ambiguity Resolution using the Quasi Ionosphere Free (QIF) strategy without ionosphere maps and introducing the resolved ambiguities in the final daily solution.

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Fig. 4 Plots of results at station CONZ day 009 with model LPIM and GIM, for periods of 2 h

Table 2 shows the number of ambiguities before and (b) Ambiguity resolution using QIF strategy and GIM from CODE (Center for Orbit Determination in after resolution. Using GIMs, slightly more ambiguiEurope) and introducing the resolved ambiguities ties could be resolved (1%) than without GIMs, which results in a small improvement on the baselines RMS. in the final daily solution. The results of the tests performed, including for the 59 stations in South America, following the options/strategies described, are presented in Tables 2 and 3.

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Fig. 5 Comparison between IGS week solution and LPIM and GIM solutions at stations BRFT, day 009

4 Discussion and Outlook

This fact is related to the fact that both models use grids that smooth the rapid spatial and time variations. Coordinates present RMS of about 25 cm for the Ionosphere maps (SA-RIM and IGS-GIM) are mandahorizontal component and 50 cm for the vertical one. tory for PPP processing using single-frequency This indicates that resorting to grids for correction receivers. Although SA-RIM has twice a time resoluis not the best solution when accuracy below a few tion of IGS-GIM and uses almost twice of the data, decimeters is desired. both maps demonstrate similar levels of performance.

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Table 2 Number of ambiguities before and after resolution, with and without CODE GIM

Table 3 RMS in 3 baselines of the final solution

QIF (without ionosphere maps)

QIF (with GIM – CODE maps)

Day

Total no of ambiguities

Total no of ambiguities after resolution

Total no of ambiguities after resolution

009 120 177 258

4454 5114 4652 4072

1728 (61.2%) 1582 (69.1%) 1536 (67.0%) 1490 (63.4%)

1628 (63.4%) 1514 (70.4%) 1480 (68.2%) 1470 (63.9%)

Day

Baseline

Length (km)

RMS (no ambiguity resolution) (m)

RMS (QIF)(m)

RMS (QIF with GIM – IGS)(m)

009

CONZ-SANT NAUS-SRNW UBER-VARG CONZ-SANT NAUS-SRNW UBER-VARG CONZ-SANT NAUS-SRNW UBER-VARG CONZ-SANT NAUS-SRNW UBER-VARG

463 1047 421 463 1047 421 463 1047 421 463 1047 421

0.0004 0.0008 0.0006 0.0003 0.0006 0.0007 0.0002 0.0005 0.0006 0.0003 0.0008 0.0008

0.0003 0.0006 0.0004 0.0002 0.0004 0.0004 0.0002 0.0004 0.0003 0.0002 0.0006 0.0005

0.0003 0.0004 0.0004 0.0002 0.0003 0.0003 0.0002 0.0003 0.0003 0.0002 0.0004 0.0005

120

177

258

An alternative solution has been tested successfully Brunini C., Meza A., Gende M. et al (2007). South American regional ionospheric maps computed by GESA: a pilot service for differential processing and as a further work must in the framework of the SIRGAS project. Submitted to Adv. be adapted to PPP (Mohino et al. 2007). Space Res. In order to implement a PPP service for the SIR- Dach, R., Hugentobler U., Fridez P., Meindl M. (2007). Bernese GAS community, transformation parameters between GPS Software Version 5.0, Astronomical Institute University SIRGAS2000 and ITRF must also be estimated. of Berne, Berne. Acknowledgments The work described here has been carried out under the scope of the Projeto de Mudanc¸a de Referencial Geod´esico and the National Geospatial Framework Project http://www.pign.org, funded by the Canadian International Development Agency (CIDA). The authors would like to thank NRCan GSD for providing PPP software to IBGE.

References Azpilicueta, F., Brunini, C., Radicella, S.M. (2005) Global ionospheric maps from GPS observations using modip latitude. Adv. Space Res. 36, 552–561. Brunini, C., Van Zele M.A., Meza A. et al (2004) Quiet and perturbed ionospheric representation according to the electron content from GPS signals. J. Geophys. Research, 108, doi: A2 10.1029/2002JA009346.

Davies, K., Hartmann G.K. (1997) Studying the ionosphere with the Global Positioning System. Radio Science 32(4): doi: 10.1029/97RS00451. issn: 0048-660. Feltens, J., Schaer S. (1998). IGS Products for the Ionosphere. Proc. 1998 IGS Analysis Centres Workshop. Darmstadt, 285–297. Gende M., Brunini C., Meza A. (2007) Inter comparison and validations of ionospheric VTEC maps for the South American Region. IUGG XXIV General Assembly. Perugia, Italy. Mohino, E., Gende M., Brunini C. (2007). “A method for ionospheric bias mitigation in differential GPS positioning using multiple reference stations.” J. Surv. Eng. 133(1), 1–5. Rawer, K. (1984). Encyclopedia of Physics, Geophysics III, Part VII, Springer-Verlag, 389–391. T´etreault P., Kouba J., H´eroux P., Legree P (2005) CSRS-PPP: An Internet Service for GPS – User Access to The Canadian Spatial Reference Frame, Geomatica Magazine, 59(1), 17–28.

Current State of Precise Point Positioning and Future Prospects and Limitations S. Bisnath and Y. Gao

Abstract The Precise Point Positioning Working Group within the Next Generation RTK SubCommission of IAG Commission 4 has been involved with Precise Point Positioning (PPP) developments for the past few years. The information presented here summarizes the Working Group’s findings concerning the state of PPP technology, and discusses the probable near-term future potential and limitations of the technique. The broad question of the place of PPP within the future spectrum of space geodetic measurement techniques is addressed by investigating specific aspects of the method.

The paper begins with a review of the current state of PPP, covering performance and usage. Current technical limitations of the approach are then discussed, including solution convergence period, accuracy and integrity of solutions. The next section considers potential improvements upon the current approach, in terms of integer ambiguity resolution, integration with other data, e.g., from RTK solutions and inertial navigation systems (INSs), and the use of other external modelling data, e.g., atmospheric refraction models. Equally important are PPP infrastructure challenges, including the availability of precise satellite orbits and clock offsets, precise Keywords GPS · Precise Point Positioning · orbit and clock prediction, real-time dissemination of predicted obits and clocks, and reference frame Navigation realizations. Given the great changes due to GPS Modernization and the development of other Global Navigation Satellite Systems (GNSSs), we would be 1 Introduction remised not to speculate on the potential significant positive impacts of these added signals on future PPP The main goal of this paper is to describe the current performance. Finally we end with a discussion of the performance of what has become known as the Precise potential of PPP to perform in a similar manner as Point Positioning (PPP) technique, and to discuss the RTK. future potential of the technique, along with its technical limitations. The basic methodology and algorithms of the approach are assumed known, and hence will not be discussed. Interested readers can review, e.g., Zum- 2 Current State of PPP berge et al. (1997) and Kouba and H´eroux (2001). This section is designed to summarize current PPP performance using a number of metrics, and to set S. Bisnath the technique’s impact within the context of the wider Department of Earth and Space Science and Engineering, York field of positioning and navigation. What is meant by University, 4700 Keele St., Toronto, Ontario, Canada, M3J 1P3 PPP is the state space solution to the processing of Y. Gao pseudorange and carrier phase measurements from Department of Geomatics Engineering, University of Calgary, a single GNSS receiver, utilizing satellite constella2500 University Drive N.W., Calgary, Alberta, Canada, T2N 1N4 tion precise orbits and clock offsets determined by M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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separate means. Typically, a dual-frequency GNSS receiver is utilized with dual-frequency code and phase measurements linearly combined to remove the first-order effect of ionospheric refraction and the realvalued carrier phase ambiguity terms estimated from the measurement model. The tropospheric refraction is also estimated along with the position and ambiguity parameters from the measurements. PPP using a single-frequency GNSS receiver has also been investigated with great promise for certain applications which however will not be further exploited in this paper (see, e.g. Gao et al., 2006; Le and Tiberius 2006). To achieve the best position accuracy possible from PPP, effects such as carrier phase wind-up and transmitter antenna phase offset must be corrected using models. Residual terms such as receiver noise and multipath are generally ignored or handled minimally via stochastic means. Refer to, e.g., Kouba and H´eroux (2001) for a full parameterization discussion. A somewhat unique aspect of PPP is that it is an area of research being actively pursued by academia, government and industry, in concert and individually. As is typical, early development occurred in research settings for scientific goals. Governments, as service providers, have in some cases engaged in providing PPP services to the public, given the socioeconomic benefits. Industry has embraced and advanced the technology to better serve its clients. The results are: (a) rapid development and use of PPP in a variety of application areas, and (b) significant overlap between the three groups in terms of research and development, and service models. The latter point will be discussed further in the infrastructure section.

S. Bisnath and Y. Gao

real-time (see, e.g., Dixon, 2006; Bisnath et al., 2003; Muellerschoen et al., 2001; and Gao and Shen, 2002). The convergence period, namely the length of time required from a cold start to a decimetre-level positional solution, is typically about 30 min under normal conditions and will be significantly longer before the position solution can converge to the few centimetre-level, if at all (Gao and Shen, 2001; H´eroux et al., 2004). This period is determined by the measurement strength of the observables for a GPS-only solution, the geometry of the problem, and the redundancy available for the estimation problem. Initial solutions rely almost exclusively on noisy pseudorange measurements, the uncertainties of which are magnified via the ionosphere-free linear combination. The availability of solutions is usually high, given that application areas for this technique are open sky, continuously unobstructed environments. Otherwise PPP would typically not be used. Finally, the availability of integrity measures for the PPP solution is considered. Aside from filter covariance estimates, quantitative quality measures of the obtained results are limited. For example, knowledge of biases in correctors (e.g., precise transmitter orbit and clock products), the potential for biases in estimated coordinates, and measurement outliers are typically not considered (that is, rigorously specified and accounted for) in solutions.

2.2 Usage and Applications

PPP can be used for processing of static and kinematics data, both in real-time, if the dissemination mechanisms are in place to construct, transmit, receive and process precise satellite orbit and clock products, and 2.1 Performance Specifications in post-processing mode. The caveat for all such usage is that there needs to be uninterrupted GNSS sigThe standard metrics used here to describe the perfor- nal availability, as loss of tracking lock on a minimance of conventional PPP services are: accuracy, pre- mum number of satellites requires processing filter recision, convergence period, availability and integrity. initialization, resulting in tens of minutes of greater With PPP solutions showing very little in the way of than decimetre resolution positioning, until filter rebiases – typically a few centimetres for the most (Bis- convergence. This constraint severely limits the utility nath et al., 2003), there is very little difference between of PPP, in so far as it can only be robustly (i.e., successthe accuracy and precision metrics. In terms of north, fully) used in environments with continuous open sky east and up component accuracies at the 1σ level, coverage. One of the first uses of a prototype PPP approach PPP is able to provide few centimetre-level results in static mode and decimetre-level results in kinematic was for the post-processing of static geodetic data mode, both could be achieved in either post-mission or for, e.g., rapid processing of GNSS tracking station

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1 data, or crustal deformation monitoring (Zumberge 1/Jul/04 et al., 1997). Other scientific uses have included pre- 0.9 2/Jul/04 0.8 3/Jul/04 cise orbit determination of Low Earth Orbiters (e.g., 0.7 4/Jul/04 Bisnath and Langley, 2002). The main commercial 0.6 5/Jul/04 6/Jul/04 applications of PPP have been in agricultural industry 0.5 7/Jul/04 0.4 for precision farming (see, e.g., Dixon, 2006), marine 0.3 applications, for sensor positioning in support of 0.2 seafloor mapping and marine construction (see, e.g., 0.1 0 Bisnath et al., 2003, and Arroyo-Suarez et al., 2005), 0:00 0:15 0:30 0:45 1.00 1.15 1.30 1.45 2.00 2.15 2.30 2.45 3.00 and airborne mapping (e.g. Gao et al., 2005). Time from Session Start Further growth in current active areas is underway, Fig. 1 Convergence time of different days (H´eroux et al., 2005) and the technique is making inroads into other application areas such as atmosphere remote sensing, precise 1 ALBH(L3S) time transfer, land surveying, construction and military 0.9 NRC1(L3S) (e.g., Gao et al., 2004, and Dixon, 2006). Fundamen- 0.8 YELL(L3S) tally, wherever precise positioning and navigation is 0.7 STJO(L3S) required in isolated locations or expansive areas, and 0.6 reference station infrastructure is not available, or very 0.5 costly to temporarily erect, for, e.g., natural resource 0.4 mapping and retrieval, PPP is being strongly consid- 0.3 0.2 ered as a positioning and navigation solution.

0.1

0 0.00

3 Technical Limitations 3.1 Convergence Period

1.00

2.00 3.00 4.00 Time from Session Start

5.00

6.00

Fig. 2 Convergence time of different user locations (H´eroux et al., 2005)

show a high degree of day-to-day variability, despite the similar satellite geometry one would expect given Although the PPP approach presents definite advan- a common daily session start time. For example, the tages for many applications in terms of operational solution crosses the 10 cm threshold within 30 min on flexibility and cost-effectiveness, it requires a long ini- several days, but also takes more time to do so on other tialization time as phase ambiguities converge to con- days. Shown in Fig. 2 are the average weekly converstant values and the solution reaches its optimal preci- gence time series at four different IGS tracking stations sion, taking full advantage of the precise but ambigu- (separated by 1000 km or more and therefore subject to ous carrier phase observation. PPP convergence de- different satellite geometries). Significant differences pends on a number of factors such as the number and in positioning accuracy and convergence time exist, geometry of visible satellites, user environment and likely caused by varying satellite geometry and / or stadynamics, observation quality and sampling rate. As tion specific tracking conditions, such as multipath enthese different factors interplay, the period of time re- vironment. quired for the solution to reach a pre-defined precision level will vary. Shown in Fig. 1 are the convergence times with respect to different position accuracies from processing 3.2 Accuracy data at an IGS tracking station over seven consecutive days (H´eroux et al., 2005). The station data collected The primary factors that limit the accuracy of PPP are at a 30 s interval for each 24 h session was processed the limited precision of current precise orbit and clock using IGS precise orbits and 5 min clocks, and the po- products and the effects of unmodelled error sources. sition error was computed every 15 min. The results PPP is currently able to provide few centimetre-level

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S. Bisnath and Y. Gao Table 2 Accuracy statistics for kinematic positioning results after filter convergence (in cm) RMS BIAS STD

Latitude

Longitude

Height

2.8 −0.2 2.8

6.8 −1.5 6.7

4.9 −1.5 4.6

baseline, double-differenced, ambiguity-fixed position solutions were used as ground-truth and the positioning accuracy statistics are given in Table 2. The results indicate that it takes about 20–30 min for the positioning solution to converge to the decimetre-level. Fig. 3 Static Positioning Using IGS Dataset In addition to further improvement of the precise orbit and clock products, the ability to exploit the interesults in static mode and decimetre-level results in ger property of phase ambiguities can further improve kinematic mode. Shown in Fig. 3 are the positioning the obtainable position accuracy of PPP. Minor error results of a high-quality, static dataset from an IGS sta- sources including initial satellite and receiver phase bition (ALGO) using the JPL real-time orbit and clock ases would need to be estimated and removed (as they products. It can be seen that the coordinate estimates cannot be eliminated in undifferenced processing) in could converge to the centimetre-level within 30 min. the measurement model. After convergence, all position coordinate components are accurate at the sub-centimetre level and the positioning accuracy statistics are given in Table 1. Shown 3.3 Integrity in Fig. 4 are the positioning results of a kinematic dataset acquired from an airborne platform flying at an Integrity monitoring is an essential component of any altitude of approximately 250 m above the ground at 50 positioning/navigation system. Given that in PPP proknots. The precise orbit and clock correction are again cessing some parameters are estimated, while others from JPL’s real-time orbit and clock product. The short are eliminated via estimates derived from a separate Table 1 Accuracy statistics for static positioning results after filter convergence (in cm) RMS BIAS STD

Latitude

Longitude

Height

0.9 0.8 0.3

1.0 0.3 0.9

0.7 0.0 0.7

Fig. 4 Positioning with aircraft dataset

process, without multiple solutions (e.g., as in the case with network RTK), providing integrity information for PPP single receiver estimates is all that more important. An industrial example is that at least one service provider has clients who are willing to pay for two independent solutions: PPP and long-range RTK (with float ambiguities). The independent solutions are used to judge accuracy of the solutions. Obviously, post-fit residuals from a PPP epoch solutions can be analyzed to detect individual measurement outliers, or more significant problems. This should be standard practice. More complex examples of integrity monitoring exist in other GNSS applications and should be considered for PPP processing. Potentially, Receiver Anonymous Integrity Monitoring (RAIM)-type of screening can be implemented for PPP estimates. This would be a straightforward design, and would provide users with additional confidence in their PPP solutions, beyond covariance estimates and post-fit residuals. More elaborate, it may be possible to contemplate a Wide Area Augmentation System

Current State of Precise Point Positioning

(WAAS)-type of space-space grid error approach be used to evaluate PPP orbit and clock corrector products, which could be specified when generated from tracking stations and integrated into the PPP processing.

4 Potential for Improvement 4.1 Ambiguity Resolution Convergence time can be improved as demonstrated in Gao and Shen (2001), through stochastic model refinement and higher sampling rate. The improvement level however was found insignificant. Exploiting the integer property and the application of integer resolution techniques to PPP have the potential to reduce convergence time to several minutes or even several seconds. In double-differenced GPS processing, where integer ambiguity resolution has been widely utilized, the double-differenced ambiguity parameters are integers, which can render the position solution accuracy to the few centimetre-level or better after they have been fixed to their correct integer values. For PPP with undifferenced observations, the ambiguity parameters are not integers as they are corrupted by the initial fractional phase bias in the GNSS satellites and receivers. Significant progress is being made in understanding the characteristics and the estimation of the abovementioned initial phase biases (e.g., Ge et al., 2006; Wang and Gao, 2007). This research will shed light on the recovery of the integer property of the ambiguities.

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4.3 Integration with INS The integration of stand-alone and double-differenced GNSS and INS has been extensively investigated in the past as the coupling has benefits such as improved cycle slip detection, smoothed trajectory and increased reliability. An integrated PPP GPS/INS system has been developed by Zhang and Gao (2005) to support geo-referencing in airborne mapping, which offers similar performance to a differential GPS/INS system. An integration of PPP with INS can also reduce re-initialization time, since INS can supply accurate position and velocity information during short periods and subsequently reduce the position convergence time. This is particularly important for real-time kinematic applications, as frequent signal blockages are common in the field.

4.4 Ingestion of Precision Atmospheric Models

A tropospheric parameter unknown is usually estimated along with the position and ambiguity parameters from the measurements in dual-frequency GNSS receiver-based PPP, while several ionosphere parameters are estimated in single-frequency GNSS receiver based PPP (Gao et al., 2006). Ingestion of precision atmospheric models can reduce the total number of unknown parameters that need to be estimated from the measurement model, potentially remove the need for noise propagating linear combinations of observables, and potentially improve positioning performance (e.g., Dodd et al., 2006; Keshin et al., 2006). As a result, it can reduce the convergence time of 4.2 Integration with RTK PPP. The challenge is that range corrections from such measurement and physics-based atmospheric Integration of PPP with network RTK techniques may models have to be very accurate, e.g. better than a few lead to improved position accuracy and performance, centimetres. particularly reduction in convergence time. A concept of network RTK-based PPP was described in Wubbena et al. (2005). A hybrid system of optimal integration of 5 Infrastructure Issues PPP and RTK solutions has also been implemented into a global differential positioning system (Dixon, 2005). Infrastructure refers to the satellite orbit and clock inAs PPP can be an efficient alternative to RTK in certain formation (products) being generated and used in PPP applications, it is expected that more work will be car- parameter elimination schemes, and the information ried out to investigate the seamless integration of PPP and processes related to the collection, generation and and RTK methods. dissemination of these products.

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5.1 Availability of Precise Orbits and Clock project (IGS real-time, 2007). A number of other institutes are now providing predicted orbits and clocks for Offsets Precise orbit and clock products have improved significantly in recent years. And post-processed and predicted products are freely available over the Internet. As such, PPP processing can be performed by anyone who can develop the processing software. This comment is particularly applicable to post-processing of single receiver data sets. The phenomena which impact satellite orbit and clock estimates include transmitter antenna models, antenna phase centres, satellite radiation models and eclipse periods.

5.2 Precise Orbit and Clock Prediction As can be seen in Table 3, post-processed (final) orbit and clock IGS products are produced at quiet a high level of accuracy. However, improvements in IGS predicted products for real-time usage is desirable, in order to marginally improve PPP solutions. The greatest disparity between the final and predicted products is the 50 times worsening of the clock product coupled with a 3 times enlarging of the data rate. Comparable positioning accuracy, however, has been demonstrated using IGS final products and JPL real-time orbit and clock correction products by Gao and Chen (2005). Table 3 IGS ephemeris products (IGS products, 2007) Product

Accuracy

Interval

Latency

Final

Orbits Clocks

<5 cm <0.1 ns

15 min 5 min

∼13 days

Predicted

Orbits Clocks

∼10 cm ∼5 ns

15 min 15 min

Real-time

real-time PPP use. Dissemination of products in the form of corrections can be done in a variety of ways that can primarily be grouped into satellite-based or Internet-based. Satellite-based transmission is the usual choice of commercial service providers, as the correct signals can be received by an antenna built into the GPS receiver package (see Bisnath et al., 2003). Internet-based correction transmission can be made with much lower cost and therefore has been used as the model for academic prototypes, with great potential to be widely adopted in applications (see, e.g., NTRIP, 2007). The manner of development of PPP and PPP infrastructure leads to an interesting question of provider model: free versus paid real-time corrections. That is, an academic / government model of infrastructure support, or a commercial model of paid service provider.

5.4 Reference Frame Satellite orbit and clock products refer to a particular realization of ITRF. This realization depends on data from numerous GNSS, VLBI, and SLR stations distributed heterogeneously around the world, and the non-uniform weighting of the varying-length data records from each station. When a new version of ITRF is established and published, and data products redefined with respect to that frame, PPP user coordinates are affected (altered). The end-user must be aware of this link, and furthermore, that the coordinates generated by PPP are referred to ITRF and a coordinate transformation is necessary to bring the PPP solution into the “flavour” of coordinates the user requires.

5.3 Real-time Dissemination of Predicted Orbits and Clocks

6 Effects of GNSS Evolution

The production of orbit and clock information for realtime processing, hence prediction of quantities, is a major focus of current research efforts. The IGS, for example, has been studying the generation and dissemination of real-time data products for the past few years and has recently begun a real-time data products pilot

GPS-only PPP has its limitations, such as insufficient number of visible satellites due to signal blockages and insufficient reliability for ‘safety of life’ applications. An integration of GPS with other navigation systems such as the Russian GLONASS, European Galileo and Chinese COMPASS could provide many

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more observations and is expected to have a significant impact on position accuracy, reliability and convergence time of PPP. Shen and Gao (2006) have compared GPS-only and GPS/Galileo-based PPP and the simulation results demonstrate that combined system can reduce convergence time by half over GPS-only PPP. In Cai and Gao (2007) it was demonstrated that combined GPS/GLONASS PPP greatly improves positioning accuracy and reduces convergence time (see Figs. 5 and 6). These improvements are dependent on the enhanced level of satellite availability and geometry for position determination. Issues such as interoperability and compatibility however must be addressed for integration of data from hybrid navigation systems.

Given the previous discussions about the capabilities and potential of PPP, an obvious place to termination this paper is with a comparison of PPP and the industry standard RTK technique. Or more precisely, the question: Can PPP ever replace RTK? To address this overall issue, we pose a series of logically ordered questions. Some queries may be answered quite readily, while others can only be partially addressed.

7.1 Can PPP Algorithms, Data and Operation be Improved to the Point Where the Technique Obtains the Same Level of Performance as RTK? And if so, What Specific Improvements are Required for PPP to Perform like RTK?

3 East North Up

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Fig. 6 Simulated GPS/GLONASS PPP convergence period and accuracy performance

This result can be seen as a potential final objective of PPP algorithm research – and a recasting of the overall question. All of the measurement strength of the undifferenced PPP observables is used to estimate: (i) fewcentimetre-level position estimates, (ii) with a few seconds worth of measurements, (iii) without the need for a reference station. The utility of such a solution would be significant, and is the subject of the last question asked here. The third characteristic (of removing the need for reference stations) is the one that would make PPP so appealing as compared to RTK. The first characteristic (of attaining centimetre-level positioning accuracy) may be possible, given recent research results related to ambiguity resolution of undifferenced observables. The potential exists to isolate and estimate initial fractional phase biases in order to isolate true integer ambiguities, without over-parameterizing the processing model. The second characteristic (of attaining the desired accuracy with a few seconds of data) is perhaps the most challenging. Given the inherent weaker measurement models of PPP versus RTK (that is, less data in PPP), it will be difficult estimating the correct biases and ambiguities, aside from performing this resolution quickly. Meeting the initialization and re-initialization challenge will be the most difficult hurdle for the PPP

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technique to receive greater industrial acceptance for real-time applications.

S. Bisnath and Y. Gao

7.5 How Would Science, Industry and Society be Affected/Impacted by Such a Result?

7.2 Can this Objective be Reached in the Near Future? In a Cost-Effective Way? In a Practical Manner?

RTK-like PPP would positively impact all facets. Relieved of RTK baseline constraints, users would be able to perform few centimetre-level positioning almost anywhere. Though performance would be similar If the objective of parity between techniques can be in urban areas, where RTK networks are already estabreached, it will required significant further algorithm lished. That said, a two system solutions would enable development and perhaps more observables to reduce new applications. processing filter convergence period. This latter requirement would delay implementation of an RTK-like Acknowledgments The authors acknowledge financial supPPP processor. Significant infrastructure overhead may port from the Canadian Geomatics for Informed Decisions be encumbered in order to estimate receiver and satel- (GEOIDE) Network Centres of Excellence (NCE) and the lite code and phase biases, the added complexity of second author, the Natural Sciences and Engineering Research Council of Canada (NSERC). which could increase costs to service providers. If as some research suggests, few minute convergence periods are possible, we are hesitant to state that this limitation will be tolerated by many industries other than References the ones currently using PPP.

7.3 Do We Even Want PPP to Work Like RTK? Though it may seem like an odd question, given the context of this paper, it is a reasonable question if theoretically PPP can perform at the level of RTK. The two approaches have been developed independently, for different purposes. This fact makes them very useful as independent, mutual integrity checkers for some scenarios. Though the benefit obtained from the removal of the constraint for the need for reference stations with PPP has obliged researcher to seek out a potential solution.

7.4 If PPP can Work Like RTK, Would it Replace RTK/Network RTK/DGPS? It can be speculated that PPP would first be used as a complimentary solution to RTK in positioning and navigation work. But as the approach gained acceptance, it could replace RTK. The only caveat here is that a significant level of integrity would have to be issued along with accurate PPP solutions to gain industrial acceptance.

Arroyo-Suarez, E.N., J.L. Riley, G.F. Glang and D.L. Mabey (2005). “Evaluating a global differential GPS system for hydrographic surveying.” ION GPS 2005, 13–16 September, Long Beach, CA, pp. 1500–1506. Bisnath, S. and R.B. Langley (2002). High-precision platformindependent positioning with a single GPS receiver. Navigation, Vol. 29, No. 3, pp. 161–169. Bisnath, S., D. Wells and D. Dodd (2003). Evaluation of commercial carrier phase-based WADGPS services for marine applications. Proceedings of ION GPS/GNSS 2003, 9–12 September, Portland, Oregon, pp. 17–27. Cai, C. and Y. Gao (2007). Positioning model and accuracy analysis of Precise Point Positioning with GPS and GLONASS. Poster paper, Geoide Annual Meeting, 6–8 June, Halifax, Nova Scotia. Dixon, K. (2006). StarFire TM: A global SBAS for subdecimetre precise point positioning. Proceedings of ION GNSS 2006, Fort Worth, Texas, pp. 2286–2296. Dodd, D., S. Howden and S. Bisnath (2006). Implementation of Ionosphere and Troposphere Models for High-Precision GPS Positioning of a Buoy During Hurricane Katrina. Proceedings of ION GNSS 2006, Fort Worth, Texas, pp. 2006–2016. Gao, Y. and X. Shen (2001). Improving convergence speed of carrier phase based Precise Point Positioning. Proceedings of ION GPS 2001, 12–14 September, Salt Lake City, Utah, pp. 1532–1539. Gao, Y. and X. Shen (2002). A new method for carrier phase based Precise Point Positioning. Navigation, Vol. 49, No. 2, pp. 109–116. Gao Y., S. Skone, K. Chen, N.A. Nicholson and R. Muellerschoen (2004). Real-time sensing atmospheric water vapor using precise GPS orbit and clock products. Proceedings of

Current State of Precise Point Positioning ION GNSS 2004, 21–24 September, Long Beach, California, pp. 2343–2352. Gao, Y., A. Wojciechowski and K. Chen (2005). Airborne kinematic positioning using precise point positioning methodology. Geomatica, Vol. 59, No. 1, pp. 275–282. Gao, Y., Y. Zhang and K. Chen (2006). Development of a realtime single-frequency Precise Point Positioning system and test results. Proceedings of ION GNSS 2006, 26–29 September, Fort Worth, Texas, pp. 2297–2303. Ge, M., G. Gendt and M. Rothacher (2006). Integer ambiguity resolution for precise point positioning. Proceedings of VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy: Challenge and Role of Modern Geodesy; 29 May–2 June, Wuhan, China. H´eroux, P., Y. Gao, J. Kouba, F. Lahaye, Y. Mireault, P. Collins, K. Macleod, P. T´etreault and K. Chen (2004). Products and applications for Precise Point Positioning - Moving towards real-time. Proceedings of ION GNSS 2004, 21–24 September, Long Beach, California, pp. 1832–1843. IGS products (2007). International GNSS Service data products.http://igscb.jpl.nasa.gov/components/prods.html. IGS real-time (2007). International GNSS Service real time pilot project. http://www.rtigs.net/pilot. Keshin, M.O., A.Q. Le, and H. van der Marel (2006). Single and dual-frequency Precise Point Positioning: approaches and performances. NAVITEC 2006, Noordwijk, The Netherlands, 11–13 December 2006, 8 pp. Kouba, J. and P. H´eroux (2001). Precise point positioning using IGS orbit and clock products. GPS Solution, Vol. 5. No. 2, pp. 12–28.

623 Le, A. and C. Tiberius (2006). Single-frequency precise point positioning with optimal filtering. GPS Solutions, Vol. 11. No. 1, pp. 61–69. Muellerschoen, R.J., Y.E. Bar-Sever, W.I. Bertiger and D.A. Stowers (2001). NASA’s Global DGPS for high precision users. GPS World, Vol. 12, No. 1, pp. 14–20. NTRIP (2007). Networked Transport of RTCM via Internet Protocol. http://igs.bkg.bund.de/index ntrip.htm. Shen, X. and Y. Gao (2006). Analyzing the impacts of Galileo and Modernized GPS on Precise Point Positioning. Proceedings of ION NTM 2006, 18–20 January, Monterey, California, pp. 837–846. Wang, M. and Y. Gao (2007). An investigation on GPS receiver initial phase bias and its determination. Proceedings of ION NTM 2007, 22–24 January, San Diego, California, pp. 873–880. Wubbena, G., M. Schmitz and A. Bagg (2005). PPP-RTK: Precise Point Positioning using state-space representation in RTK networks. Proceedings of ION GNSS 2005, 13–16 September, Long Beach, California, pp. 2584–2594. Zhang, Y. and Y. Gao (2005). Performance analysis of a tightly coupled Kalman filter for the integration of undifferenced GPS and inertial data. Proceedings of ION National Technical Meeting, 24–26 January, San Diego, California, pp. 270–275. Zumberge, J.F., M.B. Heflin, D.C. Jefferson, M.M. Watkins and F.H. Webb (1997). Precise point positioning for the efficient and robust analysis of GPS data from large networks. Journal of Geophysical Research, Vol. 102, No. B3, pp. 5005–5018.

Nonstationary Tropospheric Processes in Geodetic Precipitable Water Vapor Time Series O.J. Botai, W.L. Combrinck and C.J. deW Rautenbanch

Abstract Precipitable Water Vapor (PWV) time series acquired using space geodetic techniques such as GPS (Global Positioning System) and VLBI (Very Long Baseline Interferometry) have statistical properties which exhibit spatial-temporal variations. These variations could be seen as manifestation of atmospheric structure and dynamics. Statistically, second-order moments may carry spatial-temporal information. The measures of spatial-temporal information in the PWV time series such Self-Similar (SS) and Long-Range Dependence (LRD) parameters could be utilized for both geodetic applications and meteorology. If the time series is segmented into small windows and local statistical parameters calculated, the second-order quantities derived thereof could be used to describe global and local (non-)stationary processes in the atmosphere. Results from our study show that, the PWV time series reconstructed by Singular Spectrum Analysis (SSA) has features that describe (non)stationary. The trends, spikes and excursions seen in the power spectra of a non-decimated discrete Haar wavelet transform of the PWV time series is further evidence to (non-)stationarity. Further, a wavelet based joint estimator of SS and LRD shows that the PWV time series has memory and the mean, variance, and the scaling exponents show fluctuations. These fluctuations depict nonstationarity. O.J. Botai Hartebeestshoek Radio Astronomy Observatory, P. O. Box 443, Krugersdorp 1740, South Africa, e-mail: [email protected] W.L. Combrinck Hartebeestshoek Radio Astronomy Observatory, P. O. Box 443, Krugersdorp 1740, South Africa C.J. deW Rautenbanch Department of Geography Geoinformatics and Meteorology, University of Pretoria 0002, South Africa

Keywords Precipitable water vapor · parameter · Self-similar · Wavelet transform

Scaling

1 Introduction In geodetic analyses of PWV, spatial-temporal variability is often assumed. To better understand the causal mechanism of the variability of PWV, statistical information is required; see for example in Aonashi et al. (2004) and Tesmer and Kutterer (2004). If a time series is decomposed into segments, the mean, correlations and standard deviations (local statistics) of the segments may reveal features that characterize the underlying processes (which may be (non-)stationary). Nonlinear processes exist in nature; see for example in Earth’s climate system (Rial et al. 2004), river flow and discharge (Montanari et al. 2000), meteorology (Ausloos and Ivanova 2001 and Ivanova and Ausloos 1999). The type of structure in the segments of the time series of a natural phenomenon may show a measure of nonstationarity. This may be an indication that the underlying driving forces are caused by complex processes (Verdes et al., 2001, Taqqu et al., 1995, Hu et al., 2001). Analyses of statistical properties of tropospheric parameters such as Precipitable Water Vapor (PWV) computed from geodetic techniques such as GPS and VLBI could be used to infer the presence of nonstationary processes in the troposphere. The information carried by the second order time structures of the PWV time series can be used to study stochastic mechanisms in the data (Fryzlewicz et al., 2007). Thus, inferences of stochastic processes in the atmosphere may then be used to deduce models un apriori (Abel, 2002).

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Current tropospheric modeling strategies involve the estimation of the tropospheric parameters which affect measurements of the delay observable (Bevis et al. 1994, Veedel et al. 2001, Yang et al. 1999, MackMillan 1995, Boehm et al. 2006, Hase et al. 2002). The analysis strategies involves defining models a priori (e.g. turbulence models (Nilsson et al., 2007, Nilsson et al., 2006, Emardson and Jarlemark, 1999), assuming that the atmosphere exhibits random walk processes (Tesmer and Kutterer, 2004 and references therein and Boehm et al., 2006 and references therein) and then fitting the models to the data. This approach is currently used to estimate biases in the estimated tropospheric delay. Further information on calculation of geodetic delay observable can be found in (Boehm et al. 2005) and references therein. In this paper we investigate the presence of memory in the tropospheric PWV using statistical theory. Statistical analysis of nonstationary processes in geodetic PWV is not documented anywhere in the literature (that the authors are aware of). The PWV has been wavelet transformed (Nason et al., 2000, Percival and Walden, 2000, Whitcher, 1998) and statistical techniques applied to the time series in the wavelet space in order to investigate nonstationarity as contained by

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the ‘detail’ wavelet transform at different time-scales. The correlation of PWV time series is quantified using the scaling exponent. Further, the presence of SelfSimilarity (SS) and Long-Range Dependence (LRD) properties are investigated. See Chen et al. (2002) for further details on scaling exponent. The SS and LRD in the PWV time series are computed from the secondorder statistics using wavelet-based joint estimator of parameters that characterize the LRD (Abry, 1997).

2 Data Analysis Method Using a global PWV parameter (gPWV) estimated at Hartbeestshoek Radio Astronomy ObservatorySouth Africa, the PWV time series between 2000 and 2006 was reconstructed with sliding window using Singular Spectrum Analysis (SSA). See Ghil et al. (2001) for a detailed account on SSA. The gPWV is defined (here) as the daily mean PWV value computed from two geodetic techniques (GPS and VLBI) and regional Numerical Weather Prediction (NWP) models. The Wavelet Transform (WT) of the reconstructed gPWV time series (see Fig. 1)

Fig. 1 gPWV and the SSA reconstructed gPWV time series: the reconstructed time series has dominant principal components which are projected into the linear subspace. The variance of the original time series is preserved

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Fig. 2 Haar wavelet spectra of the reconstructed gPWV at scales levels; S7 , S5 , S3 and S1 corresponding to the physical time of scales 2 days, 1 month, 4 months and 8 months respectively

was computed using non-decimated Haar wavelet. Figure 2 shows the ‘detail’ (scales: from bottom-up, [1, 3, 5, 7]) of the gPWV in the wavelet space. The scales correspond to the physical time scales 2 days, 1 month, 4 months, and 8 months respectively. More details on the non-decimated Haar WT can be found in Whitcher (1998). In order to infer (non-)stationary properties in the reconstructed time series, second order parameters were calculated as follows. Firstly, the centered spectra of the WT of gPWV time series (at different scales) was integrated and divided into segments of finite width. Then each signal in the segment was locally detrended and the root mean square fluctuation calculated. This process was repeated for different window widths and root mean square fluctuations of the reconstructed gPWV over all segments averaged. The relation between the root mean square fluctuation and width of the segment is shown in Fig. 4.

The scaling exponent of gPWV was calculated using a wavelet based joint estimator using the Discrete Wavelet Transform (DWT). A linear regression was fitted on the power spectra of WT. The temporal variation of the mean, variance, the scaling exponent (α) of gPWV is shown in Fig. 5.

3 Results and Discussion Using the SSA, the reconstructed gPWV shown in Fig. 1 was computed from leading principal components projected onto a reduced subspace with minimum redundancy. In this way, the gPWV was smoothed and any noise in the time series eliminated through filtering. This method of filtering conserves the second order statistical parameters which characterize nonstationarity and/or local stationarity. Using the Maximum

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Fig. 3 The power spectral density of gPWV at different scales (Si shown). The SS and LRD measures are dependent on ‘detail’ of WT. The temporal scales S7 , S5 , S3 of the reconstructed gPWV shows scale dependent correlations

Overlap Discrete Wavelet Transform (MODWT) technique, the non-stationary in the gPWV time series is visualized (in the wavelet space) as trends, jumps and spikes at different time-scales as shown in Fig. 2. The ‘detail’ of the WT in the wavelet space (not shown here) confirmed that gPWV exhibits temporal nonstationary characteristics. gPWV times-series was also segmented and a local linear least square fit was used to standardize the integrated time series. The average fluctuation for each segment was used to compute the scaling exponent. Figure 3 show that a single linear fit on the gPWV power spectral density has different values of β at different scales. An increase in the scale of the wavelet coefficient leads to an increase β. This indicates that the SS and LRD properties are dependent on the scale of the time series in the wavelet space. Figure 4 show that the root mean square fluctuations and the sliding window width have different scaling exponents (α). This shows that the long-range power-law correlation in the time series (i.e. α > 0.5) is dependent of the physical time scales. This means that the signals in the original time series could be identified

based on power spectra of the WT. These results indicate that atmospheric processes manifested in the PWV fluctuations have memory. This is also confirmed from the recurrence plots (not shown here). The temporal variation in the mean and variance (and also correlations) can be used to measure the presence of SS and/or LRD. In our analysis using the wavelet estimator of Abry (1998) of second-order parameters the test for SS and/or LRD properties in gPWV time-series showed that the mean and variance are not time invariant as shown in Fig. 5. This shows that tropospheric PWV has SS and LRD properties.

4 Conclusions For the first time, tropospheric parameters and the geodetic PWV time series in particular has been decomposed into segments and (non-stationary processes investigated using statistical information. We discover that, in the wavelet space the tropospheric PWV has trends and jumps indicating nonstationarity.

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Fig. 4 Scaling behavior of reconstructed gPWV time series. The scaling exponent is dependent on the inherent type of signals in the original time series

Fig. 5 Wavelet estimator for SS and LRD at the third scale: The LRD properties (mean, variance, α, cf ) fluctuate based on the gPWV sub series sizes

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Further, the second-order statistical properties (means, variances and correlations) show that they are time variants. This is additional evidence that that PWV variability is driven by forces that are nonstationary. The PWV time series also has self-similar and longrange dependence properties which are dependent on the WT ‘detail’. However, further analyses are required to investigate the variability of the scaling parameter in the PWV using alternative filtering technique(s). Acknowledgments The authors are grateful to the reviewers for their useful comments. The Institute of Geodesy and Geophysics (IGG) TU-Wien is acknowledged for providing the OCCAM 6.2 software for VLBI analyses. The authors acknowledge IAG/IUGG for the partial travel grant given to the first author to attend the IUGG assembly. This work received support from HartRAO Space Geodesy Program and the University of Pretoria.

References Abel, M. (2002). Nonparametric modeling and spatiotemporal dynamical systems. ArXiv Nonlinear Sci. e-prints. Abry, P., and D. Veitch (1987). Wavelet analysis of long range dependence traffic. IEEE Trans. Inform. Theory. Aonashi, K., T. Iwabuchi, and S. Shoji (2004). Statistical study on precipitable water content variations observed with ground-based microwave radiometers. J. Meteo. Soc. Japan, 82(1B), pp. 269–275. Ausloos, M., and K. Ivanova (2001). Power-law correlations in the southern-oscillation-index fluctuations characterizing E1 Nino. Phys. Rev. E 63, p. 047201. Bevis, M., T. A. Herring, R. A. Anthes, C. Rocken, and R. H. Ware (1994). GPS meteorology: mapping zenith wet delays onto precipitable water. J. Appl. Meteo., 33, p. 379. Boehm, J., M. Ess, and H. Schuh (2005). Asymmetric Mapping Functions for CONT02 from ECMWF. In: Proceedings of the 17th Working Meeting on European VLBI for Geodesy and Astrometry, M Vennebusch, A Nothnagel (Eds.), Instituto di Radioastronomie, Noto Italy, pp. 64–68. Boehm, J. and H. Schuh (2007). Tropospheric gradients from the ECMWF in VLBI analysis. J. Geodesy, doi: 10.1007/s00190007-0144-2. Chen, Z., P. Ch. Ivanov, K. Hu, and H. E. Stanley (2002). Effects of nonstationarities on detrended fluctuation analysis. Phys. Rev. E, 65, p. 041107. Emardson, T. R., and P. O. J. Jarlemark (1999). Atmospheric modelling in GPS analysis and its effect on the estimated geodetic parameters. J. Geodesy, 73, pp. 322–331. Fryzlewicz, P., V. Delouille, and G. P. Nason (2007). GOES-8 X-ray sensor variance stabilization using the multiscale datadriven Haar-Fisz transforms. Appl. Stat.

O.J. Botai et al. Ghil, M., M. R. Allen, M. G. Dettinge, K. Ide, D. Kondrasshov, M. E. Mann, A. Robertson, A. Sounders, Y. Tian, F. Varadi, and P. Yiou (2001). Advanced spectral methods for climatic time series. Rev. Geophys., 40(1), pp. 1-1–1-41. doi: 10.1029/2001RG000092. Hu, K., P. Ch. Ivanov, Z. Chen, P. Carpena, and H. E. Stanely (2001). Effect of trends on detrended fluctuation analysis. Phys. Rev. E 64, p. 011114. Ivanova, K. and M. Ausloos (1999). Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking. Physica A, 274, p. 349. Montanari, A., R. Rosso, and M. S. Taqqu (2000). A seasonal fractional ARIMA model applied to the Nile River monthly flows at Aswan. Water Res., 36, p. 1249. Nason, G. P., R. von Sach, and G. Kroisandt (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. Series B, 62, pp. 271–292. Nilsson, T., G. Elgered, and L. Gradinarsky (2006). Characterizing atmospheric turbulence and instrumental noise using two simultaneously operating microwave radiometers. In: Proceedings of the 9th Specialist Meeting on Microwave Radiometry and Remote Sensing Applications, MicroRad 2006, pp. 270–275. Nilsson, T., R. Haas, and G. Elgered (2007). Simulations of atmospheric path delays using turbulence models. In: Proceedings of the 18th European VLBI for Geodesy and Astrometry Working Meeting, J. Boehm, A. Pany, H. Schuh (Eds.), Goewissenschaftliche Mitteilungen, Schriftenreihe der Studienrichtung Vermessung und Geoinformation, Technische Universitat Wien, 79, pp. 175–180. Percival, D. B., A. Walden (2000). Wavelet Methods for Time Series Analysis. Cambridge University, Cambridge. Rial, A. R., R. A. Pielke, M. Beniston, M. Claussen, J. Canadell, P. Cox, H. Held, N. De Noblet-Ducoudre, R. Prinn, J. F. Reynolds, and J. D. Salas (2004). Nonlinearities, feedbacks and critical thresholds within the earth’s climate. Climate Change, 65, pp. 11–38. Taqqu, M. S., V. Teverovsky, and W. Willinger (1995). Estimators for long-range dependence: an empirical study. Fractals 3(4), pp. 785–798. Tesmer, V., and Kutterer H., (2004). An advanced stochastic model for VLBI observations and its applications to VLBI data analysis. IVS 2004 General Meeting Proceedings, pp. 296–300. Verdes, P. F., P. M. Granitto, H. D. Novane, and H. A. Ceccatto (2001). Nonstationary time-series analysis: accurate reconstruction of driving forces. Phys. Rev. Lett., 87(12), p. 124101 Whitcher, B. J. (1998). Assessing Nonstationary Time Series Using Wavelets. Ph. D. Thesis, University of Washington. Zhi, C., C. L. Plamen, H. Kun, and H. E. Stanely (2002). Effects of nonstationarities on detrended fluctuation analysis. Phys. Rev. E, 65, p. 041107.

Combination of Multiple Repeat Orbits of ENVISAT for Mining Deformation Monitoring H.C. Chang, L. Ge, A.H. Ng, C. Rizos, H. Wang and M. Omura

Abstract In conventional differential radar interferometry (DInSAR), the ground surface displacement can be measured along the line-of-sight. In order to measure the vertical and horizontal displacements DInSAR results of the same deformation derived from at least two different look directions are required. This study utilised the DInSAR results generated from ENVISAT data acquired from three different look angles to determine the 3-D displacement vectors resulting from surface deformation due to underground mining. However, The ENVISAT results showed ambiguous phases due to high phase gradients in the differential interferogram. The newly acquired ALOS/PALSAR data is also used here to demonstrate how the high phase gradient problem can be overcome by using interferometric signals with longer wavelength and finer imaging resolution.

H.C. Chang School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia L. Ge School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia A.H. Ng School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia C. Rizos School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia H. Wang Department of Surveying Engineering, Guangdong University of Technology, Guangzhou 510006, China M. Omura Department of Environmental Science, Kochi Women’s University, 5-15, Eikokuji-cho, Kochi 780-8515, Japan

Keywords DInSAR · 3-D deformation vectors · ENVISAT · ALOS · Mining deformation

1 Introduction Radar interferometry techniques, including interferometric synthetic aperture radar (InSAR), differential InSAR (DInSAR) and persistent scatterer InSAR (PSInSAR), have demonstrated their capabilities for determining geodetic information on changes to the earth’s surface geometry. These techniques are considered to be more cost-effective than, and complementary to, conventional ground-based geodetic methods. The authors have been studying mine subsidence using DInSAR in the State of New South Wales, Australia for several years. For example, a mine subsidence profile extracted from a JERS-1 DInSAR result illustrated a root mean square error (RMSE) of 1.4 cm against the ground levelling data (Ge et al., 2007). In addition, sub-centimetre accuracy of DInSAR results was demonstrated using ERS-1/2 tandem pairs (Chang et al., 2005; Ge et al., 2007). The results showed that a maximum subsidence of 1 cm was developed during 24 h, which was confirmed by the surveyor from the mining company. As in most DInSAR studies for land deformation monitoring, the deformation is measured along the line-of-sight (LOS) between the radar antenna and the ground reflecting objects. Then the subsidence (vertical surface deformation) is derived under the assumption of negligible horizontal deformation. However, in-situ ground survey data shows that underground mining activity may also cause horizontal surface displacement, which may be greater than

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20–30 mm. In addition, this assumption of negligible horizontal movement is more applicable to the ERS-1/2 data as the look angle of ERS-1/2 is 23◦ . The smaller look angle makes the system more sensitive to vertical deformation than horizontal ones. In order to study both vertical and horizontal mine deformation, multiple ascending and descending orbits of ENVISAT/ASAR data are analysed in this paper. The variation of the swath mode of ENVISAT/ASAR provides the opportunity to investigate the 3-D surface deformation vectors with a range of satellite look angles ranging from 15 to 45.2◦ .

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The new spaceborne SAR sensors have various scanning modes with a range of incidence angles. ENVISAT/ASAR permits image acquisitions of the same region in seven different swath modes. Hence the same surface deformation can be measured not only in both ascending and descending orbits, but also at different viewing angles. This study examines the feasibility of combining the DInSAR results derived from images from one ascending and two descending orbits of ENVISAT to reveal the 3-D deformation vectors of ground subsidence. The 3-D deformation vectors can be calculated from the DInSAR solutions derived from 3 independent LOS results. This was demonstrated by Sircar et al. (2004) using ascending and descending Radarsat1 SAR image data. It was also demonstrated by Fialko and Simons (2001) for 1999 Hector mine earthquake using ascending and descending ERS-1/2 differential interferometric results together with azimuthal offsets of radar amplitude pixels. The deformation vector along the LOS signal (DLOS ) is a composite of up (DU ), east (DE ) and north (DN ) deformation components. The deformation measured in the differential interferogram is the sum of vertical and horizontal deformation components projected onto the LOS. The contributions of DU , DE and DN to DLOS are given in Eq. (2): ⎤ ⎡ DU [− cos(θ) sin(θ) cos(α) sin(θ) sin(α)] ⎣ DE ⎦ = [DLOS ] DN (2)

InSAR utilises two coregistered SAR images of the identical scene and measures the phase difference of the same pixels in the two images. This phase difference product, the so-called interferogram, contains topographic information, variance of water vapour in the atmosphere and ground movement that has occurred between the dates of the two SAR images acquired (if repeat-pass radar interferometry is used). DInSAR extracts the interferometric phase introduced by land surface deformation by eliminating other phase contributors in the interferogram, e.g. topography. The phase difference, φ, caused by the height deformation, d, along the LOS of the radar signal is shown in Eq. (1), where λ is the wavelength of the radar signal. So, one complete phase cycle (2π) in a differential interferogram represents λ/2 of height displacement along the where θ is the radar incidence angle and α is the azLOS: imuth of the satellite heading vector (positive clockwise from North). Note that DU is defined as negative 4π d for subsiding movement. φ = (1) Fialko and Simons (2001) utilised the azimuthal λ offsets and DInSAR to measure the displacements Usually the amplitude of vertical height deforma- along the azimuth direction and the LOS of the tion vector or subsidence is derived by assuming neg- ERS-1/2 signals. The former are mostly sensitive ligible horizontal displacement. In order to resolve the to the North-South displacements; while the latter true 3-D deformation vectors, DInSAR results derived are sensitive to the vertical, modestly sensitive to from both ascending and descending images need to be East-West, and weakly sensitive to North-South combined. If the deformation is purely vertical, the ver- displacements. Therefore, the combination provides a tical displacements derived from both orbits should be good geometry in order to separate the displacement the same, although the displacements measured along vector components DU , DE and DN . However, the the LOS may vary depending on the look angle of measurement error due to the use of azimuthal offsets radar. Otherwise, the horizontal displacement has a is of the order of a fraction of the imaging pixel. In signature. order words, with the along-track pixel size of 4 m for

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ENVISAT, the accuracy of the azimuth offsets method is at the decimetre level. It is not feasible to detect the horizontal displacement of mine deformation, which has a magnitude of a few to tens of millimetres. In this paper, three independent look angles of ENVISAT data were tested for the purpose of determining the DU , DE and DN of underground mining induced deformation. The geometry of the method is less than ideal as it is less sensitive to the displacement along the North-South orientation due to the characteristics of the satellite orbit. It is, however, an alternative when the azimuthal offsets method is not applicable.

(a)

(b)

(c)

(d)

3 Input Data and Results Three ENVISAT interferometric pairs were used for the 3-D deformation vector analysis. In addition, a recently received ALOS/PALSAR interferometric pair is tested and compared against the ENVISAT results. Some characteristics of ENVISAT and ALOS/PALSAR sensors are summarised in Table 1. The ENVISAT image acquisitions over the test site are listed in Table 2, where pairs 1, 2 and 3 were acquired in the swath modes IS4, IS3 and IS2, respectively. The maximum temporal difference is only 3 days between pairs 1 and 3. Therefore the temporal coverage of the pairs is suitable for the current purpose of 3-D deformation analysis. The DInSAR interferograms of pairs 1–3 were derived and are shown in Fig. 1. Note the high-phase gradient near the centre of the subsidence bowl. The longer baseline of pair 3 caused higher spatial Table 1 Characteristics of ENVISAT and ALOS SAR sensors λ

Sensor

Band

Altitude

Resolution

ENVISAT ASAR ALOS PALSAR

C

5.6 cm

786 km

∼ 30 m

L

23.6 cm

692 km

∼ 10 m

Table 2 Three ENVISAT interferometric pairs used in this study, all with 35 days temporal separation between master and slave images Pass desc asc desc

Pair 1 2 3

Master dd/mm/yyyy

Slave dd/mm/yyyy

8/12/2006 10/12/2006 11/12/2006

12/01/2007 14/01/2007 15/01/2007

Bperp (m) 234 238 310

Fig. 1 (a) Averaged intensity SAR image of the test site, and (b)–(d) differential interferograms of pairs 1, 2 and 3 in slant range projection

decorrelation. It is indicated by a higher level of phase noise in the interferogram. As the local topographic variation is moderate, the increased noise in pair 3 was not predominantly due to the topographic residual phases. Consequently, the unwrapped phase of pair 3 was noisier than the results of other pairs. Therefore only the deformation vectors derived from pairs 1 and 2 were used for the 3-D deformation analysis, with the assumption of small horizontal movement along the North-South direction. The geocoded deformation maps in the slant range direction are shown in Fig. 2. The estimated 3-D deformation vectors along the East, North and Up directions are shown in Fig. 3.

4 Validation By comparing the ground survey data collected at a similar period, the ENVISAT results do not reflect the true mine subsidence because the high phase gradient at the centre of the subsidence bowl cause the phase values to be ambiguous. The ambiguous phases may

634

Fig. 2 Deformation along the slant range direction of pair 1 (descending) and pair 2 (ascending)

be smoothed by filtering the interferogram. Therefore the discontinuous or wrapped phase values in the differential interferograms cannot be recovered without errors. Similar findings of underestimating mine subsidence using ERS-1/2 DInSAR results were reported in other studies (e.g. Spreckels et al., 2001). This problem can be eased by having a longer wavelength radar signals or a shorter repeat cycle of the radar image acquisition system. A previous study showed that ground subsidence measured by DInSAR using JERS-1 data can achieve an RMS error of 1.4 cm when compared to the results of ground survey (Ge et al., 2007). In the comparison,

H.C. Chang et al.

the ground survey line was not across the centre of the subsidence bowl, therefore the maximum subsidence measured along this survey line was about 16 cm. According to the ground survey data provided by the mining company (BHPBilliton), the mining activity would cause a maximum subsidence of the order of 860–930 mm in the area. About 65% of the total subsidence (over 50 cm) becomes evident within the first 2–3 weeks after excavation. This large vertical ground movement near the centre of the subsidence bowl causes the phases in the differential interferogram to be ambiguous. This has a severe impact on the ENVISAT results during the phase unwrapping process. This problem can be solved or eased by having interferometric pairs with higher imaging resolution, signals with longer wavelength and/or a shorter site revisit cycle. This is confirmed in the next section using recent ALOSPALSAR data.

5 High Phase Gradient Problem The maximum phase gradient in an interferogram that avoids incoherence was defined in Massonnet and Feigl (1998). It is one fringe per pixel. The maximum phase gradient is also constrained when phase unwrapping the differential interferogram. The common phase unwrapping techniques assume the phase difference between the two adjacent pixels in the interferogram

Fig. 3 3-D deformation vectors due to mining-induced subsidence over 35 days in the easting, northing and up directions

Combination of Multiple Repeat Orbits of ENVISAT

is less than π. Otherwise the wrapped phase in the interferogram becomes ambiguous and cannot be unwrapped to produce continuous errorfree phase results. In the application of DInSAR for monitoring deformation due to underground coal mining,, the maximum subsidence usually occurs along the centre of the longwall panel. At this test site the longwall panel has a width of about 200–250 m. Due to the rectangular layout of the longwall, the subsidence gradient across the longwall panel is much greater than along the longwall. Hence the phase gradient in the differential interferogram is at its maximum in the direction across the longwall panel. Due to the longwall geometry, the centre of the subsidence bowl normally lies along the centre of the panel. For example, for a longwall panel with a width of 250 m, the centre of the subsidence bowl should be about 125 m from the edge of the panel (pillar). That is equivalent to about 4 pixels for ENVISAT data (with a spatial resolution of 30 m). In order to avoid phase unwrapping errors, the maximum height displacement over these 4 pixels has to be less than 4π. Therefore, better imaging resolution of SAR sensor improves the maximum detectable height displacement without increasing phase unwrapping errors. Two recently received interferometric pairs of ALOS/PALSAR data are used in this analysis. ALOS/PALSAR can operate in five different imaging modes: fine beam single polarisation (FBS), fine beam double polarisation (FBD), direct downlink (DSN), ScanSAR and polarimetry (PLR). These two ALOS/PALSAR pairs were acquired in single and dual-polarisation modes. For DInSAR processing the authors only used the data acquired in the FBS mode with 46 days separation (single repeat cycle of ALOS), as shown in Table 3. The differential interferogram of pair 4 shown in Fig. 4 clearly indicates the fringes caused by ground subsidence at the same mine site. These are more distinguishable than the results derived from ENVISAT data. Due to the wavelength of

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Fig. 4 ALOS differential interferogram of mining-induced subsidence generated from pair 4: 27/12/2006 – 11/02/2007

23.6 cm and an incidence angle of 38.7◦ , the height displacement map in Fig. 5 shows the maximum height displacement near the centre of the subsidence bowl as being over 40 cm. The georeferenced DInSAR

Table 3 Interferometric characteristics of two pairs of ALOSPALSAR ascending data with an incidence angle of 38.7◦ Pair 4 5

Master Slave dd/mm/yyyy dd/mm/yyyy Bperp (m) 27/12/2006 11/02/2007 530 14/08/2007 29/09/2007 501

Btemp (days) 46 46

Fig. 5 Georeferenced mine subsidence map based on the ALOS differential interferogram of pair 4 in Fig. 4

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Fig. 7 Ground survey data at the upper mine in Fig. 6 for the period of 10/04/2007-28/05/2007 (48 days) and 16/04/20074/06/2007 (49 days) respectively. The maximum subsidence occurred during the time span is about 30 cm

(a)

6 Further Work

As illustrated in the previous section, using ALOS/PALSAR data (or SAR data with longer wavelength) may ease the problem of high phase gradient in interferograms caused by large and/or rapid ground deformation such as that induced by underground mining. Other options for avoiding high phase gradients are to use (if available) interferometric pairs with finer imaging resolution and/or shorter revisit times. Two new SAR satellites, TerraSAR-X and COSMO-SKYMED, were launched in June 2007. Both satellites have SAR sensors operating in the Xband frequency band. Their finest imaging resolution is less than 3 m. The repeat cycles of TerraSAR-X and (b) a single COMSMO-SKYMED are 11 and 16 days, respectively. In future studies, X-band data will be Fig. 6 Georeference ALOS/PALSAR results of (a) DInSAR interferogram overlaid with mine plan, and (b) colour coded sub- used to test their performance for mining-induced sidence map overlaid on ALSO/PALSAR intensity image with subsidence monitoring. mine plan, derived from pair 5: 14/08/2007 – 29/09/2007. The It may be challenging to apply the SAR pixel matchmine plan for the lower subsidence bowl is not yet available ing technique (Tobita et al., 2001; Werner et al., 2001) to measure the horizontal ground deformation with the interferogram and subsidence map of pair 5 are shown current ENVISAT and ALOS/PALSAR data due to the large imaging resolution compared to the amount in Fig. 6. Figure 7 shows the ground survey data (from BH- of horizontal displacement. But it may be applicable PBilliton) at the upper mine in Fig. 6. The maximum when the 1 m resolution of TerraSAR-X and COSMOsubsidence during a time span of 48–49 days is about SKYMED become available. 30 cm. The ALOS/PALSAR DInSAR subsidence map in Fig. 6 shows a maximum subsidence of about 35– 40 cm over 46 days, which is similar to the ground 7 Concluding Remarks survey result. More detailed validation is needed, with ground truth data with the same or similar spatial and This study illustrated how 3-D deformation vectors due temporal coverage. to mining-induced subsidence can be measured using

Combination of Multiple Repeat Orbits of ENVISAT

the DInSAR technique, from a combination of ascending and descending image data collected with different look angles (swath modes) by ENVISAT/ASAR. When three deformation vectors with different line-of-sight SAR geometry are available, it is possible to derive the deformation components in the Up, East and North directions. However, this method is less sensitive to the displacement in the North-South direction. The proposed method may be useful when a third displacement map along an independent look angle cannot be achieved, e.g. using azimuthal offsets. However, this method is limited by requiring three interferometric pairs (the triplets) with good coherence covering a similar period of time, or two ascending and descending pairs with a good estimation of the deformation in the third direction. Unfortunately, this study found that high phase gradient in the differential interferogram caused by mining-induced deformation is unavoidable using ENVISAT data. This is because of the comparatively coarse imaging resolution and long revisit orbit cycle. Therefore, the true height deformation cannot be recovered from the ambiguous interferometric phases. Consequentially errors are propagated into the final height displacement map. ALOS/PALSAR data have longer wavelength and also finer imaging resolution than ENVISAT/ASAR. The result derived from ALOS/PALSAR demonstrates the problem caused by high phase gradient can be mitigated. The new TerraSAR-X and COSMO-SKYMED data, with shorter site revisit cycle and finer imaging resolution, will be tested for their capability of monitoring ground deformation. Acknowledgments This research work is supported by the Cooperative Research Centre for Spatial Information (CRC-SI) Project 4.2, whose activities are funded by the Australian Commonwealth’s Cooperative Research Centres Programme. This study was also conducted in collaboration with the Strata Control Technology, supported by the Australian Coal Association Research Program. The authors wish to thank the European Space Agency for providing ENVISAT data. ALOS/PALSAR interferogram was processed under CRCSI project 4.2, including material copyright METI and JAXA (2006, 2007) L-1.1 product: produced by ERSDAC. METI and JAXA retain ownership of the ALOS/PALSAR data. ERSDAC

637 produced and distributed the PALSAR L-1.1 product. The PALSAR L-1.1 products were distributed to the IAG Consortium for Mining Subsidence Monitoring. The authors are grateful to ERSDAC for their support of the consortium. The authors wish to thank BHPBilliton for providing the ground survey data.

References BHPBilliton, Mine subsidence ground survey report. (private communication). Chang, H.C., L. Ge and C. Rizos, 2005. ERS tandem DInSAR: the change of ground surface in 24 hours. IGARSS ‘05, Seoul, Korea, 25–29 July, vol. 7. pp. 5265–5267. Fialko, Y. and M. Simons, 2001. ‘The complete (3-D) surface displacement field in the epicentral area of the 1999 Mw 7.1 Hector Mine earthquake, California, from space geodetic observation’ Geophys. Res. Lett., vol. 28 no. 16: pp. 3063–3066. Ge, L., H.C. Chang and C. Rizos, 2007. ‘Mine Subsidence Monitoring Using Multi-source Satellite SAR Images’ Photogrammetric Eng. Remote Sensing, vol. 73 no. 3: pp. 259–266. Massonnet, D. and K.L. Feigl, 1998. ‘Radar interferometry and its application to changes in the Earth’s surface’ Rev. Geophys., vol. 36 no. 4: pp. 441–500. Michel, R., J.-P. Avouac and J. Taboury, 1999. ‘Measuring ground displacements from SAR amplitude images: application to the landers earthquake’ Geophys. Res. Lett., vol. 26 no. 7: pp. 875–878. Sircar, S., D. Power, C. Randell, Y. Youden and E. Gill, 2004. Lateral and subsidence movement estimation using InSAR. Geoscience and Remote Sensing Symposium, 2004. IGARSS ‘04. Anchorage, Alaska, 20–24 Sept. vol. 5. pp. 2991–2994. Spreckels, V., U. Wegmuller, T. Strozzi, J. Musiedlak and H.-C. Wichlacz, 2001. Detection and observation of underground coal mining-induced surface deformation with differential SAR interferometry. ISPRS Workshop “High Resolution Mapping from Space 2001”, Hannover. pp. 227–234. Tobita, M., M. Murakami, H. Nakagawa, H. Yarai, S. Fujiwara and P.A. Rosen, 2001. ‘3-D surface deformation of the 2000 Usu eruption measured by matching of SAR images’ Geophys. Res. Lett., vol. 28 no. 22: pp. 4291–4294. Werner, C., T. Strozzil, A Wiesmann, U. Wegmuller, T. Murray, H. Pritchard and A. Luckman, 2001. Complimentary measurement of geophysical deformation using repeat-pass SAR. IGARSS ‘01. Sydney, Australia, 9–13 July. vol. 7. pp. 3255–3258.

An Investigation into Robust Estimation Applied to Correlated GPS Networks R.C. Erenoglu and S. Hekimoglu

Abstract High precision positioning can be achieved using GPS satellites. However, there exist many error sources. For example, receiver locations have to be chosen on the basis of task requirements rather than by optimality of the environment with respect of GPS signal propagation. Hence the effect of unfavorable error sources on precise GPS positioning is a typical problem. Least squares estimation (LSE) yields results of low accuracy in the presence of outliers in GPS measurements. Different weight models have therefore been suggested in order to account for such errors. In this study the authors use robust estimators which clearly identify outlier observations. They perform significantly better than LSE. Furthermore, GPS observations are fundamentally correlated, hence the robust estimators are applied to heterogeneous and correlated observations. The mean success rate (MSR) statistic is employed as a practical tool for measuring the abilities of different methods. Many different correlated GPS networks based on IGS sites were used and the robust methods were applied to simulated corrupted samples and the degree of corruption is varied. Performance of the estimators was further demonstrated using real GPS data. Keywords GPS network · Robust Correlation · Mean success rate (MSR)

methods

R.C. Erenoglu Yildiz Technical University, Faculty of Civil Engineering, Department of Geodesy and Photogrammetry Engineering, 34349, Istanbul, Turkey S. Hekimoglu Yildiz Technical University, Faculty of Civil Engineering, Department of Geodesy and Photogrammetry Engineering, 34349, Istanbul, Turkey

·

1 Introduction GPS provides many advantages for geodetic positioning. These are accuracy, speed, simplicity and low cost. The technique of precise point positioning requires ideal observating conditions. The standard computational procedure in GPS positioning is least squares estimation (LSE). Even though the errors in the GPS observations are considered to be normally distributed, there are many gross error sources which may cause outliers in the measured data (e.g. multipath, atmospheric effects, etc.) (Xu, 2003; Weiser, 2002; Yang et al., 2001; Santos et al., 1997; Hofmann-Wellenhof et al., 1994). As is well known, outliers are observations that deviate significantly from the majority of observations. They may be generated by a different mechanism to “normal” data, such as sensor noise, process disturbances, instrument degradation, and/or human-related errors. It is futile to carry out data-based analysis when data are contaminated with outliers because outliers will lead to model misspecification, biased parameter estimation, and incorrect analysis results (Awange and Aduol, 1999). Experiments on geodetic networks show that correlated observations are frequently encountered. In this study, they will be taken into account. By using different factorization schemes the authors will transform correlated observations into uncorrelated ones. Subsequently robust estimators will be used to identify outlying observations. The mean success rate (MSR) statistic is employed as a practical tool for measuring abilities of the various factorization methods. The correlated GPS networks based on IGS sites were used, and the robust

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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methods were applied to simulated corrupted samples. 3 Robust Methods The performance of four decorrelation algorithms will be analyzed. The methods collected under the term M-estimator were first introduced by Huber (1964) to replace the squared residuals in LSE by another function 2 Decorrelation Methods of the residuals. The normal equation system of the M-estimation is non-linear. To solve it, the method Decorrelation is basically the transformation of a pos- of Iteratively Reweighted Least Squares (IRLS) itive definite matrix into an almost diagonal form. In estimation is used (Koch, 1999; Wilcox, 1997). this study four different methods will be applied to The principle of IRLS is to obtain a weight for each decorrelate the observations in GPS networks. observation that is dependent on the size of the individIn order to investigate the behavior of these algo- ual residual. Such weights make it possible to isolate rithms for decorrelate the observations several experi- the effect of outliers on the final estimate. The scale ments were performed. The procedures that were used factor is necessary to achieve the affine equivariance can be summarized as follows: for the estimators. There exist about seventy weight functions. Herein, the following robust methods are chosen: M2.1 Gauss Algorithm estimation of Andrews, M-estimation of Beaton and Tukey, the Danish method, M-estimation of Hampel, The basic Gaussian elimination (GE) algorithm is a M-estimation of Huber, L1-norm, IGGIII method (see fundamental tool of linear algebra for solving systems, Huber, 1981; Hampel et al., 1986; Andrews, 1974; computing determinants, decorrelating data and deter- Kraup et al., 1980; Barradole and Roberts, 1974; mining the rank of a matrix. Koch, 1996).

2.2 Cholesky Algorithm 3.1 M-Estimation of Andrews The Cholesky algorithm can decompose a symmetric positive matrix into a lower triangular matrix and the transpose of the lower triangular matrix. The lower one is the Cholesky triangle of the original positive definite matrix.

Andrews (1974) proposed that the weight function is given as:  W(vi ) =

−1

|vi | c

 sin

|vi | c



|vi | ≤ cπ |vi | > cπ

0

2.3 Schur Algorithm

(1)

where c = 1.5.vi are the standardized residuals. The Schur algorithm computes a factorization of a square matrix. A Schur factorization is a pair of matrices. A complex matrix is in the Schur form if it is 3.2 M-Estimation of Beaton and Tukey upper triangular with its eigenvalues on the diagonal of the matrix. The weight function for the Beaton–Tukey method (Beaton and Tukey, 1974) is as follows:

2.4 LLL Algorithm A new method is the LLL algorithm developed by Lenstra (1982). The main aim is to obtain Gauss– Schmidt decorrelation. The LLL method may produce an almost orthogonal form.

 W(vi ) =

1−

 vi 2 2 c

0

|vi | ≤ c |vi | > c

(2)

where c is taken as 1.5 or 2. vi are the standardized residuals.

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641

3.3 Danish method

The L 1 -norm has been commonly used for two purposes: the estimation of parameters and for outlier detection (Amiri-Simkooei, 2003).

The Danish method’s weight function is:  f(vt ) =

1 exp

|vi | c

 |vt | < c |vt | ≥ c

(3)

3.7 IGGIII Scheme

Yang et al. (2001) proposed the following weight funcwhere c = 3. This method was proposed by tion: Kraup (1973), and used for geodetic computations as ⎧ |vi | ≤ c0 the standard method at the Danish Geodetic Institute. ⎪ ⎨ 1c c (7) W(vi ) = 0 < |vi | ≤ c1 ⎪ ⎩ |vi | 0 |vi | > c1

3.4 M-Estimation of Hampel

where c0 and c1 are constant values which can be taken as 2.0 ∼ 3.0 and 4.5 ∼ 8.5, respectively. vi are the standardized residuals.

In Hampel’s M-estimation the weight function is: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

1 0 < |vi | ≤ a a a < |vi | ≤ b |v | W(vi ) = a (c −i |vi |) ⎪ ⎪ b < |vi | ≤ c ⎪ ⎪ ⎪ ⎩ |vi | (c − b) |vi | > c 0

4 Mean Success Rate (4)

where a = 1.5, b = 3 and c = 6 (Hampel et al., 1986). vi are the standardized residuals.

3.5 M-Estimation of Huber

To enable a comparison among the different robust methods for correlated and uncorrelated cases, measures of performance are required, such as:

r r

The Breakdown Point by Hampel et al. (1986). The Mean Success Rate (MSR) by Hekimoglu and Koch (1999, 2000).

Donoho and Huber (1983) give the following definition of a finite sample version of the breakdown point: the breakdown point is, generally speaking, the smallest Huber’s weight function (Huber, 1981) is: amount of contamination that may cause an estimator  to take on arbitrarily large aberrant values (Rousseuw 1 |vi | ≤ c c (5) W(vi ) = and Leroy, 1987). Outliers can occur in both observa|vi | > c |vi | tion space and also in the design space. The gross outliers in both observation space (i.e. influential points) where c is a constant that takes the value of 1.5. vi are and design space (i.e. leverage points) are called here the standardized residuals. extreme outliers (Hekimoglu, 2007; Hekimoglu 1997). What is the ability of the robust methods to detect small outliers in both observation space and design 3.6 L1 -norm space with a certain percentage of data contamination? Further, can the breakdown point of any robust method The L 1 -norm is known for being very robust to out- be checked practically? To answer these questions the liers. It is based on minimization with a linear pro- success rates of robust estimators and tests for outliers gramming of a modified simplex method. The simplex were introduced by Hekimoglu and Koch (1999, 2000). method is a technique designed to solve linear pro- The same approach was applied to define the reliability gramming problems: of robust methods in the geodetic networks by Berber and Hekimoglu (2001). Hekimoglu (2007) investigated Ax = b, x ≥ 0 (6) the efficiency of the MSR for geodetic networks. The

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MSR does not try to develop an alternative criterion for identifying the finite sample breakdown point. Note that the MSR approach is secondary to the breakdown point determination, and can do nothing more than provide a tool to confirm it. Let us take a good sample o of observations in a geodetic control network. A contaminated sample o¯ is obtained by randomly replacing any m of the original measurements from the sample o with arbitrary values. After applying a robust estimator denoted as T to the contaminated sample o¯ , the contaminated residual vector v¯ is derived. Let o include m outliers. The robust estimator T applied to a certain corrupted sample o¯ is defined as being successful if the residual v¯ i of each contaminated observation from o¯ 1 is greater than 3σ1 . |¯vi | > 3σ, i = 1, 2, . . . , m

where int(σ) represents both intervals int(σ1 ) and int(σ2j ), q is the number of success, t is the number of contaminated samples o¯ , j denotes a certain good sample o and i denotes a certain kind of outliers. There is a random error vector in a good sample o. Many good samples can be obtained by replacing the random error vector with the other ones. Thus, the SRs can be computed in the number p of these good samples o. They will change from one sample o to another one. As a result, the mean value of these SRs can be computed as: γimean (T, o¯ , int(σ), m, n) =

⎧ p ⎨ ⎩

j=1

⎫ ⎬ γij (T, o, o¯ , int(σ), m, n) / p ⎭

(12)

5 Experiments and Results (8)

The authors have simulated a GPS network based on where σ denotes the standard deviation of the direc- active hourly IGS (International GNSS Service) stations. The outlier region for o is defined by: tions. The data are obtained from the Scripps Orbit and Permanent Array Center (SOPAC). The network 3σ < b < ∞ (9) is composed of 10 stations with HUEG fixed as reference and 21 baseline observations. The baselines are The subsets of these outliers are called an outlier correlated. The network is shown in Fig. 1. interval int(σ) for o: In order to analyse different algorithms and make a comparison among them, the covariance matrix and int(σ) = 1σ − kσ, k > 3, 1 > k (10) the positions are obtained using the BERNESE v.4.2 data processing software. The baselines Sij are easily computed. where k and 1 denote integers. Random observation errors (normally distributed) Many contaminated samples o¯ may be generated from the good sample o by changing the positions and are generated using MATLAB functions. By using four magnitudes of outliers randomly. Let the samples o¯ contain m outliers of any kind (i.e. random and nonPTBB BOR1 random outliers) and a magnitude in the given outlier BRUS intervals of int(σ). A robust estimator T is accepted here as being successful when it can identify all of GOPE these m outliers exactly. The partial reliability of a robust estimator T applied to this contaminated samples GANP o¯ is defined as a success rate (SR), where the number HUEG of successful identifications is divided by the number BZRG GRAZ of experiments. Therefore the SR for this good sample o can be defined depending on the given intervals int(σ), on a certain kind of outliers, as: γij (T, o, o¯ , int(σ), m, n) = (the SR for a certain kind of outliers)

q t

GENO

(11) TLSE

Fig. 1 GPS network based on active hourly IGS stations

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643

decomposition algorithm, correlated true error vectors are obtained for each case. Adding these errors to the baselines Sij , observation with errors are obtained. Then, simulated outliers are added to Sij . It is assumed that one and two outliers are between 3σ–6σ and also 6σ–12σ, separately. The authors simulated a hundred good samples, i.e. random error vectors, and a hundred contaminated samples for each good sample. Thus there are 10,000 different samples. These will allow the MSRs to be computed, and also measure the effectiveness of the methods. Correlated errors and the covariance matrix for the simulated baseline components are known. Hence one can readily analyse and compare the methods to detect outliers using the MSRs. First an uncorrelated GPS network is designed (Case A). Then, four decomposition algorithms are used for obtaining the samples, and the results obtained from these experiments are arranged in the following order:

r r r r

Table 3 The MSRs of robust methods for two outliers between 3σ and 6σ Robust method

(A)

(B)

(C)

(D)

(E)

Andrews Beaton and Tukey Danish Hampel Huber L 1 norm IGGIII

50.3 48.8 55.7 46.1 42.6 49.2 50.1

34.0 33.0 37.8 30.9 28.6 33.3 33.9

37.8 36.7 42.0 34.5 30.9 37.0 37.7

36.1 35.1 40.1 32.9 30.4 35.4 36.0

40.8 39.6 45.3 37.2 34.4 39.9 40.6

Table 4 The MSRs of robust methods for two outliers between 6σ and 12σ Robust method

(A)

(B)

(C)

(D)

(E)

Andrews Beaton and Tukey Danish Hampel Huber L 1 norm IGGIII

36.6 35.8 78.8 85.3 86.5 79.7 76.8

23.1 22.6 53.4 61.2 62.1 56.6 52.4

29.3 28.7 63.6 67.8 68.7 62.7 58.1

27.9 27.3 60.8 64.9 65.8 60.0 55.6

31.7 31.1 68.4 72.9 73.9 67.5 62.5

Gauss – Case B Cholesky – Case C Schur – Case D LLL – Case E

outlier between 3σ and 6σ, one outlier between 6σ and 12σ,two outliers between 3σ and 6σ, two outliers between 6σ and 12σ respectively. The following conclusions can be drawn. The MSRs The MSRs of seven robust methods are computed change depending on the number and magnitude of for each experiment by taking account also the the outliers. The uncorrelated case, i.e. Case (A), has non-diagonal versions for the Cases (B), (C), (D) and the biggest MSRs of all the methods. Decomposition (E). Tables 1, 2, 3 and 4 present the results for one algorithms impact differently to robust methods. The Cholesky and LLL algorithms attract attention in reTable 1 The MSRs of robust methods for one outlier between spect of the others. The most successful methods to detect outliers are the Danish, Hampel, L 1 norm and 3σ and 6σ IGGIII methods. Note that if the data includes two outRobust method (A) (B) (C) (D) (E) Andrews 55.0 38.8 42.6 40.9 45.5 liers between 3σ and 6σ, Andrews methods is successBeaton and Tukey 54.4 38.4 42.2 40.5 45.1 ful than the others, but not than Danish and IGGIII Danish 60.8 42.9 47.0 45.2 50.3 methods. Hampel Huber L 1 norm IGGIII

51.0 47.8 54.5 57.6

36.0 33.7 38.4 40.6

39.5 35.9 42.1 44.5

37.9 35.5 40.5 42.8

42.2 39.6 45.1 47.7

Table 2 The MSRs of robust methods for one outlier between 6σ and 12σ Robust method

(A)

(B)

(C)

(D)

(E)

Andrews Beaton and Tukey Danish Hampel Huber L 1 norm IGGIII

45.8 44.9 89.9 94.8 95.8 88.5 85.6

32.3 31.7 63.4 66.9 67.6 62.4 60.4

35.4 34.8 69.6 73.4 74.2 68.5 66.2

34.1 33.4 66.9 70.5 71.3 65.8 63.7

37.9 37.2 74.4 78.5 79.3 73.3 70.9

6 Conclusions The robust estimation methods are techniques used commonly in geodetic networks to detect outliers in observations. In this study, the algorithms and implementations of decorrelation functions for GPS networks were investigated. In order to demonstrate the methods for detecting outliers, some examples were presented from uncorrelated and correlated GPS networks. The results show that the robust methods

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are more efficient than the classical approaches when Hampel, F., E. Ronchetti, P. Rousseeuw, and W. Stahel (1986). Robust Statistics: The Approach Based on Influence Functhere are outliers in the data. tions, John Wiley and Sons, New York. Moreover, the reliability of the methods is indicated Hekimoglu, S. (1997). The Finite Sample Breakdown Points of by the mean success rates. The authors discussed how the Conventional Iterative Outlier Detection Procedures. J. the effectiveness of the methods changes depending on Surv. Eng., 123(1), pp. 15–31. various key parameters of the networks, i.e. the num- Hekimoglu, S., and K.R. Koch (1999). How can Reliability of the Robust Methods be Measured? In: Altan MO, Gr¨undig ber of outliers and the magnitude of the outliers. SimL (eds) proceedings Third Turkish-German Joint Geodetic ulation results indicate that the Mean Success Rates Days, Istanbul, 1, pp. 179–196. (MSRs) of the methods for geodetic networks change Hekimoglu, S., and K.R. Koch (2000). How can Reliability depending on the number of outliers and the magnitude of Tests for Outliers be Measured? AVN. All. Vermessungsof the outliers. Nachrichten, 107/7, pp. 247–254. It seems that the most successful robust methods Hekimoglu S., and R.C. Erenoglu (2007). Effect of Heteroscedasticity and Heterogeneousness on Outlier Detection are the Danish, Hampel, L 1 norm and IGGIII methods. for Geodetic Networks. J. Geodesy, 81(2), pp. 137–148. To detect outliers in GPS network the robust methods Hofmann-Wellenhof, B., H. Lichtenegger, and J. Collins (1994). should be used since their MSRs are usually greater Global Positioning System, Theory and Practice, 3rd Ed., than the ones of the Baarda’s test (Baarda, 1968) (based Springer-Verlag, New York. Huber, P.J. (1964). Robust Estimation of a Location Parameter. on the authors’ previous studies).

References Amiri-Simkooei, A.R. (2003). Formulation of L1 Norm Minimization in Gauss- Markov Models. J. Surv. Eng., 129(1), pp. 37–43. Andrews, D. (1974). A Robust Method for Multiple Linear Regression. Technometrics, 16, pp. 523–531. Awange, J.L., and F.W.O. Aduol (1999). An Evaluation of Some Robust Estimation Techniques in the Estimation of Geodetic Parameters. Surv. Rev., 35(273), pp. 146–162. Baarda, W. (1968). A Testing Procedure for Use in Geodetic Networks. Publication on Geodesy, New series 2, no. 5., Netherlands Geodetic Commission, Delft. Beaton, A., and J. Tukey (1974). The Fitting of Power Series, Meaning Polynomials, Illustrated on Band-Spectroscopic Data. Technometrics, 16, pp. 146–185. Berber, M., and S. Hekimoglu (2001). What is the Reliability of Robust Estimators in Networks? In: Proceedings of the 1st International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS, ETH, Zurich, pp. 61–66. Donoho, D.L., and P.J Huber (1983). The Notion of Breakdown Point. In: Bickel PJ, Doksum KA, Hodges JL Jr (eds) A Festschrift for Erich L. Lehmann Wadsworth, Belmont, pp. 157–184.

Ann. Math. Stat., 35, pp. 73–101. Huber, P.J. (1981). Robust Statistics, John Wiley and Sons, New York. Koch, K.R. (1996). Robuste Parameterschaetzung. AVN. All. Vermessungs-Nachrichten, 103, pp. 1–18. Koch, K.R. (1999). Parameter Estimation and Hypothesis Testing in Linear Models, 2nd Ed., Springer-Verlag, Berlin. Lenstra, A.K., H.W. Lenstra, and L. Lovasz (1982). Factoring Polynomials with Rational Coefficients. Math. Ann. 261, pp. 515–534. Rousseeuw, P.J., and A.M. Leroy, (1987). Robust Regression and Outlier Detection, John Wiley and Sons, Inc., New York. Santos, M.C., P. Vanicek, and R.B. Langley (1997). Effect of Mathematical Correlation on GPS Network Computation. J. Surv. Eng., 123(3), pp. 101–111. Wieser, A. (2002). Robust and Fuzzy Techniques for Parameter Estimation and Quality Assessment in GPS, Shaker Verlag, Aachen. Wilcox, R.R. (1997). Introduction to Robust Estimation and Hypothesis Testing, Academic Press, New York. Xu, G. (2003). GPS – Theory, Algorithms and Applications, Springer Heidelberg, in English. Yang Y., L. Song, and T. Xu (2001). Robust Estimator for the Adjustment of Correlated GPS Networks. In: Carosio A, Kutterer H (Eds.), First International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS, IGPBericht Nr. 295, pp. 199–207.

Geometry-Based TCAR Models and Performance Analysis Yanming Feng and Chris Rizos

Abstract The paper presents a methodology that combines geometry-based models and geometric constraints for improved three carrier ambiguity resolution (TCAR). First of all, a general modelling strategy using three or more phase-based ranging signals is presented. The strategy can generally identify three best “virtual” signals to allow for more reliable AR under certain observational conditions characterised by ionospheric and tropospheric delay variability, level of phase noise and orbit accuracy. The selected virtual signals often have minimum or comparatively low ionospheric effects, and thus are known as ionospherereduced virtual signals. As a result, the ionospheric parameters in the geometry-based observation models can be eliminated for long baselines, typically those of length tens to hundreds of kilometres. Secondly, the general formulation of the equation system for the ambiguity resolution problem is examined. This general formulation enables the combination of apriori position and ambiguity and ionosphere information, to achieve reliable integer solutions or high availability of Real-Time Kinematic (RTK) solutions. Thirdly, numerical experiments have been performed and three sets of dual-frequency GPS data have been collected over baselines of length 21, 56 and 74 km and the performance benefits of the proposed algorithms have been investigated. Results confirm that the AR with the ionosphere-reduced NL signals, instead of the original L1 and L2 signals, performs better over

Yanming Feng Faculty of Information Technology Queensland University of Technology, GPO Box 2434, Q4001, Australia Chris Rizos School of Surveying and Spatial Information Systems The University of New South Wales, Sydney, NSW 2052, Australia

longer baselines. For instance, the AR success rate for the 74 km data set is improved from 88 to 93%. Introducing geometric constraints, particularly, apriori position and ionospheric corrections predicted from the previous epochs, AR reliability is further improved, to 98% for the above data set. Keywords Three carrier ambiguity resolutions real-time kinematic · Ionosphere-reduced signals · GPS and Galileo

1 Introduction Significant research efforts have been made towards carrier phase ambiguity resolution (AR) and position estimation (PE) using three or more GPS or Galileo signals in the past decade. The earlier studies by Forssell et al. (1997) described the Three Carrier Ambiguity Resolution (TCAR) method, and De Jonge et al. (2000) and Hatch et al. (2000) proposed the Cascade Integer Resolution (CIR) method. Both early TCAR and CIR methods use essentially the same geometry-free bootstrapping procedure. They start with the extra-widelane (EWL: λ ≥ 2.93 m) signal of the two closer L-band carriers, for instance L2 and L5 , using the combined code measurements on the same carriers. The ambiguity is resolved by rounding the “float” value to its nearest integer quite reliably. With this information the widelane (WL, 0.75 m ≤ λ2.93 m) combination of the two other L-band carriers, e.g. L1 and L2 , can be resolved through rounding. With the first two ambiguity-resolved signals, the ambiguity of the third signal, e.g. L1 , is resolved with the same rounding process. Han and Rizos (1999)

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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identified some more efficient signals for AR in multiple-frequency cases. Teunissen et al. (2002) compared TCAR, CIR and the Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) methods for triple-frequency cases, suggesting that the TCAR and CIR methods are designed for use with geometry-free models and the LAMBDA method be used for the geometry-based model. Werner and Winkel (2003) used the same TCAR geometry-free analysis, and presented success-rate results for the Galileo code and frequencies on L1 , E6, E5A and E5B. In more recent efforts by Feng and Rizos (2005), Feng and Moody (2006), Hatch (2006) and Vollath et al. (2005), the concepts and algorithms of TCAR or Multiple Carrier Ambiguity Resolution (MCAR) were proposed for use with the geometry-based models as well. There are two widely recognised problems in triple- and dual-frequency based AR: (1) distancedependent problem; and (2) low overall AR success rate problem. The former is related to the effects of orbital, ionospheric and tropospheric errors in the double-differenced carrier phase measurements. To address this problem, a traditional approach is to introduce additional state parameters in the measurement equations. This will increase the computational burden on the system (Vollath et al., 2005) and require measurements collected over a longer period in order to overcome the under-determined problem and/or obtain stable solutions. Han and Rizos (1999) proposed a geometry-free procedure resolving three virtual signals without distance constraints. Feng and Rizos (2005) outlined a distance-independent, geometry-free method for performing ambiguity resolution with three GNSS signals, comprising three major processes, including resolving two WL ambiguities and one medium-lane ambiguity. Similar studies were conducted by Hatch (2006), focusing on two of the three WL carrier phase differences formed from the difference of pairs of the fundamental phase measurements. These differences are ambiguity-resolved using ionospheric-matching code measurements and are then combined into a refraction-corrected composite measurement. While this WL composite is quite noisy, it can be smoothed with a refraction-corrected, composite measurement with much lower noise. It was confirmed that although the smoothing process may require some minutes to attain the optimal accuracy level, the result should significantly extend the ranges over which Real-Time Kinematic (RTK) positioning

Y. Feng and C. Rizos

results can be obtained without requiring modelling of the ionosphere. Feng (2007) presented a general modelling strategy for improved AR and positioning estimation (PE) using three or more phase and code ranging signals. Given three L-band frequencies, the proposed strategy can generally identify the three best virtual signals to allow for more reliable AR under certain observational conditions, as characterised by differing ionospheric and tropospheric delay variability, level of phase noise and orbit accuracy. The selected virtual signals often have minimum or comparatively low ionospheric effects, and thus are known as ionosphere-reduced virtual signals. As a result, the effects of the first-order ionospheric delays in the geometry-based observation models can be eliminated for AR process over the base-rover distances as long as tens to hundreds of kilometers. For position estimation, ionosphere-free or ionosphere-corrected signals should be used to eliminate the ionospheric effects. The low AR success rate problem becomes more serious when using a geometry-free method to resolve WL and medium-Lane (ML, 0.19 m ≤ λ < 0.75 m) ambiguities, or using a geometry-based method for AR over long distances. Low overall AR success rate means that rapid resolution of all ambiguities may not be possible. This problem was first examined by Teunissen et al. (1999), introducing a method known as Partial Ambiguity Resolution (PAR), albeit for the dual-frequency case. Teunissen et al. (2000) and De Jonge et al. (2000) also applied PAR methods to the triple-frequency case. Jonkman et al. (2001) proved analytically that full ambiguity resolution would not likely be possible over long baselines. AR reliability problems have also been extensively studied by Verhagen (2006a, b). This paper will address both the distance-dependent and the low AR success rate problems in the context of TCAR or MCAR. Section 2 outlines the geometrybased TCAR models using the ionosphere-reduced virtual signals based on proposals by Feng (2007). Section 3 examines the general formulation of the equation system for the AR problem. This general formulation enables the combination of apriori positions, partially resolved or known ambiguities and geometric constraints for apriori ionospheric information. In Sect. 4, experimental results that demonstrate the performance benefits of different modelling strategies are presented.

TCAR Models and Performance Analysis

647

2 TCAR Models with IonosphereReduced Virtual Signals

Note that the magnitude of the phase delay is no longer equal to that of the group delay if the higher order effects are included. With three GPs signals, the linear combination of The geometry-based TCAR models in Feng (2007) three fundamental signals can be generally formulated perform AR with two parallel or separate processes. as (Han and Rizos, 1999; Feng and Rizos, 2005): One identifies the two EWL signals and resolves the integer ambiguities quite reliably with a geometry-based i · f1 · φ1 + j · f2 · φ2 + k · f5 · φ5 φ(i,j,k) = (3) estimator over long distances. Another identifies an ini · f1 + j · f2 + k · f5 dependent narrow-lane (NL) signal which is used together with a WL to resolve both of the ambiguities where i, j, k are integer coefficients,. This linearly comwith a geometry-based integer estimation and search bined signal has the virtual frequency: algorithm. It is envisaged that these two AR processes can support RTK positioning over baselines of tens to f(i,j,k) = i · f1 + j · f2 + k · f5 (4) hundreds of kilometres in length, providing decimetre and centimetre accuracy results respectively. the virtual wavelength: To introduce the abovementioned TCAR models, the definitions of the virtual signals and related pac (5) λ(i,j,k) = rameters are first given. Considering the satellite and f(i,j,k) receiver clock biases and various propagation delays, the observation equations of the carrier phase measure- and the cycle ambiguity: ments, φi , for the satellite S, the receiver R, and the carrier signal Li (e.g. φi ) can be expressed as: N(i,j,k) = i · N1 + j · N2 + k · N5 (6) With any three GNSS signals, it can be assumed that the same conditions for frequencies: f1 > f2 > f5 , and (1) f2 − f5 < f1 − f2 , apply. + δtrop − λi (ϕ0S − ϕ0R + N)i + εφi Whilst the frequency-independent errors remain the same in the combination (3), the frequency-dependent where ρ is the geometric distance between satellite S errors will change. The magnitude of the first-order and and receiver R antennas; c is speed of radio waves second-order ionospheric effects on the linear combiin vacuum; δorbit is the satellite orbital error in units nation can be expressed as: of metres, δtR is the receiver clock error of all comf2 (i/f1 + j/f2 + k/f5 ) K1 ponents in units of seconds; δtS is the satellite clock δI(i,j,k) = 1 − i · f1 + j · f2 + k · f5 f21 error of all components in units of seconds; δtrop is the tropospheric propagation delay (in metres); λi is f3 (i/f21 + j/f22 + k/f25 ) K2 − 1 the wavelength of the carrier Li (in metres) with frei · f1 + j · f2 + k · f5 2f21 quency fi in Hz; ϕ0S is the initial phase of the satelK1 K2 lite oscillator (cycle), which is satellite-dependent; ϕ0R = − β(i,j,k) 2 − θ(i,j,k) 3 (7) f1 2f1 is the initial phase of the oscillator (cycle), which is receiver-dependent; Ni is the integer ambiguity of the phase measurement in units of cycles; εϕI is the re- where K1 is a constant depending on the slant total ceiver code noise (in metres), including phase multi- electron content; β(i,j,k) is known as the Ionospheric path errors. δIφI is the ionospheric phase delay on the Scale Factor (ISF) defined with respect to the firstorder ionospheric delay on the L1 carrier: K21 ; θ(i,j,k) carrier Li , given as (Klobuchar, 1996): f1 is the second-order ISF, defined with respect to the second-order delay K22 . Assuming that the noise level   2f1 K1 K2 δIφi = − 2 − 3 (2) is the same for all the carrier phase measurements, i.e. fi 2fi σ = σφ1 = σφ2 = σφ5 , the phase noise variance of φi = ρ + δorbit + c(δtS − δtR ) + δIφi

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the combined signal (3) can also be given relative to the L1 signal:

σ2 (φ(i,j,k) ) = =

)2

(i · f1 )2 · σ2φ1 + (j · f2 )2 · σ2φ2 + (k · f5 )2 σ2φ5 (i · f1 + j · f2 + k · f5 )2 )2

(i · f1 + (j · f2 + (k · f5 )2 2 σφ1 ≡ μ2(i,j,k) σ2φ1 (i · f1 + j · f2 + k · f5 )2

(8)

The equation for a virtual phase measurement is: ∇φi,j,k = ∇ρ + ∇δorbit + ∇δtrop ∇K1 ∇K2 − β(i,j,k) 2 − θ(i,j,k) f1 2f22

(9)

− λ(i,j,k) ∇N(i,j,k) + ε ∇φ(i,j,k) where the symbol “ ∇” represents the DD operation applied to the quantity immediately to the right; ∇ρ is the theoretical value of the DD range; ∇δorbit and ∇δtrop , respectively are the satellite orbit error and the residual tropospheric error in the DD observables. Considering all the effects of the first and second order ionosphere errors as given in Eq. (2), the phase noise εφ1 , on the doubled differenced (DD) phase measurements, the Total Noise Level (TNL) can be defined as

σTN

  2 2 2 2  β  (i,j,k) σ ∇δI,φ1 + θ(i,j,k) σ ∇δI2,φ1     +μ2(i,j,k) σ2 ∇φ1 + σ2 ∇δtrop + σ2 ∇δorbit = λ2(i,j,k) (10)

where the uncertainty terms, σ ∇δI,φ1 , σ ∇δI2,φ1 , σ ∇δtrop , σ ∇δorbit in Eq. (10) generally represent the effects of respective bias or modelling/correction errors. Therefore, given a set of error budget parameters, it is possible to find signals with the smallest σTN . Clearly, signals with the minimal value in the same category should be considered the best choice for AR. Based on this analysis criterion, the two best EWL/WL signal from all possible combinations in the category of two independent EWL/WL signals can be identified. The first one is always the EWL φ(0,1,−1) . In each GNSS three-frequency service there is one more superior EWL that has a minimal (or near minimal) ISF. For instance, in the GPS case the signal is φ(1,−6,5) ,

where the value of ISF is −0.0744 and wavelength is 3.256 m. This signal was first identified by Han and Rizos (1999). Notably this signal has a large PNF equal to 103.8. The TNL criteria (11) can consider the effects of all the factors. Theoretically, the best signal choice depends on the observational conditions in the real world, which may vary significantly from time to time. However, it is also observed that the TNL characteristics that distinguish the best and next best EWL/WL signals are not significantly different. It is therefore more practical to choose one virtual signal which has overall the lowest total noise levels for all observation conditions. For instance, in the GPS case, the best second EWL signal could be φ(1,−6,5) . For the Galileo L1 , E6 and E5A signals, the best EWL signal φ(1,−3,2) apparently outperforms all other EWL/WL choices. It is possible to also identify the third combination from a new category, in which each virtual signal is linearly independent of the previous two EWL/WL signals or their derivatives. The most useful NL signals are identified in both two-frequency and three-frequency cases. In all two-frequency cases, including GPS L1 and L2 , GPS L1 and L5 , Galileo tropospheric error, δtrop , the orbital error, δorbit , and L1 and E6, Galileo L1 and E5A, it is found that the best NL signals include φ(2,−1,0) , φ(2,0,−1) for shorter distances, and φ(4,−3,0) , φ(4,0,−3) for longer distances. In other words, AR with the chosen NL signals (with WL φ(1,−1,0) or φ(1,0,−1) ) is more effective than using ML signals φ1 or φ2 or φ5 . The linear equation for least squares estimation of any of the above EWL integers from a single data epoch, for instance ∇N(1,−6,5) for φ(1,−6,5) , can be expressed as: ⎡

⎤ ⎡ ⎤⎡ ⎤ ∇PIF − ∇ρ0 A 0 0 δX ⎣ ∇φ(1,−6,5) − ∇ρ0 ⎦ = ⎣ A −λ(1,−6,5) ⎦ ⎣ 0 ∇N(1,−6,5) ⎦ ∇φ(0,1,−1) − ∇ρ0 ∇N(0,1,−1) A 0 −λ(0,1,−1) ⎡ ⎤ ε ∇P(1,1,0) + ⎣ ε ∇φ(1,−6,5) ⎦ ε ∇φ(0,1,−1)

(11) where ∇PIF is the DD ionosphere-free (IF) code measurement; ∇ρ0 is the computed DD range; δX is the 3-by-1 position vector to be estimated (with respect to the approximate position vector X0 ) and A is the design observational matrix. Resolution of the ambiguity of the third virtual signal can be performed, whether or not the first two EWL ambiguities have been resolved. But a ML/NL signal,

TCAR Models and Performance Analysis

for instance, φ(4,0,−3) , has to be used along with a WL signal to enable the estimation problem to be overdetermined with measurements from a single epoch. The deterministic models with one WL signal and one NL signal respectively, are: ⎡

⎤ ⎡ ⎤⎡ ⎤ ∇P(1,0,1) − ∇ρ0 A 0 0 δX ⎣ ∇φ(1,0,−1) − ∇ρ0 ⎦ = ⎣ A −λ(1,0,−1) ⎦ ⎣ ∇N(1,0−1) ⎦ 0 A 0 −λ(4,0,−3) ∇φ(4,0,−3) − ∇ρ0 ∇N(4,0−3) ⎤ ⎡ ε ∇P(1,0,1) + ⎣ ε ∇φ(1,0,−1) ⎦ (12) ε ∇φ(4,0,−3)

649

system, which includes ionospheric and tropospheric parameters and treats apriori information of each state parameter as a virtual observation. It is clear that the observation equations for combined code and phase measurements discussed in the previous section are the special case where the parameters for the ionospheric and tropospheric terms were treated as known, despite that the fact that the effects of the residual errors are left to be accounted for in the stochastic models.

3.2 Modelling the Apriori Position and is the DD combined code measureAmbiguity Information

where ∇P(1,1,0) ment with similar definition as in Eq. (3). The AR problem with Eq. (12) involves L1 and L5 only, which is actually a dual-frequency based AR problem using a WL and a NL signal. In this work, the φ(1,−1) and φ(4,−3) or φ(2,−1) are used together to resolve the ambiguities for GPS L1 and L2 signals. The linear equations are explicitly expressed as:

In kinematic positioning one important goal is to compute the current position using the ambiguityparameters only, whereas the effects of distancedependent ionospheric and tropospheric errors are either reducible or negligible under certain medium to long distance circumstances. Given an adequate variance-covariance matrix (vc-matrix) for the error ⎡ ⎤ ⎡ ⎤⎡ ⎤ ∇P(1,1) − ∇ρ0 A 0 0 δX term in each case, AR can then be performed using ⎣ ∇φ(1,−1) − ∇ρ0 ⎦ = ⎣ A −λ(1,−1) ⎦ ⎣ ∇N(1,−1) ⎦ 0 an available well-defined AR algorithm such resolved A 0 −λ(4,−3) ∇φ(4,−3) − ∇ρ0 ∇N(4,−3) phase measurements of the current epoch only. This ⎡ ⎤ ε ∇P(1,1) + ⎣ ε ∇φ(1,−1) ⎦ (13) can avoid the cumulative effects of incorrect posiε ∇φ(4,−3) tion solutions from previous epochs and inaccurate assumptions of user dynamics. However, the meaThe linear observation Eqs. (11) and (12) allow three surements from previous epochs, or positional and phase ambiguities to be resolved in two separate or ambiguity solutions from previous epochs, should be parallel processes, while the dual-frequency ambigu- used somehow to improve AR at the current epoch. To make use of phase measurements from multiple ity resolution problem described by the linear Eq. (13) epochs to improve AR solutions one can assume the is solved in one process. The models (11) and (12) or set of ambiguity parameters remain constants over the (13) take the simplest forms that include position and period of the samples, and allow the positional states ambiguity as LAMBDA. to vary in the AR equation system. This has been a standard practice in kinematic positioning. The idea of taking advantage of apriori position solutions is based 3 Modelling Strategies for Improving on the assumption that the user’s position state of the previous epoch has been precisely determined and can AR Performance be transferred to the current position using the deltaposition between two epoch determined with 3 or more 3.1 General Formation of DD Observation cycle-slip free DD phase measurements. In the singleEquations for AR base RTK RTK case where there are (m + 1) satellites commonly in view, the observation Eqs. (12) or (13) of In general, the observation equations for an AR prob- the current epoch tk are rewritten in the overall vector lem can be formed with additional state parameters and format as: measurements, if needed to improve AR performance. Table 1 gives a general formulation of the AR equation δLk = Ak δXk + Bk ∇N + εk (14a)

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Table 1 General formulation of observation equations for an AR problem Observation type

State parameters Position √

Virtual code

Ambiguity

√

Virtual phase Apriori Position Apriori Ambiguity Apriori Ionosphere Apriori Troposphere

Ionosphere √

√

√

√

√ √ √ √

where δLk is the 3m-by-1 measurement vector; Ak is the 3m-by-3 designed matrix for the 3-by-1 user state vector δXk to be estimated (with respect to approxiˆ 0 ; Bk is the 2m-by-2m diagonal mate position vector X k matrix for the 2m-by-1 ambiguity vector ∇N. The stochastic model of Eq. (14a) is described as E(εk ) = 0,

Troposphere √

Cov(εk ) = Rk

Noise terms in Stochastic models Orbital, code noise & multipath Orbital, phase noise & multipath Error in the apriori position Error in the ambiguity observational equation Error in the apriori ionosphere Error in the apriori troposphere

where Qk is the m1 -by-m1 vc-matrix for the error vector ε(δk −δk−1 ) . The apriori position of the current epoch is then propagated from the previous epoch, that ˆ k−1 + δX ˆ k,k−1 , whose vc-matrix is obtained is, X0k = X as follows: −1 −1 CX˜ k = (ATk−1 R−1 + (aTk Q−1 k−1 Ak−1 ) k ak )

(18)

(14b) The normal equation system for the ambiguity estimation is then given as:

Given the 3m-by-3m vc-matrix Rk , the normal equation for AR estimation using measurements of the current epoch would become: ⎡ ⎣

ATk R−1 k Ak

ATk R−1 k Bk

BTk .R−1 k Ak

BTk R−1 k Bk

⎤ ⎦





δXk = ∇N



ATk R−1 k δLk



BTk R−1 k δLk

(15) The user state is updated every epoch with the current integer-fixed NL or IF phase measurements, that ˆk = X ˆ 0 + δX ˆ k. is, X k With the m1 -by-1 NL or IF cycle-slip free phase difference vector between two consecutive epochs, δk − δk−1 , and the m1 -by-3 design matrix ak , being ˆ k−1 , the computed with respect to the last epoch state X linear equation δk − δk−1 = ak δXk,k−1 + εδk −δk−1

(16)

where δXk,k−1 = δXk − δXk−1 and the effect of ak ˆ k−1 on the solubeing approximately computed from X tion is ignored. The least-square delta-position vector is estimated as follows: ˆ k,k−1 = (aTk Q−1 ak )−1 aTk Q−1 (δk − δk−1 ) (17) δX k k

⎡ ⎣

⎡ T −1 ⎤

Ak Rk δLk δX k ⎦ ⎦ = ⎣ T −1 ∇N BTk R−1 Bk Rk δLk k Bk

−1 T −1 ATk R−1 k Ak + C ˜ Ak Rk Bk Xk

BTk .R−1 k Ak

⎤



(19)

The LAMBDA method can be used to complete the ambiguity resolution process with Eq. (19). Comparing Eqs. (19) and (15), the apriori position information adds constraints to the equation system, thus improving AR. The user position state can then be updated with ambiguity-fixed phase measurements: ˆk = X ˆ 0 + δX ˆk X k

(20)

It is important to note that the AR and PE process in Eq. (19) is based on the measurements of the current ˆ k,k−1 provides a epoch, whereas the delta-position δX time update to the apriori state, whose errors, due to approximation of Eq. (16) do not accumulate over times. Taking advantage of apriori known ambiguities is based on the fact that a subset of integer ambiguities can be estimated from a geometry-free process using code measurements. For instance, we have the observation equation for the WL ambiguity in Eq. (13)

TCAR Models and Performance Analysis

651

1 ∇P(1,1) − ∇φ(1,−1) = λ(1,−1) ∇N(1,−1) +ε ∇P(1,1) A DD ionospheric bias on L1 , δI = ∇K , can f2 (21) be estimated using the ambiguity-fixed DD1 phase A general formation of a geometry-free observation measurements for L and L measurements. For each 1 2 equation for the 2m-by-1 ambiguity vector can be ex- satellite-receiver pair, the prediction value δ˜I at k pressed as follows: the epoch tk is based on the 1st-order or 2nd-order polynomial fitted with the observed DD ionospheric δNk = k ∇N + εNk (22) biases over the past 60 s or generally n epochs, δIk−n , δIk−n+1 , . . . , δIk−1 , the WL measurement at ˜ where δNk is the 2m-by-1 apriori ambiguity vectors, the current epoch tk is then corrected with δ(1,−1) δIk . In order to be able to take advantage of apriori ionoincluding a subset of partially-resolved ambiguities; k is the 2m-by-2m diagonal matrix with elements of sphere information obtained from the past epochs as wavelengths and zeros; εNk is the 2m-by-1 noise vec- discussed above, a necessary condition must be sattor for the apriori ambiguity δNk . Given the vc-matrix isfied that the integer ambiguities in the previous n Dk for their uncertainty that would be smaller through epochs were resolved correctly so that the DD ionoaveraging processing over multiple epochs, the normal spheric delays could be precisely observed with phase measurements. Ionospheric prediction to the current Eq. (19) would become epoch does not need to assume continuation of phase   measurements to the current epochs.

−1 ATk R−1 ATk R−1 δXk k Ak + CXk k Bk ˜ T −1 ∇N BTk .R−1 BTk R−1 k Ak k Bk + k Dk k

ATk R−1 k δLk = (23) 4 Experimental Analysis −1 T Bk Rk δLk + Tk D−1 δN k k

An easy way to keep the partially-fixed ambiguity subset unchanged in the estimation and integer search process is to assign small values to the respective elements of the vc-matrix Dk .

3.3 Refining WL Phase Measurements with Apriori Ionospheric Information While apriori position and ambiguity solutions can provide constraints for AR resolution at the current epoch, refining a subset of phase measurements can lead to an improvement of AR performance for the full set of phase measurements. For instance, refined WL in Eq. (13) will improve the AR success rates of both the WL and NL signals. The major error source in the WL signal is the effect of the ionosphere. Therefore, introduction of apriori ionospheric corrections to the equation system is the measure to take ed. Apriori ionospheric knowledge may be obtained through interpolation of a regional/local ionospheric grid. In the following, the authors describe an alternative method which can predict the current ionospheric delay using the ionospheric estimates obtained from several previous epochs.

With the ionosphere-reduced signals, the AR success rate for the two EWL signals can be almost 100% over a distance of hundreds of kilometres. This performance is quite predictable from the low Total Noise Level defined by Eq. (10), as firstly shown by Feng (2007). The major concern is now the AR performance of the ionosphere-reduced NL signals, whose ambiguity can be resolved along with the WL signal over medium distances, e.g. 20–100 km. Carrier-phase AR involves modeling of distance-dependent biases, dealing with distance-independent errors, and estimation and search algorithms, regardless of whether two or three carrier GNSS signals are used. Failure in any of these processes would affect the performance. However, it is possible to compare performance improvement with/without the proposed strategies or constraints. Using the GPS dual-frequency data it is possible to demonstrate the performance improvement of the proposed NL model in Eq. (13). It is generally the case that reliable and successful AR can be achieved with data from single epochs over short baselines by commercial RTK systems, typically up to 20 km in length. On the other hand, long range AR is also achievable if the data period is sufficiently long. Therefore our interest is the AR performance of the NL signals for longer baselines and when the AR is

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Table 2 AR performance results from Scenario I and Scenario II CORS Station Total number of IDs epochs

Scenario I: P(1,1) · ϕ(1,0) , ϕ(0,1)

Scenario II: P(1,1) , ϕ(1,−1) , ϕ(4,−3)

RTK AR success # of wrong RTK AR success # of wrong availability (%) rate (%) epochs/ integers availability (%) rate (%) epochs/ integers P474–P478 (21 km) P473–P478 (57 km) P473–P474 (74 km)

5760

98.02

99.04

114/699

99.96

99.93

4/28

5760

77.68

91.98

1285/5806

87.57

96.44

716/2578

5758

65.79

88.24

1976/8370

75.69

93.16

1400/4943

Table 3 AR performance results from Scenario III and Scenario IV CORS Station Scenario III: P(1,1) · ϕ(1,−1) , ϕ(4,−3) with apriori position ID

P474–P478 (21 km) P473–P478 (57 km) P473–P474 (74 km)

# of wrong epochs/integers

Scenario IV: P(1,1) , ϕ(1,−1) , ϕ(4,−3) with apriori position and ionosphere

Overall RTK accuracy

RTK availability (%)

AR success rate (%)

RMS at N, E, Up (cm)

100

RTK availability (%)

AR success rate (%)

99.98

99.99

1/7

99.99

97.74

99.59

130/297

98.51

88.18

97.65

681/1782

92.39

# of wrong epochs/integers 1/3

0.8, 1.5, 3.3

99.73

85/193

1.1, 1.0, 2.3

98.07

438/1397

1.1, 1.3, 3.0

performed instantaneously with measurements from is obtained from ambiguity-fixed, ionosphere-free single epochs. phase measurements. Three GPS data sets for the January 1, 2007 col- r Scenario II: AR performed with φ(1,−1) , φ(4,−3) lected from the Continuously Operating Reference Staphase measurements from a single epoch and PE is tions (http://www.ngs.noaa.gov/CORS) were used for obtained from ambiguity-fixed φ(4,−3) phase meaperformance analysis. Stations and baseline informasurements. tion is given in Column 1 of Tables 2 and 3. Other r Scenario III: Based on Scenario II, the apriori columns provide the AR success rate and RTK availposition constraints are added to the AR equation ability results for four different solution scenarios. The system. r AR success rate, or AR reliability, is defined as the raScenario IV: Based on Scenario III, the ionospheric tio of the correct integers to the total number of DD correction predicted from the previous four epochs phase measurements. RTK availability (in terms of AR is added (60 s span). For the given date sets the prereliability) is defined as the percentage of time during diction can effectively remove the ionospheric bias which the integers are correctly fixed for all integers at term, thus refining the WL phase measurements to each epoch, and all the ambiguity-fixed solutions will a high precision. give the required centimetre-level accuracy (Feng and From these results, the following comments can be Wang, 2007). The data were processed using a research version of the QUT kinematic GPS positioning soft- made: ware using the standard LAMBDA ambiguity search r AR (Scenario II) with φ(1,−1) , φ(4,−3) meaalgorithms. The software can resolve integer ambigusurements outperforms AR (Scenario I) with ities using single or multiple data epochs, and recover φ(1,0) , φ(0,1) in all the cases in terms of RTK the correct integers on L1 and L2 simultaneously. For availability and AR success rate, especially for long the given data sets, the results were obtained from four baselines. AR scenarios: r Introducing apriori position information transferred r Scenario I: AR is conducted with φ(1,0) , φ(0,1) from the previous epoch can efficiently improve the AR success rate and RTK availability. phase measurements from a single epoch and PE

TCAR Models and Performance Analysis

r

653

Adding ionospheric corrections can slightly Feng Y and Rizos C (2005) Three Carrier Approaches for Future Global, Regional and Local GNSS Positioning Services: improve the AR success rate and RTK availability.

In general, in the GPS dual-frequency case, the AR with the ionosphere-reduced NL signals instead of the original ML signals perform better over longer baselines. The proposed constraints with apriori position and ionosphere derived from the previous epochs can efficiently improve the AR performance.

5 Concluding Remarks This paper has outlined an efficient three carrier ambiguity resolution (TCAR) modelling strategy. The TCAR modelling strategy can efficiently identify the best virtual signals for AR under certain defined criteria. In a three-frequency scenario, AR with the 2 best EWL and 1 NL signals would outperform others over long distances. In a dual-frequency case, 1 WL + 1 NL choices would perform better over medium distances. Experimental results with GPS dual-frequency data sets have demonstrated that AR with the ionospherereduced NL signals instead of the original ML signals perform better over longer baselines in terms of AR success rate and RTK availability. The paper has also examined the general formulation of the equation system to permit the combination of apriori positions, partially determined or apriori ambiguities and apriori ionospheric information. Theoretically this generalisation improves AR reliability on an epoch-by-epoch basis, instead of direct combination of measurements from previous epochs. Results from the same data sets have demonstrated that the proposed modelling constraints with apriori position and ionosphere information derived from previous epochs can efficiently improve the AR performance. Acknowledgments The work was carried out with financial support from the Cooperative Research Centre for Spatial Information (CRCSI) project 1.4:- “Deliver precise positioning services in regional areas”, 2007–2010.

References De Jonge PJ, Teunissen PJG, Jonkman NF, Joosten P (2000) The Distributional Dependence of the Range on Triple Frequency GPS Ambiguity resolution, Proceedings of ION-NTM 2000, pp. 605–612, 26–28 January, Anaheim, CA, USA.

Concepts and Performance Perspectives, Proceedings of ION GNSS 2005, pp. 2277–2787, 13–16 September, Long Beach, CA, USA. Feng, Y (2005) Future GNSS Performance, Predictions Using GPS with a Virtual Galileo Constellation, GPS World, March, pp. 46–52. Feng Y, Moody M (2006) Improved Phase Ambiguity Resolution Using Three GNSS Signals, PCT/AU2006/000492, http://www.wipo.int/pctdb, April 2006. Feng Y (2007), Three Carrier Ambiguity Resolution Using Ionosphere-Reduced GNSS Signals, submitted to Journal of Geodesy. Feng Y and Wang J (2007) Exploring GNSS RTK Performance Benefits with GPS and Virtual Galileo Measurements, Proceedings of ION NTM 2007, pp. 218–226, 22–24 January, San Diego, CA. Forssell B, Martin-Neira M, Harris RA (1997) Carrier Phase Ambiguity Resolution in GNSS-2. Proceedings of ION GPS97, pp. 1727–1736, 16–19 September Kansas City. Han S, Rizos C (1999) The Impact of Two Additional Civilian GPS Frequencies on Ambiguity Resolutions Strategies, Proceedings of ION Annual Technical Meeting, pp. 315–321, 28–30 June, Cambridge, MA, USA. Hatch R, Jung J, Enge P, Pervan B (2000) Civilian GPS: The Benefits of Three Frequencies, GPS Solution, Vol 3, No 4, pp. 1–9. Hatch R (2006) A New Three-Frequency, Geometry-Free Technique for Ambiguity Resolution, Proceedings of ION GNSS 2006, pp. 309–316, 26–29 September, Fort Worth, TX. Klobuchar JA (1996) Ionospheric Effects on GPS, Global Positioning System: Theory and Application, Vol. I, B. Parkinson, J. Spilker, P. Axelrad, and P. Enge (Eds.), American Institute of Aeronautics and Astronautics, pp. 485–515. Mirsa P, Enge P (2004) Global Positioning Systems, Signals, Measurements and Performance, Ganga-Jamuna Press, pp. 227–254. Teunissen PJG, Joosten P, Tiberius CCJM (1999) Geometry-free ambiguity success rates in case of partial fixing, Proceedings of ION-NTM 1999, pp. 201–210, 25–27 January, San Diego, CA, USA. Teuniessen PJG, Jonkman NF (2001) Will Geometry-Free Full Ambiguity Resolution be Possible At All for Long Baselines, Proceedings of ION-NTM 2001, pp. 271–280, 22–24 January, Long Beach, CA, USA. Werner WJ, Winkel (2003) TCAR and MCAR Options, With Galileo and GPS, Proceedings of ION GPS/GNSS 2003, pp. 201–210, 9–11 September, Portland, OR, USA. Volltah U (2005) Ambiguity estimation of GNSS Signals for Three and More Carriers”, US patent application 2005010248, pp. 1–40. Verhagen S (2006a) How Will the New Frequencies in GPS and Galileo Affect Carrier Phase Ambiguity Resolution? Inside GNSS, 2. Verhagen S (2006b) Improved Performance of Multi-Carrier Ambiguity Resolution Based on the LAMBDA Method. In Proceedings of Navitec 2006, ESA-ESTEC, Noordwijk NL, p. 8.

Real-Time Kinematic OTF Positioning Using a Single GPS Receiver Y. Gao and M. Wang

Abstract This paper will investigate the potential to conduct Real-Time Kinematic (RTK) On-The-Fly (OTF) positioning using a single GPS receiver. OTF refers to fast ambiguity resolution methodology to resolve the carrier phase ambiguities to their integer values. The paper will first analyze the characteristics of satellite and receiver initial phase biases and how to recover the integer property of the carrier phase ambiguities in Precise Point Positioning (PPP) mode and then investigate the feasibility of OTF ambiguity resolution in PPP. A zero-baseline test scheme has been developed to assess the characteristics of the receiver initial phase biases. The test results indicate that the receiver initial phase bias may not be calibrated since it changes as the receiver powers on and off and experiences full loss of lock. The bias however can be removed through observation differencing between satellites. The satellite initial phase bias can be calibrated since it is spatio-temporally stable. Based on the above, a PPP-based RTK OTF positioning system has been proposed using differential observations between satellites. Simulation tests indicate that OTF can reduce the initialization time by 65% on average compared to the float ambiguity position solutions.

Y. Gao Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N.W., Calgary, AB, Canada, T2N 1N4 M. Wang Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N.W., Calgary, AB, Canada, T2N 1N4

Keywords Global Positioning System (GPS)· Initial phase bias· Ambiguity resolution· Precise point positioning (PPP)

1 Introduction Precise Point Positioning (PPP) is a positioning method that performs precise position determination using a single GPS receiver (Zumberge et al., 1997; Kouba and H´eroux, 2001; Muellerschoen et al., 2001; Bisnath and Langley, 2002; Leandro and Santos, 2006). The method arose from the advent of widely available precise GPS orbit and clock products which can be applied to substantially reduce the errors in GPS satellite orbits and clocks, two of the most significant error sources in GPS positioning. PPP is different from double-difference based RTK positioning that requires access to observations from one or more base stations with known coordinates. Since no local base receivers are required, PPP is being considered as an efficient option wherever precise positioning and navigation is required but a base station is not available or very costly to temporarily erect. PPP has also found application in other areas such as atmosphere remote sensing and precise time transfer (Heroux et al., 2004). PPP requires access to precise orbit and clock products with centimeter level accuracy. Such precise orbit and clock products are now widely available, in postmission and real-time, from a number of organizations such as the International GNSS service (IGS), Natural Resources Canada (NRCan) and the Jet Propulsion Laboratory (JPL). IGS, for example, offers precise orbit and clock products with varying accuracy levels and latency from the Final, Rapid to Ultra-Rapid products.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

655

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Driven by significant demand to use PPP form realtime applications, IGS has recently begun a real-time data products pilot project. Several institutes are now providing predicted orbits and clocks for real-time PPP use. In Gao and Chen (2005), a comparable positioning accuracy has been demonstrated using IGS Final products and JPL real-time orbit and clock correction products, which is creating increased interest to develop PPP-based RTK systems to support real-time precise positioning applications. Note that although there are already services providing PPP solutions in real time such as StarFire (Dixon, 2006), they are not able to support real-time RTK function since their position determination is in ambiguity float mode without OTF ability. Since current PPP methods can provide only ambiguity float position solutions, they require a significant initialization time period, typically in the range of 30 min, before the solutions approach decimeter to centimeter accuracy. Re-initialization is necessary in the presence of interrupted GNSS signal availability due to loss of tracking lock on a minimum number of satellites, resulting in a long time period of degraded positioning results until float solution convergence. Such long initialization and re-initialization time is becoming the major challenge to the development of a PPPbased RTK system. There exist different methods that can be applied to reduce the PPP initialization time such as using improved stochastic model as shown in Gao and Shen (2001). Since the long initialization time is due to significant time period needed for the float phase ambiguity estimates to converge to constant values, OTF fast ambiguity resolution to resolve the carrier phase ambiguities to their integer values has the potential to solve the long initialization time problem in PPP. OTF has been successfully applied in double-difference RTK systems which can significantly reduce the initialization time to the level of a few seconds. This paper will investigate the feasibility of OTF ambiguity resolution in PPP and the related challenge. Section 2 will discuss the major challenge to applying the OTF technique to PPP. Section 3 investigates the estimation method of receiver and satellite initial phase biases and examines their stabilities. Sections 4 and 5 investigate the integer ambiguity property recovery and the development of a PPP-based RTK OTF positioning system based on differential observations between satellites. Section 6

Y. Gao and M. Wang

presents simulation test results to demonstrate the reduction in initialization time.

2 PPP OTF RTK: The Challenge The un-differenced pseudorange and phase observation equations with a dual-frequency GPS receiver can be written as: Pi = ρ + c(dT − dt) + dor b + dtr op + dion/L i + ε(Pi )

(1)

i = ρ + c(dT − dt) + dor b + dtr op − dion/L i + λi [Ni + φr (t0 , L i ) − φs (t0 , L i )] + ε(i ) (2) where i = 1, 2 denote frequency L1 and L2 respectively; Pi is the measured pseudorange on L i (m); i is the measured carrier phase on L i (m); ρ is the true geometric range (m); c is the speed of light (m/s); dT is the receiver clock error (s); dt is the satellite clock error (s); dor b is the satellite orbit error (m); dtr op is the tropospheric delay (m); dion/L i is the ionospheric delay on L i (m); λi is the wavelength of L i (m); Ni is the integer phase ambiguity on L i (cycle); φr (t0 , L i ) is the initial phase bias of receiver on L i (cycle); φs (t0 , L i ) is the initial phase bias of satellite on L i (cycle); t0 is the time of phase lock to satellite signal; ε(.) is the noise including residual multipath (m). Here we assume that the multipath effect can be mostly eliminated by in-receiver mitigation algorithm or using a chock-ring antenna. In double-difference RTK processing, position determination is based on the following doubledifferenced observations: ∇ Pi = ∇ρ + ε( ∇ Pi )

(3)

∇i = ∇ρ + λi ∇ Ni + ε( ∇i )

(4)

where ∇ is the double-differencing operator. Here we have assumed a short baseline length between the base and rover receiver so that the residual tropospheric and ionospheric terms can be ignored. Since the doubledifferenced ambiguity parameter ∇ Ni in Eq. (4) is an integer, OTF ambiguity resolution can render the position solution accuracy to centimeter-level after the ambiguity parameters have been fixed to their integer values.

Real-Time Kinematic OTF Positioning

657

f 1 1 − f 2 2 In Gao and Shen (2002), a model known as “UofC WL = = ρ + c(dT − dt) f1 − f2 model” has been proposed to facilitate the ambiguity f1 resolution in PPP. The related observation equations + dor b + dtr op + dion/L1 are given below: f2 + λWL [N1 + φr (t0 , L 1 ) − φs (t0 , L 1 )] − λWL [N2 + φr (t0 , L 2 ) − φs (t0 , L 2 )]

PPi,i = 0.5[Pi + i ] = ρ + c(dT − dt) + dor b + dtr op

GF = 1 − 2

− φs (t0 , L i )] + ε(PPi,i )

=

f 12 1 − f 22 2

−

f 22

dion/L1

f 12 λ1 [N1 + φr (t0 , L 1 ) − φs (t0 , L 1 )] f 12

−

f 22

(8)

− λ2 [N2 + φr (t0 , L 2 ) − φs (t0 , L 2 )] + ε(GF )

= ρ + c(dT − dt) + dor b + dtr op

f 22 λ2 [N2

f 12 − f 22

+ λ1 [N1 + φr (t0 , L 1 ) − φs (t0 , L 1 )]

f 12 − f 22

+

+ ε(WL )

(5)

+ 0.5λi [Ni + φr (t0 , L i )

IF =

(7)

(6)

+ φr (t0 , L 2 ) − φs (t0 , L 2 )] f 12 − f 22

+ ε(IF ) where PPi,i is the average of pseudorange and carrier phase observations on L i (m); IF is the ionosphere-free carrier phase combination (m); f i is the carrier frequency of L i (Hz). In this model, the ambiguity term contains three components, namely, Ni , φr (t0 , L i ), φs (t0 , L i ) and it is not an integer because it has been corrupted by the initial fractional phase bias in the GPS satellite and receiver. These biases need to be removed or calibrated so that the integer property of the undifferenced ambiguity parameters can be recovered. The existence of initial fractional phase biases is the major challenge for OTF application to PPP. In the rest of the Section, the characteristics of the initial fractional biases will be investigated to answer the question whether such biases can be calibrated or not.

3 Initial Phase Bias Analysis 3.1 Receiver Initial Phase Bias Analysis To analyze the property of the receiver initial phase bias, the Wide-Lane (WL) and Geometry-Free (GF) combinations are formulated as follows:

where WL is the WL carrier phase combination (m); GF is the GF carrier phase combination (m); λWL is the carrier phase WL combination wavelength (m). To eliminate the ionosphere error in Eqs. (7) and (8), a zero-baseline will be used with two GPS receivers connected to a common antenna using a signal splitter. A Receiver-Receiver Single Difference (RRSD) results in the following differential observation equations for P1 code, WL and GF carrier phase combinations: P1 = c dT + ε( P1 )

(9)

WL = c dT + λWL [ N1 + φr (t0 , L 1 )] − λWL [ N2 + φr (t0 , L 2 )] + ε( WL ) (10) GF = λ1 [ N1 + φr (t0 , L 1 )] − λ2 [ N2 + φr (t0 , L 2 )] + ε( GF ) (11) where is the RRSD operator. Note that the differential code bias between the two receivers will be included in the receiver clock term in Eq. (9). Since we have used the same dT term for both code and carrier observables in Eqs. (9) and (10), the estimated initial phase will then include the effect of the differential code bias. Differencing between Eqs. (9) and (10) leads to the following equation: WL − P1 = λWL [ N1 + φr (t0 , L 1 )]− λWL [ N2 + φr (t0 , L 2 )] + ε( WL − P1 )

(12)

A combination of Eqs.(11) and (12) provides an estimator for the differential integer ambiguity terms on L1 and L2 as shown below:

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Nˆ 1 Nˆ 2



 = round

1 −1 λ1 −λ2

−1  WL − P1  λWL

GF (13)

Applying the obtained integer ambiguity estimates from Eqs. (13) to (11) results in a RRSD inter-frequency receiver initial phase bias estimate as follows (Wang and Gao, 2007): λ1 φr (t0 , L 1 ) − λ2 φr (t0 , L 2 )   = GF − λ1 Nˆ 1 − λ2 Nˆ 2

(14)

In the following, the test results with two receivers set up in a zero-baseline configuration will be described. The used receivers include two NovAtel OEM4 receivers and two Javad Legacy receivers. Figure 1 shows the receiver configuration. To ensure all four receivers to operate under the same condition, one additional NovAtel receiver was used to provide the power for the common antenna. To assess the stability of receiver initial phase bias under different circumstances, three experiments were conducted at different times of the day (22 June, 2006 morning and midnight and 23 June, 2006 midnight local time in Calgary), each lasting for about 10 h. All receivers were powered off before the start of each experiment to assess the bias stability with respect to receiver’s power on and off. After eight hours of continuous data acquisition for each experiment, a signal full Loss of Lock (LL) was simulated by disconnecting

the antenna input cable to the splitter to assess its impact on the receiver initial phase bias. The antenna disconnection lasted for about one minute and followed with two hours of GPS data acquisition. Figures 2 and 3 show the receiver initial phase bias estimates from the three experiments. For both Javad and NovAtel pairs, the receiver initial phase bias estimate requires about one hour to be stabilized to a constant value. The initial phase bias estimates are different for the three experiments which indicate that the receiver initial bias value would change after each receiver’s power off and on operation. The receiver initial phase bias is also receiver dependent since the bias estimates are different between the NovAtel receiver pair and the Javad receiver pair. Figure 4 and Table 1 show the receiver initial phase bias estimates over ten hours from the three experiments, the full LL is simulated at the end of the eight hour. The results indicate that different receiver pairs react differently to the occurance of a full LL. The stabilized receiver initial phase bias estimate for the Javad receiver pair stays the same after the full LL but it has changed for the NovAtel pair. This is likely due to differences in receiver design. Based on the results from the three zero-baseline experiments, the preliminary conclusion is that the receiver initial phase bias is not stable in response to receiver’s power on and off operations and signal full LL. As a result, we conclude that it is not feasible to calibrate a receiver initial phase bias.

3.2 Satellite Initial Phase Bias Analysis To investigate the characteristics of satellite initial phase bias, the Narrow-Lane (NL) phase combination, the NL and WL code combinations are needed and are given as follows: NL =

f 1 1 + f 2 2 = ρ + c(dT − dt) f1 + f2 f1 + dor b + dtr op − dion/L1 f2 + λNL [N1 + φr (t0 , L 1 ) − φs (t0 , L 1 )] + λNL [N2 + φr (t0 , L 2 ) − φs (t0 , L 2 )] + ε(NL )

Fig. 1 Zero-baseline configuration

(15)

Real-Time Kinematic OTF Positioning

Fig. 2 RRSD inter-frequency receiver initial phase bias of NovAtel pair

Fig. 3 RRSD inter-frequency receiver initial phase bias of Javad pair

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Fig. 4 RRSD inter-frequency receiver initial phase bias before and after full LL

Table 1 RRSD inter-frequency receiver initial phase bias stabilized values before and after full LL NovAtel pair (m) Before LL 22 June, Morning 22 June, Midnight 23 June

After LL

Javad pair (m) Before LL

After LL

−0.021

0.008

−0.023

−0.023

−0.002

−0.018

0.014

0.012

0.020

−0.013

−0.015

−0.015

f 1 P1 − f 2 P2 PWL = = ρ + c(dT − dt) + dor b f1 − f2 f1 + dtr op − dion/L1 + ε(PWL ) f2 (16) f 1 P1 − f 2 P2 PNL = = ρ + c(dT − dt) + dor b f1 + f2 f1 + dtr op − dion/L1 + ε(PNL ) f2 (17) where NL is the NL carrier phase combination (m); λNL is the effective wavelength of the NL combination

(m); PWL is the WL pseudorange combination (m); PNL is the NL pseudorange combination (m). In Ge et al. (2006), a method has been proposed to estimate the WL satellite initial phase bias using NL code and WL phase measurements. Satellite-Satellite Single Difference (SSSD) is also applied in this method to remove the receiver-related errors or biases such as receiver initial phase bias, resulting in: 

∇WL − ∇ PNL λWL

 ∇WL − ∇ PNL −round (18) λWL

∇φs (t0 , L WL ) = −

where ∇ is the SSSD operator. The NL satellite initial phase bias could be estimated in a similar way using WL code and NL phase measurements as follows: 

∇NL − ∇ PWL λNL

 ∇NL − ∇ PWL −round (19) λNL

∇φs (t0 , L NL ) = −

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Once the WL and NL satellite initial phase biases 2006, starting from Week Of Year (WOY) 25 and acare known, the satellite initial phase biases on L1 and quired at IGS station FAIR, has been used to analyze the bias’ weekly stability. L2 frequencies can be obtained as follows: The daily stability of SSSD WL satellite initial phase bias is fairly good as shown in Fig. 7, where the ∇φs (t0 , L 1 ) = 0.5[∇φs (t0 , L NL ) + ∇φs (t0 , L WL )] bias variation over the ten days is less than 0.05 cycle ∇φs (t0 , L 2 ) = 0.5[∇φs (t0 , L NL ) + ∇φs (t0 , L WL )] or about 4 cm. Even over ten weeks, the bias variation (20) is still under 0.1 cycle or about 8 cm, as shown in Fig. 8. The satellite initial phase bias is therefore The stability of the satellite initial phase bias will be considered temporally stable. analyzed in both spatial and temporal domains. A total At present, the satellite initial phase bias for the of 46 globally distributed IGS 1 Hz high-rate tracking narrow-lane is still difficult to estimate using the stations have been used to analyze the spatial stability of the satellite initial phase bias. Figure 5 illustrates Table 2 Station IDs and names of 46 IGS high-rate stations the geographic distribution of these 46 tracking stations ID Station ID Station ID Station and Table 2 lists their IDs and names. 1 AMC2 17 KELY 33 PERT Shown in Fig. 6 are the satellite initial phase bias 2 AREQ 18 KIRU 34 PETS estimates for 3 satellites. The results indicate that the 3 ARTU 19 KOKB 35 PIE1 SSSD WL satellite initial phase bias for a particular 4 BOGT 20 KOUR 36 PIMO BREW 21 MADR 37 POL2 satellite is spatially stable since the difference between 5 CRO1 22 MALI 38 QUIN bias estimates derived from the 46 stations is generally 6 7 DGAR 23 MASI 39 REDU less than 0.2 cycle (about 17 cm). A similar bias stabil- 8 FAIR 24 MBAR 40 SANT ity has been demonstrated for other satellites. 9 GLPS 25 MCM4 41 SEY1 GOLD 26 MKEA 42 TIDB The temporal stability of the satellite initial phase 10 GOPE 27 MOBN 43 USN2 bias has been analyzed on a daily and weekly basis. 11 12 GUAM 28 MSKU 44 USUD Data over ten consecutive days in 2006, starting from 13 HRAO 29 NNOR 45 VILL Day Of Year (DOY) 170 and acquired at a randomly se- 14 IISC 30 NRIL 46 YAKT IRKM 31 NRL1 lected IGS station (FAIR), has been used to analyze the 15 ISPA 32 OKC2 bias’ daily stability. Data over ten consecutive weeks in 16

Fig. 5 46 IGS tracking stations distribution

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Fig. 6 Spatial stability of satellite initial phase bias

Fig. 7 Daily stability of satellite initial phase bias

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Fig. 8 Weekly stability of satellite initial phase bias

method of Eq. (19) due to the high noise level of the WL code combination and the short wavelength of the NL carrier phase combination. Although only the stability of the satellite initial phase bias in wide-lane can currently be assessed, the bias behavior for the narrow-lane is expected to be similar to the wide-lane, but it requires further investigation in the future to confirm this.

4 Ambiguity Integer Property Recovery Before bias calibration, accurate SSSD satellite initial phase bias estimates are required. Figure 9 shows an example of the estimated SSSD WL satelite initial phase biases for all GPS satellites. The reference satellite for SSSD is PRN 1. The biases are calculated using one day of data on DOY 170 of 2006 from three IGS tracking stations, namely FAIR, BREW and USUD. Shown in Figs. 10 and 11 are the WL float ambiguities for the seven visible satellites after SSSD and

bias calibration. The processed data is ten-minute long and collected at the FAIR station on DOY 171 of 2006. After the SSSD with PRN 9 as the reference satellite which has the highest elevation angle and bias calibration using the estimated satellite initial phase biases in Fig. 9, the float ambiguities of all seven satellites have converged to integers. This confirms that the integer property of the ambiguity term has been successfully recovered. The precise ephemeris and clock products and the estimated satellite initial phase bias are required to be transmitted to the receiver via a communication link. After processing the observations in a single receiver using the model in Eqs. (21) and (22), the PPP float solution and biased float ambiguities are obtained, and then the bias calibration is applied on the biased float ambiguities to recover the integer property of the undifferenced ambiguity terms. PPP fixed solution can therefore be obtained after the implementation of an ambiguity resolution algorithm to fix the integer ambiguity terms based on the satellite-satellite single difference mode using a single GPS receiver. A PPP-based

664 Fig. 9 Estimated satellite initial phase biases for calibration

Fig. 10 SSSD WL float ambiguities after bias calibration, PRN 1, 5, 11 and 17 observed at FAIR

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Fig. 11 SSSD WL float ambiguities after bias calibration, PRN 23, 24 and 30 observed at FAIR

RTK OTF system will make precise position solution available within a much shorter time period, such as a few minutes or even a few seconds, compared to the float solution.

∇IF =

f 12 ∇1 − f 22 ∇2 f 12 − f 22

=∇ρ − c∇dt + ∇dor b + ∇dtr op +

5 PPP-Based RTK OTF System

−

f 12 λ1 [∇ N1 − ∇φs (t0 , L 1 )] f 12 − f 22 f 22 λ2 [∇ N2 − ∇φs (t0 , L 2 )] f 12 − f 22

+ ε(∇IF ) (22)

A PPP-based RTK OTF positioning system has been developed based on the “UofC model” described in Gao and Shen (2002). The UofC model allows simultaneous estimation of both L1 and L2 ambiguities and further support the application of OTF ambiguity resolution in PPP. To allow the removal of the receiver initial phase bias and the calibration of the satellite initial phase bias in order to recover the integer property, the SSSD is introduced into the UofC model as given below: ∇ PPi,i = 0.5[∇ Pi + ∇i ] = ∇ρ − c∇dt + ∇dor b + ∇dtr op + 0.5λi [∇ Ni − ∇φs (t0 , L i )] + ε(∇ PPi,i ) (21)

Figure 12 shows the system configuration of the PPPbased RTK OTF system.

6 Simulation Results and Analysis Since only the wide-lane initial phase bias can be successfully calibrated using the method described in this paper and there still has difficulty on narrowlane initial phase bias calibration, only software simulation results were presented to assess the initialization time reduction using a PPP-based RTK OTF system (Wang and Gao 2006). Different random error and systematic bias sources, including receiver clock error, tropospheric error,

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Fig. 12 Diagram of PPP based RTK OTF system

ionospheric error, multipath noise level, true integer ambiguity value, initial phase bias, and observation noise level, are generated via simulation to support various tests. The receiver location for the simulation is at the pillar S5 on the roof of the Engineering Building at University of Calgary. Given in Figs. 13, 14 and Table 3 are results of the position and tropospheric delay solutions in both float

Fig. 13 3D positioning accuracy comparison in one scenario

and fixed modes. The simulated coordinates of the antenna and the zenith tropospheric delay are applied as the truth for the accuracy analysis. The results indicate that it took about 1000 seconds for the float solution to converge to a 3D position accuracy of 5 cm. With the initial phase biases removed and therefore the integer property of the ambiguity terms recovered, it took about 200 s to fix the ambiguity parameters to their

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Fig. 14 Tropospheric zenith wet delay estimation accuracy in one scenario

Table 3 Accuracy statistics in one scenario Estimated results

Float (after converged)

Fixed (after TTFF)

Mean (cm)

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0.47 0.19 −2.43 2.71 0.12

0.73 0.42 1.43 1.26 0.11

−0.01 0.00 −0.06 0.99 0.01

0.48 0.42 0.90 0.49 0.04

Fig. 15 Convergence time improvement in 50 scenarios

integer values and obtain centimeter accurate position solutions. Ambiguity Time To First Fix (TTFF) means the time taken for the ambiguity terms to be firstly correctly fixed. To assess the overall reduction of the initialization time, ambiguity resolution simulation is repeated fifty times, each with half hour long span of data. The obtained results are shown in Fig. 15 which indicates that it took 198 s on average to fix the ambiguity parameters

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to their integer values and provide position solutions at Gao, Y. and Chen, K. (2005). “Performance analysis of precise point positioning using real-time orbit and clock prodthe centimeter level. But it would take the float solution ucts.” Journal of Global Positioning Systems, Vol. 3, No. 1–2, 669 s on average to converge to 5 cm accuracy. OTF pp. 95–100. ambiguity resolution has reduced the PPP initialization Gao, Y. and Shen, X. (2001). “Improving convergence speed of time by 65% on average and up to as much as 90%. carrier phase based precise point positioning.” Proceedings

7 Conclusions The development of a PPP-based RTK OTF system is challenged by the existence of the satellite and receiver initial phase biases which must be removed or calibrated in order to recover the integer property of the ambiguity term in the un-differenced observations used by PPP. The receiver initial phase bias is not stable but can be removed by differencing between satellites, and the satellite initial phase bias is stable so it can be estimated for calibration. OTF can be implemented in a PPP-based RTK system after the removal of the initial phase bias. In a PPP-based RTK OTF system, OTF ambiguity resolution can significantly reduce the PPP initialization time. Acknowledgments The financial support for this research from GEOIDE (Canadian Geomatics for Informed Decisions) NCE (Networks of Centres of Excellence) and NSERC (Natural Sciences and Engineering Research Council) of Canada is greatly acknowledged.

References Bisnath, S. and Langley R.B. (2002). “High-precision platformindependent positioning with a single GPS receiver.” Navigation, Vol. 29, No. 3, pp. 161–169. Dixon, K. (2006). “Star FireTM : a global SBAS for subdecimeter precise point positioning.” Proceedings of ION GNSS 2006, Fort Worth, September 26–29, 2006, pp. 2286–2296.

of ION GPS 2001, Salt Lake City, September 12–14, 2001, pp. 1532–1539. Gao, Y. and Shen, X. (2002). “A new method for carrier phase based precise point positioning.” Navigation, Journal of the Institute of Navigation, Vol. 49, No. 2, pp. 109–116. Ge, M., Gendt, G., and Rothacher M. (2006). “Integer ambiguity resolution for precise point positioning.” Presented in Proceedings of VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy: Challenge and Role of Modern Geodesy, May 29–June 2, 2006, Wuhan, China. H´eroux, P., Gao, Y., Kouba, J., Lahaye, F., Mireault, Y., Collins, P., Macleod, K., T´etreault, P., and Chen, K. (2004). “Products and applications for precise point positioning – moving towards real-time.” Proceedings of ION GNSS 2004, September 21–24, 2004, Long Beach, pp. 1832–1843. Kouba, J. and H´eroux, P. (2001). “Precise point positioning using IGS orbit and clock products.” GPS Solution, Vol. 5, No. 2, pp. 12–28. Leandro, R.F and Santos, M.C. (2006). “Wide area based precise point positioning.” Proceedings of ION GNSS 2006, Fort Worth, September 26–29, 2006, pp. 2272–2278. Muellerschoen, R., Bar-Sever, Y., Bertiger, W., and Stowers, D. (2001). “NASA’s global DGPS for high-precision users.” GPS World, Vol. 12, No. 1, pp. 14–20. Wang, M. and Gao, Y. (2006). “GPS un-differenced ambiguity resolution and validation.” Proceedings of ION GNSS 2006, Fort Worth, September 26–29, 2006, pp. 292–300. Wang, M. and Gao, Y. (2007). “An investigation on GPS receiver initial phase bias and its determination.” Proceedings of ION NTM 2007, San Diego, January 22–24, 2007, pp. 873–880. Zumberge, J.F., Heflin, M.B. Jefferson, D.C. Watkins, M.M. and Webb F.H. (1997). “Precise point positioning for the efficient and robust analysis of GPS data from large networks.” Journal of Geophysical Research, Vol. 102, No. B3, pp. 5005–5018.

Concept of a Multi-Scale Monitoring and Evaluation System for Landslide Disaster Prediction M. Haberler-Weber, A. Eichhorn, H. Kahmen and B. Theilen-Willige

Abstract In 2006, OASYS, an EU funded project on a multi-scale monitoring concept for landslides as a basis for an alert system was completed. Twelve institutes from 6 countries tried to merge their multidisciplinary knowledge in the field of landslides and disaster management. The main goal of the research was to develop a cost saving concept for landslide disaster prediction in areas with a higher density of landslides. The present paper reports about the innovative steps and about some highlights of the research, emphasising mainly three tasks: – GIS integrated geological evaluations of remotesensing data to delineate the high-risk areas in regions with a larger number of landslides – geometrical analysis of the monitoring data by fuzzy techniques as a basis for the design of the sensor network and – geomechanical modelling of the landslide by Finite Difference-methods as a basic information for an alarm system.

M. Haberler-Weber Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29 / E128-3, 1040 Vienna, Austria A. Eichhorn Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29 / E128-3, 1040 Vienna, Austria H. Kahmen Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29 / E128-3, 1040 Vienna, Austria B. Theilen-Willige B¨uro f¨ur Angewandte Geowissenschaftliche Fernerkundung (BAGF), Birkenweg 2, 78333 Stockach, Germany

At the moment the achieved results are separately focussed on the main tasks. A combined solution must be realized in subsequent investigations. Keywords Landslides· Risk assessment· Alert system· Fuzzy techniques· Kalman filtering techniques

1 Introduction One main challenge of the research was to develop cost saving solutions which can especially be used in areas with a larger number of landslides. A first proposal of a concept was described in (Kahmen and Niemeier, 2005). During the last years, an advanced model was investigated in the project, based on large scale monitoring and evaluation as a first step, regional monitoring as a second step, culminating in a multicomponent knowledge-based alert system.

1.1 Detection of Potential Landslides (Large Scale Monitoring and Evaluation) - Step 1 To get the borders of the moving areas and information about the long-term geodynamical processes a large scale evaluation has to be performed, see Sect. 2, e.g. (Theilen-Willige, 1998). This includes e.g. the historical data and remote sensing data, such as aerial photographs, optical and radar images from satellites. Remote sensing techniques (e.g. In-SAR), differential GPS and tacheometric measurements can be used to obtain additional information about the deformation

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process. The measurements are usually performed only three or four times a year, and the results are vector fields describing the displacements and velocities. Based on all displacement information available, an advanced analysis algorithm (see Sect. 3) performs the detection of the so-called taking-off-domains, which are areas where the landslides have their origins, on or below the surface. In these zones, deformation can be detected at an early stage. Additionally, the takingoff-domains can give a first insight into the possible progress of the landslides.

1.2 High Precision Permanent Measurements in the Taking-off-Domains - Step 2 High precision relative measurement systems (borehole tiltmeters, extensometers, hydrostatic levelling systems, etc.) can be installed in a cost saving way in the area of the taking-off-domains to obtain online information about the geodynamical process (see Sect. 4). This multi-sensor system can measure continuously and can therefore support the real time alert system in this application.

1.3 Impact and Risk Assessment; Development of Strategies for Knowledge-Based Alert Systems - Step 3 The integration of hazard and vulnerability analysis leads to an estimation of the actual risk situation of the affected population. The risk management measures also depend heavily on the specific conditions and include landuse planning, technical measures (e.g. building drainage systems), biological measures (e.g. afforestation) and temporary measures in case of danger. Three tasks of the project are described below in more detail.

2 GIS Integrated Geological Evaluations of Remote-Sensing Data The goal of this step is the delineation of high-risk areas in regions with a greater number of landslides.

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Remote sensing technology embedded in a GIS database can be used as a complementary tool for existing landslide hazard studies, (Theilen-Willige, 2006; Theilen-Willige et al., 2007). In the OASYS project, LANDSAT ETM, ENVISAT, ERS imageries and aerial photographs are used to produce maps within a ‘Landslide Hazard Information System’ as layers in a GIS data base with the aim to create user-defined computations of landslide hazard maps. The various data sets as LANDSAT TM data, topographic, geological and geophysical data from the investigation areas are integrated as layers into GIS using the software ArcGIS of ESRI. By using earth observation data it is possible to detect traces of past or even recent mass movements which could be source areas for future landslides. In order to establish a cost effective method for getting a quick overview of determining factors influencing environment and potential damage intensity in landslide prone areas it is recommended to start analysing the causal/preparatory factors and their complex interactions first based on remote sensing and GIS methodologies and later step by step, going into details. This approach enables to asses the mass movements in respect to their complex dependencies. The GIS integrated evaluation of the georeferenced satellite imageries allows the storage of the results in a standard form such as vector-formats (point-, line- or polygon shapefiles). Various digital image processing tools delivered by ENVI Software/CREASO have been tested, as for finding the best suited LANDSAT Enhanced Thematic Mapper (ETM) band combinations or contrast stretching parameters. The imageries merged with the panchromatic Band 8 of LANDSAT ETM reach the spatial resolution of 15 m. Standard approaches of digital image processing with regard to the extraction of landslide hazard relevant information used are methods like classification for land use inventory and vegetation information, and processing of the thermal LANDSAT Band 6 for deriving surface temperature information. For the investigation of vegetation anomalies that might be related to subsurface structures and landslides the NDVI (Normalized Difference Vegetation Index) based on the available LANDSAT ETM Bands 3 and 4 is used. This is demonstrated by an example (see Fig. 1) from the Lake Constance area in SW-Germany where anomalies on the NDVIimage can be clearly correlated with landslides in the field.

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Fig. 1 Correlation of low NDVI intensities (dark-blue) with landslides near Bodman, Lake Constance area in SW-Germany

Especially ‘Lineament Analysis’ based on LANDSAT ETM and radar images can help to delineate those local fracture systems and faults that might influence dynamics and shape of landslides. In this case digital image enhancement is an important step followed by visual interpretation in an interactive manner. Aim of these techniques is to detect linear or curve linear features in order to find traces of possible slope failure. Combining lineament maps for example with slope degree maps, derived from digital terrain models, helps to delineate areas with higher risk. The term lineament is used for all linear, rectilinear or slightly bended image elements being extracted by image enhancement. Lineaments are symbols for e.g. linear valleys, linear zones of abundant watering, drainage network, peculiar vegetation, landscape and geological anomalies. The definition of the borders of landslide areas can be based on the assumption that lineament systems in a satellite image are closely connected to deformation which is caused by a change in the Earth’s crust stress field (see Fig. 2).

The evaluation of digital topographic data is of great importance as it contributes to the detection of the specific geomorphologic/topographic settings of landslide prone areas. Data of the Shuttle Radar Topography Mission (SRTM, 2000) can be used worldwide free of charge to provide digital elevation information at 90 m spatial resolution. A systematic GIS approach is recommended extracting geomorphometric parameters based on Digital Elevation Model (DEM) data as part of a landslide-hazard information system. The causal factors for landslide hazards have to be extracted systematically: From slope gradient maps those areas with the steepest slopes and from curvature maps the areas with the steepest slopes and from curvature maps the areas with the highest curvature are extracted as these are more susceptible to landslides than smoother slopes; from height level maps the highest areas are derived that are generally more susceptible to

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Fig. 2 Remote sensing contribution to landslide hazard assessment

landslides than flat areas. Flow direction maps provide displacement vectors for distinct points being observed information considering the potential moving direction in each epoch. So an advanced analysis method is necessary, using the displacement vectors for a grouping of landslides. of observed points into blocks of consistent movement pattern. Within the classical quasi-static geodetic 3 Geometrical Analysis of Monitoring deformation analysis only single point movements can be assessed. A block detection and assessment is Data with Fuzzy Systems only possible manually, i.e. the user can define models describing different blocks, which are assessed by One aim within the integration of different measurestatistical properties (e.g. Welsch, 1983). This strategy ment methods was the precise, continuously monitorof ‘trial and error’ is not useful for a large number of ing of the taking-off-domains with geotechnical senobject points. So an automated block detection method sors. This information can be used in the alert system was necessary. to analyse the actual situation. A pure mathematical approach cannot fully achieve So one task was the detection of block boundaries this goal due to the inherent fuzziness of this probbetween stable and unstable areas or between unstalem. A human expert solves this task by looking at the ble areas (moving with different velocities in different graphical representation of the displacement vectors. directions) out of the displacement vectors given by Keeping in mind other nonmathematical relevant inforthe geodetic deformation measurements. These block mation like geological facts or some properties of the boundaries are the optimal places for the installation of measurement process, he or she can separate the blocks geotechnical sensors, which give a very precise relative by a combination of mathematical facts and human exinformation on the movement in this area. perience. This kind of processing can be achieved with Classical deformation measurements are usually help of fuzzy systems, where the human way of thinkdone in several epoch; for processing the geodetic ing can be imitated by a rule based expert knowledge. deformation analysis is used. This analysis results in

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The task is to find groups of points with a similar pattern of movement, so that the boundaries between these blocks can be identified. Within the automated block detection process, the smallest starting block of 4 points is identified due to some indicators. Then the block is expanded by an iterative algorithm, where the best fitting point is added to the block in each iteration step until no neighbouring points with a similar pattern of movement exist. Two different types of parameters are used as indicators for this block separation (for more details see e.g. Haberler, 2005): 1. Geodetic parameters: an overdetermined affine coordinate transformation and the derived strain parameters are used to assess the movement pattern of the observed points. If the points of the block under investigation are lying on one block (showing a similar pattern of movement), e.g. the strain parameters and the standard deviation of unit weight of the coordinate transformation are rather small. If in the next iteration step a neighbouring point with another movement pattern is added to this block, these indicators usually are increasing significantly. But for a fully automated analysis, these geodetic indicators are not sufficient. So a second type of parameters is needed, using human expert knowledge in a fuzzy system. 2. Visual parameters: Human experts do the block detection by looking at the pattern of displacement vectors, selecting all points and the corresponding vectors which show a similar length and a similar direction. This is a typical example for the application of fuzzy systems because no sharp definition of ‘similar length’ or ‘similar direction’ can be found. One example will be given here for the indicator ‘similarity of direction’: If the vectors under investigation are within a range of approximately 20 gon, they can be assessed as ‘similar’. The greater the difference in the azimuth, the smaller the indicator ‘similarity’ will be within the processing, according to the human way of thinking.

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OASYS project. Based on this information of the block boundaries, in the next step of the project the high precision geotechnical sensors could be installed in these areas.

4 FD-Modelling of the Test Slope for a Knowledge-Based Alarm System 4.1 Basic Concept of the Knowledge-Based Alarm System The basic concept for a knowledge-based alarm system includes two complementary strategies for the analysis of the current state of the slope and the alarm level decision: the data- and the knowledge-based system analysis. Its architecture is described in detail in (Kahmen et al., 2007). Besides the collection and investigation of geotechnical and geodetic monitoring data one central task for the data-based system analysis will be the provision of calculation results from calibrated numerical slope models which quantify the inner structure of the slope and the mechanisms of possible failure events. Numerical models can principally be represented by DE (distinct element), FD (finite difference) or FE (finite element) codes. Standard software packages are available e.g. from (HCltasca, 2007). The slope models will aim at the calculation of the behaviour of the slope under time variable loads (e.g. mass extraction, changes in the groundwater table). In the alarm system the calculation results shall be mainly used for

– the comparison of model predictions and measured displacements to detect deviations from normal behaviour, – the realistic simulation of failure events as re-action to critical trigger influences (possibility of prevention) and for – the calculation of the local and global inner stress distribution of the slope which shall be used for The analysis algorithm was implemented in  the quantification of numerical safety factors (e.g. Matlab . The fuzzy toolbox provides the basic methFS = factor of safety, Wittke, 2002) which enable ods like standard membership functions, rulebased the evaluation of the risk potential. inference algorithms and calculation of the output parameters. One goal of OASYS was to accomplish a feasability Some examples confirmed the applicability of the block separation algorithm in the first part of the study for numerical slope modelling.

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4.2 Selected Test Slope

4.3 Creation of a FD Model for Data-Based Analysis

The monitoring system of TU Vienna (together with Geodata company) was installed at a test slope in a large open cast lignite mine in Germany. This study site provides nearly lab conditions for the development and evaluation of investigation methods for slope mechanisms, this means well known main trigger events (mass excavation process) and an also well known geological structure. A cross section of the test slope is shown in Fig. 3. The mass extraction is realised by huge bucket wheel excavators which dig out discrete stages, so called ‘berms’. The installed monitoring system for regional scale consists of two parts: a geodetic monitoring system (Fig. 3b) with GPS (GOCA = GNSS/LPS/LS-based online Control and Alarm System, see Jager et al., 2006) and robot tacheometers (GEOROBOT, see Kahmen and Suhre, 1983). In addition a geotechnical monitoring system was installed in the vicinity of the current digging area (top of berm 12, see Prader, 2003). The system consists of 1 piezometer, 1 inclinometer (length ≈ 20 m), 6 accelerometers (2 3D-systems), 1 magnetostrictive extensometer and 2 tiltmeters (see Fig. 3a and TU Braunschweig et al., 2005).

(a)

In the feasability study the slope model was restricted to static loading and the 2D-space (to avoid too extensive computing time). The idea of static loading was motivated by mass excavation as dominant trigger event and the observation of new balanced states of the slope as reaction. The numerical model was realised with the FD software FLAC3D 2.10 from HCItasca (see Kampfer, 2005). Its basic geometrical and physical features are – a horizontal extension of approximately 1500 m and a vertical extension of approximately 500 m, – a discretisation with more than 100,000 finite meshes (mean extension <10 m), – a density due to the geological situation and special areas of interest, – seven physical parameters (e.g. density, Poisson ratio and bulk modulus) per mesh and – mass extraction as main trigger event. The numerical model was calibrated using ‘trial and error methods’, (Kampfer, 2005). The material parameters were derived from geological plans and

(b)

Berm 10 Georobot & GOCA

Sand Berm 11

–280 m so me ter Ex ten

Berm 12

Clay

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–308 m

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Fig. 3 Test-slope with (a) geotechnical and (b) geodetic monitoring system (Prader, 2003)

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Fig. 4 FD modelling of the test-slope with FLAC3D: slope points 60–74 (ref. to Kampfer, 2005)

literature and adapted within a realistic range (a priori assumed uncertainty of 30%). For this purpose calculated displacements in selected slope points (see Fig. 4) were compared with independent control measurements (e.g. GPS). In Fig. 4 it is also shown how the [x, z] slope model coordinate system is defined. The x-axis defines the horizontal and the z-axis the vertical direction of the 2D slope. Positive x shows into the slope and positive z up to the zenith.

4.4 Calculation Results from the Calibrated FD Model In the years 2004–2005 trigger events were performed at the test slope by 13 excavation steps from top ground surface (tgs) down to the lignite layer (Figs. 3 and 4). The resulting berms have a mean height of approximately 30 m. In the FD model this mass extraction could be quantified by the reduction of the associated meshes and the calculation of the resulting states of equilibrium. In Fig. 5 the comparison between predicted and monitored displacements is exemplarily shown for the sum of excavation steps 12 and 13 as trigger input. The steps were performed in mid and end of January 2005. Control point 325 is situated on the top of berm 12 next to the geotechnical monitoring system and was not used for the calibration process. Its horizontal (Δx)

Jan

Feb

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–0,1

dx ≈ 1 cm –0,15

x

Δx Δz

0

–0,05

Berm 13

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0,05

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13

Fig. 5 Comparison of FD prediction and measured displacements in control point 325

and vertical (Δz) displacements are measured by a GPS receiver from the GOCA monitoring system (see Sect. 4.2). The available time series (original coordinates from GOCA output, not smoothed) start in January 2005 and end in February with a scanning rate of Δt = 10 min. They are reduced to a (nearly) static state of the slope at the beginning of January just before excavation step 12. As a reaction to steps 12 and 13 the point rises in total with Δz ≈ 5–6 cm and performs a horizontal move ’out of the slope’ with Δx ≈ −11 cm. In both cases the movements are not finally stabilised but show an asymptotic behaviour. The related static FD prediction of the total displacements at the end of step 13 shows a contradiction of only d x ≈ 1 cm in the horizontal and dz ≈ 2–3 cm in the vertical component. Predicted and real behaviour of point 325 are fitting together within a range of some cm. These first results can be stated as promising for further attempts to predict the at least normal behaviour of the slope. Using the calibrated FD model it is also possible to simulate realistic failure events in instable slope areas (Kampfer, 2005; Eichhorn, 2006). One simulation result is presented in Fig. 6. In the simulation the real gradient angle of the excavated berms was increased with 15%. Figure 6 shows the static calculation of the total horizontal (x) and vertical (z) movement of control point 70 (also on top of berm 12) as reaction to the 13 excavation steps beginning at tgs with 0 m. The failure event starts when the lignite layer is reached by excavation and causes a horizontal move out of the slope and a vertical sagging with ≈ 1.5 m in each direction.

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Static calculation k+j

0,5

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0 –300 m

– 200 m

–100 m

0m

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Sediment layers

Fig. 6 Simulation of a failure event: local slope sagging in control point 70 (Ref to Kampfer, 2005)

Finally the movement stabilises itself again. These results could be verified by interviews with local experts which classified this combination of slope gradient, excavation quantity and mass movement as realistic. This example shows the potential of the calibrated numerical slope model for the simulation of failures in case of critical triggers and how to perform prevention measures. In this case one possible solution could be a reduction of the gradient angle.

5 Conclusions A multi-scale monitoring and analysis system for detecting landslides was developed and combined with an efficient alarm system. The design of a common GIS database structure – always open to new data – can greatly contribute to the homogenisation of methodologies and procedures of landslide hazard risk management. Meanwhile, the so-called Free-GIS software fulfilling the basic GIS requirements, as for example DIVA-GIS, Map Window GIS, SAGA GIS, etc., can be used without costs. Additional Free-GIS software is available also for the spatial analysis of DEM data. Basic LANDSAT ETM and SRTM data are provided free of charge for scientific research purposes for example by the University of Maryland/USA. Therefore the use of the remote sensing and GIS technology for natural hazard site assessment and

for the elaboration of hazard maps according to the presented approach can be recommended as low cost approach that could be achieved by local communities in every country as contribution to a GIS data base for disaster preparedness. GIS integrated geological evaluations of remotesensing data and geometrical analysis of monitoring data with fuzzy systems enables to define the more active zones of the landslides and to install there wireless sensor networks. This concept supports cost-saving solutions especially in regions with a larger number of landslides. A concept for a knowledge-based alarm system was presented which aims to produce alarm levels by the interaction of a data- and a knowledgebased system analysis. The data-based system analysis is based on FD-modelling of the slopes which was successfully applied within the OASYS project. It can be stated that the cooperation with partners from different disciplines was the right way to investigate this complex topic. Of course further investigations are necessary to develop a functional alarm system; according research proposals are currently in evaluation. Satellite and airborne data will be part of the selection of landslide-prone areas which is the basis for the installation of a functional early warning system. Fuzzy methods and slope modelling represent the next steps and are directly related to geodetic and geotechnical monitoring data. They represent the data-based part of the knowledge-based alarm system.

Multi-Scale Monitoring and Evaluation System Acknowledgments This work was partially supported by the European Commission, Research DG, Environment Programme, Global Change and Natural Disasters, and by the Austrian Ministry of Education, Science and Culture. We thank the OASYS-Team, especially Albrecht Prader (TU Vienna, Geodata) for the good cooperation and the interesting discussions.

References Braunschweig, T.U. T.U. Vienna, Geodata GmbH (2005). Instrumentation Data Analysis and Decision Support for Landslide Alarm Systems. Report of the project OASYS. Eichhorn, A. (2006). Geomechanical Modelling as Central Component of a Landslide Alert System Prototype: Case Study ‘Opencast Mine’. Proceedings XXIII FIG Congress, Munich, TS29.4. Haberler, M. (2005). Einsatz von Fuzzy Methoden zur Detektion konsistenter Punktbewegungen. Geowissen-schaftliche Mitteilungen, Schriftenreihe der Studienrichtung Vermessungswesen und Geoinformation der Technischen Universit¨at Wien, Nr. 73. HCltasca (2007) http://www.itascacg.com/, last access 03/07. J¨ager, R., S. K¨alber, M. Oswald, M. Bertges (2006) GNSS/GPS/LPS Based Online Control and Alarm System (GOCA) – Mathematical Models and Technical Realisation of a system for Natural and Geotechnical Deformation Monitoring and Analysis. In: Kahmen, H., Chrzanowski, A. (Eds.): 3rd IAG Symp. on Geodesy for Geotechnical and Structural Engineering/12th FIG Symp. on Deformation Measurements, Baden. Kahmen, H., A. Eichhorn, M. Haberler-Weber (2007). A MultiScale Monitoring Concept for Landslide Disaster Mitigation. In: Proceedings of the IAG, IAPSO and IABO joint Conference Dynamic Planet 2005, Cairns, International Association of Geodesy Symposia, Volume 130, Springer, BerlinHeidelberg, pp. 769–775. Kahmen, H., W. Niemeier (2005). OASYS – Integrated Optimization of Landslide Alert Systems. In: Sans`o, F. (Ed.): A Window on the Future of Geodesy, Sapporo, Japan, June 30 – July 11, 2003. International Association of Geodesy Symposia, Volume 128, Springer, BerlinHeidelberg, pp. 101–104.

677 Kahmen, H., H. Suhre (1983). Ein lernf¨ahiges tachymetrisches ¨ Vermessungssystem zur Uberwachung kinematischer Vorg¨ange ohne Beobachter. ZfV 108, pp. 345–351. Kampfer, G. (2005). Zusammenfassung der Ergebnisse der geotechnischen Modellierung der RWE Testb¨oschung. Report of the project OASYS. Prader, A (2003). Measurement Methods and Sensors for Geodetic and Geotechnical Information. OASYS Deliverable 1.3, Vienna. Theilen-Willige, B. (1998). Seismic Hazard Localization Based on Lineament Analysis of ERS- and SIR-C Radar-Data of the Lake Constance Area and on Field Check. Proceedings of the European Conference on Synthetic Aperture Radar. 25–27 May 1998 in Friedrichshafen, VDE-Verlag, Berlin, pp. 447–550. Theilen-Willige, B. (2006). Detection of Landslides and HighRisk Areas. Contribution to the Final Report of OASYS, 2006 (Integrated Optimization of Landslide Alert Systems(OASYS), EVG1-2001-00061), pp. 11–24. Theilen-Willige, B., H. Wenzel, B. Neuh¨auser, M. PapathomaKohle (2007). Remote Sensing and GIS Contribution to Natural Hazard Assessment in the Vienna Area. II International Congress of Environmental Planning and Management, 5–10. August 2007, TU Berlin, Germany, in: Landschaftsentwicklung und Umweltforschung, Schriftenreihe der Fakult¨at Planen-Bauen-Umwelt, TU Berlin, Band S 20, Berlin, pp. 137–140. Welsch, W. (Hrsg.) (1983). Deformationsanalysen ’83 – Geometrische Analyse und Interpretation von Deformationen Geod¨atischer Netze. Beitr¨age zum Geod¨atischen Seminar. Schriftenreihe Wissenschaftlicher Studiengang Vermessungswesen Hochschule der Bundeswehr M¨unchen, Heft 9. Wittke, W., C. Erichsen (2002). Stability of Rock Slopes. Geotechnical Engineering Handbook, Ernst & Sohn.

Remote Sensing Data free of charge in the WEB: SRTM: http://srtm.csi.cgiar.org/SELECTION/input Coord. asp LANDSAT: http://glcfapp.umiacs.umd.edu:8080/esdi/ index.jsp Shapefiles, DEM and satellite data: http://biogeo.berkeley.edu/bgm/gdata.php

Ionospheric Modelling in the North of Algeria H. Dekkiche, S. Kahlouche, C.B. Kadri and R. Mir

Abstract The knowledge of the ionospheric behaviour is an important factor in GPS positioning. Data of five ALGEONET (Algerian GEOdynamical NETwork) and a hundred IGS stations are processed with the Bernese GPS software, which is based on the use of spherical harmonic expansion, to provide an ionospheric mapping of the Total Electronic Content (TEC). The used model supposes that the whole free electrons are concentrated on a thin spherical layer to an altitude varying between 250 and 450 km. Results are obtained for several heights (350, 400 and 450 km) with a 2-hourly resolution. The maximal value of the obtained TEC in the north Algerian area is about 20 (midnight) to 60 (midday) TECU, such values are obtained for the day of April 30, 2001, which refers to a high solar activity period with a sunspot number of 112. Calculated TEC values are thereafter applied to correct the single frequency solution. The comparison between the corrected L1 and ionosphere-free solutions shows that the ionospheric effects involve a network contraction. Also, the corrected L1 solution is close to the ionosphere-free one; for baselines of several hundreds of kilometers, deviations are about 10 cm (in horizontal

H. Dekkiche Division of Space Geodesy, Centre of Space Techniques, Arzew 31200, Algeria, e-mail: [email protected] S. Kahlouche Division of Space Geodesy, Centre of Space Techniques, Arzew 31200, Algeria C.B. Kadri Division of Space Geodesy, Centre of Space Techniques, Arzew 31200, Algeria R. Mir Division of Space Geodesy, Centre of Space Techniques, Arzew 31200, Algeria

coordinates), they are reduced by about 50% in vertical component. Keywords GPS · ALGEONET · TEC · Ionospheric maps · Positioning

1 Introduction The ionosphere is the ionised region of the atmosphere which is situated between 50 and 1000 km (Vesna Ducic, 2004). Its superior limit is difficult to estimate; nevertheless, beyond 1000 km the electronic density is very weak. It is a plasma obtained by different mechanisms of ionization of the neutral atmosphere. The two main sources of ionization are the solar radiation absorption (ultraviolet and X rays) and the energising particle impact on the atmosphere, its rate is sufficient to affect the electromagnetic wave propagation. Total Electron Content (TEC) values may be considered as a key to monitor the behaviour of the ionospheric medium. Rapid or abrupt TEC variations caused by the space weather often affect our daily lives, such as satellite communication, navigation, positioning, and other microwave systems. Therefore, it is important to precisely estimate the Total Electron Content of the ionosphere. Nowadays, continuous GPS observations can provide an efficient tool to monitor timely ionospheric irregularities (Otsuka et al., 2002; Jakowski et al., 2002; Jin et al., 2004; Afraimovich et al., 2000 and 2004). As the ionospheric conditions vary with the time, location, solar activities etc., the differences of ionospheric effects are obvious in different areas. Many scientists have investigated the local space weather with global ionospheric models

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on the basis of different observations data (Mannucci et al., 1998; Afraimovich et al., 2004). For example, IGS Ionosphere Working Group yielded daily TEC maps for user services from GPS data (Feltens and Schaer, 1998). In addition, the International Reference Ionosphere (IRI) models were widely used to predict regional or global ionospheric behaviour of the ionosphere (Ezquer et al., 1998; Bilitza, 2001). These ionospheric models give the average behaviour of the ionosphere. The aim of this paper is to investigate the Algerian regional ionosphere using semi-continuous GPS data. Bernese GPS software (developed by the University of Berne-Switzerland) is used in this study. The implemented model in the GPS Bernese software is based on the use of spherical harmonic expansion, it supposes that the set of the free electrons are concentrated on a thin spherical layer to an altitude varying between 350 and 500 km (Schaer, 1999; Hugentobler et al., 2001). In our current study we also investigate the practicability and efficiency of using the calculated TEC maps to provide ionospheric range error corrections for single frequency GPS users.

Fig. 1 Single-layer model

GPS-derived ionosphere models are usually based on the so-called Single-Layer Model (SLM) as shown in Fig. 1. This model assumes that all free electrons are concentrated in a shell of an infinitesimal thickness. The SLM mapping function FI may be written as: FI (z) =

2 TEC Modelling

with sin z =

R sin z. R+H

Where

2.1 Principle of GPS Mapping TEC The equations of code and phase observations can be expressed as follows: s Prs = ρr,g + dρ + c(dtr − dts ) s s + dr,ion + dr,tr o + c(br + bs ) s rs = ρr,g + dρ + c(dtr − dts ) s s s − dr,ion + dr,tr o + λBr

1 cos z

(1)

s is the true distance between the satellite Where ρr,g s and the receiver r . dρ is the orbital error, c is the speed of light, dts and dtr are the receiver and satellite s clock errors, respectively, dr,ion is the ionospheric des lay, dr,tr o is the tropospheric delay, bs and br are the receiver and satellite instrumental bias, λ is the wavelength of the carrier phase, and Brs is the sum of unknown ambiguities Nrs and the instrumental bias (i.e. λBrs = λNrs + c(br + bs )).

z, z are the zenith distances at the height of the station and the single layer, respectively, R is the mean radius of the Earth, and H is the height of the single layer above the Earth’s surface. From Fig. 1, it can be easily verified that the geocentric angle α equals z − z . The height of the idealized layer should be set to some value close to the effective height of the electron density distribution, which is usually higher than the maximum electron density height, due to the asymmetry of the electron density profiles and to the plasmaspheric component. Furthermore, the Total Electron Content E (the surface density of the layer) is assumed to be a function of geographic or geomagnetic latitude β and sun-fixed longitude s. According to [Hugentobler et al., 2001], the geometry-free code and phase observables (P4 and L4 ) may be written as:

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P4 = P1 − P2 = +a

L 4 = L 1 − L 2 = −a Where:

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frame is adopted in this study; the position of a pierce point on the thin shell is described using sun-fixed lon(2) gitude s, which is related to the earth rotation. The ! conversion between geomagnetic and geographic ref1 1 − 2 FI (z)E(β, s) + B4 erence frames can be easily achieved by knowing the f 12 f2 precise geomagnetic polar position in the geographic (3) reference frame. The regional TEC model may be written as: 1 1 − 2 f 12 f2

FI (z)E(β, s) + b4

a = 4.03 · 1017 ms−2 TECU−1 is a constant, f 1 , f 2 are the frequencies associated with the carriers L 1 and L 2 , F1 (z) is the mapping function evaluated at the zenith distance z 0 , E(β, s) is the vertical TEC (in TECU) as a function of geographic or geomagnetic latitude β and sunfixed longitude s, b4 is the differential code (b1 − b2 ), and B4 = λ1 B1 − λ2 B2 is a constant bias (in meters) due to the initial phase ambiguities B1 and B2 with their corresponding wavelengths λ1 and λ2 .

E(β, s) =

n n max 

P˜ nm (sin β)

n=0 m=0

×(anm cos ms + bnm sin ms)

(4)

Where n max is the maximum degree of the spherical harmonic expansion; P˜ nm (sin β) are the normalised associated Legendre functions of degree n and order m; anm and bnm are the unknown TEC coefficients of the spherical harmonics, which should be estimated; β is the geomagnetic latitude of the pierce point (intersection of the receiver-satellite line with the ionospheric shell) and s is the sun-fixed longitude which is related Equations 2 and 3, which are obtained from Eq. 1, to the local solar time (LT) according to: are analyzed. They principally contain ionospheric s = LT − 12 h ≈ UT + λ − 12 h information.

2.2 TEC Shell and Spherical Harmonic Function The ionosphere is constituted of charged particles (ions) that are present above the earth surface, between 50 and 1000 km altitude. However, for simplicity reasons, the ionosphere is described as a thin spherical shell model located at an altitude comprised between 250 and 450 km (Ping, 2002), this altitude varies from one region to an other. Three altitudes (350, 400 and 450 km) were tested in this work, and the obtained corrections are applied to a single frequency GPS solution. In general, the impact of the used reference for a model does not play an important role as long as we consistently use the consistent reference to estimate and apply the model. As the geomagnetic frame is not flatly independent of the physics of the ionosphere (it approximates the earth’s magentic field), the difference between the magnetic and geomagnetic frame is too small to have a significant impact on the obtained ionosphere maps. In this context, the geomagnetic reference

UT is the universal time and λ denotes the geographical longitude of the intersection point.

3 TEC Maps from ALGEONET and IGS Observation The observation data collected by the dual frequency GPS receivers of 05 ALGEONET stations (Fig. 2) and 108 IGS stations during April 30, 2001 are processed and analysed. The processing method consists of several steps; smoothing the code by phase observations, the estimation of the Differential Code Biases (DCB), the estimation of the spherical harmonic coefficients and the generation of ionospheric maps. Before code smoothing, the phase observations are pre-processed to remove the cycle slip. The estimated TEC of every 2 h are expanded by using a spherical harmonic function. Models with different values of the maximum degree and maximum order of the spherical harmonics were tested, and finally the parameter estimation was accomplished with a degree

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Fig. 2 ALGEONET network

and order equal to 6, which showed best results (results obtained from the comparison between the ionospherefree solution and the corrected single-frequency solution). This part is not detailed because it is not the aim of this study. Regional ionosphere model (ALGEONET) was created and compared with the different solutions delivered by the five IGS Ionosphere Associate Analysis Centers (IAACs): CODE (Center for Orbit Determination in Europe, Astronomical Institute, University of Berne, Switzerland), ESOC (European Space Operations Center of ESA, Darmstadt, Germany), JPL (Jet Propulsion Laboratory, Pasadena, California, U.S.A), NRCan/EMR (Natural Resources Canada, Ottawa, Ontario, Canada) and UPC (Technical University of Catalonia, Barcelona, Spain). The different IAAC TEC maps were computed with different approaches but with a common formal resolution of 2 h, 5◦ and 2.5◦ in UT, longitude and latitude (details can be found in e.g. Schaer 1999; Feltens, 1998; Mannucci et al., 1998; Gao et al., 1994; Hern´andez-Pajares et al., 1999).

For illustrative purpose, we chose two stations (Algiers and Laghouat which have latitudes of 36◦ 44 15.75. and 33◦ 48 05.52. respectively) from ALGEONET stations, used to generate ionospheric maps, to display and compare the obtained results with the different IAAC TEC maps. In Figs. 3 and 4, TEC values were plotted in TECU for the period of 30 April 2001 using the five different IAAC TEC maps solutions and the obtained ALGEONET one. We also plotted the differences between the ALGEONET and the five IAAC TEC maps (see Figs. 5 and 6). The differences between the different models are about 10 TECU. The general trend is that the regional model (ALGEONET) differs less from the CODE one, which is also computed with spherical harmonics; differences may come from the fact that when CODE TEC maps are estimated, the data from ALGEONET stations were not used. The ALGEONET TEC maps agree very well with the UPC solutions, too.

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Fig. 3 ALGEONET and IACC TEC over the Algiers station for the day of April 30, 2001

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The variation of TEC values is function of the position of the sun. Around 14 h (local time), where the solar activity is at the maximum, the value of the TEC can reach 60 TECU over the north of Algeria.

An other validation procedure, based on the use of the calculated ionospheric maps to correct the single frequency GPS solution, is presented subsequently. In this work, we have also studied the dependency of the ionosphere model from the chosen height of the single layer. TEC maps were computed by fixing the height to 350, 400 and 450 km. The influence of the single layer height causes a variation in the regional models less than one TECU. The absolute mean differences between the three models (Fig. 7) indicate that the TEC values delivered from ALGEONET are not sensitive to the height of the layer. Hence, the choice of the single layer height is not important when modelling the ionosphere over the north of the Algerian region. The ionosphere maps, of which the coefficients are calculated for a shell height of 450 km, are visualised in Fig. 8. TEC snapshots taken at 01:00, 03:00, 05:00, . . ., 23:00 UT, with a spatial resolution of 2.5◦ × 2.5◦ , are shown. Contour lines are given for every 25 TECU. A set of 12 maps describe the ionospheric evolution during the day of April 30, 2001.

4 Impact of the Ionosphere on Single Frequency GPS Solutions In GPS positioning, the ionosphere-free combination, which needs the use of the dual frequency, is usually employed to eliminate the ionospheric effect. When only single frequency observations are available, the use of ionospheric corrections, calculated by a GPS network observing on both frequencies, is suitable. A network of 10 GPS stations (Fig. 9) is used to study and analyse the ionospheric effect on the single frequency GPS solution. L1 observations, with and without modelled ionosphere are processed and the

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Fig. 9 Stations of the network used for the study

obtained results are compared to the ionosphere-free solution (L3 ), which is considered as a reference solution. The processed baselines vary from 300 to 1500 km. Ionospheric corrections are calculated from 108 IGS stations with and without the 5 ALGEONET stations, and they are applied to correct and compare the computed baselines. Figure 10 shows that the not modelling the ionosphere induces deviations (with respect to L3 solution)

in baselines of about 50 cm–1.5 m; these deviations are correlated with the baseline length. It can be shown that differences between baselines obtained from L3 combination and those calculated by L1 single frequency are still positive. Hence, the impact of the not modelled ionosphere on single frequency GPS solution causes an apparent contraction of the network. When we include the ALGEONET stations in the computation of the ionospheric corrections, a

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significant improvement is obtained; deviations from L3 solution vary from 1 to 7 cm, while the obtained results from only the IGS stations permits to achieve a deviation of about 7–28 cm.

Results related to the ionospheric effect on geographic coordinates are described in Figs. 11, 12 and 13, which give deviations from the L3 reference solution.

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Figure 11 shows that the not modelling of the ionosphere may cause an error of more than one meter on the horizontal coordinates, while the error on the vertical component can reach 2.5 m (case of ZIMM and WTZR stations). Figure 12 shows that the use of ionospheric corrections, obtained from ALGEONET maps, allow to reduce the deviations from L3 solution with a very efficient manner. A deviation better than 10 cm in horizontal coordinates is obtained, while deviations in vertical component are reduced by about 50%. Results obtained from applying IGS ionospheric maps, and described in Fig. 13, are slightly less good than those obtained for ALGEONET’s ionospheric corrections.

5 Conclusion In this paper, we investigated the use of dual frequency GPS observations provided by a regional network (ALGEONET) to calculate TEC maps in the north of Algeria. The obtained ionospheric corrections are, thereafter, used to improve the quality of the single frequency GPS solution. The obtained results show that the maximal TEC value measured over the north of Algeria is about 60 TECU, this value corresponds to the noon period, where the sun is close to the zenith. The minimal TEC value varies around 20 TECU, it corresponds to the midnight period. The use of ionospheric corrections, deduced from calculated ionospheric maps permit to reduce the deviation with respect to the Ionosphere-Free solution, which is considered as a reference solution in the sense where it permits to eliminate the quasi-totality of the ionospheric effect. A deviation of 10 cm in horizontal coordinates is obtained, where the deviation in the vertical component is reduced by about 50%. We note that the effect of the non-modelled ionosphere on the single frequency GPS solution causes an apparent contraction of the network. The study of the ionosphere model dependency from the chosen height of the single layer has shown that the ALGEONET TEC maps are not sensitive to the tested heights of the layer (300, 400 and 450 km). All findings emitted through this work are based on the analysis of results obtained from the processing of one day test data (April 30, 2001). For the future

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works, it is recommended to use data from long period, which will permit to better understand and characterise the ionospheric behaviour and its impact on single frequency GPS positioning.

References Afraimovich E.L., Kosogorov E.A., Leonovich L.A., et al. Determining parameters of large scale travelling ionospheric disturbances of auroral origin using GPS arrays. Journal of Atmospheric and Solar Terrestrial Physics, 2000, 62, 553–565. Afraimovich E.L., Astafieva E.I., voyeikov S.V. Isolated ionospheric disturbances as deduced from global GPS network. Annales Geophysicae; 2004, 22, 47–62. Bilitza D. International reference ionosphere 2000. Radio Science, 2001, 36, 261–270. Ezquer R.G., Jadur C.A., Mostert M. IRI-95 TEC predictions for the South American peak of the equatorial anomaly. Advance in Space Research, 1998, 22(6), 811–814. Feltens J., Schaer S. IGS products for the ionosphere. Proceedings of the IGS Analysis Center Workshop, ESA/ESOC Darmstadt, Germany, 1998, 225–232. Feltens J. Chapman Profile Approach for 3-d Global TEC Representation. IGS Presentation, in proceedings of the 1998 IGS Analysis Centers Workshop, ESOC, Darmstadt, Germany, February 9–11, 1998, pp. 285–297. Gao Y., Heroux P., Kouba J. Estimation of GPS receiver and satellite L 1 /L 2 signal delay biases using data from CACS. In: Proceedings of KIS94, the International Symposium on Kinematic Systems in Geodesy, Geomatics, and Navigation, Banff, Alberta, Canada, 30 August-2 September. Department of Geomatics Engineering, The University of Calgary, Calgary, Alberta, Canada, 1994, pp. 109–117. Hern´andez-Pajares M., Juan J.M. and Sanz, J. New approaches in global ionospheric determination using ground GPS data. Journal of Atmospheric and Solar Terrestrial Physics, 1999, 61, 1237–1247. Hugentobler U, Schaer S, Fridez P. Bernese GPS Software Version 4.2. Users Manual, February 2001. Jakowski N., Wehrenpfennig A., Heise S., Reigber C. GPS radio occultation measurements of the ionosphere from CHAMP: Early results. Geophysical Research Letters, 2002, 29(10): 10.1029/2001GL014364. Jin S.G., Wang J., Zhang H.P., Zhu W.Y. Real-time monitoring and prediction of Ionosphere Electron Content by means of GPS. Chinese Astronomy and Astrophysics, 2004, 28, 331–337. Mannucci A.J., Wilson B.D., Yuan D.N., Ho C.H., Lindqwister U.J. and Runge T.F. A global mapping technique for GPSderived ionospheric total electron content measurements. Radio Science, 1998, 33, 565–582, 1998. Otsuka Y., Ogawa T., Saito A., Tsugawa T., et al. A new technique for mapping of total electron content using GPS network in Japan. Earth Planets Space, 2002, 54, 63–70.

Ionospheric Modelling in the North of Algeria Ping J. et al., Regional ionosphere map over Japanese Islands. Earth Planets Space, 2002, 54, e13–e1. Schaer S. Mapping and Predicting the Earth’s Ionosphere Using the Global Positioning System. Ph.D dissertation, Astronomical Institute, University of Berne, Switzerland, 1999.

689 Vesna Ducic. Tomographie de l’ionosph`ere et de la troposph`ere a` partir des donn´ees GPS denses. Applications aux risques naturels et am´elioration de l’interf´erom´etrie SA. Doctoral thesis. Paris VII University, Denis Diderot, March 2004.

Kinematic Precise Point Positioning During Marginal Satellite Availability N.S. Kjørsvik, O. Øvstedal and J.G.O. Gjevestad

Abstract Precise Point Positioning (PPP) is a technique where observations from a single GNSS receiver are used to estimate coordinates with precision ranging from centimeters to decimeters. PPP does not rely on data from dedicated reference receivers and is logistically a very competitive alternative to differential GNSS methods. In areas with reduced satellite availability the geometrical strength of the solution will be reduced and observations from such periods should be treated with special attention. In marine applications the use of additional information in the form of height constraints will enhance PPP during periods with marginal satellite availability. Additional observations from GLONASS satellites or Inertial Navigation System (INS) will also to a certain degree compensate for reduced availability of GPS satellites. This paper addresses the contribution from height aiding and from current GLONASS satellites. Based on formal precision and reliability as well as accuracy both height aiding and GLONASS improve the quality of estimated coordinates. The contribution to reliability is more significant than the contribution to precision and accuracy. In periods with very weak GPS constellation, height aiding gives the most valuable contribution. The paper verifies that even under marginal satellite availability, PPP processing is capable of producing coordinates with precision and accuracy at the subdecimeter level and with reliability at the one meter level. N.S. Kjørsvik TerraTec AS, P.O. BOX 513, N-1327 Lysaker, Norway O. Øvstedal Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, N-1432 As, Norway J.G.O. Gjevestad Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, N-1432 As, Norway

Keywords GNSS · PPP · Marine positioning

1 Introduction Precise Point Positioning (PPP) using GPS satellites was first developed for static applications (e.g. Zumberge et al., 1997), but has in recent years also been used in kinematic mode (e.g. Kouba and Heroux, 2001). In PPP undifferenced carrier phase and code phase observations are used to estimate coordinates and nuisance parameters like receiver clock errors, carrier phase ambiguities and residual tropospheric delay. The initial carrier phase offsets in satellites and receiver make the identification of integer carrier phase ambiguities a very challenging task, and in the standard PPP model the ambiguities are estimated as real numbers in a float solution. Compared to differential processing with fixed ambiguities, the geometrical strength of the adjustment in a float solution is weaker, requiring longer convergence times. When processing a sufficient time span of dual frequency code- and carrier phase observations from a single GPS receiver, sub-decimeter accuracy is nevertheless readily available. Prerequisites are more than 20–60 min of continuous observations of good quality as well as a sufficient number of well-distributed satellites. This exposes the need to treat periods with marginal satellite availability with special attention. In applications such as marine surveys in narrow and steep fjords, a significant number of satellites might be masked by the terrain. Satellites are typically only available at high elevation angles, and the geometrical strength of the solution will be severely weakened. A typical survey pattern will consist of a number of parallel fjord crossings. Hence, a sufficient

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number and distribution of satellites will only be available near the middle of the fjord. The situation is further complicated by the fact that different satellites are visible at each side of the fjord, leading to short time spans of continuous observations to some satellites. For other investigations of PPP in marine environments see e.g. Kjørsvik et al. (2006) and Øvstedal et al. (2002). In the proposed test design, almost 40 days of 1 Hz GNSS observations were collected at a shuttle ferry under good conditions. A reference trajectory was computed using a local (<2 km) reference station with fixed double-differenced ambiguities. Using a Digital Terrain Model (DTM) the topography of a nearby fjord was simulated, and the artificial horizon was applied to the actual data to limit the satellite availability. We thus have real observations in a scenario with marginal satellite availability but with a precise and reliable reference trajectory. Several approaches to improve PPP accuracy during marginal and changing satellite availability are possible. In this work we first investigate the contribution of current GLONASS satellites. The International GNSS Service (IGS) (Dow et al., 2005) provides precise orbit and clock corrections for both GPS and GLONASS, however the GLONASS products are currently not on par with the GPS products. The GLONASS constellation has also suffered from poor maintenance. Hence, under normal conditions GLONASS gives a very small contribution. The orbital inclination is however favorable at high latitudes, and even a few extra satellites may be highly beneficial during marginal conditions. In a second step we investigate the effect of height constraints during periods of marginal satellite availability. During regular surveys a reference (“truth”) trajectory will not be available. Instead, the quality of the positioning must be inferred solely from the data. The formal precision of the estimates provides some information, but high precision does unfortunately not guarantee high accuracy. In addition to precision, reliability theory (e.g. Teunissen and Salzmann, 1989) has proven to be a useful tool for quality assessment. The reliability theory uses rigorous statistics to quantify a system’s ability to detect and remove erroneous observations. This statistic is often denoted Minimal Detectable Bias (MDB). In addition, the effect of an undetected erroneous observation on the parameter estimates can also be computed. The latter is often denoted Minimal Detectable Error (MDE).

N.S. Kjørsvik et al.

2 Height Support A marine survey vessel will usually have a moderate vertical motion (heave). Heave is due to e.g. ocean tides and waves. A reference point (e.g. GNSS antenna) displaced from the vessel’s center of mass will also experience some vertical motion due to roll and pitch. The astronomical ocean tide can be modeled fairly accurate, provided that phases and amplitudes of the various constituents are known. The phases are however subject to large variations due to the coastline and local seabed topography. The actual tide will in addition be affected by local and regional weather phenomena.

2.1 Theoretical Development Assuming that the actual tide and wave motion can not be accurately predicted, the preferred solution should constitute a simple and robust model with well-defined error characteristics. As a first step it can be assumed that height coordinates can be accurately determined during periods with adequate satellite availability. A simple approach may then be to extrapolate this height to the periods with marginal satellite availability. Obviously heights related to the geoid should be considered. The error caused by the height extrapolation is expected to grow with time, and be a function of e.g. tidal amplitudes and local weather conditions. In the following development it is assumed that the tide consists of one dominating constituent with known amplitude and period, but unknown phase. It is assumed that a constant height (above the mean sea surface) can be estimated and removed. A predicted height using constant projection will then have a mean square error given by  2π 1 σt2 (θ ) = [ A cos(α + θ ) − A cos(α)]2 dα 2π 0 = A2 [1 − cos(θ )],

(1)

where A denotes the amplitude and θ = 2π t/T denotes the prediction step (in radians) and T is the period of the tidal wave. The period of the wave is assumed to be the semidiurnal period of the moon (M2 ), i.e. approximately 12 h 25 m. It can be assumed that the short period waves

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are uncorrelated with the tidal wave. As a first approximation we propose to model this contribution as an uncorrelated noise process with variance σw2 . The final expression for mean square error of the constant height assumption may then be formed as a sum of the independent contributions, and reads σ 2 (t) = A2 [1 − cos(2πt/T )] + σw2 .

(2)

2.2 Implementation Representative amplitudes were derived from the NAO99b model (Matsumoto et al., 2000) using an empirical approach. The ocean tide was predicted in a global grid with with 0.5◦ resolution using data from a random period of 30 days. Software from the NAO99 website1 was used to predict the tides. The amplitude was estimated for each node as 1/2 of the mean daily peak to peak value. In our PPP software the reference heights used for height support are estimated in a separate Kalman filter, working in parallel to the master filter. To compensate for geoid undulations, heights are converted to orthometric heights using the EGM96 model (NIMA, 2000). The variance of the waves (σw2 ) are estimated as the variance of the post-fit residuals from the height Kalman filter. The height support is implemented by adding a constraint (pseudo-observation) directly on the estimated height using an appropriate variance, as determined by the model in Eq. (2). Activation of the height support is triggered by testing a “modified” vertical dilution of precision (VDOP) against a predefined threshold of 3.0. The modified VDOP is computed using only satellites with a precisely determined phase bias parameter. A precision threshold of 2 cm is currently used. To avoid rapid and repeated switches, the height support is always kept enabled for at least 5 min.

3 GLONASS Modeling

between GPS and GLONASS. The parameter is estimated as a random walk process, with variance 3 (ns)2 /h. GLONASS precise orbits and clocks are often estimated in a separate process from GPS, and both GPS and GLONASS clock products are aligned to their respective broadcast clocks on a daily basis. Hence, the estimated inter-system bias must be reset each midnight to account for discontinuities between the GPS and GLONASS timescales. Inter-frequency biases in the carrier phase observations will be fully absorbed in the phase bias estimates, provided they are stable over timescales of hours. Similar biases in the code phases are currently ignored ignored in the functional model. The mean bias is absorbed in the inter-system bias estimate, and remaining biases are assumed to be small compared to the observation noise and code multipath. Problems might arise during rapid constellation changes, but should be alleviated by the stochastic model. GLONASS satellites have a Cesium-based (Cs) frequency standard and will consequently have a slightly worse short-time stability than Rubidium-based (Rb) satellites (e.g. all GPS block IIR and approx. 50% of GPS block II and IIA satellites). Short-time frequency instability leads to increased errors of interpolated satellite clock corrections, yielding increased noise in the corrected range observations. In the analyzed data set two satellites (R17 and R21) were manually excluded throughout the processing. The receiver experienced severe tracking anomalies, with several thousand cycle slips and a vast amount of outliers even when observed at high elevation angles without signal obstruction. Both satellites were flagged as healthy during the test period (cf. Table 1). A satellite with unhealthy observations has the potential of causing divergence in the estimate of the inter-system bias parameter, in particular shortly following a parameter reset (at midnight) in combination with few observed GLONASS satellites.

4 Test Data

From March to May 2006 the Norwegian HydroA system specific bias is estimated to account for graphic Service made a full scale test of their newly receiver hardware biases and time system biases acquired PPP software TerraPos from TerraTec AS, Norway. A shuttle ferry traveling between Lauvvik 1 http://www.miz.nao.ac.jp/staffs/nao99/index En.html and Oanes outside Stavanger, Norway was equipped

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Table 1 Status of GLONASS satellites during the test period Slot

Statusa

R01 R02 R03 R04 R05 R06

OK OK OK OKb UNHb OKb

R07 R08 R17 R18 R19 R20 R21 R22 R23 R24

OK OK OKb UNH OKb OK OK OKb UNH UNH

Excluded

Notes

62)N Unhealthy 2 h 30 m Healthy for 6 days Incomplete ephemeris, unhealthy 6 days

60)N

Oanes Yes

Yes

Tracking anomalies No ephemeris / obs. Unhealthy 10 h 30 m

58 N )

4)E

Tracking anomalies Unhealthy 30 m Incomplete ephemeris

a According

to the Russian Space Agency ftp://ftp.glonassianc.rsa.ru/MCC/STATUS/2006. b Unhealthy in parts of the dataset.

km 0

0.5

12)E

8)E

Lauvvik 1

Fig. 2 Map showing where the original data were collected

with one Topcon Legacy GNSS receiver. An identical receiver serving as reference station was deployed close to Lauvvik. Both stations were equipped with identical geodetic antennas. The ferry route is approximately 1.5 km long and the ferry repeats the route every half an hour. The terrain in the area is relatively flat, giving very good satellite availability. During this full scale test more than 40 days of continuous observations at 1 Hz were collected, with a one day break in the middle due to repairs in Stavanger. Real-Time Kinematic (RTK) solutions were recorded along with raw data. Figure 1 shows the ferry used to collect the data, with a map of the area shown in Fig. 2.

4.1 Reference Trajectory A reference trajectory was computed post mission using Geogenious version 2.11 from Spectra Precision Terrasat in 24 h batches. The post-processed solutions were then compared to the RTK solutions. For each epoch, the average of the two solutions were accepted if the 3D discrepancy was less than 2 cm. The average solutions constitute the reference trajectory.

4.2 Terrain Simulation

Fig. 1 Shuttle ferry carrying the GNSS receiver

In this study a subset consisting of 16 days (DoY 103–119) is analyzed. In order to simulate a marginal satellite availability with this real dataset, the more challenging topography of a nearby fjord called Lysefjorden has been utilized. Lysefjorden consists of very high and steep mountain sides and would generate severe obstructions to a GNSS receiver in the area. In this study a digital terrain model (DTM) of Lysefjorden is relocated to the ferry route between Lauvvik and Oanes, thus providing an artificial horizon to the ferry trajectory.

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lites is computed by assuming an open sky. Using the actual vessel position, a horizon is interpolated using a nearest neighbor strategy. A new list of visible satellites is constructed, and unavailable satellites are masked and used as information to reduce the real observation file.

5 Data Processing 5.1 Software Fig. 3 Oblique view of Lysefjorden, viewed towards east

Figures 3 and 4 are included to give an impression of the challenging topography in the simulation. The starting point of the terrain reduction is YUMA format almanacs for both GPS and GLONASS. The artificial horizons are based on a 3 × 6 arc sec. DTM (i.e. approximately 100 m grid size). A horizon is calculated for each grid position at sea level. Each horizon is stored with 2◦ azimuthal resolution. After the satellite almanacs are loaded the positions of the vessel trajectory are read epoch by epoch. For each vessel position a list of potentially visible satel-

Fig. 4 Map showing the terrain used in the simulation

The data were processed using the TerraPos software from TerraTec AS, Norway. TerraPos is compatible with the latest IERS conventions (McCarthy and P´etit, 2004), as well as IGS recommendations. TerraPos utilizes an optimal Kalman filter and smoother combination (e.g. Gelb, 1974). The data were processed in a single filter/smoother run, ensuring optimal estimation of all parameters. Effects that are currently handled include, e.g. solid earth tides, pole tide displacements, receiver and satellite antenna offsets and phase center variations and phase wind-up.

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Satellites in eclipses and noon-turn maneuvers are included in the processing. Satellite attitude is normally determined using the model from BarSever (1996). For periods or satellites where the attitude cannot be reliably predicted, the phase bias is assigned a random walk process. This is the case for all GLONASS satellites, as well as GPS satellites where the shadow entry or exit times can not be determined. No corrections are applied for ocean tide loading displacements. Tropospheric delay is predicted using the UNB3 model (Collins and Langley, 1997) in combination with the Global Mapping Function (Boehm et al., 2006). Residual tropospheric zenith delay is estimated as a 1st order Gauss-Markov process with a large time constant, i.e. a process approaching random walk. Table 2 gives an overview of the spectral amplitudes associated with the white noise driving functions of the various state parameters.

Table 2 Filter parameters Process

Spectral amplitude

Cross-track velocity Along-track velocity Vertical velocity Tropospheric ZD Inter-system bias

(2.0)2 m2 /s3 (2.0)2 m2 /s3 (0.2)2 m2 /s3 (3.0)2 mm2 /h (3.0)2 ns2 /h

Dual-frequency code and carrier phases as well as Doppler observations are employed using their respective ionosphere-free linear combinations.

5.2 Ephemeris and Satellite Clock Corrections In this study we use final ephemerides and satellite clock corrections from the Center of Orbit Determination in Europe (CODE) for the GPS satellites, due to the availability of high-rate (30 s) satellite clock corrections. Unfortunately, only broadcast clock corrections are included for GLONASS satellites. Hence, GLONASS ephemerides and clock corrections from the European Space Agency/European Space Operations Center (ESA/ESOC) are used. These files do however only have 15 min resolution, which is not optimal for high-rate kinematic positioning. Also, only the quality of the ephemeris is stated in the SP3-files as overall values for each day. The precision of the clock corrections are assumed to be at the same level as that of the orbits, i.e. 0.5 ns, corresponding to 0.15 m in terms of range error.

6 Results 5◦

An elevation cutoff angle of is used throughout this study. Minimal Detectable Errors (MDE) are computed using a Minimal Detectable Bias (MDB) approach as described in e.g. Teunissen and Salzmann (1989). The MDBs are computed using α ≈ 0.1% (probability of committing a type I error) and β = 80.0% (probability of committing a type II error). For a discussion of type I and II errors see e.g. Teunissen (2000). The reliability computations are performed during the forward filter run. Although somewhat suboptimal, steady-state is reached within minutes. Efficient implementation of an optimal computation of reliability is considered to be a non-trivial task. The absolute level of reliability is to a large extent governed by the parameters characterizing the dynamic model, the observation noise as well as the chosen levels of confidence. We therefore base most of the analysis of performance relative to that of a GPS-only system.

A total of 14 days of continuous observations at 1 Hz are merged into one observation file and processed in kinematic PPP-mode using the TerraPos software. The observations are processed using four different strategies: ◦ ◦ ◦ ◦

GPS only GPS with height support GPS and GLONASS GPS and GLONASS with height support

6.1 Overall Statistics In Tables 3, 4, 5 and 6 quality indicators from the four processing are presented. Precision and MDE are based on formal statistics from the processing software while the accuracy is based on differences between positions

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Table 3 95%-quantile for the GPS-only solution, using 1294820 epochs Hor. Vert.

STD (m)

MDE (m)

Error (m)

0.064 0.068

1.43 0.75

0.067 0.100

◦ Formal precision (1σ ) below 0.10 m horizontally and 0.15 m in height. ◦ MDE better than 0.30 m horizontally and 0.45 m in height.

Epochs where the quality criteria were not met in the GPS-only solution were therefore extracted and Table 4 95%-quantile for the GPS solution with height support, used to compute another set of quality indicators. using 1299197 epochs Table 7 shows formal precision, MDE, accuracy, STD (m) MDE (m) Error (m) HDOP and VDOP as well as total number of epochs Hor. 0.067 1.46 0.067 for the marginal set of data. The contribution from Vert. 0.069 0.75 0.098 height aiding, GLONASS and GLONASS+height aiding is more significant for the marginal set of the Table 5 95%-quantile for the GPS/GLONASS solution, using data. 1299766 epochs The results in Table 7 are based on epochs where STD (m) MDE (m) Error (m) a marginal GPS-only solution is possible. Due to an Hor. 0.063 1.14 0.074 insufficient number of GPS-satellites, there are 5237 Vert. 0.064 0.60 0.103 epochs where the GPS-only processing is not possible. Table 6 95%-quantile for the GPS/GLONASS solution with The extra information from height aiding, GLONASS and GLONASS+height aiding respectively leads to esheight support, using 1299767 epochs STD (m) MDE (m) Error (m) timated positions for a substantial part of these missing epochs. Table 8 presents quality indicators for the Hor. 0.062 1.14 0.073 epochs where a GPS-only solution is not possible. The Vert. 0.064 0.59 0.100 formal precision as well as accuracy are at the decimeter level for the height aided strategies while the MDE from the PPP processings and the reference trajectory. is at the 2–3 m level. The contribution of GLONASS The statistics are given as 95% quantiles. is not as significant as the contribution from height For all strategies the precision and accuracy are at aiding. the sub-decimeter level horizontally and at the decimeter level in height. The MDEs are approximately at Table 7 Statistics (95%) for the epochs rejected in a GPS-only the meter level. Compared to the GPS-only solution, solution GPS GPS GPS GPS the improvements in formal precision and accuracy GLO GLO are at the millimeter level when including height HS HS aiding, GLONASS and GLONASS+height aiding. 0.086 m 0.084 m 0.081 m 0.081 m The improvement in MDEs are at the level of several H prec. V prec. 0.094 m 0.092 m 0.088 m 0.086 m decimeters. H rel. 2.11 m 2.08 m 1.70 m 1.70 m The apriori process noise for vertical coordinates is V rel. 1.03 m 0.97 m 0.73 m 0.69 m 0.080 m 0.079 m 0.085 m 0.084 m smaller than for horizontal coordinates, leading to bet- H err V err. 0.123 m 0.116 m 0.125 m 0.119 m ter MDEs for vertical coordinates.

6.2 Quality of Marginal Solutions It should be noted that Tables 3, 4, 5 and 6 are based on a very large number of epochs where the majority of estimated positions are of reasonably good quality. In order to assess the epochs with somewhat marginal satellite availability, quality indicators are also computed based on epochs where a GPS-only solution does not meet the following quality criteria:

HDOP VDOP # epochs

6.1 8.1 392854

6.1 8.1 392854

3.1 5.0 392854

3.1 5.0 392854

6.3 Post-Fit RMS Statistics During the filter run an overall RMS of the post-fit residuals is computed for each observable and system separately. These overall values are hence averaged

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Table 8 Statistics (95%) for the epochs not available in a GPSonly solution GPS

GPS GLO

GPS GLO HS

1.659 m 4.141 m 3.35 m 2.67 m 2.159 m 6.205 m 5.1 14.2 4951

0.174 m 0.223 m 3.43 m 1.92 m 0.108 m 0.113 m 5.1 14.2 4951

HS H prec. V prec. H rel. V rel. H err V err. HDOP VDOP # epochs

0.211 m 0.236 m 3.54 m 2.82 m 0.111 m 0.100 m ∞ ∞ 4377

Table 9 Overall post-fit RMS statistics. All observables are used with their respective ionosphere-free linear combinations Observable

GPS

GLONASS

Carrier phase Code phase Doppler

0.01 m 2.5 m 0.01 m/s

0.15 m 3.5 m 0.01 m/s

over all elevation angles and satellites in a system. The results are shown in Table 9. The carrier phase errors are dominated by satellite clock errors. A value of 0.01 m for GPS demonstrates the excellent quality of the IGS GPS products. A value of 0.15 m for GLONASS carrier phases thus verifies the assumed quality of the GLONASS satellite clock corrections. The slightly higher RMS of the GLONASS code phase observations as compared to GPS is to some extent explained by the lower chip rate, but also suggests the presence of small inter-frequency biases. The effect of satellite clock errors (<0.15 m, as seen from the carrier phase statistics) is negligible compared to the combined level of multipath and noise. Both GPS and GLONASS Doppler observations seem to have comparable performance.

7 Summary and Conclusion We have optimized the memory usage of our software in order to process huge kinematic datasets without loss of optimality. The software is now capable of processing several months of 1 Hz kinematic GNSS data in an integrated and optimal adjustment on a standard 32 bit PC-based workstation, with only modest memory requirements. This achievement can be important in

the study of several geophysical signals observed by GNSS. The software’s ability to process GLONASS observations has been verified, and valuable experience with multi-system modeling and parameter estimation has been gained. We have identified issues with timescales and inter-system biases that arise when using clock products spanning several day and week boundaries. These artifacts are due to current analysis and data combination strategies of the analysis centers that contribute to the IGS. In light of the current state of GLONASS ephemeris and clock products the current handling of code phase inter-frequency biases seems adequate, as justified by the post-fit RMS statistics. A refined model for handling of inter-frequency biases (in satellites and in receivers) must be used by all IGS analysis centers to ensure a high degree of consistency in satellite clock and calibration products. No such convention is currently available. The current contribution of GLONASS to PPP is hampered by the number of available satellites and the quality of ephemeris and satellite clock corrections. The quality of the satellite clock corrections is partly due to the temporal resolution, but also to the shorttime stability of the Cesium frequency standard. The net result is that the low precision of the corrected ranges limits the impact on the parameter estimates and their precision. Redundancy is nevertheless improved, yielding improved MDE. In this study we found that the MDE was improved by approximately 20%. In cases where GPS alone was unable to provide a solution, the combined solutions were accurate only to the meter-level. In contrast to the addition of a GLONASS observation, the height support constrains the height component directly. A GLONASS observation has a functional relation to many parameters, including position, receiver clock bias, inter-system bias, tropospheric zenith delay etc. The activation of height support was found to consistently improve the statistics of the marginal solutions, in particular by improving the MDE by approximately 5%. The high-resolution terrain simulation provides a detailed model of the topography within Lysefjorden, yielding a very realistic obstruction scenario. However, one should keep in mind that the quality of the observations, e.g. multipath and cycle slips, might be too optimistic when applying artificial terrain obstructions. In a real-world degraded environment a receiver

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would occasionally experience low signal power and Kjørsvik, N., Gjevestad, J. G. and Øvstedal, O. (2006). Handling of the troposheric delay in kinematic Precise Point Positioncycle slips. One should also note that in situations with ing. In Proceedings of ION GNSS 2006, Fort Worth, TX. Inmarginal satellite availability observation domain erstitute of Navigation. rors may have severe impact on estimated parameters. Kouba, J. and Heroux, P. (2001). GPS Precise Point Positioning Acknowledgments The authors greatly acknowledge the International GNSS Service for its continuing efforts. The artificial horizons were simulated by Lars K. Nesheim and Arne E. Ofstad of the Norwegian Hydrographic survey.

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using IGS orbit products. GPS Solutions, 5(2), pp. 12–28. Matsumoto, K., Takanezawa, T. and Ooe, M. (2000). Ocean tide models developed by assimilating TOPEX/POSEIDON altimeter data into hydrodynamical model: A global model and a regional model around Japan. Journal of Oceanography, 56, pp. 567–581. McCarthy, D. D. and P´etit, G. (editors) (2004). IERS Conventions (2003). Verlag des Bundesamtes f¨ur Kartographie und Geod¨asie, Frankfurt am Main. IERS Tecnical Note No. 32. NIMA (2000). Department of Defense World Geodetic System 1984. Technical Report TR8350.2, National Imagery and Mapping Agency. Third edition, Amendment 1. Øvstedal, O., Ofstad, A., Haustveit, K. and Kristiansen, O. (2002). An empirical comparison between absolute satellite positioning methods and differential methods in a maritime environment. In Proceedings of ION GPS 2002, Portland, OR. Institute of Navigation. Teunissen, P. J. G. (2000). Testing Theory. An Introduction. Series on Mathematical Geodesy and Positioning. Delft University Press. ISBN 9040719756. Teunissen, P. J. G. and Salzmann, M. A. (1989). A recursive slippage test for use in state-space filtering. Manuscripta Geodaetica, 14, pp. 383–390. Zumberge, J. F., Heflin, M. B., Jefferson, D. C. and Watkins, M. M. (1997). Precise point positioning for the effecient and robust analysis of GPS data from large networks. Journal of Geophysical Research, 102(B3), pp. 5005–5017.

Establishing a GNSS Receiver Antenna Calibration Field in the Framework of PROBRAL ¨ C.P. Krueger, J. Freiberger, B. Heck, M. Mayer, A. Knopfler and B. Sch¨afer

Abstract PROBRAL - Precise positioning and height determination by means of GPS: Modeling of errors and transformation into physical heights is the name of a joint venture between the Department of Geomatics (DGEOM), Federal University of Paran´a (UFPR), Curitiba (Brazil) and the Geodetic Institute (GIK), University Karlsruhe (TH), Karlsruhe (Germany). The aim of this research project, which started in 2006 and is founded by the Brazilian academic exchange service CAPES and the German academic exchange service DAAD, is to validate and to improve the quality of GNSS-based positioning, especially concerning the height component. Therefore, the close cooperation between the DGEOM and the GIK was intensified. One main objective of the first year of this three years lasting

C.P. Krueger Department of Geomatics, Federal University of Paran´a, Centro Polit´ecnico, CP 19.001, CEP 81531-990, Curitiba, Paran´a, Brazil, e-mail: [email protected] J. Freiberger Department of Geomatics, Federal University of Paran´a, Centro Polit´ecnico, CP 19.001, CEP 81531-990, Curitiba, Paran´a, Brazil B. Heck Geodetic Institute, University of Karlsruhe (TH), Englerstraße 7, D-76131 Karlsruhe, Germany e-mail: [email protected] M. Mayer Geodetic Institute, University of Karlsruhe (TH), Englerstraße 7, D-76131 Karlsruhe, Germany A. Kn¨opfler Geodetic Institute, University of Karlsruhe (TH), Englerstraße 7, D-76131 Karlsruhe, Germany B. Sch¨afer Geodetic Institute, University of Karlsruhe (TH), Englerstraße 7, D-76131 Karlsruhe, Germany

cooperation was to establish a receiver antenna calibration field for GNSS instrumentations on the roof top of the so-called LAGE (http://www.lage.ufpr.br/). In the framework of PROBRAL case studies were carried out in close collaboration at the DGEOM as well as at the GIK concerning site-specific effects (e.g. multipath, receiver antenna modeling), especially. The status of the establishment of a GNSS receiver antenna calibration field, called First Baseline Calibration Station for GNSS Antennas in Brazil (1a . BCALBR), is going to be presented within this paper. The very first results derived on 1a . BCALBR showed that the pillar 1000 is affected by carrier phase multipath less than pillar 2000. The IBGE (Brazilian Institute of Geography and Statistics) reference station PARA is slightly affected by carrier phase multipath. Keywords GNSS · GPS · Antenna calibration field · PROBRAL · 1a . BCALBR

1 Introduction The subject of GNSS antenna calibration has been investigated and analyzed by different international working groups, for nearly two decades. Various studies, for example on receiver and satellite antenna phase center variations and on the influence of the dominant reflectors in the vicinity of the receiving antenna, have been carried out by the GNSS community. Due to the continuous improvement of the functional GNSS model, concerning orbit determination and atmospheric modeling for example, site-specific effects are playing nowadays a more important role in GNSS positioning.

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Since 2006 studies dealing with these site-specific effects have been performed in Brazil by the Federal University of Paran´a (UFPR), Curitiba (Brazil) in the framework of the international project PROBRAL - Precise positioning and height determination by means of GPS: Modeling of errors and transformation into physical heights, founded by the Brazilian academic exchange service CAPES and the German academic exchange service DAAD. Within this project the Department of Geomatics (DGEOM), UFPR, Curitiba (Brazil) and the Geodetic Institute (GIK), University of Karlsruhe (TH), Karlsruhe (Germany) are cooperating. One main objective of the first year of this cooperation is to establish a receiver antenna calibration field for GNSS instrumentation. This receiver antenna calibration field was developed in Brazil in order to fulfill two main goals:

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Sensitization: The error source receiver antenna has to be realized as limiting effect in GNSS positioning from Brazilian surveyors. This intention is easier to fulfill, when a receiver antenna calibration field is existing. It was decided to built up a relative calibration site in order to derive phase center offset (PCO) and phase center variations (PCV) values using the software Wal/Kalib (Wanninger, 2006). In order to guarantee high-quality calibration values site-specific characteristics, like multipath effects, obstructions, have to be checked carefully. Because every calibration field has to be completely independent from any multipath effects (W¨ubbena et al., 2002) Verification: Development of testing methods for geodetic instrumentations used in GNSS real time applications.

In order to fulfill these goals three milestones were set concerning

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distance to LAGE, power supply, and security aspects.

One important milestone was fulfilled when the First Baseline Calibration Station for GNSS Antennas in Brazil (1a .BCALBR) on the roof top of the Astronomical Laboratory Camil Gemael near the Laboratory of Space Geodesy (LAGE – Laborat´orio de Geod´esia Espacial, http://www.lage.ufpr.br/) on the Polythecnical Campus was built.

2 Construction of 1a . BCALBR Pillars An important aspect of PROBRAL projects is to transfer technology between the two cooperating institutes. Hence, three stable pillars could be constructed consisting of material with a long expected life time based on experiences concerning the establishment and monumentation of geodetic network points gained at the GIK. The 1a . BCALBR is actually equipped with two pillars 1000 and 2000 on the roof of the Astronomical Laboratory Camil Gemael. The construction is based on a core of steel and concrete with a 1.5 m height, measured from the base of the block. The weight of each pillar is approximately 350 kg. A mantle of concrete with space in between, in order to guarantee thermal isolation and to minimize thermal expansion, surrounds the centre of the pillars. The pillars are equipped with a metal thread to enable the application of a geodetic tribrach following the rules of the responsible surveying and mapping authorities in Brazil - IBGE (Brazilian Institute of Geography and Statistics) (IBGE, 2007). The marker used guarantees submillimeter accuracy. The circular form was chosen

feasibility studies and development of an appropriate calibration method, construction of pillars of 1a . BCALBR, and first GPS measurements on the pillars of 1a . BCALBR.

In the beginning, studies concerning an appropriate location of the establishment of 1a . BCALBR on the Polytechnic Campus were carried out. The decision concerning the location of the calibration filed was based on indicators like

Fig. 1 Pillar 1000 of the 1a . BCALBR and the cross-section AA’

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Fig. 2 Horizon mask including obstructions of pillar 1000 resp. 2000

to reduce pillar reflected multipath effects. A picture of pillar 1000 is presented in Fig. 1. Figure 1 also shows a horizontal cross-section of the pillars. The horizontal mask of both pillars is given in Fig. 2. Please note that some trees exist above the antenna horizons. The roof of the Astronomical Laboratory itself has to be considered as potential reflectors. In the case of pillar 1000 (West), the roof will affect the signals received from all elevation angles from 50◦ to 110◦ in azimuth range, especially. At pillar 2000 (North) there are fewer obstructions.

Six control points (P1 to P6) fixed to the laboratory were established and observed by means of leveling, see Fig. 3. Until now, the differences between the leveling runs were considerably small. The maximum value of 0.6 mm was observed at control point P4 (Fig. 4) (Freiberger, 2007).

3 Monitoring of 1a . BCALBR Until now two high-precision geometrical leveling runs have been carried out to monitor 1a . BCALBR. One was realized before the construction of the pillars and another nearly 50 days after that. It is planned to repeat the leveling runs regularly. The reference for the leveling consists of three reference points which are located near the GPS station PARA, which is part of the IGS (International GNSS Service) as well. The distance from PARA to the calibration pillars is less than 100 m. This station is also part of the Brazilian Continuous Monitoring GPS Network (RBMC) of the Brazilian Institute of Geography and Statistics IBGE. The coordinates of PARA are given in the SIRGAS2000 reference frame (Brunini and Sanches, 2007).

Fig. 3 Location of the pillars of the 1a . BCALBR and the six control points

4 Quality of the First GPS Measurements at 1a . BCALBR In this initial phase of the project, additional objectives were to detect and to quantify carrier phase multipath effects of the pillars and to carry out quality investigations based on signal-to-noise ratio values in order to guarantee precise and reliable calibration values.

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The first GNSS measurements were realized with different dual-frequency GPS receivers. The data were collected in several sessions of 24 h with a data rate of 30 s. The software WaSoft/Multipath 3.2 was used to detect and to localize carrier phase multipath (Wanninger, 2005). Therefore, various GPS networks consisting of a maximum of five reference stations were analyzed. WaSoft/Multipath 3.2 is a post-processing software package, which was developed at the Geodetic Institute of the University of Dresden, Germany (Wanninger, 2003). According to Wanninger and May (2000) the multipath detection algorithm is based on the following characteristics and assumptions:

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In detail, the software is based on undifferenced residuals of the ionosphere-free linear combination L3, calculated for 20 minute blocks of data of low elevated (below 50◦ ) satellites. Using undifferenced residuals double-differenced residuals of simultaneous observations of the highest elevated satellite at the same station and the corresponding observations at the other stations in the network of at least three (better four or up) stations are generated. By means of averaging techniques carrier phase ambiguities, remaining tropospheric effects, and the influence of orbit errors are reduced. The resulting data set is analyzed using correlation coefficients.

The program WaSoft/Multipath 3.2 needs 24 h, with an interval not bigger than 60 s. In this case study RINEX data sets collected with different dualfrequency GPS intrumentations were used. Broadcast ephemerides were introduced, too. The network limits should be smaller than 100 km. The coordinates of the stations have to be known in the WGS84 reference frame with a precision in the range of a few centimeters (Wanninger, 2003). Figure 5 shows an example of a multipath detection grid map. The used symbols are relating to multipath impact

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blank: no observation data, dots: no multipath effects detected (standard deviation based on L3 <5 mm), small squares: minor multipath effects (standard deviation based on L3:5–15 mm),

Dominant multipath periods range from 10 to 45 min and dependent mainly on the distance between reflector and receiver antenna. Ionospheric refraction is eliminated by forming ionosphere-free linear combinations. The software was developed to check multipath effects of GNSS reference stations. Due to the location of the receiver antennas, such as top of roofs, it is assumed that all reflectors are situated below the antenna horizon and thus it seems appropriate to expect that satellite signals with low elevation angles (e.g. below 50◦ ) are affected and signals received from higher elevations (e.g. above 50◦ ) show little or no multipath effects. Due to the fact that multipath effects are sitespecific, multipath effects of different stations are Fig. 5 Example of multipath detection grid map; Source: http://www.wasoft.de/e/mltp/index.html uncorrelated.

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completely black: major multipath effects (standard daily multipath maps were analyzed an compared to maps created by Wasoft/MP. deviation based on L3 >15 mm). Based on this analysis it was clearly visible that The interpretation of Fig. 5 is major carrier phase there are only weak multipath effects affecting the sigmultipath effects disturb signals incident from east nals received at the stations PARA, 1000 and 2000. directions and from an elevation of 10◦ –30◦ . All But the signals received at the stations UNIC and 0003 other signals show minor or no multipath errors are distorted strongly due to multipath effects (UNIC: strong multipath effects, 0003: very strong multipath (Wanninger, 2005). effects). The antenna of UNIC is located at a metallic pillar mounted on a flat roof. The antenna type used was developed for real time kinematic GPS, therefore 4.1 Multipath Analyses Based on a and because of the installation on the roof, strong multipath effects result. 0003 is characterized by a lot of Network Solution of Five Stations potential reflectors in the vicinity of the antenna such Firstly, a network consisting of five stations UNIC, as walls and roof tops. Based on Wanninger and May (2000) the behavior 0003, PARA, 1000, and 2000 was chosen (Fig. 6) shown by PARA could be described as “multipathand analyzed by means of WaSoft/Multipath 3.2 free”. In another investigation (Mayer et al., 2004) (Sch¨afer, 2007). stations with about the same carrier phase multipath The station UNIC is part of the so-called Network effects as PARA were called “good” behaving stations. for Continuous GPS Monitoring operated by Manfra, which is a private Brazilian company (Manfra, 2007). The IBGE reference station PARA showed throughout This station is accepted by the IBGE and the coordi- the whole experiment just a few multipath effects. This also was expected, because the antenna is a chokering nates are given in the SIRGAS2000 reference frame. The local company Digital Mapas in Pinhais estab- antenna that has a good protection against multipath effects and no obstructions or reflector exists above the lished the station 0003. The multipath detection method described above antenna horizon. was applied to a local network (Fig. 6). The resulting

Fig. 6 GPS network for multipath analyses

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The intention was to derive more reliable results related to multipath effects of the pillars 1000 and 2000. In order to guarantee reliable results GPS data of the stations 1000 and 2000 was collected on six days using different antennas (“CR”: Leica chokering; “GP”: Trimble geodetic L1/L2 compact with groundplane; “no GP”: Trimble geodetic L1/L2 compact without 4.2 Multipath Analyses Based on a groundplane). The environment was interchanged beNetwork Solution of Three Stations tween the pillars. Figure 7 presents the resulting multipath maps of PARA, 1000 and 2000 for all sessions. In this section the evaluation described in Sect. 4.1 A comparison with respect to the multipath impact is repeated without the stations 0003 and UNIC, of the two pillars was possible as well, using the results because the detection and localization algorithms of the same antenna/receiver combination at different require a majority of unaffected stations (Wanninger sessions. Pillar 1000 seems to be slightly more affected and May, 2000).

The pillars 1000 and 2000 would be ranked as “good” or “medium”. The results change due to the used antenna.

Fig. 7 Maps describing the multipath effects of PARA, 1000 and 2000, GPS Week 1393; legend: blank: no observation data, dots: no multipath effects, small squares: minor multipath effects, completely black: major multipath effects

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by multipath effects than pillar 2000, analyzing the results of the CR antenna at 1000 (doy 261) and at 2000 (doy 262). The results obtained using the GP antenna (2000: doy 261; 1000: doy 262) confirm this conclusion. As expected, the multipath effects are highly correlated with the horizon masks shown in Fig. 2. For pillar 1000 multipath effects can be seen between 50◦ and 100◦ of azimuthal direction in all experiments carried out for this pillar. This agrees very well with the existing obstructions, see Fig. 2. On pillar 2000 the situation is different, the potential reflector (roof top) is located between 140◦ and 160◦ (azimuthal direction), but because of the southern hole, less GPS signals arrive from these azimuths, so the multipath effects due to the roof are significantly smaller. In the multipath detection grid map (cf. Fig. 7) this effect can be seen clearly. Based on these results it could be stated that the multipath effects at pillar 1000 are always stronger than at pillar 2000. Looking at the multipath maps for the same pillar and different antennas (cf. Fig. 7), CR gives the best results. A significant difference between the geodetic antenna with groundplane (GP) and without groundplane (no GP) could not be detected using multipath maps.

5 Future Work Further investigations are planed including

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detailed analysis of the carrier phase multipath effects of 1a . BCALBR based on signal-to-noise ration values, analysis of the influence of meteorology on carrier phase multipath effects, carry out the first calibration on the 1a . BCALBR in November 2007, and building of a third pillar on the 1a . BCALBR.

707 Acknowledgments The GPS reference station observations were made available by the IBGE, MANFRA and DIGITAL MAPAS. The pillars 1000 and 2000 were built by the support from CAPES. The research project was founded by CAPES and DAAD.

References Brunini, C., Sanches, L. (2007): Sistema de Referencia Geoc´entrico para las Am´ericas SIRGAS. Boletin Informativo n◦ . 12, Source: http://www.sirgas.org/fileadmin/docs/ Boletines/Boletin SIRGAS 12.pdf. Freiberger Jr., J. (2007): Investigac¸o˜ es para a calibrac¸a˜ o relativa de antenas de receptores GNSS. Thesis (Doutorado em Ciˆencias Geod´esicas). Curso de P´os-Graduac¸a˜ o em Ciˆencias Geod´esicas, Departamento de Geom´atica, Universidade Federal do Paran´a, Curitiba, Paran´a, Brazil. IBGE (2007): Instituto Brasileiro de Geografia e Estat´ıstica. Source: http://www.ibge.gov.br/home/geociencias/geodesia. Manfra (2007): Rede Manfra de Estac¸o˜ es de Monitoramento Continuo de GPS. Source: http://www.manfra.com. br/conteudo.php?id=31. Mayer, M., Wanninger, L., Dick, H.-G., Derenbach, H., Heck, B. (2004): Mehrwegeeinfl¨usse auf den SAPOS-Stationen Baden-W¨urttembergs. Geod¨atische Woche, Stuttgart, 12–15 October. Sch¨afer, B. (2007): Investigations on multipath effects of GPS reference stations in Paran´a and the First Baseline Calibration Station of Brazil (1a . BCALBR) in Curitiba, PR, Brazil. Study work (unpublished). Wanninger, L., May, M. (2000): Carrier Phase Multipath Calibration of GPS Reference Stations In: Proceedings of ION GPS2000, Salte Lake City, UT, pp. 132–144. Wanninger, L. (2003): Anleitung Wasoft/Multipath 3.2, Ingenierub¨uro Wanninger, http://www.wasoft.de, 09/2003. Wanninger, L. (2005): Carrier Phase Multipath Detection and Localization in GPS Reference Station Networks. http://www.wasoft.de/e/mltp/index.html. Wanninger, L. (2006): Anleitung Wal/Kalib. Source: http://www.wasoft.de, 03/2006. W¨ubbena, G., Schmitz, M., Seeber, G. B¨oder, V., Menge, F. (2002): Developments in Absolute Field Calibration of GPS Antennas and Absolute Site Dependent Multipath. IGS Workshop “Towards Real-Time”, Poster-Session, April 8–11, Ottawa, Canada.

Trimble’s RTK and DGPS Solutions in Comparison with Precise Point Positioning ¨ Herbert Landau, Xiaoming Chen, Soren Klose, Rodrigo Leandro and Ulrich Vollath

Abstract Differential GPS (DGPS) and Real-TimeKinematic (RTK) solutions are widely accepted methods for accurate positioning and navigation. Initially these methods were based on single reference stations. A big breakthrough in performance and accuracy was achieved by the introduction of network solutions. Today, the use of the Virtual Reference Station method is a standard technology applied in a large number of regional RTK networks all over the world, to provide a positioning service with centimetre accuracy. DGPS techniques are used for marine, airborne and land applications and provide accuracies in the several decimetre range. The DGPS and RTK methods are based on regional reference station networks or nearby single stations. In contrast, the Precise Point Positioning (PPP) method is providing position accuracy of several centimetres to decimetres based on a globally distributed tracking network. Precise satellite orbit and clock information is derived from the network processing. The achievable accuracy with PPP is impressive, but this method suffers from long convergence times of the order of more than 20–30 min to achieve acceptable

accuracies and therefore might not be the method of choice for all applications. This paper describes the current state-of-the-art in RTK and DGPS technology and compares it with the performance of PPP applications today. Performance numbers are presented for various baseline lengths and networks including accuracy, convergence, and reliability. Keywords PPP · DGPS · RTK · Network RTK · Virtual Reference Station (VRS)

1 RTK and Network-RTK RTK (Real-Time-Kinematic) positioning was introduced in the early nineties; RTK is basically the real-time implementation of methods used since the early eighties (e.g. Remondi, 1984). Code and carrier phase data are used in RTK positioning. The following simplified observation equations can be used to describe GPS code/carrier phase data (noise and multipath effects are ignored in this description):

Herbert Landau Trimble Terrasat GmbH, Haringstrasse 19, 85635 H¨ohenkirchen, Germany

λφrs (ti ) = ρrs (ti ) + c · (τr (ti ) − τ s (ti ))

Xiaoming Chen Trimble Terrasat GmbH, Haringstrasse 19, 85635 H¨ohenkirchen, Germany

Prs (ti ) = ρrs (ti ) + c · (τr (ti ) − τ s (ti ))

S¨oren Klose Trimble Terrasat GmbH, Haringstrasse 19, 85635 H¨ohenkirchen, Germany Rodrigo Leandro Trimble Terrasat GmbH, Haringstrasse 19, 85635 H¨ohenkirchen, Germany Ulrich Vollath Trimble Terrasat GmbH, Haringstrasse 19, 85635 H¨ohenkirchen, Germany

+ αrs (ti ) − Irs (ti ) + (βr (ti ) − β s (ti )) + λNrs

(1)

+ αrs (ti ) + Irs (ti ) + (γr (ti ) − γ s (ti )) where λ φrs Prs

nominal carrier phase wavelength, carrier phase observation for receiver r and satellite s, Pseudorange observation for receiver r and satellite s,

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ρrs τr , τ s αrs Irs βr , β s γr , γ s Nrs c

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geometric range from receiver r to satellite s, receiver and satellite clock bias terms respectively, non-dispersive atmospheric and other biases, dispersive atmospheric (ionospheric) bias, receiver and satellite carrier bias terms respectively, receiver and satellite code bias terms respectively, integer ambiguity term, and speed of light.

RTK is based on differencing techniques; mostly the double-difference technique is applied. Differencing allows elimination, or at least a significant reduction, of all major error sources, as shown in Eq. (2): ab ab ab ∇ λφ ab jk (ti ) = ∇ ρ jk (ti ) − ∇ I jk (ti ) + λ∇ N jk ab ab ∇ P jk (ti ) = ∇ ρ ab jk (ti ) + ∇ I jk (ti )

(2) where ∇ is the double difference operator introducing differences between 2 satellites a and b and 2 receivers j and k. It is important to note that the clock bias terms τ and the code and carrier bias terms β, γ are completely eliminated, while the non-dispersive (tropospheric) effects and the biases α, which are playing a dominant role in PPP, are greatly reduced for short baselines of a few kilometres when applying a standard tropospheric model. The ultimate accuracy and productivity of RTK is derived from the carrier phase ambiguity resolution and fixing process. Once the ∇ N are determined, the carrier phase measurements provide optimum accuracy for the position solution. The ionospheric effects are either neglected (on very short baselines) or eliminated by using the standard dual-frequency combination. Today, RTK is a standard technique implemented in various commercially available GNSS systems. While in the past mostly GPS was the only satellite system used, GLONASS is used now in a variety of RTK implementations as well. We’d like to mention at this point that all the analyses we have done for this paper were carried out using GPS data only, and therefore we will use the term DGPS instead of DGNSS in this paper. The biggest advantages of RTK are definitely its

accuracy and productivity (the time needed to achieve the high accuracy with the positioning system). Disadvantages are its dependency on a reference station and its inability to work reliably over long distances from the reference station. Commercial RTK systems today are achieving 99.9% reliability or better on distances of up to approximately 20 km. On longer distances this kind of reliability is not routinely reached under standard conditions. The main limiting factors are the differential atmospheric (ionospheric and tropospheric) effects, which limit the ability to fix ambiguities reliably. In order to get RTK systems working over longer distances, the Virtual Reference Station (VRS) method for post-processing was developed and introduced by Wanninger (1997). Later on VRSTM (trademarked by Trimble) became the standard technology of real-time network processing for RTK positioning. Instead of “single Base RTK”, VRS or “Network-RTK” uses a fixed network of reference stations. An example of such an RTK network is the Trimble “VRSNowTM ” UK system covering England and most of Scotland (Fig. 1). The network consists of more than 90 stations. While the number of commercial service networks is increasing, the majority of the networks are public and operated mainly by state land survey authorities. Typical distances between reference stations are 50– 70 km. After its introduction in the late nineties, NetworkRTK technology based on the VRS approach became an accepted and proven technology, and is widely used today in a large number of installations all over the world. Developments over the past years (Chen et al., 2003, 2004, 2005, 2007; Kolb et al., 2005; Landau et al., 2007) have resulted in the software server solution GPSNetTM . Today this solution handles up to 100 reference stations on one single personal computer (PC) and provides optimized correction streams to RTK users via cell phone technology or radio. In comparison with traditional single base RTK technology, network-RTK removes a significant amount of spatially correlated errors dueto the troposphere, ionosphere and satellite orbit errors, and thus allows RTK positioning in reference station networks with inter-receiver distances of 40 km or more from the nearest reference station, while still providing the performance of short baseline positioning. Currently more than 3000 reference stations are operating in networks in more than 30 countries using the

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Fig. 1 An example VRS Network (Trimble VRSNowTM in the UK)

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Trimble GPSNet solution (Fig. 2). Data processing in GPSNet utilizes the mathematically optimal Kalman filter technique to process data from all network reference stations. This models all relevant error sources, including satellite orbit and clock errors, reference station receiver clock errors, multipath and particularly ionospheric and tropospheric effects. Today RTK and VRS/Network-RTK provide high accuracies (centimetres, and in some cases sub-centimetres) in kinematic applications. Typical accuracies for a 16 km baseline are shown in Figs. 3 and 4. Landau et al. (2007) have shown that these results can be achieved by optimizing the use of VRS information in the rover receiver. A summary of typical single base RTK accuracies can be found in Fig. 5. Note that the vertical error increases with baseline length The error increase in VRS/RTK (network-RTK) solutions is much more moderate than in the single base RTK solution (Fig. 6). As can be seen in Fig. 7, RTK provides today initialization times of less than 10 s on short baselines of up to approximately 5 km. Even on longer baselines the average initialization time does not increase dramatically. However, we find that the 90% TTF (a measure which defines that 90% of the initializations are done within that time) are increasing above approximately

Fig. 2 Countries in which Trimble VRS systems are operating and providing RTK services

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Fig. 3 Position errors in East direction for the L1/L2 optimized VRS corrected solution in red (2 mm RMS) and the standard ionospheric-free solution in blue (6 mm RMS)

15 km (Fig. 8). This is expected since the residual systematic effects are also increasing. RTK solutions in VRS networks provide a lower increase in TTF with respect to the baseline length. RTK does initialize routinely on distances of up to 35 km in less than 20 s. Due to the fact that network-RTK (VRS) reduces significantly the atmospheric effects, the average as well as the 90% TTF can be kept consistent over the whole range of baseline distances up to 35 km.

Fig. 4 Position errors in Height direction for the L1/L2 optimized VRS corrected solution in red (13 mm RMS) and the standard ionospheric-free solution in blue (21 mm RMS)

Today most of developed countries in the world either have already existing dense reference station infrastructure for providing RTK services, or have started to build up such systems. This development is going hand in hand with the increasing coverage of GPRS/UMTS/3G digital cell phone communication, which allows the transfer of VRS corrections to large customer groups. VRS/RTK positioning without a local reference station is a given for dual-frequency users

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Fig. 5 RTK RMS Position errors for different baseline lengths Fig. 8 Typical RTK initialization performance in a VRS network on different baseline lengths. Statistics are based on at least 24 h of data with several hundreds of re-initializations

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Fig. 6 VRS/RTK RMS Position errors for different baseline lengths

Fig. 7 Typical RTK initialization performance on different baseline lengths. Statistics are based on at least 24 h of data with several hundreds of re-initializations

today. Such systems are used for various applications in surveying, machine control, precision agriculture, and precision GIS. While RTK provides the highest accuracies and shortest convergence times, it requires a dense infrastructure of reference stations and dual-frequency receivers to achieve high reliability.

In contrast to RTK, which is mainly based on carrier phase measurements, differential GPS traditionally has been based on differential pseudorange (code) measurements, where the code measurements usually were smoothed with carrier. However, the results we are going to show in the following are based on a differential float carrier solution, using some assistance of code measurements, but having carrier phase as the main observable in the solution. The major difference to the RTK approach is that this DGPS technique does not fix the carrier phase integer ambiguities Eq. (2). The technique was developed for GIS/mapping applications and is today used in the “Trimble Pathfinder Office” software. As with RTK, DGPS suffers from the necessity of using a reference station data stream. However it can be applied on baseline distances of several thousand kilometres. The stringent requirement for DGPS is that identical satellites are tracked on the reference and rover receivers. This requirement therefore limits the baseline length in DGPS. To demonstrate the performance of the DGPS technique we have used datasets from two stations in Germany. One station at the Terrasat office in H¨ohenkirchen was used as the rover station and using a reference station which is 70 km away. Both stations are equipped with Trimble NetR5 reference station receivers, but we have used only the GPS observations for our analysis in this paper, and not the GLONASS data. Multiple days of data have been analyzed to ensure that the results are representative, but the results were

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finally derived from 24 h of data. The data was analyzed for static positioning performance to answer the question “How much static occupation time is required to reach a predefined accuracy?”. Multiple data periods of three hours have been analyzed in steps of one hour resulting in 21 sessions. Results were compared with the true ITRF position of the Hoehenkirchen station and RMS values were derived from the various runs for the horizontal and vertical component (Figs. 9 and 10).

Fig. 9 Typical horizontal DGPS and PPP RMS accuracy from 24 h of data for station H¨ohenkirchen, Germany

Fig. 10 Zoomed in view of Fig. 9 (showing only the first 20 m)

H. Landau et al.

DGPS, as applied here, can provide horizontal position accuracies of better than 30 cm after two minutes convergence time. The maximum horizontal accuracy is achieved (in our example) after approximately one hour (5 cm RMS). The vertical component requires a longer convergence time but finally reaches also 5 cm accuracy. It is interesting to note that the L1-only DGPS solution is really not very much worse than the dual-frequency DGPS solution, in vertical it even shows better performance due to the lower noise of the basic observable.

Trimble’s RTK and DGPS Solutions

3 Precise Point Positioning In contrast to DGPS and RTK Precise Point Positioning (PPP) does not require a local reference station. Besides the local measurement data the primary input for PPP are precise orbits and clocks, which are usually derived from a global tracking network. A number of authors have developed PPP algorithms and published promising results over the past years (Gao and Shen, 2001, 2002, 2006; Leandro and Santos 2006; Rho and Langley, 2005). For comparison of PPP with DGPS performance in this paper we used IGS final orbits (900 s intervals, RMS <3 cm per component) and clocks (30 s interval, RMS <0.1 ns). Looking again at Eq. (1) we can say that the uncertainty in the geometric range is corrected by the availability of precise orbital information, the satellite clock τ s is also provided by the IGS. Carrier phase bias terms are absorbed by the ambiguity for the satellite-dependent bias and by the receiver clock error unknown for the receiverdependent bias. Since PPP is not based on differential techniques the method requires precise modelling of all effects included in α (Eq. 1), which do not necessarily need to be modelled for RTK and most of the DGPS applications. These are in detail:

r r

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r r r r

Ocean tide loading displacements (typically to 50 mm in height), Pole tide displacements (up to 25 mm in height), Satellite and receiver antenna phase centre offset and variation, Tropospheric delays.

Some of the above effects require complex models to be implemented in GNSS receivers for PPP, which could potentially lead to substantial CPU load. The implementation for field positioning systems will require careful decisions on model simplifications. It is important to note that, while most of the publications (including ours) on PPP are using dual-frequency data to eliminate the ionospheric effects, some authors are discussing the use of singlefrequency code and carrier data to do PPP (Shen and Gao, 2002; Muellerschoen et al., 2004).

3.1 Static PPP

In Figs. 9, 10 and 11 we show typical results of a static PPP solution for station Hoehenkirchen and compare them with DGPS. The identical data for Hoehenkirchen was applied as for the DGPS analysis above. The PPP processing was done by a software implementation developed at Trimble Terrasat. The Relativistic effect (up to approx. 8 m), Solid earth tide displacements (up to 300 mm in process uses a sequential filter estimating receiver clock errors, positions, carrier ambiguities and a zenith height),

Fig. 11 Typical vertical DGPS and PPP RMS accuracy from 24 h of data for station H¨ohenkirchen, Germany

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tropospheric total delay. Ionospheric-free code and carrier measurements are used. Our analysis shows that the basic difference between our DGPS approach and PPP is not in the ultimate achievable accuracy, it is in the convergence of the solution. Typically DGPS achieves a horizontal accuracy of better than 30 cm after 2 min convergence time. PPP required 12 min in the example above to achieve this. It is interesting to see that the accuracy achieved in the vertical after 3 h with PPP is even better than with the DGPS solution. In average static PPP provides 50 mm horizontal accuracy after one hour of convergence (Table 1), after 12 h the maximum accuracy of 20 mm (that we were able to achieve) was reached.

3.2 Kinematic PPP Above we presented some typical results for static convergence and accuracy for PPP. Next, we investigated the kinematic performance using the same data as above. The overall positioning errors are shown in Fig. 12. If we ignore the first 2 h convergence time, the

H. Landau et al.

RMS values for North, East and Up are 48, 26, and 87 mm. The ultimate accuracy we were able to achieve for a 30 minute segment is shown in Fig. 13. The RMS values for North, East and Up are 20, 10, and 41 mm. Looking at the RMS values and by comparing with Figs. 3 and 4 we can clearly see that we are still far away from the accuracy we are able to achieve with VRS/RTK today. An average 40 mm horizontal accuracy was achieved by using kinematic positioning when ignoring the first 2 h convergence time.

3.3 “Stop and Go” PPP Another analysis we were interested in was the application of PPP for stop-and-go surveys with the aim to survey points. With this analysis we wanted to answer the question “what kind of accuracy we can achieve by occupying specific points for a specified time once the solution has converged?”. We therefore used the first two hours for the convergence and then started to have two hour segments of alternating static occupation

Fig. 12 Kinematic position errors in North, East and Up with respect to the true ITRF position for a 21 h period

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Fig. 13 Zoomed in view on the best 30 minute segment of kinematic positioning

Table 1 Typical horizontal accuracies achievable by PPP Method

Time

Typical Accuracy (mm)

Static

10 min 30 min 1h 2h 12 h 1 day after 2 h 5 min 30 min 2h

500 100 50 30 20 20 40 50 40 30

Kinematic Stop-and-go

One way to speed up the convergence time is to resolve the carrier phase ambiguities. However, in order to do this it is necessary to derive the satellite and receiver hardware dependent biases with very high accuracies. Some authors have started to work on this challenge and are publishing first results (Shen and Gao, 2002; Ge et al., 2007; Wang and Gao, 2007).

4 Summary and Outlook

Today commercial state-of-the-art RTK implementations are providing highly reliable and precise positioning. Especially in combination with established VRS networks and digital cell phone technology, RTK users are achieving routinely RMS accuracies of 10 mm in the horizontal and 20 mm in the vertical with average initialization times of 20 s or less. While VRS networks are common in developed countries, the coverage area is increasing all the time. We can expect that within 3.4 Reducing Convergence Time a few years all relevant parts (from a market perspective) of the world will be covered with such networks. The key to making PPP competitive with respect to The “drawback” of RTK/VRS is that it requires dualRTK and DGPS is to improve the convergence time. frequency observations. and kinematic movement periods. From the static occupations we derived typical average occupation times, which are summarized in Table 1. The results we obtained for stop-and-go PPP are promising. Once the solution was converged we could actually achieve 50 mm horizontal positioning accuracy after 5 min static occupation.

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DGPS solutions are providing similar levels of accuracies as PPP but possibly with shorter convergence times. Such solutions can be generated from the VRS networks easily, and will target special markets such as GIS mapping. DGPS will be mainly of interest for the single-frequency market. PPP provides accuracies in the cm-range but still requires long convergence times. In order to compete with RTK the accuracy and especially the convergence time will have to improve significantly. From a market perspective PPP as it is at the moment will be mainly suited for areas without established reference station networks. When talking about PPP we have to emphasize once more that we have used GPS data only. Additional satellites (GLONASS, Galileo. . . ) and additional frequencies (like GPS L5) will also help significantly to reduce the convergence time (Shen and Gao, 2006). However, these will also be of benefit to RTK and DGPS. Acknowledgments We thank our colleague Jens B¨ohme for gathering the data used in the DGPS and PPP analysis.

References Chen, X., H. Landau, U. Vollath (2003). New tools for Network RTK integrity monitoring, In: Proceedings of IONGPS/GNSS 2003, Portland, Oregon, Sept. 9–12, 2003, pp. 1355–1361. Chen, X., U. Vollath, H. Landau (2004). Will GALILEO/ modernized GPS obsolete Network RTK, In: Proceedings of ENC-GNSS 2004, Rotterdam, May 2004, Netherlands. Chen, X., A. Deking, H. Landau, R. Stolz, U. Vollath (2005). Correction formats on Network RTK performance, In: Proceedings of ION-GNSS 2005, Long Beach, California, USA, Sept. 13–16, 2005, pp. 2523–2530. Chen, X., A. Kipka, J. K¨ohler, H. Landau, U. Vollath (2007). GNSS modernization and its consequences for reference station network solutions, In: Proceedings of ENC-GNSS 2007, May 29–June 1, 2007, Geneva, Switzerland. Gao, Y., X. Shen (2001). Improving ambiguity convergence in carrier phase-based Precise Point Positioning, In:

H. Landau et al. Proceedings of ION GPS-2001, Sept. 11–14, 2001, Salt Lake City, Utah, pp. 1532–1539. Gao, Y., X. Shen (2002). A new method for carrier phase based Precise Point Positioning. navigation, In: Journal of the Institute of Navigation, Volume 49 Number 2. Ge. M., G. Gendt, M. Shi, C. Liu (2007). Resolution of GPS carrier-phase ambiguity in Precise Point Positioning, Paper presented at EGU Assembly 2007, Vienna, April 16–20, 2007. Kolb, P.F., X. Chen, U. Vollath (2005). A new method to model the ionosphere across local area networks, In: Proceedings of ION-GNSS 2005, pp. 705–711. Landau, H., X. Chen, A. Kipka, U. Vollath (2007). Latest developments in Network RTK modeling to support GNSS modernization, In: Proceedings of the ION National Technical Meeting, San Diego, California, January 22–24, 2007, pp. 851–859. Leandro, R.F., M.C. Santos (2006). Wide area based Precise Point Positioning, In: Proceedings of ION GNSS 2006, Fort Worth, Texas, Sept. 26–29, 2006, pp. 2272–2278. Muellerschoen, R.J., B. Iijima, R. Mayer, Y. Bar-Sever, E. Accad (2004). Real-time Point-Positioning Performance, evaluation of single-frequency receivers using NASA’s Global Differential GPS System, In: Proceedings of ION GNSS 2004, Long Beach, California, Sept. 21–24, 2004, pp. 1872–1880. Remondi, B.W. (1984). Using the Global Positioning System (GPS) phase observable for relative geodesy: Modeling, processing, and results, PhD thesis, the University of Texas at Austin, May 1984. Rho, H., R.B. Langley (2005). Dual-frequency GPS Precise Point Positioning with WADGPS corrections, In: Proceedings of ION GNSS 2005, Long Beach, California, Sept. 13–16, 2005, pp. 1470–1482. Shen, X., Y. Gao (2002). Ambiguity pseudo-fixing in Precise Single Point Positioning, In: Proceedings of the ION 58th Annual Meeting, Albuquerque, NM, USA June 24–26, 2002. Shen, X., Y. Gao (2006). Analyzing the impacts of Galileo and modernized GPS on Precise Point Positioning, In: Proceedings of the ION National Technical Meeting, San Diego, California, January 2006. Wang, M., Y. Gao (2007). An investigation on GPS receiver initial phase bias and its determination, In: Proceedings of the ION National Technical Meeting, San Diego, California, January 22–24, 2007, pp. 873–880. Wanninger, L. (1997). Real-time differential GPS-error modeling in regional reference station networks, in: Brunner, F.K. (Ed.): Advances in Positioning and Reference Frames, In: Proceedings of the IAG Scientific Assembly, Rio de Janeiro, 1997, pp. 86–92.

The Precision and Accuracy of Shanghai VRS Network Lizhi Lou and Yi Chen

Abstract To overcome the limitations of standard real-time kinematic (RTK) GPS systems, a multiplereference-station network can be used to achieve larger service coverage, increase robustness, and higher positioning accuracy compared with using a single reference station. The virtual reference station (VRS) approach is an efficient method of transmitting corrections through a data link to network users for RTK positioning. The Shanghai Virtual Reference Station (VRS) Network serves as infrastructure for the “Digital Shanghai” information system, and will provide a high precision and fast positioning service. Concerning different applications, a systematic test has been performed at different locations in Shanghai to assess the achievable accuracy and reliability for VRS-RTK positioning. The results indicate that Shanghai VRS Network positioning achieved satisfactory accuracies in both the horizontal and vertical position components, with the corresponding values being within 1.5 and 3 cm respectively. The average reliability was within 3 mm.

Keywords GPS network · Virtual Reference Station (VRS) · Precision · RTK

Lizhi Lou Department of Surveying and Geo-Informatics, Tongji University, Siping Road 1239, 200092 Shanghai, China Yi Chen Department of Surveying and Geo-Informatics, Tongji University, Siping Road 1239, 200092 Shanghai, China

1 Introduction Real Time Kinematic (RTK) positioning using Global Navigation Satellite Systems (GNSS) and in particular the Global Positioning System (GPS) is typically implemented in a single-reference-station mode. However, a single reference station framework lacks data redundancy and stations must be located within a distance of 10 ∼ 15 km to the user because GPS errors become less spatially correlated over longer baselines, causing a degradation in the resulting positioning accuracy (Chen et al., 2001; Hu et al., 2003). Previous work has shown that a network approach offers several advantages over the standard single-reference- station approach. One significant advantage is the improvement in reliability and availability of the service (Retscher, 2002). Furthermore, the distance dependent or spatially correlated errors can be modelled, such as ionospheric, tropospheric and satellite orbit effects (Wanninger, 1999; Lachapelle et al., 2000). A direct result of modelling the spatially correlated errors is an improvement in the efficiency of the resolution of carrier phase ambiguities, which is particularly important for geodesists and engineers who would like to perform high-precision RTK surveys. However, little research has been conducted on the distribution of the network corrections to users in real time. An efficient method of transmitting corrections to network users for RTK-GPS positioning is the virtual reference station (VRS) concept (Landau et al., 2002; Hu et al., 2003). Recently, the VRS concept has been proposed by many research groups as a more feasible approach for relaying network correction information to the network-RTK users (Wanninger, 1999; Vollath et al., 2000; Landau et al., 2001; H¨akli, 2004). The

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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VRS-RTK technique does not need an actual physical reference station (with GPS receiver and data link). Instead, it allows users to access data from a non-existent or “virtual” reference station at any location within the network coverage area (Vollath et al., 2000; Landau et al., 2001, 2002; H¨akli, 2004). In addition, the VRS approach is more flexible in terms of permitting users to operate their current receivers and software, without the need for special software, to manage corrections from a series of reference stations simultaneously. The Shanghai Virtual Reference Station Network (http://www.shsmi.cn/rtk/) is operated by the Shanghai Surveying and Mapping Institute and will serve as infrastructure for the “Digital Shanghai” information system. “Digital Shanghai” information system uses digital information processing and network communication technologies and in particular Geographic Information System (GIS) as the core technology, Global Positioning System (GPS), Remote Sensing (RS) and other key technologies. It consists of the information infrastructure and utilization of a system, which incorporates all kinds of digital information and integration of information resources, services in urban planning, construction and operation management and realizes a virtual representation of the daily life Shanghai social reality. The purpose of the “Digital Shanghai” information system is to play key roles in economic and social sustainable development, environmental protection, disaster mitigation, natural resources conservation and improvement of the citizens’ living standard. Integrating with the city’s intelligent transportation system, and network and communication system, the VRS network will provide a high-precision and fast positioning service for all users. The Shanghai VRS Network currently includes nine continuously operating reference stations (CORS) that operate with Trimble GPSNetTM software (Version 2.4). There are seven R Trimble NetRS and two Ashtech Micro Z-CGRS GPS reference stations, all the receivers with choke ring antennas mounted on roof tops. The height differences between CORS nodes can amount up to 55 m. The average spacing of CORS nodes is about 35 km and all the raw CORS data are transmitted via digital data network (DDN) or frame relay (FR) to the central processing facility. In addition there are many different kinds of GPS user/rover receivers (Trimble, Leica, Topcon, Thales, etc.) in this system. To address different applications, the accuracy and reliability of the Shanghai VRS Network needs to be evaluated.

L. Lou and Y. Chen

2 Test and Methods The measurements were carried out in Shanghai at the beginning of June 2006, when the level of ionospheric activity was normal. There were twenty-two test points distributed across the city. At every test point, five independent VRS fixed solutions were collected in order to investigate the repeatability of VRS-RTK, which means every time before collection of VRS result a complete re-initialization from receiver cold start was performed. In order to improve the reliability of the test, eleven locations were selected (see Fig. 1, the black points with the name are the test locations and the grey areas mean the land of Shanghai city), well distributed with respect to the coverage of the VRS network (see Fig. 1, the network related to the star locations identifies the CORS nodes of Shanghai VRS network, see http://www.shsmi.cn/rtk/). There were two points at every location (01 and 02 represent the points

WS

ML

LC

TG TJ JQ

CY ZY WZ KQ CQ

Fig. 1 The test locations and VRS nodes distributed across Shanghai: The grey areas denote the land of Shanghai city, inclusive the islands in the north of figure; the stars are the VRS nodes, the corresponding network is the coverage of the VRS network; the black points with point name are the test locations

The Precision and Accuracy of Shanghai VRS Network

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3 Results and Discussions At every test point five independent VRS-RTK results were obtained. Every time from receiver cold start to VRS fixed solution, the time of initialization was less than 40 s. Furthermore, the mean results of two points at the same location were checked by total station survey. Leica TCA2003 was used and its distance measuring accuracy of 1 mm + 1 ppm was achieved. From Table 1, the standard deviations of the repeatabilities of VRS can be deduced: 2.2, 2.4 and 3.8 mm in the north, east and height components respectively. Simultaneously, the differences between the VRS and total station results were less than 2 mm. Comparing the VRS results with those obtained from the GPS network, some useful results are obtained: geodetic coordinates have an average bias of 0.582 m in the longitude direction with 9 mm root mean square (rms), 0.363 m in the latitude direction with 12 mm

Table 2 Statistics of the differences (meters) between the VRS results and the results obtained from the GPS network (geodetic coordinate system) Longitude Latitude Height

STD

Mean

Max

Min

0.009 0.012 0.030

0.582 0.363 −0.534

0.603 0.402 −0.494

0.561 0.339 −0.600

rms, and −0.534 m in height with 30 mm rms. The biases are caused by inaccurate transformation parameters due to the fact that Shanghai VRS network and the traditional geodetic network are two independent networks; there is a small difference between their data. The local coordinates can also be calculated from parameters of the Shanghai projection. Compared with the results obtained from the GPS network, an average bias of 60 cm was found. Then, three common points (three permanent stations with known local coordinates in the Shanghai datum) were used to calculate the transformation parameters, and the test points were measured another time. In the following part, both fixed WGS-84 coordinates and local grid coordinates were applied to calculate the transformation parameters. 1. When the WGS-84 coordinates were fixed, the transformation parameters were calculated from all points and the differences in position from those obtained from the GPS network are shown in Figs. 2, 3 and 4:

0.05 Differences in X-direction

at the same location in the discussion part) and they were visible from each other for checking by total station survey. In the centre, the base station of Tongji University (TJBS), with known WGS84 and local coordinates, is situated. In order to compare with the results of VRS, the GPS network using the same points was constructed. To avoid any errors due to apparent deformations of the GPS network, only TJBS was fixed as the known point and TJBS-SHAO (IGS station in Shanghai) as the known direction, and all other points were regarded as unknown points. All baselines were observed for at least two hours in static mode. Within this test Ashtech ZXtreme dual-frequency receivers were used at all stations, its logging interval was 15 s and the mask elevation was 10 degree. GPS processing software TGPPS (Version 6.0) developed by Tongji university was adopted and used to obtain the adjusted results – 3D geodetic coordinates, and the local coordinates using the Shanghai projection parameters.

0.04 0.03 0.02 0.01 0.00 –0.01 –0.02

Table 1 Statistics of the differences (meters) between the VRS results and the corresponding mean value North East Height

STD

Max

Min

0.0022 0.0024 0.0038

0.0067 0.0065 0.0096

−0.0065 −0.0068 −0.0095

JQ01 JQ02 ZY01 ZY02 LC01 LC02 WS01 WS02 ML01 ML02 CY01 CY02 WZ01 WZ02 TG01 TG02 KQ01 KQ02 CQ01 CQ02

–0.03

Test Points Fig. 2 Differences (meters) in X-direction between the VRS results and the results obtained from the GPS network (WGS-84 coordinate system)

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L. Lou and Y. Chen

Differences in north direction

0.02

0.02 0.00 –0.02 –0.04 –0.06

0.01

0.00

–0.01

JQ01 JQ02 ZY01 ZY02 LC01 LC02 WS01 WS02 ML01 ML02 CY01 CY02 WZ01 WZ02 TG01 TG02 KQ01 KQ02 CQ01 CQ02

–0.02

Test Points

Test Points

Fig. 5 Differences (meters) in north direction between the VRS results and the results obtained from the GPS network (local coordinate system) 0.04

Test Points Fig. 4 Differences (meters) in Z-direction between the VRS results and the results obtained from the GPS network (WGS-84 coordinate system)

The results indicate that the differences are quite small with a standard deviation of 17.1, 27.2 and 20.9 mm in the X-direction, Y-direction and Zdirection, respectively. 2. When local grid coordinates were employed to calibrate the transformation parameters, the results shown in Figs. 5, 6 and 7 were obtained.

Differences in east direction

0.04 0.03 0.02 0.01 0.00 –0.01 –0.02 –0.03 –0.04 –0.05

JQ01 JQ02 ZY01 ZY02 LC01 LC02 WS01 WS02 ML01 ML02 CY01 CY02 WZ01 WZ02 TG01 TG02 KQ01 KQ02 CQ01 CQ02

Differences in Z-direction

Fig. 3 Differences (meters) in Y-direction between the VRS results and the results obtained from the GPS network (WGS-84 coordinate system)

0.03 0.02 0.01 0.00 –0.01 –0.02

JQ01 JQ02 ZY01 ZY02 LC01 LC02 WS01 WS02 ML01 ML02 CY01 CY02 WZ01 WZ02 TG01 TG02 KQ01 KQ02 CQ01 CQ02

–0.08

JQ01 JQ02 ZY01 ZY02 LC01 LC02 WS01 WS02 ML01 ML02 CY01 CY02 WZ01 WZ02 TG01 TG02 KQ01 KQ02 CQ01 CQ02

Differences in Y-direction

0.04

Test Points

Fig. 6 Differences (meters) in east direction between the VRS results and the results obtained from the GPS network (local coordinate system)

4 Conclusion

From the experiment, repeatabilities of VRS-RTK of 2.2, 2.4 and 3.8 mm in the north, east and height components respectively have been obtained, with the differences between VRS and total station results being less than 2 mm. Furthermore, comparing the VRS reAfter the calibration procedure, the new differences sults with those obtained from the GPS network, the between the VRS results and the reference values from geodetic coordinates have an average bias of 0.582 m the GPs network were calculated. The standard devi- in latitude, 0.363 m in longitude and −0.534 m in elations of the VRS local coordinates were 11.2, 13.2 lipsoidal height, and the local coordinates have an avand 26.2 mm in the north, east and height components erage bias of 60 cm. When three common points were respectively. used to calculate the transformation parameters, the

The Precision and Accuracy of Shanghai VRS Network 0.05

Acknowledgments This work was financially supported by Key Laboratory of Advanced Engineering Surveying of SBSM, No. ES-SBSM-(06)-08 and the Natural Science and Technology Development Foundation of Tongji University. We appreciate the valuable comments of the reviewers, which contributed to the improvement of this article.

0.04 Differences in height direction

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0.03 0.02 0.01 0.00 –0.01 –0.02

References

–0.03 –0.04 –0.06

JQ01 JQ02 ZY01 ZY02 LC01 LC02 WS01 WS02 ML01 ML02 CY01 CY02 WZ01 WZ02 TG01 TG02 KQ01 KQ02 CQ01 CQ02

–0.05

Test Points

Fig. 7 Differences (meters) in height direction between the VRS results and the results obtained from the GPS network (local coordinate system) Table 3 Standard deviations of the differences (millimeters) between the VRS results (after point calibration) and the results obtained from the GPS network (in WGS-84 coordinate system and local grid coordinate system) WGS-84 coordinate STD

Local grid coordinate

X

Y

Z

North

East

Height

17.1

27.2

20.9

11.2

13.2

26.2

accuracies of VRS are 17.1, 27.2 and 20.9 mm in the X-direction, Y-direction and Z-direction in the WGS84 coordinate system, corresponding to 11.2, 13.2 and 26.2 mm in the north, east and height components in the local grid coordinate system, respectively. Thus, the Shanghai VRS Network has a good internal accuracy but a 60 cm bias with respect to the GPS network. Hence when the local coordinate system must be used for surveying purposes, a calibration of the transformation parameters is required. From Figs. 5, 6 and 7 it becomes obvious that there is no area in Shanghai, which has significant big or small accuracy. Thus, using the VRS station concept, similar accuracies can be achieved across the whole Shanghai area. The results also exhibited 1.2 cm precision for the horizontal components and 2.6 cm precision for the height component.

Chen H.Y., C. Rizos, S.W. Han (2001). From simulation to implementation: Low-cost densification of permanent GPS networks in support of geodetic applications, Journal of Geodesy, Vol. 75, 515–526. H¨akli P. (2004). Practical test on accuracy and usability of virtual reference station method in Finland, FIG Working Week, Athens, Greece, 22–27 May, 1–16. Hu G.R., H.S. Khoo, P.C. Goh, C.L. Law (2003). Development and assessment of GPS virtual reference stations for RTK positioning, Journal of Geodesy, Vol. 77, 292–302. Lachapelle G., P. Alves, L.P. Fortes, M.E. Cannon, B. Townsend (2000). DGPS RTK positioning using a reference network, Proceedings of the 13th International Technical Meeting of the Satellite Division of the Institute of Navigation, Salt Lake City, UT, 19–22 September, 1165–1171. Landau H., U. Vollath, A. Deking, C. Pagels (2001). Virtual reference station networks – recent innovations by Trimble, GPS Symposium, Tokyo, Japan, November, 39–52. Landau H., U. Vollath, X.M. Chen (2002). Virtual reference station systems, Journal of Global Positioning Systems, Vol. 1, No. 2, 137–143. Retscher G. (2002). Accuracy performance of virtual reference station (VRS) networks, Journal of Global Positioning Systems, Vol. 1, No. 1, 40–47. Vollath U., A. Buecherl, H. Landau, C. Pagels, B. Wagner (2000). Multi-base RTK positioning using virtual reference stations, Proceedings of the 13th International Technical Meeting of the Satellite Division of the Institute of Navigation, Salt Lake City, UT, 19–22 September, 123–131. Vollath U., H. Landau, X.M. Chen (2002). Network RTKconcept and performance, Proceedings of the GNSS Symposium, Wuhan, China, November 2002. Wanninger L. (1999). The performance of virtual reference stations in active geodetic GPS-networks under solar maximum conditions, Proceedings of the 12th International Technical Meeting of the Satellite Division of the Institute of Navigation, Nashville, TN, 14–17 September, 1419–1427.

Improving the Stochastic Model of GNSS Observations by Means of SNR-based Weighting X. Luo, M. Mayer and B. Heck

Abstract In many GNSS software packages a simplified observation weighting model is used which is merely based on the satellite elevation angle and valid under the assumption of azimuthal symmetry. This elevation-dependent weighting model is only suitable for undisturbed GNSS signals based on the existing strong correlation between signal quality and satellite elevation angle. However, for high-precision geodetic applications this geometry-related weighting model becomes obsolete if observations are strongly affected by multipath effects, signal diffraction as well as receiver characteristic under non-ideal observation conditions. An improved observation weighting model based on signal-to noise power ratio measurements has been developed and experimentally implemented in the Bernese GPS software 5.0. Tests indicate that when this weighting model is used for low elevation data additional 10% ambiguities can be resolved and the accuracy of the estimated site-specific neutrosphere parameters can be improved by nearly 25% compared with the standard elevation-dependent weighting model.

1 Introduction

Global Navigation Satellite Systems (GNSS) have become an efficient and reliable tool for a wide range of applications in geodesy and surveying engineering, e.g. deformation monitoring. Some of these activities require highly accurate positions. Since its introduction to civilian users satellite positioning techniques have experienced continuous improvements, not only in hardware technology but also in data processing. The GNSS carrier-phase measurements are generally processed by means of least-squares methods. However, these algorithms deliver reliable estimates of unknown parameters and associated accuracy measures only if both the functional and the stochastic model applied in GNSS data processing are realistic. The functional model describes the mathematical relationship between the GNSS phase measurements and the unknown parameters, while the stochastic model formulates the statistical properties of the measurements expressed by an appropriately defined variance-covariance matrix. In the past decades, the functional model of GNSS Keywords GNSS · Stochastic modelling · carrier-phase observations has been continuously improved; see e.g. Hofmann-Wellenhof et al. (2001) for Observation weighting · Signal-to-noise ratio (SNR) an overview. The deficiencies in the stochastic model are due to unrealistic observation weighting, neglecting physical (temporal and spatial) correlations among the X. Luo raw observations, and disregarding mathematical corGeodetic Institute, University of Karlsruhe (TH), Englerstr. 7, relation arising from the differencing process, when it D-76128 Karlsruhe, Germany, e-mail: [email protected] is applied. M. Mayer Concerning satellite configuration, site-specific Geodetic Institute, University of Karlsruhe (TH), Englerstr. 7, factors as well as atmospheric effects, the quality D-76128 Karlsruhe, Germany of GNSS observations may become inconsistent. In B. Heck Geodetic Institute, University of Karlsruhe (TH), Englerstr. 7, addition, observation weighting plays a dominant role D-76128 Karlsruhe, Germany

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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in comparison to other factors impacting GNSS data processing, e.g. baseline length, multipath effects, neutrospheric modelling and ambiguity resolution strategies (Luo et al., 2007a). Numerous observation weighting models using different approaches have been developed. Euler and Goad (1991) suggested the first elevation-dependent weighting scheme which is based on an exponential function and empirically determined parameters for original GNSS observations. Jin and de Jong (1996) validated this model by applying it to both code and carrier-phase observations. Han (1997) investigated the effects of such an elevation-dependent weighting model on the parameter estimation and found significant improvements in ambiguity resolution. Using signal-to-noise power ratio values (SNR), Hartinger and Brunner (1999) developed the SIGMA-ε model to handle the opposite impact of atmospheric phase delay effects and satellite geometry. Moreover, based on the SIGMA-ε model Brunner et al. (1999), Wieser and Brunner (2000) and Wieser (2001) introduced the SIGMA- model which is able to detect and mitigate the error due to signal diffraction automatically. Both elevation-dependent and SNR-based weighting models result in significant improvements with respect to parameter estimation (Barnes et al., 1998; Wang et al., 1998). Compared to observation weighting, an appropriate consideration of physical correlations is still a difficult task, because these correlations originate from different sources which can hardly be controlled or determined accurately (Kim and Langley, 2001). Nevertheless, attempts have been made to model the temporal correlation by applying autocorrelation functions (Howind, 2005) or low order autoregressive models (Wang et al., 2002). Any misspecification within the stochastic model will inevitably produce unreliable positioning results. Therefore, the precision of GNSS positions delivered by processing software is considered to be overoptimistic (Ananga et al., 1994; Han and Rizos, 1995; Tiberius and Kenselaar, 2000). In this paper, an alternative, SNR-based weighting model for long-term static observations is presented. In Sect. 2, the standard elevation-dependent weighting model as applied in the Bernese GPS Software 5.0 (BS5) (Dach et al., 2007) is validated. In Sect. 3, the realisation of the SNR-based weighting model and its implementation in the BS5 are briefly described. Finally, in Sect. 4, the improved weighting model is exemplarily illustrated by processing GNSS data from

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the Antarctic Peninsula and Germany considering ambiguity resolution, double difference residuals, sitespecific neutrosphere parameters and repeatability of coordinates.

2 The Elevation-Dependent Weighting GNSS receivers calculate positions by measuring pseudo-distances to transmitting satellites. How well a GNSS receiver performs mainly depends on the signal power in the receiver’s tracking loops (Langley, 1997). According to Butsch and Kipka (2004) the signal power delivered by GNSS receivers is affected, among others, by:

r r r r

antenna gain of satellite antenna antenna gain of receiver antenna distance between satellite and receiver signal damping due to atmospheric absorption.

These factors vary with the elevation angle of the transmitted GNSS signals and therefore, for undisturbed signals the signal power normally grows with increasing satellite elevation angle. Based on the strong correlation with signal quality, the elevation angle e is usually used as indicator for the quality of GNSS observations, and a simple observation weighting model (e.g. p = sin2 e) is implemented in GNSS processing software. However, this elevationdependent weighting model which is indirectly related to the signal quality is not appropriate for those observations which are strongly affected by multipath, signal diffraction as well as receiver characteristic. Under these circumstances the correlation between signal quality and satellite elevation angle is attenuated. Figure 1 shows the correlation situation between the signal quality given in manufacturer-dependent units AMU (Arbitrary Manufacturer Units) and the satellite elevation angle on the site Punta Spring (SPR1) situated on the Antarctic Peninsula. As the graphs show, the elevation-dependent weighting model is suitable for the observations of satellite PRN 02, but inappropriate for satellite PRN 30 due to the absence of the required strong correlation between signal quality and satellite elevation angle. Additionally, depending on antenna design and receiver processing technique, the registered signal quality measures on L1 and L2 may differ significantly.

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lack of observations due to local effects like obstruction at high geographic latitudes), it is necessary to include low elevation data in order to stabilise the parameter estimation and to improve the accuracy of the height component (e.g. Mayer, 2006). It is commonly known that low elevation data are much more disturbed by atmospheric effects, as well as being more subject to multipath effects. An appropriate handling of low elevation data requires a realistic observation weighting model which is directly related to signal quality measurements.

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3 The SNR-Based Weighting Model This section contains a discussion of the conversion of different SNR-related measures followed by the description of the new SNR-based weighting model.

3.1 SNR Conversion In addition to pseudo-range and carrier-phase observations, a geodetic GNSS receiver also records signal-tonoise power ratio (SNR) data. The signal power can be estimated based on the power of the replica signal in the correlation process between received and replica signal (Butsch and Kipka, 2004). The main part of the noise power stems from both the receiver electronics itself and the heat radiation of the objects in the surrounding area. Since many signals have a very wide dynamic range, SNR-values are usually expressed in terms of the logarithmic decibel scale and normalised to signal-to-noise power density ratio (SNR0) in 1 Hz bandwidth. In practice, sometimes the so-called Arbitrary Manufacturer Units (AMU) are used to quantify the signal quality (e.g. Trimble 4000SSI receivers). Using manufacturer-specific formulas, AMU can be converted into SNR0 in [dBHz]. Taking Trimble 4000SSI receiver as example, Eq. (1) is used to convert AMU to SNR0 for both frequencies (Trimble, 1999): SNR01,2 = 30 + 10 · log10 (AMU1,2 2 ) − 3.

(1)



3.2 Realisation of the SNR-Based Weighting

f (SNR0 iL1,2 )

SNR-based weighting of GNSS observations was considered first by Talbot (1988). Langley (1997) presented a functional model which expresses the variance of the original phase observations σPLL 2 in [mm2 ] as a function of the measured SNR0 values in [dBHz] 

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residuals. It should be noted that a weighting scheme based on Eq. (2) ignores any contribution to the phase noise from the local oscillator and is only strictly valid for relatively strong signals well above the tracking threshold of the receiver (Collins and Langley, 1999). In contrast to the SIGMA-ε model, the SNR-based weighting model presented in this paper is independent from Eq. (2) and based on the measured SNR-values directly. In comparison to the standard elevation-dependent weighting in BS5, the SNR-based weighting model takes advantage of using original signal quality information obtained by the same measurement process and time period as the GNSS phase observations. The SNR-based weighting model is realised in two steps, schematically shown in Fig. 2. In the first step, AMU/SNR0-values are extracted from RINEX files which are previously obtained from the binary raw observation data by means of the program TEQC (Estey and Meertens, 1999). The extracted AMU-values are converted into SNR0 using modeland manufacturer-dependent formulas. In the second step, for all antenna-receiver (ANT-REC) combinations within the network the minimum and the maximum of the obtained SNR0 values are searched on both frequencies and over the entire project. This procedure guarantees that the found extreme values cover the complete GNSS project (network, observation period). After that, for each L1 and L2 observation an individual weight value (WGT) is calculated by forming the minimum-related ratio between the actual and the corresponding maximum SNR0. The empirically derived formula

(2)

where the subscript i corresponds to L1 or L2. This formula was applied by Hartinger and Brunner (1999) in the SIGMA-ε model and the parameter Ci was determined based on the variance of double difference

=

SNR0 iL1,2 − SNR0 min L1,2 min SNR0 max L1,2 − SNR0 L1,2

2 (3)

can be used to calculate L1 and L2 weight values for a certain antenna-receiver combination. After forming linear combinations, these frequency-related weight values are used to obtain the variance-covariance matrix (COV) of double difference observations by means of variance propagation law. Within the framework of this case study this SNR-based weighting model has been implemented in the BS5. The data preprocessing, for example, search for the minimum and maximum SNR0 values, is performed in MATLAB. In Fig. 3, for the site Arturo Prat (PRA1) on R the Antarctic Peninsula and the SAPOS (Satellite

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Fig. 2 Realisation of the SNR-based weighting model and its contribution to advanced stochastic modelling in BS5

Fig. 3 Comparison of the weight values of L1 observations related to weighting model and GNSS equipment; upper: site PRA1 (Trimble 4000SSI, 4000ST. L1/L2 GEO), DOY1998: 025, lower: site HLBR (Leica SR 520, LEIAT503), DOY2004: 186

Positioning Service of the German State Survey) site Heilbronn (HLBR) in Baden-W¨urttemberg, the calculated weight values of L1 observations are visualised related to weighting model and GNSS equipment. The Trimble receiver on PRAI records AMUvalues with a resolution of 0.25, while the Leica receiver on HLBR delivers SNR in integer values R with a lower resolution due to SAPOS internal data processing. The larger interval between neighbouring SNR values produces a stair structure in the corresponding weight values. Additionally, in com-

parison to the elevation-dependent weighting model, the SNR-based weight values show a much larger bandwidth to be taken into account within GNSS data processing. This property brings the SNR-based weighting model closer to reality. Furthermore, in both graphs of Fig. 3, not only low but also medium elevation observations (30–50◦ ) are downweighted by the standard elevation-dependent weighting model. Above e ≈ 50◦ the elevation-dependent weighting model seems to be a satisfying approximation of the reality represented by measured signal quality values.

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4 Results using SNR-Based Weighting

4.1 Effects on Ambiguity Resolution In this case study the ambiguities were resolved at first on the wide-lane linear combination (L5) and then on the ionosphere-free linear combination (L3) using the so-called SIGMA strategy provided by the BS5 (Dach et al., 2007). To resolve the L3 ambiguities, the L5 ambiguities obtained in the previous step were fixed. In order to illustrate the effects of the SNR-based weighting model on the ambiguity resolution, the baseline SIGMR SCHA formed between the SAPOS sites Sigmaringen (SIGM) and Schw¨abisch Hall (SCHA) is shown. The baseline length is about 120 km and the site SIGM was fixed for the ambiguity resolution with L3. Figure 4 presents the resolved ambiguities in percent with different satellite elevation cut-off angles (10◦ , 3◦ ).

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In order to validate the SNR-based weighting model, daily (24 h) datasets of the Antarctic Peninsula and R the SAPOS network in Baden-W¨urttemberg were processed. In contrast to past experiments (e.g. Hartinger and Brunner, 1999), in which mainly short baselines were formed, the average baseline length in this case study amounts to more than 100 km. Besides the quality of the obtained coordinates, other aspects, e.g. ambiguity resolution, double difference residuals and site-specific neutrosphere parameters, are taken into account to analyse the effects of the SNR-based weighting model on GNSS data processing. In Table 1 the most important parameter settings for the data processing are listed.

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Table 1 Selected parameter settings of the GNSS data processing using BS5 Parameter

Characteristics

Observations Interval Observation weighting model R network) Elevation cut-off angle (SAPOS Elevation cut-off angle (Antarctic Peninsula network) Satellite orbits and earth rotation parameters Neutrospheric prediction model Mapping function Time span of the neutrosphere parameters Ambiguity resolution strategy R network) Antenna calibration (SAPOS Antenna calibration (Antarctic Peninsula network)

GPS carrier phase observations; double differences 15 s sin2 e, f (SNR0) 3◦ , 10◦ 10◦ Precise final IGS products Model Niell (dry) Niellwet (Niell, 1996) 2h SIGMA strategy (L5, L3) Individual absolute calibration Relative calibration

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By decreasing the elevation cut-off angle from 10 to 3◦ , the average number of ambiguities to be resolved increases from 64 to 98 (approx. 53%). Due to more cycle slips by including low elevation observations, new ambiguities must be introduced in the data processing. Setting the elevation cut-off angle to 10◦ , the SNRbased weighting model does not essentially improve the ambiguity resolution, whereas with an elevation cut-off angle of 3◦ more than 10% of the ambiguities – both on L5 and L3 – can be additionally resolved using the advanced SNR-based weighting model. In addition to observation weighting, ambiguity resolution is affected by other factors, such as weather situation. This influence can be recognised by the less percentage of the resolved ambiguities on day 190, when heavy precipitation was registered in different areas of BadenW¨urttemberg.

4.2 Effects on Double Difference Residuals High accuracy GNSS applications commonly use double differences to reduce satellite and receiver clock errors as well as atmospheric effects for short baselines. The unknown parameters, such as ambiguities, site-specific neutrosphere parameters and coordinates, are estimated by means of the least-squares (LS) method. After the adjustment, normalised double difference residuals (NDDR) can be computed by dividing the LS residuals by the corresponding estimated standard deviations. In comparison to LS residuals NDDR are more sensitive to observation weighting and therefore can be used to test the appropriation of the applied observation weighting models within GNSS data processing. In general, smaller magnitude and homogenous variation of NDDR indicate more appropriate weighting models applied in GNSS data processing (Howind, 2005). Considering the corresponding signal quality, Fig. 5 shows an example of the temporal behaviour of the NDDR depending on satellite elevation angle. Observations of the two sites Punta Spring (SPR1) and Esperanza (ESP1) situated on the Antarctic Penisula and two representative satellites (PRN: 25, 06) are contributing. Using the advanced SNR-based weighting model, both the magnitude and the variation of the NDDR shown in Fig. 5 are significantly reduced in comparison to the elevation-dependent weighting model. This improvement is caused by a more appropriate weighting

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of the observations with low signal quality. By comparing the corresponding AMU-values on L1 with respect to the site quality, site SPR1 is obviously worse than site ESP1. This statement agrees with results of Mayer (2006): site SPR1 is strongly affected by multipath effects. Under such non-ideal observation conditions in which the correlation between signal quality and satellite elevation angle is absent due to multipath effects, signal diffraction as well as receiver characteristic, a realistic observation weighting model is mandatorily required.

4.3 Effects on Neutrosphere Parameters The GNSS signals are delayed by propagating through the Earth’s neutral atmosphere. This delay term plays an important role in GNSS data processing and can be modelled in BS5 by estimating site-specific neutrosphere parameters (SSNP). However, the accuracy of the estimated parameters is affected by several factors in different magnitudes (e.g. Mayer, 2006; Luo et al., 2007b). For validating the effects of the advanced SNRbased weighting model on the SSNP, the baseline R OFFE-HEID formed by the SAPOS sites Offenburg (OFFE) and Heidelberg (HEID) is particularly suitable for emphasizing the advantages of the SNR-based weighting model, because these two sites are strongly affected by multipath effects (Mayer et al., 2004). In Fig. 6, the standard deviations of the estimated SSNP are compared with respect to the observation weighting model. The average difference of the estimated SSNP using these different weighting models is about 1.5 cm, corresponding to approx. 10% of the mean SSNP value. Considering the standard deviations in the same way, an improvement by nearly 25% can be achieved if the advanced SNR-based weighing model is applied to the data processing. The stronger a site is affected by multipath, the more significant is the improvement.

4.4 Effects on Repeatability of Coordinates R In this case study, using SAPOS data covering eight days (DOY2004: 186–193), the SNR-based weighting model is incapable to significantly improve the

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repeatability of site coordinates determined from daily baseline solution. However, this fact can be explained: firstly, the observation data of reference sites are generally of good quality. In this case, due to the existing strong correlation between signal quality and satellite elevation angle, the SNR-based weighting model shows no significant advantages over the elevationdependent model. Secondly, because of the long observation period (24 h), the effects of satellite geometry are largely reduced. Thirdly, for long baselines, the atmospheric effects cannot be totally eliminated

using differential techniques. The performance of the atmospheric models used in GNSS data processing decreases if low elevation data (below 10◦ ) is used (Davis and Herring, 1984).

5 Conclusions and Outlook In this paper an alternative SNR-based weighting model to improve the stochastic modelling of long-term static GNSS observations was presented.

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and the computation is time-consuming. Furthermore, its current realisation is sensitive to outliers in the SNR values which may result in a shift of the whole weighting level. The future work will concentrate on the verification of the SNR-based weighting model using different datasets considering data quality, observation period and GNSS equipment. In addition, physical correlations among the raw observations must be taken into account within data processing. Furthermore, a comparison of the results with other studies is planned.

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Fig. 6 Comparison of standard deviation of estimated neutrosphere parameters, site: HEID, baseline: OFFE-HEID (114.8 km), Leica SR 520, LEIAT503

This approach is independent from the functional model described in Langley (1997) and is based on normalised SNR-weighting related to antenna-receiver combination. Within the framework of a case study, the SNR-based weighting model has been experimentally implemented into the Bernese GPS Software 5.0 and enables a weighting for each observation on both frequencies. With example datasets (Antarctic Peninsula; state Baden-W¨urttemberg (Germany)), the performance of this SNR-based weighting model has been demonstrated considering ambiguity resolution, double difference residuals, site-specific neutrosphere parameters (SSNP) and repeatability of coordinates. Using a minimum elevation cut-off angle of 3◦ more than 10% ambiguities can be additionally resolved and an improvement of nearly 25% in accuracy of the estimated SSNP can be achieved by means of the SNR-based weighting model. Additionally, the magnitude and the temporal fluctuation of the normalised double difference residuals of baselines including more low quality observations are significantly reduced. No essential improvement in repeatability of coordinates is detectable for long baselines due to the good quality of R the SAPOS reference sites and the long observation period (24 h) as well as deficiencies in atmospheric modelling. Compared with the elevation-dependent weighting model, the SNR-based weighting model is more realistic, but its implementation is more complicated

Acknowledgments We would like to thank the department 31 of the state survey office Baden-W¨urttemberg for providing the GNSS data and absolute antenna calibration values. The first author gratefully acknowledges the support of the Landesgraduiertenf¨orderung of Baden-W¨urttemberg for the scholarship during this work.

References Ananga N, Coleman R, Rizos C (1994) Variance-covariance estimation of GPS-networks. Bulletin G´eod´esique 68: 77–78 Barnes B, Ackroyd N, Cross P (1998) Stochastic modelling for very high precision real time kinematic GPS in an engineering environment. Proceedings FIG XXI International Conference, Brighton: 61–76 Brunner FK, Hartinger H, Troyer L (1999) GPS signal diffraction modelling: the stochastic SIGMA- model. Journal of Geodesy 73: 259–267 Butsch F, Kipka A (2004) Die Bedeutung des Signal- zu Rauschleistungsverh¨altnisse und verwandter Parameter f¨ur die Messgenauigkeit bei GPS. Allgemeine VermessungsNachrichten (AVN) 111 (2): 46–55 Collins JP, Langley RB (1999) Possible Weighting Schemes for GPS Carrier Phase Observations in the Presence of Multipath, Geodetic Research Laboratory, University of New Brunswick, Canada, Report to the United Sates Army Corps of Engineers Topographic Engineering Center Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS Software Version 5.0, Astronomical Institute, University of Bern, Switzerland Davis JL, Herring TA (1984) New atmospheric mapping function. Center of Astrophysics, Cambridge, Massachusetts, USA Estey LH, Meertens CM (1999) TEQC: The Multi-Purpose Toolkit for GPS/GLONASS Data, GPS Solution 3 (1): 42–49 Euler H, Goad C (1991) On optimal filtering of GPS dual frequency observations without using orbit information. Bulletin G´eod´esique 65: 130–143 Han S (1997) Quality-control issues relating to instantaneous ambiguity resolution for real time GPS kinematic positioning. Journal of Geodesy 71: 130–143

734 Han S, Rizos C (1995) Standardization of the variancecovariance matrix for GPS rapid static positioning. Geomatics Research Australasia 62: 37–54 Hartinger H, Brunner FK (1999) Variances of GPS phase observations: The SIGMA- model. GPS solutions 2 (4): 35–43 Hofmann-Wellenhof B, Lichtenegger H, Collins J (2001) GPS – theory and practice, 5th edition. Springer, Wien New York. Howind J (2005) Analyse des stochastischen Modells von GPSTr¨agerphasenbeobachtungen. Deutsche Geod¨atische Kommission, Reihe C, No. 584, M¨unchen, Germany Jin X, de Jong C (1996) Relationship between satellite elevation and precision of GPS code observations. The Journal of Navigation 49 (2): 253–265 Kim D, Langley RB (2001) Quality control techniques and issues in GPS applications: Stochastic modelling and reliability testing. Proceedings of Tutorial and Domestic Session, International Symposium on GPS/GNSS, Jeju, Korea, 7–9 November 76–85 Langley R (1997) GPS Receiver System Noise. GPS World 8: 40–45 Luo X, Mayer M, Heck B. (2007a) Quantifizierung verschiedener Einflussfaktoren in GNSS-Residuen. Zeitschrift f¨ur Geod¨asie, Geoinformation und Landmanagement (ZfV) 132 (2): 97–107 Luo X, Mayer M, Heck B (2007b) Bestimmung von hochaufl¨osenden Wasserdampffeldern unter Ber¨ucksichtigung von GNSS-Doppeldifferenzresiduen. Schriftenreihe des Studiengangs Geod¨asie und Geoinformatik, 2/2007, University of Karlsruhe, Karlsruhe, Germany Mayer M (2006) Modellbildung f¨ur die Auswertung von GPS-Messungen im Bereich der Antarktischen Halbinsel, Deutsche Geod¨atische Kommission, Reihe C, No. 597, M¨unchen, Germany

X. Luo et al. Mayer M, Wanninger L, Dick HG, Derenbach H, Heck B (2004) R -Stationen BadenMehrwegeeinfl¨usse auf den SAPOS W¨urttembergs, Poster, Geod¨atische Woche 2004, Stuttgart 12–15, October 2004 Niell AE (1996) Global mapping functions for the atmosphere delay at radio wavelengths. Journal of Geophysical Research 101, 3227–3246 Talbot N (1988) Optimal weighting of GPS carrier phase observations based on the signal-to-noise ratio. In Proceedings of the International Symposium on Global Positioning Systems, Queensland, Australia, 17–19 October, V.4.1–V.4.17 Tiberius C, Kenselaar F (2000) Estimation of the stochastic model for GPS code and phase observables. Survey Review 35: 441–454 Trimble (1999) Trimble: “Mail from Trimble Support” from Jul-13-1999, Subject: SNR Conversions, E-mail: Trimble [email protected] Wang J, Satirapod C, Rizos C (2002) Stochastic assessment of GPS carrier phase measurements for precise static relative positioning, Journal of Geodesy 76(2): 95–104 Wang J, Steward M, Tsakiri M. (1998) Stochastic modelling for static GPS baseline data processing. Journal of Surveying Engineering 124(4): 171–181 Wieser A (2001) Robust and fuzzy techniques for parameter estimation and quality assessment in GPS. Ph.D. dissertation, Graz University of Technology, Shaker Verlag, Aachen Wieser A, Brunner FK (2000) An extended weight model for GPS phase observations. Earth, Planets, Space 52: 777–782

Evaluating the Brazilian Vertical Datum Through Improved Coastal Satellite Altimetry Data Roberts Teixeira Luz, Wolfgang Bosch, Silvio Rog´eris Correia de Freitas, Bernhard Heck and Regiane Dalazoana

Abstract The integration of the Brazilian Fundamental Vertical Network (BFVN) with satellite altimetry data and multiple tide gauges (TG) is one way for an independent validation of the Brazilian Vertical System. Satellite altimetry (Sat-Alt) gives precise information on the sea surface topography (SSTop) over a great part of the oceans. However, SSTop is characterized by large differences at deep, open sea and at coastal or shallow waters. The evaluation/validation of the studies on BFVN could be achieved through the connection of the reference levels from TGs and SatAlt data, allowing SSTop “correction” at some selected TGs along the Brazilian coast. Initial tests were performed with SSTop estimates from the application of a 1D-filter to the EIGEN-GL04C geoid model and to the altimetric data from Jason-1 and TOPEX/Poseidon extended mission (T/P-EM). Tracks of these two satellites over deep, open oceans were taken as reference for connecting three selected TG reference levels. SSTop Roberts Teixeira Luz Geodetic Sciences Program, Paran´a Federal University (UFPR), CP 19001, 81531-990, Curitiba, Brazil, e-mail: [email protected] Wolfgang Bosch Deutsches Geod¨atisches Forschungsinstitut (DGFI), Alfons-Goppel-Str. 11, D-80539, M¨unchen, Germany, e-mail: [email protected] Silvio Rog´eris Correia de Freitas Geodetic Sciences Program, Paran´a Federal University (UFPR), CP 19001, 81531-990, Curitiba, Brazil, e-mail: [email protected] Bernhard Heck Geod¨atisches Institut (GIK), Universit¨at Karlsruhe (UniKA), Englerstr. 7, D-76131, Karlsruhe, Germany, e-mail: [email protected] Regiane Dalazoana Geodetic Sciences Program, Paran´a Federal University (UFPR), CP 19001, 81531-990, Curitiba, Brazil, e-mail: [email protected]

time series were obtained for Sat-Alt points near those TGs as well as for the crossing points between the reference tracks and those passing the TGs. These time series show good correlation along each TG-track but have low correlation to the ref-tracks. This points out the need for additional ref-tracks in order to improve that correlation. Keywords Sea surface topography · SIRGAS · Heights · Man sea level · Vertical Datum

1 Introduction The improvement of Brazilian heights is subject of several studies carried out since 1995, initially focusing on local effects at the Brazilian Vertical Datum at Imbituba, BVD-I (Freitas et al., 2002). After establishing the SIRGAS Working Group “Vertical Datum” (WG3) in 1997 (Drewes et al., 2002; S´anchez, 2006), those studies coincide with the aims of the SIRGAS project. A systematic discussion on the particular situation of the Brazilian Fundamental Vertical Network (BFVN) was started only in 2004 (Luz et al., 2006) emphasizing above all the lack of associated gravity data. In the past, the evaluation of BFVN’s internal errors was very difficult due to the absence of an independent, suitable tool for this purpose. However, this is important because different kinds of measured and interpolated gravity data have been used for the corrections of levelling heights. The mean sea level (MSL) at tide gauges (TG) along the coast could be used for such a control if the sea surface topography (SSTop) is known (Torge, 2001, p. 78; Bosch, 2002). Figure 1 shows the differences between

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coastal TGs and the Sat-Alt tracks. Reliable estimates of SSTop derived from the standard AVISO Sat-Alt data at deep waters are suitably propagated along the satellite tracks towards coastal TGs by considering local patterns. First attempts to such an approach at the Brazilian coast were performed at BVD-I itself (Dalazoana, 2006, p. 99). These studies allowed the comparison of spectral characteristics of altimetric open sea data and coastal tide gauge observations, as well as their use for the extrapolation of Sat-Alt data and its trend. This paper describes the continuation of these first investigations. The extrapolation of SSTop estimates Fig. 1 Differences between local mean sea level at tide gauges towards selected coastal TGs is performed as an atalong Brazilian coast and the corresponding heights referred to tempt to unify their reference levels. These “reduced” BVD-I, from three different adjustments references will then be used in the analysis of the heights of a sub-network of BFVN after its processing heights referred to BVD-I and local MSL along the with the integration of gravity data. Brazilian coast. The heights in all three sets of BFVN adjustments were computed without gravity measurements – only normal-orthometric corrections were ap- 2 Profile Approach to SSTop plied. The last adjustment (“LPL12”) was performed to investigate partitioning problems in the previous, official adjustment (“AAGP”). On the other hand, SSTop The present studies are based on the basic concept of has very distinctive behaviours at each location along SSTop being the deviation of the sea surface height the coast, depending on the general circulation, the (SSH) relative to the geoid height (N): bathymetry, prevailing winds, etc. Both quantities, i.e. SSTop = SSH − N (1) MSL and SSTop, are time-dependent, so for any comparison they must refer to the same epoch and period. Using satellite altimetry (Sat-Alt) data for the Restrictions and details regarding the use of Eq. (1) homogenization of the tide gauge reference levels is (tidal system, ellipsoid, etc.) were discussed by one way for an independent validation of the BFVN. Bosch (2002). The present studies apply an approach Sat-Alt gives precise information on SSTop over suggested by Bosch et al. (2007) in which geoid a great part of the oceans, thus contributing for a heights from EIGEN-GL04C geoid model were reliable realization of the “Global Geoid” concept extracted for each SSH along-track position in each (Heck and Rummel, 1990; Heck, 2004). However, mission cycle. A 1-D filter with 250 km radius is the quality of altimeter data is degraded over coastal applied to reduce the effects of the orbit signatures or shallow waters (Bosch et al., 2007; Deng and (“stripes”) shown by the EI-GEN-GL04C, as well Featherstone, 2006). Global ocean tide models do not as to ensure the spectral consistency between SSH properly account for coastal upwelling and non-linear and N. Their approach also includes an implicit SSH tides (Andersen et al., 2005). The wet troposphere extrapolation within the filling-in procedure for the correction is corrupted if the altimeter approaches the Sat-Alt tracks over land, to avoid the loss of data coast. The standard waveform analysis is less suited involved in the use of the mentioned filter. For the for coastal and shallow waters (Andersen and Knud- present analysis only SSTop estimates for Jason-1 and sen, 2000; Deng and Featherstone, 2006) and can be TOPEX/Poseidon extended mission (T/P-EM) were improved by dedicated retracking of raw observations used. with higher sampling rate. The SSH were derived from AVISO’s MGDR, The present study relies on a profile approach tak- Version C data, upgraded by the FES2004 ocean ing advantage of the approximate colinearity of some tide model (Lyard et al., 2006), cross-calibrated with

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Fig. 2 Tide gauge locations and Sat-Alt tracks (solid lines, left panel) used for this investigation (ERS/Envisat tracks suited to strenghten the link mainly to the Imbituba tide gauge are shown

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as dotted line). The right panel shows the shelf extension with 200 and 1000 m isobaths. Tracks 061 and 137, running over deep waters, approx. parallel to the coast, are taken as reference

replaced those given by the partial, “historical” ones between 1959 and 1975 (Luz and Guimar˜aes, 2001). AAGP was designed for solving regional inconsistencies and for validating data-bases and procedures for future more rigorous adjustments. With completion of AAGP, the need of sea level stations specifically established for geodetic purposes became evident. This led to the so-called RMPG (“Rede Maregr´afica Permanente para Geod´esia”, Permanent Geodetic Tide Gauge Network), presently with four stations: Imbituba, Maca´e, Salvador (shown in Fig. 2) and Macap´a (at the Amazonas River’s mouth). Each station is equipped with at least two different sea 3 The Geodetic Tide Gauge Network level sensors, complementary meteorological devices, specially designed stilling wells and a dual frequency The heights computed in the 1993 BFVN adjustment, GPS receiver (Luz and Guimar˜aes, 2003). The sites are known as AAGP (“Ajustamento Altim´etrico Global maintained with frequent datum checks and geodetic Preliminar”, Preliminary Global Vertical Adjustment), control surveys. other missions by a multi-mission analysis (Bosch and Savcenko, 2006) and re-organized according a network of cells (“bins”) along the nominal ground tracks of each satellite. Figure 2 shows the pair of Jason-1 and T/P-EM tracks over deep ocean which are taken as initial reference for checking the results of cross-over analysis and the propagation of Sat-Alt information towards the coast – using also pairs of crossing tracks of the same Sat-Alt missions.

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Fig. 3 Weekly sea level variations at RMPG stations Imbituba, Maca´e and Salvador. Large meteorological effects are present at the first two stations. The low variation at Salvador and the delayed signal from Imbituba to Maca´e reflecting the northward propagation of cold fronts are also present in the data from RMPG’s meteorological sensors (IBGE, 2002)

The RMPG stations play an essential role in the present studies due to: (1) their intrinsic connection to BFVN and SIRGAS; and (2) the natural data homogeneity achieved with the same operational standards. The records from the RMPG stations and Sat-Alt data will be used as constraints for the adjustment of a temporally homogeneous, gravity corrected sub-network of BFVN (Luz et al., 2006). The extrapolation of Sat-Alt data towards the coast will be tested at the RMPG station “Salvador”. It has the narrowest shelf (Fig. 2) and rather low meteorological effects in its sea level observations (Fig. 3). This makes the tide gauge ideal for a link with altimetry. Figure 3 also suggests the possibility of correcting individual Sat-Alt observations with the weekly relative MSL before the tide reduction and the cross-over analysis. Such a correction could therefore improve the Sat-Alt data smoothness needed by the extrapolation procedures. The weekly relative MSL displayed in Fig. 3 was computed from hourly data with a 168 h filter (Pugh, 1987, p. 416).

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Fig. 4 Depths (GEBCO) and SSTop estimates along the reftracks – Jason-1 061 (top) and T/P-EM 137 (bottom). The vertical lines mark crossing with the descendent tracks coming from the selected tide gauges. SSTop time series for these crossing points are shown in Fig. 6, extracted from all tracks

Fig. 5 Depths (GEBCO, grey) and SSTop profiles (black) along the descendent tracks 152, 202 and 100, passing the TGs of Imbituba, Maca´e and Salvador respectively (Fig. 2). The vertical lines mark the points where these tracks cross the ref-tracks

tracks from TGs towards the ref-tracks. The large range, probably coming from seasonal variability as shown in Fig. 6, was intentionally kept to allow an appreciation of the general behavior of these first results. The increased variability of SSTop south of 30◦ S 4 Preliminary Results is common to both ref-tracks and is understood because the tracks pass the central area of the Brasil curSSTop estimates with the 250 km filter approach along rent. However, from 25◦ S to 15◦ S the parallel tracks the reference tracks (“ref-tracks”) are displayed in show a totally opposite behavior – Jason-1 track 061 Fig. 4, while Fig. 5 shows results for the descending has a minimum of 0.50 ∼ 0.65 m around 19◦ S/20◦ S

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imately 80 km offshore, at the bins 257, 395 and 554. This was enforced by increasing data gaps for Jason-1 tracks (Salvador and Imbituba). To avoid the problems of referring TG data to the same level of Sat-Alt results, instead of the TG hourly observations we estimated the SSTop by the previously described 168h-smoothed “short period mean sea level” (MSL168h), interpolated for the Sat-Alt SSTop epochs. Figure 6 also shows this MSL168h time series. First the mean values of SSTop estimates only from the TG-tracks were compared, to check the profile consistency of each track. Along the TG-tracks at the three extraction points the SSTop estimates show good agreement for Salvador, with mean values around 0.75 ± 0.03 m for the three locations. With respect to Maca´e, the mean values range from 0.56 ± 0.03 m (at the near-TG bin) to 0.74 ± 0.03 m (at track 137 crossing point). Along Imbituba track, the mean values vary from 0.72 ± 0.06 m to 0.60 ± 0.10 m at the equivalent locations. These values are reflected in Fig. 5. The series provided by the Jason-1 track 100 at the bin near Salvador TG (257) and at the ref-tracks crossing points are in good agreement, with correlation factors of 0.90 and 0.96 (Table 1). For Maca´e (T/P-EM track 202) and Imbituba (Jason-1 track 152) there is good agreement only between the series at the reftracks crossing points – correlation 0.81 for Maca´e and 0.91 for Imbituba. Even with the relatively small distance between the near-TG bin and the track 061 crossing, the correlation is only 0.66 for Maca´e. For ImFig. 6 Comparison of time series of the SSTop estimates: (a) bituba the large distance and the temporal variability only from the Maca´e track (202) at the three extraction points; shown in Fig. 5 probably are the causes for the 0.45 (b) from 202 and 061 tracks at their crossing point; (c) from 202 correlation, introducing difficulties if the SSTop estiand 137 tracks at their crossing point; and (d) between 202 track mates from the reference tracks are to be applied as a at bin 395 and the smoothed MSL168h kind of “correction” to the TG-tracks – in this case, the track 152. Table 2 shows the correlation between the time sewhereas T/P-EM track 137 shows an SSTop maximum of 0.90 ∼ 1.05 m. This could be an effect of the ries from the tracks crossing each other. The agreesubmarine mountain chain located between 19◦ S and ment is not so good, most probably due to the fact 21◦ S (Fig. 2). Further north, from 15◦ S to 14◦ S both tracks have approximately the same level (around 0.80 m), as does Table 1 Correlation of SSTop along the TG-tracks [Near-TG bin] × [061 crossing] × the Jason-1 track 100, running from Salvador RMPG [061 crossing] [137 crossing] station (Fig. 5). Figure 6 shows examples for Maca´e of the SSTop TG-track ρ dist. (km) ρ dist. (km) time series at those locations of the TG-oriented tracks 100 (Salvador) 0.90 413 0.96 206 0.66 265 0.81 180 indicated by vertical lines in Figs. 4 and 5. The SSTop 202 (Maca´e) 0.45 684 0.91 144 time series for the TGs were actually extracted approx- 152 (Imbituba)

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tracks has to be strengthened and actual TG observations are to be used instead of filtered data. [TG-track × 061] × [TG-track × 137] × Parallel efforts involve the improvement of DGFI’s [061 × TG-track] [137 × TG-track] Sat-Alt database in the vicinities of selected RMPG Time diff. Time diff. stations, using corrections based on their own weekly TG-track ρ (days) ρ (days) MSL (Fig. 3). The well known deficiencies of the 100 (Salvador) 0.59 +1.5 0.69 −1.4 global ocean tide model FES2004 are under investi202 (Maca´e) 0.27 −4.4 0.34 +2.6 gation and residual tide corrections can most proba152 (Imbituba) 0.39 +3.6 0.49 +0.6 bly explain part of the observed inconsistencies. A local hydro-dynamical model around Salvador area also that the two tracks observe the crossing at different could be used to validate or improve the tidal reduction. times. There are reasonable agreements for the Imbituba track, in spite of more than 3 days time differ- Acknowledgments The authors acknowledge the Brazilian and ence, and good agreements for Salvador. At the track German Education agencies (CNPq, CAPES and DAAD) for the financial support; DGFI and IBGE for the data made available; passing Maca´e the time differences are even larger and the Directors of GIK/UniKA and DGFI for the exchange periods; the agreement is very poor, indicating the need for IBGE for the Doctorate license; the reviewers who provided very choosing other ref-tracks, with smaller time differences constructive comments; and IAG for the Grants. regarding the TG-oriented tracks. Regarding mean values, along ref-track 061 SSTop is 0.59 ± 0.07, 0.74 ± 0.04 and 0.66 ± 0.10 m for the crossing points with Salvador, Maca´e and Imbituba References tracks. Along ref-track 137, values for equivalent points are 0.97 ± 0.03, 0.82 ± 0.03 and 0.61 ± 0.11 m. Andersen OB, Knudsen P (2000) The role of satellite altimetry in gravity field modelling in coastal areas. Phys Chem Earth So, there are reasonable agreements only for Maca´e (A), 25(1), pp 17–24. and Imbituba tracks, also introducing difficulties for Andersen OB, Knudsen P, Berry PAM, Mathers EL, Trimmer R, “correcting” the TG-tracks with the SSTop estimates Kenyon S (2005) Initial Results from Retracking and Reprofrom ref-tracks. cessing the ERS-1 Geodetic Mission Altimetry for Gravity The correlations between near-TG Sat-Alt SSTop Field Purposes. In: Jekeli C et al. (eds) Gravity, Geoid and Space Missions (IAG Series, v. 129). Springer, pp 1–5. estimates and the 168h-smoothed TG observations are very poor −0.31, 0.22 and 0.33, for Salvador, Maca´e Bosch W (2002) The Sea Surface Topography and its Impact to Global Height System Definition. In: Drewes H et al. (eds) and Imbituba. This fact allows the conclusion on the VeReS (IAG Series, v. 124). Springer, pp 225–230. need of more investigation on the way the TG observa- Bosch W, Savcenko R (2006) Satellite Altimetry: Multi-Mission tions should be included in the analysis. Cross Calibration. In: Tregoning P, Rizos C (eds) Dynamic Table 2 Correlation of SSTop from TG- and ref-tracks at their crossing points

5 Final Remarks In order to link TGs along the Brazilian coast a first analysis of SSTop profiles has been performed. The study takes advantage of the approximate colinearity of coastal TGs and the Sat-Alt tracks of Jason1 and T/PEM. A pair of Jason-1 and T/P-EM tracks over deep waters were taken as reference. The analysis is complicated due to inconsistencies between the parallel reftracks obviously caused by the high sea level variability identified even beyond the continental shelf. This prevented their use for linking the SSTop estimates derived along the TG-tracks. The network of crossing

Planet (IAG Series, v. 130). Springer, pp 51–56. Bosch W, Savcenko R (2007) Estimating the sea surface topography – profile approach with error examination. Presented in the XXIV IUGG Gen. Assembly, Perugia. Bosch W, Fenoglio-Marc L, W¨oppelmann G, Marcos M, Novotny K, Savcenko R, Karpytchev M, Nicolle A, Becker M, Liebsch G (2007) Coastal Sea Surface Topography – a Synthesis of Altimetry, Gravity, and Tide Gauges. In: Aviso Newsletter, n. 11, CNES, in press. Dalazoana R (2006) Studies on the temporal analysis of the Brazilian Vertical Datum. PhD Thesis, Universidade Federal do Paran´a, Curitiba. Available in Portuguese at , accessed 23.08.2007. Deng X, Featherstone WE (2006) A coastal retracking system for satellite radar altimeter waveforms: Application to ERS-2 around Australia. J Geophys Res, 111, C06012, doi:10.1029/2005JC003039. Drewes H, S´anchez L, Blitzkow D, Freitas S (2002) Scientific Foundations of the SIRGAS Vertical Reference System. In:

Evaluating the Brazilian Vertical Datum Drewes H et al. (eds) VeReS (IAG Series, v. 124). Springer, pp 297–301. Freitas S, Schwab SHS, Marone E, Pires AO, Dalazoana R (2002) Local Effects in the Brazilian Vertical Datum. In: Adam J, Schwarz KP (eds) Vistas for Geodesy in the New Millennium (IAG Series, v. 125). Springer, pp 102–107. Heck B (2004) Problems in the Definition of Vertical Reference Frames. In: Sans`o F (ed) V Hotine-Marussi Symposium (IAG Series, v. 127). Springer, pp 164–173. Heck B, Rummel R (1990) Strategies for Solving the Vertical Datum Problem Using Terrestrial and Satellite Geodetic Data. In: S¨unkel H, Baker T (eds) Sea Surface Topography and the Geoid (IAG Series, v. 104). Springer, New York, pp 116–128. IBGE (2002) Short analysis of the RMPG first results – the propagation of a cold front through the southern Brazilian coast. Internal report, Geodesy Coordination. Available in Portuguese: , in 19.11.2007. Luz RT, Guimar˜aes VM (2001) Status and perspectives of the Brazilian Fundamental Vertical Network. In: Proceedings (CD) of the II Brazilian Geodetic Sciences Colloquium, Curitiba. Available in Portuguese at , accessed in 23.08.2007.

741 Luz RT, Guimar˜aes VM (2003) Ten years of sea level monitoring at IBGE. In: Proceedings (CD) of the III Brazilian Geodetic Sciences Colloquium, Curitiba. Available in Portuguese at , accessed in 23.08.2007. Luz RT, Freitas SRC, Dalazoana R, Baez JC, Palmeiro AS (2006) Tests on Integrating Gravity and Leveling to Realize SIRGAS Vertical Reference System in Brazil. In: Tregoning P, Rizos C (eds) Dynamic Planet (IAG Series, v. 130). Springer, pp 646–652. Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dynamics, 56, pp. 394–415, DOI 10.1007/s10236-006-0086x. Pugh DT (1987) Tides, Surges and Mean Sea-Level. John Wiley, Chichester. pp xiv+472. S´anchez L (2006) Definition and Realisation of the SIRGAS Vertical Reference System within a Globally Unified Height System. In: Tregoning P, Rizos C (eds) Dynamic Planet (IAG Series, v. 130). Springer, pp 638–645. Torge W (2001) Geodesy. 3rd edn., Walter de Gruyter, Berlin, pp xv+416.

Land Subsidence Monitoring in Australia and China using Satellite Interferometry Alex Hay-Man Ng, Hsing-Chung Chang, Kui Zhang, Linlin Ge and Chris Rizos

Abstract Continued excessive extraction of groundwater may lead to significant land subsidence, which causes economic loss. The aim of this research is to investigate ground subsidence in several Australian capital cities using the Persistent Scatterer Interferometry approach. Such studies may help improve our understanding of the deformation of terrain and built structures, and evaluate the effectiveness of any measures taken to ease the problem. The Persistent Scatterer Radar Interferometry approach is an alternative technique for land subsidence monitoring to conventional surveying methods. The Persistent Scatterer Interferometry approach first identifies all the stable point scatterers and a deformation analysis is then applied to these points. Persistent Scatterer InSAR techniques can generate deformation time-series with an accuracy down to the millimetre level depending on the number and quality of SAR images. The Persistent Scatterer InSAR results are validated with other spatial data such as groundwater extraction bore sites and groundwater level. Six cities are selected for these analyses, and the results show that the deformation Alex Hay-Man Ng School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia Hsing-Chung Chang School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia Kui Zhang School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia Linlin Ge School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia Chris Rizos School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia

rate from cities with groundwater significant levels of water extraction are much larger than for those cities with low levels of groundwater extraction. Keywords Radar Interferometry · Urban subsidence monitoring · Persistent Scatterer Interferometry · Australia

1 Introduction Land subsidence can be the result of many factors, for example, groundwater extraction, high-rise building construction, metro construction, underground tunnel excavation, and so on. Although land subsidence may not directly threaten human life, ground subsidence usually occurs in urban areas, which could lead to serious economic loss. For example, it can lead to damage to buildings, roads, gas/water pipes, utilities, and even telecommunications cables. Australia is the driest inhabited continent on the Earth, with highly variable rainfall patterns. Every year only about 12% of rainfall over Australia actually runs off into streams and rivers or is retained as groundwater. The other 88% evaporates back into the atmosphere. Hence water shortages have always been a problem for Australia. Australia has approximately 5% of the world’s land area; however it only has 1% of the water carried by the world’s rivers. Groundwater is one of the most important natural resources in Australia. It provides water to the population, thereby supporting agriculture, households, electricity and gas generation, manufacturing industry, the mining industry, just to name only a few beneficiaries. In the past two decades the amount of water

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usage has increased rapidly in Australia. As a result the degree of groundwater usage has increased dramatically in recent years due to the limited surface water supplies as a result of severe drought conditions. In addition, the future sustainability of groundwater resources is at risk from overuse in many capital cities in Australia. Many areas now suffer from excessive groundwater extraction, which has worsened because of the severe drought. One typical example is the major inland groundwater systems in the state of New South Wales, which represents approximately 80% of the total groundwater extraction in New South Wales, have already been over extracted since 2001 (CRC for Water Quality, 2006). Continued groundwater overextraction can lead to ground subsidence. Ground subsidence in major Australian cities is the topic of this study. Several methods can be used to monitor urban ground subsidence. These include extensometers, levelling (Bitelli et al., 2000), total stations, vertical extensometers (Leake, 1997), GPS surveys (Bitelli et al., 2000), and spaceborne radar interferometry (Ferretti et al., 2000, 2001). Spaceborne radar interferometry is the technique used in this study for the following reasons: (1) it is very difficult to establish a dense network of ground survey marks for a large subsidence basin using conventional surveying methods, (2) it is less time consuming, (3) it requires less labour effort, and (4) the cost is much lower compared to the conventional methods. Differential InSAR (DInSAR) and Persistent Scatterers InSAR are the most common radar interferometry techniques for monitoring ground deformation. DInSAR is typically used to measure the short term deformation, while Persistent Scatterers InSAR is used to measure long term deformation.

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images are acquired over the same area, the phase difference between the images can be written as: φ = φT opo + φ De f o + φ Atmos + φ Or bit + φ N oise (1) where φ is the phase difference between the two images, φT opo is the phase due to topography, φ De f o is the phase due to the geometric displacement of the point, φ Atmos is the phase due to atmospheric disturbances, φ Or bit is the phase due to orbit error, and φ N oise is the phase due to decorrelation noise. If other components are carefully removed it is possible to determine the change in geometry (or deformation) between the two radar images on a pixel-by-pixel basis. Spaceborne radar interferometry has already proven its potential for ground deformation monitoring applications due to its comparatively high precision and spatial resolution. If long time series of SAR images are available it is possible to measure the deformation in the time domain. Recent studies have focussed on estimating long term deformation using a large number of radar images. The basic principle of carrying out measurements on image pixels corresponding to point-wise stable reflectors using long times series of images was first suggested in the late 1990s (Ferretti et al., 2000, 2001). It is possible to measure long term deformation rate with an accuracy down to the millimetre per year level depending on the size of the image stack. This technique was developed by Ferretti et al. (2001), which is protected by a patent, and the term “Permanent Scatterer Technique” is trademarked. Nevertheless many methods have been developed based on this concept, such as Interferometric Point Target Analysis (Werner et al., 2003), Stanford Method for PS (Hooper et al., 2005), Coherent Target Monitoring, Stable Point Network Analysis (Arnaud et al., 2003) and Small Baseline Subset Approach (Berardino et al., 2002, 2003). In this study the name Persistent Scatterer InSAR is used.

Radar measures the received strength and time of return of the microwave signals that are emitted by a radar antenna, reflected off a target point, and returns to the receiving antenna. The received signal can be 3 Persistent Scatterer InSAR Approach represented as a complex value, where the strength is represented by the amplitude of the signal and the time Figure 1 shows the main processing steps for the Peris represented by the phase of the signal. If two radar sistent Scatterer approach.

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Image coregistration

Persistent scatterer point Identification

DEM

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3.2 Interferogram A stack of N interferograms are generated with a common master image. N-1 interferograms can be generated with respect to the master image. A high resolution DEM (Digital Elevation Model) is used to remove the topographic phase and to generate the initial differential interferogram stack.

Initial phase model estimation

3.3 PS Selection PS selection (Remove unreliable points)

Network refinement

The unreliable PS points are removed in this step. The PS points that appear to be incorrectly estimated and incoherent, and those that appear to be dominated by scatterers in adjacent PS pixels are removed.

Phase model correction

3.4 Displacement Estimation Final product

Fig. 1 Persistent Scatterer InSAR processing flow

3.1 Persistent Scatterer Candidate Identification The persistent scatterer (PS) point is the pixel corresponding to the stable reflector, which is coherent in time in all images in the stack. If this assumption is true, then the effect of random phase is minimised at the PS point. Hence the phase in that individual pixel is mainly a function of topography and deformation. All the SAR images are co-registered with a common master prior to identifying the PS candidates. All the SAR images that are not precisely coregistered are removed in this process. The PS candidate identification process was conducted to identify all the stable natural reflectors based on the amplitude dispersion index (Ferretti et al., 2001) of the point targets and its intensity factor in a time series of SAR images.

The final step is to estimate the displacement on the PS candidates. This involves estimation for the initial phase model using an integer least squares estimator, network refinement, and correction for deformation which considers the DEM error and atmospheric bias. The method and model used in this study are mainly based on Kampes’ method and model (Kampes and Adam, 2003; Kampes and Hansesen, 2004; Kampes, 2006):

i i φx,y = −2π · ax,y + βxi h

−

D 4π  (αd(x,y) ) · pd (t i ) + n ix,y λ

(2)

d=1

where ai is the integer ambiguity for interferogram i. βxi h is the residual topographic phase due to an inacD  curacy in the reference DEM. 4π (αd(x,y) )· pd (t i ) λ d=1

represents the target motion with respect to the master image. n i is the decorrelation noise. A linear displacement model is used in this study and the atmospheric phase contribution is treated as noise.

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4 Test Sites

Table 2 Deformation rate of cities selected for Persistent Scatterer Analysis

Six cities were selected and divided into two groups. Group 1 consists of Australian cities where groundwater extraction is not excessive. The cities selected were Brisbane, Sydney, Newcastle and Canberra. Group 2 concentrates on those cities where there is believed to be excessive groundwater extraction. The Perth Basin in Western Australia has one of the highest rates of groundwater extraction in Australia. Another Group 2 site is located in northern China. Serious over-extraction of groundwater has been reported in that city. Data from the ERS-1 and ERS-2 satellites, from the European Space Agency, were used in this study of the first five cities, while ENVISAT data (also from the European Space Agency) were used in the analyses for the Chinese city in order to map the linear deformation rate in each city (Table 1). Without both ascending and descending data, it is not possible to derive a 3-D deformation vector without making certain assumptions. Over-extraction of groundwater from aquifers is the major cause of subsidence (Bell, 1997). The land deformation due to groundwater extraction is predominantly in the vertical component. Therefore in this study, horizontal deformation is not considered because the level of horizontal deformation is much less than the vertical deformation (or sinking).

Task

City

μ Deformation (mm/yr)

σ Deformation (mm/yr)

Group 1

Sydney Newcastle Brisbane Canberra Perth City in China

−5 −8 −1 −1 −24 −18

2 3 2 1 12 6

Group 2

μ Deformation = Mean of deformation rate σ Deformation = Standard deviation of deformation rate.

within each city are calculated and listed in Table 2. The results show that there is almost no deformation in most areas of these test sites. Less than on average 5 mm subsidence per year is observed in Brisbane, Canberra and Sydney, and less than on average 1 cm subsidence per year is measured in Newcastle. In the Sydney area (Fig. 2), the highest linear deformation rate, up to −9 mm/yr, is observed in the Eastern Suburbs, especially inside the Botany Sands Aquifer. The Botany Aquifer (Fig. 3) is the largest aquifer in Sydney and the measured subsidence may be due to groundwater extraction. The locations of the groundwater extraction bore holes (triangle) are overlaid on the Sydney deformation map. There is some

Table 1 Cities selected for Persistent Scatterer Analysis Task

City

Country

Image

Date (mm/yyyy)

Group 1

Sydney Newcastle Brisbane Canberra Perth City

Australia Australia Australia Australia Australia China

18 22 20 15 10 9

04/1992∼04/1997 08/1992∼01/2001 08/1992∼01/2001 08/1992∼12/1996 06/1992∼12/1996 12/2003∼06/2006

Group 2

5 Results 5.1 Group 1 Urban Subsidence The linear deformation rates (mm/yr) in Brisbane, Canberra, Sydney and Newcastle are derived from the Per- Fig. 2 Sydney deformation map from Persistent Scatterer Insistent Scatterer Approach. The mean and standard de- SAR analysis (period 04/1992∼04/1997; number of PS point: viation of the linear deformation rates of PS points 73093)

Land Subsidence Monitoring in Australia and China

Fig. 3 Botany aquifer map. The blue line represents the boundary of the Botany Aquifer. The use of groundwater is banned in the area coloured in red. The residential use of groundwater is banned in the area coloured yellow, especially for drinking water, irrigation, washing cars, bathing, or to fill swimming pools. Centennial Park is located in the white area in the middle right of the map

correlation between areas of groundwater extraction and areas of high subsidence. Although it is possible to see that the rate of subsidence is somehow related to the groundwater extraction from Fig. 2, the area with the higher density of groundwater extraction bore holes does not essentially correspond to areas of highest subsidence. Highest deformation is observed in the area around Centennial Park, which is the source of the Botany Aquifer. This might due to the minerals in the area around Centennial Park being carried away by the groundwater along the groundwater flow paths. Another reason is due to the varying rate of groundwater extraction at different bole holes. For instance, a race course and a university are located in the area with highest deformation area, where the groundwater is used for irrigation at both locations. From the Brisbane deformation map (Fig. 4) it is observed that the Brisbane city is comparatively stable. Most of the areas have almost no observed deformation. Canberra is the capital city of Australia (Fig. 5). An average of 1 mm/yr deformation, with a standard deviation of 1 mm/yr, has been observed. There are only two groundwater bore holes in Canberra.

747

Fig. 4 Brisbane deformation map from Persistent Scatterer InSAR analysis (period 08/1992∼01/2001; number of PS point: 5111)

Fig. 5 Canberra deformation map from Persistent Scatterer InSAR analysis (period 08/1992∼12/1996; number of PS point: 10576)

It is observed that Newcastle has a mean linear deformation rate of −8 mm/yr, with a standard deviation of 3 mm/yr. This is the highest mean deformation rate amongst the four cites in Group 1. It is interesting to note that the groundwater resources in Newcastle

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are indeed much closer to being over-extracted compared to the groundwater resources in Sydney and Canberra (Department of Environment and Conservation NSW, 1995).

5.2 Group 2 Urban Subsidence Very high deformation rates have been observed in Perth (Australia) and a city in China under study (the authors cannot name the city). Up to 50 millimetre subsidence per year have been observed in both cities. The deformation rates of both cities are therefore much higher than the four cities in Group 1 (Table 2). The linear deformation rate in Perth is shown in Fig. 6 and the change in mean groundwater levels is shown in Fig. 7, from an independent study by the CSIRO (Smith et al., 2005). The linear deformation rate in Perth is compared to the change in groundwater levels, which shows that high deformation rates are found in the area which has a high fall in groundwater level. The drop in groundwater level in Perth has been identified as being mainly due to groundwater extraction (Department of Environment and Water Resources, 2001). The city in northern China has suffered from a groundwater over-extraction problem since the 1970s.

Fig. 6 Perth deformation map from Persistent Scatterer InSAR analysis (period 06/1992∼12/1996; number of PS point: 48745)

Fig. 7 Change in mean groundwater level (Smith et al., 2005) (period 1995∼2004)

Ground surveys have measured 10.8–36.6 mm subsidence per year from 1976 to 1986 (Toalin Zhao, 2006). The deformation map (Fig. 8) shows that the major subsidence areas are located around the city hall in the middle of the map, and in the industrial area in the northwest. Very high subsidence rates have been

Fig. 8 Deformation map for a city in north China (period 12/2003∼06/2006; number of PS point: 9469)

Land Subsidence Monitoring in Australia and China

detected in the two subsidence bowls. Up to 4 cm deformation per year have been observed in both subsidence bowls, which are very close to historical maximum levels. This suggests that the city has continued to have high subsidence over the past 30 years. However the locus of maximum subsidence has shifted northward slightly from the city centre.

749 The Australian Research Council (ARC) has also supported InSAR research at UNSW over a number of years. The authors wish to thank European Space Agency and the Australian Centre for Remote Sensing for providing the C-band ERS-1/2 and ENVISAT SAR images.

References 6 Conclusions By overlaying the Persistent Scatterer InSAR results on the Landsat images, groundwater levels, and groundwater extraction sites in a GIS it is possible to speculate on the causes of the subsidence. This can provide more information for urban monitoring, for example, the effect of groundwater extraction on land subsidence, and hence to improve our understanding of the deformation of the terrain and buildings within urban environments. In addition, accurate monitoring of land subsidence is also necessary to give best recommendations for sustainable use of the underground resources. The average deformation rates of the six cities studied are listed in Table 2. The deformation rate for cities in Group 2 (cities with excessive groundwater extraction) are significantly larger than the cities in Group 1.

7 Future Study This study only focusses on the effect of groundwater extraction on urban subsidence. However, there are many other factors which may also affect urban ground subsidence, for example, soil erosion, presence of underground structures, etc. For example, the high subsidence area in the right of Fig. 1 is not fully explainable. It may be due to groundwater extraction as well as other factors such as soil erosion. A future focus is the generation of a soil erosion model based on soil type and slope data. In addition it is planned to overlay the Persistent Scatterer InSAR results with a road structure map in order to investigate other causes of subsidence. Acknowledgments This research work has been supported by the Cooperative Research Centre for Spatial Information project 4.2, whose activities are funded by the Australian Commonwealth’s Cooperative Research Centres Programme.

Arnaud, A., N. Adam, R. Hanssen, J. Inglada, J. Duro, J. Closa, and M. Eineder (2003). ASAR ERS interferometric phase continuity. In: International Geoscience and Remote Sensing Symposium, Toulouse, France, 21–25 July. Bell, J. W. (1997). Title of subordinate document. In: Las Vegas Valley: Land subsidence and fissuring due to groundwater withdrawal. Nevada Bureau of Mines and Geology. Impact of Climate Change and Land Use in the Southwestern United States. Available via world wide web. http://geochange.er.usgs.gov/sw/impacts/hydrology/ vegas gw/ of subordinate document. Cited 1 Dec 2003. Berardino, P., F. Casu, G. Fornaro, R. Lanari, M. Manunta, A. Pepe, and E. Sansosti (2003). Small baseline DIFSAR techniques for earth surface deformation analysis. In: Third International Workshop on ERS SAR Interferometry ‘FRINCE03’, Frascati, Italy 1–5 December, pp 6. Berardino, P., G. Fornaro, R. Lanari, and E. Sansosti (2002). A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms. IEEE Transactions on Geoscience and Remote Sensing, Vol. 40(11), pp 2375–2383. Bitelli, G., F. Bonsignore, and M. Unguendoli (2000). Levelling and GPs networks to monitor ground subsidence in the Southern Po Valley. Journal of Geodynamics, Vol. 30(3), 24 February 2000, pp 355–369. CRC for Water Quality (2006). A consumer’s guide to drinking water May. The Cooperative Research Centre for Water Quality and Treatment Australia’s national drinking water research centre. Department of Environment and Conservation NSW (1995). NSW State of the Environment 1995. Department of Environment and Water Resources (2001). Australia State of the Environment Report 2001, Department of Environment and Water Resources, Australia. Ferretti, A., C. Prati, and F. Rocca (2000). Nonlinear subsidence rate estimation using permanent scatterers in differential SAR interferometry. IEEE Transactions on Geoscience and Remote Sensing, Vol. 38, pp 2202–2212. Ferretti, A., C. Prati, and F. Rocca (2001). Permanent scatterers in SAR interferometry. IEEE Transactions on Geoscience and Remote Sensing, Vol. 39, pp 8–20. Hooper, A. (2006). Persistent Scatterer Radar Interferometry for crustal deformation studies and modelling of volcanic deformation. PhD thesis. Stanford University. Kampes, B. M (2006). Radra Interferometry: Persistent Scatterer Technique. Book, Springer, ISBN 140204576X.

750 Kampes, B. M. and N. Adam (2003). Velocity field retrieval from long term coherent points in radar interferometric stacks. In: International Geoscience and Remote Sensing Symposium, Toulouse, France, 21–25 July. Kampes, B. M. and R. F. Hansesen (2004). Ambiguity resolution for permanent scatterer interferometry. IEEE Transactions on Geoscience and Remote Sensing, Vol. 42(11), pp 2446–2453. Leake, S. A. (1997). Title of subordinate document. In: Land Subsidence From Ground-Water Pumping. U.S. Geological Survey. Impact of Climate Change and Land Use in the Southwestern United States. Available via world wide web. http://geochange.er.usgs.gov/sw/changes/anthropogenic/ subside/ of subordinate document. Cited: 6 Jan 2004.

A.H.-M. Hg et al. Smith, A., D. Pollock, and D. McFarlane (2005). Opportunity for Additional Self Supply of Groundwater from the Superficial Aquifer beneath Metropolitan Perth. Client Report, CSIRO: Water for a Healthy Country National Research Flagship Canberra. Toalin Zhao (2006). The change in groundwater environment and the strategy of improvement. Department of Hydrology information, China, 2006. Werner, C., U. Wegmuller, T. Strozzi, and A. Wiesmann (2003). Interferometric point target analysis for deformation mapping. In: International Geoscience and Remote Sensing Symposium, Toulouse, France, 21–25 July.

Analysis on Temporal-Spatial Variations of Australian TEC Gary Ouyang, Jian Wang, Jinling Wang and David Cole

Abstract While using GPS data to calculate TEC (Total Electron Content), the bias due to GPS hardware errors (including satellite and receiver clocks), tropospheric error and multipath error must be removed, or at least significantly reduced. In this paper the authors use carrier phase smoothed pseudo-range data within a complementary Kalman filter. Under the assumption that there is a linear variation of TEC with respect to geographical latitude and longitude, the Kalman filter has been used to estimate the TEC with less bias. Also an ionospheric electric potential model is used for smoothing the TEC. Taking Australian GPS continuous observation station data as an example, the authors have analysed the ionospheric TEC daily variations in 2006, and hourly variations in a sample day in Spring, Summer, Autumn and Winter. In addition the Australia TEC spatial distribution during a certain time has been determined using the data from 14 Australia GPS continuous observation stations. The result shows that the TEC values in Summer and Spring are higher, and have more obvious variability, than during Autumn and Winter. Gary Ouyang School of Surveying and Surveying and Spatial Information Systems, University of New South Wales, Sydney NSW 2052, Australia Jian Wang School of Surveying and Surveying and Spatial Information Systems, University of New South Wales, Sydney NSW 2052, Australia Jinling Wang School of Surveying and Surveying and Spatial Information Systems, University of New South Wales, Sydney NSW 2052, Australia David Cole IPS radio and Space Services, Department of Industry Tourism and Resources, Sydney, Australia

Keywords GPS · TEC · Carrier phase smoothing · Kalman filter

1 Introduction When extreme ultraviolet (EUV) light and x-rays from the Sun act on the gaseous atoms and molecules in the Earth’s atmosphere ionisation occurs and positively charged ions and negatively charged free electrons are formed in the atmospheric layer located from 50 to 2500 km above the Earth’s surface. Cosmic radiation also contributes to this ionisation (Heelis and William, 1999). This atmospheric layer is the ionosphere. The ionospheric spatial variability is influenced by the Earth’s magnetic field. The temporal-spatial variations of ionospheric TEC have an impact on GPS static and kinematic applications, and navigation. Therefore, the analysis of temporal-spatial variations of TEC is an important topic of GNSS application research. There are many methods to obtain electron density distributions or TEC profiles. Ionosondes are an effective instrument to collect data for this purpose. In addition, empirical and mathematical models have been developed. The IRI (International Reference Ionosphere) model, the NeQuick model, the PIM (the Parameterized Ionospheric Model) are examples of empirical models. The SLIM (the Semi-Empirical Low-Latitude Ionospheric Model) is a combination of empirical and mathematical model. Since the early dates of GPS, more and more accurate models have been developed, such as the Klobuchar Model (Klobuchar, 1987) and the GIM (Global Ionospheric Maps) (Liu et al., 2005a,b). Some innovative models

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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using LEO (Low Earth Orbit) satellite GPS data to generate near real time 3D ionospheric distribution have also been developed (Hajj et al., 2002; Coster et al., 1992). In this paper, the authors describe a combined method to calculate less bias and smoothed TEC. These methods include (1) smoothing pseudo ranges by carrier phases through Kalman filter; (2) computing TEC by GPS measurement data; (3) using TEC linear model assumption to correct TEC bias; (4) smoothing TEC by an ionospheric electric potential model. After explaining the theory, some experiments with real Australian GPS data are presented.

2 Mathematical Models to Calculate TEC Values 2.1 Kalman Filter for Smoothing Pseudo-Ranges with Carrier Phases The GPS pseudo-range measurement model can be expressed as: j

j

j

j

j

j

j

Pi = ρi +c·(dt j −dti )+Ii +Ti +dri +dm i +ei (1) j

− + PSt = PSt−1 + (L t − L t−1 ) + pt− = pt−1 +q

kt = pt− ( pt− + r )−1

+ PSt pt+

=

− PSt

+ kt (Pt −

= (1 − kt ) pt−

(3) − PSt )

− where PSt is the estimated smoothed pseudo-range + at epoch (t), PSt−1 is the smoothed pseudo-range at epoch (t − 1), L t is the measured carrier phase at epoch (t), L t−1 is the measured carrier phase at epoch (t − 1), pt− is the estimated error covariance matrix at + epoch (t), pt−1 is the error covariance matrix at epoch (t − 1), q is the process noise covariance matrix, kt is the Kalman gain at epoch (t), r is the measurement + noise covariance matrix, PSt is the smoothed pseudorange at epoch (t), Pt is the measured pseudo-range at epoch (t), and pt+ is the error covariance at epoch (t) (Hajj et al., 2004).

2.2 Vertical TEC Inversion Model Equation (1) is rewritten with the smoothed pseudorange generated from Eq. (3) as:

where Pi is the pseudo-range from receiver i to satelPs = ρ + I + Bias (4) j lite j, ρi is the topocentric distance between receiver i and satellite j, c is the speed of EMR, dt j is the satel- where P is the smoothed pseudo-range, ρ is the s j lite clock error, dti is the receiver clock error, Ii is the topocentric distance (Garcia-Fernandez, 2004), and j ionospheric delay, Ti is the tropospheric delay, dr ij is j the satellite orbit error, dm i is the multipath error, and ∼ 40.3 TECslant I = (5) j f2 ei is the pseudo-range measurement error (Gao and Liu, 2002). where TECslant is the slant total electron content in The GPS carrier phase measurement model is: TEC units (1 TECU = 1016 /m2 ), which is the accuj j j j j j j j mulated TEC value along the ray path from the GPS j Pi = ρi +c·(dt −dti )+Ni −Ii +Ti +dri +dm i +ei (2) satellite to the ground based GPS receiver, and f is the GPS signal frequency (unit: MH). j where L i is the carrier phase from receiver i to satellite Using a dual-frequency receiver and substituting j j and Ni is the integer carrier phase ambiguity. All Eq. (5) into Eq. (4), then: other terms are the same as in Eq. (1). j In Eq. (1), because Pi may contain a bias of more TECslant P1s = ρ + 40.3 + Bias1 (6) than 0.1 m, in this paper carrier phase data is used to f 12 smooth pseudo-ranges with a complementary Kalman TECslant filter (Kaplan, 1996). The algorithm is described below. P2s = ρ + 40.3 + Bias2 (7) 2 f 21 For convenience the subscripts i and j are omitted.

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where P1s , P2s are the smoothed pseudo-ranges on the After the Bias in Eq. (11) is fixed, T EC(θ, λ) is L1 carrier (frequency f 1 = 1575.42 MH) and L2 car- computed by Eq. (10). The method of solving Bias is rier (frequency f 2 = 1227.6 MH), Bias1 and Bias2 described as below. Define function F(θ, λ) to represent the ideal ionoare defined as Bias in Eq. (4) with respect to f 1 and f 2 , respectively. By differencing Eqs. (6) and (7), the spheric model, assumed to be a linear function of variTECslant can be calculated (Gaussiran et al., 2004; ables geographical latitude θ and longitude λ, or Lin, 1997; Lin and Rizos, 1996). It is assumed that ionosphere is an infinite thin layer F(θ, λ) = A0 + A1 θ + A2 λ (12) located at 350 km altitude above the Earth. When GPS signals pass through the ionosphere and reach ground where F(θ, λ) is the TEC value at latitude θ and longibased receivers, these signals and the ionosphere layer tude λ, and A0 , A1 , A2 are unknown coefficients. define points known as IPP (ionospheric pierce point). Given F(θ, λ) = TEC(θ, λ), combining Eqs. (10), Assume the IPP coordinate is latitude θ , longitude λ (11) and (12), one can obtain: and altitude 350 km. Then the TEC (vertical TEC) response to this point (θ, λ) is defined as TECslant along f 12 f 22 (P1s − P2s ) A + A θ + A λ = 0 1 2 the direction connecting the Earth’s centre and the IPP. 40.3( f 22 − f 12 ) (13) TEC is calculated as: × Obliquity + Bias TEC(θ, λ) = TECslant (θ, λ) × Obliquity

(8)

where Obliquity =



f 12 f 22 1 − a2

and a=

R cos(e) R+h

where R = 6378137 m and h = 350000 m (assuming ionosphere only located at this altitude). By differencing Eqs. (6) and (7), one can obtain: TECslant (θ, λ)

Rewriting Eq. (13) as:

f 12 f 22 = 40.3( f 22 − f 12 )

40.3( f 22 − f 12 )

(P1s − P2s ) × Obliquity

⎛ ⎞ 1 ⎜1⎟ ⎟ = (−BiasA0 A1 A2 ) ⎜ ⎝θ ⎠ λ

(14)

where there are four unknown coefficients: Bias, A0 , A1 and A2 . The term (in the left of the equation) f 12 f 22 40.3( f 22 − f 12 )

(P1s − P2s ) × Obliquity

(9) can be calculated using GPS measurements. Equation (14) can be solved by Kalman filter × (P1s − P2s ) + Bias12 (Strang and Borre, 1997). Through tuning the error where Bias12 is equal to (Bias1 − Bias2 ). Substituting covariance matrix p, the process noise covariance matrix q, and the measurement noise covariance Eq. (8) into (9): matrix r , it makes four unknown coefficients Bias, A0 , A1 and A2 reach near constant values, which are the f 12 f 22 TEC(θ, λ) = (P1s − P2s ) results we need. Finally, the TEC is corrected by this 40.3( f 22 − f 12 ) solved Bias with Eq. (10). × Obliquity + Bias12 × Obliquity The ionosphere early in the morning (local time (10) around 2:00 am) is at its most stable, and thus the bias at that time, which is obtained by Eq. (13), is more acLet curate than the values at other times during the day. Therefore we can use the early morning bias (local time (11) 2:00–3:00) to represent the whole day bias. Bias = Bias12 × Obliquity

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It is assumed that the above calculated bias is a con- 3 Generating Australian TEC Maps stant only related to one GPS station, one GPS satellite, in a small local area and in a short period of observa- The data processing procedure can be summarized in tion (i.e. less than one hour). Therefore, a few of biases the following flowchart (Fig. 1). need to be calculated for generating the Australian region TEC maps. Read pseudo- range Smooth P1, P2 P1 , P2 and carrier using L1, L2 phase L1, L2 By Kalman filter

2.3 Ionospheric Electric Potential Model After the bias is computed, it adjusts the TEC value. Then the ionospheric electric potential model, which uses interplanetary magnetic field (IMF) to describe the TEC, is adopted for smoothing the TEC and generating the TEC maps. The model is given by the spheric harmonic function as below:

TEC(θ, φ) =

n max

n 

Using linear model correct bias by Kalman filter

Calculate slant TEC and vertical TEC

Ionospheric electric potential model

Generating temporal and special TEC maps

Pnm Fig. 1 The scheme for Australian TEC inversion

n=0 m=0 ∗

× (sin θ ) (anm cos(mφ) + bnm ) sin(mφ)) (15)

First, the RINEX files of GPS data were processed to extract pseudo-range and carrier phase measurements. Secondly, the pseudo-ranges were smoothed Where θ is the solar fixed longitude of the IPP; φ is the with the carrier phase measurements. Thirdly, the GPS geomagnetic latitude of the IPP; Pnm are the normalsatellite positions were obtained, and the IPP (ionoized associated Legendre functions (Ping et al., 2002; spheric pierce point) coordinates were calculated, and Weimer, 1996). then the TEC was computed. Fourthly, the TEC was corrected using a linear function model. Fifthly, solving the coefficients of the ionospheric electric potential model, and smoothing TEC, Finally, Australian 2.4 Validation Estimation region TEC temporal-spatial distribution maps were generated. The standard deviation of the TEC from the smoothed pseudo ranges is estimated as below.

σTEC =

f 12 f 22 40.3 × 1016 ( f 22 − f 12 )

 σ p21s + σ p22s

4 Experimental Results (16)

Australia has 12 permanent GPS reference stations including Ceduna, Cocos Island, Darwin, Hobart, where σ p1s and σ p2s are the standard deviations of the Karratha, Melbourne, Mount Stromlo, Tidbinbilla, smoothed pseudo range measurements related to L 1 Townsville, Alice Springs, Yaragadee and Jabiru, and 4 permanent GPS stations, Mawson, Casey, Davis and and L 2 . In each epoch, considering the σ p1s and σ p2s are Macquarie Island, in Antarctica. These stations supply roughly equal to the deviations in the error covariance continuous GPS data. In these experiments this data matrix P in Kalman filter (Eq. 3), then the σTEC was used to generate 4 ionospheric distribution maps, in each epoch can be calculated by Eq. (16) (Liu including hourly spatial distribution maps, seasonal and yearly distribution maps. et al. 2005a,b).

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4.1 Smoothing Pseudo-Ranges with Carrier Phases and Combining Ionospheric Electric Potential Model Figures 2 and 3 show the TEC differences before and after carrier phase smoothing using a Kalman filter (with a sampling interval of 30 s). The TEC data from the smoothed pseudo range measurements (Fig. 3) are further processed by an ionospheric electric potential model. Then, the smoothed TEC data are generated (Fig. 4).

4.2 Spatial Variations of the Australian One Hour Average TEC Hourly data sets from 14 GPS stations (Cocos, Darwin, Hobart, Karratha, Melbourne Mount Stromolo, Tidbinbilla, Townsville, Alice Springs, Yaragadee, Mawson, Fig. 3 TEC daily variations (with smoothed pseudo-ranges) Casey, Davis and Macquarie Island) were used. The sampling interval was 30 s. Figure 5 shows the TEC variations corresponding to different latitudes and longitudes from UT 14:00 to 14:59 on May 3, 2007. The TEC map can be also be viewed in a 3D angle, as shown in Fig. 6. In processing these data, the satellite

Fig. 4 TEC daily variations (ionospheric electric potential model)

Fig. 2 TEC daily variations (with pseudo-ranges)

elevation cut-off angle was 20◦ . Normally, the higher the elevation angle is, the more accurate the TEC value is generated. Furthermore, in Fig. 5 or 6, given the fact that there are only 12 permanent GPS stations on the

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from June to August. Hence 15th October, 2006, 15th January 2006, 20th April, 2006 and 15th July, 2006 were selected as sample points for Spring, Summer, Autumn and Winter, respectively. Figure 7 shows that the TEC in Summer is the highest, followed by Spring, Autumn and Winter. In Summer, the TEC variation from day to night is also the most obvious compared to other seasons.

4.4 TEC Temporal Variations During 2006 Fig. 5 Australian region TEC spatial variation (in plan)

One year’s GPS data (in 2006) from the Alice Springs station (with sampling interval of 30 s) were analyzed to generate TEC daily variations, as shown in Fig. 8. The result shows that the daily average TEC value is between 13 TECU and 50 TECU. On three individual days which locate in Summer and Spring, the TEC values went up to more than 45 TECU. The average TEC in Summer (335th–59th day) is the highest, Autumn (60th–151th) and Spring (244th–334th) are close, Winter (152th–243th) is the lowest. Also there are more significant TEC variations from day to day in Summer and Spring.

Fig. 6 Australia region TEC spatial variation (viewed in a 3D angle)

Australian continent, the TEC values for the points far away from these stations may have large biases. The TEC values outside the Australian continent are not accurate because no GPS data are available there.

4.3 Comparison of Daily TEC Variation from Different Seasons The Alice Springs GPS station is located at the centre of the Australian continent. Therefore this station can be considered representative of the Australian region TEC average distribution. In Australia, Spring is from September to November, Summer is from December to February, Autumn is from March to May and Winter is Fig. 7 TEC hourly variation in one day in different seasons

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These two kinds of methods and their comparison are still under investigation. In the paper, TEC is assumed to vary with a linear model. However, the TEC may vary with a quadratic or cubic function of latitude and longitude. This issue will be further investigated in a future study so as to improve the accuracy of the TEC.

References

Fig. 8 TEC daily variation in 2006

5 Conclusion In this paper, carrier phase measurements have been used to reduce the noise in the pseudo-ranges, while a Kalman filter has been used to estimate the TEC bias and ionospheric electric potential model further smoothes TEC. Based on these data processing strategies, TEC temporal-spatial variation maps in the Australian region have been generated. Nowadays, there are two kinds of methods to smooth the TEC, including (1) computing TEC by pseudo range measurements smoothed by carrier phases measurement (single frequency smoothing). (2) deriving two TEC data from pseudo range measurements and carrier phase measurements separately as a first step, and then performing the smoothing process by an algorithm. Based on the validation test from other researchers (Liu et al., 2005a,b), there is no significant difference between the results of the above two methods. The TEC data generated by Method (1) have better accuracies than by Method (2), especially when using a small number of smooth epochs. Also there exists a turning point, after this turning point, the TEC precision starts to diverge. The divergence becomes more severe accompanying with the increase of the ratio of pseudo range noise level to carrier phase one.

Coster, A.J., E.M. Gaposchkin, L.E. Thornton, 1992. Real-time ionospheric monitoring system using GPS. Navigation, 39 (2), 191–204 Gao, Y., Z.Z. Liu, 2002. Precise ionosphere modeling using regional GPS network data. Global Positioning Systems, 1 (1), 18–24. Garcia-Fernandez, M., 2004. Contribute to the 3D Ionospheric Sounding with GPS data, PhD thesis, Universitat Politecnica de Catalunya (UPC), Spain Gaussiran, T., D. Munton, B. Harris, B. Tolman, 2004. An Open Source Toolkit for GPS Processing, Total Electron Content Effects, Measurements and Modeling. International Beacon Satellite Symposium, Trieste, Italy. October 22, 2004 Hajj, G.A., E.R. Kursinski., L.J. Romans, W.I. Bertiger, S.S. Leroy 2002. A technical description of atmospheric sounding by GPS occultation. Journal of Atmospheric and SolarTerrestrial Physics, 64, 451–469 Hajj, G.A., B.D. Wilson, C. Wang, X. Pi, I.G. Rosen, 2004. Data assimilation of ground GPS total electron content into a physics-based ionospheric model by use of the Kalman filter. Radio Science, 39, RSIS05 doi:10.1029/2002RS002859. Heelis, R.A., B. William, 1999. Ionization layers observed at middle latitudes by atmosphere explorer-C. Journal of Atmospheric and Solar-Terrestial Physics, 61 (5), 407–414 Kaplan, E.D., 1996. Understanding GPS: Principles and Applications. Artech House Publishers, Boston Klobuchar, J.A., 1987. Ionospheric time-delay algorithm for single-frequency GPS users. IEEE Transactions on Aerospace and Electronic Systems, AES-23 (3), 325–331 Lin, L.S., 1997. Real-time estimation of ionospheric delay using GPS measurements. UNISURV rept. S-51, School of Geomatic Engineering, The University of New South Wales, 199pp Lin, L.S., Rizos, C., 1996. An algorithm to estimate GPS satellite and receiver L1/L2 differential delay biases and its application to regional ionosphere modelling. Geomatics Research Australasia, 65, 1–26 Liu, Z., Y. Gao, S. Skone, 2005a. A study of smoothed TEC precision inferred from GPS measurement. earth planets space, 57, 999–1007 Liu, Z., S. Skone, Y. Gao, A. Komjathy, 2005b. Ionospheric modeling using GPS data. GPS Solution, 9, 63–66 Ping, J., Y. Kono, K. Matsumoto, Y. Otsuka, A. Saito, C. Shum, K. Heki, N. Kawano, 2002. Regional ionosphere map over Japanese islands. Earth Planets Space, 54, e13–e16

758 Strang, G., K. Borre, 1997. Linear Algebra, Geodesy and GPS. Wellesley-Cambridge Press, America Weimer, D.R., 1996. a flexible IMF dependent model of highlatitude electric potentials having “space weather” applications. Geophysical Research Letter, 23, 2549–2553

G. Ouyang et al.

Use of Global and Regional Ionosphere Maps for Single-Frequency Precise Point Positioning A.Q. Le, C.C.J.M. Tiberius, H. van der Marel and N. Jakowski

Abstract This paper compares global and regional ionosphere maps in different aspects, from vertical TEC and slant TEC in the range domain to their application in single-frequency Precise Point Positioning. Under quiet ionospheric conditions, the mean of the difference in vertical TEC between the global (GIM) and regional (SWACI) map, in Europe, was found to be less than 1 TECU, and the RMS generally in the order of 1–2 TECU. Both static data and kinematic (flight) data are analysed. Although being limited in its coverage for the moment, the real-time regional map SWACI (from DLR) provides promising results also with highly kinematic data. It is shown that for the dataset under investigation the SWACI map can bring the vertical positioning accuracy to the same level as the horizontal one, at 2–3 km (95% level).

1 Introduction

Precise Point Positioning (PPP) is a positioning technique in which only a standalone receiver is used to achieve decimeter to centimeter level accuracy anywhere on the world. Major error sources in standalone positioning, including satellite orbit and clock errors, ionosphere and troposphere delays, and site displacement effects, are accounted for by using corrections from an augmentation system on the basis of a global tracking network or a priori models. Correction products can be obtained from publicly available sources, in particular the International GNSS Service (IGS). In our PPP implementation, position parameters are estimated recursively together with the ambiguities of the precise carrier phase measurements in a filter known as the phase-adjusted pseudorange algorithm. This is a purely kinematic approach and well suited for cheap Keywords GPS · Precise Point Positioning (PPP) · single-frequency receivers, with relatively fast convergence and easy to implement. Single-frequency · Ionosphere For single-frequency PPP, the ionospheric delay is the most critical source of error. Global Ionosphere Maps (GIM) from the IGS are a popular product to compensate for the ionospheric delay. With the GIMs, Vertical Total Electron Content (VTEC) values are provided with an accuracy of about 2–8 TECU every two A.Q. Le hours on a 5◦ × 2.5◦ longitude and latitude grid with a Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, The Netherlands latency of 1 day (Beutler et al., 1999). The remaining C.C.J.M. Tiberius ionospheric error mainly affects the vertical positionDelft Institute of Earth Observation and Space Systems ing component. (DEOS), Delft University of Technology, The Netherlands Since mid 2004, regional ionosphere maps have H. van der Marel been made available for Europe under the Space Delft Institute of Earth Observation and Space Systems Weather Application Center – Ionosphere (SWACI) (DEOS), Delft University of Technology, The Netherlands project. It is a joint project of the DLR Institute N. Jakowski Institute of Communications and Navigation, German of Communications and Navigation (IKN) and the Aerospace Center (DLR), Germany

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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German Remote Sensing Data Center (DFD). The SWACI ionosphere maps are produced from a GPS tracking network in Europe, using techniques similar to the global GIM. A linear combination of dualfrequency GPS observations is used to determine the slant TEC from receiver to satellite. The slant TEC data are then mapped onto the vertical axis by applying a mapping function which is based on a single layer approximation at 400 km height. Finally, to produce regional TEC maps over Europe, the measured and calibrated TEC data obtained from 27 ground stations are assimilated into an empirical TEC Model (SWACI, 2007). The amount of detail in the regional maps, with an update interval and latency of 5 min, and a grid size of 1 by 1 degree, is much larger than that provided by the GIMs. Therefore, using a more detailed regional ionosphere map could help to improve the positioning accuracy of single-frequency PPP. In this paper, the impact of global and regional ionosphere maps on single-frequency PPP performance is analysed by (post) processing data from static GPS stations in Europe as well as from a kinematic flight experiment. The comparison between different ionosphere maps is made in the error, range and (user) position domain.

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The coordinates of the receiver are estimated recursively together with the receiver clock error and the ambiguities of the precise carrier phase measurements in a filter known as the phase-adjusted pseudorange algorithm (Teunissen, 1991). The recursion, or epochwise updating, concerns the estimation of the ambiguities, which, in the absence of cycle slips, are assumed to be constant over time. It is a purely kinematic approach as it estimates new coordinate (and receiver clock error) unknowns for each observation epoch. The algorithm allows for an optimised use of the GPS carrier phase in PPP as it puts all the observations (code and phase) on one side and all the unknown parameters on the other side with the design matrix – describing the relations of observations and unknowns – in between. The phase-adjusted pseudorange algorithm can be seen as a recursive least-squares solution or a Best Linear Unbiased Estimation (BLUE) when the proper stochastic model is used (W = Q −1 y ). The algorithm as applied to single-frequency PPP is described in detail in Le and Teunissen (2006).

3 TEC Model 3.1 Global Model – GIM

2 Single-Frequency PPP Concept

Global Inosphere Maps (GIM) in the IONEX format are a regular product of the International GNSS Service Single-frequency PPP aims at using only a standalone (IGS) since 1998. The GIM’s provide the ionosphere receiver to achieve a high positioning accuracy. To Total Electron Content (TEC) on a 5 by 2.5 degrees obtain this, external information from an augmentation grid in longitude and latitude with a 2-h time resolusystem is needed to compensate for every major error tion, and daily sets of GPS satellite (and receiver) hardsource including satellite orbits and clocks, ionosphere ware differential code bias (DCB) values. Currently four IGS Ionosphere Analysis Centers and troposphere delays. Corrections for these errors can be obtained from publicly available sources, in contribute to the IGS ionosphere products (CODE, particular the IGS. Other effects, such as site displace- ESA, JPL and UPC). The computation of these TEC ment effects due to solid Earth tides, are compensated maps and DCB sets is based on the routine evaluation using state of the art models. Products and models used of GPS dual-frequency tracking data recorded by for single-frequency PPP are (a) Final IGS satellite the global IGS tracking network. The geometry-free orbits and clocks (including satellite antenna phase linear combination of one-way carrier phase and and relativistic corrections); (b) Ionospheric delays code observations is used as the basic observation. from either Final IGS GIM or real-time SWACI; (c) Instrumental biases, so-called differential code biases IGS satellite DCBs; (d) Saastamoinen troposphere (DCB), for all GPS (and GLONASS) satellites and model with Ifadis mapping functions and MOPS ground stations are estimated as constant values for input (RTCA/DO-229A, 1998); (e) Phase-adjusted each day, simultaneously with the parameters used to pseudorange algorithm (fully kinematic); (f) Solid represent the global VTEC distribution. To convert line-of-sight TEC into vertical TEC, a single-layer Earth tide corrections.

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model mapping function is used. For the computation of the ionospheric pierce points, a spherical layer at around 400 km is assumed. The algorithms and stations selection used by each individual ionosphere analysis center are different. CODE is the only analysis center currently including GLONASS data. The AC ionosphere products are routinely compared, combined and validated with TOPEX altimeter data, resulting in the final IGS global maps. The final IGS global maps, computed with a latency of about 11 days and with a resolution of 2 h − 5◦ − 2.5◦ in UT − Longitude − Latitude, are stored in IONEX format and available from several IGS data centers, together with the individual AC products. A comparison of the individual GIM’s with the final IGS GIM is given in Fig. 1 for the period under consideration in this paper (11–17 October 2006) for a location in The Netherlands. The RMS differences between the individual products are below 0.5 TECU, but the biases can be as large as one TECU. In December 2003 the IGS started with the generation of a rapid IGS ionosphere product (with a latency of less than 24 h). The corresponding rapid combined global TEC maps have the same resolution as the final ones. The same four analysis centers contributing to the final maps contribute to the rapid maps as well. A comparison of the rapid maps with the IGS final maps is shown in Fig. 1. The performance of the rapid maps is just slightly worse (5–10%) than the final maps (more

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details can be found in Hern´andez-Pajares (2004)). The RMS difference between the rapid and final IGS map is only 0.11 TECU. It is worthwhile to note that the CODE analysis center is also providing predicted ionosphere maps. For more information on the prediction process see Schaer (1999). In order to use the GIM for single frequency PPP, firstly, the VTEC at the ionospheric pierce point has to be computed by (typically bi-linear) interpolation from the surrounding grid points. Secondly, the VTEC value needs to be interpolated in time. Interpolation in time can be done by linear or cubic interpolation in universal time, but also the maps themselves can be rotated and interpolated in an hour-angle system. Thirdly, Slant TEC (STEC) is computed from VTEC using a mapping function. Typically, each of these steps, is a contributor to the error budget of the PPP application. To get an idea of the quality of the maps and the used interpolation method one could look at the day boundaries in Fig. 1. The daily IONEX files provide 13 maps, starting at 0 h UT and ending at 24 h UT. Clearly, some analysis centers, but not all, provide continuous data at the day boundaries for their products. For those that do not provide continuity, the day jumps in the rapid products are obviously larger than the day jumps in the final products. However, all products have a discontinuity in the first time derivative at the day boundaries. This is related to the interpolation method. In Fig. 1 cubic interpolation was used, but interpolation

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measurements, useable ionospheric corrections can still be delivered to the user. To enable the operational service, the instrumental satellite-receiver biases are estimated 24 h before use. Post-processed TEC maps over the European and polar area (http://www.kn.nz.dlr.de/daily/tec-eu or . . . /tec-np) are mainly based on data from IGS. The operational TEC maps provided by SWACI are derived from 27 real-time stations of EUREF and ascos GPS networks. These stations provide 1 s sampled data in streaming mode via the Bundesamt f¨ur Kartographie und Geod¨asie (BKG) in Frankfurt. The current update rate of the maps is 5 min, but can be reduced to 1 min 3.2 (European) Regional Model – SWACI or even less. The map covers the area 10◦ W–25◦ E and 35◦ N–65◦ N at a spatial grid resolution of 1 × 1 DLR established a novel operational space weather degree spacing in latitude and longitude following monitoring service as part of the Space Weather project user needs. Due to the uneven and permanently Application Centre Ionosphere (SWACI) project. The changing distribution of ionospheric piercing points objective is to provide a permanent service, based on of the satellite links tracked by the 27 GPS stations, GNSS and space weather observations. The system the achievable accuracy at each grid point changes provides in the generation of value-added products permanently. The corresponding TEC data distribution such as TEC maps and derivative products covering is provided by SWACI together with TEC maps. the European and Polar regions, post-processing and Further improvements are expected when more GPS analysis of ionospheric/space weather information, stations are included in the data base. A novel in-house analysis of ionospheric/space weather effects, user procedure has been developed for monitoring the TEC mapping quality by estimating the so-called grid ionobenefit analysis, etc. A system for regularly processing ground-based spheric vertical delay and the grid ionospheric vertical GPS measurements from the IGS network and produc- error (Klaehn et al., 2003). The typical value for errors ing TEC maps over the European region (20◦ W–40◦ E; around the centre area is about 1 TECU or less (RMS 32.5◦ N–70◦ N) has been operating at DLR since value based on internal empirical validation). 1995 (Jakowski, 1996). The GPS data allow the determination of slant TEC values along numerous satellite-receiver links with 30 s time resolution. The instrumental biases are separated from the ob- 3.3 GIM-SWACI Comparison servations by assuming a second-order polynomial approximation for TEC variations over the observing The real-time SWACI maps have been compared with GPS ground station. Both the TEC and the instrumen- the final post-processed GIM. Due to the low resolutal satellite-receiver biases (modelled as constants for a tion (in both space and time) of the GIMs, the SWACI whole day) are estimated simultaneously by a Kalman maps were first resampled to the 5◦ ×2.5◦ grid at the 2filtering process over 24 h. The calibrated slant TEC h interval, so that vertical TEC values from both maps data are then mapped onto the vertical by applying can be compared. Figure 2 shows the mean and RMS a mapping function which is based on a single layer values of the differences between the GIM and SWACI approximation at 400 km. To provide a value for each TEC values at the 5◦ × 2.5◦ grid for one week (Octogrid point and to ensure higher reliability of the maps, ber 11–17, 2006). This week was chosen for a practiparticularly in cases of sparse measurements in certain cal reason of a kinematic experiment carried out during areas, the available TEC measurements are combined the week and the availability of SWACI maps obtained with values from an empirical model (Jakowski from DLR. During the week, the ionospheric condiet al., 1998). The advantage of applying such an tions were quiet with the k p -index ranging from 0 to 4. assimilation is that even in cases of low numbers of Time-series of the GIM-SWACI comparison for a point was restricted to within a single day only (and therefore did not crossday boundaries). The topic of interpolation is not further elaborated upon in this paper. From the above analysis and Fig. 1, taking everything into account, the internal accuracy of the IGS products is probably better than 1 TECU (16 cm) for Europe for the period under consideration, but systematic biases of about the same magnitude may still exist. These errors are therefore at the lower bound compared to the quoted accuracy of 2–8 TECU at IGS (2006).

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near the centre and a point at the edge of the area are plotted in Fig. 3. The biases between the global and regional maps are within 1 TECU (approximately 16 cm on L1) when averaged over a week. The differences do not seem to be larger at the edge of the SWACI domain than those near the centre. However, the RMS of the differences is definitely larger at the Southern edge of the SWACI domain, and smaller near the centre of the domain where both maps agree better. The time-series of VTEC values in Fig. 3 again show that the global and regional maps closely match in the center of the SWACI domain, whereas bigger discrepancies (5–10 TECU) can

be observed at the edge of the domain around local noon, potentially due to poorer accuracy of the SWACI map at its boundaries. So far, the comparisons do not say much about the qualities of the individual maps. In order to get more insight in the quality of the individual maps, the maps are compared with slant ionosphere delays computed from dual-frequency GPS data. The slant ionosphere delay is computed from the geometry-free linear combination (L4) of code and carrier phase data. The code data is corrected for satellite specific differential code biases (DCB) provided by CODE (2006). The receiver differential code bias can be estimated along with the

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ionospheric delay or taken from CODE (2006) in case the station is an IGS station. For carrier phase measurements, the (float) ambiguities are also estimated. Station GRAZ (Graz, Austria [47,15]), part of the IGS network, has been chosen for the slant delay comparison as it is close to the centre of the SWACI coverage area. The satellite elevation cut-off angle is set at 15◦ . Data were collected in the same week as above, at 30-s interval. The VTEC values from GIM and SWACI were mapped on to slant delays using the ionospheric Single-Layer Model (SLM) at 400 km height. According to Schaer (1999), a layer height of 450 km is more appropriate to use with the GIM. However, the difference between 400 and 450 km is hardly visible at not-too-low elevation angles. Particularly in this case, we made an experimental comparison and with the cut-off angle of 15◦ , no significant differences can be found. A bivariate linear interpolator was used to obtain the VTEC value at a desired location (or more precisely at the ionospheric pierce point). Before interpolation, the maps were rotated as described in Schaer (1999). Slant delays were computed from L4, GIM and SWACI at 5-minute interval, meaning that the IGS maps were interpolated in time but the SWACI maps were not. Ionospheric slant delays differences from GIM and SWACI with respect to the ‘ground-truth’ (GRAZ) are shown in Fig. 4 for 24 h. Each (receiver to satellite) channel is plotted separately, where continuous lines

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Fig. 5 Slant delay errors from GIM and SWACI wrt L4 for station GRAZ at 5-min interval for all satellites in view (1 week, 15-degree cut-off elevation angle). Mean and RMS values (plotted with continuous and dotted lines, respectively) are average per elevation bin. Receiver DCB is estimated based on CODE’s satellite DCBs. Unites are in Fig. 4

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represent the GIM and dotted lines are from the SWACI. Similar behaviour of GIM and SWACI can be seen in this graph with some outliers from both of them. Figure 5, in which the mean and RMS of the difference (over the full week) are plotted against elevation angle, seems to indicate that the global map performs slightly better than the regional maps. Apparently, the better accuracy of a post-processed product just wins it in a close finish from the higher density and update rate of the real-time regional maps. At first sight, the slant delay errors from GIM and Slant delay differences wrt L4 2 SWACI in Fig. 5 seem to be much closer than they 10 1.6 are in the VTEC comparison in Fig. 2. Actually, that 8 1.2 is not true. Remember that station GRAZ is more or 6 0.8 less at the centre of the SWACI coverage area. Since 4 0.4 GPS satellites only go up to 56 latitudinal degrees, 2 0 0 all the ionospheric pierce points (at 400 km height) –2 roughly stay in the bottom half of the area of Fig. 2. –0.4 –4 In this region, VTEC differences between GIM and –0.8 –6 SWACI are relatively small, from 0 to 0.8 TECU and –1.2 –8 about 0.3 TECU on average. Therefore, the VTEC dif–1.6 –10 ferences are in line with the slant delay differences –2 0 4 16 24 8 12 20 of about 5 cm. Although the VTEC differences are Time [hours] magnified by the mapping function, using 15-degree Fig. 4 Slant delay errors from GIM and SWACI wrt L4 for sta- cut-off elevation angle keeps the scaling factor small tion GRAZ at 5-min interval for all satellites in view (24 h, 15(up to 2.3). deg cut-off elevation angle). Continuous and dotted lines repreA similar reasoning can be applied for the RMS valsent GIM and SWACI, respectively. Receiver DCB is estimated based on CODE’s satellite DCBs. Units are in meters delay on ues in general except for the lower edge of the area, L1 frequency on left axis and in TECU on right axis where relatively large RMS differences can be seen

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in Fig. 2. The large differences can also be spotted in Fig. 3 in which time-series of VTEC differences at alocation on the lower edge were plotted. As the comparison was done solely at the grid points of both GIM and SWACI (5◦ × 5◦ × 2-h) and the RMS values provided together with the VTEC in the GIM are relatively small for the European region (less than 1 TECU), it is likely that the large differences (RMS) come from the SWACI maps. One of the reasons is that in this lower edge area, SWACI does not have many ground stations to cover it really well. However, one cannot see the differences in the STEC comparison since a 15-degree cut-off elevation angle was used and hence all the ionospheric pierce points do not go lower than 4◦ below the latitude of the station (47◦ ), and therefore do not fall in the 35-degree latitude row of the area in Fig. 2.

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4.1 Static Data

After having compared the GIM and SWACI in the error domain (Sect. 3.3), we will now consider the impact of the ionosphere maps in the position domain. The same station GRAZ was used in static test with the same data set (1 week of 30-s interval data October 11–17, 2006). The standard deviations of code and phase are assumed to be 20 and 10 cm respectively for a satellite at zenith (they increase as a function of elevation angle). Please note that although static data were used, the processing approach is fully kinematic (new coordinate unknowns for each epoch). Due to the current limited geographical coverage of the SWACI map, its availability over time for positioning is not 100%. To avoid poor satellite geometry and outliers at low elevation angles, several selection criteria based on the number of satellites and cut-off elevation angle were considered. A minimum of 5 satellites with 15-degree cut-off elevation angle 4 Application in Single-Frequency PPP was chosen for the balance between availability and satellite geometry. It results in approximately The ionosphere is the most challenging error source 70% availability of time (similar to the minimum of in single-frequency PPP. The accuracy of ionosphere 6 satellites with 10-degree cut-off elevation angle maps is the last barrier which keeps single-frequency situation). Under this condition at GRAZ, the SWACI PPP away from centimetre accuracy as it can be can provide the ionospheric delay when the satellite achieved with dual-frequency PPP (Kouba, 2003). The is above 15 degrees practically all the time. Both the final product of GIM produces promising results for GIM and SWACI were processed under this condition PPP (Le and Tiberius, 2007), but it only can be used in for a fair comparison. The PPP results are summarised post processing and not in real-time or near real-time. in Table 1. The normal PPP-GIM result without the The predicted, real-time GIM from several Analysis above-mentioned constraint but with 5-degree cut-off Centre (AC) of IGS, e.g. CODE, is worse than the elevation angle is also included for reference. This final one. With SWACI, a real-time, high resolution solution has 100% availability. regional map, hopefully single-frequency PPP can be The PPP result using the GIM with exactly the same improved. In this section, the SWACI regional map in satellite geometry (as with the SWACI) is about 30% single-frequency PPP is applied and compared to the performance of the well-known IGS global map. Both Table 1 Static results of station GRAZ [m]. Mean, RMS and a static and kinematic test are presented. 95%-value of position errors in local North, East and Up coordiBoth the ionosphere maps are brought into PPP us- nates, one week of data at 30-s interval (1 day missing). GIM∗ is ing exactly the same interpolation method and mapping normal PPP-GIM solution with 5-degree cut-off elevation angle function. These are the standard steps recommended by North East Up IGS and its ACs, which were introduced by CODE. GIM∗ Mean −0.02 −0.17 0.09 First the VTEC values are interpolated to the ionoRMS 0.19 0.21 0.35 95% 0.38 0.40 0.69 spheric pierce point location using the bivariate interMean 0.20 −0.14 −0.12 polation. After that, VTEC values are mapped onto the GIM RMS 0.35 0.44 1.32 slant delays according to the SLR model at 400 km 95% 0.59 0.46 1.12 height. The map rotating method is applied to com- SWACI Mean 0.35 −0.25 −0.18 RMS 0.51 0.56 1.60 pensate for the high correlation with the Sun position 95% 0.85 0.61 1.38 (Schaer, 1999).

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A flight test was carried out at the end of May 2005 with the Cessna Citation aircraft owned by Delft University of Technology and NLR (Dutch National Aerospace Laboratory). During the fight, the K p -index 10 was between 2 and 3, which implied quiet ionospheric condition. Dual-frequency data were collected for about two hours with a NovAtel EUR04 receiver 5 (Dorne-Margolin antenna element) at 1 Hz sampling rate. The reference trajectories were computed separately with commercial software, Trimble Geomatics 0 0 5 10 15 20 25 Office (version #1.62), through (dual-frequency carrier phase, ambiguity fixed) differential positioning. A Time of day [hour] station in Delft equipped with a Septentrio PolaRx2 Fig. 6 Formal standard deviations from PPP solution with geometry constraint for 1 day (similar for other days). The poor receiver and a (choke ring) antenna Leica AT504 was used as the reference station. The station (antenna geometry epochs can be seen around 4:00 and 16:00 phase centre) coordinates are known, and the station is only a few kilometres away from the starting point of the flight trajectory. The air-craft was even below better than that using SWACI. The biases with SWACI, 1000 m when flying around the reference station. Then although slightly larger than those with the GIM, stay the aircraft climbed to 3000 m and continued to the within 2–4 dm. The results with the geometry conNorthern part of the Netherlands. The speed of the straint are poor (especially vertically) due to small aircraft was generally between 100–150 m/s. The flight number of satellites (and bad geometry) at a number produced about 1 h of kinematic data. of epochs, see Fig. 6. It does not affect much the other The single-frequency PPP results using GIM and numbers though. SWACI ionosphere maps are shown in Fig. 7. StatisEven though SWACI is out performed by the final tics of the results is summarised in Table 2. GIM when applied to PPP, the fact that the SWACI map It can be seen from the graphs as well as from is produced in real-time and it is originally not meant the Table 2 that horizontal accuracies are similar in for PPP applications makes it valuable asset. For PPP, both cases, approximately 2 dm (at the 95% level). A the domain should be, however, enlarged. Formal STD [m]

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North(95%): 0.18 East (95%): 0.21 Up (95%): 0.18

1.5 1 0.5 0 –0.5 –1 –1.5

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Fig. 7 Kinematic flight results using IGS GIM (left) and SWACI (right) maps [m]. 95%-value of position errors in local North, East and Up coordinates, 1 h of data at 1-s interval

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Table 2 Kinematic flight results [m]. Mean, RMS and 95%value of position errors in local North, East and Up coordinates, 1 h of data at 1-s interval GIM

SWACI

Mean RMS 95% Mean RMS 95%

North

East

Up

0.14 0.07 0.24 0.02 0.10 0.18

−0.02 0.06 0.11 0.13 0.05 0.21

−0.34 0.08 0.51 −0.01 0.09 0.18

0.4

0.8

0.3

0.7

0.2

0.6

0.1

0.5

STD [m]

Ambiguities [m]

significant improvement is achieved using the SWACI map pulling the vertical error to the same level as the horizontal ones. This is probably due to smaller biases in SWACI in comparison to the GIM. It is not really contradictory to the static results knowing that 9 satellites were available within the SWACI coverage during the kinematic test and hence it kept the main problem of SWACI – availability – away. The SWACI shows its advantages of high resolution and high rate in this kinematic test. If the problem of coverage can be solved in the future, SWACI would be a great asset for real-time PPP. The single-frequency PPP features a recursive estimation of the carrier phase ambiguities. The kinematic test provides a good opportunity to look at the convergence of the filter. Both the solutions give equally fast convergence within 1–2 min. Actually, the convergence time does not depend much on the ionosphere maps but more on the (pseudorange) measurement noise. As an example for SWACI, Fig. 8 shows the estimated

ambiguities and their formal standard deviations for all satellites in view (the mean over the full hour period of each ambiguity was left out for better visualisation). It is to be noted that the aircraft is always on the move. There are no cycle slips, hence initialisation takes place only at the beginning. It can be seen that the standard deviations reach 10 cm after about 1 min and 5 cm after 2–3 min. The variations of the estimated ambiguities confirm this statement. This is the reason why single-frequency PPP performs superior to standard dual-frequency PPP using the ionosphere-free linear combination on the aspect of convergence time as the noise in the dual-frequency PPP is about 3 times amplified (Keshin et al., 2006). Static data collected at a permanent station nearby (in Delft, Trimble 4700 receiver with choke ring antenna) were also processed for the same hour during the flight. Results using the GIM and SWACI maps are shown in Fig. 9. A similar improvement by going from the GIM to the SWACI can be seen in the vertical component. Although the static result using SWACI is a bit poorer in the North component, its behaviour is quite similar to that of the flight result. The reason for better kinematic result is that during the flight, satellite availability (under SWACI coverage) was very good with 9 satellites. For the static station in Delft, due to its location (near the edge of the SWACI coverage area), the number of satellites under SWACI coverage went down from 9 to 7. Moreover, the large sampling interval (30 s) of the static data resulted in slower convergence.

0 –0.1

0.4 0.3

–0.2

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0.1

–0.4

11.15 11.2 11.25 11.3 11.35 11.4 11.45 Time of day [hours]

0

11.15 11.2 11.25 11.3 11.35 11.4 11.45 Time of day [hours]

Fig. 8 Estimated ambiguities and their formal standard deviations [m] from the 1-s interval flight data with SWACI maps (similar results for GIM). The first epoch of measurements is at 11.12 h

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North(95%): 0.17 East (95%): 0.14 Up (95%): 0.57

1.5 1 0.5 0

–0.5 –1

–1.5 –2

2

North(95%): 0.35 East (95%): 0.26 Up (95%): 0.31

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Fig. 9 Static results during the same period as the flight for a permanent station nearby (in Delft) using IGS GIM (left) and SWACI (right) maps [m]. 95%-value of position errors in local North, East and Up coordinates, 1 h of data at 30-s interval

5 Conclusions and Recommendations

over different seasons, different times of day and different conditions of the ionosphere. This cannot be done About half a metre position accuracy has been demon- in the paper due to the limited amount of data provided strated using the IGS Global Ionosphere Maps (at the so far by DLR. In this paper, the standard procedure for the iono95% level). In one case (under quiet ionospheric conspheric maps was used, including linear interpolation ditions), the SWACI maps showed an improvement to and single-layer height mapping function. Application 2–3 dm in the vertical component, bringing the vertiof more sophisticated interpolation methods and a betcal accuracy to the same level as the horizontal one ter mapping function might help to improve the perfor(2–3 dm – 95%). Moreover, the very small latency of mance with the regional maps since it is available with the regional maps allows application in real-time kinehigher resolution, both in time and in space. matic positioning. A regional ionospheric map can be a useful product for the future of real-time single-frequency PPP. Although the current position accuracy is limited mainly due to the poor geographical coverage of the SWACI maps, a regional map has a number of advantanges over global ionosphere maps. The SWACI maps provide a much higher temporal and spatial resolution. And the maps are available in real-time. The SWACI coverage is already planned to be extended on the long term, not in the near future. At the current level of accuracy (a few dm), the remaining ionospheric error is still one of the largest errors in comparison to other error sources and it is comparable to the pseudorange code noise. However, single-frequency data with either maps out perform the ionosphere-free linear combination without ionospheric maps in terms of its fast convergence, while being slightly less accurate in terms of position accuracy. A longer time span of data should be used to further analyse the regional ionospheric maps and compare

Acknowledgments Our colleague Yahya Memarzadeh is thanked for providing scripts and help to generate the slant delays ‘ground truth’. The Control and Simulation division of our faculty provided the data of the flight test.

References Beutler, G., Rothacher, M., Schaer, S., Springer, T. A., Kouba, J., and Neilan, R. E. (1999). The International GPS Service (IGS): An interdisciplinary service in support of Earth sciences. Advances in Space Research, 23(4): 631–635. CODE (2006). Center for Orbit Determination in Europe, Astronomy Institute (AIUB), University of Berne. http://www.aiub.unibe.ch/ionosphere.html. Hern´andez-Pajares, M. (2004). IGS Ionosphere WG Status Report: Performance of IGS Ionosphere TEC Maps. In Technical IGS Meeting, Bern. IGS (2006). International GPS Service. http://igscb. jpl.nasa.gov. Jakowski, N. (1996). TEC Monitoring by Using Satellite Positioning Systems, pages 371–390. European Geophysical

Use of Global and Regional Ionosphere Maps Society. In: Modern Ionosphere Science (Eds H. Kohl, R. Rster and K. Schlegel). Jakowski, N., Sardon, E., and Schlueter, S. (1998). GPS-based TEC observations in comparison with IRI95 and the European TEC model NTCM2. Advances in Space Research, 22(6): 803–806. Keshin, M. O., Le, A. Q., and van der Marel, H. (2006). Single and dual-frequency Precise Point Positioning: Approaches and performances. In Proceedings of the 3rd ESA Workshop on Satellite Navigation User Equipment Technologies, NAVITEC 2006, December 11–13, Noordwijk, The Netherlands. Klaehn, D., Schlueter, S., and Jungstand, A. (2003). On the implementation and performance of the ESTB ionospheric corrections processor. In European Navigation Conference, April 22–25, Graz, Austria. Kouba, J. (2003). A Guide to Using IGS Products. IGS Publication, Ottawa, ON. Le, A. Q. and Teunissen, P. J. G. (2006). Recursive least-squares filtering of pseudorange measurements. In European Navigation Conference, May 7–10, Manchester, U.K.

769 Le, A. Q. and Tiberius, C. C. J. M. (2007). Single-frequency Precise Point Positioning with optimal filtering. GPS Solutions, 11(1): 61–69. RTCA/DO-229A (1998). Minimum Operational Performance Standards for Global Positioning System/Wide Area Augmentation System Airborne Equipment. Number RTCA/DO229A. Schaer, S. (1999). Mapping and predicting the Earth’s ionosphere using the Global Positioning System. PhD thesis, University of Berne. SWACI (2007). Space Weather Application Center – Ionosphere. http://w3swaci.dlr.de. Teunissen, P. J. G. (1991). The GPS phase-adjusted pseudorange. In Proceedings of the 2nd International Workshop on High Precision Navigation, pages 115–125, Stuttgart/Freudenstadt, Germany.

Quality Control for Building Industry by Means of a New Optical 3D Measurement and Analysis System A. Reiterer, M. Lehmann, J. Fabiankowitsch and H. Kahmen

Abstract In the past, high-precision online 3D measuring required artificial targets defining the points on the objects to be monitored. For many tasks such as monitoring of displacements of buildings, artificial targets are not desired. Today’s image-assisted measurement systems can perform their measurements even without targeting. Such systems use the texture on the surface of the object to find “interesting points” which can replace the artificial targets. However, well-trained “measurement experts” are required to operate such a measurement system. In order to make such systems easy to use even for non-experts, it can be extended by an appropriate decision system which supports the operator. At the Institute of Geodesy and Geophysics of Vienna University of Technology a new kind of image-based measurement system is being developed (research project “Multi-Sensor Deformation Measurement System Supported by Knowledge-Based and Cognitive Vision Techniques”). This system can be used for measuring, analysing and interpreting deformations for the task of quality A. Reiterer Researchgroup Engineering Geodesy, Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstr. 27-29/E128-3, A-1040 Vienna, Austria M. Lehmann Researchgroup Engineering Geodesy, Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstr. 27-29/E128-3, A-1040 Vienna, Austria J. Fabiankowitsch Researchgroup Engineering Geodesy, Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstr. 27-29/E128-3, A-1040 Vienna, Austria H. Kahmen Researchgroup Engineering Geodesy, Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstr. 27-29/E128-3, A-1040 Vienna, Austria

control in civil engineering. The work-flow is based on new techniques originally developed in the field of Artificial Intelligence. Keywords Deformation classification · Image-based measurement system · Automated point detection · Point cloud

1 Introduction In civil engineering, highly accurate 3D representations and/or monitoring of objects are required. Deformation measurement enables the early detection of damage, failure or an injury to the safe operation of an object in order to be able to react appropriately and in time. Several factors, like changes of ground water level, tidal phenomena or tectonic effects, can be the reason for deformation. For monitoring of an object subjected to a deformation process the object and its surrounding has to be modelled, which means dissecting the continuum by discrete points in such a way that the points characterise the object, and that the movements of the points represent the movements and distortions of the object as a whole. Modelling the deformation of an object means to observe the characteristic points at certain time intervals by means of a suitable measurement system in order to properly monitor the temporal course of the movements. A classical type of deformation network is shown in Fig. 1. A great variety of optical 3D measurement techniques, such as laser scanners, photogrammetric systems, or image-based measurement systems, are available for deformation monitoring.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Fig. 1 Classical type of a deformation-network – taken from (Pelzer, 1985)

An example of such a monitoring task is the stability control during the whole construction process of engineering structures such as bridges, high-rise buildings, dams, etc. In comparison with laser scanners, imagebased systems measure objects with higher accuracy; and compared with photogrammetric systems they can be easier used for on-line measurement processes. This will especially be the case if the measurements can be performed with a high degree of automation. Recently, research interest in the area of imagebased measurement systems has increased. Leica Geosystems developed a prototype of an “imageassisted total station” (IATS) in order to define a hybrid or semi-automatic way to combine the strength of the traditional user-driven surveying mode with the benefits of modern data processing (Walser, 2003; Walser and Braunecker, 2003). Furthermore, Topcon introduced a prototypical tacheometer which provides focused colour images (TOPCON, 2007). At the Technische Universit¨at M¨unchen, an image-based measurement system for object recognition was developed (Wasmeier, 2003). An important step for the development of an automated image-based measurement system is the integration of so-called feedback techniques (e.g. precise line scanning (Buchmann, 1996), corner detection (Scherer, 2003), etc.). Such techniques are summarised under the concept of Intelligent Tacheometry or Intelligent Scanning (Scherer, 2004). The central topic of all image-based measurement systems is the calculation of 3D object coordinates from 2D image coordinates for subsequent processing

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steps such as deformation analysis or object reconstruction (details about the transformation of image coordinates into the object space can be found in Reiterer 2003). The Institute of Geodesy and Geophysics of Vienna University of Technology has a great deal of experience in the field of development of total stations, imageassisted theodolites and IATS. Fabiankowitsch (1990) was the first researcher who experimented with a new measurement system on the basis of the Leica TM3000 video-theodolite. His work resulted in special measurement methods (filtering techniques) for active targets. In 1994, the research project “Stereovideometry and Spatial Object Recognition” commenced. One of the milestones of this project was the work done by Roic (1996). The aim of his work was to prepare images of unsignalised targets by using image processing methods, in order to make interactive and automatic spatial surface measurement possible. A few years later Mischke (1998) developed a powerful measurement system based on two video-theodolites. The system was able to measure active or passive targets and non-signalised points, such as the intersections of edges or lines. Mischke has implemented the “F¨orstner Interest Operator” to select all remarkable but non-signalised points. Interest operators (IOPs) were well-known for off-line applications in photogrammetry, but had not been used for videotheodolites before. From 2001 to 2005 the research project “Theodolite-based and Knowledge-based Multi-Sensor-System” was undertaken, with the main goal of developing a knowledge-based measurement system for 3D-sensing of static or moving scenes by computer-controlled rotating cameras and 3D information extraction (Reiterer, 2004). In a current research project (“Multi-Sensor Deformation Measurement System Supported by Knowledge-Based and Cognitive Vision Techniques”) a new kind of image-based measurement system will be developed. This system is based on new techniques (originally developed in the area of Artificial Intelligence) which shall be used for the task of quality control (deformation measurement, analysis and interpretation). The authors report on the state-of-the-art of such a new measurement system, its functionality and development stage.

Quality Control for Building Industry

2 Concept for a New Kind of Measurement System An IATS measurement system is a combination of different components: image sensors (e.g. IATS or videotheodolites), a computer system, software (e.g. control system, image processing, etc.), and accessories. IATS and video-theodolites have a CCD camera in their optical path. The images of the telescope’s visual field are projected onto the camera’s CCD chip. The camera is capable of capturing mosaic panoramic images through camera rotation, if the axes of the measurement system are driven by computer controlled motors. With appropriate calibration these images are accurately georeferenced and oriented as the horizontal and vertical angles of rotation are continuously measured and fed into the computer. An optical device for such a system (see Fig. 2) was developed by Leica Geosystems (Walser, 2003). It is reduced to a two lens system consisting of a front and a focus lens. Instead of an eyepiece a CCD sensor is placed in the intermediate focus plane of the objective lens. The image data from the CCD sensor are fed into a computer using a synchronised frame grabber. For the transformation of the measured image points into the object space the camera constant must be known (in an optical system with a focus lens the camera constant, however, changes with the distance of the object).

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The camera constant can be derived from the focal length. This can be performed automatically if an encoder measures the focus lens position relative to an origin, which is chosen when focusing to infinity. Now the optical mapping model includes not only the measurement system axis errors and the vertical index error, but also errors resulting from a displacement of the projection centre from the intersection of the measurement system axes and from the optical distortions for field points. Consequently calibration of an imagebased measurement system has to account for all these errors. The new system is based on a combination of the IATS prototype developed by Leica Geosystems and a terrestrial Laser Scanner (TLS). The data of the different sensors have to be merged by a special data fusion process, which will be designed and implemented in the next months (an adoption of the development done by Eisenbeiss et al. (2005), should be possible). Such a system provides an immense number of 3D data, both from the IATS system and from the laser scanner. This point cloud may be reduced by filtering, even if not very effectively. The authors’ approach builds on cognitive vision techniques. These methods can be used for finding regions of interest as well as for point filtering. The development of a combined system of this kind will be a great challenge in the future. The basic concept of the proposed measurement system/procedure can be formulated as follows:

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Fig. 2 Cross section of the telescope developed by Leica Geosystems (Walser, 2003)

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IATS and TLS are used in a common way for data capture – probably the TLS will be used to produce an overview of the scene. The data of the different sensors have to be merged by a suitable data fusion process (as mentioned above). The captured data (images and laser scanner data) are used by a data interpretation tool (which is working on the basis of image understanding and/or cognitive vision techniques) to produce a description of the scene/object. This can be done by recognising different objects and assigning them to proper categories together with information about the object and relevant parameters. This process results in a special kind of “object information system”. The information that is obtained is used as input for a decision system, to produce a list of actions (e.g.,

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ranking of suitable point detection algorithms or the context between different object parts and suitable sensors). Furthermore, on the basis of the generated scene description, regions of interest (ROI) are selected. One ROI is the smallest measurement area in the current framework. For provision of a suitable data base the order of measurements can be predefined either as individual measurements, as repetitive measurements (each hour at same minute, each day at same hour, and each month at same day, etc.) or can be created automatically. On the basis of the captured measurements a new kind of deformation analysis will be performed – this process will be referred to as deformation assessment. This step results in an appropriate report.

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of the captured image or on values which represent this image (image analysis).

2.1 Image Analysis and Decision Making

Using the whole image for decision-making is not suited for an on-line measurement system (because of the heavy processing time). For this reason appropriate values are used as input for the decision process. The process of extracting these values from the image is r called image analysis and is based on two categories of features: histogram feature extraction and Haralick feature extraction. Histogram Features: The histogram of an image is a plot of the grey-level values versus the number of pixels at that value. It can be utilised to generate a class of In the following the focus is on the image-based part image features (histogram features). The shape of an of the system. image histogram provides a large amount of informaThe process of image-based measurement consists tion about the character of the image; e.g. a narrowly of several steps, including image capture, image predistributed histogram indicates a low-contrast image, processing, point detection, calculation of 3D point cowhile a bimodal histogram suggests regions of differordinates, etc. A simplified overview of this process is ent brightness (Pratt, 1978). shown in Fig. 3 – and details about the measurement Based on the histogram, several features can be forsequence can be found in Reiterer (2004). All these mulated. The authors use mean (M1 ), variance (M2 ) steps have been automated by means of a suitable deand skewness (M3 ) to describe the shape of the image cision system – decisions have to be made on the basis histogram and for subsequent decision-making. Mean (M1 ) is correlated to the brightness of the image; variance (M2 ) is a measure of the average distance between each of a set of grey-levels and their mean value and is therefore correlated to the contrast of the image; skewness (M3 ) is a measure of the symmetry of distribution of grey-levels about their mean and gives information about the balance of bright and dark areas in the image. Details about the used histogram features can be found in Pratt (1978). Haralick Features: Haralick and Shapiro (1993) proposed 13 measures of textural features which are derived from the co-occurrence matrices, a well-known statistical technique for texture feature extraction. Texture is one of the most important defining characteristics of an image. The grey-level co-occurrence matrix is the two dimensional matrix of joint probabilities p(i, j) between pairs of pixels, separated by a distance d in a given direction r . It is popular in texture description and builds on the repeated occurrence of some grey-level configuration in the texture (the Fig. 3 Simplified overview of the developed process co-occurrence matrices and consequently the Haralick

Quality Control for Building Industry

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features are generated for a distance d = 1 and given directions r = 0◦ , 45◦ , 90◦ , 135◦ ). The authors have used the following Haralick features: the angular second moment (H1 ), the contrast (H2 ), the inverse difference moment (H3 ) and the entropy (H9 ). The angular second moment (H1 ) is a measure of the homogeneity of an image. Hence it is a suitable measure for the detection of disorders in textures. For homogeneous textures, the value of angular second moment turns out to be small compared to non-homogeneous ones. The angular second moment is defined as: 

H1 =

i

p(i, j)2 .

(1)

j

The contrast (H2 ) is a measure of the amount of local variations present in an image. This information is specified by the matrix of relative frequencies p(i, j) with which two neighbouring pixels occur on the image, one with grey value i and the other with grey value j. The contrast is defined as: N g −1

H2 =



n2

n=0

Ng Ng  

p(i, j); n = |i − j|.

(2)

i=1 j=1

The inverse difference moment (H5 ) measures image homogeneity too. It achieves its largest value when most of the occurrences in co-occurrence matrices are concentrated near the main diagonal. The inverse difference moment is inversely proportional to the contrast of the image. The inverse difference moment is defined as: H5 =

 i

j

1 p(i, j). 1 + (i − j)2

(3)

A description of all the 13 Haralick features can be found in Haralick and Shapiro (1993). To make the extracted numerical image features more suitable for the decision system the authors use a special fuzzification/abstraction procedure. This procedure translates the input values (image features) into linguistic concepts, which are represented by abstraction (“fuzzy”) sets. This technique is not a fuzzification in terms of definition. Only non-overlapping spring membership functions are used. The use of such an abstraction procedure makes it possible to write a decision system (rules) in terms of easily-understood word descriptors, rather than in terms of numerical values. To provide automated support, a knowledge-based (rule-based) approach has been chosen for representing the knowledge necessary for decision-making, allowing for a declarative and modular representation of a decision policy together with easy extendibility. The structure of a rule-based approach is very similar to the way people solve problems. Human experts find it convenient to express their knowledge in the form of rules (situation-action pairs). Further rules are a way to represent knowledge without complex programming constructs. Such a system consists of two components, a working memory (WM) and a set of rules (Stefik, 1998). The collected values (numerical and fuzzy values) form the working memory which is representing the input for the decision systems. The WM is a collection of working memory elements, which themselves are instantiations of a working memory type (WMT). WMTs can be considered as record declarations in PASCAL or struct declarations in C. As mentioned already, the second part of such a system is represented by the rules. Consider the following example rule. (defrule brightening (Stat Moment (M1 f very low|low)) (Stat Moment (M3 f middle positive| high positive|very high positive)) → (assert (state (brightening yes)))

The entropy (H9 ) gives a measure of complexity of the image and is related to the information-carrying capacity of the image. Entropy is maximised when the probability of each entry is the same, thus a high value for entropy means that the grey-level changes between A rule is divided into two parts, namely the leftpixels are evenly distributed, and the image has a high hand side (LHS) and the right-hand side (RHS) with degree of visual texture. The entropy is defined as: “)” separating both parts. In the LHS, one formulates the preconditions of the rule, whereas in the RHS,  H9 = p(i, j) log { p(i, j)} . (4) the actions are formulated. A rule can be applied (or fired), if all its preconditions are satisfied; the actions i j

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specified in the RHS are then executed. In his example one is checking whether there is a WME of type “Stat Moment” where the contents of M1 f-slot is “very low” or “low” and the one of slot M3 f equals “middle positiv”, “high positiv” or “very high positiv”. There are algorithms for the matching phase, i.e. the phase where all rules are checked against all working memory elements, which are efficient in practice. The result of the matching phase is the conflict set, which includes all rule instances “ready to be fired”. A conflict resolution strategy selects one rule instance which is actually fired.

2.2 Image Preprocessing and Point Detection Image preprocessing methods transform the image to a form better suited for the detection of interest points – this may not be the best form for human analysis. In this case these methods can be used to improve the interpretability (reduction of image information) of an image. The following image preprocessing methods have been implemented: histogram equalisation, greylevel scaling (image brightening/darkening), median filtering, gauss filtering, edge detection and thresholding. The implemented algorithms are very simple and widely known in the field of image processing; details about them can be found in Pratt (1978). The goal of an automated image preprocessing sequence is to choose a single algorithm or a combination of algorithms (including the necessary parameters) in order to improve the image for the subsequent application of interest operators. At critical processing steps (e.g., edge detection, median filtering) the user has the possibility to over-rule the system decision. The knowledge base (the part for the knowledgebased image processing and image enhancement system) is divided into three components, which are rules for the choice of suitable algorithms for image preprocessing, rules for the predefinition of necessary parameters, and rules to define the order of the algorithms. Up to now only a small number of image preprocessing algorithms have been implemented. Therefore the knowledge base could be kept propositionally simple and thus easily modifiable and extensible. Figures 4 and 5 show example of images before and after preprocessing and point detection, respectively.

Fig. 4 Raw image captured by the measurement system (underexposed and noisy)

Fig. 5 Image after are processing (grey-level scaling and median filtering) and point detection

After having improved the visual appearance of an image by image preprocessing, point finding in the image can follow. For point detection in the images the authors use interest operators. They highlight points which can be found easily by using correlation methods. There are a huge number of interest operators, e.g. the F¨orstner operator (F¨orstner, 1995), the Harris operator (Harris and Stephens, 1988), the Moravec operator (Moravec, 1977), or the HFVM operator (Paar et al., 2001); however no interest operator is suitable for all types of desired point detection. For this reason the authors have implemented different interest operators in the system. The algorithms implemented in the system are so-called intensity based methods. These methods go back to the development done by Moravec (1977). His

Quality Control for Building Industry

detector is based on the auto-correlation function of the signal. It measures the grey value differences between a window and a window shifted in the four directions parallel to the row and columns. An interest point is detected if the minimum of these four directions is superior to a threshold (F¨orstner, 1995). Today there are different improvements and derivatives of the Moravec operator. Among the most well-known are the F¨orstner operator (F¨orstner, 1995) and the Harris operator (Harris and Stephens, 1988), which represent two methods implemented in the system. Additionally, the authors have integrated the Hierarchical Feature Vector Matching (HFVM) operator (Paar et al., 2001), a development of the Joanneum Research in Graz (Austria). A detail description of all algorithms can be found in Reiterer (2004). The choice of a suitable algorithm, the order of application and the necessary parameters is made by the knowledge-based system – the decisions are based on image features too. In summary, image preprocessing and point detection algorithms are selected on the basis of extracted image features (histogram features and Haralick moments). The selection and combination of suitable algorithms is done in an automatic way, by a knowledge-based decision system. The knowledge which was required to be included was obtained in different ways: from technical literature, e.g. Pratt (1978), previous projects (Mischke, 1998; Reiterer, 2004; Roic, 1996), and from experiments. This knowledge was converted into “If-Then-Statements” (rules). After having detected points of interest in more than one time epoch, deformation assessment can be done.

3 Deformation Assessment for Quality Control The main goal of the developed deformation assessment process is the classification of relevant deformations in an unstructured point cloud. This process can be divided into several steps:

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First of all the surface of the object is subdivided into regions which might have deformed in the same way (this step will be processed by means of a newly developed knowledge-based image

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understanding tool – more details about this part will follow in separate publications). After having measured the points in different time epochs, a classical geodetic deformation analysis is used to find significant motions (this part is based on GOCA – a system for the detection of deformations) (GOCA, 2007; Pelzer, 1985). As a next step a process which sets up a description of the movements and distortions of the object can be executed. Currently, the developed method analyses only translations along the coordinate axes and rotations around these axes – an extension to other movements/distortions is planned for the near future. The motion of a point (x0 , y0 , z 0 → x, y, z) can be described by a similarity transformation (Dunn and Parberry, 2002). To calculate the necessary parameters (rotation around the axes: α, β, γ ; shifting along the axis: tx , t y , tz ) including their standard deviation a Gauss-Helmert equalisation is used. For the calculation a minimum of three points is necessary (the distribution of these points in the considered region is (nearly) irrelevant – see Lehmann and Reiterer, 2007). The last step is the assessment of the deformation by means of a suitable “pattern recognition sys-tem”. This process must be outlined in a framework of local-to-global information integration, by grouping locally measured deformation into a more informative deformation pattern. The identification of prototypical discriminative deformation patterns will enable the derivation of a codebook of deformation characteristics to achieve a vocabulary of prototypical patterns for future reference. Early identification of critical prototypical deformation patterns may initiate focus of attention thereon, requiring a more precise and possibly time consuming analytical process that might even involve human intervention. The calculated parameters only describe the deformation of the individual region – to make a conclusion about the deformation of the whole object, it is necessary to group these regions by their specific parameters. The parameters are fuzzified (translating the numerical values into linguistic concepts) and on the basis of these values cases are formulated (case representation) – one case represents one possible deformation scenario. By means of well known prototypical “deformation cases” a special kind of codebook of deformation characteristics can be implemented. The matching between

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A. Reiterer et al. Acknowledgments The research presented in this paper is supported by the Austrian Science Foundation (FWF) Project P18286.

References

Fig. 6 Case-based reasoning cycle (Aamodt and Plaza, 1994)

cases in the database (codebook) and a new (unknown) one will be processed by means of casebased reasoning (CBR) (see Fig. 6) (Aamodt and Plaza, 1994). The result of this process is a description of the identified deformation, consisting of the matched deformation case(s) and their appropriate fuzzy values, e.g. rotation left – 90%/translation right – 10%.

4 Conclusions In this paper, the concept of an optical 3D measurement and analysis system for quality control has been described. The main task of this development has been the automation of different decision-making processes in the course of measurement and analysis. The task of measurement (image capture, image preprocessing and point detection) is automated by a knowledge-based decision system, which is working with the help of image feature extraction. The process of deformation is based on a combination of conventional deformation analysis and case-based reasoning techniques. For an on-line system a fast execution of all algorithms and processes is necessary. Such a system presents a basic approach for an automated on-line image-based measurement and analysis system which can be used for quality control in civil engineering. The degree of automation can be very high, whereas by decision-making, human interaction remains an important part of the workflow even though the amount of decisions done by the user can be reduced considerably.

Aamodt, A. and Plaza, E. (1994). Case-based reasoning: Foundational issues, methodological variations, and system approaches. In: AICOM 7, pp. 39–59. Buchmann, P. (1996). Entwicklung eines Messsystems zur automatischen polaren Objekterfassung am Beispiel der Bauaufnahme (DGK). Dunn, F. and Parberry, I. (2002). 3D Math Primer for Graphics and Game Development. Wordware Publishing Inc. Eisenbeiss, H., Sauerbier, M., Zhang, L. and Sotoodeh, S. (2005). Kombinierte 3d-Vermessung mit terrestrischem Laserscanning und Modellhelikopter-Photogrammetrie in Palpa, (Peru). In: Internal Technical Report at IGP – ETH Zurich. Zurich. Fabiankowitsch, J. (1990). Automatische Richtungsmessung mit digitalen Differenzbildern. PhD thesis, Vienna University of Technology. F¨orstner, W. (1995). Matching strategies for point transfer In: Proceedings of the Photogrammetric Week 95, WichmannVerlag, pp. 173–183. GOCA (2007). http://www.goca.info/. Haralick, R.M. and Shapiro, L.G. (1993). Computer and Robot Vision. Addison-Wesley. Harris, C. and Stephens, M. (1988). A combined corner and edge detector. In: Proceedings of the 4th ALVEY Vision Conference, pp. 147–151. Lehmann, M. and Reiterer, A. (2007). Online Charakterisierung von Deformationen in 3Dpunktmengen – Erste konzeptionelle Ans¨atze. In: Oldenburger 3D Tage, Wichmann, pp. 206–213. Mischke, A. (1998). Entwicklung eines VideotheodoliteMesssystems zur automatischen Richtungsmessung von nicht signalisierten Objektpunkten. PhD thesis, Vienna University of Technology. Moravec, H. (1977). Towards automatic visual obstacle avoidance. In: Proceedings of the International Joint Conference on Artificial Intelligence, pp. 584. Paar, G., Rottensteiner, F. and P¨otzleitner, G. (2001). Image matching strategies. In: Digital Image Analysis, pp. 393–410. Pelzer, H. (1985). Geod¨atische Netze in Landes- und Ingenieurvermessung II. Pelzer. Pratt, W. (1978). Digital Image Processing. Wiley and Sons. Reiterer, A., Kahmen, H., Egly, U. and Eiter, T. (2003). 3D-Vermessung mit Videotheodoliten und automatisierte Zielpunkt-erfassung mit Hilfe von Interest Operatoren. AVN, pp. 150–156. Reiterer, A. (2004). A Knowledge-Based Decision System for an On-Line Videotheodolite-Based Multisensor System. PhD thesis, Vienna University of Technology.

Quality Control for Building Industry Roic, M. (1996). Erfassung von nicht signalisierten 3DStrukturen mit Videotheodoliten. PhD thesis, Vienna University of Technology. Scherer, M. (2003). Architectural surveying by combined tacheometric and photogrammetric tools about the realisation of a synthesis. Workshop “Archaeology and Computer”. Scherer, M. (2004). Intelligent scanning with robottacheometer and image processing a low cost alternative to 3d laser scanning? FIG Working Week. Stefik, M. (1998). Introduction to Knowledge Systems. Kaufmann.

779 TOPCON (2007). http://www.topcon.com/. Walser, B. (2003). Development and calibration of an image assisted total station. PhD thesis, ETH-Zurich. Walser, B. and Braunecker, B. (2003). Automation of surveying systems through integration of image analysis methods. In: Optical 3-D Measurement Techniques VI, pp. 191–198. Wasmeier, P. (2003). The potential of object recognition using a servotacheometer tca2003. In: Optical 3-D Measurement Techniques VI, pp. 48–54.

New Solutions to Classical Geodetic Problems on the Ellipsoid ¨ Lars E. Sjoberg

Abstract New solutions are provided to the direct and indirect geodetic problems on the ellipsoid. In addition, the area under the geodesic and the problem of intersection of geodesics are treated. Each solution is composed of a strict solution for the sphere plus a small correction to account for the eccentricity of the ellipsoid. The correction is conveniently determined by numerical integration. Keywords Direct geodetic problem · Indirect geodetic problem · Geodesic · ellipsoid · Area determination · Intersection

1 Introduction Textbooks on higher geodesy generally treat the forward and inverse geodetic problems on the ellipsoid by classical series solutions. The forward problem is related with the determination of a new point (P) based on the information of the coordinates of another point (Q), the azimuth at Q and the arc length along the geodesic to P. The inverse problem solves the arc length along the geodesic between P and Q and the azimuths at P and Q based on the knowledge of the positions of P and Q. Some approximate formulas for area computations with geodetic triangles can be found, e.g., in Baeschlin (1948), Kneissl (1958) and Heitz (1988). Finally, the intersection problem on the ellipsoid treated here is not directly solved in the above textbooks.

Lars E. Sj¨oberg Royal Institute of Technology, Division of Geodesy, SE-100 44 Stockholm, e-mail: [email protected]

A new interest in the classical geodetic problems evolves, e.g., from solving problems related with the implementation of the UN Law of the Sea. The common idea behind the solutions presented below is to solve each problem strictly on the sphere and to add a small correction/integral to get the ellipsoidal solution. As each such integral is not explicitly solvable, it is suggested to compute it by numerical integration.

2 Basic Equations of the Geodesic We will limit the discussion to the equations of the geodesic that goes through the points Pi (βi , λi ), where i = 1 and 2 and β1 < β2 and λ1 < λ2 . Here βi are the reduced latitudes given  √ 2 by βi = arctan 1 − e tan ϕi , where ϕi are the geodetic latitudes and e is defined below. λi are the longitudes of the two points. (Solutions for β1 = β2 were given in Sj¨oberg (2006b,c).) Also, we assume that the geometric parameters semi-major axis a and first eccentricity e of the ellipsoid are given. Of importance for each geodesic is its Clairaut constant (e.g., Schmidt, 1999): c = cos βmax = cos βi sin αi ,

(1)

where βmax is the maximum latitude attained by the geodesic, and αi is its azimuth at point Pi . Then the arc length of the geodesic and the longitude difference between the two end points are given by the equations (e.g., Heck 2003, Sect. 7.2.4)  s=a

β2

β1



1 − e2 cos2 β cos βdβ, cos2 β − c2

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

(2)

781

¨ L.E. Sjoberg

782

  ti i=2 λ2 = λ1 + arcsin + dλ, t0 i=1

and  λ 2 − λ1 = c

β2



β1

1 − e2 cos2 β dβ . cos2 β − c2 cos β

(6a)

(3) where

Equations (1), (2) and (3) are basic for the solutions of the problems that follow.

3 The Direct Problem on the Ellipsoid



x2

dx   √ √ x1 1 − c2 − x 2 1 + 1 − e2 + e2 x 2 (6b) √ Here t0 = tan βmax = 1 − c2 /c and ti = tan βi . Finally, the azimuth of the geodesic at P2 follows from Eq. (1), yielding dλ = −e2 c

Problem 1. Given point P1 , the arc length s and azimuth α1 of a geodesic through P1 , it is required to determine the point P2 and the azimuth α2 of the geodesic at the new point. Solution 1 (Sj¨oberg, 2006b). By introducing the variable substitution sin β = x, rewriting the square-root of Eq. (2) as  −1 1 − c2 − x 2 −     e2 (1 − x 2 )/ 1 − c2 − x 2 1 + 1 − e2 + e2 x 2

sin α2 = c/ cos β2 .

(7)

4 The Indirect Problem Problem 2. Given two points P1 and P2 , determine the arc length s of the geodesic between the points and the azimuths of the geodesic at P1 and P2 . Solution 2 (Sj¨oberg, 2006b).

and integrating term by term, Eq. (2) becomes  s = a arcsin √

x 1 − c2

x2 =sin β2 x1 =sin β1

+ ds,

(a) The Clairaut constant is determined by the following equation that yields t0 (the tangent of (4a) the maximum latitude) of a geodesic (Sj¨oberg, 2007a)

where t0 =

ds = −ae ×  x2 2

x1

√

(1 − x 2 )d x   √ 1 − c2 − x 2 1 + 1 − e2 + e2 x 2

(4b)



t12 + t22 − 2t1 t2 cos DL/ sin DL

(8)

where ti = tan βi and DL = λ2 −λ1 −dλ. Here dλ was given in Eq. (6b). Then the Clairaut constant becomes

 Equation (4b), of the order of e2 s, can be conveniently c = 1/ 1 + t02 (9) determined by numerical integration. Rearranging the terms of Eq. (4a) and inserting x2 = sin β2 , we arrive at (b) The arc length s follows from Eqs. (4a) and (4b). (c) The azimuths α1 and α2 follow from Clariaut’s    (s − ds)/a   Eq. (1) above: 2 √ sin β2 = 1 − c sin (5) + arcsin x1 / 1 − c2 from which expression β2 can be determined. The longitude λ2 follows from decomposing the argument of Eq. (3) in a similar way. The result is:

sin αi = c/ cos βi ; i = 1, 2.

(10)

For some further details, see Schmidt (2006) and Sj¨oberg (2006c).

New Solutions to Classical Geodetic Problems

783

5 Area Computation on the Ellipsoid

Each integral Fi is conveniently solved by the technique outlined by Eqs. (6a) and (6b), where the first equation is a function of the unknown t = tan β, and the following equation results from Eq. (13):

The area T on the ellipsoid, limited by a geodesic, the equator and the two meridians with longitudes λ1 and λ2 , can be written (Sj¨oberg, 2006a, b): y1 T = b (α2 − α1 ) + b ck 2

2

f (y) − 1 dy, g(y)

arcsin (11a)

y2

where y = cos ϕ, k =

√

t t − arcsin = k, t02 t01

(14a)

where

1 − e2 , b = ak,

k = λ1 − λ2 − dλ1 + dλ2 t2 t1 + arcsin − arcsin t02 t01

1 + ez 1 1 + ln , (11b) f (y) =  2 2 4ez 1 − ez 2 1−e z        g(y) = y 1 − e2 z 2 1 − c2 − 1 − ec2 z 2

(14b)

Here ti = tan βi are the tangents of the latitudes at the given points P1 and P2 , and t0i = tan βi,max = (11c)  1 − ci2 /ci are the tangents of the maximum latitudes  of the two geodesics through P1 and P2 , respectively. and z = 1 − y 2 . The area of an arbitrary, closed geodetic poly- Moreover, dλi are the residual longitudes as given by gon, i.e. an area on the ellipsoid limited by several Eq. (6b) and determined by numerical integration. Sj¨oberg (2007b) shows that the solutions to geodesics, can be computed by adding and subtracting Eq. (14a) are given by areas of type T above. t = ±K

t01 t02 , D

(15a)

6 Geodetic Intersection on the Ellipsoid where Problem 3. Given the coordinates of two points P1 and P2 as well as the azimuths α1 and α2 of two geodesics passing through the points P1 and P2 , respectively, it is required to determine the point of intersection P(β, λ) of the two geodesics.

(a) From the given azimuths and the latitudes of P1 and P2 the Clairaut constants ci for both geodesics are easily determined by Eq. (1). (b) By writing Eq. (3) in the abstract form λ − λi = Fi (β, βi ); i = 1, 2.

  2 + t 2 ± 2t t 2 t01 01 02 1 − K . 02

(15c)

Only one of these solutions will be realistic, and it can be found by inspection.

(12) 7 Concluding Remarks

where (λ, β) are the coordinates of P, and each Fi is an integral given by Eq. (3), the latitude β becomes the solution to the integral equation F2 (β, β2 ) − F1 (β, β1 ) = λ1 − λ2 .

(15b)

and D=

Solution 3 (Sj¨oberg, 2007b).

K = sin k

(13)

We have presented a technique to carry out ellipsoidal computations in two steps: first compute the strict spherical solution and then add a small correction to the ellipsoidal solution. The correction is conveniently computed by numerical integration.

784 Acknowledgments I am grateful to several constructive remarks to a draft version of the paper by the reviewer Dr. P. Collier.

References Baeschlin CF (1948) Lehrbuch der Geod¨asie. Orell Fuessli Verlag, Zuerich Heck B (2003) Rechenverfahren und Auswertemodelle der Landesvermessung. H Wichmann Verlag, Karlsruhe, Germany Heitz S (1988) Coordinates in geodesy. Springer, Berlin Heidelberg New York Kneissl M (1958) Handbuch der Vermessungskunde, Band IV, JB Metzlersche Verlagsbuchhandlung, Stuttgart Schmidt H (1999) L¨osung der geod¨atischen Hauptaufgaben auf dem Rotationsellipsoid mittels numerischer Integration. ZfV124: 121–128

¨ L.E. Sjoberg Schmidt H (2006) Note on Lars E. Sj¨oberg: New solutions to the direct and indirect geodetic problems on the ellipsoid (ZfV 1/2006, pp. 35–39). ZfV 131: 153–154 Sj¨oberg LE (2002) Intersections on the sphere and ellipsoid. J Geod 76: 115–120. Sj¨oberg LE (2006a) Determination of areas on the plane, sphere and ellipsoid. Surv Rev 38: 583–593 Sj¨oberg LE (2006b) Direct and indirect geodetic problems on the ellipsoid. ZfV 131: 35–39 Sj¨oberg LE (2006c) Comment to H Schmidt’s Remarks to Sj¨oberg, (ZfV 1/2006, pp. 35–39). ZfV 131:155 Sj¨oberg LE (2007a) Precise determination of the Clairaut constant in ellipsoidal geodesy. Surv Rev 39: 81–86 Sj¨oberg LE (2007b) Geodetic Intersection on the Ellipsoid. (To appear in J Geod)

GNSS Carrier Phase Ambiguity Resolution: Challenges and Open Problems P.J.G. Teunissen and S. Verhagen

Abstract Integer carrier phase ambiguity resolution is the key to fast and high-precision global navigation satellite system (GNSS) positioning and application. Although considerable progress has been made over the years in developing a proper theory for ambiguity resolution, the necessary theory is far from complete. In this contribution we address three topics for which further developments are needed. They are: (1) Ambiguity acceptance testing; (2) Ambiguity subset selection; and (3) Integer-based GNSS model validation. We will address the shortcommings of the present theory and practices, and discuss directions for possible solutions.

the very precise carrier phase data will act as pseudo range data, thus making very precise positioning and navigation possible. GNSS ambiguity resolution (AR) applies to a great variety of current and future models of GPS, modernized GPS and Galileo, with applications in surveying, navigation, geodesy and geophysics. These models may differ greatly in complexity and diversity. They range from single-baseline models used for kinematic positioning to multi-baseline models used as a tool for studying geodynamic phenomena. The models may or may not have the relative receiver-satellite geometry included. They may also be discriminated as to whether the slave receiver(s) is stationary or in moKeywords Ambiguity acceptance tests · Ambigu- tion, or whether or not the differential atmospheric deity subset selection · Integer based GNSS model lays (ionosphere and troposphere) are included as unknowns. An overview of these models can be found in validation textbooks like (Strang and Borre, 1997; Teunissen and Kleusberg, 1998; Hofmann-Wellenhoff et al., 2001; Leick, 2003; Misra and Enge, 2001). 1 Introduction Despite the key role that is played by GNSS AR in the various high-precision applications, the necessary Integer carrier phase ambiguity resolution is the key theory is far from completed. Consider the conceptual to fast and high-precision GNSS positioning and nav- steps that can be recognised in GNSS AR: igation. It is the process of resolving the unknown cycle ambiguities of the double-differenced carrier phase 1. In the first step, one discards the integer nadata as integers. Once this has been done successfully, ture of the ambiguities and performs a standard P.J.G. Teunissen Delft Institute of Earth Observation and Space systems, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, email: [email protected]/[email protected] S. Verhagen Delft Institute of Earth Observation and Space systems, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands

least-squares adjustment. As a result one obtains the so-called float solution of all the parameters (i.e. ambiguities, baseline components, and possibly additional parameters such as atmospheric delays), together with their variance-covariance matrix. In this first step, one usually also tests the data and the assumed GNSS model for possible model misspecifications, e.g. outliers, cycle slips, or other modeling errors. This can be done with the

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

785

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hypothesis testing, thereby assuming the ambiguistandard theory of hypothesis testing, (Koch, 1999; ties as known. Teunissen, 2006). 5. Once the GNSS model and integer solution are ac2. In the second step, the real-valued float solution of cepted, the last step consists of correcting the float the ambiguities is further adjusted, so as to take the solution of all other parameters by virtue of their integer constraints into account. As a result one obcorrelation with the ambiguities. As a result one obtains an integer solution for the ambiguities. The intains the so-called fixed solution. Provided a correct teger estimators mostly used in practice are: decision has been made in the third and fourth step, r Integer least squares (ILS): ILS is the optimal the fixed solution will have a precision that is in acmethod for integer estimation as it can be cordance with the high precision of the phase data. shown to maximize the probability of correct integer estimation (Teunissen et al., 1999). In this contribution we address and discuss three ILS requires a search which can be done very topics for which the theory and/or numerical impleefficiently with the LAMBDA method. The mentation need further improvements. The three topics evaluation of the ILS success rate requires sim- are: (1) Ambiguity Acceptance Tests: one of the most ulation, (Teunissen, 1998a; Verhagen, 2005a; popular such test is the ratio-test. We will discuss its Joosten and Tiberius, 2000), the use of ad- relevance and performance, and show that its current equate approximations such as based on the usage is not optimal. (2) Ambiguity Subset Selection: ADOP (Teunissen, 1997a; Ji et al., 2007; Lee since full AR may not always be possible, partial AR et al., 2005; Vollath et al., 2003), or the use of provides an alternative. We will discuss some of the upper- and lower bounds, (Teunissen, 1998a; issues of partial AR. (3) Integer-Based Model ValidaVerhagen, 2005a). tion: we will argue that the standard theory of linear r Integer Bootstrapping (IB): The advantage of IB model hypothesis testing is not quite applicable to this over ILS is that no search is needed. Although case. A first approach to perform integer-based model suboptimal, the performance of IB can come validation will be given. close to that of ILS if the appropriate decorrelating Z -transformation is used. An advantage of IB is that its success rate can be computed very 2 Ambiguity Acceptance Tests easily, (Teunissen, 1998b). r Integer Rounding (IR): This method is the simplest of all, but it also has the lowest success 2.1 Ratio-Test rate of all. Still, it can be a useful method if the strength of the underlying GNSS model is such One of the most popular ambiguity acceptance tests is that the IR success rate is large. Often, as in case the so-called ratio-test. The ratio-test is defined as folwith IB, this first requires the application of a lows. Let the float ambiguity vector and its variance decorrelating Z -transformation. matrix be given as aˆ and Q aˆ aˆ , respectively. Further3. Once the integer ambiguities are computed, they more, let aˆ be the ILS solution, i.e. the integer mini(aˆ − a), and let aˇ be the are used in the third step as input to decide whether mizer of q(a) = (aˆ − a)T Q a−1 ˆ aˆ or not to accept the integer solution. Several such integer vector that returns the second smallest value of tests have been proposed in the literature and the quadratic form q(a). Then the ratio-test reads as: are currently in use in practice, (Abidin, 1993; q(aˇ ) Chen, 1997). Examples are the ratio-test, the Accept aˇ iff : ≥c (1) distance-test and the projector-test. A review q(a) ˇ and evaluation of these tests can be found in (Verhagen, 2005b). where c is a tolerance value, to be selected by the user. 4. Knowing that the ambiguities are integer, strength- Thus only if q(aˇ ) is sufficiently larger than q(aˇ ), will ens the model and allows one, in principle, to re- the decision be made to accept the ILS solution. Othevaluate the validation of the GNSS model. In prac- erwise, the ILS solution is rejected in favor of the float tice this is done by applying the standard theory of solution.

GNSS Carrier Phase Ambiguity Resolution

Questions that need to be addressed when using the above test are: (1) What does the ratio-test actually test? (2) What errors can be made with the ratio-test? (3) What value for c should be chosen? Answers to these questions are needed, in order to have a proper understanding of the ratio-test. Let us first see how the test is executed in practice. With reference to the theory of hypothesis testing, the ratio of the two quadratic forms, q(a) ˇ and q(aˇ ), is often assumed to have a Fisher-distribution, from which c can be computed, once the level of significance has been set. The problem with this approach is, however, that the ratio of the two quadratic forms is not Fisherdistributed. Even if one was allowed to assume that q(a) ˇ and q(aˇ ) are Chi-square distributed (which is not true, since also the uncertainty of the integer vectors needs to be taken into account), then their ratio would still not be Fisher-distributed. The two quadratic forms are namely not independent. Also, in many of the existing software packages, a fixed value for c is chosen, no matter the varying strength of the underlying GNSS model. Examples of proposed c-values are given in e.g. (Wei and Schwarz, 1995; Han and Rizos, 1996; Euler and Schaffrin, 1991). It is however strange to base the testing on a fixed c-value, since one would expect that with a varying strength of the GNSS model, the use of different measurement scenarios or with varying degrees of freedom, one also would use varying values for c. The incorrect assumption of the distributional properties of the ratio-test and the often used fixed c-value approach, can be explained by the lack of a proper theory (Teunissen and Verhagen, 2007b). What does the ratio-test actually test? One motivation that is often given for the use of the ratio-test, is that it tests the correctness of the ILS solution. This is, however, incorrect, since one can add an arbitrary integer vector to the float solution, without altering the outcome of the ratio-test. Hence, biases of arbitrary size (provided they are integer) can be present in the float solution, without them ever being noticed by the ratio-test.

2.2 Relevance of the Ratio-Test

787

region. We already remarked that the outcome of the ratio-test remains unchanged when an arbitrary integer vector, say z, is added to the float solution. This implies that its acceptance region, denoted as , must be a region which is z-translational invariant. That is, if the acceptance region is translated over an arbitrary integer vector, then the same acceptance region is recovered again. Since the rejection region is complementary to the acceptance region, also the rejection region of the ratio-test is z-translational invariant. The z-translational invariance of the acceptance region, implies that it must equal the union of z-translated copies of a smaller region 0 . Thus =



z where z = 0 + z

(2)

z∈Zn

The region z is called the aperture pull-in region of the integer z (Teunissen, 2003). The aperture pull-in regions z of the ratio-test are given in (Teunissen and Verhagen, 2004; Verhagen and Teunissen, 2006). For the test, we have aˆ = z if aˆ ∈ z

(3)

Thus if the float solution resides in z , the ratio-test leads to acceptance and the ILS solution is equal to z. Hence, the ratio-test does not test the correctness of the ILS-solution. Instead, the ratio-test tests the closeness of the float solution to its nearest integer vector. If it is close enough, the test leads to acceptance of a. ˆ If it is not close enough, then the test leads to rejection in favor of the float solution a. ˆ The size or aperture of the pull-in region provides the largest distance one is willing to accept. The value for c can be used to tune this aperture. To understand in what way the ratio-test helps us in getting confidence in the outcome of ambiguity resolution, we have to realize that acceptance of the ILS solution by the ratio-test can be correct or incorrect. We therefore have to distinguish between the following three cases: aˆ ∈ a aˆ ∈  {a } aˆ ∈ /

success: correct integer estimation failure: incorrect integer estimation undecided: ambiguity not fixed to an integer

To understand what the ratio-test tests, we need to get where \a means that a is deleted from the set , a better insight into its acceptance region and rejection with a being the unknown integer ambiguity vector.

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c. It can be shown that the ratio-test is a member of the class of tests as given by the theory of integer aperture estimation developed by (Teunissen, 2003). Members from this class differ in the way the shape of the aperture pull-in region 0 is defined. Hence, within this class, one can, by fixing the failure rate, solve for the aperture pull-in region that maximizes the success rate (Fig. 1 gives a two-dimensional example of the optimal Ps Ps f = (4) aperture pull-in regions). The optimal test so obtained Ps + P f differs from the ratio-test. For a discussion of the opThe above probabilities all depend on the shape and timal test and its relation to the ratio-test, we refer to size of 0 and on the PDF of a. ˆ Thus by changing 0 (Teunissen, 2003; Verhagen and Teunissen, 2006). As and/or the PDF of a, ˆ one can influence the above prob- with the ratio-test, efficient computation of the tolerabilities. Changing the PDF will not be possible, once ance value from the user-defined failure-rate remains a the measurement scenario is given (this will be differ- challenge. ent in case one is designing a measurement scenario). Changing the shape of 0 is also not possible, since the shape is determined by the ratio-test. This leaves us with the size of 0 , which is determined by c. Hence, 3 Ambiguity Subset Selection by changing c one can influence the above probabilities. Thus through the choice of c, the user is able to 3.1 Motivation have control over the failure rate and the probability of successful fixing. This is important, because it gives the user the necessary flexibility over what he/she finds There are two reasons why we address the topic of an acceptable risk to take with integer ambiguity res- ambiguity subset selection. The first reason has to do olution. This is the relevance of having the ratio-test with ones ability to resolve all ambiguities. Full AR, i.e. the successful resolution of all ambiguities, may included in GNSS AR. The above discussion makes clear that the common not always be possible. This will occur when the unpractice of using a fixed value for c is therefore not derlying GNSS model lacks sufficient strength. When one rethe way to go. By using a fixed value for c, the user one aims at resolving all, say n, ambiguities, n is deprived from any control over the failure rate. The quires that the simultaneous event i=1 {aˇ i = ai } has failure rate will then be different for different measure- a probability close to 1. This probability has the genment scenarios. Already in a kinematic or navigation eral tendency, however, to get smaller as n gets larger. scenario where data are collected on an epoch by epoch One can speak of a dimensional curse. A way out of basis, the failure rate can change from epoch to epoch this dilemma, when it occurs, is to aim at resolving only an ambiguity subset, thus keeping the dimension if a fixed value for c is used. To use the fixed failure-rate approach, one has to bounded. The second reason has to do with the necessity of be able to compute c from the failure-rate P f as set by the user. This is a nontrivial numerical task for full AR. Full AR may, in fact, not always be needed. which a practical approach has been devised in (Verha- As an extreme example consider the case where the gen, 2005b; Teunissen and Verhagen, 2007a). A chal- float ambiguities are not correlated with the other float lenge remains however to make this approach numeri- parameters. Ambiguity resolution would then be useless, since it will then not allow for an improvement cally efficient for real-time applications. of these other parameters. This indicates that resolving a subset of ambiguities (in the extreme example, an 2.3 Is the Ratio-Test Optimal? empty subset) may lead to the same, or almost same, performance improvement of the other parameters. The The answer is no, even if the ratio-test would be used observation that the resolution of different ambiguwith a fixed failure rate instead of with a fixed value for ity subsets, has different impacts on the performance The corresponding probabilities are denoted as: Ps (success-rate), P f (failure-rate) and Pu (probability of undecidedness). The probability Ps + P f is the probability of acceptance of the ratio-test and Pu is its probability of rejection. The probability of having successful fixes, denoted as Ps f , is given by the ratio of the success rate and the probability of acceptance,

GNSS Carrier Phase Ambiguity Resolution Fig. 1 Two-dimensional example of optimal aperture pull-in regions (gray areas), together with the ILS pull-in regions (hexagons)

789

1.5

1

0.5

0

–0.5

–1

–1.5 –1.5

–1

–0.5

0

0.5

1

1.5

an application of the bootstrapping principle. Once the transformed and decorrelated ambiguity variance matrix is obtained, the construction of the subset proceeds in a sequential fashion. One first starts with the most precise ambiguity, say zˆ 1 , and computes its success rate P(ˇz 1 = z 1 ). If this success rate is large enough, one continues and determines the most precise pair of ambiguities, say (ˆz 1 , zˆ 2 ). If their success rate is still large 3.2 Partial Ambiguity Resolution enough, one continues again by trying to extend the set. This procedure continues until one reaches a point Let us now continue with the case for which it turns out where the corresponding success rate becomes unacthat full AR is impossible (i.e. the success rate is too ceptably small. When this point is reached, one can exlow). Partial AR may then be a good alternative. This pect that the previously identified ambiguities can be idea of partial ambiguity resolution was introduced in resolved successfully. This method of ambiguity subset selection has been (Teunissen et al., 1999), where it was applied to long baselines using the current GPS, see also the review shown to work very well in practice. The method is (Teunissen, 2001). The goal of partial AR is to identify applicable to any GNSS model and it is superior to the ambiguity subset which gives the largest possible ‘widelaning’ methods (Teunissen, 1997b; Teunissen success rate. Partial AR based on integer bootstrapping et al., 1999). However, the identified problem of opgoes as follows. It will be clear that the subset selection timal ambiguity subset selection has not yet been comshould be based on the precision of the ‘float’ ambigu- pletely solved with the above approach. The above apities. The more precise the ambiguities, the larger the proach, which is based on bootstrapping, identifies the ambiguity success rate. Therefore, first the decorrela- optimal subset for sequential AR and not necessarily tion step of the LAMBDA method is used, followed by for batch AR. This latter problem is not yet solved. improvements, brings the question to the fore which subset to choice. This problem of identifying the optimal ambiguity subset, in relation to the performance improvement it brings, has not yet been addressed in the literature.

790

4 Integer-Based GNSS Model Validation

P.J.G. Teunissen and S. Verhagen

function (PDF) of this predicted residual, denoted as f eˇ 0 (x), has been derived in (Teunissen, 2007). Hence, one can now test whether or not it is likely that the ob4.1 Problem Statement served eˇ 0 is a sample from the distribution f eˇ 0 (x). If it is, one accepts the observation; if it is not, one rejects Consider the following null- and alternative hypothesis the observation. But before one can execute the test, the question of determining the acceptance and rejection regions should be answered. Answering this quesH0 : y ∼ N(Aa + Bb, Q yy ), a ∈ Zn , b ∈ R p n tion is made difficult by the multimodality of f eˇ 0 (x). Ha : y ∼ N(Aa + Bb + Cc, Q yy ), a ∈ Z , Let  ⊂ R be the acceptance region with coverage (b T, c T )T ∈ R p × Rq (5) probability P eˇ ∈   = 1−α. Thus the test leads 0 to rejection if eˇ 0 ∈ / . Since we want the rejection This is the situation one will have when testing the to be rare when the underlying model is correct, the validity of a phase-based GNSS model in which the false alarm probability α is chosen as a small value. ambiguities are considered to be integer. The additional But since there are an infinite number of subsets that term Cc, with matrix C known and vector c unknown, can produce this false alarm probability, we still need models under the alternative hypothesis the supposed to determine a way of defining a proper . It seems reamodeling errors (e.g. outliers, slips, instrumental bi- sonable to define the optimal subset as the one which ases, etc). The standard theory of hypothesis testing has the smallest volume. In that case the probability would be applicable in case the vector a is real val- 1−α would be the most concentrated. This acceptance ued. However, this theory is not applicable in our case, region is given as since a is known to be integer valued (NB: stating that a is known to be integer valued, is of course not the    = x ∈ R| f eˇ 0 (x) ≥ λ (6) same as stating that a is known). In order to be able to validate the above integerbased GNSS model, one must be able to answer queswhere λ is chosen so as to satisfy the given probability tions like: constraint. Note, due to the multimodality of f eˇ 0 (x), 1. What are the appropriate test statistics for testing that the acceptance region will in general consist of a number of disconnected regions. This is illustrated in H0 against Ha ? 2. How are these test statistics distributed under H0 Fig. 2. Due to the multimodality of the PDF, it is a nonand Ha ? trivial task to decide for a certain given α whether or 3. What are the appropriate acceptance- and rejection not an observed sample of eˇ 0 leads to rejection. The regions? complication resides in the direct determination of λ However, no rigorous answers to these questions are from α. This complication can be avoided, however, if yet available. Here we will introduce a first solution, we make use of a Monte Carlo based approach. The albeit restricted to the case of data snooping. computational steps for executing the test are then as follows. Given the observed sample of the prediction error, say eˇ 0∗ , one first computes λ∗ = f eˇ 0 (ˇe0∗ ). This imof the 4.2 Data Snooping for the Integer-Based plies that the sample would lie on the boundary acceptance region if λ would be set equal to λ∗ . Hence, GNSS Model this subset is given as  ∗ = {x ∈ R| f eˇ 0 (x) ≥ λ∗ }. The next step is then to compute the value of α that would Data snooping implies that one screens the observa- correspond with λ∗ : α ∗ = 1−P[ˇe0 ∈  ∗ ]. Here the tions one at a time for potential outliers. Now, let y0 be simulation enters. Let N be the number of times a samthe observation to be tested and let eˇ 0 be its predicted ple is generated from f eˇ 0 (x) and let Ni be the number residual, i.e. the difference between y0 and the integer- of times a generated sample lies in . Then α ∗ can be based prediction of y0 based on all other observa- approximated as α ∗ = 1−Ni /N . The decision to actions (thus with y0 excluded). The probability density cept or reject the observed sample is then based on the

791

0.6

0.6

0.5

0.5 probability density

probability density

GNSS Carrier Phase Ambiguity Resolution

0.4 0.3 0.2

λ

0.3 0.2

λ

0.1 0

0.4

0.1

λ

0 –4

–2

0

2

4

–4

v

–2

0

2

4

v

Fig. 2 PDF of the prediction error and corresponding 1 − α acceptance region (grey areas) for σ = 0.31 (left) and σ = 1.00 (right)

difference between α and α ∗ . If α ∗ < α, then the test 3. Integer-Based Model Validation: We have argued that the standard theory of linear model hypotheleads to rejection, otherwise it leads to acceptance. sis testing is not applicable when validating integerbased GNSS models. Hence, a proper theory is still lacking. A first solution, albeit restricted to data snooping, has been presented. 5 Summary and Conclusions In this contribution we addressed three topics for which the theory and/or numerical implementation need further improvements. The three topics are: 1. Ambiguity Acceptance Tests: We have shown that the fixed failure-rate approach should be used when applying the popular ratio-test. Also, the ratio-test can be shown to be suboptimal. For both the optimal test and the fixed failure-rate ratio-test, the numerical difficulty lies, however, in computing the tolerance value c from the user-defined failure-rate. Although methods have been devised to solve this problem, further numerical improvements are required, in particular for real-time applications. 2. Ambiguity Subset Selection: Being able to find the optimal ambiguity subset is of importance for partial AR. In the context of integer boot-strapping an optimal solution is available. This solution, however, is sequentially oriented and not a batch solution. For the latter an optimal solution needs yet to be devised.

Acknowledgments The research of the first author was done in the framework of his ARC International Linkage Professorial Fellowship, at the Curtin University of Technology, Perth, Australia, with Professor Will Featherstone as his host. This support is greatly acknowledged.

References Abidin, H. A. (1993). Computational and geometrical aspects of on-the-fly ambiguity resolution. Ph.D. thesis, Dept. of Surveying Engineering, Techn. Report no. 104, University of New Brunswick, Canada, page 314. Chen, Y. (1997). An approach to validate the resolved ambiguities in GPS rapid positioning. Proc. of the International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Banff, Canada, pages 301–304. Euler, H. J. and Schaffrin, B. (1991). On a measure for the discernibility between different ambiguity solutions in the static-kinematic GPS-mode. IAG Symposia no. 107, Kinematic Systems in Geodesy, Surveying, and Remote Sensing, Springer-Verlag, New York, pages 285–295. Han, S. and Rizos, C. (1996). Validation and rejection criteria for integer least-squares estimation. Survey Review, 33(260):375–382.

792 Hofmann-Wellenhoff, B., Lichtenegger, H., and Collins, J. (2001). Global Positioning System: Theory and Practice. Springer-Verlag, Berlin, 5th edition. Ji, S., Chen, W., Zhao, C., Ding, X., and Chen, Y. (2007). Single epoch ambiguity resolution for Galileo with the CAR and LAMBDA methods. GPS Solutions, DOI 10.1007/s10291007-0057-9. Joosten, P. and Tiberius, C. C. J. M. (2000). Fixing the ambiguities: are you sure they’re right. GPS World, 11(5):46–51. Koch, K. R. (1999). Parameter Estimation and Hypothesis Testing in Linear Models. Springer Verlag, Berlin, 2nd edition. Lee, H. K., Wang, J., and Rizos, C. (2005). An integer ambiguity resolution procedure for GPS/pseudolite/INS integration. Journal of Geodesy, 79(4–5): 242–255. Leick, A. (2003). GPS Satellite Surveying. John Wiley and Sons, New York, 3rd edition. Misra, P. and Enge, P. (2001). Global Positioning System: Signals, Measurements, and Performance. Ganga-Jamuna Press, Lincoln, MA. Strang, G. and Borre, K. (1997). Linear Algebra, Geodesy, and GPS. Wellesley-Cambridge Press, Wellesley, MA. Teunissen, P.J.G. (1997a). A canonical theory for short GPS baselines. Part IV: Precision versus reliability. Journal of Geodesy, 71:513–525. Teunissen, P.J.G. (1997b). On the GPS widelane and its decorrelation property. Journal of Geodesy, 71(9):577–587. Teunissen, P.J.G. (1998a). On the integer normal distribution of the GPS ambiguities. Artificial Satellites, 33(2):49–64. Teunissen, P.J.G. (1998b). Success probability of integer GPS ambiguity rounding and bootstrapping. Journal of Geodesy, 72:606–612. Teunissen, P.J.G. (1999). An optimality property of the integer least-squares estimator. Journal of Geodesy, 73(11):587–593. Teunissen, P.J.G. (2001). GNSS ambiguity bootstrapping: Theory and applications. Proc. KIS2001, International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, June 5–8, Banff, Canada, pages 246–254. Teunissen, P.J.G. (2003). Integer aperture GNSS ambiguity resolution. Artificial Satellites, 38(3):79–88.

P.J.G. Teunissen and S. Verhagen Teunissen, P.J.G. (2006). Testing Theory, an Introduction. Delft University Press, Delft, 2nd edition. Teunissen, P.J.G. (2007). Least-squares prediction in linear models with integer unknowns. Journal of Geodesy, DOI:10.1007/s00190-007-0138-0. Teunissen, P.J.G., Joosten, P., and Tiberius, C. C. J. M. (1999). Geometry-free ambiguity success rates in case of partial fixing. Proc. of National Technical Meeting & 19th Biennal Guidance Test Symposium ION 1999, San Diego CA, pages 201–207. Teunissen, P.J.G. and Kleusberg, A. (1998). GPS for Geodesy. Springer, Berlin Heidelberg New York, 2nd edition. Teunissen, P.J.G. and Verhagen, S. (2004). On the foundation of the popular ratio test for GNSS ambiguity resolution. Proc. of ION GNSS-2004, Long Beach CA, pages 2529–2540. Teunissen, P.J.G. and Verhagen, S. (2007a). GNSS Phase Ambiguity Validation: A Review. Proc. Space, Aeronautical and Navigational Electronics Symposium SANE2007, The Institute of Electronics, Information and Communication Engineers (IEICE), Japan, 107(2):1–6. Teunissen, P.J.G. and Verhagen, S. (2007b). The GNSS ratio-test revisited. Submitted to Survey Review. Verhagen, S. (2005a). On the reliability of integer ambiguity resolution. Navigation, 52(2):99–110. Verhagen, S. (2005b). The GNSS integer ambiguities: estimation and validation. Ph.D. thesis, Publications on Geodesy, 58, Netherlands Geodetic Commission, Delft. Verhagen, S. and Teunissen, P.J.G. (2006). New global navigation satellite system ambiguity resolution method.compared to existing approaches. Journal of Guidance, Control, and Dynamics, 29(4):981–991. Vollath, U., Sauer, K., Amarillo, F., and Pereira, J. (2003). Three or four carriers – how many are enough? Proc. of ION GNSS2003, Portland OR, pages 1470–1477. Wei, M. and Schwarz, K. P. (1995). Fast ambiguity resolution using an integer nonlinear programming method. Proc. of ION GPS-1995, Palm Springs CA, pages 1101–1110.

Static Stress Change from the 8 November, 1997 Ms 7.9 Manyi, Tibet Earthquake as Inferred from InSAR Observation Wen Yangmao and Xu Caijun

Abstract On 8 November 1997, a Ms 7.9 earthquake occurred in Manyi, Tibet plateau. The earthquake occurred on the Manyi Fault at the northern edge of the Qiangtang Basin. The region is relatively flat with an elevation of about 5000 m, where rarefied atmosphere, cold climate, sparse vegetation coverage and lack of human habitation provide excellent conditions for Interferometric Synthetic Aperture Radar (InSAR) studies. The authors used pairs (Frame 2889/2907, Track 076/355/033) of radar images acquired by the ESA ERS-2 satellite to construct interferograms of the surface displacement field due to the earthquake. The location and extent of the co-seismic fault rupture was mapped using interferometric correlation and azimuth offset measurements. Employing a 3-D semi-analytical visco-elastic body force model and the Genetic Algorithm (GA), the authors inverted the locking depth of the co-seismic fault and slip distributions, then compared the result with the elastic dislocation model assuming variable and non-vertical dips. The result best fits the InSAR displacement. Based on the Coulomb failure criterion, the authors further calculated the static stress changes on optimal failure planes by assuming a regional compressional stress with different friction and depth. The relationship between the Manyi earthquake and aftershocks is also discussed. Most aftershocks are located within the stress increased regions (especially μ = 0.8).

Wen Yangmao School of Geodesy and Geomatics, Wuhan University, Wuhan, China, 430079 Xu Caijun School of Geodesy and Geomatics, Wuhan University, Wuhan, China, 430079

Keywords Static stress change · Manyi earthquake · Interferometric Synthetic Aperture Radar (InSAR)

1 Introduction The Kunlun Fault is one of the largest strike-slip faults in the Tibetan plateau area. It is located at the south side of the Altyn Tagh Fault and extends almost 1200 km from the northeast Tibet, separating the Qaidam Basin to the northeast from the plateau. Global Positioning System (GPS) observations and geological investigations indicate a left-lateral strike slip rate of 10– 14 mm/yr on the Kunlun fault. (Wang et al., 2001; Zhang et al., 2004). The Ms 7.9 Manyi earthquake occurred on 8 November 1997 in Manyi, Tibet, near the western end of the Kunlun Fault (Fig. 1). The region is relatively flat with an elevation of about 5000 m, where the conditions including rarefied atmosphere, cold climate, lack of boreal vegetation and the absence of human modification of the surface (for example due to agriculture) favour the application of Interferometric Synthetic Aperture Radar (InSAR). Using radar data acquired by the satellite ERS-2, Peltzer et al. (1999) obtained a nonlinear model with a ratio of 2 between compressive and tensile elastic moduli. Funning et al. (2007) and Wang et al. (2007) fitted the observations using a linear elastic dislocation model with a more complex fault geometry. In this study the authors used co-seismic pairs (Frame 2889/2907, Track 033/076/355) of radar images acquired by the ESA ERS-2 satellite to construct interferograms of the surface displacement field due to the earthquake. The location and extent of the

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Fig. 1 Shaded relief map of the epicentral area of the 1997 Manyi earthquake. The study area is depicted in the inset map. Data shown are 3 arc-second data from the Shuttle Radar Topography Mission (SRTM). The black lines indicate the coverage of three ERS-SAR tracks (T076, T355 and T033). The star is the Manyi event, and the circle

in black-white color denotes the CMT of the major event from the National Earthquake Information Center (NEIC, http://neic.usgs.gov/neis/epic). The aftershock data (M>3.0, black circles) are from the International Seismological Center (ISC, http://www.isc.ac.uk/search/bulletin). The along-track (AT) and line-of-sight (LOS) directions are marked with arrows

co-seismic fault rupture was mapped using interferometric correlation and azimuth offset measurements. Employing a 3-D semi-analytical visco-elastic body force model and the Genetic Algorithm (GA), the locking depth of the co-seismic fault and slip distributions were determined. Finally, based on the Coulomb failure criterion, the static stress changes on optimal failure planes were computed by assuming a regional compressional stress with different friction and depth.

Table 1 Descending track ERS2 data used in this study †

#

Pair

B⊥ (m)

T (day)

Track

Frame

IP1 IP2 IP3

970316–971116 970819–971202 970522–971218

65 9 −26

245 105 210

076 305 033

2889/2907 2889/2907 2889/2907

† Perpendicular baseline at image center

(Scharroo and Visser, 1998) were used. The topographic contribution to the interferometric phase was removed by using the 3 are second digital elevation model (DEM) from the Shuttle Radar Topography Mission (SRTM) (Farr and Kobrick, 2000). Finally, 2 Co-Seismic Deformation from InSAR the interferogram products were unwrapped using the branch cut method. After interferometric processing, Because the surface rupture is so long, the images three pairs of geocoded coseismic displacement maps cover three adjacent tracks 076, 305 and 033 (Table 1). covering three tracks were obtained. The final interferograms, rewrapped at 10 cm interPairs of SAR images, covering two frames from each of these tracks, were processed using the GAMMA vals, are shown in Fig. 2. The most notable feature in software (Werner et al., 2000). Tthe radar data was the data is a large cluster of fringes centred on the cenprocessed using the two-pass method (Massonnet tral track (track 305), representing the area of greatet al., 1993). The precise orbits distributed by the Delft est moment release. There is a narrow, linear band Institute for Earth Oriented Space Research (DEOS) of decorrleation that bisects this deformation pattern,

Tibet Earthquake as Inferred from InSAR Observation

795

Fig. 2 Manyi earthquake coseismic deformation. Data shown are the unwrapped interferogram data, converted to line-of-sight (LOS) displacement of the ground and rewrapped at 0.1 m intervals

surface, the correlation will be low; correlation will be high if the surface is unchanged. Typically situations which can cause decorrelation (Haseen, 2001) include the growth or movement of vegetation; bodies of water; the falling or melting of snow; human activities; long perpendicular baselines between image acquisition and/or large gradients in deformation at the surface; natural phenomena such as floods, landslides or 3 The Manyi Fault Trace earthquakes. In the Manyi region many of these effects can be neThe Manyi earthquake occurred in a remote and unin- glected, because there are no human activities, high elhabited area of the Tibetan plateau, so it is difficult to evation and lack of boreal vegetation. In the case of the map the fault rupture using field survey methods. How- co-seismic pairs of ERS2 image, most decorrelation in ever, the interferometric correlation map and azimuth the region can be interpreted as the damage caused by offset image are capable of identifingy the fault trace the earthquake, except for some presence of ground(e.g. Peltzer et al., 1999; Simons et al., 2002). The lack water or snow on the surface. A mosaic interferometof vegetation on the Tibetan plateau allows high corre- ric correlation image from 3 tracks is shown in Fig. 3. lation between radar images, which makes it relatively A prominent feature where the correlation is low can easy to identify the fault trace using coherence and az- be seen along the length of the aftershock zone, most likely representing damage to the ground and large disimuth offset information. placement gradients caused by the earthquake. cutting through the area of the highest deformation gradients, which is due to the rupturing of the surface during the earthquake. The degree of correlation in all of these interferograms is high, in many cases with clear signal right up to the fault on both sides.

3.1 Mapping Fault Trace Using Interfometric Correlation The interferometric correlation map is a by-product of InSAR processing. It provides a measure of the stability of the phase contribution of small radar scatterers within each interferogram pixel. If these scatterers are disturbed or modified by some change to the

3.2 Mapping Surface Faulting with Azimuth Offsets Another means that can be used to locate the fault trace is by the calculation of azimuth offsets, the positional shifts required to precisely align the two SAR amplitude images in the along-track direction (Peltzer et al.,

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W. Yangmao and X. Caijun

Fig. 3 Mapping fault trace using the correlation. The fault can be traced as a deep dark band of low correlation

Fig. 4 Mapping fault trace using the azimuth offset. The fault can be traced as a discontinuity between positive and negative offsets all through the area

1999; Funning et al., 2007). A mosaic image of the azimuth offsets for three adjacent tracks is shown in Fig. 4. Although the fault orientation is almost perpendicular to the track direction of satellite, obvious alongtrack offsets can be observed in the InSAR decorrelation area. Because the azimuth offset is less sensitive to surface change than the correlation map, in several areas the near-fault coverage is better than in the correlation map. The azimuth offsets provide a more comprehensive picture of the fault trace than the correlation map. The fault can be seen as a discontinuity in the azimuth offsets in Fig. 4, from red (positive) to green (negative).

Fig. 5 The mapped fault trace. There is good agreement between correlation of SAR image (dot line) and azimuth offset from magnitude (dash line). The model fault trace (solid line), including 7 segments, used in the following model. The aftershocks are from the ISC

The fault can be traced along the discontinuity of the image very well.

3.3 The Manyi Fault Trace The Manyi Fault traces extracted from the correlation map (Fig. 3) and azimuth offset image (Fig. 4) are plotted together in Fig. 5. There is an excellent agreement between the two traces along much of the length of the fault. At the eastern end, the azimuth offset image shows an even clearer fault trace than the coherence image. This segment is striking northward relative to the western segments.

Tibet Earthquake as Inferred from InSAR Observation

797

the Genetic Algorithm (GA) (Carroll, 1996). The GA technique is an effective global optimisation approach based on random iteration and evolution of nature selection and population genetics. Like other inversion methods, GA also begins with a specified initial model parameters set, then computes the misfit between observation and model, and updates the model parameters to the optimal set. Because the operation of 4 Source Parameters Using 3-D each chromosome (namely each model parameters) Semi-Analytical Visco-Elastic Model is synchronous, the GA is a parallel algorithm. The efficiency of the GA is higher than other nonlinear inversion algorithms. The authors solve the system 4.1 3-D Semi-Analytical Visco-Elastic of equations dInSAR (x, y) = dm (m, x, y), where Model dInSAR is the InSAR co-seismic displacement, dm is the modelled displacement, and m is the set of locking Smith and Sandwell (2004) have developed a semi- depth and slip parameters that minimise the root mean analytical visco-elastic model taking the visco-elastic square rms: response into account. The model consists of an elastic plate (with thickness H = 50 km) overlying a visco!  elastic half-space. The fault within the elastic plate ! N  i i 2 d − d ! m InSAR extends from the surface to a locking depth (d). The " i=1 rms = (1) model is kinematic, given the time, extent and amount f of co-seismic slip, then it can simulate visco-elastic flow in the underlying half-space. The duration of the where f is the degree of freedom and N is the number visco-elastic response, characterised by the Maxwell of InSAR observations. time, depends on the viscosity of the underlying halfTypically there is no simple way to propagate unspace and the elastic plate thickness. certainties in the data to uncertainties in the estimated The numerical aspects of this approach involve gen- model parameters. In this study the Monte-Carlo error erating a grid of vector force-couples that simulate propagation technique (Aster et al., 2004) is used to the complex fault geometry, taking that 2D horizon- obtain the variance-covariance matrix of the estimated tal Fourier Transform of the grid, multiplying by the parameters. appropriate transfer functions and time-dependent reIn order to expedite the modelling process, the aulaxation coefficients, and finally inverse Fourier trans- thors employed a uniform sampling method (Pritchard forming to obtain the desired results. The solution sat- et al., 2002) to create a more manageable dataset, i.e. isfies the zero-traction surface boundary condition and the InSAR data with a spacing of 30 arc-second (about maintains stress and displacement continuity across the 900 m), totalling 7.3 × 105 observations. base of the plate. The crust medium parameters used in the study include viscosity (η = 1.0 × 1021 ), Young’s modulus (E = 7.85 × 1010 Pa) and Poisson’s ratio 4.3 Inversion Result (ν = 0.25) (Shen et al., 2003).

In order to simplify the following inversion, the authors fitted the fault trace with a series of 7 straight line segments (Fig. 5). For the purpose of description, the fault segments are named from 1 to 7, starting with the westernmost segment.

4.2 Geodetic Inversion The relation between coseismic displacement (deformation) and source parameters (including locking depth and slip) is nonlinear, thus the unknown depths and slips of 7 fault segments were estimated using

The resampled displacements were modelled using a semi-analytical visco-elastic model. In the inversion, the authors adopted the simplified 7 segment fault geometry already described (Sect. 3.3), fixing the surface location, strike and length of each fault segment. Given that the effects of a fault rupture can be observed in the interferometric correlation along the length of the fault, it is reasonable to also fix the top-locking depth

798 Table 2 The fault geometry and best-fit fault parameters

W. Yangmao and X. Caijun # Lat.

Long.

Length (km) Locking depth (km) Slip (m)

1 2 3 4 5 6 7

86.192 86.602 86.815 87.100 87.267 87.418 87.823

46.0 25.1 31.8 19.1 16.8 45.7 19.2

35.060 35.112 35.188 35.210 35.249 35.259 35.328

of all the faults to be zero. The remaining two fault parameters for each segment (locking depth and slip) can then be determined using the GA technique. In addition, three nuisance parameters are solved for each displacement dataset – a static shift in measured displacement, and gradients of displacement in the east-west and north-south direction – to account for zero-level ambiguities and any residual long-wavelength orbital errors, respectively.

−12.2 ± 1.9 −10.8 ± 1.0 −11.3 ± 1.4 −12.9 ± 1.8 −13.7 ± 1.2 −11.8 ± 1.7 −11.2 ± 1.2

M0 (Nm)

1.62 ± 0.21 2.60 ± 0.58 3.26 ± 0.51 6.14 ± 0.67 2.51 × 1020 6.62 ± 0.57 3.54 ± 0.39 1.15 ± 0.19

rms (m)

0.045

The best-fit source parameters are listed in Table 2, and the synthetic interferograms and residuals are shown in Fig. 6. The maximum slip is 6.62 ± 0.57 m with depth 13.7 ± 1.2 km on the fifth segment. The slip is consistent with the value of 7.0 m in Peltzer et al. (1999), who use a single segment uniform slip model and fixed the width as 8 km. The rate also agrees with the value of 6.73 m in Funning et al. (2007). The estimated geodetic moment is 2.51 × 1020 Nm (Mw7.57),

Fig. 6 Seven segments model of the Manyi earthquake. (a, b, c) are model interferograms for tracks 076, 305 and 033, respectively. (d, e, f) are residual interferograms (InSAR-model) for tracks 076, 305, and 033, respectively

Tibet Earthquake as Inferred from InSAR Observation

799

similar to the seismological results (2.23 × 1020 Nm, Harvard CMT). The rms between the observations and the model displacement is 45 mm. In general, the fit to the data of the model is good, given the high level of coherence and good data coverage on both sides of the fault. The fit is especially good for segments 1 to 5, where residuals are small and local to the fault trace. Significant residuals (>0.3 m) still remain at the centre of the fault in segments 6 and 7. The characteristic of these residual features suggests that the slip geometry may be too simplified in these areas.

5 Static Stress Change Usually a Coulomb stress failure criterion is calculated to explain patterns of seismicity change (Harris and Simpson, 1998; Parsons et al., 2006). The Coulomb criterion ( τ ) is defined by: τ = | τ f | + μ( σn − p)

(2)

where | τ f | is the change in shear stress on the receiver fault (positive in the direction of fault slip), μ is the friction coefficient, σn is the change in normal

stress acting on the target fault (positive for unclamping), and p is the pore pressure change. To make Coulomb stress change calculations, it is necessary to resolve stress components on defined planes of interest; additionally, estimates of target fault friction coefficient and some means of treating pore fluid pressure must be obtained. Here the stress changes on optimally oriented fault planes were calculated, as defined by an assumed regional stress direction using the method of King (1994). The regional compressive stress of 10 MPa (100 bars) acting in the north-northeast direction (Ma, 1989) is used in the calculations. In general, Skempton’s coefficient B (varied from 0 to 1) is used to incorporate pore fluid effects, where the effective friction coefficient μ = μ(1 − B) is adjusted and used in the Coulomb failure criterion as τ = | τ f | + μ σn after the work of Reasenberg and Simpson (1992). The Coulomb stress change is shown in Fig. 7. The authors varied the target fault friction coefficient from μ = 0.0–0.8, and depths from 5.0 to 10.0 km to test effects of parameter uncertainties. It was found that there was sensitivity in the results to parameter choices, but not to the degree that altered the sign of the calculated stress change at specific target fault zones, or changed the general pattern of stress change (Fig. 7).

Fig. 7 Sensitivity of Coulomb stress change calculation to target fault friction coefficient and depth. Friction varied from μ = 0.0–0.8 and depth ranged from 5.0 to 10 km. Difference is evident, but the overall stress change pattern is consistent

800

The calculated Coulomb stress change patterns show consistent features regardless of the friction coefficient and depth values. Stress-increased zones were confined to areas near the fault; while stress-decreased areas were located southeast and northwest of the ends of the rupture (Fig. 7f). Aftershocks (M > 3.0, first 60 days) are plotted on Fig. 7, and for the most part aftershocks were located within the stress-increased region (especially μ = 0.8, Fig. 7f). This observation may not be very meaningful since aftershocks almost always are located on and near mainshock rupture planes. Perhaps more interesting is the correspondence between the stress-decreased areas and lower aftershock rates. Sixty days after the event, just 18 % of aftershocks had occurred in the stress-decreased zones (Fig. 7f).

6 Conclusions In the paper, the authors investigated the co-seismic deformation and slip distribution of the 8 November 1997 Ms7.9 Manyi earthquake using InSAR measurements. The fault trace is constrained by a combination of the interferometric coherence image and azimuth offset image. Employing a 3-D semi-analytical viscoelastic body force model and the Genetic Algorithm the authors inverted the locking depth of the co-seismic fault and slip distributions. In the best-fit model the maximum slip is 6.62 ± 0.57 m with a depth 13.7 ± 1.2 km on the fifth segment, and the estimated geodetic moment is 2.51 × 102 0 Nm (Mw7.57). The static stress change due to the co-seismic slip varied friction coefficient and depth was also calculated. Most aftershocks (nearly 82 %) are located in the stress-increased regions (trigger zones, with μ = 0.8). Acknowledgments The European Space Agency (ESA) is gratefully acknowledged for providing ERS2 SAR data under project CAT-1. The authors wish to thank B. Smith and D. Sandwell for providing source code of the 3-D semi-analytic visco-elastic model (Maxwell package). Most figures are prepared using the GMT software (Wessel and Smith, 1998). This work was supported by the Natural Science Foundation of China (No. 40574006), Program for New Century Excellent Talents in University (NCET-04-0681) and Key Laboratory of Dynamic Geodesy, Chinese Academy of Sciences (L04-02). The authors thank Ge Linlin and Ding Xiaoli for their helpful reviews.

W. Yangmao and X. Caijun

References Aster R., B. Borchers and C. Thurber (2004), Parameter Estimation and Inverse Problems, Academic Press, San Diego, CA, pp. 35–36. Carroll, D.L. (1996), Chemical laser modeling with Genetic Algorithms, AIAA. J., 34(2), pp. 338–346. Farr, M. and M. Kobrick (2000), Shuttle Radar Topography Mission produces a wealth of data, EOS Trans., 81, pp. 583–585. Funning G.J., B. Parsons and T.J. Wright (2007), Fault slip in the 1997 Manyi, Tibet earthquake from linear elastic modelling of InSAR displacements, Geophys. J. Int., doi: 10.1111/j.1365-246X.2006.03318.x. Harris R.A. and R.W. Simpson (1998), Suppression of large earthquakes by stress shadows: A comparison of Coulomb and rate-and-state failure, J. Geophys. Res., 103, pp. 24439–54452. Haseen R.F. (2001), Radar Interferometry: Data Interpretation and Error Analysis, Kluwer Academic Publishers, Dordrecht, pp. 98. King G.C.P., R.S. Stein and J. Lin (1994), Static stress changes and the triggering of earthquake, Bull. Seismol. Soc. Am., 84, pp. 935–953. Ma X. (1989), Lithospheric Dynamics Atlas of China, China Cartographic Publishing House, Beijing, pp. 23. (in Chinese). Massonnet D., M. Rossi, C. Caemona, F. Adragna, G. Peltzer, K. Feigl and T. Rabaute (1993), The displacement field of the Landers earthquake mapped by radar interferometry, Nature, 364, pp. 138–142. Parsons T., R.S. Yeats, Y. Yagi and A. Hussain (2006), Static stress change from the 8 October, 2005 M = 7.6 Kashmir earthquake, Geophys. Res. Lett., 33, L06304, doi:10.1029/2005GL025429. Peltzer G., F. Cramp´e and G. King (1999), Evidence of nonlinear elasticity of the Mw 7.6 Manyi (Tibet) Earthquake, Science, 286, pp. 272–276. Pritchard M.E., M. Simons, P.A. Posen, S. Hensley and F.H. Webb (2002), Co-seismic slip from the 1995 July 30 Mw = 8.1 Antofagasta, Chile, earthquake as constrained by InSAR and GPS observations, Geophys. J. Int., 150, pp. 362– 376. Reasenberg P.A. and R.W. Simpson (1992), Response of regional seismicity to the static stress change produced by the Loma Prieta earthquake, Science, 255, pp. 1687–1690. Scharroo R. and P. Visser (1998), Precise orbit determination and gravity field improvement for the ERS satellite, J. Geophys. Res., 103(C4), pp. 8113–8127. Shen Z., Y. Wan, W. Gan, Y. Zeng and Q. Ren (2003), Viscoelastic triggering among large earthquakes along the east Kunlun fault system, Chinese J. Geophys., 46, pp. 786–795. (in Chinese) Simons M., Y. Fialko and L. Rivera (2002), Coseismic deformation from the 1999 Mw 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations, B. Seismol. Soc. Am., 92(4), pp. 1390–1402. Smith B. and D. Sandwell (2004), A three-dimensional semianalytic viscoelastic model for time-dependent analyses

Tibet Earthquake as Inferred from InSAR Observation of the earthquake cycle, J. Geophys. Res., 109, B12, B1240110.1029/2004JB003185. Wang H., C. Xu and L. Ge (2007), Coseismic deformation and slip distribution of the 1997 Mw7.5 Manyi, Tibet, earthquake from InSAR measurements, J. Geodyn., In press. Wang Q., P.-Z. Zhang, J.T. Freymueller, R. Bilham, K.M. Larson, X. Lai, X. You, Z. Niu, J. Wu, Y. Li, J. Liu, Z. Yang and Q. Chen (2001), Present-day crustal deformation in China constrained by Global Positioning System measurements, Science, 294, pp. 574–577. Werner C.L., U. Wegm¨uller, T. Strozzi and A. Wiesmann (2000), GAMMA SAR and interferometry processing software, ERS ENVISAT Symp., Gothenburg, Sweden, Oct. 16–20.

801 Wessel P. and W.H.F. Smith (1998), New improved version of the Generic Mapping Tools Released, EOS Trans. AGU, 79, pp. 579. Zhang P.-Z., Z. Shen, M. Wang, W. Gan, R. B¨urgmann, P. Molnar, Q. Wang, Z. Niu, J. Sun, J. Wu, H. Sun and X. You (2004), Continuous deformation of the Tibet Plateau from global positioning system data, Geology, 32(9), pp. 809–812.

Application of Computer Algebra System to Geodesy P. Zaletnyik, B. Pal´ancz, J.L. Awange and E.W. Grafarend

Abstract This contribution extends the previous work of Awange and Grafarend (2005). Using Groebner basis and Dixon resultant as the engine behind Computer Algebra Systems (CAS). The authors demonstrate how 3D GPS positioning, 3D intersection, as well as datum transformation problems are solved ‘live’ in Mathematica, thanks to modernization in CAS. Mathematica notebooks containing these ‘live’ computational models and examples are available at http://library.wolfram.com/infocenter/ MathSource/6654. Keywords Computer Algebra · polynomial equations · Dixon resultant · Groebner basis

1 Introduction Nowadays computer algebra systems (CAS) integrate modern numeric and symbolic algorithms and offer the P. Zaletnyik Department of Geodesy and Surveying, Budapest University of Technology and Economics, H-1521 Budapest, P.O. Box 91, Hungary, e-mail: [email protected] B. Pal´ancz Department of Photogrammetry and Geoinformatics, Budapest University of Technology and Economics, H-1521 Budapest, P.O. Box 91, Hungary. e-mail: [email protected] J.L. Awange Western Australian Centre for Geodesy and The Institute for Geoscience Research, Curtin University of Technology, GPO Box U1987, Perth WA 6845, Australia, e-mail: [email protected] E.W. Grafarend Department of Geodesy and Geoinformatics, Stuttgart University, Geschwister-Scholl Str. 24D, Stuttgart 70174 – Germany, e-mail: [email protected]

possibility of ‘live’ interaction to users. This is in contrast to the widely held belief by most scientists, even today, that the CAS language is a programming language. CAS can be used as ‘live’ mathematics for creating, proving and evaluating algorithms and expressions in numeric or symbolic form. Most nonlinear geodetic computational problems, e.g., finding initial values for iterative algorithms, avoiding ill-conditioned numerical problems, or finding effective global or local minima are immaterial when CAS is properly employed. CAS systems are equipped with many in-built algorithms representing the state-of-the-art of mathematical methods (Wester, 1999). They provide flexibility of computing with any working precision, mixing numeric and symbolic algorithms and enable the formation of personalized algorithms. In this work the authors will demonstrate the solution of polynomial systems of equations with CAS. The geodetic computational problems are mostly nonlinear problems. Solving them with traditional methods often requires linearization, finding the approximate starting values, iterations and lots of computational time.

2 Computer Algebra System CAS integrate modern numeric and symbolic mathematical algorithms as built-in functions. Their major advantage over traditional programming languages is that they are interactive. Interactivity allows one to test different methods separately and to test their behaviour in an easy way. Besides symbolic and numeric computations, there are also good visualization tools. For solving mathematical problems with CAS, one does not need to study the theory of the algorithms,

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they can just be used in a simple way. This simplicity allows one to concentrate on the essential task and ignore peripheral matters. The most frequently used CAS systems are: Mathematica, Maple, MuPAD and Macsyma (see, e.g., Awange and Grafarend (2005)). With these systems, symbolic manipulations can be done, e.g., symbolic simplifications, differentiation, integration, matrix operations, solution of polynomial systems of equations, polynomial factorization, greatest common divisor, etc. The growing demand of speed, accuracy and reliability in engineering calculations leads to the effective merging of symbolic and numeric computations. In geodesy, CAS can be used effectively in a wide range of areas, see, e.g., Haneberg (2004).

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Here the authors provide a brief introduction of the Dixon method, previously unknown in the geodesy and surveying literature.

4 Employing the Dixon’s Method

The origin of the Dixon resultant is the Cayley’s (1865) formulation of Bezout’s method (Bikker and Uteshev, 1999) for solving two polynomial equations. Dixon (1908) generalized Cayley’s approach to systems of three polynomials with three unknowns. Employing Dixon’s method, variables from a polynomial system can be eliminated using the Dixon resultant. One way to compute the resultant of a given polynomial system is to construct a matrix with the property that whenever the polynomial system has a solution, such a matrix is a rank deficient, thereby implying that 3 Solving Polynomial Systems the determinant of any maximal minor is a multiple of of Equations the resultant. The Dixon matrix is constructed through the computation of the Dixon polynomial, using some Solution of nonlinear systems of equations is an indis- auxiliary variables. The resultant is the determinant of pensable task in all geosciences. In geodesy, ranging, the Dixon matrix. In Mathematica, the Dixon resultant was impleresection, intersection and coordinate transformation problems also lead to nonlinear systems of equations. mented by Nakos and Williams (1997, 2002). The folTechniques to solve these systems, e.g., Groebner ba- lowing function sis, Sylvester resultant and Multipolynomial resultant Dixon Resultant((poly1 , poly2 , . . .), (Macaluay, Strumfels formulations) are described in (1) (y1 , y2 , . . .)(σ, ξ, . . .)) Awange (2002). Another method, not mentioned earlier in the geodetic literature, is the Dixon (1908) refinds the Dixon resultant in which the yi variables have sultant. Selecting a proper method amongst these leads been eliminated from the polynomials and σ, ξ are auxto exact solutions of many nonlinear problems. iliary variables. The basic concept of the solution of a multivariate Considering the following system: nonlinear system of polynomial equations, when the number of equations (n) is equal to the number of varif (x, y, z) = 0 ables (m), is to reduce the number of the variables from g (x, y, z) = 0 (2) m to 1. After this reduction, the resulting monomial can h (x, y, z) = 0, be solved easily by mostly, numerical methods. Two powerful computer algebra tools for solving the Dixon polynomial is defined by: nonlinear systems of equations are the Groebner ba1 sis and Dixon resultant. Both methods have advanδ(x, y, z, σ, ξ ) = × − σ (x ) (y − ξ ) tages and disadvantages. For some problems (e.g., re⎛ ⎞ section), Groebner basis is more effective, while for (3) f (x, y, z) g (x, y, z) h (x, y, z) other problems (e.g., transformation) Dixon resultant det ⎝ f (σ, y, z) g (σ, y, z) h (σ, y, z) ⎠ . proved to be more convenient. Generally they provide f (σ, ξ, z) g (σ, ξ, z) h (σ, ξ, z) the same results, but there are considerable differences in the computational times (see e.g., Pal´ancz et al. where σ, ξ are the auxiliary variables. For every com(2007) in press). mon zero of f, g and h the Dixon polynomial is zero

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for all values of σ and ξ . Hence, at a common zero δ (x, y, z, σ, ξ ) = yz − 2x 2 yz + yξ one gets a homogeneous linear system by setting the −2x 2 yξ + zσ + ξ σ − 2x 2 ξ σ i i power products σ ξ equal to zero. The corresponding (6) determinant of the coefficient matrix is the Dixon resultant. The vanishing of the Dixon resultant is a necNow y and z variables can be eliminated. Consideressary condition for the existence of a common zero of ing that f, g and h.     Employing the method for a trivariate polynomial 2 2 δ y, z, σ, ξ = σ ξ 1 − 2x + yξ 1 − 2x (x, ) system as an example:     +zσ 1 − 2x 2 + yz 1 − 2x 2 , f = x 2 + y2 − 1 (7) (4) g = x 2 + y2 − 1 h = x 2 + y2 − 1 The corresponding homogeneous linear system of equations can be represented by the next structure: The graphical interpretation of the solution can be seen in Fig. 1. The Dixon polynomial is: δ(x, y, z, σ, ξ ) = ⎛

1 × (x − σ ) (y − ξ )

x 2 + y2 − 1 x 2 + z2 − 1 det ⎝ σ 2 + y 2 − 1 σ 2 + z 2 − 1 σ2 + ξ2 − 1 σ2 + ξ2 − 1

⎞ y2 + z2 − 1 σ 2 + z2 − 1 ⎠ σ2 + ξ2 − 1 (5)

After simplification:

y0 z0 0 0 0 1 − 2x 2

σ 0ξ 0 σ 1ξ 0 σ 0ξ 1 σ 1ξ 1

y1 z0 0 0 1 − 2x 2 0

y0 z1 0 1 − 2x 2 0 0

y1 z1 1 − 2x 2 0 0 0

The Dixon matrix is consequently given by: ⎞ 0 0 0 1 − 2x 2 ⎟ ⎜ 0 0 1 − 2x 2 0 ⎟ M =⎜ 2 ⎠ ⎝ 0 0 0 1 − 2x 2 0 0 0 1 − 2x ⎛

(8)

and its determinant, the Dixon resultant:  4  2  2 D = 1 − 2x 2 = 1 − 2x 2 1 − 2x 2 = 1 − 8x 2 + 24x 2 − 32x 6 + 16x 8

(9)

The roots of this polynomial are shown in Fig. 2.

Fig. 1 Solutions of system of polynomial equations

Fig. 2 The x coordinates of the solutions

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In this way, the solution of the trivariate polynomial system has been reduced to finding the root of a monomial, the Dixon resultant.

5 Live Application of CAS

⎛

⎞ 0 −c b S = ⎝ c 0 −a ⎠ , −b a a R = (I3 − S)−1 (I3 + S)

(10)

where I3 is a 3 × 3 identity matrix. The symbolic result is obtained using one Mathematica command (see Fig. 3).

To solve nonlinear geodetic problems the Mathematica software is used as a Computer Algebra System by the authors to demonstrate how easy it is to solve these problems using CAS. In Sect. 4, the exact mathematical solution of a trivariate nonlinear system of equations using the Dixon resultant has been given. However, the solution of this problem is much easier using Mathematica. First the Dixon resultant is activated by typing: <
$ 1 1 x → − √ , x → √ , see Fig. 2. 2 2

Fig. 3 Calculating rotation matrix in Mathematica

6 Nonlinear Problems in Geodesy 6.1 Ranging by GPS

Position determination using the Global Navigation Satellite System (GNSS) is a very frequent task. The measured pseudoranges from the receiver to the In case one needs to compute the Dixon polynomial i-th satellite (di ) are related to the unknown position Eq. (6) or the Dixon matrix Eq. (8) these can be calcu- {X, Y, Z } of the receiver by: lated with single commands also in Mathematica soft ware. di = (ai − X )2 + (bi − Y )2 + (ci − Z )2 + T, The solution of this nonlinear problem was very (11) simple using CAS. The advantage of using a CAS system is evident not only in solving nonlinear alge- where {ai , bi , ci }, i = 0, . . . , 3 are the Cartesian coorbraic problems, but also performing symbolic matrix dinates of the i-th satellite. calculations, which are also very frequent in geodesy. The measured distance is influenced also by the For example, in the coordinate transformation problem satellite and receiver’s clock bias at the time of (Sect. 6.3) the rotation matrix R needs to be calculate measurement. The satellite clock bias can be modeled with the skew symmetry matrix S. To calculated the R while the receiver’s clock bias have to be considered matrix an inverse calculation, extraction and multipli- as an unknown variable T. Denoting x1 , x2 , x3 and x4 cation are needed: as the variables for the four unknown {X, Y, Z , T },

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then the polynomial equations ( f i = 0) for i = 0 . . . 3 are given as: f i = (ai − x1 )2 + (bi − x2 )2 + (ci − x3 )2 − (di − x4 )2 (12) Which are solved algebraically to obtain three linear equations for variables x1 , x2 , x3 with parameter x4 . The relations x j = g j (x4 ), j = 1, . . . , 3 will be computed employing the Dixon method for elimination of the other two variables. For example, the Dixon re- Fig. 4 Geometrical interpretation of the 3D intersection sultant containing only the variables x1 and x4 after the elimination of x2 and x3 is: in this case measurements are taken from the known points (P1 , P2 , P3 ) to the unknown point P0 . The distance observation equations are: −x1 a2,3 b1,3 c0,3 + x1 a1,3 b2,3 c0,3 + x1 a2,3 b0,3 c1,3 − x1 a0,3 b2,3 c1,3 − 2 − 2x S cos (y ) x22 = x12 + S12 1 12 12 x1 a1,3 b0,3 c2,3 + x1 a0,3 b1,3 c2,3 − 2 2 2 (14) x3 = x2 + S23 − 2x2 S23 cos (y23 ) x4 b2,3 c1,3 d3,0 + x4 b1,3 c2,3 d3,0 + (13) 2 2 2 x4 b2,3 c0,3 d3,1 − x4 b0,3 c2,3 d3,1 − x1 = x3 + S31 − 2x3 S31 cos (y31 ) , x4 b1,3 c0,3 d3,2 + x4 b0,3 c1,3 d3,2 − where the distance S1 = x1 , S2 = x2 , S3 = x3 . This b2,3 c1,3 e0,3 + b1,3 c2,3 e0,3 + b2,3 c0,3 e1,3 − expression is converted into polynomial form (see, e.g., b0,3 c2,3 e1,3 − b1,3 c0,3 e2,3 + b0,3 c1,3 e2,3 Awange et al. (2003) as: where a0,3 = 2 (a0 − a3 ) , a1,3 = 2 (a1 − a3 ) , a2,3 = 2 (a2 − a3 ) , b0,3 = 2 (b0 − b3 ) , b1,3 = 2 (b1 − b3 ) , b2,3 = 2 (b2 − b3 ) , c0,3 = 2 (c0 − c3 ) , c1,3 = 2 (c1 − c3 ) , c2,3 = 2 (c2 − c3 ) , d3 ,0 = 2 (d3 − d0 ) , d3,1 = 2 (d3 − d1 ) , d3,2 = 2 (d3 − d2 ) ,    e0,3 = d02 − a02 − b02 − c02 − d32 − a32 − b32 − c32 ,     e1,3 = d12 − a12 − b12 − c12 − d32 − a32 − b32 − c32 ,     e2,3 = d22 − a22 − b22 − c22 − d32 − a32 − b32 − c32 , which can be solved for x1 . Similarly, one can obtain the solution for the other two variables, for x2 and x3 . By substituting these results (x1 , x2 , x3 ) into the last nonlinear equation ( f 3 ), one gets a quadratic equation for x4 , which can be solved easily.

f 1 = x12 + b1 x1 − x22 + a0 f 2 = x22 + b2 x2 − x32 + b0 f3 =

x32

+ b3 x3 −

x12

(15)

+ c0

In order to obtain a monomial for x1 , one can eliminate x2 and x3 from the system. The Dixon resultant containing only x1 can be solved numerically to give triplet solutions. Similarly, one gets three solutions for x2 and x3 . Now, in order to find the unique solution, every possible combination should be considered and all of them should be substituted into the observation equations. The proper solution is the very triplet that gives practically zero (i.e., satisfies) for all equations. Therefore, the sum of remainders of every equation should also be small. The solution is the triplet that gives the smallest sum of the remainders.

6.3 Datum Transformations

The C7 (3) Helmert transformation uses three translation parameters (X 0 , Y0 , Z 0 ), three rotation parameters (a, b, c) and a scale parameter s to transform one set of In the intersection problem, angular (direction) obser- coordinates in a given system into another coordinate vations are considered as in the resection problem, but system:

6.2 Intersection

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⎛

⎞ ⎛ ⎞ ⎛ ⎞ xi Xi X0 ⎝ yi ⎠ = sR ⎝ Y i ⎠ + ⎝ Y 0 ⎠ . zi Zi Z0

(16)

The rotation matrix R can be expressed with the skew symmetric matrix in Eq. (10). For the determination of the 7 parameters of the transformation (a, b, c, X 0 , Y0 , Z 0 , s), one needs minimally (at least) three known points in both coordinate systems. Expressing the rotation matrix with the skew symmetric matrix, the nonlinear system for three points consists of 9 polynomial equations ( f i ), i = 1, . . . , 9. This system can be reduced by subtracting the equations. In this way, the translation parameters are eliminated: r1 = f 2 − f 8 = −cx1 − cx3 − cs X 1 + cs X 3 + y1 − y3 − sY1 + sY3 + az1 − az3 + as Z 1 − as Z 3 r2 = f 2 − f 5 = −cx1 + cx2 − cs X 1 + cs X 2 + y1 − y2 − sY1 + sY2 + az1 − az2 + as Z 1 − as Z 2 r3 = f 3 − f 9 = bx1 − bx3 + bs X 1 − bs X 3 − ay1 + ay3 − asY1 + asY3 +z1 −z3 − s Z 1 + s Z 3 r4 = f 6 − f 9 = bx2 − bx3 + bs X 2 − bs X 3 − ay2 + ay3 − asY2 + asY3 +z2 −z3 − s Z 2 + s Z 3 (17) Only four unknowns (a, b, c, s) remain to be solved in Eq. (17). The second order nonlinearity is caused by the parameters. Therefore, if the value of scale s is known, the system will be linear in a, b and c. Now the reduced system can be solved, eliminating the variables a, b and c, in order to get a monomial for s. The 9-parameter 3D affine transformation is the generalization of the 7-parameter similarity transformation model, where three different scales, represented by a diagonal scale-matrix, are used in the corresponding coordinate axes instead of one scale factor. CAS provides also symbolic solution to the 9-parameter transformation via the Dixon resultant or Groebner basis techniques.

7 Conclusions Applications of CAS to solving geodetic problems, such as ranging with GPS, intersection and 3D transformations, have been demonstrated. CAS is a very effective, user friendly tool, which does not require

programming knowledge. The user can formulate the problem interactively, in the usual mathematical way, and solve it with transparent built-in functions implemented on the bases of algorithms representing the state-of-the-art of mathematical methods. Acknowledgments The first author wishes to thank the Hungarian E¨otv¨os Fellowship for supporting her visit at the Department of Geodesy and Geoinformatics of the University of Stuttgart (Germany), where this work was undertaken.

References Awange, J.L. (2002) Groebner bases, multipolynomial resultants and Gauss-Jacobi combinatorial algorithms – adjustment of nonlinear GPS/LPS observations. Ph.D. dissertation. Geod¨atisches Institut der Universit¨at Stuttgart. Awange, J.L., E. W. Grafarend., and Y. Fukuda (2003). Closed form solution of the triple three-dimensional intersection problem. Zeitschrift f¨ur Geodaesie, Geoinformation and Landmanagement 128, pp 395–402. Awange, J.L., and E. W. Grafarend (2005). Solving Algebraic Computational Problems in Geodesy and Geoinformatics, Springer, Berlin. Bikker, P., and A. Yu. Uteshev (1999). On the Bezout construction of the resultant. Journal of Symbolic Computation, Vol. 28(1), pp 45–88. Cayley, A. (1865). On the theory of elimination. Cambridge and Dublin Mathematical Journal, III, pp 210–270. Dixon, A.L. (1908). The elimination of three quantics in two independent variables. Proceedings of London Mathematical Society Vol. 6, pp 468–478. Haneberg, W.C. (2004). Computational Geosciences with Mathematics, Springer, Berlin. Nakos, G., and R. Williams (1997). Elimination with The Dixon resultant. Mathematica in Education and Research, Vol. 6, pp 11–21. Nakos, G., and R. Williams (2002). A fast algorithm implemented in Mathematica provides one-step elimination of a block of unknowns from a system of polynomial equations, http://wolfram.com/infocenter/Articles/2597. Pal´ancz, B., P. Zaletnyik, J. L. Awange, and E. W. Grafarend (2007). Dixon resultant’s solution of sytems of geodetic polynomial equations. Journal of Geodesy (in press). Wester, M.J. (1999). Computer Algebra Systems: A Practical Guide, John Wiley & Sons, Chichester, United Kingdom, ISBN 0-471-98353-5.

Crustal Velocity Field Modelling with Neural Network and Polynomials Khosro Moghtased-Azar and Piroska Zaletnyik

Abstract A comparison of the ability of artificial neural networks and polynomial fitting was carried out in order to model the horizontal deformation field of the Cascadia Subduction Zone, as determined from GPS analyses of the Pacific Northwest Geodetic Array (PANGA). One set of data was used to calculate the unknown parameters of the model (training set 75 % of the whole data set) and the other set was used only for testing the accuracy of the derived model (testing set –25 % of the whole data set). The testing set has not been used to determine the parameters of the model. The problem of overfitting (see Kecman (2001)) (i.e., the substantial oscillation of the model between the training points, the same problem than polynomial wiggle (Mathews and Fink 2004) can be avoided by restricting the flexibility of the neural model. This can be done using an independent data set, namely the validation data (one third part of the training set). The proposed method is the so-called “stopped search method”, which can be used for obtaining a smooth and precise fitting model. However, when fitting high order polynomials, it is difficult to overcome the negative effect of the overfitting problem. Different order polynomial models and neural network models with different numbers of neurons were calculated. The best fitting polynomial model was 6th order with 28 parameters. The finally used Khosro Moghtased-Azar Department of Geodesy and Geoinformatics, Stuttgart University, Geschwister-Scholl Str. 24D, Stuttgart 70174–Germany, e-mail: [email protected] Piroska Zaletnyik De`partment of Geodesy and Surveying, Budapest University of Technology and Economics, H-1521 Budapest, P.O. Box 91, Hungary, e-mail: [email protected]

neural network model contained 7 neurons in it’s one hidden layer, with radial basis activation functions, with 31 parameters. These two models, with same order of numbers of parameters, were compared. Calculating the remained errors at the training points the two models had the same fitting precision. However according to the testing point’s results, the neural network model offered more reliable results, with 2–3 times smaller errors. The computations were performed with the Mathematica software, and the results are given in a symbolic form which can be used in the analysis of crustal deformation, e.g. strain analysis. Keywords Crustal deformation · Modelling · Neural network · Polynomial

1 Introduction It has been a problem to geodesists to find efficient solutions to approximate functions that define crustal deformations by continuous velocity fields, especially when dealing with continuously monitored process. A deformation object can be considered as a dynamic system (Welsch et al., 1996). In this case displacement vectors could be considered as inputs of the system and geometrical modelling of the deformed shape and deformation analysis could be described as outputs. Both inputs and outputs may be obtained as time sequences. Several models for describing deformation process in geodesy and geophysics exist in literatures. For instance, Zaiser (1984) computes displacement field in the context of arbitrary shaped geodetic networks and three-dimensional finite elements. Eren (1984) used

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polynomial approximation for the modelling of displacement field and Miima and Niemeier (2004) used neural networks for modelling of displacement field. Polynomials can fail to give suitable approximations for strongly non-linear functional mappings. In that case, neural networks can preferably be used. Over the past decade, their popularity steadily increased within various engineering disciplines (Shuhui et al., 2001; Cherkassky and Yunqian 2003; Feng et al., 2006). In this experiment we present, an example to modelling of non-linear function, which is illustrated by comparison of horizontal velocity field of Earth’s crust by polynomials and radial bases neural networks.

K. Moghtased-Azar and P. Zaletnyik

Pacific coast of North America. There is strong evidence supporting the claim that major megathrust earthquakes have repeatedly occurred throughout history in this region (Dragert, 1987; Wang et al., 1994; Miller et al., 2001). To study the tectonic mechanism in the region, many geodetic measurements have been made in the past, ranging from conventional techniques to those using the most modern technologies. These measurements, combined with theoretical deformation models, provide the means and foundation for eathquake studies. Geodetic, GPS, and strain data provide evidence for active crustal strain build up throughout the region. Global Positioning System (GPS) measurements to determine crustal strain rates were initiated in the Cascadia region (US Pacific Northwest and 2 Data Set south-western British Columbia, Canada) more than a decade ago, with the first campaign measurements The Cascadia Subduction Zone (CSZ), located in 1986 (Kleusberg et al., 1988) and the establishbetween two migrating blocks at a triple junction, de- ment of permanent GPS stations in 1991 (Dragert forms in response to superimposed forces of the North et al., 1994, 1995). Nowadays geodesy laboraAmerica, Juan de Fuca, and Pacific plates undergoing tory, Pacific Northwest Geodetic Array (PANGA), serves continuous GPS data for the Cascadia region boundary interactions, as illustrated by Fig. 1. This area is known to be one of the most seismi- (http://www.geodesy.cwu.edu/). Data from these GPS stations is used for crustal cally active areas in western Canada and the northwest deformation, plate motion, volcano monitoring, and coastal hazards studies. The laboratory’s primary scientific role is to support high precision geodetic measurements using the Global Positioning System (GPS) satellites, particularly for the study of earthquake hazards. This organization has deployed an extensive network of continuous GPS stations that measure crustal deformation along the CSZ. The authors considered these stations, which have nearly daily solutions at least through the period January 2002–2006. Figure 2 illustrates the horizontal velocity field along the Cascadia margin assuming the North American plate to be stable. Here, we used data from 82 permanent GPS stations. In other words, Fig. 2 shows site velocities for the 2002–2005 period relative to stable North America, as defined by a set of 82 fiducial stations. Continuous GPS results in the Pacific Northwest provide a remarkably coherent view of along-strike variation in Cascadia margin deformation which is characterised by different tectonic domains. Coastal stations in the northern and central parts of the margin Fig. 1 Tectonic map of Cascadia are strongly entrained in the Juan de Fuca – North

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Fig. 3 Training set indicated with small blue triangular mark, and testing set with bigger red triangular mark

set is used for determining the fitting model parameters (62 points – the “training set”) and the other set is America convergence direction, although the north- used for testing the model (20 points – “testing set”), ward component of station velocities increases from as illustrated in Fig. 3. The efficiency of the application of soft computing north to south. Inland stations have lesser motions; consistent with their structural domains from south to methods like Artificial Neural Networks depends connorth, while the California-southern Oregon boundary siderably on the representativeness of the learning sample set employed for training the model. In this study reflects a composite velocity. the partition into training and testing set was made manually because of the low number of the dispos3 Modeling of the Velocity Field able data. In case of more data for example a method based on the Coefficient of Representativity can be To represent the continuous velocity field, an adequate used (Dubois 2000). Different order polynomial models and neural netdeformation model has to be developed. Two different models are adopted here, polynomial fitting and the work models with different numbers of neurons were calculated. The used neural network model contained neural network model. The aim was to give an analytical solution, which radial basis activation functions and was determined by could be used to compute symbolic derivations of the restricting the flexibility avoiding the overfitting probvelocity vectors for subsequent strain analysis. There- lem with the so called “stopped search method” (Kecfore local interpolation techniques like kriging were man, 2001). not used here, not only for the reason that they cannot be given in an analytical form but also because they can 3.1 Polynomial Fitting create discontinuous surfaces (Meyer 2004). The input data are the 82 velocities derived from continuously observing GPS stations. To test the effi- As a classical approximation model, the 3D polynociency and accuracy of the models one part of the data mial fitting technique is used to generate continuous

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velocity field as a function of geodetic coordinates. Displacement vectors derived from GPS observations have east, north and up components in topocentric coordinates. For modelling the horizontal displacement field only the north and the east elements are used. Accuracy of modelling is defined by differences between true values and values estimated from 3D polynomial fitting. Polynomials from order 2 to 6 were tested, each fitted using a least squares technique with 62 input points. Higher order polynomials were not assessed due to limited input dataset, so the lack of statistical degrees of freedom in the adjustment, and due to the fact that the normal equations matrices of highorder polynomials to be solved become highly ill conditioned. The 6th order polynomial was therefore the best fitting model. In this case 28 points are needed, because a two-variable 6th order polynomial has 28 parameters. When there are more points than the number of parameters there is a possibility for adjustment calculation, i.e. for polynomial fitting. In this case 62 points were used for the adjustment. The calculations were carried out with the Mathematica software. The statistical data for the 3D polynomial fitting, which describe the quality of the estimation of the 62 training points, are listed in Table 1, including maximum error, minimum error, mean value and standard deviation. The statistics of the differences between the polynomial model and the real velocities for the 20 testing points is summarized in Table 2. Comparing the results of Table 2 with Table 3 we can see big differences. While the results of the training set are quite good, the errors of the testing set are 2–9 times higher.

Table 1 The statistics of the differences between polynomial model and actual velocities in the 62 training points Differences (mm/year)

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Std. dev

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−4.1 −7.9

3.9 8.2

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Table 2 The statistics of the differences between 3D polynomial model and actual velocities in the 20 easting points Differences (mm/year)

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Std. dev

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Table 3 The statistics of the differences between neural network model and real velocities in the 62 training points Differences (mm/year)

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Std. dev

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3.1.1 Overfitting Problem The overfitting problem (see Kecman (2001)) refers to the error of the training set decreasing while the error of the testing set is growing. In other words, the network excessively fits to the training points, as illustrated in Fig. 4. The result of Tables 1 and 2 means that the determined polynomial model works well only at the training points, but between them it does not work as well. The testing set, which was not used for the determination of the model, is also needed in order to quantify the quality of the results.

Fig. 4 Overfitting problem

3.2 Neural Network Model The artificial neural network provides a very effective tool for “artificial intelligence”. The expansion of the biological knowledge, particularly the recognition of the functioning of the human neurons, led to the definition of artificial neural networks. A very important characteristic of these networks is the approximation feature. These networks can solve problems which are too difficult to solve by applying algorithms. Using closely related input and output data (training set), unknown continuous functions can be approached with the least squares method efficiently (see e.g. Zaletnyik et al. (2004), Kavzoglu and Saka (2005)). For the approximation of the continuous surface velocity model, neural network can be used. The model

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consist of the positions of the basis functions (u i , vi ), the inverse of the width of the basis functions λi , the weights in output sum wi , and the parameters of the linear part w N +1 , a1 , a2 . This kind of network has 4N +3 parameters to be determined. The parameters of the activation functions and the weights are determined during the learning procedure using iteration steps in order to determine the optimal weights and parameters. During this iteration process the so called “backpropagation” algorithm is used, as a supervised learning technique for training artificial 3.2.1 Structure of Neural Networks neural networks (Kecman, 2001). The test points are used in the second part of the proA neural network model can be imagined like a system containing several layers with nodes called “neurons” cess, to test the results of the neural network. This part in the layers. In a network there is one input layer, one is very fast, and that is the reason why neural networks can be applied very successfully to many problems. or more hidden layers and one output layer (Fig. 5). Every neuron in the hidden layer has one or more input data and one or more output data. Neurons are the basic elements of the networks. Each neuron calculates 3.2.2 Stopped Search Method a weighted sum of its inputs. The function of a neuron is to sum up all its weighted input values and then gen- A central issue in choosing the most suitable model for erate an output via a transfer (or activation) function a given problem is selecting the right structural com(Mars et al., 1996). plexity. Clearly, a model that contains too few parameThe most important difference among the networks ters will not be flexible enough to approximate imporis the type of activation function. The most frequently tant features in the data. However, if the model contains used functions are the RBF (Radial Basis Function) too many parameters, it will approximate not only the and sigmoid functions (Kecman 2001). data but also the noise in the data. Mathematically, a two dimensional RBF network, Overfitting may be avoided by restricting the flexincluding a linear part, produces an output given by ibility of the neural model in some way (Kecman, 2001). The Neural Networks package in Mathematica ofN    fers several ways of handling the overfitting problem. 2 2 2 y (x1 , x2 ) = wi2 e−λi (x1 −u i ) +(x2 −vi ) All solutions rely on the use of a second, independent i−1 data set, the so-called validation data, which has not +w2N +1 − a1 x1 + a2 x2 (1) been used to train the model. One way to handle this problem is the “stopped search” method. “Stopped search” refers to obtaining the network’s where N is the number of neurons, each containing a basis function. The parameters of the RBF network parameters at some intermediate iteration during the training process, and not at the final iteration as it is normally done. During the training phase the values of x x x Input layer the parameters are changing to reach the minimum of the mean square error (MSE). Using validation data, it is possible to identify an intermediate iteration where Hidden layer the parameter values yield a minimum MSE. At the end of the training process the parameter values at this minimum are the ones used in the final network model. In order to avoid the overfitting problem by means Output layer y y of the stopped search method, more data is needed: Fig. 5 Structure of a general neural network a learning set and a validation set. Hence, the used was carried out using the Mathematica software and its Neural Network module. The advantage of this tool is the ability to perform symbolic calculations. Other neural network software packages solve problems like a “black box”, providing no access to the analytical function determined during the learning procedure. By using the Neural Network module of Mathematica one can give input variables and derive output as an analytical expression of the function.

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Fig. 6 Illustrations of learning set with small blue triangular mark, and validation set with bigger red triangular mark

training set (62 points) has to be divided into two sets, the first will be the learning set with 42 points and the remaining 20 points will be the validation set (Fig. 6). In Fig. 7 one can see the errors of the learning and the training set during the learning procedure of the neural network model for the northing component

Fig. 7 Errors of learning (continuous line) and validation set (dashed line) during stopped search method

velocities. The errors of the learning set (continuous line) decrease during the whole procedure, but the errors of the validation set (dashed line) are decreasing only until the 262nd iteration step, from that point they are growing. The maximum number of iterations was 500, but the best parameter set is the one calculated at the 262nd iteration step. In the model, in the hidden layer 7 neurons (nodes) were used. The selection of the number of neurons is basically dependent on the number of available known points. In fact, by having more known data one can increase the number of neurons. For the calculations only 42 points were used as a learning set, therefore one cannot use more neurons. The statistics of the residuals for the whole training set (62 points) are summarised in Table 3. The statistics of the differences between the neural network model and the real velocities values at the 20 testing points is summarized in Table 4.

Table 4 The statistics of the differences between neural network model and real velocities in the 20 testing points Differences (mm/year)

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Max

Mean

Std. dev

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−6.2 −4.6

8.3 8.9

−0.5 0.2

2.8 3.8

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4 Results There are remarkable differences from comparing results of the 3D polynomial fitting and neural network model. The numbers of parameters in both cases are of the same order (28 for 6th order polynomial fitting, and 31 for the neural network model, according to Eq. (1) with 7 neurons). According to the calculation of the errors between the models and real velocities at the training points the two models have almost the same fitting precision (see Tables 1 and 3). However according to the testing point’s results, the neural network model offered more reliable results, with 2–3 times smaller errors (see Tables 2 and 4). In case of the training set (with 62 points) the 3D polynomial fitting was slightly better. The difference between the two methods in this case was not significant. However, in the case of the testing set the differences are notable. Using the polynomial method the errors in the testing points, which were not used during the model determination procedure, are increasing 2–9 times as for the training set. Using the neural network model with the stopped search method the overfitting problem can be avoided and the errors for the training and testing set are of the same order of magnitude. Using the neural network model with the stopped search method the authors obtain a smooth and good fitting model, while in the case of the high order polynomial model there are substantial oscillations between the training points. See Figs. 8 and 9.

Fig. 8 Northing velocity model from polynomial fitting

Fig. 9 Northing velocity model from neural networks

5 Concluding Remarks The adaptation of neural networks to the modelling of the deformation field offers geodesists a useful tool for describing structural deformation. The overfitting problem can occur in the case of higher order polynomials, but the neural network approach overcomes this problem thanks to the stopped search method. The greatest advantage of this method is that the solution can be given as an analytical function (see Eq. (1)), which could be used to compute symbolical derivation of the velocity vectors for subsequent strain analysis (see e.g. Fig. 10).

Fig. 10 The symbolically calculated partial derivative of the model in northing direction

816 Acknowledgments The second author wishes to thank the Hungarian E¨otv¨os Fellowship for supporting her visit at the Department of Geodesy and Geoinformatics of the University of Stuttgart (Germany), where this work was undertaken. And we would also like to thank to the geodesy laboratory, Pacific Northwest Geodetic Array (PANGA) for the used data.

References Cherkassky, V., M. Yunqian (2003). Comparison of model selection for regression, Neural Computation 15, pp. 1691–1714. Dragert, H. (1987). The fall (and rise) of central Vancouver Island: 1930–1985, Canadian Journal of Earth Science, Vol. 24, pp. 689–697. Dragert, H., R.D. Hyndman, G.C. Rogers, K. Wang (1994). Current deformation and the width of the seismogenic zone of the northern Cascadia subduction thrust, Journal of Geophysical Research, Vol. 99, pp. 653–668. Dragert, H., X. Chen, J. Kouba (1995). GPS monitoring of crustal strain in southwest British Columbia with the Western Canada Deformation Array, Geomatica, Vol. 49, pp. 301–313. Dubois, G. (2000). How representative are samples in sampling network? Journal of Geographic Information and Decision Analysis, Vol. 4/1, pp. 1–10. Eren, K (1984). Strain analysis along the North Analtolian Fault by using geodetic surveys, Journal of Geodesy, Vol. 58(2). Feng, C.X., Z.G. Yu, A. Kusiak (2006). Selection and validation of predictive regression and neural network models based on designed experiments, IIE Transactions, Vol. 38(1), pp. 11–23. Kavzoglu,. T., M.H. Saka (2005). Modelling local GPS/leveling geoid undulations using artificial neural networks, Journal of Geodesy 78, pp. 520–527. Kecman, V. (2001). Learning and Soft Computing, A Bradford Book, The MIT Press, Cambridge, Massachusetts, London. Kleusberg, A., Y. Georgiadou, H. Dragert (1988). Establishment of crustal deformation networks using GPS, A case study, CISM Journal of ACSGC Vol. 42, pp. 341–351.

K. Moghtased-Azar and P. Zaletnyik Mars, P., J.R. Chen, R. Nambiar (1996). Learning Algorithms: Theory and Applications in Signal Processing, Control and Communications, CRC Press, Boca Raton, Florida. Mathews, J.H., K.D. Fink (2004). Numerical Methods Using Matlab, Fourth Edition, Prentice-Hall Inc., Upper Saddle River, New Jersey, USA, pp. 270–273. Meyer, T.H. (2004). The discontinuous nature of kriging interpolation for digital terrain modeling, Cartography and Geographic Information Science, Vol. (4) pp. 209–216. Miima, J.B., W. Niemeier 2004. Adapting neural networks for modelling structural behavior in geodetic deformation monitoring, ZfV, Vol. 129(3), pp. 160–167. Miller, M., D.J. Johnson, C.M. Rubin, H. Dragert, K. Wang, A. Qamar, C. Goldfinger (2001). GPS-determination of alongstrike variation in Cascadia margin kinematics: Implications for relative plate motion, subduction zone coupling, and permanent deformation, Tectonics, Vol. 20(2), pp. 161–176. Rogers, G.C. (1988). An assessment of the megathrust earthquakes potential of the Cascadia subduction zone, Canadian Journal of Earth Science, Vol. 25, pp. 844–852. Shuhui, L., D.C. Wunsch, M.G. Giesselmann (2001). Comparative analysis of regression and artificial neural network models for wind turbine power curve estimation, Journal of Solar Energy Engineering, Vol. 123(4), pp. 327–332. Wang, K., H. Dragert, H.J. Melosh (1994). Finite element study of uplift and strain across Vancouver Island, Canadian Journal of Earth Science, Vol. 31, pp. 1510–1522. Welsch, W., O. Heunecke (1999). Terminology and Classification of Deformation Models. 9th International FIGSymposium on Deformation Measurements, Olsztyn, 27. 30.09.1999. Proceedings, pp. 416–429. Zaiser, J. (1984). Ein dreidimensionales geometrischphysikalisches Modell f¨ur konventionelle BeobachtungenBeobachtungsfunktionale, Parametersch¨atzung und Deformationsanalyse, Deutsche Geod¨atische Kommission, Bayerischen Akademie Wissenschaften, Reihe C, Heft Nr. 298. Zaletnyik, P., L. V¨olgyesi, B. Pal´ancz (2004). Approach of the hungarian geoid surface with sequence of neural networks, International Archives of Photogrammetry and Remote Sensing, Vol. XXXV, Part B8, pp. 119–122.

Next-Generation Algorithms for Navigation, Geodesy and Earth Sciences Under Modernized Global Navigation Satellite Systems (GNSS) Marcelo C. Santos, Richard B. Langley, Rodrigo F. Leandro, Spiros Pagiatakis, Sunil Bisnath, Rock Santerre, ´ Marc Cocard, Ahmed El-Rabbany, Ren´e Landry, Herb Dragert, Pierre Heroux and Paul Collins

Marcelo C. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 Richard B. Langley Department of Geodesy and Geomatics Engineering, University of New Brunswick P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 Rodrigo F. Leandro Department of Geodesy and Geomatics Engineering, University of New Brunswick P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 Spiros Pagiatakis Department of Earth and Space Science and Engineering, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3 Sunil Bisnath Department of Earth and Space Science and Engineering, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3 Rock Santerre D´epartement des sciences g´eomatiques, Laval University, Pavillon Casault, Quebec City, PQ, Canada G1K 7P4 Marc Cocard D´epartement des sciences g´eomatiques, Laval University, Pavillon Casault, Quebec City, PQ, Canada G1K 7P4 Ahmed El-Rabbany Department of Civil Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, Canada M5B 2K3 Ren´e Landry ´ D´epartement de G´enie Electrique, Ecole de Technologie Sup´erieure, 1100 rue Notre-Dame Ouest Montr´eal, PQ, Canada H3C 1K3

Abstract The project on “Next-generation algorithms for navigation, geodesy and earth sciences under modernized Global Navigation Satellite Systems (GNSS)” has been under development within the scope of the Geomatics for Informed Decisions (GEOIDE) Network. The GEOIDE Network is part of the Networks of Centres of Excellence program (NCE) of the government of Canada. Networks of Centres of Excellence are unique partnerships among universities, industries, government and non-profit organizations aimed at turning Canadian research and entrepreneurial talent into economic and social benefits for all Canadians. Among its objectives, the GEOIDE Network intends to drive the research and development of new geomatics technologies and methods via multidisciplinary collaboration in a fully networked environment. The GEOIDE Network, having started operations in 1998, is currently in its Phase III. In this paper the authors will present an overview of the activities which have taken place under this project. They include: (a) Processing and analysis of real modernized GNSS data (L2C) as well as simulated modernized GPS and Galileo data; (b) Performing constellation system performance and augmentation analyses of the modernized GNSS; (c) Designing algorithms for single point and relative positioning using combined signals; and (d) Integrating legacy and modernized GNSS signals.

Herb Dragert Geological Survey of Canada, Natural Resources Canada, 9860 West Saanich Road, Sidney, BC, Canada V8L 4B2

Keywords Modernized GPS · L2C · PPP · Galileo

Pierre H´eroux Geodetic Survey Division, Natural Resources Canada, 615 Booth Street, Ottawa, ON, Canada K1A 0E9

1 Introduction

Paul Collins Geodetic Survey Division, Natural Resources Canada, 615 Booth Street, Ottawa, ON, Canada K1A 0E9

Positioning from space, currently predominantly based on the Global Positioning System (GPS), has become

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nearly omnipresent in an ever growing variety of applications, both civilian and military. The global value of GPS device production is expected to increase to US$21.5 billion in 2008, up from $13 billion in 2003, according to the Taiwan-based Industrial Economics and Knowledge Center of the Industrial Technology Research Institute. Forecasts also indicate a strong annual growth and an expected market size of US $757 billion by 2017 (RNCOS, 2005). This tremendous commercial push has led to modernization plans of the GPS itself as well as the design of a European-based system known as Galileo – not to mention the revamp of the Russian GLONASS and development of for the Chinese Beidou/COMPASS system. A project is underway within the scope of the Canadian Geomatics for Informed Decisions (GEOIDE) Network of Centres of Excellence to focus on the modernization of GNSS by the new GPS classes of satellites, Block IIR-M and Block IIF, and the Galileo system. Advantages coming from a modernized GNSS include improved positioning and navigation performance by means of doubling the availability of signals, impacting single point positioning accuracy, and faster and more reliable positioning techniques, including those based on carrier-phase ambiguity resolution. The overall objective of this research endeavour is related to the modernization of GNSS and its implications for Canadian society, enhancing the current capabilities in applications related to positioning, navigation, environmental monitoring and atmospheric sciences. There are several objectives to be accomplished within this project, each one having a strong link with the other. They are: 1.

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M.C. Santos et al.

5. Integration of legacy and modernized GNSS signals. 6. Design increased capabilities of error modelling that are essential for RTK positioning and navigation, as well as for static positioning. 7. Development of robust quality control for integrity monitoring of observations from hybrid navigation systems.

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The project entails various applications: Modernization of augmentation infrastructures, by testing the satellite-delivered Canada-wide Differential GPS (CDGPS) Service and other augmentation services, including how the Galileo signals will be integrated into these services. Determination of precise orbits for space geodetic missions. Exploring existing and investigating novel ways to use GNSS for environmental monitoring and atmospheric sciences. Testing newly developed algorithms for the analyses of continuous GPS network data for improved position accuracy (especially for sub-daily samples) and resolution of crustal motions in earthquakeprone regions.

This paper presents an overview of the activities that have been taking place under the scope of this project at several of the participating institutions. This paper has been structured in such a way that it summarizes these activities highlighting the most important accomplishments so far. References are made to papers that provide a more in-depth explanation of the various topics encapsulated in this paper. The activities have been divided by university just for organizational purposes even though there are activities Study of the availability, reliability, and accuracy of that have taken place with the participation of more Galileo and GPS Block IIR-M and Block IIF mea- than one group as facilitated by the very nature of the surements and their integration. network. Development of carrier-phase ambiguity resolution techniques, involving combination of multi-frequency carrier-phase observations using GPS and Galileo signal designs in order to arrive at 2 University of New Brunswick an optimal combination that allows fast, accurate and reliable ambiguity resolution for real-time During the past two years, within the framework of the kinematic (RTK) and post-processed data. GEOIDE project, the team at the University of New Analyze the impact of sources of error in the opti- Brunswick (UNB) has concentrated most of its effort mal combination and how to minimize them. into assessing the quality of the GPS L2C signal, and Design algorithms for single point and relative po- developing the precise point positioning (PPP) softsitioning using combined signals. ware, GAPS (GPS Analysis and Positioning Software),

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deviation of multipath and noise, computed using the 24-day period, for each one of to the elevation angle bins. The spread of the standard deviations inside each bin is represented by an error bar superimposed on each mean standard deviation and expanded to twice its initial magnitude. The standard deviation is roughly inversely proportional to the sine of the elevation angle. Precise point positioning (PPP) is one of the existing techniques to determine point coordinates using a GNSS receiver. In this technique, observations carried out by a single receiver are used in order to determine the three coordinate components, as well as other parameters, such as the receiver clock error and total neutral atmosphere delay. The technique is said to be “precise” because precise information, such as satellite orbit and satellite clock errors, is required in the data processing (because of this, one should have in mind that the usage of precise orbits, clocks and other precise products is an implicit procedure whenever the term “PPP” is used). More than that, PPP is “precise” also because the resulting parameter estimates are precise (and accurate). The idea behind UNB’s present work is that PPP, and therefore precise products such as the ones provided by the International GNSS Service (IGS), can be used not only for positioning, but for a variety of other tasks, such as signal analysis. The

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(Leandro et al., 2007b), targeting the inclusion of L2C and future modernized signals. The development of GAPS allowed us to publish the first estimates of the P2-C2 bias reported in the literature (Leandro et al., 2007a). L2C data has been collected by two Trimble receivers: first a Trimble R7 and a Trimble NetR5, both on loan from Cansel, a Canadian distributor of Trimble Navigation Ltd. products. We have also used NetRS and NetR5 data from the global IGS L2C Test Network. Among several distinct analyses, we want to showcase here the one involving C/A and L2C code multipath and noise levels based on the IGS L2C Test Network (S¨ukeov´a et al., 2007). For this analysis we chose four stations: FAIC, Fairbanks, Alaska; UNAC, Boulder, Colorado; UNB3, Fredericton, New Brunswick; and GANP, G´anovce, Slovakia. The first two stations use a Trimble NetRS receiver; while the other two stations use a Trimble NetR5 receiver. The standard deviations of multipath and noise values have been calculated for each of the 10-degree elevation angle bins, from 0 to 90◦ (9 bins); separately for the C/A and L2C pseudorange observations for a period of 24 days from December 1 through 24, 2006. The computation was carried out in two ways: (i) for each day separately; and (ii) for the 24-day periods as a whole. In Fig. 1, dots are used to show the mean standard

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fact that the observation model used for accurate error modelling has to take into consideration the several effects present in GPS signals, and that observations are undifferenced (there are no differences between receivers nor between satellite measurements), makes PPP a powerful data analysis tool which is sensitive to a variety of parameters. The PPP software developed at UNB, known as GAPS, has been designed and developed in order to take advantage of precise products, resulting in a data analysis tool for determining parameters in addition to position, receiver clock error and neutral atmosphere delay. These other estimated parameters include ionospheric delays, code biases, satellite clock errors, and code multipath plus noise among others. In all cases the procedures were developed in order to be suitable for real-time as well as post-processing applications. The ionospheric delay estimation uses a spherical ionospheric shell model, in which the vertical delays are described by means of a zenith delay at the station position and two horizontal gradients. This estimation makes use of carrier phase measurements only. The use of precise orbits and clocks is a key element for the quality control of the data which goes into the ionospheric estimation filter. The code multipath estimation is based on the assumption that the several effects present in code measurements are dealt with within PPP, but strongly based on carrier phase measurements. Based on this, these effects can be removed from pseudorange measurements, and the residual effect is essentially the code multipath plus receiver noise. The advantage of this technique is that it potentially can retrieve the mean multipath effect of a satellite arc, as opposed to other multipath retrieval techniques. This is possible only under the assumption that satellite orbit and clock errors are negligible for this application, which is achievable when using precise IGS products. Another effect which afflicts pseudorange measurements is the code bias. The code biases are important because satellite clock data products are computed using a certain arbitrary convention of observation type, such as P1 code measurements rather than C/A code. If the user’s receiver employs a different observation type than the one which was used to generate the satellite clock error corrections, he (or she) has to apply an offset to the correction, equivalent to the bias between the observations, to be able to use these clock products. One of the analysis tools of GAPS produces values of the satellite code biases,

M.C. Santos et al.

based on a positioning observation model, as opposed to being based on a satellite clock estimation observation model as is usually the case when bias values are provided to users. This estimation is made possible by means of a pseudo-observable, which reflects the mismatch between the precise clock reference code (e.g. P1 and P2 for IGS products), and the code the receiver is actually using (e.g. C/A). Regarding satellite clock error estimates, GAPS was enhanced in order to provide estimates of satellite clock offsets. This tool was created to provide a suitable approach for real-time carrier-phase-based satellite clock estimation. Once again, within the clock estimation filter; the assumption that precise orbit errors are reasonably small is made. It is worth mentioning that both code bias and clock error estimations can be improved by using data from several receivers. This is mainly due to multipath effects present in each individual receiver estimate, which is averaged out to a certain degree when combining uncorrelated (site-dependent) multipath effects from different stations, an obvious advantage of having data from a global network of receivers promptly available. However, even in a case where a combination of several stations is used, single-receiver data processing is still used as an initial step. This is an important aspect in terms of data processing performance, because solving receiver-dependent parameters in a single receiver step and solving network-dependent parameters (only) in a network processing step speeds up the data processing. GAPS is available online via a web interface, (http://gaps.gge.unb.ca) through the UNB Research and Learning Resources website, which can be easily run from anywhere, producing all data analysis results mentioned in this paper. In addition to signal analysis outputs, GAPS provides state-of-the-art PPP results, including position, receiver clock errors, and neutral atmosphere delays, in static or kinematic mode. All aspects briefly mentioned above make GAPS a novel application, with innovations mainly in the field of GPS data analysis, available to the user community. One of the main accomplishments in GAPS development is that it takes advantage of precise satellite products made available by the IGS, coupled with a very complete observation error modelling to make possible a variety of analyses based on GPS data. Future enhancements to GAPS will include the ability to process modernized GPS observables as well as those from other GNSS constellations.

Modernized Global Navigation Satellite Systems

´ 3 Laval University and Ecole ´ de Technologie Superieure During the last two years, within the framework of the GEOIDE project, Laval University’s team has been mainly working on two research themes. The first one was related to the contribution of modernized GPS signals and the addition of the future Galileo signals for the improvement of the instantaneous ambiguity resolution success rate (Boukhecha, 2006). Plans have been made to add GLONASS into this study. The second project dealt with the development of calibration methods for the receiver and satellite phase biases (Banville, 2007) – a key element for PPP ambiguity resolution. The current GPS constellation limits RTK solutions to relatively short baselines. This limitation is mainly due to the disturbing influence of the ionosphere on the resolution of the initial phase ambiguities, which is indispensable to reach the centimetre accuracy level. In the near future, GPS will be modernized by the addition of a third frequency and the European Union will deploy its own satellite navigation system Galileo. The interoperability of both systems and the availability of hybrid receivers which are able to simultaneously track GPS and Galileo satellites on all three frequencies will lead to substantial improvements. In order to investigate the performance of the future GNSS in terms of instantaneous ambiguity resolution, the upcoming constellations are simulated based on a Keplerian representation of the orbits. Through the Gauss-Markov model, the normal equation matrix

Fig. 2 Percentage of success of instantaneous ambiguity resolution for modern GNSS triple-frequency combinations as a function of the ionospheric noise

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for the unknowns containing the 3D-coordinates, the clock bias, the phase biases, the ambiguities, and zenith ionosphere biases is built. The latter are treated as an ionosphere-weighted model by applying constraints to these unknowns. Different values for the ionospheric constraint were adopted allowing for simulation of different levels of ionospheric disturbance present in the data. A modified search algorithm, adapted to the a priori case, is applied, leading to a theoretical discrimination factor used as a primary performance indicator. Figure 2 is an example of results obtained. It presents the percentage of success of instantaneous ambiguity resolution for modern GNSS triple-frequency combinations as a function of the ionospheric noise. The results were obtained by simulation using standard deviations of 30 cm and 3 mm for code and phase measurements, respectively. The standard deviation of the assumed ionospheric noise refers to a nominal wavelength of 1 m (Boukhecha, 2006). The simulation-based investigations showed that introducing Galileo and modernizing the GPS by adding a 3rd frequency will have a major impact on integer ambiguity resolution. In the mono-frequency case, a hybrid solution will guarantee a success of nearly 100 % in the absence of ionospheric noise. In the dualfrequency case, a hybrid solution will allow the resolution of the ambiguities even in the presence of moderate ionospheric noise. As expected, the best results were obtained for the hybrid triple-frequency case. One can reasonably expect that the future hybrid receivers which are able to measure on all 3 frequencies will become available. With such receivers, even over longer

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Fig. 3 Receiver widelane phase biases for three calibration sessions using a Spirent STR4760 GPS simulator (Banville, 2007)

baselines, instantaneous integer ambiguity resolution will become feasible. This work was done in collaboration with Prof. Ren´e Landry from Department of Elec´ trical Engineering at Ecole de Technologie Sup´erieure de Montreal. In PPP one currently refrains from integer ambiguity resolution. Real-valued biases are estimated instead. This allows the use of the ionosphere-free combination as input data. In addition, these biases absorb unmodelled errors such as receiver and satellite phase biases. The drawback, however, is a long convergence time of several hours before reaching cm-level accuracy. In order to solve for integer ambiguities in the absolute mode explicit modelling of different error sources becomes mandatory. This research concentrated on the calibration of the receiver and satellite phases biases. Methods to calibrate the satellite phase bias have been proposed during the last few years. A modified method has been suggested by Banville (2007) to improve the compatibility with PPP’s functional model. This part of the research will be reported in a future paper. Next, a method to calibrate the receiver phase bias will be discussed. For the calibration of the receiver phase biases, a simulator was used to generate errorless code and phase signals. Data from three 3-h calibration sessions have been gathered on two consecutive days using the Spirent STR4760 simulator at UNB and a NovAtel ProPack V3 dual-frequency receiver connected to the hardware simulator, collecting phase and code observations on L1 and L2. Between each calibration session the simulator and the receiver were turned off.

Figure 3 presents the estimated receiver widelane phase biases for the three sessions. The L1 and L2 receiver phase bias values can also be found in Banville (2007). The results show that the receiver phase biases were similar for each satellite over one session, but were different from session to session. As already mentioned, the simulated signals did not contain satellite phase biases. The results also indicated the presence of unmodelled, absolute receiver code biases affecting the estimated values of the receiver phase biases. In addition, clear evidence for a thermal sensitivity of the phase biases was found. Further work needs to be done to control the potential sources of electronic bias delays in the hardware simulator. Other investigations and tests for the calibration of the receiver phase biases will be conducted in collaboration with colleagues from the Department of Electri´ cal Engineering at Ecole de Technologie Sup´erieure de Montr´eal.

4 Ryerson University The Ryerson group has been working on high-rate precise orbital prediction and stochastic modelling of residual tropospheric delay. Precise real-time GPS orbit information is required for a number of applications, including real-time PPP, long range RTK and GPS weather forecasting. At

Modernized Global Navigation Satellite Systems

present, users may take advantage of the predicted part of the IGS ultra-rapid orbit for real-time applications. Unfortunately, however, the accuracy of the predicted part of the ultra-rapid orbit is limited to about 10 cm (the 24-h predicted part), which is not sufficient for the above applications. In this research, a 6-h predicted orbit is generated by extrapolating a concatenated group of previous precise ephemerides for 5 days. RINEX observation files corresponding to the same period of precise ephemeris are collected from 32 globally distributed tracking stations. Using the Bernese GPS data processing software those observation files were utilized to make further improvements of the prediction. The resulting prediction is finally refined by implementing a modular, three-layer feed-forward back-propagation artificial neural network (ANN). It is shown that the obtained precision of ANN-based prediction is less than 3 cm, which is superior to that of the IGS ultrarapid orbit (Yousif and El-Rabbany, 2007). Future research will enhance the orbital prediction through the use of 35 well-distributed IGS tracking stations (Yousif, 2007). Real-time and near-real-time precise GPS positioning requires shorter GPS solution convergence time. Residual tropospheric delay, which exists as a result of the limitations of tropospheric correction models, represents a limiting factor for rapid GPS solution convergence. In this research a stochastic modelling of the residual tropospheric delay is proposed. The NOAA tropospheric delay model is used to generate daily time series of zenith total tropospheric delays (ZTD) at ten IGS stations spanning North America for many days in 2006. The NOAA-based ZTD is then compared with the new IGS tropospheric delay product to obtain daily residual tropospheric delay time series at 5 min intervals. Estimates of a set of autocovariance functions of the unmodelled residuals are obtained, which are used to develop the empirical covariance functions, in the least-squares sense. Several empirical covariance functions are examined, including first- and secondorder Gauss-Markov processes, and exponential cosine function. Fitting results show that the exponential cosine function gives the best fit most of the time, while the second-order Gauss-Markov model gives the worst fit. The first-order Gauss-Markov fits are close to those of the exponential cosine. Additionally, the model coefficients seem to be season independent, but change with geographical location (Ibrahim and

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El-Rabbany, 2007). In future research, the developed covariance model will be used to stochastically model the residual tropospheric error through a modification of the observations’ covariance matrix. The effect of implementing the stochastic model on the solution convergence and accuracy estimation will then be examined.

5 York University The York team has been working on improving the measurement accuracy of atmospheric constituents as derived from GPS satellites and occultation measurements. The ultimate goal is to obtain 3D maps of refractivity and eventually vertical profiles of atmospheric parameters locally and/or globally. This can be achieved through: (1) improving the ionospheric calibration (i.e. accounting for higher order ionospheric effects), and (2) extracting atmospheric refractivity from bending angle profiles. As far as the refractivity profile determination is concerned, the basic software code has been written and tested to estimate bending angle profiles, through measurements of the atmospheric Doppler shift observed at a LEO satellite. Solution of a non-linear system of equations provides an estimate of the bending angle as a function of impact parameter, which is inverted with the Abel inversion algorithm (with spherical symmetry) to give refractivity profiles. The position and velocity vectors of both the transmitter (GNSS) and receiver (LEO) need to be obtained with high accuracy. The Bernese GPS software v5.0 is employed for the precise orbit determination of both satellites. For the ionospheric calibration we use the Bernese software for the estimation of the ionospheric total electron content (TEC) along the GPS-LEO raypath. This is required because higher order ionospheric effects need accurate estimates of the TEC. We have also been making progress on the modification of the Bernese software to account for higher order ionospheric corrections in order to examine their impact on the retrieved refractivity profiles. The leading role of York University in this project transpires through the modelling of the ionosphere and its impact on the retrieved atmospheric constituents, as well as the improvement of the 3D mapping of the atmospheric refractivity locally and/or globally by accounting for the ellipsoidal shape of the atmospheric layers.

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York’s contributions for the coming two years includes the development of a multi-GNSS constellation blended performance software covariance propagation simulator, designed to simulate the performance of various combinations of systems when particular processing methodologies are employed.

6 Concluding Remarks In this paper we have presented an abridged description of activities taking place under Project 31 of the GEOIDE Network Phase III. These activities involve research on L2C noise assessment, development of new tools for data analysis, development of PPP software packages, performance analysis of future GNSS with respect to ambiguity resolution, calibration methods for receiver and satellite biases, experiments pushing the limits of solution convergence time in real-time and near-real-time positioning, enhancing the quality of real-time orbits, improving the modelling of the atmosphere, and covariance analysis of various system combinations. The interested reader may find details of project activities on its website: gge.unb.ca/Research/GRL/GNSS/index.htm. As major contributions so far we can call attention to calibration of receiver phase biases with a simulator, the first determination of P2-C2 bias and the development of an artificial neural network-based orbit prediction model. Acknowledgments Primary financial assistance for the research described in this paper has been provided by the GEOIDE Network of Centres of Excellence. Additional funding was provided by the Natural Sciences and Engineering Research Council of Canada and the Canadian International Development Agency.

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References Banville, S. (2007). Aspects li´es a` la r´esolution des ambigu¨ıt´es de phase dans le positionnement ponctuel de pr´ecision (PPP) par GPS, M.Sc. Thesis, D´epartement des sciences g´eomatiques, Universit´e Laval, April, 149 pp. Boukhecha, M. (2006). R´esolution instantan´ee des ambigu¨ıt´es de phase pour des syst`emes hybrides GNSS modernes, M.Sc. Thesis, D´epartement des sciences g´eomatiques, Universit´e Laval, December, 132 pp. Ibrahim, H. and A. El-Rabbany (2007). Stochastic models for NOAA-based residual tropospheric delay, Proceedings of the International CIG/ISPRS joint conference on Geomatics for Disaster Management, May 23–25, Toronto, Ontario, Canada. CD-ROM. Leandro, R.F., R.B. Langley, and M.C. Santos (2007a). Estimation of P2-C2 biases by means of Precise Point Positioning, Proceedings of the ION 63rd Annual Meeting, Cambridge, Massachusetts, April 23–25, pp. 225–231. Leandro, R.F., M.C. Santos and R.B. Langley (2007b). GAPS: the GPS analysis and positioning software, Proceedings of the ION 20th International Technical Meeting of the Institute of Navigation Satellite Division, Fort Worth, Texas, September 25–28. RNCOS (2005). Global Positioning Services: Emerging Technological Trends, Research and Markets, RNCOS, Delhi, India. S¨ukeov´a, L., M.C. Santos, R.B. Langley, R.F. Leandro, O. Nnani, F. Nievinski (2007). GPS L2C signal quality analysis, Proceedings of the ION 63rd Annual Meeting, Cambridge, Massachusetts, April 23–25. Yousif, H. (2007). Prediction of hight rate GPS satellite orbit using artificial neural networks, M.Sc. E. Thesis, Ryerson University. Yousif, H. and A. El-Rabbany (2007). Precise orbital prediction using artificial neural networks. Ontario Professional Surveyor, Vol. 50, No. 3, pp. 6–10.

Linear Combinations for Differential Radar Interferometry L. Ge, H. Wang, H.C. Chang and C. Rizos

Abstract Synthetic aperture radar interferometry (InSAR) has been widely used over the past two decades for such geodetic applications as topographic mapping and ground deformation monitoring. In particular, in contrast to other geodetic techniques, differential InSAR has the capability of measuring surface deformation with a measurement precision of a few millimetres and a spatial resolution of a few tens of metres or less, over areal extents of thousands of square kilometers. However, the ground surface may deform by a large amount within a small area due to localised effects such as earthquakes, underground mining, groundwater extraction, and others, which may cause severe damage to manmade surface and underground structures. Current satellite radar interferometry, because of its single-frequency signal structure, cannot measure deformations with large horizontal gradients as they produce very dense fringe patterns. The upper limit of the deformation gradient is determined by the signal wavelength and pixel spacing. Although the longer wavelength of the radar signal is less susceptible to

high deformation gradients, loss of correlation often still occurs even when L-band imagery is used. In this paper, the authors propose a method to effectively monitor such large-gradient deformation using differential InSAR by linear combinations of interferograms acquired from dual-frequency sensors. Some simulated interferograms are generated through the combination of the commonly used radar wavelengths, for instance C, L1 , L2 , S and even P-band. Using appropriate linear combinations, the virtual wavelength of the combined interferogram is flexible enough to be used for different applications. From the simulated results, it is found that the linear combination technique is powerful and can improve the correlation, as well as making the phase unwrapping process much easier.

Keywords InSAR · Dual-frequency SAR · Linear combination · Large-gradient deformation

1 Introduction L. Ge School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia H. Wang School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia; Department of Surveying Engineering, Guangdong University of Technology, Guangzhou 510006, China H.C. Chang School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia C. Rizos School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia

Synthetic aperture radar interferometry (InSAR) has been widely used over the past two decades for such applications as topographic mapping and ground deformation monitoring (see, e.g., Massonnent and Feigl, 1998; Rosen et al., 2000). In particular, in contrast to other geodetic techniques, differential InSAR (DInSAR) has the capability of measuring surface deformation with a measurement precision of a few millimetres and a spatial resolution of a few tens of metres or less, over areal extents of hundreds to thousands of square kilometres.

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Coherence between master and slave images is the most important requirement for InSAR/DInSAR, as it directly affects the quality of the final interferometric products. However, coherence is often reduced by a number of factors, such as thermal noise decorrelation, geometric decorrelation, temporal decorrelation, volume decorrelation and data processing decorrelation (Zebker and Villasenor, 1992; Hanssen, 2001). In particular, the interferogram will also completely lose coherence when the maximum deformation gradient exceeds half a cycle per pixel, because only wrapped phase is recorded in the radar imagery (Massonnent and Feigl, 1998; Hanssen, 2001). In other words, DInSAR measures ground deformation by counting the number of interferometric fringes from the stable region to the most deformed region of the imaged area. When a large deformation occurs within a small area, producing a high deformation gradient, the fringes will be so dense that it is impossible to count them. The upper limit of the deformation gradient is determined by signal wavelength and pixel spacing. Although the longer wavelength of the radar signal is less susceptible to the high deformation gradient, loss of correlation often occurs due to the high coseismic strain near the epicentre (in the case of displacement due to seismic activity) or high subsidence gradient near the centre of a mine panel (in the case of subsidence from underground coal mining), even when L-band imagery are used (e.g., Ge et al., 2004). Figure 1 shows mine subsidence at Tahmoor in Australia (left Fig. 1a, Ge et al., 2007) and the coseismic deformation field of the 1997 Mw 7.5 Manyi, Tibet, earthquake from InSAR measurements (right Fig. 2b, Wang et al., 2007). Decorrelation occurred in the subsidence bowl (left) and the epicentre (right) due to large-gradient deformation.

(a)

Fig. 1 (a) Mine subsidence at the Tahmoor Colliery in Australia from C-band Radarsat-1 images, and (b) coseismic deformation of the 1997 Mw 7.5 Manyi, Tibet, earthquake from C-band ERS1/2 images. Decorrelation occurred in the subsidence bowl and

In this paper the authors propose a method to effectively monitor such large-gradient deformation in DInSAR results by linear combination of interferograms acquired from a dual-frequency sensor. Some interferograms are simulated through combination of the commonly used radar wavelengths, for instance C, L1 , L2 , S and even P-band.

2 Methodology As already mentioned, the maximum detectable deformation gradient depends on the signal wavelength and pixel spacing of the imagery. Larger maximum detectable deformation gradient is permissable for a longer wavelength and smaller scale pixel spacing. The relevant parameters for the currently available SAR satellites are summarised in Table 1. For ERS-1/2 imagery with a wavelength of 5.6 cm and pixel spacing of 7.9 m, decorrelation will occur if the deformation gradient is larger than about 1.8 × 10−3 . From Table 1 it can be seen that the upper limit is less than 1 × 10−2 for almost all of the satellites. In other words, a largegradient deformation limitation does exist for all satellites if using the conventional InSAR method. The concept of linear combinations for InSAR is similar to that developed for processing GPS dualfrequency range and phase observations (Parkinson and Spilker, 1996). Assume the same satellite is carrying two SAR sensors. The phases measured by these sensors to the same feature on the ground are ϕ1 and ϕ2 . The linear combination of the two phases is defined by:

(b)

epicentre (white pixels) due to large-gradient deformation. The bold black line in (b) indicates the location of the seismic fault trace

Linear Combinations

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Table 1 Relevant SAR satellite parameters Satellite Seasat ERS-1 JERS-1 ERS-2 Radarsat-1 Envisat ALOS/PALSAR Radarsat-2 TerraSAR-X

Launch date (dd/mm/yy)

Mission end date (dd/mm/yy)

Wave length (cm)

27/06/78 17/07/91 11/02/92 21/04/95 04/11/95 01/03/02 24/01/06 – 15/06/07

10/10/78 10/03/00 11/10/98 – – – – – –

23.5 5.66 23.5 5.66 5.66 5.66 23.5 5.66 3.1

ϕ = n 1 ϕ1 + n 2 ϕ2

(1)

where n 1 and n 2 are arbitrary numbers, ϕ is the combined phase. The substitution of the relations ϕi = f i t (t is the time between the transmission and return of the radar pulse) for the corresponding frequencies f 1 and f 2 gives:

ϕ = n1 f1t + n2 f2t = f t

Band L C L C C C L C X

Range pixel spacing (m)

Max deformation gradient (10−3 )

7.9 7.9 9.9 7.9 1.5–12.9 5.0–13.6 5.4–10.7 1.5–12.9 1

7 1.8 5.9 1.8 9.4–1.1 2.8–1.0 10.9–5.4 9.4–1.1 7.75

Note that the combined phase error will be changed as per Eq. (5), i.e., 

n 21 σ12 + n 22 σ22 > min(σ1 , σ2 )

(6)

To magnify the wavelength, the factors must satisfy the following conditions:

(2) n 1 f 1 + n 2 f 2 < min( f 1 , f 2 )

Therefore

(7)

Table 2 shows the wide-lane ( f w ) and narrow-lane ( f n ) combinations of the commonly used SAR frequencies. Using the dislocation model associated with the 1997 Manyi earthquake (Wang et al., 2007), a comis the frequency and parison can be made between the synthetic interferograms generated by single-frequency imagery and a c c (4) linear combination wavelength (Fig. 2). λ= = f n1 f1 + n2 f2 From Fig. 2 it can be seen that L1/L2 (λ = 19,24 cm) and S/L dual-frequency combinations are is the wavelength of the linear combination. more suitable for InSAR, which is similar to the case From Eq. (4) it can be seen that the wavelength with GPS. For the L1/L2 wide-lane combination the of the interferogram after linear combination will be wavelength is magnified from around 20 to 86 cm. either magnified, e.g. wide-lane combination (n 1 = Therefore the density of fringes in the combined in1, n 2 = −1), or reduced, e.g. narrow-lane combina- terferogram would be greatly reduced and the maxition (n 1 = 1, n 2 = 1). Therefore, the wavelength of mum detectable deformation gradient, which is prothe combined interferogram is flexible, and for differ- portional to the wavelength, becomes about four times ent applications the appropriate linear combination fac- the original detectable gradient. On the other hand, the tors can be selected. wavelength is reduced from around 20 to 10.7 cm for The resultant combined phase error σϕ can be de- the narrow-lane combination. Hence subtle deformarived using the error propagation law: tion can be detected for small events. In addition to wide-lane and narrow-lane combinations, other factors  can be taken into account to select the most appropriate σϕ = n 21 σ12 + n 22 σ22 (5) linear combination for a specific case study. f = n1 f1 + n2 f2

(3)

828 Table 2 Linear combinations of commonly used microwave SAR frequencies

L. Ge et al. ID

f 1 (MHz)

f 2 (MHz) λ1 (cm) λ2 (cm)

f w (MHz)

f n (MHz) λw (cm) λn (cm)

X&L X&S X&C C&S C&L S&L L1 & L2

9650 9650 9650 5331 5331 1544 1575

1270 1544 5331 1544 1270 1270 1228

8380 8106 4319 3787 4061 274 347

10920 11194 14981 6875 6601 2814 2803

3.11 3.11 3.11 5.62 5.62 19.42 19.03

23.76 19.42 5.62 19.42 23.76 23.76 24.41

3.58 3.70 6.94 7.92 7.38 109.41 86.40

2.75 2.68 2.00 4.36 4.54 10.65 10.70

Fig. 2 Synthetic coseismic interferograms associated with the Manyi earthquake with wide-lane and narrow-lane linear combinations of L1, L2, C, S, and X-band frequencies

3 Concluding Remarks The authors have proposed a linear combination method for differential radar interferometry. From the simulated coseismic interferograms it has been demonstrated that it is feasible to use this method to overcome the decorrelation due to large-gradient deformation. Comparing the interferograms generated by different combinations, it appears that the L1/L2 and L/S combinations are the best for InSAR. Large-gradient and subtle deformation can be detected using wide-lane and narrow-lane linear combinations respectively. In addition to improving coherence, the linear combination method is also useful for phase

unwrapping since sparse fringes result from using the wide-lane combination (Xu et al., 1994). Baseline/wavelength limits set how large a jump can be tolerated. However, in the event of either very large deformation (>> wavelength) or in the case of true layover, there will be phase discontinuities that cannot be resolved by sampling considerations. Futture work will focus on developing a phase unwrapping algorithm to integrate the original singlefrequency and linearly-combined interferograms. Once validated using real imagery acquired by the multifrequency SIR-C/X-SAR sensor, this method may be useful for the design of next generation satellite radar interferometry systems.

Linear Combinations Acknowledgments This research work is supported by the Cooperative Research Centre for Spatial Information (CRC-SI) Project 4.2, whose activities are funded by the Australian Commonwealth’s Cooperative Research Centres Programme. This study is also sponsored by a Hong Kong Research Council Grant in collaboration with the Chinese University of Hong Kong and a China 863 High-Tech Grant in collaboration with the Wuhan University.

References Ge L, Chang H C, Rizos C (2004) Satellite radar interferometry for mine subsidence monitoring. In: 22nd Seminar of Australian Institute of Mine Surveyors (AIMS), Wollongong, Australia Ge L, Chang H C, Rizos C (2007) Mine subsidence monitoring using multi-source satellite SAR images. Photogramm Eng Remote Sens 73(3):259–266

829 Hanssen R F (2001) Rader Interferometry: data interpretation and error analysis. Kluwer, Dordrecht Massonnet D, Feigl K L (1998) Radar interferometry and its application to changes in the Earth’s surface. Rev Geophy 36(4):441–500 Parkinson B, Spilker J (1996) Global positioning system: theory and applications. American Institute of Aeronautics & Astronautics, Washington, DC Rosen P A, Hensley S, Joughin I R, et al. (2000) Synthetic aperture radar interferometry. Proc IEEE 88(3):333–382 Wang H, Xu C, Ge L (2007) Coseismic deformation and slip distribution of the 1997 Mw 7.5 Manyi, Tibet, earthquake from InSAR measurements. J Geodyn 44:200–212 Xu W, Chang E C, Kwoh L K, et al. (1994) Phase-unwrapping of SAR interferogram with multi-frequency or multibaseline. In: Proceedings of the IGARSS’94, Pasadena, CA, 730–732 Zebker H A, Villasenor J (1992) Decorrelation in interferometric radar echoes. IEEE Trans Geosci Remote Sens 30(5): 950–959

Symposium GS005

The Global Geodetic Observing System (GGOS) Conveners: H.-P. Plag, M. Rothacher, R. Neilan

Recent Progress in the VLBI2010 Development ¨ D. Behrend, J. Bohm, P. Charlot, T. Clark, B. Corey, J. Gipson, R. Haas, Y. Koyama, D. MacMillan, Z. Malkin, A. Niell, T. Nilsson, B. Petrachenko, A. Rogers, G. Tuccari and J. Wresnik

D. Behrend NVI, Inc./NASA GSFC, Code 698, Greenbelt, MD 20771, USA J. B¨ohm Institute of Geodesy and Geophysics, University of Technology Vienna, AT-1040 Wien, Austria P. Charlot Observatoire de Bordeaux – CNRS/UMR 5804, BP 89, FR-33270 Floirac, France T. Clark NVI, Inc./NASA GSFC, Code 698, Greenbelt, MD 20771, USA B. Corey MIT Haystack Observatory, Westford, MA 01886-1299, USA J. Gipson NVI, Inc./NASA GSFC, Code 698, Greenbelt, MD 20771, USA R. Haas Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, Sweden Y. Koyama Kashima Space Research Center, NICT, 893-1 Hirai, Kashima, Ibaraki 314-8501, Japan D. MacMillan NVI, Inc./NASA GSFC, Code 698, Greenbelt, MD 20771, USA Z. Malkin Pulkovo Observatory, St. Petersburg 196140, Russia A. Niell MIT Haystack Observatory, Westford, MA 01886-1299, USA T. Nilsson Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, Sweden B. Petrachenko Geodetic Survey Division, Natural Resources Canada, Ottawa, ON K1A 0E9, Canada

Abstract From October 2003 to September 2005, the International VLBI Service for Geodesy and Astrometry (IVS) examined current and future requirements for geodetic VLBI, including all components from antennas to analysis. IVS Working Group 3 “VLBI 2010”, which was tasked with this effort, concluded with recommendations for a new generation of VLBI systems. These recommendations were based on the goals of achieving 1 mm measurement accuracy on global baselines, performing continuous measurements for time series of station positions and Earth orientation parameters, and reaching a turnaround time from measurement to initial geodetic results of less than 24 h. To realize these recommendations and goals, along with the need for low cost of construction and operation, requires a complete examination of all aspects of geodetic VLBI including equipment, processes, and observational strategies. Hence, in October 2005, the IVS VLBI2010 Committee (V2C) commenced work on defining the VLBI2010 system specifications. In this paper we give a summary of the recent progress of the VLBI2010 project. Keywords Geodetic VLBI · Next generation VLBI2010 · IVS · Monte Carlo simulations

1 Introduction Very Long Baseline Interferometry (VLBI) is a vital technique in the realization and maintenance of

A. Rogers MIT Haystack Observatory, Westford, MA 01886-1299, USA G. Tuccari Istituto di Radioastronomia INAF, IT-96017 Noto, Siracusa, Italy

J. Wresnik Institute of Geodesy and Geophysics, University of Technology Vienna, AT-1040 Wien, Austria

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the terrestrial and celestial reference frames (TRF and CRF). It uniquely provides the ICRF and the link between the ICRF and ITRF by providing the full set of Earth orientation parameters (EOP), in particular DUT1 and nutation. The current VLBI system was conceived and constructed mostly in the 1960’s and 1970’s. Aging antennas, increasing interference problems (RFI), obsolete electronics, and high operating costs make it increasingly difficult to sustain the current levels of accuracy, reliability, and timeliness. To alleviate this situation, in 2003 the International VLBI Service for Geodesy and Astronomy (IVS) (Schl¨uter and Behrend, 2007) initiated Working Group 3 (WG3) to define the next generation geodetic VLBI system, which has come to be called VLBI2010. Over a two-year period, WG3 looked into the conceptual aspects of a new VLBI system. In particular, the observing strategies, frequencies and timing, backend systems, data acquisition and transport, correlation and fringe-finding, data analysis, and data archiving and management were investigated. The WG3 work culminated in a “vision paper” (Niell et al., 2006) on the future VLBI system; with this final report the WG was closed in 2005. At the same time, in order to encourage the implementation of the recommendations of WG3, the IVS established the VLBI2010 Committee (V2C) as a permanent body of IVS. The V2C takes an integrated view of VLBI and evaluates the effectiveness of proposed system changes based on the degree to which they improve IVS’s final products. The V2C work went hand-in-hand with the gradual establishment of the Global Geodetic Observing System (GGOS) (Pearlman et al., 2006) of the International Accociation of Geodesy (IAG). The IVS aligned the design goals for the VLBI2010 system with the GGOS goals and strives for global baselines to be accurate to 1 mm and stable to 0.1 mm/yr. To reach these goals, a new set of criteria to measure the next generation geodetic VLBI system was established, based on:

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r

the science driven geodetic goals outlined in the NASA Solid Earth Science Working Group Report (SESWG)

To realize the demanding goals, the IVS VLBI2010 Committee investigated the various facets that the new system needs to have in order to fulfill the requirements. The major focus right now is on reducing the major error sources stemming from the atmosphere, the instrumentation, and the structure of the radio sources. Furthermore, the following strategies are investigated:

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a reduction of the random component of the delayobservable error, e.g., the per-observation measurement error, the stochastic properties of the clocks, and the unmodeled variation in the atmosphere a reduction of systematic errors, e.g., the thermal and gravitational deflection of the antenna, drifts of the electronics, and radio source structure an increase of the number of antennas for geodetic VLBI and an improvement of their geographic distribution an increase of observation density, i.e., the number of observations per unit time a reduction of susceptibility to external radiofrequency interference

2 Increasing the Number of Observations per Day Increasing the number of observations per day depends on three factors:

r r r

decreasing the length of time to slew from source to source decreasing the time to acquire data while on a source improving the scheduling algorithm

Although the steps taken to increase the number of observations have been successful, the effort has shown that more work needs to be done to optimize the sky the recommendations for future IVS products de- coverage in the short (< 10 min) intervals correspondtailed in the IVS Working Group 2 Report (WG2) ing to the atmospheric variability, perhaps at the expense of the total number of observations. Figure 1 (Schuh et al., 2002) the requirements of the Global Geodetic Observing shows the number of observations as a function of System project (GGOS) of the International Asso- slew speed of the antennas and different observing strategies. ciation of Geodesy (IAG)

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Observations (1000's)

200

NM-NL-NS

175

NM-N

150

NM-OL-OS

125

RDV42

NM-OL-NS

Table 1 Specifications of the current VLBI system and the VLBI2010 system

OM-OL-OS

100 75 50 25 0 0

3

6

9

12

Parameter Antenna size Slew speed Sensitivity Frequency range Recording rate Data transfer

AZ Slew Rate (deg/sec)

Fig. 1 Number of observations as a function of slew speed of the antennas. All schedules were generated assuming identical 12 m antennas. Improvements in hardware, software and source selection all contribute to an increase of the number of observations. Shown are the effects of OM (old mode, i.e. standard observing mode where data are recorded as they are observed), NM (new mode, i.e. “burst mode” where data are buffered and then written to a disk), OL (old list, i.e. old geodetic radio soure list), NL (new list, i.e. new extended geodetic radio source list), OS (old scheduling, i.e. standard scheduling strategy), and NS (new scheduling, i.e. new scheduling strategy). For comparison we also include the results for RDV42, a recent VLBI session involving 10 VLBA stations (slew speed 1.5 deg/s) and 9 other stations (various slew speeds) as a red dot. The remaining curves (from bottom to top) are: (1) OM–OL–OS; (2) NM–OL–OS; (3) NM–OL–NS; (4) NM–NL–OS; (5) NM–NL–NS

3 Decreasing the per Observation Measurement Error Recent technological advances have made it economically feasible to simultaneously use several (four or more) frequency bands spread across a wide frequency range, e.g. 2–15 GHz. Analysis shows that, with such systems, VLBI phase ambiguities can be reliably resolved, resulting in typical delay precision of about 2.5 ps (<1 mm) (Petrachenko, 2007). For comparison, with current group delay systems, the per-observation delay measurement error is typically about 10–30 ps. Table 1 gives an overview of the key-parameters of the current VLBI system and the planned VLBI2010 system. To demonstrate that this broadband delay concept will work in practice, NASA is supporting an R&D project to build a proof-of-concept “broadband” system and to evaluate its performance under real-world conditions. Major challenges for the concept include successful operation in the presence of RFI, realistic radio source structure, and uncalibrated instrumental offsets.

Current VLBI system 5–100 m dish ∼20–200 deg/min SEFD 200–15 000 S/X-band 128, 256 Mbps Usually disk shipping some e-transfer

Planned VLBI2010 system ∼12 m dish ≤360 deg/min SEFD ≤ 2 500 ∼2–15 (18) GHz 8–16 Gbps e-transfer, e-VLBI disk shipping when required

The principal components of the new system are:

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Broadband dual linearly polarized feeds covering 2 to ∼15 GHz (see e.g. Fig. 2) Broadband low noise amplifiers for 2–15 GHz Digital backends with four 512 MHz channels each and 2 bits/sample Mark5B+ recorders (Whitney, 2007) with 2 Gbps

Initial tests will be performed on a ∼600 km long baseline using the Westford (near Boston) 18 m antenna and the GGAO (at Goddard Space Flight Center, Washington, DC) 5 m antenna. The proof-of-concept system will record both linear polarizations from four 500 MHz bands that can be independently positioned in the frequency range from 2 to 12 GHz. The aggregate data rate at each site will be 8 Gbps using four Mk5B+’s (Whitney, 2007) and two Digital Back Ends (DBEs) (Whitney et al., 2007).

Fig. 2 The ETS-Lindgren quad-ridge broad band feed-horn (ETS-Lindgren, 2007) inside a heat shield attached to the 15 K station of the cryogenic refrigerator. The dewar is on the right hand

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Detailed designs are at an advanced stage, parts are The clock values are simulated as the sum of a ranbeing acquired and assembled, and the first tests are dom walk plus an integrated random walk (Herring expected in autumn of 2007. et al., 1990). These processes are driven by white noise with a PSD (power spectral density) of 0.3 ps2 /s for the random walk and 0.1 (fs/s)2 /s for the integrated ran4 Simulations dom walk. The wet zenith delay values are simulated with a turbulence model that is described in detail in Nilsson As part of the development of specifications for the et al. (2007). This model has been empirically verified new VLBI2010 observing system, the IVS has been with microwave radiometry by Nilsson et al. (2005) performing simulations to evaluate the geodetic perand Nilsson et al. (2006). The turbulence model uses formance of VLBI2010 networks. Monte Carlo simulations with different system specifications have been as input station dependent structure constants Cn , troperformed by geodetic analysis using simulated data pospheric height h, saturation scale length L and wind consisting of troposphere, clock, and thermal noise speed and direction. The wind speed and direction and delay contributions. The o−c vector (observed minus the tropospheric height for the VLBI2010 simulations are taken from the ERA40 numerical weather model computed group delay) is set up as follows: (ERA40, 2007) of the European Center for MediumRange Weather Forecast (ECMWF, 2007). The struc  o − c = WZD2 · mfw2 () + CL2 (1) ture constants are derived by an empirical fit to high− (WZD1 · mfw1 () + CL1 ) + WNBsl resolution radiosonde data, and the saturation scale length is fixed to the values proposed by Treuhaft and For each observable, WZD1,2 and CL1,2 are the sim- Lanyi (1987). Once the o−c vector is created the simulations alulated wet zenith delay and clock values at station 1 and 2, respectively, and mfw1,2 () are the wet mapping low us to investigate optimal network antenna locafunctions for the elevation angle . The wet mapping tions, antenna sensitivities, slew rates and observing functions are assumed to be without error in our stud- schedules. Figure 3 compares the results of an analysis of real ies. For each baseline a white noise contribution WNBsl CONT05 data (which is the best VLBI data set to date), is added. 25 CONT05 real data CONT05 simulated data 16 station network simulated data wzd: turbulence model

20

Length Repeatability [mm]

Fig. 3 Baseline length repeatability as a function of baseline length: The blue dots show the analysis of real CONT05 data. The red crosses show the analysis of simulated CONT05 data using the Monte Carlo simulator with clock parameters of 1 × 10−14 at 50 min, a turbulence model for the tropospheric parameters, and delay measurement errors based on the real CONT05 observation errors. The green triangles show the simulation of a 16 station network of VLBI2010 antennas. Here the clock and atmosphere delays are simulated as for the CONT05 simulation, and a delay measurment error of 4 ps was used

15

10

5

0

0

2

4

6

8

Baseline Length [1000 km]

10

12

14

Recent Progress in the VLBI2010 Development

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a Monte Carlo simulation of the CONT05 data, and a Monte Carlo simulation of an anticipated VLBI2010 observing scenario. Both simulations were carried out with parameterizations of the atmospheres, clocks and measurement errors intended to be as realistic as possible. In both cases, the wet zenith delays are simulated using the turbulence model while the clocks are simulated using the random walk plus integrated random walk model constrained by an Allan Standard Deviation (ASD) of 1 × 10−14 at 50 min. In the case of the CONT05 simulation, the delay measurement errors are based on the observing errors of the real observations, which have a mean value over all observed baselines of Fig. 5 Example of a 16 station test network of VLBI2010 29 ps. For the VLBI2010 simulation, a delay measure- antennas ment error of 4 ps, which is realistic for phase delays, is used for all observations. stations. Simulation and analysis of the dependence of EOP quality on the network size have shown an ex5 Network Size pected twofold improvement in both EOP precision and accuracy compared to existing regular IVS netA series of Monte Carlo runs were performed to eval- works. Figure 5 shows an example of a test network uate the relative performance of networks of different with 16 stations with VLBI2010 antennas. size. For this purpose, a series of theoretical networks were generated that have roughly uniform global coverage and use plausible locations (e.g. the sites are on 6 Source Structure Corrections land and are co-located with GPS). These simulations indicate that the VLBI2010 network could improve the As we approach 1 mm precision, source structure bedetermination of the TRF scale to roughly several parts comes an ever more significant error component for in 1010 in a single session, compared to the current geodetic VLBI. Furthermore, it significantly degrades level of determination of the TRF scale of about less our potential to use the proposed “broadband delay” than one part in 109 . technique to access VLBI’s precise phase delay observFigure 4 shows the precision of the relative scale able. As a result, it will be important to apply appropriparameter as a function of the number of network ate modeling to correct for such source structure errors.

Relative Scale Precision

As an example, Fig. 6 (Charlot, 2002) shows in the top a contour plot of the X-band radio emission for a structure index 3 radio source, and in the bottom the 1.0 corresponding absolute values of structural delays for 0.8 this source for baselines up to the length of one Earth diameter. 0.6 There is also temporal variation of radio source 0.4 structure as for example shown in Fig. 7 (Charlot, 2002) for a typical X-band radio source that is regularly 0.2 observed by the IVS. This phenomenon is impossible 0.0 to address with the current VLBI system. With the new 40 0 20 30 10 VLBI2010 observing scenarios that include larger netNumber of Network Sites works and a manifold increase in the number of obserFig. 4 Terrestrial Reference Frame scale parameter precision from Monte-Carlo simulations. Precision is expressed relative to vations per session, UV-coverage will improve to the the √ precision of the 8-site network. The line represents a simple point where precise VLBI images of the ICRF sources (1/ n)-law can be constructed on a daily basis directly from the

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geodetic observations, therefore enabling source structure corrections to be calculated. Simulations are currently underway to evaluate the potential of this approach. Figure 8 shows the UV-coverage of a typical IVS experiment in today’s R1-series compared to the UV-coverage of a VLBI2010 schedule.

7 Summary As part of the IVS VLBI2010 renewal process studies are carried out to develop requirements for achieving 1 mm position accuracy on global length baselines. This is a work in progress. The highlights of this work include:

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Fig. 6 Top: Contour plot of the X-band radio emission for a structure index 3 source. Structure index 3 indicates that the median structure error over all possible baselines is between 10 and 30 picoseconds (ps). About 35% of the current ICRF defining sources are in this structure class. Bottom: The corresponding absolute values of structural delays for this source. The structural delay is plotted as a function of the length and orientation of the VLBI baseline projected onto the sky, expressed in millions of wavelengths (u, v coordinates), and the color coded scale ranges from 0 to 100 ps. The circle drawn in this plot has a radius equal to one Earth diameter, corresponding to the longest baselines that can be theoretically observed with Earth-based VLBI

Studies show that faster slewing antennas, burst mode recording and improved scheduling algorithms yield a quantum increase in VLBI observations per session. Proof-of-concept tests are in preparation to demonstrate the broadband delay technique for achieving an order of magnitude improvement in delay precision. Monte Carlo simulators have been developed to evaluate the benefits of improved instrumentation and scheduling strategies. The Monte Carlo simulators are being calibrated against CONT05 real data. Monte Carlo results indicate that increased observation density and delay precision lead to precisions of positions in the range 1.5–3.5 mm. Better scheduling and analysis strategies are expected to lead to further improvements. Initial studies indicate that the increase in observation density and larger networks will enable active source structure corrections. Further simulations are underway.

Acknowledgments J. B¨ohm and J. Wresnik are grateful to the Austrian Science Fund (FWF) for supporting this work within Project P18404-N10.

References Charlot, P. (2002). Modeling Radio Source Structure for Improved VLBI Data Analysis. In: International VLBI Service for Geodesy and Astrometry 2002 General Meeting Proceeding, N. Vandenberg and K. Baver (Eds.), NASA/CP-2002210002, pp. 233–242, ftp://ivscc.gsfc.nasa.gov/pub/generalmeeting/2002/pdf/charlot.pdf

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Fig. 7 Charlot (2002) VLBI images of 2200+420 at X-band for 10 successive epochs, spanning the period 1997.1–1998.6, as available from (Fey et al., 2002)

Fig. 8 Top: UV-coverage of a typical R1-experiment. Bottom: UV-coverage of a VLBI2010 schedule

ECMWF (2007). http://www.ecmwf.int/ ERA40 (2007). http://data.ecmwf.int/research/era/ ETS-Lindgren horn (2007) http://www.ets-lindgren.com/ page/?i=3164-05 Fey, A.L, D.A. Boboltz, R.A. Gaume, and K.A. Kingham (2002). USNO Analysis Center for Source Structure Report. In: International VLBI Service for Geodesy and Astrometry 2001 Annual Report, N. Vandenberg and K. Baver (Eds.), NASA/TP-2002-210001, pp. 249–252. ftp://ivscc.gsfc.nasa. gov/pub/annual-report/2001/pdf/acusnoss.pdf Herring, T.A., J.L. Davis, and I.I. Shapiro (1990). Geodesy by Radio Interferometry: The Application of Kalman Filtering to the Analysis of Very Long Baseline Interferometry Data. J. Geophys. Res., 95(B8), pp. 12561–12581. Niell, A., A. Whitney, B. Petrachenko, W. Schl¨uter, N. Vandenberg, H. Hase, Y. Koyama, C. Ma, H. Schuh, and G. Tuccari (2006). VLBI2010: Current and future requirements of geodetic VLBI systems. In: International VLBI Service for Geodesy and Astrometry 2005 Annual Report, D. Behrend and K. Baver (Eds.), NASA/TP-2006214136, pp. 13–40. ftp://ivscc.gsfc.nasa.gov/pub/annualreport/2005/pdf/spcl-vlbi2010.pdf Nilsson, T., L. Gradinarsky, and G. Elgered (2005). Correlations Between Slant Wet Delays Measured by Microwave Radiometry. IEEE Trans. Geosci. Remote. Sens., GE-43(5), pp. 1028–1035. Nilsson, T., G. Elgered, and L. Gradinarsky (2006). Characterizing Atmospheric Turbulence and Instrumental Noise Using Two Simultaneously Operating Microwave Radiometers. In: Proc. 9th Specialist Meeting on Microwave Radiometry and Remote Sensing Applications, MicroRad 2006, pp. 270–275. Nilsson, T., R. Haas, G. Elgered (2007). Simulations of Atmospheric Path Delays Using Turbulence Models. In: Proc. of the 18th European VLBI for Geodesy and Astrometry Working Meeting, J. B¨ohm, A. Pany, H. Schuh (Eds.), Geowissenschaftliche Mitteilungen, Schriftenreihe der Studienrichtung Vermessung und Geoinformation, Technische Universit¨at Wien, 79, pp. 175–180. http://mars.hg. tuwien.ac.at/∼evga/proceedings/S64 Nilsson.pdf Pearlman, M., Z. Altamimi, N. Beck, R. Forsberg, W. Gurtner, S. Kenyon, D. Behrend, F.G. Lemoine, C. Ma, C.E. Noll, E.C. Pavlis, Z. Malkin, A.W. Moore, F.H. Webb, R.E. Neilan, J.C. Ries, M. Rothacher, and P. Willis (2006). Global Geodetic Observing System–Considerations for the Geodetic Network Infrastructure. Geomatica, 60(2), pp. 193–204. Petrachenko, B. (2007). Achieving a Quantum Leap in Observation Density. In: International VLBI Service for Geodesy

840 and Astrometry 2006 General Meeting Proceedings, D. Behrend and K. Baver (Eds.), NASA/CP-2006214140, pp. 67–71, ftp://ivscc.gsfc.nasa.gov/pub/generalmeeting/2006/pdf/petrachenko.pdf Schl¨uter, W., and D. Behrend (2007). The International VLBI Service for Geodesy and Astrometry (IVS): Current Capabilities and Future Prospects. J. Geodesy, 81(6–8), pp. 379–387. Schuh, H., P. Charlot, H. Hase, E. Himwich, K. Kingham, C. Klatt, C. Ma, Z. Malkin, A. Niell, A. Nothnagel, W. Schl¨uter, K. Takashima, and N. Vandenberg (2002). IVS Working Group 2 for Product Specification and Observing Programs. In: International VLBI Service for Geodesy and Astrometry 2001 Annual Report, N. Vandenberg and K. Baver (Eds.), NASA/TP-2002210001, pp. 13–45, ftp://ivscc.gsfc.nasa.gov/pub/annualreport/2001/pdf/wg2 report 2.pdf Treuhaft, R.N. and G.E. Lanyi (1987). The Effect of the Dynamic Wet Troposphere on Radio Interferometry Measurements. Radio Sci., 22(2), pp. 251–265.

D. Behrend et al. Whitney, A. (2007). The Mark 5 VLBI Data System. In: Proc. of the 18th European VLBI for Geodesy and Astrometry Working Meeting, J. B¨ohm, A. Pany, H. Schuh (Eds.), Geowissenschaftliche Mitteilungen, Schriftenreihe der Studienrichtung Vermessung und Geoinformation, Technische Universit¨at Wien, 79, pp. 33–38. http://mars.hg. tuwien.ac.at/∼evga/proceedings/S21 Whitney.pdf Whitney, A.R, S.S Doeleman, B. Fanous, H.F. Hinteregger, and A.A.E. Rogers (2007). A Wide-Band VLBI Digital Backend System. In: International VLBI Service for Geodesy and Astrometry 2006 General Meeting Proceedings, D. Behrend and K. Baver (Eds.), NASA/CP-2006214140, pp. 72–76, ftp://ivscc.gsfc.nasa.gov/pub/generalmeeting/2006/pdf/Whitney2.pdf

The Contribution of GGP Superconducting Gravimeters to GGOS David Crossley and Jacques Hinderer

Abstract The network of more than 24 superconducting gravimeters (SGs) of the Global Geodynamics Project (GGP) is available as a set of reference stations for studies related to time-varying gravimetry. The inherent stability of the SG allows it to detect signals from a sampling time of 1 s up to periods of several years with a time-domain accuracy of 0.1 μGal or better. SGs within the GGP network comprise a valuable set of stations for geodetic and geophysical studies that involve Earth’s surface gravity field. Experience has shown that SGs can be calibrated to an accuracy of 0.01–0.1 %, and that most instruments have a low, but well-modeled, drift of a few μGal/yr. For most purposes except the determination of an absolute gravity reference level, the SG is the best observation-style instrument we have today. SG data is now freely available, much of it going back to the early 1990’s, from the GGP database at ICET (International Centre of Earth Tides, in Brussels, Belgium) and GFZ (Potsdam, Germany). Frequently it is combined with other datasets such as atmospheric pressure and hydrology for studies of ground deformation and tectonics. One of the most interesting new ideas within GGOS (Global Geodetic Observing System) is the determination of the geocenter using a combination of satellite and ground-based gravimetry. The GGP network can provide a unique contribution through continuous data at the stations where absolute gravimeters (AGs) will be deployed. The combination David Crossley Department of Earth and Atmospheric Sciences, Saint Louis University, 3642 Lindell Blvd. St. Louis, MO 63108, USA, e-mail: [email protected] Jacques Hinderer EOST/Institute de Physique du Globe, 5 Rene Descartes, 67084 Strasbourg Cedex France

of the two instruments is necessary to ensure that AG measurements are referencing the mean station gravity and not short-term gravity perturbations due for example to hydrology or meteorology. Another promising application is the use of SG sub-networks in Europe and Asia to validate time-varying satellite gravity observations (GRACE, GOCE). Keywords Superconducting · Gravimetry · GGP · GRACE

1 Introduction The GGOS Reference Document (Plag and Pearlmanm 2007) is a draft document and hence cannot be cited directly, but it is the basis of the issues discussed in this article. The reference document describes, in great detail, the rationale for a future ideal network of high quality observing site for a number of important issues in geoscience such as mass changes at the Earth’s surface due to climate change. These observing sites are complemented by an array of satellites that measure the Earth’s gravity field (e.g. gravity recovery and climate experiment, GRACE) and observe the terrestrial reference frame (e.g. satellite laser ranging, SLR). GGOS, as a full component of the IAG, is an umbrella for the many geophysical and geodetic systems that are either in place, or need to be developed in the next decade (the target date is 2020). GGP is identified within GGOS as one of the groups responsible for monitoring the relative changes in the gravity field. GGP currently has about 25 operating superconducting gravimeters (SGs), a few of which will probably be replaced or relocated within the next few years (Fig. 1). If all existing and planned stations remain operational,

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Fig. 1 Distribution of SG stations: light circles are operational, dark circles are stopped, diamonds are new installations, and squares are planned

System). GGP groups have now become expert on the treatment of data and searching for small signals and interpreting many of the field variations. Such information is circulated in GGP Newsletters and the proceedings of GGP Workshops, as described in Crossley and Hinderer (2007). Many of the personnel doing work with SGs will inevitably be involved in 2 GGP Operations GGOS measurements, and thus be able to draw on GGP has been operating officially for just 10 years their GGP experience for GGOS projects. There are new aspects of GGP operations that can (Crossley et al., 1999; Crossley and Hinderer, 2007), be exploited for GGOS benefit. The first is to add GPS but has data archived back to the 1980’s for some indata from GGP sites that are perhaps not otherwise struments. This data base is the primary achievement of GGP, and is freely available 1 year after collection available. The second is to provide access to absolute by logging on to http://ggp.gfz-potsdam.de. From the gravimeter (AG) measurements at SG sites, those mea1 min uncorrected data it is relatively simple to make surement having been made with respect to the calibraobvious ‘corrections’ for missing data, or to make gaps tion of the SGs, or establishing the drift level of the SG for bad data. This is done on a regular basis by ICET (frequently both). These are tasks that GGP has long personal to provide ‘1 min corrected’ data that is partic- discussed (particularly the archiving of AG data), and ularly suitable for tidal analysis. The 1 min data is very if they deemed important to GGOS, this gives us useful good for short period studies such as tides (up to diur- incentive. nal periods), and studying atmospheric effects in gravity and connections with hydrology. The difficulty is to extract accurate long-timescale field variations where 2.1 SGs at Fundamental Stations the inevitable drift (however small) and disturbances and offsets need to be carefully determined. GGOS defines a fundamental station as a site where The latter is an aspect where GGP experience can at least 3 independent space-geodetic techniques are contribute to GGOS (Global Geodetic Observing co-located. Additionally, as stated in the Reference there could be as many as 30 GGP stations available to GGOS by about 2010. Here we consider to what use these stations can be put for the GGOS requirements to be realized by 2020.

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Table 1 GGP sites co-located with two or more space geodetic techniques Location

Country

Latitude

CA CB ES MC ME NY SU SY TC WU WE

Cantley Canberra Esashi Medicina Metsahovi Ny-Alesund Sutherland Syowa Concepcion Wuhan Wettzell

Canada Australia Japan Italy Finland Norway S. Africa Antarctica Chile China Germany

45.5850 −35.3206 39.1511 44.5219 60.2172 78.9306 −32.3814 −69.0067 −36.8437 30.5159 49.1440

284.1929 149.0077 141.3318 11.6450 24.3958 11.8672 20.8109 39.5857 286.9745 114.4898 12.8780

y y y y y y y y y y y

na na

Sunspot NM Tahiti

USA France

32.7660 −17.5769

−105.8200 −149.6063

y y

∗ Continuous

Longitude

GPS∗

Code

VLBI

y y y y y

SLR

OTHER

y

satellite DORIS tiltmeters

y

y

y y y

y

y

DORIS DORIS optical DORIS DORIS LLR, DORIS laser-gyro LLR, DORIS DORIS

recording.

Document, there ‘should be’ AG and SG observations and tide gauges ‘where possible’. GGOS lists 25 such stations in current operation. Only a few SGs are located at current fiducial geodetic sites, i.e. those with multiple large instruments such as VLBI, SLR, and DORIS, for geodetic work (Table 1). These sites are: Medicina, Metsahovi, Canberra, Ny-Alesund, Syowa, Tigo-Concepcion, and Wettzell. All GGP sites have GPS and meteorological equipment and many have some way to assess local hydrological variables such as groundwater. There is an increasing use of soil moisture meters within GGP, based on the need for such measurements in the demonstrated success of global hydrology models such as GLDAS (Rodell et al., 2004). We note also that many of the GGP sites have other geodynamical equipment such as seismometers and tiltmeters. A subset of new stations in Table 1 is being planned by individual groups. In the future, station Sutherland – that is now co-located with a number of optical telescopes – may become a fully operational GGOS site with SLR, LLR and possibly VLBI. A probable future GGP location will be Tahiti that is now a geodetic fiducial station. To date there has been no master plan for the development of the GGP network, and no declaration that fiducial stations need to include SGs. This is one of the items we hope that GGOS will be able to clarify in the near future. Thus under the present planning, the proportion of all GGP sites that are fiducial stations will remain at about 30 %. The interesting, and by no means easy-to-answer, question to be addressed within the gravity and geodetic communities is to what extent

does knowledge of time-variable gravity assist in the fundamental geodetic work of the fiducial stations? Continuous gravity variations (arising from local mass changes) are considered important by GGOS to improve the geodetic solutions, which is why an AG and SG are specified at each of the future GGOS fundamental stations. If the current GGOS plan to extend the number of fundamental stations to 30–40 by 2020 is feasible, then we need to extend the GGP network to more of the future GGOS stations. A question that arises with any observing system is its lifetime, defined through the performance of the instruments and the goals and resources of the institutions that support them. Within GGP the majority of the stations have continued recording for up to 10 years or more. The issue of the possible closure of ground stations, whether from changes in personnel or policies, has affected GGP operations, but not so far compromised the viability of the network. It would be a good idea if vulnerable ground stations that are essential to GGOS should be identified as early as possible so that appropriate countermeasures could be taken. GGP has been built from the goals of individual projects, but in future if GGP could have a global mission directly connected with GGOS, this may discourage established stations from fading away. Fiducial stations, or at least stations with multiple instruments, have a heavy investment, which is both an advantage and a disadvantage. Initial and continuing costs are substantial, and the investment demands continued support. On the other hand, the same investment and costs can be justified by a wider variety of projects, and such sites are probably less likely to

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lose instruments because they use resources more effi- are able to generate models at the 10-day time resolution (http://bgi.cnes.fr:8110/geoid-variations/ ciently than isolated installations. README.html), so the GGOS time specification appears realistic in this respect.

3 GGOS Gravity Field Specifications 3.1 Resolution

3.2 Accuracy and Noise Levels

Of central importance to GGOS is the question of what resolution (i.e. what time and distance scales) is required of the gravity field in order to address the important scientific questions. We agree with the GGOS emphasis on hydrology at regional scales as the most problematic of the tasks facing accurate geodetic measurements, and that spacecraft tracking is the most demanding external goal. GGOS suggests that the surface gravity field should be available at length and time scales of 50 km and 10 days. This resolution is not currently available by any combination of techniques (satellite or ground). Satellite gravimetry has vastly superior spatial coverage, but it is difficult to extract useful information at wavelengths as short as 50 km. GRACE can measure sub- μGal levels, but only at wavelengths of a 400–1000 km. Traditional GRACE solutions have been limited to 1 month field updates, but some recent GRACE-derived solutions

The accuracy also is important, but it is not simply an instrumental problem. GGOS rightly frames the question in terms of the need to know the geoid to a 1 mm level with a stability of 0.1 mm in order to have a TRF that is an order of magnitude more precise than the phenomena to be monitored. SGs and AGs can do spot ground measurements of the gravity field itself at accuracies of about 0.1 and 1 μGal respectively (the precision of the SG is much better, about 1–10 nGal, but this is difficult to verify observationally). Thus in SG-AG combination we can claim to be measuring the gravity field at accuracies that are typical (but not an order of magnitude below) ground noise levels at periods of seconds to years. The ambient noise levels are set by seismic energy release, ocean circulation, the atmosphere and different contributions to hydrology, all of which compete at about the 1 μGal level at a wide range of periods (Fig. 2). In the seasonal and secular

Fig. 2 Amplitude of known effects on surface gravity. Some signals are at discrete periods (lines), and others are equivalent normalized amplitude spectra (atmosphere, groundwater) or are represented by blocks

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period ranges, tectonics competes mainly with hydrology and atmospheric effects at the several- μGal level. We demonstrate these statements in Fig. 3 that shows a 10-year series of residual data (1 day samples) from Strasbourg, both in its original form, and with a linear trend removed. The purpose is to show that continuous variations of gravity occur at all timescales at typical levels of 1–3 μGal over periods of weeks with larger fluctuations of 5–10 μGal at longer periods known to be caused by the annual hydrology cycle. The inset of 6 months of data (at 1 h samples) shows variations at the few μGal level that are typical of SG data at SG stations, but are not generally identified with any one cause. These high accuracy recording are point measurements, and are dependent on the gravimeter location with respect to soil moisture (see later). From a GGOS perspective, it would be very difficult to establish the gravity field at 50 km spatial resolution with this type of SG precision.

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There are many examples (too many to be cited here) of the combined use of AG and SG measurements in the literature because almost all SGs are calibrated by an AG, often several times a year. The calibration aspect is required a few times after the SG is installed, but repeated calibrations have rarely yielded significant changes to the SG scale factor, as can be seen in Fig. 4 (Amalvict and Hinderer, 2007). Their overall weighted mean value is −79.14±0.30, which is within 0.07 % of the nominal value first measured and listed in the GGP headers for this station. It is interesting to note that the calibration is independent of whether individual drop means or set means are used. The optimum time to establish a calibration is about 5 days. The SG calibration also does not depend on which FG5 (the most common model of AG) is used for the calibration (Francis and van Dam, 2002), and a host of experiments suggests that the calibration can be done successfully to 0.1 % or better. Other than calibration, repeated AG measurements 4 Examples of GGP Applications to GGOS serve instead two other important functions – first as a check on major offsets in an SG series of a nonWe present some GGP results that are relevant to instrumental nature, (e.g. power supply problems or GGOS, primarily in the areas of combined AG-SG disturbances during He refills) and as a check on inobservations and hydrology, as the combination of strument drift, and second, to verify that the AG is instruments is best suited to long-term changes in the functioning as expected. This latter issue may seem gravity field at seasonal periods and longer.

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Fig. 4 Scale factor for SG CO26 in Strasbourg, calibrated by FG#206 over 8 years. (Amalvict and Hinderer, 2007)

surprising, but in reality AGs are subject to occasional offsets and level problems that must be verified either by intercomparison campaigns (e.g. Francis et al., 2005) or using an SG (e.g. Imanishi et al., 2002). An example of the latter is shown in Fig. 5 in which we show at Strasbourg SG continuous gravity,

Fig. 5 Gravity increase at Strasbourg from SG, AG, and hydrology models (Amalvict et al., 2006)

hydrology from two different models (LaD and GLDAS) and several AG ‘points’ from 3 different FG5s. The SG series is the same as the slightly longer data set in Fig. 3 (black curve). Note that part of the trend in Fig. 3, and all of it in Fig. 5 (the difference being SG drift), is real because the overall increase

The Contribution of GGP

in gravity at Strasbourg is verified by AG measurements. All 3 series agree to first order, suggesting that hydrology can explain most of the AG trend; the remainder is tectonically related. At the end of the series in 2007 (Fig. 5) the gravity loses the upward trend, for reasons that are still being considered. We note the AG measurements sometimes deviate from the SG series for no apparent reason, i.e. when there are no obvious offsets or disturbances in the SG series and when the AG set values seem ‘good’. Stray single measurements are expected, but note the cluster of AG values around day 53,004 that are several μGal high than the SG series. We end this brief review of AG-SG combinations by referring to our German colleagues who reported in a series of papers (e.g. Wziontek et al., 2006; Wilmes et al., 2007) on the combination of 4 AGs and a dualsphere SG in Bad Homburg, with strict processing and impartiality of the treatment of the results. Interestingly, they find that the SG calibration can be made successfully and consistently only when offsets of about 1–4 μGal are introduced into the AG series (after allowing for monument offsets and all other possible causes) over time spans of a year. We cannot pursue discussion on the precision of AGs, see the many other papers such as Van Camp et al. (2005), but simply

Fig. 6 Observed absolute gravity stations in 2004 occupied by the absolute gravimeters FG5-220 (IfE), FG5-221 (FGI). FG5-226 (UMB) (Timmen et al., 2006).

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stress the advantages of an SG to check for potential problems. Early on, Okubo et al. (1997) pointed out that an SG can be used to verify AG measurements to 1–2 μGal (the anticipated and oft-quoted accuracy) under controlled conditions. When so checked, this allows the AG data to be usefully combined with tide gauge and GPS data to determine the cause of sea level changes, an issue of central importance to GGOS. Nevertheless, it seems unavoidable to conclude that: (a) without frequent AG measurements at a site, questionable (i.e. not obvious) disturbances or offsets in the SG cannot be identified, and obviously no check on instrumental drift, and (b) without an SG (or a second AG) it is possible that unknown offsets may have crept into an AG determination. So SGs and AGs are complementary – they serve to check each other by entirely independent observations.

4.2 Tectonic Uplift Using AGs One example of a comprehensive AG campaign has been described by Timmen et al. (2006). The proposed network is shown in Fig. 6, consisting of many established stations and many others not yet occupied,

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particularly in the Eastern section, to complete the coverage. The campaign will establish 30 stations to be surveyed annually, with an estimated 3 μGal accuracy from a single determination. The authors strongly recommend parallel recording by two AGs where possible to improve the reliability of the network. Combined with GPS, such a network is a powerful tool for geodynamics even though the effort needed to continuously deploy the 3–4 AGs involved is considerable. Only one of the stations is tied to an SG, at Metsahovi. The target here is to validate GRACE measurements that are at the 1 μGal accuracy at wavelengths of 1000 km. Post-glacial uplift is associated with changes of about 0.6 mm yr−1 in the geoid and 2 μGal yr−1 changes in gravity, at the maximum amplitude. Uplift projects inevitably raise the question of the gravity/height ratio. Wahr et al. (1995) showed that the traditional free air effect of −0.31μGal mm−1 is modified by viscous vertical displacement, reducing the value to −0.15μGal mm−1 . A recent paper by de Linage et al. (2006) shows that the ratio is strongly modified by surface hydrology loading effects, with a global mean value of −0.86μGal mm−1 determined using the LaD model (Milly and Shmakin, 2002). The ratio varies according to the loading and basin size. Naturally the intercomparison of GRACE and surface gravity involves also knowing the ellipsoidal height changes of the AG from GPS height observations. Fukuda et al. (2007) showed some interesting contradictory results on uplift in Antarctica between GRACE solutions (positive gravity trend) and ground AG measurements (negative), where they suggest that satellite data may be used to assess the regional hydrology, leaving the ground results to deal with the local hydrology and height variations. It is more usual that AG projects such as that outlined in Antarctica by Rogister et al. (2007) must proceed without SG support.

4.3 Geocenter Variation by AG and GPS One of the projects identified by GGOS is to determine the geocenter position to the same accuracy as the geoid, i.e. 1.0 mm stable to 0.1 mm. Typically this parameter is a solution from SLR and similar techniques (e.g. Wu et al., 2006) but it is also possible to use ground gravimetry at stations where AG and GPS data are available to tie the center of mass (CM) to

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the the origin of the TRF. The accurate estimation of the gravity / height ratio (Plag and Pearlmanm, 2007) again involves the small amplitude of the desired signal (secular gravity) to noise levels of similar size (hydrology, atmosphere), at long periods. The best way to improve the gravity part of the gravity/height ratio is the co-location of different gravity techniques (ASSG) at a single station, or the use of close-packed arrays of instruments. Interestingly, the initial results of Plag et al. (2007) confirm the −0.86μGalmm−1 gravity/height ratio of de Linage et al. (2006), and this leads to a shift in the Z-component of the CM of about 1.8 mm/yr, similar to that found by other methods.

4.4 Practicality on AG-SG Combinations It is impractical to insist that an SG can be placed at every AG site, even for GGOS. Not every site is suitable for a continuous installation, particularly as SG measurements need to continue for some months or even years generally to be useful. It is reasonable, however, to suggest that as many SG stations as possible be included in GGOS projects that involve AG measurements, particularly as the latter are done regularly in any case. There are obvious advantages to GGOS if the GGP network can be expanded to more of the key AG sites where there is infrastructure to support an SG. For some of the GGOS projects (e.g. the ground-based estimation of the geocenter determination) it is not necessary to have the gravity/GPS measurements at space geodetic sites.

5 Hydrology Effects There are now many studies of the gravity effect of soil moisture and groundwater using a variety of instruments. All of them confirm the complexity of finding a simple admittance between surface gravity and a variable such as groundwater.

5.1 Gravity/Groundwater Admittance This relationship becomes especially complicated at sites where the gravimeter is not just close to the surface, but below it; for example in an underground bunker or the basement of a building. The problem

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was recognized in the analysis of SG data with the predictions of hydrological models such as LaD (Milly and Shmakin, 2002) and GLDAS (Rodell et al., 2004) in the studies of GGP stations in Europe (e.g. Crossley et al., 2006, Boy and Hinderer, 2006). Stations such as Strasbourg, Membach, and Moxa show a mixed hydrological signal that is more complicated than at Wettzell for example (Fig. 7), where the correlation is much higher and a Bouguer slab approximation for groundwater storage is more reasonable (e.g. Crossley, Xu and Van Dam, 1998). Recent studies at these stations have been done by Longuevergne et al. (2007) for Strasbourg, Meurers et al. (2007) for Membach, and Kroner and Jahr (2006) for Moxa). They all show

that detailed modeling can be used in explaining how the soil moisture and groundwater signal is divided both above and below the instrument. Only when a detailed model has been shown to work is it possible to consider the actual ‘correction’ of gravity data for hydrology (e.g. Meurers et al., 2007). Longuevergne et al. (2007) showed the agreement at the Strasbourg GGP station between rainfall, soil moisture, and gravity changes at the 1–5 μGal level (Fig. 8). Note the sudden reduction in gravity caused by soil moisture, here in a layer only a few m thick above the station. It is especially useful to see the AG measurements that are able to track the hydrology variations, and confirm the SG (and vice-versa). Without the SG,

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Fig. 8 Contribution of soil moisture to gravity at an SG below ground level (Longuevergne et al., 2007)

one might easily just ascribe the AG discrepancies to measurement errors. Clearly the continuous SG sampling, (1–5 s for gravity and 1 min for hydrology), and the accuracy of 0.1 μGal or better is ideally suited to study these effects. Together with readily available rainfall data, the response of a particular site to meteorological forcing can be studied in detail. The problem becomes more complicated if one wants to assess the contribution of local hydrology to gravity, say in a 1 km2 area surrounding the instrument. This type of study out of practicality uses portable spring gravimeters due to their ease of set up and relatively modest costs (cf. SG, AG), but they have other problems associated with limited accuracy and high drift. Smith et al. (2007) note the difficulties of measuring hydrology with the Scintrex CG-3M, whose nominal accuracy is only 5 μGal. They review the strategies for making corrections to find the desired signal whose size is at the ground noise level and accuracy limit of the instrument. Similar studies reported by Dijksma et al. (2006) using the same type of instrument located only 0.75 m below the surface, found gravity signals in the range 1–3 μGal that would have been closer to 10 μGal if the instrument had been located at ground level, with all hydrology below the instrument. A comprehensive study is currently underway in Moxa (Naujoks et al., 2006) using a set of 4 wellcalibrated and maintained Lacoste and Romberg ET (spring) gravimeters to monitor gravity at intervals of several months at 390 stations (!). The SG is near the bottom of a valley with steep, vegetation-dominated, sides, and water flow is all around the gravimeter. In general the porosity and permeability of the overburden

layers must be well known to do the modeling accurately. Their second paper (Naujoks et al., 2007) included detailed modeling and comparison with the ET network observations. They showed that 80 % of the gravity variations could be modeled by hydrological changes within a 0.5 × 0.5 km area around the SG. Further, the gravity variations were in the range 2–10 μGal, above the estimated 1 μGal accuracy of the Lacoste & Romberg ET (spring gravimeter) observations (after all corrections), and therefore sufficient to be compared with the models.

6 Conclusions Our main conclusions for AG measurements are thus: 1. hydrological variations – primarily in soil moisture and groundwater, but also in ice mass changes for stations such as NY and SY – are normally the largest unmodeled effect on AG measurements, 2. the conversion of rainfall to gravity effect at a station rarely can be done with a simple admittance; it requires detailed modeling using a high accuracy gravimeter such as an SG, and 3. when AG measurements are require to detect geophysical effects at the limit of AG precision (1–2 μGal), especially by measurements that are spaced more than a few days apart, the use of a co-located SG will significantly enhance the quality of the detection of long-term changes. In contributing to GGOS goals, we have seen that GGP can:

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1. share experience in use of the GGP database with Francis, O., T. van Dam, M. Amalvict, M. de Andrade Sousa, M. Bilker, R. Billson, G. D’ Agostino, S. Desogus, R. Falk, GGOS representatives (through Newsletters, GGP A. Germak, O. Gitlein, D. Jonhson, F. Klopping, J. KostWorkshops, etc.), elecky, B. Luck, J. M¨akinen, D. McLaughlin, E. Nunez, C. 2. record and report on all GPS measurements at GGP Origlia, V. Palinkas, P. Richard, E. Rodriguez, D. Ruess, stations – these are observed as height variations D. Schmerge, S. Thies, L. Timmen, M. Van Camp, D. van that contribute to gravity variations, Westrum and H. Wilmes, 2005, Results of the international ccomparison of absolute gravimeters in Walferdange (Lux3. record and report all AG measurements made at the embourg) of November 2003, in: IAG Symposia, Gravity, GGP sites – these would be benchmark measureGeoid, and Space Missions 129, 272–275, Springer, Berlin. ments (one point with error bar and supplementary Fukuda, Y., T. Higashi, K. Keiko Yamamoto, and S. Shuzo Takeinformation), moto, 2007, A strategy for detecting temporal gravity varia4. participate in future campaigns to inter-compare tion using AG, SG and space gravimetry, paper presented at First Asian Workshop on SGs, Taiwan, March 2007. AGs at a site, where there is an SG, and Hinderer, J. and D. Crossley, 2004, Scientific achievements 5. join initiatives in geodesy or tectonics where the use from the first period (1997–2003) of the Global Geodynamof an SG would significantly improve the interpreics Project using a worldwide network of superconducting tation of measurements from other instruments. gravimeters, J. Geodyn. 38, 237–262.

References Amalvict, M., J. Hinderer, and S. Rozsa, 2006, Crustal vertical motion along a profile crossing the Rhine graben from the Vosges to the Black Forest Mountains: Results from absolute gravity, GPS and leveling observations, J. Geodyn., 41, 358–368 Amalvict, M. and J. Hinderer, 2007, Is the calibration factor stable in time? paper presented at First Asian Workshop on SGs, Taiwan, March 2007. Boy, J.-P. and J. Hinderer, 2006, Study of the seasonal gravity signal in superconducting gravimeter data, J. Geodyn., 41, 227–233. Crossley, D., H. Xu, and T. Van Dam, 1998, Comprehensive analysis of 2 years of data from Table Mountain, Colorado, Proc. 13th Int. Symp. Earth Tides, 659–668., eds. Ducarme, B., and Paquet, P., Brussels. Crossley, D. J. et al., 1999, Network of superconducting gravimeters benefits several disciplines, EOS, 80, 121–126. Crossley D., J. Hinderer, J.-P. Boy, C. de Linage, 2006, Status of the GGP Staellite project, Bull. d’Inf. Marees Terr, 142, 11423–11432. Crossley, D. and J. Hinderer, 2007, Report of GGP activities to Commission 3, completing 10 years for the worldwide network of superconducting gravimeters, submitted to IAG Proceedings of the IUGG Meeting, Perugia 2007. de Linage C., J. Hinderer, J.-P. Boy, 2006, A search on the gravity/height ratio induced by surface loading: theoretical investigation and numerical applications, Bull. d’Inf. Marees Terr, 142, 11451–11460. Dijksma, R., G.H. Ros, B. de Jong, and P.A. Troch, 2006, Local water storage changes affect in-situ gravity observations at Westerbork, the Netherlands, submitted to PEPI. Francis, O. and T. van Dam, 2002, Evaluation of the precision of using absolute gravimeters to calibrate superconducting gravimeters, Metrologia, 39, 485–488.

Imanishi, Y., T. Higashi, and Y. Fukuda, 2002. Calibration of the superconducting gravimeter T011 by parallel observation with the absolute gravimeter FG5 210 – a Bayesian approach, Geophys. J. Int. 151, 867–878. Kroner, C. and T. Jahr, 2006, Hydrological experiments around the superconducting gravimeter at Moxa Observatory, J. Geodyn. 41, 268–275. Longuevergne, L., G. Ferhat, P. Ulrich, J.-P. Boy, N. Florsch, and J. Hinderer, 2007, Towards physical modeling of local-scale hydrological contribution of soils for precise gravimetric corrections in Strasbourg, paper presented at First Asian Workshop on SGs, Taiwan, March 2007. Meurers, B., M. van Camp, and T. Petermans, 2007, Correcting superconducting gravity time-series using rainfall modeling at the Vinna and Membach stations and application to Earth tide analysis, J. Geodesy, doi: 10.1007/s00190-007-137-1. Milly, C., and A. Shmakin, 2002. Global modeling of land water and energy balances. Part I: the land dynamics (LaD) model, J. Hydrometeorol. 3, 283–299. Naujoks, M., C. Kroner, T. Jahr, and A. Weise, 2006, From a disturbing influence to a desired signal: hydrological effects in gravity observations, Bull. d’Inf. Marees Terr. 142, 11359–11360 (Abstract only). Naujoks, M., C. Kroner, T. Jahr, P. Krause, and A. Weise, 2007, Gravimetric 3D modelling and observation of timedependent gravity variations to improve small-scale hydrological modeling, poster, session HW2004, IUGG XXIV General Assembly, Perugia, Italy. Okubo, S., S. Yoshida, T. Sato, Y. Tamura, and Y. Imanishi, 1997, Verifying the precision of a new generation absolute gravimeter FG5 – comparison with superconducting gravimeters and detection of oceanic loading tide, Geophys. Res. Lett. 24, (4), 489–492. Plag, H.-P. and M. Pearlmanm, 2007, The Global Geodetic Observing System: meeting the requirements of a global society on a changing planet in 2020; available on GGOS website, but not for direct citation. Plag, H.-P., C. Kreemer, and W. Hammond, 2007, Combination of GPS-observed vertical motion with absolute gravity changes constrain the tie between reference frame origin and

852 Earth center of mass, in: Report of the Seventh SNARF Workshop, held in Monterrey, California, 28 March, 2007 (also Geophys. Res. Lett., submitted). Rodell, M., P.R. Houser, U. Jambor, J. Gottschalck, K. Mitchell, C.-J., Meng, K. Arsenault, B. Cosgrove, J. Radakovich; M. Bosilovich, J.K. Entin, J.P. Walker, D. Lohmann, and D. Toll, 2004, The global land data assimilation system, Bull. Am. Meteor. Soc. 85 (3), 381–394. Rogister, Y., M. Amalvict, J. Hinderer, B. Luck, and A. Memin (2007), Absolute gravity measurements in Antarctica during the International Polar Year: in Antarctica: A Keystone in a Changing World – Online Proceedings of the 10th ISAES, edited by A.K. Cooper and C.R. Raymond et al., USGS Open-File Report 2007-1047, Extended Abstract 173, 4 p. Smith, A.B., J.P. Walker, A.W. Western, K.M. Ellett, R.B. Grayson, and M. Rodell, 2007, Factors affecting the detection of a soil moisture signal in field relative gravity measurements, EOS, Transactions American Geophysical Union, Western Pacific Meeting Supplement, Abstract H23C-02. Timmen, L., J. Gitlein, J. Muller, H. Denker, J. Makinen, M. Bilker, B.R. Pettersen, O.C.D. Omang, J.G.G. Svendsen, H. Wilmes, R. Falk, A. Reinhold, W. Hoppe, H.-G. Scherneck, B. Engen, B.G. Harsson, A. Engfeldt,

D. Crossley and J. Hinderer M. Lilje, G. Strykowski, and R. Forsberg, 2006, Observing Fennoscandian gravity change by absolute gravimetry, in: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles, IAG Sympoisia 131, 193–199. Van Camp, M., S.D.P. Williams, and O. Francis (2005), Uncertainty of absolute gravity measurements, J. Geophys. Res. 110, B05406, doi:10.1029/2004JB003497. Wahr, J.H. DaZhong, and A. Trupin, 1995, Predictions of vertical uplift caused by changing polar ice volumes on a vscoelastic earth, Geophys. Res. Lett. 22, 977–980. Wilmes, H., H. Wziontek, R. Falk, and P. Wolf, 2007, Investigations of gravity variations and instrumental properties by combing superconducting and absolute gravity measurements, paper presented at First Asian Workshop on SGs, Taiwan, March 2007. Wu, X., M.B. Heflin, E.R. Ivins, and I. Fukumori, 2006, Seasonal and interannual global surface mass variations from multisatellite geodetic data, J. Geophys. Res. 111, B09401, doi:10.1029/2005JB004100. Wziontek, H., R. Falk, H. Wilmes, P. Wolf, 2006, Rigorous combination of superconducting and absolute gravity measurements with respect to instrumental properties, Bull. d’Inf. Marees Terr. 142, 11417–11422.

Combined Analysis of Earth Orientation Parameters and Gravity Field Coefficients for Mutual Validation ¨ A. Heiker, H. Kutterer and J. Muller

Abstract The determination of the temporal variations of the Earth orientation parameters (EOP) and spherical harmonic coefficients of the gravity field are geodetic contributions to the analysis of global geodynamic processes. As the Earth’s tensor of inertia is functionally related both to the EOP via the Euler-Liouville equations and directly to the gravity field coefficients of degree 2 it allows the mutual validation of these two sets of parameters. This paper proposes a statistically founded method of combining the temporal variations of the EOP and of the gravity field coefficients of degree 2 by a leastsquares estimation based on the Gauss-Helmert model (condition equations with unknowns). Thereby statistically founded values for the unknown parameters can be derived, together with residuals for the observations, checkability and accuracy measures. The EOP and gravity field coefficients of degree 2 are introduced as observations together with correction terms such as excitation functions of atmosphere and ocean. The results of the Gauss-Helmert model computed with gravity field coefficients from GRACE and the C04 series from the International Earth Rotation and Reference Systems Service are shown in this paper. If given standard deviations are taken into account and covariances are neglected the following results are A. Heiker Geod¨atisches Institut, Leibniz Universit¨at Hannover, Nienburger Straße 1, 30167 Hannover, Germany, e-mail: [email protected] H. Kutterer Geod¨atisches Institut, Leibniz Universit¨at Hannover, Nienburger Straße 1, 30167 Hannover, Germany J. M¨uller Institut f¨ur Erdmessung, Leibniz Universit¨at Hannover, Schneiderberg 50, 30167 Hannover, Germany

obtained: lod is coupled with C20 and vice versa, C21 and S21 are coupled with the polar motion, but the polar motion itself and the gravity field coefficients C22 and S22 are not checked by any other parameter. Keywords Earth orientation parameters · Gravity field coefficients · Temporal variations · Validation

1 Introduction The variations of the rotation axis and gravity field of the Earth are caused by mainly the same geophysical influences, such as changes in mass distribution inside and outside the Earth’s body. The tensor of inertia connects the Earth rotation and the gravity field. It can be used for the mutual validation of the parameters describing Earth rotation and the gravity field. The Earth’s rotation is observed since 1962 using modern space-geodetic techniques. The state of the art in the determination of Earth rotation data is presented on the website of the International Earth Rotation and Reference Systems Service (IERS, 2007). The temporal variations of the gravity field are regularly observed by the satellite missions CHAMP since July 2000 and GRACE since March 2002 and by satellite laser ranging (SLR). The present status concerning gravity field determination can be obtained from, e.g., the webpage of the GFZ Potsdam (2007). The Earth rotation vector can be derived from the Earth orientation parameters. This vector is linked with the tensor of inertia by the Euler-Liouville equations (Moritz & Mueller, 1987). The gravity field coefficients of degree 2 are linearly dependent of the Earth’s tensor of inertia (Munk & MacDonald, 1960).

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, c Springer-Verlag Berlin Heidelberg 2009 

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Although the two sets of parameters are connected by the tensor of inertia, this link is presently not taken into account for the determination of the EOP and gravity field coefficients. That may lead to inconsistencies between the EOP and gravity field coefficients. This natural link can be used for mutual validation of the EOP and the gravity field coefficients (Chao & Gross, 1987; Seitz, 2004). The tensor of inertia can be (partially) derived from the EOP and from the gravity field coefficients of degree 2. These two kinds of condition equations lead to a redundancy and therefore to an adjustment problem. The chosen solution for the adjustment problem is the Gauss-Helmert model described below. The adjustment reveals also possible inconsistencies between the EOP and gravity field coefficients and allows the identification of weakly determined parameters. This paper presents an example for the adjustment of EOP and gravity field coefficients by using a GaussHelmert model. An analysis of the numerical results shows a statistically founded reliability measure of each Earth rotation parameter and each gravity field coefficients of degree 2. These reliability measures indicate the checkability of one parameter by the other parameters.

2 Gauss-Helmert Model The Gauss-Helmert model is a further development of the well-known Gauss-Markov model (Koch, 1987). If the observations and unknown parameters of an adjustment problem cannot be separated by transforming the functional relation between the observations and unknowns, as it is the case for the Euler-Liouville equations, the Gauss-Helmert model allows an adjustment by solving an optimization problem with condition equations. In order to work with a linear GaussHelmert model, non-linear condition equations are linearized using a proper set of approximation values. In general a least-squares adjustment requires the minimization of the sum of weighted squared residuals of the observations. If the residuals are denoted with the vector v and the matrix Q denotes the regular cofactor matrix of the observations, there exists the following minimization problem vT Q−1 v → min .

(1)

All (linearized) condition equations of an adjustment problem can be represented using the following vector/matrix notation BV + Ax + w = 0,

(2)

where x denotes the unknown parameters. The matrices A and B result from differentiating the (non-linear) conditions equations with respect to the observations and unknowns. The vector w denotes a misclosure vector. Solving the optimization problem (1) in consideration of the functional model model (2) leads to the following system of equations, where the vector kˆ contains the adjusted Lagrange factors and xˆ the adjusted unknown parameters: %

% & &% & w BQBT A kˆ = − . T 0 A 0 xˆ () * '

(3)

N

A left multiplication of Eq. (3) with the inverse of the normal equations matrix N leads to the solution of xˆ . The inverse of N exists, if N is regular. There are two reasons for a possible singularity of the normal equations matrix: First the condition equations in Eq. (2) are not independent and second the unknowns are not uniquely described by the functional model (2). Estimators in the Gauss-Helmert model with a singular normal equations matrix are given in Sj¨oberg (1985). If the singularity of the normal equations matrix is due to a singularity in the functional model, the nonsingularity can be achieved by introducing additional linear constraints for the unknowns. These constraints together with the functional model (2) must lead to a unique determination of the unknowns. If the additional constraints are described by the equation RT x = w E ,

(4)

the normal equations matrix in Eq. (3) is expanded by introducing Eq. (4) in the Gauss-Helmert model. With kˆ E denoting the new Lagrange factors and w E the misclosure of the additional constraints, the expanded Gauss-Helmert model can be written as ⎡ ⎤ ⎤ ⎤−1 ⎡ kˆ w BQBT A 0 ⎣ xˆ ⎦ = −⎣ AT 0 R⎦ ⎣ 0 ⎦. ˆk E wE 0 RT 0 () * ' ⎡

N

(5)

Combined Analysis of Earth Orientation Parameters

3 Geophysical Parameters Describing the Earth Rotation The variation of the conventional intermediate pole (CIP) and the variation of the length of day ( lod) are part of the EOP provided by the IERS. The Earth rotation vector ω k of epoch k can be computed from the pole coordinates and lod and vice versa (Gross, 1992). The direction of the instantaneous rotation axis is described by the direction of ω k and the rotation velocity by its norm. Taking the gravitional effect of Sun and Moon into account expressed by the lunisolar torque l and the relative angular momentum h due to the deformation of the nonrigid Earth, the rotation of the Earth can be described by the nonlinear Euler-Liouville equations (Moritz & Mueller, 1987)

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The term a 2 Me describes the product of the squared radius of the Earth and its mass. The elements of the main diagonal of the tensor of inertia are not uniquely determined by Eq. (7) which leads to some consequences which are explained in Sect. 3.

4 Combining EOP and Gravity Field Coefficients in a Gauss-Helmert Model

The equation systems (6) and (7) describe the dependence of the EOP and gravity field coefficients of degree 2 on the Earth’s tensor of inertia. For each epoch one particular system of equations is formulated for Eqs. (6) and (7) which leads to condition equations in the Gauss-Helmert model. Two problems have to be considered: Firstly the ωk ×Ik ω k +ω ωk ×hk −Ik = 0. (6) various time derivatives of the Euler-Liouville equa˙ k +h˙ k +ω I˙k ω k +Ik ω tion are not observed and secondly the Euler-Liouville Ik is the unknown Earth’s tensor of inertia, described equations are non-linear. As long as the time derivaby a symmetric 3×3-matrix. The Euler-Liouville equatives of the rotation vector and the relative angular motions contain the values of the tensor of inertia, the romentum are not available as observed data series, they tation vector and relative angular momentum as well as have to be computed by numerical differentiation. In their time derivatives. principle there are two possibilities to introduce the The relative angular momentum h is caused by the time derivative of the tensor of inertia. Introducing the motion of masses inside and outside of the Earth. Vartime derivative of the tensor of inertia as additional unious subsystems of the Earth such as atmosphere and known parameters would lead to 12 unknown parameoceans contribute to the relative angular momentum. ters at each epoch (6 for the tensor of inertia itself and Hence h can be computed by adding the influences 6 for its time derivative). But from Eqs. (6) and (7) a of the subsystems. hatm and hoce due to atmosphere system of only 8 condition equations for each epoch reand ocean are directly proportional to the motion terms sults (3 from the Euler-Liouville equations and 5 from of the atmospheric and oceanic excitation functions. the gravity field coefficients of degree 2). Hence in this Barnes et al. (1983) derived atmospheric excitation case the Gauss-Helmert model is underdetermined and functions from wind velocities and pressure data of a validation is not possible. So it is reasonable to rethe atmosphere. Oceanic excitation functions are comduce the unknowns and replace the time derivative of puted analogously. the tensor of inertia by a numerical differentiation, too: The lunisolar torque can be computed from Earth⎫ fixed coordinates of Moon and Sun (Moritz & Mueller, ω 1 ω2 ˙ 1 = t (ω − ω 1 ) ⎪ ⎬ 1987), which can be derived from ephemerides such as ˙ 1 = 1 (h2 − h1 ) first epoch, h t JPL ephemeris DE405/LE405 (Standish, 1989). ⎪ ⎭ 1 1 2 1 I˙ = t (I − I ) The gravity field coefficients are linearly dependent ⎫ 1 ωk+1 from the Earth’s tensor of inertia (Lambeck, 1980): ω ˙ k = 2 t (ω − ω k−1 ) ⎪ ⎡ k ⎤ ⎬ ⎡ ⎤ i k ⎤ 11 ⎡ 1 ˙ k = 1 (hk+1 − hk−1 ) C20 epoch k = 2, . . . , n − 1, h 1 ⎥ ⎢ k / 0 0 − / 0 1 − t 2 2 ⎪ ⎢ k ⎥ ⎢ i 12 ⎥ ⎭ 1 ⎢ C21 ⎥ ⎢ 0 0 1 0 0 0⎥⎢ k ⎥ I˙k = 2 t (Ik+1 − Ik−1 ) ⎢ k ⎥ ⎥⎢ ⎥ ⎢ ⎢ S ⎥ + 1 ⎢ 0 0 0 0 1 0 ⎥ ⎢ i 13 ⎥ = 0. ⎫ 1 ωn ⎢ 21 ⎥ a 2 M ⎢ ⎥ ⎢ ik ⎥ ˙ n = t (ω − ω n−1 ) ⎪ ω e ⎣ 1 ⎢ k ⎥ ⎬ ⎥ ⎢ ⎦ 1 22 /4 0 0 − /4 0 0 ⎢ ⎥ ⎣ C22 ⎦ k ⎦ ˙ n = 1 (hn − hn−1 ) last epoch. h ⎣ i 1 t 0 /2 0 0 0 0 23 k ⎪ S22 ⎭ k 1 n n n−1 ˙ i 33 I = t (I − I ) (7) (8)

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The time derivatives of one epoch are directly dependent on the previous and next epoch. This induces a tight functional dependence of the different epochs. Due to the numerical differentiation the EulerLiouville equations of one epoch are dependent on the Euler-Liouville equations of the other epochs. So it is necessary to drop at least the Euler-Liouville equations for one epoch to reach a regular normal equation matrix. In this study the Euler-Liouville equations of the first and last epoch are dropped because there is no reason to prefer only the first or last epoch. The lack of the Euler-Liouville equations of the first and last epoch leads to a weak redundancy of these two epochs because their tensor of inertia is only indirectly described by numerical differentiation in the second and second last epoch. Replacing the time derivatives of the EulerLiouville equation of the epochs k = 2, . . . , n − 1 by Eq. (8) leads to non-linear equations, which have to be linearized. The six elements of the unknown tensor of inertia are compiled in a vector i. If E(ω) denotes the expectation value of the rotation vector and an approximated value i0 for the tensor of inertia is introduced, the Taylor expansion of the Euler-Liouville equations can be written as: δϕ (i0 , ω ) ω)) ≈ ϕ EL (i0 , ω ) + EL ω) − ω ) ϕ EL (i, E(ω (E(ω () * ω *' ' () * ' δω () WEL

+...

vEL

BEL

δϕ EL (i0 , ω ) (i − i0 ) = 0 ' () * δi ' () * x

AEL

⇒ BEL vEL + AEL x + wEL = 0. (9)

Due to the numerical differentiation the matrices AFL and BEL show a band structure (Fig. 1). Equation (7) is transformed in a similar way: Bgfc vgfc + Agfc x + wgfc = 0

ω1 ω 2 ω 3 ω 4

(10)

ω n-2 ω n-1 ω n

epoch

2

×

×

×

0

0

0

0

epoch

3

0

×

×

×

0

0

0

epoch n-1 0

0

0

0

×

×

×

Fig. 1 Band structure of the matrix Bω . x indicates non-zero elements

The Eqs. (9) and (10) are introduced as condition equations in the Gauss-Helmert model. First analyses of the Gauss-Helmert model. First analyses of the Gauss-Helmert model have shown a rank deficiency of the normal equations matrix. It is caused by the undetermined trace of the tensor of inertia. Hence the Gauss-Helmert model is expanded by additional constraints: k k k x11 + x22 + x33 = 0.

(11)

While the rank deficiency of the normal equation matrix is 2 it is sufficient to introduce only 2 additional trace constraints, e.g., for the first and the last epoch. It is possible to introduce more additional constraints which would lead to a higher redundancy of the solution.

5 Numerical Studies The Gauss-Helmert Model described above is numerically evaluated by using the following datasets. The rotation vector was derived from the C04 time series provided by the IERS. The time series is referred to the IAU2000A precession/nutation model (Mathews et al., 2002). A detailed description can be found in the IERS conventions 2003 (McCarthy & Petit, 2003). The a priori variance-covariance matrix of the observed rotation vector is composed of the given standard deviations of the EOPs obtained by simple variance propagation. Three processing centers, the GeoForschungsZentrum Potsdam, the Center for Space Research (University of Texas, Austin) and the Jet Propulsion Laboratory provide different gravity field solutions from the satellite mission GRACE. These solutions differ in the models used for tides, atmosphere, ocean etc. Details can be obtained from the GRACE product description available from the Information System and Data Center for geoscientific data (ISDC, 2007). The processing centers provide monthly gravity field coefficients. During the processing of the GRACE data the influence of atmospheric and oceanic mass is reduced by using atmospheric/oceanic models. The gravity field coefficients for the static gravity field (abbreviated with GSM) are computed from GRACE data only and the GCM product contains the gravity field coefficients for the static gravity field computed from GRACE data and terrestrial gravity information. To get

Combined Analysis of Earth Orientation Parameters

the gravity field of the whole system Earth the effect of ocean and atmosphere has to be added. The ISDC provides two kinds of average atmospheric/oceanic background models. The GAC product contains the non-tidal atmospheric and oceanic influences and the GAD product contains the so-called bottom pressure products. For more information see the GRACE product description (ISDC, 2007). The following results are based on the gravity field model EIGEN (release 04). The GSM and GAC products from 34 months (February 2003 to November 2006) are added and introduced as observed gravity field coefficients. The Eq. (7) is valid for non-normalized gravity field coefficients. Schmidt et al. (2007) point out that the standard deviations of the gravity field coefficients from GRACE are too optimistic. They suggest so-called normalized standard deviations instead. The a priori variance-covariance matrix of the denormalized gravity coefficients is derived from the proposed normalized standard deviations by simple variance propagation. The atmospheric relative angular momentum is derived from global motion terms of atmospheric excitation functions which result from the NCEP/NCAR reanalysis (GGFC, 2007). The 6-hourly values of NCEP/NCAR are averaged over one month. The relative angular momentum of the atmosphere is introduced as fixed correction factor in the Euler-Liouville equations. Contributions to the relative angular momentum caused by other influences are neglected as well as the lunisolar torque. Hence the results given in Sect. 6 are due to this less detailed geophysical model. A variation of the numbers of epochs, for which the trace constraint (11) is introduced, reveals that the solution is the more stable the more constraints are introduced. The results presented in Sect. 6 are computed with introducing the trace constraint for all epochs.

6 Results The results of the Gauss-Helmert model are not only the residuals of the observations and the unknown parameters but also accuracy and reliability measures which are derived by variance-covariance propagation based on Eq. (5).

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Fig. 2 Residuals of the observations

The residuals of the polar motion in Fig. 2 with a maximum of 0.4 μas are small compared to the a priori standard deviation of the polar motion (0.2 mas). The residuals from lod as well as the residuals from C20 seem to show a linear trend overlayed by a semi-annual period. The residuals from C21 and S21 are nearly as big as the values themselves. In contrast the residuals from C22 and S22 are very small compared to their values of about 10−6 . The residuals of the first and last epoch of C21 , S21 , C22 and S22 are all zero. The partial redundancy can be derived from the cofactor matrix of the adjusted observations. They are a measure of the checkability of one observation by the other observations. A partial redundancy of 1 means on the one hand that this observation is completely checked by other observations and on the other hand that this observation does not contribute to the adjustment. A partial redundancy of 0 means that this observation is not checked by with other observations and furthermore this observation is essential for the particular adjustment problem.

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Fig. 3 Partial redundancies in year 2005

As shown in Fig. 3 the polar motion, C22 and S22 are not checked by other observations. This explains the smallness of the residuals of these four parameters. They are not overdetermined and hence it is not possible to derive reliable residuals for them. C20 and lod are coupled with each other. C21 and S21 of the mid epochs k = 2, . . . , n−1 are fully checked by polar motion. Hence their residuals are reliable. The size of the

Fig. 4 Analysis of the eigenvalues and eigenvectors

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residuals indicates that the polar motion on one hand and C21 and S21 on the other hand do not fit together very well in this particular Gauss-Helmert model. Due to the lack of the Euler-Liouville equations in the first and last epoch the partial redundancies of C21 and S21 of these epochs are also zero and therefore the residuals are small. An eigenvalue decomposition of the cofactor matrix of the adjusted unknowns leads to some additional information about the unknowns. The eigenvalues describe the variances of the linear combinations expressed by the eigenvectors. Figure 4 shows the eigenvalues. The linear combinations of parameters associated with the eigenvalues are described in the legend of the figure. The linear combination i11 −i22 describes the deviation from the rotational symmetry of the Earth and is roughly by the factor 100 more inaccurate than the linear combination i11 + i22 − 2i33 defining C20 . i12 is proportional to S22 and less accurately determined than i13 and i23 of the mid epochs which are proportional to C21 and S21 . The null space describes the additional trace constraints. As the trace constraints are introduced as hard constraints (i.e. a priori standard deviations for the constraints are zero) their a posteriori standard deviations are also nearly zero.

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7 Conclusions and Outlook

References

The results presented in this paper are highly dependent on the geophysical model implemented in the Gauss-Helmert model and on the a priori cofactor matrices of the observed EOP and gravity field coefficients. As long as not all components of the Euler-Liouville equations are considered, the results are only intermediate. The big residuals of C21 and S21 found in Sect. 6 may be caused by this less detailed geophysical modeling. The geophysical model needs some improvement to be addressed in near future. The consideration of the lunisolar torque is basically solved (Seitz, 2004) and requires only to be coded. The use of the complete relative angular momentum and its time derivative is a more difficult task. As long as the motion of masses inside and outside the Earth is not completely known there remain some inconsistencies due to not considered relative angular momentum. On the other hand discovered inconsistencies may contribute for improving the modeling of the mass transport of the Earth. The a priori cofactor matrices of the observations are simply built from the given standard deviations. Because the covariances are not available hitherto, they are neglected, which leads to an incorrect a priori stochastic model. Furthermore there are some indications that the improvement of the accuracy in the gravity field coefficients leads to other reliability measures. A variance component estimation is scheduled to determine those a priori cofactor matrices which allow homogeneous partial redundancies for all observations. If the time derivatives of the gravity field coefficients were available, the Gauss-Helmert model could be expanded by differentiating Eq. (7) and the trace constraint (11), which leads to 6 additional constraint equations for each epoch. It would then be possible to introduce the time derivatives of the Earth tensor of inertia as unknowns. Hence the different epochs would not be correlated by a functional dependency and the constraints induced by the numerical differentiation would not play a role.

Barnes, R. T. H., Hide, R., White, A. A., Wilson, C. A.: Atmospheric angular momentums fluctuations, length-of-day changes and polar motion. Proc. R. Soc. Lond. A., 387, 31–78, 1983 Chao, B., Gross, R.: Changes in the earth’s rotation and lowdegree gravitational field induced by earthquakes. Geophys. J. Int., 91, 569–596, 1987 Earth Rotation Portal: http://www.erdrotation.de/ (31.08.2007) GFZ Potsdam: http://www.gfz-potsdam.de/html/projects/ index.html (31.08.2007) GGFC: http://www.ecgs.lu/ggfc/ (31.08.2007) Gross, R.: Correspondence between theory and observations of polar motion. Geophys. J. Int., 109, 162–170, 1992 IERS: www.iers.org (31.08.2007) ISDC: http://isdc.gfz-potsdam.de/ (31.08.2007) Koch, K. R.: Parameter Estimation and Hypothesis Testing in Linear Models. Springer, Berlin, 1987 Lambeck, K.: The Earth’s Variable Rotation: Geophysical Causes and Consequences. Cambridge University Press, Cambridge, 1980 Mathews, P. M., Herring, T. A., Buffett, B. A.: Modelling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth’s interior. J. Geophys. Res., 107 (B4), 10.1029/2001B000390, 2002 Moritz, H., Mueller, Ivan I.: Earth Rotation. Ungar, New York, 1987 McCarthy, D. D., Petit G. (eds.): IERS Conventions. IERS Technical Notes No. 32, International Earth Rotation and Reference Systems Service (IERS). Verlag BKG, Frankfurt (Main), 2003 Munk, W. H., MacDonald, G. J. F.: The Rotation of the Earth. A Geophysical Discussion. Cambridge University Press, 1960 Schmidt, R., Flechtner, F., K¨onig, U., Neumayer, K. H., Reighber, C., Rothacher, M., Petrovic, S., Zhu, S. Y. G¨untner, A. GRACE Time-Variable Gravity Accuracy Assessment, in Tregoning, P. Rizos, P. (eds.): Dynamic planet, Monitoring and Understanding a Dynamic planet with Geodetic and Oceanographic Tools. IAG Symposium Cairns, Australia, 22–26. August 2005, 237–243, Springer, 2007 Seitz, F.: Atmosph¨arische und ozeanische Einfl¨usse auf die Rotation der Erde. DGK Reihe C, Nr. 578, 2004 Sj¨oberg, L. E.: Adjustment and variance-covariance component estimation with a singular covariance matrix. Zeitschrift f¨ur Vermessungswesen, 110, 145–151, 1985 Standish, E. M.: JPL Planetary and Lunar Ephemerides DE405/LE405, Techn. Ber. IOM 312.F-98-048, JPL; Pasadena, 1989

Acknowledgments The results presented have been derived within the work on the project “Mutual validation of EOP and gravity field coefficients” within the framework of the research unit FOR 584 “Earth Rotation and global Geodynamic Processes” funded by the German Research Foundation (Earth Rotation Portal, 2007). This is gratefully acknowledged.

Index

A Abd-Elmotaal, H. A., 203 Abidin, H. Z., 595 Abreu, M. A., 65 Alberts, B., 263 Altamimi, Z., 24, 32 Andersen, C. J., 135 Andreas, H., 595 Angermann, D., 21 Antoni, M., 293 Avalos, D., 421 Awange, J. L., 387, 639 Awange, J. L., 804, B Bariˇsi´c, B., 169 Barriot, J.-P., 177 Barzaghi, R., 132, 148 Baˇsi´c, T., 169 Behrend, D., 833 Benveniste, J., 431 Berio, P., 39 Berry, P. A. M., 431 Beutler, G., 113, 330, 569 Bezdek, A., 307 Bisnath, S., 615, 817 Blewitt, G., 463 Blitzkow, D., 187 B¨ohm, J., 833 Bonnefond, P., 39 Bosch, W., 735 Botai, O. J., 625 Boucher, C., 42 Boudin, F., 533 Boy, J. P., 516 Brockmann, E., 47 Brunini, C. A., 605 Bruyninx, C., 47 Buongiorno, A., 329 C Camargo, P. O., 71 Campos, I. O., 187 Caporali, A., 47 Caramiello, C., 329

Chang, H. C., 631, 743, 825 Charlot, P., 833 Chen, X., 710 Cheng Pengfei, 89, 315 Cheng, Y., 89 Chu Yonghai, 161, 379 Clark, T., 833 Clarke, P. J., 463 Cocard, M., 817 Cole, D., 751 Collilieux, X., 31 Collins, P., 696 Combrinck, W. L., 625 Coren, F., 427 Corey, B., 833 Costa, S. M. A., 65, 71, 605 Coulot, D., 32 Creati, N., 427 Crossley, D., 511, 841 ˇ Cunderl´ ık, R., 229 D da Silva, A. L., 605 Dach, R., 113, 353, 569 Dalazoana, R., 735 Davis, J. L., 557 de Freitas, S. R. C., 735 de Matos, A. C. O. C., 187 de Moura, Jr. N. J., 605 Dehant, V., 455 Dekkiche, H., 679 Deleflie, F., 39 Delikaraoglou, D., 245 Denker, H., 177, 346, 406 deW Rautenbanch, C. J., 625 Ditmar, P., 307, 323 Dragert, H., 810 Drewes, H., 3 Ducarme, B., 513 E Eichhorn, A., 669 Eicker, A., 99 Elhabiby, M., 211, 212, 275 Ellmann, A., 155, 187, 253

861

862 El-Rabbany, A., 817 Encarnac˜ao, J., 323 Erenoglu, R. C., 639 Even-Tzur, G., 581 Exertier, P., 39 F Fabiankowitsch, J., 771 Fairhead, D., 177 Falk, R., 301 Fastellini, G., 549 Feng, Y., 645 F´eraudy, D., 39 Ferentinos, G., 245 Fern´andez, L. I., 493 Floberghagen, R., 329 Florsch, N., 529 Folgueira, M., 455 Forsberg, R., 135, 177 Fortes, L. P. S., 65 Freeman, J. A., 431 Freiberger, Jr. J., 701 G Galo, M., 71 Gamal, M., 595 Gao, Y., 615, 655 Ge, L., 631, 743, 825 Gende, M. A., 605 Ghazavi, K., 397 Gipson, J., 833 Gjevestad, J. G. O., 691 G¨ottl, F., 439 Grafarend, E. W., 803 Grgi´c, I., 169 Gross, R. S., 441 Gruber, Th., 105 Guangbin, Z., 315 Gudmundsson, G. H., 569 Gurtner, W., 47 H Haas, R., 833 Haberler-Weber, M., 669 Habrich, H., 47 Hanjiang, W., 315, 242 Heck, B., 701, 725, 735 Heiker, A., 853 Hekimoglu, S., 639 Hense, A., 447 Hern´andez, A., 421 H´eroux, P., 655 Hinderer, J., 456, 841 Hirt, C., 413 Holota, P., 219 Hornik, H., 47 Hunegnaw, A., 397 I Ihde, J., 47, 81, 135, 177, 263 Ilk, K. H., 99

Index J J¨aggi, A., 113, 353 Jakowski, N., 759 Jarecki, F., 363 Jiang Weiping, 161, 379 Jingnan Liu, 541 Johansson, J. M., 557 J´unior, N. J. M., 65 J¨urgenson, H., 125 K Kadri, C. B., 679 Kahlouche, S., 679 Kahmen, H., 669, 771 Keller, W., 293, 387 Kenyeres, A., 47, 177 Kjørsvik, N. S., 691 Klees, R., 237, 263, 323 Klose, S., 709 Kn¨opfler, A., 701 Kotsakis, C., 371 Koyama, Y., 833 Krot, A. M., 283 Krueger, C. P., 701 Kuhn, M., 387 K¨uhtreiber, N., 203 Kusche, J., 323 Kusuma, M. A., 595 Kutterer, H., 853 L Lambert, S. B., 455 Landau, H., 709 Landry, R., 817 Langley, R. B., 817 Laurain, O., 39 Lavall´ee, D. A., 463 Le, A. Q., 759 Leandro, R. F., 71, 655 Leandro, R., 709 Lehmann, M., 771 Li Jiancheng, 161, 379 Lidberg, M., 557 Liebsch, G., 135, 263 Liibusk, A., 125 Liker, M., 169 Liu, X., 307 Lizhi Lou, 719 Longuevergne, L., 533 Luˇcic, M., 169 Luo, X., 725 Luz, R. T., 735 Lysaker, D. I., 397 M MacMillan, D., 833 Maggi, A., 195 Magna, Jr. J., 71 Maia, T. B., 71 Makhloof, A. A., 143 M¨akinen, J., 47, 81

Index Malkin, Z., 833 Manoussakis, G., 245 Marchenko, A. N., 473, 483 Marson, I., 427 Marti, U., 177 Maso, M., 427 Mayer, M., 701, 725 Mayer-G¨urr, T., 143, 353 Mervart, L., 113, 353 Mi Jinzhong, 89 Migliaccio, F., 337 Mikula, K., 229 Milne, G. A., 557 Mir, R., 679 Moghtased-Azar, K., 809 Monico, J. G., 65, 71 Monjoux, E., 329 Mueller, F., 143 M¨uller, H., 21 M¨uller, J., 363, 387, 501, 853 N Nahavandchi, H., 397 Nesvadba, O., 219 Ng, A. H., 631, 743 Niell, A., 833 Nievinski, F. G., 71 Nilsson, T., 836 Nov´ak, P., 155 O Oliveira, L. C., 75 Omang, O. C. D., 397 Omura, M., 631 Oudin, L., 533 Ouyang, G., 751 Ovstedal, O., 692 P Pagiatakis, S., 504 Pal´ancz, B., 804 Peters, Th., 105 Petrachenko, B., 835 Prange, L., 116, 353 Prutkin, I., 237, 264 R Radicioni, F., 549 Rambaux, N., 461 Reguzzoni, M., 338 Reiterer, A., 772 Repani´c, M., 169 Rizos, C., 645, 646, 726, 753, 787 Roberts, C., 11 Rogers, A., 833 Rosat, S., 516 S Sampietro, D., 211 Sans`o, F., 181, 199 Santerre, R., 817 Santos, M. C., 65, 187, 421, 639, 655, 715

863 Santos, M. F., 71 Sarrailh, M., 177 Sch¨afer, B., 705 Sch¨afer, U., 135 Scherneck, H.-G., 557 Schirmer, U., 135, 263 Seeger, H., 47 Seitz, F., 439, 854 Seung-Ki Min, 447 Shahar, L., 581 Sharifi, M. A., 387 Sideris, M. G., 196, 212 Silva, A. L., 65 Silva, C. U., 71 Simek, J., 47 Sj¨oberg, L. E., 781 Skourup, H., 135 Smith, R. G., 431 Solheim, D., 397 Sørensen, S. L. S., 135 Stangl, G., 47 Steffen, H., 501 Sterzai, P., 427 Stoppini, A., 549 Sumintadireja, P., 595 Sun Heping, 523 T Tassa, A., 329 Tenzer, R., 155, 246, 253, 263 T´etreault, P., 68 Teunissen, P. J. G., 646 Theilen-Willige, B., 669 Tiberius, C. C. J. M., 765 Torres, J. A., 47 T´oth Gy, 405 Tselfes, N., 195, 338 Tsoulis, D., 195 Tuccari, G., 833 Tziavos, I. N., 178, 195 V Vajda, P., 155 van der Marel, H., 759 Vandercoilden, L., 523 Van´ıcˇ ek, P., 156, 188 Vergos, G. S., 196 Verhagen, S., 646 Vigione, G., 329 Voigt, C., 417 V¨olksen, C., 61 Vollath, U., 709 W Wang Jian, 11, 751 Wang, H., 631, 793 Wang, M., 619 Weber, G., 52 Weigelt, M., 296 Wilmes, H., 303 Winkelnkemper, T., 447 Wittwer, T., 237

864 Wolf, K. I., 346 Wolf, P., 301 Wresnik, J., 833 Wziontek, H., 301 X Xiaohui Zhou, 541 Xiaotao, C., 315 Xu Caijun, 793 Xu Jian-Qiao, 523 Xu Xinyu, 161, 379

Index Y Yangmao, W., 793 Yi Chen, 719 Z Zaletnyik, P., 803, 809 Zapreeva, E., 323 Zenner, L., 105 Zhang, K., 743 Zhao, Q., 307 Zou Xiancai, 161, 379