Modern Approaches to Data Assimilation in Ocean Modeling (Elsevier Oceanography Series)

MODERN A PPROA CHES TO DATA ASSIMILATION IN OCEAN MODELING

Elsevier Oceanography Series Series Editor." David Halpern (1993-) FURTHER TITLES IN THIS SERIES Volumes 1-7, 11, 15, 16, 18, 19, 21, 23, 29 and 32 are out of print. 8 E. LISITZIN SEA-LEVEL CHANGES 9 R.H. PARKER THE STUDY OF BENTHIC COMMUNITIES 10 J.C.J. NIHOUL (Editor) MODELLING OF MARINE SYSTEMS 12 E.J. FERGUSON WOOD and R.E. JOHANNES TROPICAL MARINE POLLUTION 13 E. STEEMANN NIELSEN MARINE PHOTOSYNTHESIS 14 N.G.JERLOV MARINE OPTICS 17 R.A. GEYER (Editor) SUBMERSIBLES AND THEIR USE IN OCEANOGRAPHY AND OCEAN ENGINEERING 20 P.H. LEBLOND and L.A. MYSAK WAVES IN THE OCEAN 22 P. DEHLINGER MARINE GRAVITY 24 F.T. BANNER, M.B. COLLINS and K.S. MASSIE (Editors) THE NORTH-WEST EUROPEAN SHELF SEAS: THE SEA BED AND THE SEA IN MOTION 25 J.C.J. NIHOUL (Editor) MARINE FORECASTING 26 H.G. RAMMING and Z. KOWALIK NUMERICAL MODELLING MARINE HYDRODYNAMICS 27 R.A. GEYER (Editor) MARINE ENVIRONMENTAL POLLUTION 28 J.C.J. NIHOUL (Editor) MARINE TURBULENCE 30 A. VOIPIO (Editor) THE BALTIC SEA 31 E.K. DUURSMA and R. DAWSON (Editors) MARINE ORGANIC CHEMISTRY 33 R.HEKINIAN PETROLOGY OF THE OCEAN FLOOR 34 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF SEMI-ENCLOSED SEAS 35 B. JOHNS (Editor) PHYSICAL OCEANOGRAPHY OF COASTAL AND SHELF SEAS 36 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF THE EQUATORIAL OCEAN 37 W. LANGERAAR SURVEYING AND CHARTING OF THE SEAS 38 J.C.J. NIHOUL (Editor) REMOTE SENSING OF SHELF-SEA HYDRODYNAMICS 39 T.ICHIYE (Editor) OCEAN HYDRODYNAMICS OF THE JAPAN AND EAST CHINA SEAS 40 J.C.J. NIHOUL (Editor) COUPLED OCEAN-ATMOSPHERE MODELS 41 H. KUNZENDORF (Editor) MARINE MINERAL EXPLORATION 42 J.C.J NIHOUL (Editor) MARINE INTERFACES ECOHYDRODYNAMICS 43 P. LASSERRE and J.M. MARTIN (Editors) BIOGEOCHEMICAL PROCESSES AT THE LANDSEA BOUNDARY 44 I.P. MARTINI (Editor) CANADIAN INLAND SEAS

45 J.C.J. NIHOUL (Editor) THREE-DIMINSIONAL MODELS OF MARINE AND ESTUARIN DYNAMICS 46 J.C.J. NIHOUL (Editor) SMALL-SCALE TURBULENCE AND MIXING IN THE OCEAN 47 M.R. LANDRY and B.M. HICKEY (Editors) COASTAL OCENOGRAPHY OF WASHINGTON AND OREGON 48 S.R. MASSEL HYDRODYNAMICS OF COASTAL ZONES 49 V.C. LAKHAN and A.S. TRENHAILE (Editors) APPLICATIONS IN COASTAL MODELING 50 J.C.J. NIHOUL and B.M. JAMART (Editors) MESOSCALE SYNOPTIC COHERENT STRUCTURES IN GEOPHYSICAL TURBULENCE 51 G.P. GLASBY (Editor) ANTARCTIC SECTOR OF THE PACIFIC 52 P.W. GLYNN (Editor) GLOBAL ECOLOGICAL CONSEQUENCES OF THE 1982-83 EL NINO-SOUTHERN OSCILLATION 53 J. DERA (Editor) MARINE PHYSICS 54 K. TAKANO (Editor) OCEANOGRAPHY OF ASIAN MARGINAL SEAS 55 TAN WEIYAN SHALLOW WATER HYDRODYNAMICS 56 R.CHARLIER and J. JUSTUS OCEAN ENERGIES, ENVIRONMENTAL, ECONOMIC AND TECHNOLOGICAL ASPECTS OF ALTERNATIVE POWER SOURCES 57 P.C. CHU and J.C. GASCARD (Editors) DEEP CONVECTION AND DEEP WATER FORMATION IN THE OCEANS 58 P.A. PIRAZZOLI WORLD ATLAS OF HOLOCENE SEA-LEVEL CHANGES 59 T.TERAMOTO (Editor) DEEP OCEAN CIRCULATION-PHYSICAL AND CHEMICAL ASPECTS 60 B. KJERFVE (Editor) COASTAL LAGOON PROCESSES

Elsevier Oceanography Series, 61

MODERN APPROACHES TO DATAASSIMILATION IN OCEAN MODELING Edited by P. M a l a n o t t e - R i z z o l i

Physical Oceanography Massachusetts Institute of Technology Department of Earth, Atmospheric & Planetary Sciences, Cambridge, MA 02139, USA

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Modern approaches to data a s s i m i l a t i o n in ocean modeling / e d i t e d by P. M a ] a n o t t e - R i z z o ] i . p. cm. - - ( E l s e v i e r oceanography s e r i e s ; 61) I n c l u d e s index. ISBN 0 - 4 4 4 - 8 2 0 7 9 - 5 ( a c i d - F r e e paper) 1. O c e a n o g r a p h y - - M a t h e m a t i c a l models. I. Halanotte-Rizzo]i, Pao]a, 1946II. Series. GC10.4.M36M65 1996 551.46'001'5118--dc20 96-3901 CIP ISBN 0 4 4 4 8 2 0 7 9 - 5 (hardbound) ISBN 0 4 4 4 8 2 4 8 4 - 7 (paperback) 91996 Elsevier ScienceB.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, w i t h o u t the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers Copyright Clearance Center Inc. can be obtained from the CCC publication may be made in the outside of the U.S.A., should be otherwise specified.

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Preface

The motivation that prompted the structure of this book was provided by the growing perception that the field of oceanographic data assimilation had not only reached the mature stage but also the point of revisiting its major objectives. Clearly, the delayed development of data assimilation in oceanography with respect to meteorology was primarily due to the lack of a unifying focus such as the need of weather prediction. But equally important for this delay was the lack of adequate oceanographic datasets, with space/time coverage comparable to the meteorological ones. The observational oceanographic revolution of the 90s capitalized on the promises of E1 Nifio prediction and of satellite altimetry that has been so successful with TOPEX/POSEIDON, both of which spurred the beginning and rapid growth of the field. By and large the fundamental motivation of oceanographic data assimilation has been insofar the necessity for systematic model improvement and for ocean state estimation. In this respect, the research activity has by now reached the mature state where "real" observations of different types (altimetric, hydrographic, Eulerian and Lagrangian velocity measurements, etc.) are being currently assimilated into complex and realistic ocean general circulation models (OGCM). Also, after the initial phase of using very simple assimilation methods such as optimal interpolation and nudging, the most sophisticated techniques are now being implemented in the OGCM's such as the Kalman filter/smoother and the variational adjoint approach. Recently, moreover, a turning point has been reached and the need for ocean prediction has been emerging as a legitimate goal p e r se. The oceanographic applications in which prediction is not only timely but necessary cover a broad range of space/time scales, from hundreds of years in climate problems to a few weeks in regional nowcasting/forecasting. These two motivations lie at the foundation of the present book which is not meant to be a "pedagogical" book. Rather, it wants to present a picture as exhaustive as possible even though obviously far from complete, of the state-of-the-art of data assimilation in oceanography in the mid 90s. Hence the philosophy of the book. First, it reviews the present panorama of models and observations from the data assimilation perspective. Second, for each oceanographic application, from the global to the regional scale, it offers reviews and new results of fundamental assimilation methodologies and strategies as well as of the state-of-the-art of operational ocean nowcasting/forecasting. Finally, the last chapter presents a first example of interdisciplinary modeling with data assimilation components, a direction into which the field is also evolving. All manuscripts were prepared in 1995. Each manuscript of the book underwent anonymous peer review, most often by two reviewers, and authors modified the manuscript in accordance with reviewers' comments. David Halpern, Editor of the Elsevier Oceanography Series, took upon himself to organize the review process. I cannot adequately express my deep gratitude and appreciation for all his efforts. We are truly thankful to the reviewers who generously gave of their time and contributed their expertise to improve the manuscripts. As a small token of appreciation, each reviewer will receive a copy of the book. Reviewers were: Andrew Bennett, Oregon State University James Carton, University of Maryland Ching-Sang Chiu, Naval Postgraduate School Michael Clancy, Fleet Numerical Meteorology and Oceanography Center Bruce Cornuelle, Scripps Institution of Oceanography John Derber, National Meteorological Center Martin Fischer, Max-Planck Institut fiir Meteorologie Philippe Gaspar, Collect Localisation Satellites David Halpern, Jet Propulsion Laboratory Zheng Hao, Scripps Institution of Oceanography

Frank Henyey, University of Washington Eileen Hofmann, Old Dominion University Greg HoUoway, Institute of Ocean Sciences Lakshmi Kantha, University of Colorado Aaron Lai, Los Alamos National Laboratory Christian Le Provost, Institut de MEcanique de Grenoble Florent Lyard, Proudman Oceanography Laboratory Jochem Marotzke, Massachusetts Institute of Technology John Marshall, Massachusetts Institute of Technology Robert Miller, Oregon State University Nadia Pinardi, Istituto per 1o Studio Delle Metodologie Geofisiche Ambientali Stephen Rintoul, Division of Oceanography Albert Semmer, Naval Postgraduate School Julio Sheinbaum, Centro de Investigacion Cientifica y de Educacion Superior de Ensenada Ole Smedstad, Planning Systems Incorporated Neville Smith, Bureau of Meteorology Research Detlef Stammer, Massachusetts Institute of Technology Carlisle Thacker, Atlantic Oceanographic and Meteorological Laboratories Eli Tziperman, Weizmann Institute of Science Leonard Walstad, Horn Point Environmental Laboratory Dong-Ping Wang, State University of New York, Stony Brook Francisco Werner, University of North Carolina Li Yuan, Oregon State University Finally, it is with great pleasure that I acknowledge the National Aeronautics and Space Administration under the auspices of Dr. Donna Blake for the generous financial contribution made towards the publication of this book. Paola Malanotte-Rizzoli Cambridge, Massachusetts October 1995

vii LIST OF CONTRIBUTORS Dr. Frank Aikman NOAA, National Ocean Service Office of Ocean & Earth Sciences N/OES333, Room 6543, SSMC4 1305 East-West Highway Silver Spring, MD 20910-3281 Dr. Laurence A. Anderson Division of Applied Sciences Harvard University Pierce Hall 29 Oxford Street Cambridge, MA 02138 Dr. Hernan G. Arango Rutgers University Institute of Marine & Coastal Sciences P.O. Box 231 Cook Campus New Brunswick, NJ 08903-0231 Prof. Andrew Bennett College of Oceanic & Atmospheric Sciences Oregon State University Oceanography Administration, #1 04 Corvallis, OR 97331-5503 Dr. R. Bosley Sayre Hall Princeton University, POB CN 710 Princeton, NJ 08544-0710 Dr. Antonio Busalacchi Laboratory of Hydrospheric Processes NASA Goddard SFC MC 972 Building 22 Greenbelt, MD 20771 Dr. Mark Cane Lamont-Doherty Geological Observatory Columbia University Route 9W Palisades, NY 10964 Dr. Antonietta Capotondi NCAR-UCAR P.O. Box 3000 Boulder, CO 80307 Dr. Michael Carnes Naval Research Laboratory Code 7323 Stennis Space Center, MS 39522

Dr. Bruce Cornuelle Department 0230 Scripps Institution of Oceanography 9500 Gilman Drive La Jolla, CA 92093-0230 Dr. Gary Egbert College of Oceanic & Atmospheric Sciences Oregon State University Oceanography Administration, #104 Corvallis, OR 97331-5503 Dr. Tal Ezer Program in Atmospheric & Oceanic Science POB CN 710 Princeton University Princeton, NJ 08544-0710 Dr. Michael Foreman Institute of Oceanic Sciences POB 6000 Sidney, British Columbia V8L 4B2 CANADA Dr. Daniel Fox 123 D'Evereux Slidell, LA 70461 Dr. Lee-Leung Fu Jet Propulsion Laboratory 300-323 4800 Oak Grove Drive Pasadena, CA 91109 Dr. Ichiro Fukumori Jet Propulsion Laboratory 300-323 4800 Oak Grove Drive Pasadena, CA 91109 Dr. Avijit Gangopadhyay Jet Propulsion Laboratory California Institute of Technology Mail Stop 300-323 4800 Oak Grove Drive Pasadena, CA 91109-8099 Dr. R. Gudgel Geophysical Fluid Dynamic Laboratory Princeton University P.O.B. 308 Princeton, NJ 08540

viii Dr. Patrick J. Haley Division of Applied Sciences Harvard University Pierce Hall 29 Oxford Street Cambridge, MA 02138 Dr. Nelson Hogg Clark 301A Woods Hole Oceanographic Institution Woods Hole, MA 02543 Dr. William Holland NCAR POB 3000 Boulder, CO 80307 Dr. Harley Hurlburt Naval Research Laboratory MC7320 Stennis Space Center, MS 39539-5004 Dr. Greg Jacobs Naval Research Laboratory Stennis Space Center Code 321 Bay St. Louis, MS 39529-5004 Dr. Ming Ji National Center for Environmental Prediction 5200 Auth Road, Room 807 Camp Springs, MD 20746 Mr. Wayne G. Leslie Division of Applied Sciences Harvard University Pierce Hall 29 Oxford Street Cambridge, MA 02138 Dr. Carlos J. Lozano Division of Applied Sciences Harvard University Pierce Hall 29 Oxford Street Cambridge, MA 02138 Prof. Paola Malanotte-Rizzoli Dept. of Earth, Atmospheric & Planetary Sciences M.I.T., Room 54-1416 Cambridge, MA 02139 Prof. George Mellor Sayre Hall Princeton University POB CN 710 Princeton, NJ 08544-0710

Dr. Arthur J. Miller Scripps Institution of Oceanography Climate Research Division La Jolla, CA 92093-0224 Dr. Robert Miller College of Oceanic & Atmospheric Sciences Oregon State University Oceanography Administration, # 104 Corvallis, OR 97331-5503 Dr. James Mitchell Code 322 Naval Ocean Research & Development Activity NSTL Station, MS 39529 Dr. Kikuro Miyakoda Geophysical Fluid Dynamics Laboratory Princeton University POB 308 Princeton, NJ 08540 Dr. Desiraju Rao National Meteorology Center BIAA, W/BNC21 WW Building, Room 204 Washington, DC 20233 Dr. R.C. Rhodes 436 Pine Shadows Slidell, LA 70458 Prof. Allan Robinson Division of Applied Sciences Harvard University Pierce Hall 29 Oxford Street Cambridge, MA 02138 Dr. A. Rosati Geophysical Fluid Dynamics Lab Princeton University POB 308 Princeton, NJ 08540 Dr. D. Sheinin Sayre Hall Princeton University POB CN 710 Princeton, NJ 08544-0710 Dr. Ziv Sirkes Naval Rcsearch Labortory Code 7322 Stennis Space Center, MS 39522

ix Dr. N. Quincy Sloan Division of Applied Sciences Harvard University Pierce Hall 29 Oxford Street Cambridge, MA 02138 Dr. Ole Martin Smadsted Planning Systems, Inc. 115 Christian Lane SlideU, LA 70458 Dr. Eli Tziperman Dept. of Environmental Sciences Weizmann Institute of Science Rehovot, 76100 ISRAEL Dr. Alex Warn-Varnas Naval Research Laboratory Code 7322 Stennis Space Center, MS 39529 Dr. Peter Worcester IGPP (0225) Scripps Institution of Oceanography La Jolla, CA 92093-0225 Dr. Roberta Young Dept. of Earth, Atmospheric & Planetary Sciences MIT, Room 54-1410 Cambridge, MA 02139

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xi

Contents

Preface P. Malanotte-Rizzoli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction The Oceanographic Data Assimilation Problem: Overview, Motivation and Purposes P. Malanotte-Rizzoli and E. Tziperman

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Models and Data Recent Developments in Prognostic Ocean Modeling W.R. Holland and A. Capotondi

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Oceanographic Data for Parameter Estimation N. G. Hogg

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A Case Study of the Effects of Errors in Satellite Altimetry on Data Assimilation L.-L. Fu and I. Fukumori

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Ocean Acoustic Tomography: Integral Data and Ocean Models B.D. Cornuelle and P.F. Worcester

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Gobal Applications Combining Data and a Global Primitive Equation Ocean General Circulation Model using the Adjoint Method Z. Sirkes, E. Tziperman and W. C. Thacker ......................

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Data Assimilation Methods for Ocean Tides G.D. Egbert and A.F. Bennett

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xii Global Ocean Data Assimilation System A. Rosati, R. Gudgel and K. Miyakoda

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Tropical Ocean Applications Tropical Data Assimilation: Theoretical Aspects R.N. Miller and M.A. Cane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Data Assimilation in Support of Tropical Ocean Circulation Studies A.J. Busalacchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Ocean Data Assimilation as a Component of a Climate Forecast System A. Leetmaa and M. Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Regional Applications A Methodology for the Construction of a Hierarchy of Kalman Filters for Nonlinear Primitive Equation Models P. Malanotte-Rizzoli, I. Fukumori and R.E. Young . . . . . . . . . . . . . . . . . .

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Data Assimilation in a North Pacific Ocean Monitoring and Prediction System M.R. Carnes, D.N. Fox, R.C. Rhodes and O.M. Smedstad . . . . . . . . . . . . .

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Towards an Operational Nowcast/Forecast System for the U.S. East Coast F. Aikman IH, G.L. Mellor, T. Ezer, D. Sheinin, P. Chen, L. Breaker and D.B. Rao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Real-time Regional Forecasting A.R. Robinson, H.G. Arango, A. Warn-Varnas, W. Leslie, A.J. Miller, P.J. Haley and C.J. Lozano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Interdisciplinary Applications An Interdisciplinary Ocean Prediction System: Assimilation Strategies and Structured Data Models C.J. Lozano, A.R. Robinson, H.G. Arango, A. Gangopadhyay, Q. Sloan, P.J. Haley, L. Anderson and W. Leslie . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction

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Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

The O c e a n o g r a p h i c Data A s s i m i l a t i o n P r o b l e m : O v e r v i e w , M o t i v a t i o n a n d Purposes Paola Malanotte-Rizzoli a and E.li Tzipermanb aDepartment of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology Cambridge, MA 02139 bWeizmann Institute of Science Rehovot, Israel Abstract A brief non-technical overview is given of the data assimilation problem in oceanography. First, a historical perspective is presented that illustrates its main motivations and discusses the objectives of combining fully complex ocean general circulation models (OGCM) and oceanographic data. These objectives are divided into three main categories: model improvement, ocean state estimation and ocean/climate forecasting. Forecasting applications vary from global climate change simulations on a time scale of 50-100 years; through decadal and interannual climate variability, such as the E1 Nino-Southern Oscillation and the Atlantic thermohaline variability; to extended seasonal forecasts and finally to regional forecast of ocean frontal systems on a time scale of a few weeks. Appropriate assimilation methodologies for each class of oceanographic applications are discussed. For each ocean prediction problem on different time/space scales the needs for data assimilation approaches are pointed out where these are still lacking as they might overcome some of the present deficiencies of the related modeling efforts.

1. INTRODUCTION The terminology "data assimilation" developed in meteorology about 30 years ago as the methodology in which observations are used to improve the forecasting skill of operational meteorological models. In the practice of operational meteorology, all the observations available at prescribed times are "assimilated" into the model by melding them with the model-predicted values of the same variables in order to prepare initial conditions for the forecast model run. When used in the oceanographic context, the name data assimilation has acquired a much broader meaning, as reflected in the chapters of this book. Under this general denomination a vast body of methodologies is collected, originating not only in meteorology but in solidearth geophysics inverse theories and in engineering control theories. All of these methods attempt to constrain a dynamical model with the available data. Moreover, the purposes of oceanographic data assimilation are also often very different from the meteorological case, and three main objectives can be distinguished. One goal is to quantitatively use the data in order to improve the ocean model parameterizations of subgrid scale processes, boundary conditions etc. A second goal is to obtain a four-dimensional realization (the spatial description coupled with the time evolution) of the oceanic flow that is simultaneously

consistent with the observational evidence and with the dynamical equations of motion; the resulting realization can be used for detailed process studies. A third major motivation of ocean data assimilation, the closest to the meteorological one, is to provide initial conditions for predictions of the oceanic circulation. Such predictions are needed in very diverse problems and on very different time scales, from 100 years in climate problems, through interannual climate variability and extended seasonal weather forecasting, to a few weeks in regional ocean forecasting. In this paper we wish to provide a brief and non-technical overview of the various assimilation problems and methodologies used in oceanography as an introduction to the more specific and technical chapters that follow. Our main focus here is the objectives of oceanographic data assimilation, rather than the methodologies used, and we try to concentrate on what still needs to be done rather than on a review of the existing body of work. Here, as well as in the following chapters, attention is limited to the use of oceanographic data with the most realistic and sophisticated tools presently available to simulate oceanic flows, the ocean general circulation models (OGCM), where one assumes the future of oceanographic data assimilation must lie. There are many detailed technical references for the various assimilation methodologies used in oceanography, some of them we would like to list here for the reader interested in more technical background information. At the most fundamental levels, inverse methods in oceanography are rather similar to those used in geophysics. Some comprehensive textbooks for this mature field are "Geophysical Data Analysis: Discrete Inverse Theory" by Menke (1984) and "Inverse Problem Theory" by Tarantola (1987). However, these reviews do not meet the requirement of oceanography, that is an analysis of these methods for their application to nonlinear, time-dependent dynamical models of the three-dimensional ocean circulation. From the point of view of the complexity of the physical systems, and of the associated dynamical models, the analysis and application of these methods discussed in Daley's (1991) book, "Atmospheric Data Assimilation", is perhaps the most relevant for oceanographers. Two major differences still prevent the simple "borrowing" of techniques from meteorology. The first is the motivation for oceanic data assimilation which, as discussed further in the next section, is not as narrowly focused towards short term prediction as are most meteorological efforts. Although it must be added that the motivation for ocean forecasting is rapidly emerging as legitimate and important per se. This book in fact provides important examples of oceanographic operational forecasting. The second reason resides in the major difference between the meteorological and the oceanographic data sets, as further discussed in the next section. This implies that these methodologies, far from being blindly applied to oceanic dynamical problems, must be revisited and sometimes profoundly modified to make them feasible and successful for physical oceanography. Recent reviews and synthesis of data assimilation methods for oceanographic applications can be found, for example, in the lectures by Miller (1987); in the special issue of Dynamics of Atmospheres and Oceans devoted to Oceanographic Data Assimilation, Haidvogel and Robinson, eds. (1989); and in the review paper by Ghil and Malanotte-Rizzoli (1991). The latter one provides also a very comprehensive review of the literature up to the early 90's. A very recent, thorough synthesis of oceanographic assimilation methodologies is given in Bennett (1992). 2. H I S T O R I C A L PERSPECTIVE Over the past 25 years or so, since the initial efforts to develop three dimensional ocean circulation models (Bryan, 1969), ocean modeling has made a very significant progress. Chapter 2.1 by Holland and Capotondi provides a review of the milestones in the development and advancement of OGCM's, up to the complexity and sophistication of the present generation of models, capable of most realistic simulations on the global scale.

Chapter 2.1 also offers a perspective of the future possibilities and trends of ocean modeling. In parallel, oceanic observational techniques have been thoroughly revolutionized. However, the lack of a single focusing motivation of oceanic data assimilation such as provided by the need for Numerical Weather Predi6tion (NWP) in meteorology, caused ocean models and observational techniques to develop quite independently from each other. When oceanic models and observations started converging, it happened in different paths, depending on the specific objective of each effort. The early days of oceanography saw dynamic calculations as the main quantitative tool to combine data (temperature and salinity) with "models" (the thermal wind relations). From this modest beginning, relying on highly simplified models and on no formal assimilation procedure, the next step was to introduce a formal least square inverse methodology imported from solid earth geophysics and add the tracer conservation constraints in order to solve the problem of the level of no motion (Wunsch, 1978; Wunsch and Grant, 1982; Wunsch, 1989a,b). This was done in the framework of coarse resolution box models whose dynamics was still very simple although the inverse methodology used was very general. Much of the work done at present on the combination of OGCMs and data stems from the experience obtained in the pioneering work on oceanographic box inverse models. At the other extreme of model complexity versus sophistication of the assimilation method, efforts began with the "diagnostic models" in which temperature and salinity data were simply inserted into the dynamical equations of fairly complex ocean models in order to evaluate the velocity field (Holland and Hirschman, 1972). The results were very poor due to model-data-topography inconsistencies, and at the next stage, a very simple assimilation methodology was introduced into OGCMs and became known in the oceanographic context as the "robust diagnostic" approach (Sarmiento and Bryan, 1982). The same approach had actually been introduced earlier in meteorology as the "nudging" technique (Anthes, 1974) and the term "nudging" has by now become commonly used also in oceanography. In this approach there is no effort to introduce least-square optimality, and the data are just used to nudge the model solution towards the observations at each time step through a relaxation term added to the model equations. The result is far superior to simple diagnostic models, but leaves much to be desired due to the inability to use information about data uncertainty or to estimate the errors in the solution obtained (Holland and Malanotte-Rizzoli, 1989; Capotondi et al., 1995a,b; Malanotte-Rizzoli and Young, 1995). As the objectives of modeling and observational oceanography began to converge, more formal least square methods taken from meteorology were also used in ocean models, in particular the Optimal Interpolation (OI) method (Robinson et al., 1989; Derber and Rosati, 1989; Mellor and Ezer, 1991). OI may be viewed as a nudging technique in which the amount of nudging of the model solution towards the observations depends on the data errors, while also allowing to make error estimates for the solution. This approach, developed in meteorology for NWP, is not capable of improving model parameters or parameterizations, nor is it capable of fitting the entire four dimensional distribution of observations simultaneously to the model solution. However, due to the relatively low computational cost of OI, it is appropriate for higher resolution, short term prediction and state estimation purposes. Carrying the least square approach for a time dependent model to its rigorous limit, leads to the "Kalman filter/smoother" assimilation methodology, which is capable of assimilating data into a time dependent model while assuring least-square optimality, full use of a priori error estimates, and calculation of the covariance error matrix for the model outputs. Apart from the fact that the Kalman filter is a formally optimal technique in the least-square sense only for linear models, its high computational cost limits its use at present to simple models, or very coarse OGCMs. Recent efforts are directed at developing efficient even though sub optimal variants of the Kalman filter that allow the use of a full nonlinear OGCM with this method (e.g. Fukumori and Malanotte-Rizzoli, 1995). The ultimate goal of combining a formal least-square optimization approach with a full complexity OGCM requires the simultaneous solution of hundreds of thousands of coupled

nonlinear equations (the model equations at all grid points and all time steps), and therefore requires an efficient approach which can be found in the "optimal control" engineering literature. This approach, also known as the "adjoint method", is capable of model improvement, parameter estimation and true four dimensional data assimilation. It is equivalent in principle to the Kalman filter (Ghil and Malanotte-Rizzoli, 1991), except that it allows to give up the use and calculation of full covariance matrices, and therefore is more computationally feasible for higher resolution nonlinear OGCMs (Tziperman and Thacker, 1989; Tziperman et al., 1992a,b; Marotzke, 1992; Marotzke and Wunsch, 1993; Bergamasco et al., 1993). The covariance information may be added to the calculation if the computational cost can be afforded. The development of assimilation methods in physical oceanography seemed to always trail behind meteorology by a few years. This lag is in spite of the fact that the ocean and atmosphere, even though characterized by some important differences, are at the same time similar enough that they can be treated with the same theoretical approaches and methodologies. It is important, therefore, for the ocean modeler to try and understand the reason for this difference in rate of development of data assimilation methodologies in order to be able to isolate potential obstacles for their future use in oceanography. Clearly a primary reason for the delayed development of oceanic data assimilation was the lack of urgent and obvious motivation such as the need of forecasting the weather and of producing better and longer forecasts as necessary in meteorology. This situation has been rapidly changing in recent years as further discussed in the following section, and ample motivation for ocean data assimilation now exists due to the need for systematic model improvement and for ocean state estimation. The need for ocean prediction is also arising now on various temporal and spatial scales, from climate change predictions, through regional forecasts of the large scale ocean climate variability, e.g. of the North Atlantic thermohaline circulation or E1 Nino in the Pacific Ocean, to a few weeks regional mesoscale ocean forecasts in frontal regions such as the Gulf Stream system that are required for example by various Naval applications. The most profound limitation on the development of oceanic data assimilation may have been, however, the lack of adequate data sets. The number of available oceanographic observations is far smaller than the number of meteorological observations, especially when the different temporal and spatial scales are considered. It is estimated, in fact, that the number of presently available oceanographic observations is smaller than its meteorological counterpart by several orders of magnitude (Ghil and Malanotte-Rizzoli, 1991). New oceanographic deta sets, nearly comparable to the meteorological one, i.e. synoptic and with global coverage, are however becoming available. This oceanographic observational revolution of the 90's has been made possible by the advent of satellite oceanography. Already --.40,000 sea surface temperatures are now available daily on a global scale, measured by the NOAA satellites that have been flying since the 80's. In addition, two satellite altimeters are now providing observations of the ocean surface topography that is tightly coupled to ocean currents. The first is TOPEX/POSEIDON, launched in 1992, that is currently producing global maps of sea surface height with a horizontal resolution of---300 km x 300 km at mid-latitudes every 10 days, and at an impressive accuracy of 5cm (Wunsch, 1994; Fu, 1994; Stammer and Wunsch, 1994). The European satellite ERS-1 is also measuring sea surface topography with higher spatial resolution that resolves the mesoscale eddy field. It also measures the surface wind field on the global scale, at a 1 degree resolution, hence providing information about a crucial driving force for the oceanic circulation. Chapter 2.3 by Fu and Fukumori gives a review of the effects of errors in satellite altimetry for constraining OGCM's through data assimilation. In order to be able to use the altimetric data to study the large scale oceanic circulation, it is however necessary to filter out the effects of tides on the altimetric measurements. The evaluation of global ocean tides can be formulated as an inverse problem and Chapter 3.2 by Egbert and Bennett discusses the possible data assimilation methods.

It is worthwhile to mention two other novel sources of oceanic observations that should help the development of oceanographic data assimilation. The first is the relatively new observational technique of ocean acoustic tomography. Tomography exploits the fact that the ocean is transparent to sound waves and, like in the medical application, the tomographic technique scans the ocean through two-dimensional (vertical or horizontal) slices via sound waves. The difference and novelty of ocean tomography with respect to more traditional point-wise oceanographic measurements lies in the integral nature of the tomographic datum (Worcester et al., 1991). The implications and needs for the assimilation of such integral data into OGCM's are discussed by Comuelle and Worcester in Chapter 2.4. A second worldwide major source of oceanographic observations is the World Ocean Circulation Experiment (WOCE) that, through basin wide hydrographic sections, meridional and zonal, should provide a zero-order picture of the large scale global circulation in the 90's. Because hydrographic sections are not synoptic, and are mostly carded out only once, no data of the time evolution will be available and very large water bodies between adjacent sections still remain void of data. Hence the great importance of numerical models endowed with data assimilation capability to act as dynamical interpolators/extrapolators of the oceanic motions. Clearly ocean models and assimilation methods can make better use of the various new and traditional sources of oceanographic data when reliable error estimates are available. Particularly important is the possibility of obtaining estimates of the non-diagonal terms of the error covariance matrices, for which only the diagonal terms, i.e. the data standard deviations, are usually specified. The efforts to obtain such estimates of the full error covariances of traditional oceanographic datasets are discussed by Hogg in Chapter 2.2. The above brief discussion of the arising needs for ocean data assimilation and the new data sets that are becoming available indicates that possible obstacles to the development of oceanic data assimilation methods have been overcome. Oceanographic data assimilation should now become a fully developed research field. Hence the timeliness of developing modern oceanographic assimilation methods for the OGCM's and the oceanographic data set of the 90's. 3. OBJECTIVES OF OCEANOGRAPHIC DATA ASSIMILATION Efforts to combine fully complex OGCMs and oceanographic data may roughly be divided into three main categories: model improvement, study of dynamical processes through state estimation, and, finally, ocean/climate forecast. Let us now consider these objectives in some detail, as well as the relevant assimilation methodologies for each of them. Even the highest resolution ocean circulation models cannot resolve all of the dynamically important physical processes in the ocean, from small scale turbulence to basin scale currents. There will always be processes that are not represented directly, but rather are parameterized. These parameterizations are sometimes simple, often complicated, and always quite uncertain both in form and in the value of their tunable parameters. Very often, the uncertainty in these parameterizations is accompanied by an extreme sensitivity of the model results to slight variations in them. An obvious though not unique example is the parameterization of small scale vertical mixing in the ocean interior for which many forms have been proposed, and which drastically affects the strength of the thermohaline circulation and the estimate of meridional heat flux of OGCMs (Bryan, 1987). A few other examples are the parameterizations of mesoscale eddies in coarse ocean models used for climate studies (Boning et al., 1995), of mixed layer dynamics (Mellor and Yamada, 1982), and of deep water formation (Visbeck et al., 1994). Another set of uncertain yet crucial parameters corresponds to the poorly known surface forcing by wind stress, heat fluxes and evaporation and precipitation, all of which are subject to typical uncertainties of 30-50% (Trenberth et al., 1989; Schmitt et al., 1989; Trenberth and Solomon, 1993). Although observations of most of the above unknown model parameters are not available, and many of these parameters are not even directly measurable, there is a wealth of other

oceanographic data that can be used to estimate the unknown parameters. In fact, a most important goal of oceanographic data assimilation is to use the available data systematically and quantitatively in order to test and improve the various uncertain parameterizations used in OGCMs. It is important to understand that by model improvement we refer to the use of data for the determination of model parameters or parameterizations in a way that will result in better model performance when the model is later run without data assimilation. There are typically thousands of poorly known internal model parameters, such as viscosity/diffusivity coefficients at each model grid-point, and many thousands if the surface forcing functions are included at every surface grid point (Tziperman and Thacker, 1989). The estimation of these parameters therefore becomes an extremely complicated nonlinear optimization problem which needs to be carried out using efficient methodologies and powerful computers. An assimilating methodology which seems to have the potential to deal with these estimation problems is the conjugate gradient optimization using the adjoint method to calculate the model sensitivity to its many parameters (Hall and Cacuci, 1983; Thacker and Long, 1988). Due to the extreme nonlinearity and complexity of the problem, it is possible however that gradient based methods will not suffice and will need to be combined with some sort of simulated annealing approach to assist in finding a global optimal solution in a parameter space filled with undesired local solutions (Barth and Wunsch, 1990). The adjoint method, while efficient, still requires a significant computational cost when applied to a full OGCM, and is therefore probably limited at present to medium to low resolution ocean models. The resolution of coupled ocean atmosphere models is also limited due to the high computational cost of running them. It is feasible, therefore, that the adjoint method can be used for improving the ocean component of course coupled ocean atmosphere models. A step in this direction is presented in Chapter 3.1 by Sirkes et al., who use the adjoint method with a global primitive equation ocean model of a resolution and geometry similar to that used in several recent coupled ocean-atmosphere model studies. To demonstrate the above general discussion of model improvement by data assimilation, let us now briefly consider two examples of well known difficulties with ocean models that could potentially benefit from data assimilation methodologies. The first is the very strong artificial upwelling in the mid-latitude North Atlantic, in the region inshore of the Gulf Stream (Toggweiler et al., 1989) and in mid-latitudes either using the GFDL (Geophysical Fluid Dynamics Laboratory) model (Sarmiento, 1986; Suginohara and Aoki, 1991; Washington et al., 1993) or using the Hamburg large-scale geostrophic model (Maier-Reimer et al., 1993). Boning et al., (1995) show that this upwelling is concentrated in the western boundary layer, roughly between 30* to 40~ and significantly reduces the amount of deep water carried from the polar formation region toward low latitudes and the equator. This strong upwelling is also responsible for the underestimated meridional heat transport in the subtropical North Atlantic which is reduced by about 50% and is due to the deficiency of the parameterization of tracer transports across the Gulf Stream front through the usual eddy diffusivity coefficient. By improving the mixing parameterization using an isopycnal advection and mixing scheme recently proposed by Gent and McWilliams (1990), Boning et al. are able to obtain very substantial improvements in the southward penetration of the NADW (North Atlantic Deep Water) cell and consequently in the meridional heat transport in the subtropical North Atlantic. The parameterization used by Boning et al is but one of many possible forms, and one would like to see the work of Boning et al. done in an even more thorough and systematic manner, by putting all possible parameterizations into a model, and letting a systematic data assimilation/inverse procedure choose the parameterization and parameters that result in the best fit to the available data. A second example concerns the difficulty of high resolution ocean models to reproduce the correct separation point of the Gulf stream from the North American continent. This may be due to insufficient model resolution, yet may also be due to imperfect model parameterizations or poor data of surface boundary forcing (Ezer and Mellor, 1992). It is

foreseeable that an improved set of surface boundary conditions may be found through data assimilation, that may eliminate this model problem. In both of the above examples, the improvement of internal model parameters and of surface boundary conditions via data assimilation may be complemented by a second data assimilation activity, the "state estimation". In this case, model deficiencies are compensated for by using data to force the model nearer to observations during the model run. Thus the strong upwelling found in most simulations of the North Atlantic circulation in the region inshore of the Gulf Stream that results in the shortcut of the thermohaline circulation may be corrected by running the model in a data assimilation mode, rather than as a purely prognostic model. Such a calculation has been carried out by Malanotte-Rizzoli and Yu (private communication) using the fully nonlinear, time-dependent GFDL code (Cox 1984) and its adjoint first used by Bergamasco et al. (1993) which has been adapted to the North Atlantic ocean to carry out assimilation studies of North Atlantic climatologies (Yu and MalanotteRizzoli, 1995). The model is forced by the Hellermann and Rosenstein winds (1983) and the adjoint calculation provides the steady state optimal estimate of the North Atlantic circulation consistent with the Levitus (1982) climatology of temperature and salinity. The assimilation partially corrects for the deficiencies of the analogous purely prognostic calculation. A more realistic meridional thermohaline cell is obtained that protrudes southward much more significantly with -2/3 of the production rate of 16 Sverdrups (SV.) crossing the equator, more closely to the observational figure o f - 1 4 Sv (Schmitz and McCartney, 1993) than in the prognostic simulation. The strong upwelling at 30~ inshore of the Gulf Stream observed by Boning et al. (1995) is in fact eliminated. On the other side, the horizontal wind-driven circulation of the subtropical gyre reconstructed by the adjoint is still too weak, with a maximum Gulf Stream transport o f - 6 0 Sv compared to the value o f - 1 2 0 Sv found after detachment from Cape Hatteras when encompassing the Southern Recirculation gyre transport (Hogg, 1992). This is due to the smoothed nature of the Levitus climatology showing a "smeared" Gulf Stream front with a cross-section o f - 6 0 0 km as compared to the realistic values of 200-300 km (Hall and Fofonoff, 1993). In the case of the Gulf Stream separation point, altimetric and other data can be used to constrain the model to the fight separation point (Mellor and Ezer, 1991; Capotondi et al., 1995a,b), and then the resulting model output may be used to study the dynamical processes acting to maintain this separation point. The improved understanding of the dynamics obtained through such uses of data assimilation should eventually result in improved model formulation and more realistic model results. In spite of the extensive data sets that are becoming available through the new remote sensing methods and the extensive global observational programs mentioned in section 2, the ocean is still only sparsely observed. Most of the interior water mass, and especially the abyssal layers, will still remain unmonitored. Hence a second aspect of state estimation is the one in which numerical models are constrained by the data to reproduce the available observations, and act as dynamical extrapolators/interpolators propagating the information to times and regions void of data. An especially important example concerns the use of satellite data. It has been shown that ocean models are indeed able to extrapolate instantaneous surface altimetric observations to correctly deduce eddy motions occurring as deep as the main thermocline, at approximately 1,500m (Capotondi et al., 1995a,b; Ghil and MalanotteRizzoli, 1991). Clearly this strengthen the case for both the need for data assimilation developments and for satellite altimetry as a global observational system. The ocean state dynamically interpolated by data assimilation may serve several important goals. On a global scale, unobservable quantifies such as the meridional heat flux and the airsea exchanges can be continuously monitored from the assimilation output to infer possible changes due to climate trends. The knowledge of the natural variability of these quantities is essential for us to be able to differentiate between natural climate variability and a maninduced climate change. On a more regional scale, the high resolution, eddy resolving interpolation of remote sensing data by the models (Mellor and Ezer, 1991; Capotondi et al.,

10 1995a,b) provides a four dimensional picture of the eddy field which can then be used to study detailed dynamical processes of eddy-mean flow interaction, equatorial wave dynamics, ring formation and ring/jet interactions in the energetic western boundary currents. Such studies, even though they can be done based on the sparse data alone or on model output alone, will gain considerably when carried out on the "synthetic" oceans obtained through data assimilation in dynamical models. Many of the chapters of this book concern the problem of oceanic state estimation through data assimilation. The global applications of the already mentioned chapters 3.1 and 3.2 are related to the estimate of the steady state global circulation (Sirkes et al., Chapter 3.1) and of global ocean tides (Egbert and Bennett, Chapter 3.2). In the tropical ocean, Chapter 4.2 by Busalacchi illustrates how the unique physics of the low-latitude oceans and the wealth of observational data from the Tropical Ocean Global Atmosphere program have been a catalyst for tropical ocean data assimilation. Among these tropical ocean assimilations are some of the first applications of the Kalman filter and adjoint methods to actual in situ ocean data. These methodologies and related theoretical considerations are discussed in Chapter 4.1 by Miller and Cane. In the context of regional applications, Chapter 5.1 by Malanotte-Rizzoli et al. discusses the development of an efficient and affordable Kalman filter/smoother for a complex, fully nonlinear Primitive Equation model suitable for studies of nonlinear-jet evolutions, model used for realistic simulations of the Gulf Stream system, albeit until now with only a simple nudging assimilation scheme (Malanotte-Rizzoli and Young, 1995). The third distinct objective of oceanic data assimilation, i.e. ocean and climate nowcasting and prediction, has not been until recently a subject of interest to mainstream oceanography. At present, however, there are more and more specific oceanographic applications in which prediction is not only timely but necessary. It is convenient to classify the oceanographic prediction problems by their time scale, as each of them requires different methodologies of approach and different data. The problem of climate change is a prediction problem, and therefore needs to be treated as such. Simulation studies of climate change, on a time scale of 50 to 100 years, due to CO2 increase and the greenhouse effect, have recently begun to use coupled ocean-atmosphere models. A very recent study has extended such coupled models simulations to a multiple century time scale (Manabe and Stouffer, 1994). The inclusion of full ocean models in these studies is obviously a step in the fight direction considering the significant effect of the ocean on climate on time scales of decades and longer. The use of coupled models is also an important progress from a few years ago when such studies were based on atmospheric models alone, or coupled to a simple mixed-layer ocean models (Wilson and Mitchell, 1987; Schlesinger and Mitchell, 1987; Wetherald and Manabe, 1988; Washington and Meehl, 1989a), or coupled to a model parameterizing heat transport below the mixed layer as a diffusive process (Hansen et al., 1988). Recent studies using fully coupled atmosphere-ocean GCM's have taken one of two routes in initializing greenhouse warming simulations. The first approach is to initialize the simulation with steady state solutions of the separate ocean and atmosphere sub models obtained by running the two models separately (Stouffer et al., 1989; Manabe et al., 1991; Cubasch et al., 1992; Manabe and Stouffer, 1994). In this procedure the atmospheric model is spun up to a statistical steady state using prescribed SST climatology, such as the Levitus (1982) analysis. The ocean model is then spun-up using boundary conditions which restore the surface temperature and salinity to a similar climatology. The difference in the diagnosed heat and fresh water fluxes from the separate ocean and atmosphere spin-up runs is used to calculate "flux correction" fields. The two models are then coupled, and the flux correction fields are added to the ocean surface forcing at every time step during the subsequent long coupled integration. This correction, while clearly artificial and often of undesirably large amplitude, prevents the quite substantial drifts of the coupled system from the present climate occurring due to the fact that the ocean steady solution is incompatible with the heat and fresh water fluxes provided by the atmospheric model. The initialization of coupled models with steady ocean solutions that are

11 obtained by restoring the surface model fields of temperature and salinity to climatological data averaged over the last 40 years or more clearly leaves room for significant improvements. This initialization procedure ignores most of the available data which are data from the ocean interior. In addition, the use of many year averaged surface data sets results in a very artificial smoothing and therefore distortion of many important observed features of the oceanic circulation. The second approach to greenhouse warming simulations is to initialize the model with the observed ocean climatology averaged over tens of years, normally without applying flux correction to avoid a climate drift of the coupled system (Washington and Meehl, 1989b). This approach, while avoiding the artificial flux adjustment procedure, suffers from a serious drawback. It is well known from numerical weather prediction that initializing a forecast with the raw data without any weight given to the model dynamics, leads to severe initial "shocks" of the forecast model while it is adjusting to the initial conditions. Such a violent response may be expected in the climate prediction context as well and may severely affect the model response to the greenhouse signal. What is needed for the climate prediction problem is an assimilation approach that will initialize the prediction simulation using a blending of the data and model results. The initialization should prevent initial shocks, yet constrain the initial condition using the available four dimensional oceanic data base, without the artificial smoothing resulting from the temporal averaging procedure. Such an initialization may also reduce the need for the artificial flux correction procedure. For such an initialization, a four dimensional global coverage of the ocean is required, as may be provided by programs such as WOCE. Synoptic eddy resolving ocean data are most probably not necessary, as the models used for climate simulations are at this stage far from being eddy resolving, and a precise mapping of the eddy field is not essential for the dynamics in question, but only an overall knowledge of the eddy statistics. Because climate models are fairly coarse due to the high computational cost of these simulations, the assimilation problem can probably be carried out using the more sophisticated assimilation methods, such as the extended Kalman filter or the variational adjoint method. It is important to note, however, that practically nothing has been done so far to address this assimilation/prediction problem which is clearly of paramount interest and importance. Another coupled climate problem in which prediction is needed is the decadal climate variability problem in which the ocean plays the major role. There are indications, for example, that variability of the North Atlantic thermohaline circulation affects the northern European climate on time scales of 10 to 30 years (Kushnir, 1994). The resulting climate and weather variability has important implications on atmospheric temperature and precipitation over vast regions, is mostly controlled by oceanic processes, and its prediction is of obvious value. The forecasting of decadal climate variability, like that of the global greenhouse problem, needs to be carried out using coupled ocean-atmosphere models and appropriate data sets and assimilation methodology. The mechanisms of the thermohaline variability are still under investigation, with very diverse explanations offered so far, from strongly nonlinear mechanisms (Weaver et al. 1991) suggested using ocean-only model studies to gentler, possibly linear mechanism, based on coupled ocean-atmosphere model studies (Delworth et al, 1994; Griffies and Tziperman, 1994). As the mechanism of this variability is not yet clear, data assimilation could be used to interpolate the little data that exist for this phenomenon, and perhaps clarify the unresolved dynamical issues. The physical mechanisms of decadal climate variability that results from fluctuations of the thermohaline circulation may have important implications concerning the predictability of this variability. Preliminary efforts to examine the predictability of such decadal climate variability are underway (Griffies and Bryan, 1995), yet practically no work has been carried out so far to address this issue as an assimilation and prediction problem, nor are the appropriate data available at present. An ocean/climate forecasting problem which presents a successful example of the application of data assimilation methods to ocean/climate problems is the occurrence of E1

12 Nino-Southern Oscillations (ENSO) in the Pacific equatorial band every three to six years. The profound global socio-economic consequences of this phenomenon have attracted considerable attention in terms of both pure modeling, data collection, and assimilation/ forecasting studies. Barnett et al. (1988) discussed three different approaches used to successfully predict the occurrences of ENSO. One such forecasting scheme uses statistical models that rely on delayed correlations between various indicators in the Equatorial Pacific and the occurrence of ENSO (Barnett, 1984; Graham et al. 1987). A second scheme uses a linear dynamical ocean model that is driven by the observed winds. In the forecast mode, the winds are assumed to remain constant beyond the last time when observations are available, and the ocean model is integrated ahead for a few months to produce the forecast (Inoue and O'Brien, 1984). The third ENSO forecast scheme uses a simple coupled ocean atmosphere model with linear beta plane dynamics, and a nonlinear equation for the SST evolution. The model is again initialized by running it with the observed winds, and then is integrated further to obtain the forecast (Cane et al, 1986). Using these various schemes, ENSO occurrences can now be forecast a year in advance with reasonable accuracy. Yet the existing schemes clearly leave room for improvements. Even models that are used now quite successfully for ENSO prediction (Cane et al, 1986) are still fairly simple, with the background seasonal cycle specified in both the atmosphere and the ocean, with linearized dynamics, and with very simplified atmospheric parameterizations. Improvements are needed in the form of fuller models with more realistic parameterization of the oceanic and atmospheric physics, that can simulate both the mean seasonal state and the interannual variability. In addition, the present forecasting schemes do not make full use of the available data, and rely mostly on the observed winds. Better performance may be achieved using more complete assimilation methodologies that use all the available data, including interior ocean data for the temperature, salinity and currents as demonstrated in the recent work by Ji et al. (1995). Indeed, work is underway to apply the most advanced models and assimilation schemes to the ENSO prediction problem. Until very recently, simple coupled ocean-atmosphere models seemed to be more successful in ENSO forecasting, and fuller primitive equation models had serious difficulties in simulating, not to mention forecasting, ENSO events. This situation is changing now, and full three dimensional primitive equation ocean models coupled to similar atmospheric models are now catching up with the simpler models. Miyakoda et al (1989), for example, have been using such a PE (Primitive Equation) coupled model together with an OI assimilation method to forecast ENSO events. Another direction in which progress has been made is the development of more advanced assimilation methods such as Kalman filtering for this application. As in other applications discussed above, the ENSO prediction problem requires its own variant of these assimilation methodologies, based on the apparently chaotic character of ENSO dynamics (Burger and Cane, 1994; Burger et al., 1995). Chapter 3.3 by Rosati et al. provides an important example of an oceanic fourdimensional data assimilation system developed on the global scale for use in initializing coupled ocean-atmosphere general circulation models (GCM) and to study interannual variability. The model used is a high resolution global ocean model and special attention is given to the tropical Pacific ocean examining the E1 Nino signature. Chapter 4.3 by Leetma and Ji also provides an example of an ocean data assimilation system developed as a component of coupled ocean-atmosphere prediction models of the ENSO phenomenon, but only for the tropical Pacific configuration. The assimilation system combines various datasets with ocean model simulations to obtain analyses used for diagnostics and accurate forecast initializations. These improved analyses prove to be essential for increased skill in the forecast of sea surface temperature variations in the tropical Pacific. On a yet shorter time scale, we find the problem of extended seasonal weather prediction, in which again the ocean plays a crucial role. There are many situations in which a seasonal forecast of the expected amount of precipitation, for example, can have a significant impact on agricultural planning, especially in semi-arid regions, but not only there. The application

13 of coupled ocean-atmosphere GCMs to this problem is at its infancy, and the obvious need for such work can be expected to result in more efforts in this direction in the near future. It is interesting to note that all the ocean forecasting problems surveyed so far involve using a coupled ocean-atmosphere model, rather than an ocean-only model. There are, however, situations in which ocean-only models can be utilized for relevant short term assimilation and forecasting studies. A first example for the ocean component alone is given in Chapter 5.2 by Carnes et al. who discuss an ocean modeling-data assimilation monitoring and prediction system developed for Naval operational use in the North Pacific Ocean. Results are presented from three-months long pseudo-operational tests in the effort to address, among other issues, the problem of extended ocean prediction. A further example of forecasts on a very short time scale is given in Chapter 5.3 by Aikman et al., in which a quasi-operational East Coast Forecast system has been developed to produce 24-hour forecasts of water levels, and the 3dimensional fields of currents, temperature and salinity in a coastal domain - 24 hour forecasted and observed fields are compared to improve the basic system itself before implementing it with a data assimilation capability. Finally, an important example is the interest of navies in ocean frontal systems on a time scale of two to four weeks, such as the prediction of the Gulf Stream front and of its meandering. The operational prediction of such synoptic oceanic motions is therefore a primary objective "per se" and a new professional, the ocean forecaster, is rapidly emergingl Like the east coast forecast system of Chapter 5.3, this application is the closest to the meteorological spirit of real-time assimilation and prediction. It involves real time processing and assimilation of remote sensing data, and the production of timely forecasts of front locations and other eddy features in the ocean. A significant body of work already exists for this purpose, and development of such operational forecasting systems is fairly advanced. See, for instance, the issue of Oceanography, Vol. 5, no. 1, 1992 for a review of such operational forecasting systems in the world ocean, with a general discussion of the Navy Ocean Modeling and Prediction Program (Peloquin, 1993) and the interesting DAMEE-GSR effort in the Gulf Stream System involving the assessment of 4 different models through prediction evaluation experiments (Leese et al., 1992; see also Ezer et al., 1992 and Ezer et al., 1993). Chapter 5.4 by Robinson et al. discusses real-time regional forecasting carried out in different areas of the world-ocean. The use and limitations of this methodology are illustrated with practical examples using both a primitive equation and an open ocean quasi-geostrophic model. The latter one constitutes by itself a flexible and logistically portable open-ocean forecasting system, that has been tested in 11 sites of the world ocean comprising frontal systems. All the tests were real-time forecasts, and for six of them the forecasts were carried out aboard ships (Robinson, 1992). Finally, Chapter 6 by Lozano et al. presents one of the first interdisciplinary applications in developing an ocean prediction system. 4. CONCLUSIONS Having considered some of the objectives of ocean data assimilation, it is quite surprising to realize how much work is still required to meet them. Much of the effort presently invested in oceanographic data assimilation is in the development of appropriate methodologies, in preparation to approaching the objectives discussed above. The diverse set of objectives discussed here clearly points out that no single assimilation methodology can address all of the needs. It is more likely that several techniques, such as the Kalman Filter, Adjoint Method and Optimal Interpolation will be the main candidates for addressing the future needs of oceanographic assimilation. Each of these methodologies will be used for the specific goals to which it is best suited. With ample motivation for the combination of fully complex Ocean General Circulation Models and oceanic data, and with new observational techniques and global observational

14 programs being developed, further developments in oceanic data assimilation are essential. Clearly the needs in this area surpass the invested efforts at this stage, and a significant growth of this research field is needed and may be expected to occur in the very near future. 5. ACKNOWLEDGEMENTS This research was carried out with the support of the National Aeronautics and Space Administration, Grant #NAGW-2711 (P. Malanotte-Rizzoli). 6. REFERENCES

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17 Trenberth, K.E., J.G. Olson, and W.G. Large, 1989: A global ocean wind stress climatology based on ECMWF analysis, NCAR/TN-338+STR, pp. 93. Trenberth, R.E. and A. Solomon, 1995: The global heat balance: heat transports in the atmosphere and ocean, Clim. Dyn., submitted. Tziperman, E. and W.C. Thacker, 1989: An optimal control/adjoint equation approach to studying the oceanic general circulation, J. Phys. Oceanogr., 19, 1471-1485. Tziperman, E., W.C. Thacker, R.B. Long and S.-M. Hwang, 1992a: Oceanic data analysis using a general circulation model, Part I: Simulations, J. Phys. Oceanogr., 22, 1434-1457. Tziperman, E., W.C. Thacker, R.B. Long and S.-M. Hwang and S.R. Rintoul, 1992b: Oceanic data analysis using a general circulation model, Part II: A North Atlantic model, J. Phys. Oceanogr., 22, 1458-1485. Visbeck, M., J. Marshall and H. Jones, 1995: On the dynamics of convective "chimneys" in the ocean, J. Phys. Oceanogr., submitted. Washington, W.M. and G.A. Meehl, 1989a: Seasonal cycle experiments on the climate sensitivity due to a doubling of CO2 with an atmospheric general circulation model coupled to a simple mixed layer ocean model, J. Geophys. Res., 89, 9475-9503. Washington, W.M. and G.A. Meehl, 1989b: Climate sensitivity due to increased CO2: experiments with a coupled atmosphere and ocean general circulation model, Clim. Dyn., 4, 1-38. Washington, W.M., G.A. Meehl, L. VerPlant, and T.W. Bettge, 1995: A world ocean model for greenhouse sensitivity studies: resolution intercomparison and the role of diagnostic forcing, Climate Dynamics, in press. Weaver, A.J., E.S. Sarachik, and J. Marotzke, 1991: Freshwater flux forcing of decadal and interdecadal oceanic variability, Nature 353, 836-838. Wetherald, R.T. and S. Manabe, 1988: Cloud feedback processes in a general circulation model, J. Atmos. Sci., 45, 1397-1415. Wilson, C.A. and J.F.B. Mitchell, 1987: A doubled CO2 climate sensitivity experiment with a global climate model including a simple ocean, J. Geophys. Res., 92, 13,315-13,343. Worcester, P.F., B.D. Cornuelle, and R.C. Spindel, 1991: A review of ocean acoustic tomography: 1987-1990, Rev. Geophys., Supplement, 557-570. Wunsch, C.I., 1978: The general circulation of the North Atlantic west of 50~ determined from inverse methods, Rev. Geophys. Space Phys., 16, 583-620. Wunsch, C.I., 1994: The TOPEX/POSEIDON data, International WOCE Newsletter, No. 15, 2224. Wunsch, C.I., 1989a: Using data with models, ill-posed and time-dependent ill-posed problems in "Geophysical Tomography", Y. Desaubies, A. Tarantola and J. Zinn-Justin, eds., Elsevier Publ. Company, pp. 3-41. Wunsch, C.I., 1989b: Tracer inverse problems, in "Oceanic circulation models: combining data and dynamics", D.L.T. Anderson and J. Willebrand, eds., Kluwer Academic Publ., pp. 1-78. Wunsch, C.I. and B. Grant, 1982: Towards the general circulation of the North Atlantic ocean, Progr. Oceanogr., 11, 1-59. Yu, L and P. Malanotte-Rizzoli, Analysis of the North Atlantic climatologies through the combined OGCM/Adjoint Approach, to be submitted.

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Models and Data

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Modern Approaches to Data Assimilation in Ocean Modeling

edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

21

Recent Developments in Prognostic Ocean Modeling William R. Holland and Antonietta Capotondi National Center for Atmospheric Research, Boulder, Colorado 80307-3000 Abstract Prognostic ocean circulation models have developed rapidly in the past two decades. Global general circulation models are now capable of reproducing reasonably well the large scale features of the temperature and salinity fields representative of today's climatic state. Both eddy and non-eddy models are actively being used to address a wide variety of issues concerning the oceanic circulation on a variety of time scales. Here we describe a sampling of this work and then turn toward the remaining difficulties in malting further progress in this class of models. In particular, the necessity for and ability to parameterize fast time-scale and small space-scale behavior- the internal mixing of heat, salt and momentum- is outlined and the tools and recent steps for validating such parameterizations discussed. Better prognostic models of the ocean system are a necessary step toward models that can accurately combine observational data with a high quality model for prediction purposes.

1. I N T R O D U C T I O N Numerical models of the general circulation of the ocean are important research tools for understanding the oceanic circulation, the role of the ocean in climate change and the biogeochemical processes occuring within the ocean interior. An adequate understanding of the dynamics and thermodynamics of the physical system and the important geochemi~ry is necessary for the eventual prediction of climate change, an ability that is urgently required of the scientific community. The climate system is in an ever-changing state with vast impact on mankind in all his activities. Both short and long-term aspects of climate variability are of concern, and the unravelling of "natural" variability from "man-induced" climate change is necessary to prepare for and ameliorate, if possible, the potentially devastating aspects of such change. In this paper, we shall discuss p r o g n o s t i c ocean models, rather than models that assimilate observations. The quality of the basic physical model when no data constraints are applied is the first step for successful assimilation experiments, where the prognostic ocean model supplies a reliable dynamical behavior while the assimilated observations can help constructing realistic initial conditions. Without validated prognostic models, one can have little confidence in companion assimilation models. There are several difficulties in developing realistic models of the ocean system.

22 Firstly, the physical laws that describe the large-scale ocean circulation are quite complex and highly nonlinear. The global oceanic problem requires that a broad range of space and time scales be properly represented in the calculations, either by explicitly computing them or by representing the effects of smaller spatial scales and faster time scales in some validated way. This is the closure problem. However, present parameterizations of physical processes known to have some importance may not be adequate, and there are a whole host of different eddy-mean flow interactions that might be important. As we shall discuss, ocean model results are highly dependent upon the choice of these parameterizatious. This in fact will be a major theme of the present paper. Many ocean modeling studies of the past have laid the foundations for understanding the oceanic circulation. The quest for a complete, high quality, three-dimensional solution to the global ocean circulation, including realistic temperature/salinity properties (the ocean's mean climatic state) predicted only on the basis of calculated interior circulations and oceanic surface boundary conditions, is the first step toward a validated ocean model. Much has been accomplished in this regard in the past few years, and a brief review and discussion of the next steps will be given here. A second issue of importance in applications, whether assimilative or not, is the high computational cost of such models. The choice between eddy-resolving and non-eddy-resolving models and the ability to examine solutions with a true climatic equilibrium for numerical experiments in the global context are questions of importance. The rule of thumb for increasing horizontal resolution is that it costs (computationaUy) a factor of eight or so to halve the horizontal grid interval (four times as many points in the horizontal domain and a factor of two in time step size) so that there can be a factor of several thousand times the computational cost in carrying out a 1/6 ~ resolution model (a "typical" eddy-resolving case) and a 2 ~ resolution model (a typical non-eddyresolving case) for a comparable length of time. Consequently the eddy models have not been run to equilibrium but only for a small fraction of the time necessary to establish deep ocean temperature/salinity characteristics from the surface boundary conditions. Even in coarse resolution calculations, the equilibrium state is often not reached; the experiments are terminated early. The use of acceleration techniques associated with artificially lengthening the tracer time step relative to the momentum time step and even maldng this lengthening a function of depth (Bryan, 1984; Danabasoglu and McWilliams, 1995) is a useful and workable approach to finding a final deep ocean equilibrium in the non-eddy models. Its potential role in eddy problems has not been assessed. This paper does not intend to be a complete review of all the work that has been done or is presently being carried out to improve the large variety of ocean models available within the ocean modeling community. Neither will it try to deal with the somewhat bewildering variety of model types that have proliferated in the past decade. The subject is simply too vast to be undertaken here. Instead, we shall focus upon what we consider to be the central issue that is of concern for all model types that are intended to be used for studying the general circulation of the ocean, the parameterization of subgrid scale processes. For our purposes and because the authors are most familiar

23 with this class of models, we shall make use of results from the primitive equation GFDL model (in its several forms) to make our points. In addition, because non-hydrostatic B o u a ~ e s q models are paving the way for understanding important convective problems, they too will be d i s c u a ~ . Nevertheless, it is important to emphasize that there are a variety of other kinds of ocean models under development besides the GFDL model and the non-hydrostatic models mentioned below. In particular, models that make use of isopycnic vertical coordinates (Bleck and Boudra, 1981, 1986; Oberhuber, 1993a,b) have some very exciting characteristics that might make them particularly useful in long term simulations. These characteristics involve a more direct control of the processes that occur in isopycnal layers (eddy mixing of heat and salt, parameterized or explicit), simpler handling of the parameterization of cross isopycnal mixing processes, and minimal numerical errors. Some model developers have focused upon numerical aspects of the model while retaining the primitive equation form; others have begun to examine alternative "physics," for example the so-called planetary geostrophic equations. As such models are used in a variety of applications, the value of these variants of the basic GFDL primitive equation model used so extensively to date will emerge. Our approach in this chapter will be to show some results from recent eddy and non-eddy model calculations based upon the primitive equations (Section 2), to discuss the variety of parameterizations needed in such models to include the effects of subgridscale processes not explicitly included in the model (Section 3), to discuss other important problems that remain to be solved in order to achieve realistic simulations of large scale ocean circulation (Section 4), and finally to conclude with a summary (Section 5). 2. E D D Y A N D N O N - E D D Y R E S O L V I N G M O D E L S We examine here some issues concerning the progress in and future directions for modeling the large-scale oceanic circulation with both eddy and non-eddy resolutions. First, we will describe some results from a relatively coarse resolution global ocean model that is being developed at NCAR to attack long term variability issues in a coupled atmosphere-ocean-sea ice model of the global climate. Then we will show some sample results from a comprehensive, high resolution model of the global ocean carried out at NCAR (Semtner and Chervin, 1992) to demonstrate the feasibility of making such calculations with eddy resolution. These two ends of the spectrum of potential horizontal resolutions in large scale models highlight the need to include some parameterizations explicitly in coarse resolution and to also carry out numerical experiments that explicitly include these processes, for the purpose of testing the formulation of the parameterizations. We shall come back to this issue later. Finally, we show some results based upon the CommunityModeling Experiment (CME) models in which both eddy and non-eddy model calculations have been carried out for the same physical problem, i.e the large scale circulation in the North Atlantic Ocean. For our purpose here, we shall describe this variety of results that are based upon the primitive equation GFDL Ocean Model (Bryan, 1969a,b; Bryan and Cox, 1968; Semtner, 1974; Cox, 1984; Pacanowski et. al., 1991). Other models will have exactly

24 similar issues concerning the inclusion of various subgridscaie physical parameterizations. Due to supercomputers of the CRAY Y-MP class and beyond, numerical experiments that include both a complete representation of the thermodynamic processes responsible for water mass formation as well as sufficient horizontal resolution to allow the hydrodynamic instabilities responsible for eddy formation have become feasible, at least in limited areas. Basin scale and global calculations are beginning to be carried out that explicitly include the horizontal mixing processes associated with mesoscale eddying processes. There is a large literature describing results that make use of the basic GFDL model, as a result of 25 years of use in a wide variety of applications. See, for example, the papers by K. Bryan (1979), Bryan and Cox (1968), Cox (1975; 1984; 1985; 1987a,b; 1989), Bryan and Lewis (1979), Toggweiler et. al. (1989a,b), Toggweiler and Samuels (1993), Danabasoglu et al. (1994), Danabasoglu and McWilliams (1995a,b), Killworth et al. (1991), Killworth and Nanneh (1994), Stevens and Killworth (1992), Holland (1971; 1973; 1975), Holland and Bryan (1993a,b), Philander et al. (1986; 1987), F. Bryan (1986; 1987), F. Bryan et al. (1995), Bryan and Holland (1989), B6ning (1989), B6ning and Budich (1992), B6ning et al. (1991a,b; 1995a,b), Marotzke and Willebrand (1991), Sarmiento (1986), Schott and B6ning (1991), Spall (1990; 1992), Stammer and B6ning (1992), Weaver and Sarachik (1991a,b), Weaver et al. (1991; 1993), Webb (1994), Semtner and Chervin (1988, 1992), Dukowicz et al. (1993), Dukowicz and Smith (1994), Smith et al. (1992), Hirst and Godfrey (1993), Hirst and Cai (1994), Power and Kleeman (1993) and many more. Here only a sampling of basin and global examples have been listed and the authors apologize to the many, many investigators whose works have not been cited. They number in the hundreds and, for the interested reader, further citations can be found in the references of the above papers. The vast majority of these numerical experiments have been carried out with non-eddy resolution except for those by B6ning, Bryan and Holland and their collaborators (using the high resolution Community Modeling Effort [CME] model), Semtner and Chervin (with a GFDL model variant called the Parallel Ocean Climate Model [POCM]), the FRAM GROUP (1991) (Killworth, Webb and Stevens and collaborators, using the Fine Resolution Antarctic Model [FRAM]), and Dukowitz and Smith (1994) and Dukowitz et al. (1993) (using another variant of the GFDL code call the Parallel Ocean Program [POP]). All of these codes have the same physical basis; they vary according to restructuring the numerics to carry out calculations in parallel fashion, or to attack certain regional problems with maximal resolution. Some of these experiments have been "demonstration" experiments to show the possibility that eddy simulations could be done (on a variety of computer architectures); others have explored the physical issues associated with high resolution, eddying processes in basin and global situations. However, none of these experiments has been carried out for long enough to reach an equilibrium state that one might call the "climate equilibrium" of the model ocean. Only the considerably coarser resolution model studies have approached a true global equilibrium (see particularly Danabasoglu and McWilliams, 1995b). First, let us examine a basic global numerical experiment with resolution inadequate to include mesoscale eddy processes. This experiment has nominal 2~ horizontal

25 resolution but with variable (increasing) resolution at high latitudes, where the latitudinal resolution increases to about 1~. There are 45 layers in the vertical. The model is forced with mean monthly E C M W winds and the surface temperature and salinity fields are relaxed to the monthly Levitus climatology. The model has been run for the equivalent of 6000 years (deep) and 600 years (shallow) using a distorted time step procedure suggested by Bryan (1984). After the spin-up to near equilibrium, the model run was switched to synchronous time stepping to achieve a final, seasonally varying equilibrium state. Lateral mixing processes have been parameterized using the Gent/McWilliams isopycnal approach [see Gent and McWilli~ms (1990), Gent et al. (1995), Danabasoglu et al. (1994) and Danabasoglu and McWilliams (1995a,b)]. Figure 1 shows the vertically averaged mass transport streamfunction at a single instant at the end of the run, figure 2 shows maps of temperature at two levels, and figures 3 and 4 show the temperature and salinity patterns in noah-south sections from the Antarctic to the Arctic. Note, in the latter, that the vertical coordinate is stretched (i.e. it is the level number, not depth, so the upper layers are expanded relative to the deeper ones) and that both the model fields and the Levitus climatological fields are shown for 90"N ,=cO =

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Figure 1. An instantaneous map of the vertically averaged, mass transport field in the NCAR 2X global ocean model. The contour interval is 10 Sverdrups (one Sverdrup

equals 106m3 /s.

26

Year 600.,5 Temperature 2" model (Cl=

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Figure 2. Instantaneous maps of the potential temperature field at two levels in the NCAR 2X global ocean model: (upper panel) surface layer (6m), contour interval 2~ and (lower panel) layer 21 (542m), contour interval 1~

27

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600.54 Mod_Lev.. T (Cl=

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Figure 3. Instantaneous maps of the potential temperature field in north-south sections in the NCAR 2X global ocean model: (upper) Atlantic section at 23~ (lower) Pacific section at 179~ The contour interval is 4~ in all plots. Note that the vertical coordinate is the layer number- since the layer thicknesses increase downwaxd by a factor of 20, the upper ocean is expanded relative to the deep ocean. Note also that the Levitus climatology (dotted lines) is shown for comparison on the same grid.

28

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Figure 4. Instantaneous maps of the salinity field in north-south sections in the NCAR 2X global ocean model: (upper) Atlantic section at 23~ and (lower) Pacific section at 179~ The contour interval is 0 . 5 % 0 in both plots. Note that the vertical coordinate is the layer number- since the layer thicknesses increase downward by a factor of 20, the upper ocean is expanded relative to the deep ocean. Note also that the Levitus climatology (dotted lines) is shown for comparison on the same grid.

29 this time (mid-July). Note the quite realistic representation of the T-S fields; there is still a somewhat deep thermocline in equatorial regions and a somewhat too cold bias in the deep Atlantic temperature field but, in general, the results show a quite remarkable ability to capture the large-scale climatology. Such models ought to be quite capable of addressing issues of climate change on the decadal and longer time scale, under realistically fluctuating atmospheric conditions or in coupled models. Both natural

Figure 5a. Instantaneous map of the upper ocean, seasonally-forced, vertically-averaged (0-135m) horizontal velocity field from the eddy-resolving numerical experiment of Semtner and Chervin (1992). A vector length of 2 ~ equals 18.5 cm/s.

30 variability and man-induced (Greenhouse driven) change are under study with such models. Next, as an example of an early, global, eddy-resolving calculation, we show some results from the POCM model calculations of Semtner and Chervin (1992). The model is just barely in the eddy-resolving regime, with a horizontal resolution of 0.5 ~ in latitude and longitude and having 20 levels in the vertical. A weak restoring of the T

Figure 5b. Instantaneous map of the seasonally-forced, vertically-averaged (10003300m) horizontal velocity field from the eddy-resolving numerical experiment of Semtner and Chervin (1992). A vector length of 2 ~ equals 2.2 cm/s.

31 and S fields to Levitus climatology is included below about 700m, so the fields are not precisely conservative in the main thermocline and deeper. A number of fairly short runs have been carried out and we show here the results in the seasonally-forced case after 10 years adjustment from a previous annually-forced case. Figure 5 shows just a part of the global domain (the Atlantic sector) in order to highlight the eddy activity in the instantaneous velocity fields at two levels, near surface (0-135m) and deep (1000-3300). With a carefully chosen biharmonic friction, the eddies are found to be quite vigorous and to be active virtually everywhere in the domain. Semtner and Chervin remark that the resolution is somewhat coarse to adequately resolve the eddy field but this is the first demonstration in the global domain of the capability to explicitly include the eddying processes therein. Studies at higher resolution but in limited parts of the global system (the FRAM study of the Southern Ocean and the CME studies of the North Atlantic; see above) have also shown this capability. It should be emphasized however that these studies cannot yet be carried out for the thousand year time-scale of the thermocline and deep ocean equilibration. There is still a strong memory of their initial conditions at the ends of the respective experiments, in contrast to the coarse resolution case described above. Perhaps one of the best uses of such calculations for long term climate purposes will be to examine the possibility of adequate eddy parameterizations in terms of large-scale properties of the solutions. Although parameterizations exist (e.g. Gent and McWilliams, 1990), they have not yet been tested by such analyses. As a final example of eddy versus non-eddy circulations, we show some results from the basic CME simulations of the wind- and thermohaline-driven circulation in the North Atlantic basin. This is useful because this is the only case where many experiments have been carried out in both eddy and non-eddy resolutions, using the same domain, boundary conditions, etc. We shall briefly describe some results (similar fields at similar depths) at both higher and lower resolutions for the same domain. The high resolution simulations of the general circulation of the North Atlantic Ocean were carried out using the GFDL model with horizontal resolution of 2/5 ~ zonal by 1/3 ~ meridional grid size. This is just sufficient to explicitly include the hydrodynamic instability processes responsible for eddy formation. The model has a quite high vertical resolution (30 layers). The computational domain is the North Atlantic basin from 15~ to 65~ latitude, including the Caribbean Sea and Gulf of Mexico but excluding the Mediterranean Sea. The model is forced with climatological, seasonally-varying wind stresses and surface heat and freshwater fluxes based upon the restoring principle. An instantaneous map of the SST field for January 1 is shown in Figure 6. The basic meridional gradient is apparent in the eastern Atlantic and the complex eddying and meandering of the Gulf Stream is clear in the northwestern quadrant of the basin. Upwelling of cold water along the coast of South America and off northwestern Africa is caused by along-shore winds in those locations. A map of the annually-averaged sea-surface height field for the whole CME domain is shown in Figure 7. This field is determined diagnostically from the surface pressure field calculated in this rigid lid model of the oceanic circulation. Note the signature of the Gulf Stream as it passes through the Florida Strait, up along the coast of North America, finally turning

32

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50N 40N

30N 20N ION

10S 100W

90W

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70W

60W

50W

40W

30W

20W

10W

0

10E

Longitude

Figure 6. An instantaneous map of the sea surface temperature field from the HR CME model (with resolution 2/5 ~ by 1/3~ The contour interval is 1 ~ C.

60N 50N 40N 30N ~

20N ION

10S

90W

75W

60W

45W

30W

15W

0

Longitude

Figure 7. An annually-averaged map of the sea surface height field from the Hit CME model (with resolution 2/5 ~ by 1/3~ The contour interval is 10 cm.

33

60N 50N 40N 30N .,-4

20N 1ON

10S 90W

75W

60W

45W

30W

15W

0

Longi[ude

Figure 8. The BMS deviation of the sea surface height from the mean, showing one measure of the variability found in the Hit CME model (with resolution 2/5 ~ by 1/3 ~ All time scales, seasonal, mesoscale and interannual, are included in this structure of the variability field. The contour interval is 1 cm.

eastward between 35~ and 40~ Part of the Stream feeds the subtropical gyre and part continues northeastward to the northern boundary. Figure 8 shows the root-meansquare deviation in the height field based upon a five year time series from the model run as one measure of the variability found in this type of experiment. In this map the seasonal as well as the mesoscale variability is included. Note the large amplitudes in the region of the Gulf Stream and its extension and the much weaker variability in the oceanic interior. Secondary maxima occur along the north coast of South America and in the Caribbean/Gulf of Mexico regions. This model has been extended (Holland and Bryan, 1993a,b) from the basic 1/3 ~ by 2/5 ~ horizontal resolution to include both higher and lower horizontal resolutions (1/6 ~ x 1/5 ~ and 1~ x 1.2~ These experiments can be considered as high resolution (HR), very high resolution ( H R ) , and medium resolution (MR) cases, respectively. The turbulent nature of the flow increases as one passes from 1~ to 1/3 ~ to 1/6 ~ horizontal resolution, and the eddy kinetic energy levels in the VHR case are higher and in better agreement with observational estimates at the sea surface (Richardson, 1983; Le Traon et al., 1990). Many of the larger scale features of the solutions, however,

34

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are not much changed (e.g., northward heat transports), other factors being kept the same. As an illustration we show maps (figures 9-11) of the surface horizontal velocity fields from these same experiments. In these diagrams we focus upon a very local region off Labrador to compare the details of the flows. Note the increasingly fine scales and the intensification of the currents as the horizontal resolution increases. Finally, figure 12 shows the horizontal velocity fields through the Florida Strait for 1/3 ~ and 1/6 ~ cases, emphasizing the point that resolution (both horizontal and vertical) can be of great importance in describing flows in narrow passages- and it is not clear how to parameterize such effects in coarse resolution experiments.

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The experiments described here represent the current state-of-the-art in ocean general circulation modeling. It is clear that the physical processes responsible for both water mass and eddy formation can to a certain degree be successfully simulated in models of this class. A question remains about the role of mesoscale eddies in the general circulation, in particular their interactions with thermodynamic processes such as poleward heat transport and thermocline ventilation. In the CME experiments described above, we have not detected any significant change in the northward heat transport by increasing the meridional resolution from 1/3 ~ to 1/6 ~ However the duration of the ~ simulations is much too short for achieving a complete spin-up of the model thermohaline circulation, so that no definite conclusion can yet be drawn.

36

Figure 11. A map of the instantaneous surface velocity field from the H R CME model with resolution 1/5 ~ by 1/6 ~. Only the region off" Labrador is shown. The maximum velocity scale is 150 cm/s.

3. P A R A M E T E R I Z A T I O N

ISSUES

Large scale ocean circulation models must parameterize several key mixing processes in order to close the mathematical problem and to include all the physical processes known to be important. For "tracer" fields, like temperature and salinity, these include quasi-horizontal (or isopycnal) mixing due to mesoscale eddy processes, vertical mixing at or near the upper ocean surface due to atmospherically-induced turbulent boundary layer effects (the mixed layer), vertical mixing in the deep interior of the ocean due to internal wave brealdng and double ditfusion processes (those

37

Figure 12. Instantaneous maps of the surface velocity fields in the vicinity of the Florida Strait, from (upper) the VHB. CME model (with resolution 1/5 ~ by 1/6 ~ and (lower) the HR CME model (with resolution 2/5 ~ by 1/3~ The maximum vector length is 130 cm/s in both plots.

38 processes responsible for Munk's (1966) fabled "Abyssal Recipes"), "deep" vertical mixing due to convective effects associated with buoyancy driving at the sea surface (often parameterized as convective adjustment), and other as yet poorly known mixing effects associated with eddy-mean flow-topographic interactions and shear flow instabilities that influence the large scale temperature and salinity fields. In addition, there is a need for the parameterization of momentum mixing effects (and hence vorticity) due to horizontal and vertical turbulent effects (horizontal Reynolds stresses; boundary layer turbulence- Ekman layer parameterizations near the sea surface and near the ocean's bottom; and other eddy-mean flow- topographic interactions that may affect the large scale circulation). In the basic GFDL model, this requires the specification of a number of eddy diffusivities and viscosities that control the supposed influence of small-spatialscale/fast-time-scale events that the model cannot include explicitly. The success or failure of such parameterizations can determine the success or failure of the large scale simulations. Each of the basic physical processes associated with each of these kinds of eddy influence on the mean flow must somehow be understood, observationally and theoretically (e.g. with models), in order to create sensible and validated subgridscale parameterizations. 3.1. M i x i n g by mesoscale eddies; baroclinic instability As discussed above, high horizontal resolution experiments are orders of magnitude more costly to run than low resolution ones, making clear the need for a high quality subgridscale parameterization that will allow us to use lower resolutions. Such a paraIneterization is needed to describe the effects of mesoscale eddy processes on the horizontal (isopycnal) mixing of heat and salt. A major improvement in achieving a "realistic" northward heat transport in non-eddying models has been obtained by modeling the subgridscale eddy mixing processes according to the formulation of Gent and McWilliams (1990), hereafter called GM. See also Gent et al. (1995) and Neelin and Marotzke (1994) for further discussion. The effect of this parameterization on the model transport processes is twofold. First of all mixing of heat and salt occurs only along isopycnals, so that frontal structures such as the Gulf Stream are not smeared out and weakened by a somewhat artificial cross-isopycnal horizontal diffusion. This effect has been included in earlier model studies (Redi, 1982; Cox, 1987b) but with the need for some extra horizontal diffusion for numerical reasons. Second, a paraIneterization of an effective eddy advective velocity is introduced, a term that represents the effects of baroclinic instability processes which tend to flatten isopycnal surfaces, releasing potential energy, without changing the density of individual water parcels. The latter has proved very effective in enhancing the poleward heat transport in ocean model calculations to values closer to the observations (Danabasoglu et al., 1994; BSning et al., 1995a) Figures 13-15 show some aspects of the improvements derived from this parameterization. Figure 13 (from the study by Danabasoglu et al., 1994) shows zonal mean meridional sections of the difference between model and Levitus climatological values of potential temperature and salinity for cases differing only by the use of horizontal ditSmion (HOR) or isopycnal GM mixing (ISO). The errors are reduced considerably for both fields in the ISO case, almost everwhere. Figure 14 shows improvement achieved

39

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40

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41 in the northward heat transport in these two experiments, for the global domain and for the North Atlantic. Also shown is the observational estimate of Trenberth and Soloman (1994). Figure 15 shows a similar improvement in the the northward heat transport in CME model calculations at 1~ resolution carried out at NCAR (see BSning et al., 1994a). Experiment N1-12.0 is a case with the standard horizontal diffusive parameterization, while experiment N1-26.0 makes use of the GM parameterization. For comparison the observational heat transport estimates of Trenberth and Solomon (1994) and of Isemer et al. (1989) are included, showing the greatly enhanced and more realistic values at 25 ~ N for the GM case.

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The GM parameterization does not address the issue of momentum mixing by the mesoscale eddy field (Gent et al., 1995). Due to the very large tendency for geostrophic balance in the horizontal momentum equations, it may be that the most important effects of such an eddy field are already incorporated in its influence on the density field (the layer thickness diffusion inherent in GM). However, it may be necessary in future parameterizations to develop a Reynold's stress parameterization that goes beyond the very simple inclusion of "friction" in present day, large scale models. Again, high horizontal resolution model calculations, with resolutions even higher than those carried out in the CME, FRAM and POP efforts, should allow us to evaluate the importance of such effects and to develop an accurate and validated parameterization of subgridscale viscous processes.

42 Important future work regarding this parameterization (or any other) is a set of numerical experiments to show how well the eddy heat transports found in very high horizontal resolution eddy models can be reproduced by the new parameterization. In the case of the mesoscale mixing issue, the basic primitive equation physics of the GFDL model is adequate to describe the eddy-mean flow interactions as long as enough horizontal resolution is included. But just how much resolution will really be required is still not clear; virtually all of the so-call eddy-resolving experiments have just barely rear~hed the Rossby radius of deformation. It could easily be that 1/12 ~ or 1/24 ~ horizontal resolutions will be required to capture the finer scales of baroclinic instability! We are far from able to run global calculations at such a resolution, at least for sufficiently long a time as to allow us to learn something about climate change. 3.2. M i x i n g in t h e oceanic p l a n e t a r y b o u n d a r y layer Several other developments are required in this class of models. For example, a more sophisticated planetary boundary layer (mixed layer) is needed; the parameterization of the truly small scale processes responsible for diapycnal mixing and momentum dissipation is of vital importance for coupling to the atmosphere. Models of the mixed layer have had a long history and several of them have been incorporated into the GFDL model. Examples are the bulk mixed-layer model of Krans and Turner (1967) and the second-order moment closure model of Mellor and Yamada (1982). A recent effort (Large et al., 1994) has taken a new look at the problem, based upon experience gained from large eddy simulations of the atmospheric boundary layer, and has created a noulocal K-profile parameterization of boundary layer mixing for inclusion in large scale ocean models. The model has been extensively tested in one-dimensional form and is currently being incorporated into NCAR's GFDL-type ocean model. It seems likely that the direct simulaton (so-called large eddy simulations- LES) of the oceanic boundary layer will help in the validation of the such models (see discussion in section 3.4 below). 3.3. I n t e r n a l v e r t i c a l m i x i n g

One of the issues being considered in the development of the NCAR non-eddy model is the parameterization of vertical (croas isopycnal) mixing of heat and salt in the oceanic interior. Measurements and inferences from inverse theories (Garrett, 1993) suggest that in the main thermocline the vertical diffusivity should be at a rather low value, lower than has usually been used in coarse resolution models. A vertical tracer diffusivity of 0.1 cm2/s is cited today as the canonical value (Garrett, 1993; Davis, 1994; Ledwell et. al., 1993; Ledwell and Watson, 1994), compared to a value of 1.0 cm2/s suggested by Munk (1966). This seems to imply that we may well need much finer vertical resolutions than previously thought in GFDL type models; the NCAR model now uses 45 layers in the vertical. At such resolution, we can carry out numerical experiments with quite small vertical diffusivities without difficulties. McWilliams et al. (1995) discuss some calculations at somewhat coarser resolution than in the NCAR calculations shown in Section 2 above; in that paper, the implications of such a small vertical diffumvity on the shape of the main thermocline and upon certain global constraints are examined. The actual vertical resolution required is still an issue under

43

study.

As yet, no model of the turbulent nature of the ocean interior exists that can relate the large scale properties to the turbulent diffusivity, a model that contains explicit effects of internal wave bre~idng as well as double diffusion, processes that might be responsible for the vertical mixing. Moreover it is completely unclear whether such processes are uniform in space and time or occur, for instance, mainly near lateral boundaries. At the present time, such "physics" must be based upon direct observations, fits to large scale properties, or theoretical/inverse model calculations (Garrett, 1993). In this respect, the parameterization problem is a more dit~icult one than for mesoscale eddy processes, which can be included in the same model context when finer horizontal resolution is used. Whether a constant small diifusivity is adequate for determining the large scale structure of the oceanic interior remains to be demonstrated. Attempts to model the details of such phenomena and then to test parameterization schemes are sorely needed. 3.4. C o n v e c t i o n at high l a t i t u d e s It is well known that buoyancy-driven convection will require non-hydrostatic models with extremely high resolution in both the vertical and horizontal directions. Such models have existed for a long time in non-oceanic applications of rotating, stratified convection, but recently a number of investigators have begun to examine the results of such models (Jones and Marshall, 1993; Legg and Marshall, 1993; Denbo and Skillingstad, 1994; Paluszkiewitz et al., 1994; Garwood et al., 1994) with emphasis on oceanographic applications. These results and ones yet to come may form the basis for better parameterizations of these processes, which are treated very simply in present models like the GFDL archetype. In that model, static instability is usually treated by a "convective adjustment" process in which effectively infinite vertical mixing is introduced when a part of the water column is statically unstable (Bryan and Cox, 1968; Cox, 1984). As an example of the explicit convection simulations, Figures 16-18 show results from a non-hydrostatic model calculation of surface buoyancy-driven flow in a tiny piece of an idealized ocean (from Jones and Marshall, 1993). The case is one of weak stratification in a 32 km square, doubly-periodic domain that is 2000km deep. At time zero, a strong buoyancy flux is turned on in a circular region and convective plumes begin to fall from the surface and mix downward. Aspects of the horizontal velocity fields (panels a,c), the vertical velocity field (panel b) and the temperature contours and velocity pattern in a vertical section through the region (panel d) are shown at day 0.5 (figure 16), day 2.0 (figure 17) and day 3.0 (figure 18). Note particularly the very small scales of the plumes (order of one kilometer) but also the ultimate restructuring of the density field on the size of the forcing patch. After a few days, baroclinic instability processes have become important and horizontal mixing effects begin to manifest themselves. Such experiments are key to unravelling the parameterization of convectively-driven mixing in the vertical.

3.5. E d d y effects due to e d d y - t o p o g r a p h i c interactions Other eddy effects than those discussed above might be of importance in large

44

Figure 16. Day 0.5 in the simulation of chimney formation and adjustment in a small, bouyancy-forced, weakly stratified region of the ocean. The doubly-periodic domain is 32kin square and 2000m deep. (a) and (c) show horizontal currents at 200m and 1000m respectively, (b) shows the vertical velocity field, and (d) shows temperature contours and the velocity field in the plane of an east-west section through the domain. [From Jones and Marshall, 1993].

45

Figure 17. Day 2.0 in the simulation of chimney formation and adjustment in a small, bouysncy-forced, weakly stratified region of the ocean. The doubly-periodic domain is 32km square and 2000m deep. (a) and (c) show horizontal currents at 200m and 1000m respectively, (b) shows the vertical velocity field, and (d) shows temperature contours and the velocity field in the plane of an east-west section through the domain. [From Jones and Marshall, 1993].

46

Figure 18. Day 3.0 in the simulation of chimney formation and adjustment in s small, bouyancy-forced, weakly stratified region of the ocean. The doubly-periodic domain is 32kin square and 2000m deep. (a) and (c) show horizontal currents at 200m and 1000m respectively, (b) shows the vertical velocity field, and (d) shows temperature contours and the velocity field in the plane of an east-west section through the domain. [From Jones and Marshall, 1993].

47 scale ocean circulation problems. For example, Holloway has suggested that eddy interactions with bathymetry might have a role to play in establishing large scale mean flow characteristics (Holloway, 1992; Eby and Holloway, 1994). The theory is based upon theoretical ideas from statistical mechanics and so far there are no direct simulations with eddy resolution that unequivocally show this to be important. The eddy calculations carried out thus far could easily have some enhancements of the "mean" state by the eddies that are present due to such interactions. Carefully designed numerical experiments will be needed to find out. This highlights an important issue in parameterizing unresolved eddy processes in large scale models; there is no inherent reason that the process under discussion will be simply parameterizable. Many eddy-mean flow interactions are quite complex and it may well be the case that the interaction is not related simply to the mean flow characteristics, except in very idealized circumstances. We can expect success only up to a point from this array of parameterizations. But we can definitely do a much better job that heretofore. 4. O T H E R M O D E L I N G

ISSUES

There are many possible directions to go in improving the present class of ocean solutions. Some of these have to do with the numerical formulation of the model equations since the discrete numerical equations are only approximate analogues to the continuous equations. There are order 6X and 6T (space and time discretization) errors involved that may not be small. And even when such errors are formally small, they may lead to substantial difficulties upon long time integrations by accumulating errors. By way of example, consider the old problem of representing the advection of a tracer by the motion field. In the oceanic case, better (more accurate- higher order) numerical algorithms for the advection of heat and salt, particularly for the less well-resolved models, can have substantial benefits (see as examples Gerdes et al., 1991 and Hecht et al., 1995). Other numerical accuracy issues are aLqo under investigation and have led to models that use a variety of discrete mathematical formulations (examples are the use of finite elements [Iskandarani et al., 1995] or the semi-spectral representation [Haidvogel et al., 1991]). Although we shall not dig deeper into such issues here, the reader should be aware that the question of numerical accuracy sterning from discrete mathematics is fundamental in computational fluid problems. A much more substantial issue for calculations of the oceanic state in oceanonly studies (as contrasted with coupled atmosphere-ocean studies) is the accurate specification of the heat and freshwater fluxes at the sea surface. We simply do not know them accurately enough to correctly determine the 3D properties of the ocean interior even if the model physics were perfect! In addition, even more accurate surface climatological fluxes might still lack some aspects of the ocean-atmosphere interactions. Here we shall discuss this question at some length. Many ocean-only simulations have been carried out using "mixed boundary conditions" for temperature and salinity: the surface ocean temperature is restored toward a prescribed temperature field (a climatology), while the surface boundary condition

48

for salinity is in the form of a specified freshwater flux. Since sea-surface salinity is known more accurately than surface freshwater fluxes, the implementation of mixed boundary conditions is usually preceded by a spin-up phase in which the model surface salinity is also restored toward a given salinity field. At the end of this spin-up phase, a freshwater flux is diagnosed from the restoring term and used to force the model surface salinity. At this point the sea-surface salinity is free to evolve. Although the freshwater flux used in this type of experiment should be consistent with the model equilibrium state at the end of the spin-up, small deviations from a perfect steady state are often sufficient to destabilize the model. The associated instability, first described by Bryan (1986), has been called the "polar halocline catastrophe" because it manifests itself as a spontaneous formation of a negative salinity anomaly at high latitudes. The formation of the polar halocline is associated with a rapid evolution of the model thermohaline circulation from a state of continuous deep water formation at high latitudes to a state in which the sinking is almost absent and the meridional overturning has ceased. F. Bryan's (1986) seminal work has shown the sensitivity of the thermohaline component of the ocean circulation to the specification of the surface fluxes of heat and freshwater. Typical restoring constants for SST are of the order of 30-60 days for an ocean mixed layer of about 25m depth. These values have been derived (Haney, 1971) by considering local air-sea heat exchange and assuming an unchangeable atmosphere, an atmosphere with an infinite heat capacity. The ocean, on the other hand, has a much larger thermal inertia than the atmosphere, whose entire heat capacity is equivalent to that of the upper 2.5 meters of the ocean. Consequently, the atmospheric time scales are much shorter than the oceanic ones, resulting in an almost instantaneous adjustment of the atmosphere to any perturbation produced at the ocean surface. This implies that from the point of view of the "slow" oceanic evolution the surface fluxes of heat and freshwater are non-local or, equivalently, that they are scale dependent. Several authors (Power and Kleeman, 1993; Zhang et al., 1993; ~ t o r f and Willebrand, 1995) have d i s c ~ the possible effects of an interactive atmosphere on the stability characteristics of the thermohaline circulation. At high latitudes the atmospheric temperature is lower than the oceanic temperature and the ocean releases heat to the atmosphere. If the model ocean SST is relaxed toward a fixed temperature on a relatively short time scale (as in the traditional restoring boundary condition), very little variability in the SST distribution is allowed. A decrease in the northward oceanic heat transport due to a weakening of the meridional overturning will simply result in a decreased heat loss to the atmosphere, thus leaving the SST field at high latitudes almost unaffected. If, on the other hand, the ocean surface temperatures were allowed to deviate from the fixed temperature distribution used in the restoring boundary conditions, a possible decrease in SST temperature associated with a reduced northward oceanic heat transport would act as a negative feedback on the thermohaline circulation. These ideas have been tested (Zhang et al., 1993) by introducing an energy balance model of the atmosphere with a fixed meridional transport. Heat flux anomalies from the ocean surface can be removed only through radiation to space, a rather

49 inefficient process. As a result the associated relaxation time scale used in restoring type boundary conditions should be longer and the SST is freer to deviate from the prescribed temperature field. The possibility of achieving colder temperatures at high latitudes proves to be, as expected, a stab'flizing factor, leading to the suppression of the tendency for the polar halocline catastrophe mentioned above to occur. The dissipation of SST anomalies through radiation to space can be supported only for anomalies with global spatial scales. SST anomalies at smaller spatial scales, in fact, can be removed more efficiently by winds. The rate of heat exchange between ocean and atmosphere is scale dependent. Rahmstorf and Willebrand (1995) have introduced a scale dependence in the surface heat flux by considering an atmospheric energy balance model with a scale dependent transport in the form of a diffusion law. The surface temperature boundary condition they propose includes two terms: the first term has the form of a weak restoring to an equivalent atmospheric temperature and represents the slow process of radiation to space for the global scale SST anomalies as in Zhang et al. (1993). The second term has the form of a diffusion operator, thus enhancing the ocean-atmosphere heat exchange at smaller spatial scales. The application of this new type of boundary condition to the idealized global model used by Marotzke and Willebrand (1991) leads to a "conveyor belt" equilibrium with somewhat different characteristics of the thermohaline circulation. The stability characteristics of this equilibrium are also modified, since the "conveyor belt" appears to be less affected by freshwater perturbations at high latitudes. This study, however, does not consider changes in the atmospheric freshwater transport that might be responsible for important positive feedbacks on the strength of the meridional overturning (N~kamura et al., 1994), thus acting as a destab'flizing factor. The alternative thermal boundary conditions use a very crude model of the atmosphere (an energy balance model). Capotondi and Saravanan (1995) have shown that the diffusive assumption for the divergence of the vertically integrated atmospheric transport as in Rahmstorf and Willebrand (1995) can mimic quite well some of the atmospheric feedbacks that are found in an idealized coupled atmosphere-ocean model (Saravanan and McWilliams, 1995), feedbacks associated with the atmospheric meridional transport processes. However, the implementation of these boundary condition to more realistic three dimensional models requires a more in-depth analysis of the large scale atmospheric response to SST anomalies. Due to the large sensitivity of the ocean circulation and its stability to the specification of the surface thermohaline boundary conditions it seems crucial to implement these new ideas in ocean-only models that are to be used for climate studies, especially if we are interested in understanding possible transitions between different modes of behavior of the ocean circulation.

5. D I S C U S S I O N Ocean circulation models have continued to develop rapidly over the past few years. They are now being incorporated into complete models of the climate system that will allow us to understand the ocean/atmosphere/sea ice interactions that lead to decadal and longer term variability.

50 A high-quality ocean model is central to doing such calculations well, but all components need further development. For example, present day stand-alone atmospheric models need more sophistication and realism in their planetary boundary layer physics to produce accurate air-sea fluxes of momentum, heat, and moisture. The same can be said for the ocean in that oceanic mixed layers now have to focus upon the issues of " p ~ g through" the correct amount of heat, salt, and momentum from the atmosphere to the bottom of the mixed layer (Large et al., 1994). These problems become difficult ones when the feedbacks from the two components are allowed to occur in a fully interacting climate model with no flux correction. Here we have tried to show the powerful tools that numerical modeling can bring to issues of global ocean modeling. Mesoscale eddy models can be used to create and test valid parameterizations of mesoscale eddy processes for non-eddy models. Such work has yet to be done. This approach simply requires that standard models, like the GFDL model, be run with very high horizontal resolution. The Gent/McWilliams parameterization of baroclinic instability processes and the ideas of Holioway and collaborators concerning eddy/topographic interactions can both be tested in present day models (but using very fine resolution). In a somewht different way, new models may be necessary to examine other eddy phenomena that the primitive equations cannot handle. In this regard, non-hydrostatic models of deep convection and large eddy simulations of the planetary boundary layer can give us the tool needed to be able to describe the role played by very fine scale eddy events on the larger circulation. The works of Legg and Marshall (1993), Jones and Marshall (1993), Garwood et al. (1994), Denbo and Skillingstad (1994) and Paluszkiewitz et al. (1994) are steps along that path. Mixing processes that axe examined and understood in models built to study them can then be sensibly incorporated into models that cannot deal with such fine details explicitly. The work of Large et al. (1994) at creating a mixing parameterization of such processes in GFDL type models is an exciting step in this direction. While there is yet much to be done, it should be emphasized that progress has been and will continue to be rapid on developing models capable of unravelling the climate puzzle. This is of vital importance to mankind so that changes in the climate, be they catastrophic or gradual, can be planned for and even ameliorated as man seeks a sensible balance with the natural world. 6.

REFERENCES

Bleck, R. and D.B. Boudra, 1981: Initial testing of a numerical ocean circulation model using a hybrid (quasi-isopycnic) vertical coordinate, J. Phys. Oceanogr., 11,

755-770. Bleck, R. and D.B. Boudra, 1986: Wind-driven spin-up in eddy-resolving ocean models formulated in isopycnnic and isobaric coordiantes. J. Geophys. Rea., 91, 76117621.

B~ning, C.W., 1989: Influences of a rough bottom topography on flow kinematics in an eddy-resolving circulation model. J. Phys. Oceanogr., 19, 77-97. BSning, C.W., and R. G. Budich, 1992: Eddy dynamics in a primitive equation

51 model: Sensitivity to horizontal resolution and friction. J. Phys. Oceanogr., 22, 361381. B6ning, C.W., R. Doscher and H.J. Isemer, 1991a: Monthly mean wind stress and Sverdrup transports in the North Atlantic: A comparison of the Hellerman-Rosenstein and Isemer-Hasse climatologies. Jour. Phys. Ocean., 21, 221-235. B6ning, C.W., R. Doscher, and tL Budich, 1991b: Seasonal transport variation in the western subtropical North Atlantic: Experiments with an eddy-resolving model. Jour. Phys. Ocean., 21, 1271-1289. BSning, C.W., W. R. Holland, F. O. Bryan, G. Danabasoglu, and J. C. McWilliams, 1995a: An overlooked problem in model simulations of the thermohMine circulation and heat transport in the Atlantic Ocean. J. Climate, 8, 515-523. B6ning, C. W., F. O. Bryan, W. tL Holland and R. D6scher, 1995b: Deep water formation and meridional overturning in a high-resolution model of the North Atlantic.

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52 tory/NOAA., 141pp. Cox, M.D., 1985: An eddy-resolving numerical model of the ventilated thermocline. J. Phys. Oceanogr., 15, 1312-1324. Cox, M.D., 1987a: An eddy-resolving numerical model of the ventilated thermocline: time dependence. J. Phys. Oceanogr., 17, 1312-1314. Cox, M.D., 1987b: Isopycnal diffusion in a z-coordinate ocean model. Ocean Modelling, 74, 1-5. Cox, M.D., 1989: An idealized model of the World Ocean. J. Phys. Oceanogr., 19, 1703-1752. Danabasoglu, G., J.C. McWilliams, and P.R. Gent, 1994: The role of mesoscale tracer transports in the global ocean circulation, Science, 264, 1123-1126. Dansbasoglu, G., and J.C. McWilliams, 1995a: Sensitivity of the global ocean circulation to parameterizations of mesoscale tracer transports. J. Climate, submitted. Danabasoglu, G., and J.C. McWilliams, 1995b: Approach+ to equilibrium with tracer acceleration. J. Ocean. Arm. Tech,, submitted. Davis, R.E., 1994: Diapycnal mixing in the ocean: equations for large-scale budgets. J. Phys. oceanogr., 24, 777-800. Denbo, D.W. and E.D. Skillingstad, 1994: An ocean large eddy model with application to convection in the Greenland Sea. J. Phys. Oceanogr., submitted. Dukowicz, J. K., R. D. Smith, and R. C. Malone, 1993: A reformulation and implementation of the Bryan-Cox-Semtner ocean model on the connection machine, jr. Atmos. Oceanic Technol., 10, 195-208. Dukowicz, J. K. and R. D. Smith, 1994: Implicit free-surface method for the Bryan-Cox-Semtner ocean model, J. Geophys. Res, 99, 7991-8014. Eby, M and G. Holloway, 1994: Sensitivity of a large-scale ocean model to a parameterization of topographic stress. J. Phys. Oceanogr., 24, 2577-2588. FRAM Group, 1991: An eddy-resolving model of the Southern Ocean. EOS, Trans. Amer. Geophys. Union, 7'2, 169-175. Garrett, C., 1993: Dispersion and mixing in the ocean. Ocean Processes in Climate D~amics: Global and Mediterranean Examples. P. Malanotte-Rizzoli and A. R. Robinson (eds.), NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 61-77. Gaxwood, B..W., S.M. Isakari, and P.C. Gallacher, 1994: Thermobaric convection. In: The Role of the Polar Ocean in Shaping the Global Environment. O. Johannessen, R. Muench and J. Overland, eds. AGU Monograph 85. Gent, P.R., and J.C. McWilliams, 1990: Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20, 150-155. Gent, P.R., J. Willebrand, T. J. McDougall and J.C. McWilliams, 1995: Par rametefizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr., 25, 463-474. Gerdes, R., C. K6berle, and J. Willebrand, 1991: The influence of numerical advection schemes on the results of ocean general circulation models. J. Climate, 5, 211-226. Haidvogel, D.B., J. Wilkin and R. Young, 1991: A semi-spectral primitive equation

53 ocean circulation model using vertical sigma and horizontal orthogonal curvilinear coordinates. Journal of Computational Physics, 94, 151-185. Haney, R.L., 1971: Surface thermal boundary conditions for ocean circulation models. J. Phys. Oceanogr., 23, 1057-1086. Hecht, M. W., W. R. Holland, and P. J. ~ , 1995: Upwind-weighted advection schemes for ocean tracer transport: An evaluation in a passive tracer context. J. Geophys. Res., in press. Hirst, A.C. and J. S. Godh~y, 1993: The role of Indonesian Throughflow in a global ocean GCM. J. Phys. Oceanogr., 23, 1057-1086. Hirst, A.C. and W. Cai, 1994: Sensitivity of a World Ocean GCM to changes in subsurface mixing parameterization. J. Phys. Oceanogr., 24, 1256-1279. Holland, W.tL, 1971: Ocean tracer distributions. I. A preliminary numerical experiment. Tellus, 23, 371-392. Holland, W.R., 1973: Baroclinic and topographic influences on the transport in western boundary currents, Geophysical Fluid Dynamics, 4, 187-210. Holland, W.tL, 1975: Energetics of baroc!inic oceans. In Numerical Models of Ocean Circulation, Nat. Acad. of Sciences, Wash. D. C., pp. 168-177. Holland, W. R., and F. O. Bryan, 1993a: Sensitivity studies on the role of the ocean in climate change. Ocean Processes in Climate Dynamics: Global and Mediterranean Ezamples. P. Malanotte-Rizzoli and A. R. Robinson (eds.), NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 111-134. Holland, W. tL, and F. O. Bryan, 1993b: Modeling the wind and thermohaline circulation in the North Atlantic Ocean. Ocean Processes in Climate Dynamics: Global and Mediterranean Ezamples. P. Malanotte-Rizzoli and A. R. Robinson (eds.), NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 135-156. Holloway, G., 1992: Representing topographic stress for large-scale ocean models. J. Phys. Oceanogr., 22, 1033-1046. Iskandarani, M., D.B. Haidvogel and J.P. Boyd, 1995: A staggered spectral finite element model with application to the oceanic shallow water equations. International Journal for Numerical Methods in Fluids, in press. Isemer, H-J., L. Hasse and J. Willebrand, 1989: Fine adjustment of large scale air-sea energy flux parameterizations by direct estimates of ocean heat transport. J. Climate, 2, 1173-1184. Jones, H. and J. Marshall, 1993: Convection with rotation in a neutral ocean: a study of open-ocean deep convection. J. Phys. Oceanogr., 23, 1009-1039. Killworth, P.D., D. Stainforth, D.J.Webb, and S.M. Paterson, 1991: The development of a free-surface Bryan-Cox-Semtner ocean model. J. Phys. Oceanogr., 21, 1333-1348. Killworth, P.D., and M. Nanneh, 1994: Isopycnal momentum budget of the Antarctic Circumpolar Current in the Fine Resolution Antarctic Model. J. Phys. Oceanogr., 24, 1201-1223. Krauss, E.B. and J.S. Turner, 1967: A one-dimensional model of the seasonal thermocline, II. The general theory and its consequences. Tellus, 19, 98-105. Large, W.G., J.C. McWilliams, and S.C. Doney, 1994: Oceanic vertical mixing: A

54 review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32, 363-403. Ledwell, J.R., A.J. Watson, and C.S. Law, 1993: Evidence for slow mixing across the pycnocline from an open-ocewan tracer-release experiment. Nature, 364, 701-703. Ledwell, J.R. and A.J. Watson, 1994: North Atlantic tracer release experimentnewest results. WOCE Notes, 6, 1-4. Legg, S. and J. Marshall, 1993: A heton model of the spreading phase of openocean deep convection. J. Phys. Oceanogr., 23, 1040-1056. LeTraon, P.Y., M.C. Rouquet and C. Boissier, 1990: Spatila scales of mesoscale variability in the North Atlantic as deduced from Geosat data. J. Geophys. Res., 95, 20267-20285. Marotzke, J., and J. Willebrand, 1991: Multiple equilibria of the global thermohaline circulation, J. Phys. Oceanogr., 21, 1372-1385. McWilliams, J.C., G. Danabasoglu and P.R. Gent, 1995: Tracer budgets in the Warmwasser- sphere, Tellus, submitted. MeUor, G.L. and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys., 20, 851-875. Munk, W., 1966: Abyssal recipes. Deep Sea Res., 13, 707-730. N~kamura, M., P.H. Stone, and J. Marotzke, 1994: Destabilization of the thermohaline circulation by atmospheric eddy transports. J. Climate, 7, 1870-1882. Neelin, J. D. and J. Marotzke, 1994: Representing ocean eddies in climate models. Science, 264, 1099-1100. Oberhuber, J. M., 1993a: Simulation of the Atlantic circulation with a coupled sea ice- mixed layer- isopycnal general circulation model, part I: model description. J. Phys. Oceanogr., 23, 808-829. Oberhuber, J. M., 1993b: Simulation of the Atlantic circulation with a coupled sea ice- mixed layer- isopycnal general circulation model, part II: model experiment. J. Phys. Oceanogr., 23, 830-845. Pacanowski, R., K. Dixon, and A. Rosati, 1991: The GFDL Modular Ocean Model users guide version 1.0. GFDL Ocean Group Tech. Rep. No. 2, 46pp. Paluszkiewitz, T., R.W. Garwood, and D.W. Denbo, 1994: Deep convective plumes in the ocean. Oceanography, 7, 37-44. Philander, S.G.H., W.J. Hurlin, and R.C. Pacanowski, 1986: Properties of long equatorial waves in models of the seasonal cycle in the Tropical Atlantic and Pacific Oceans. J. Geophys. Res., 91, 14207-14211. Philander, S.G.H., W.J. Hurlin and A.D. Siegel, 1987: Simulation of the seasonal cycle of Tropical Pacific Ocean, J. Phys. Oceanogr., 17, 1986-2002. Power, S.B., and R. Kleeman, 1993: Multiple equilibria in a Global General Circulation Model, J. Phys. Oceanogr., 23, 1670-1681. Rahmstorf, S., and J. Willebrand, 1995: The role of temperature feedback in stabilizing the thermohaline circulation, J. Phys. Oceanogr, 25, 787-805. Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr., 12, 1154-1158.

55 Richardson, P.L., 1983: Eddy kinetic energy in the North Atlantic from surface

drifters. J. Geophys. Res., 43, 83-111.. Saravanan, P~ and J.C. McWilliams, 1995: Multiple equilibria, natural variability, and climate transitions in an idealized ocean-atmosphere model. J. Climate, 8, 22962323. Sarmiento, J.L., 1986: On the North and Tropical Atlantic heat balance, J. Geophys. Res., 91, 11677-11689. Schott, F.A. and C. B6ning, 1991: The WOCE model in the western equatorial Atlantic: Upper layer circulation. J. Geophys. Res., 96, 6993-7004. Semtner, A. J., 1974: A general circulation model for the World Ocean. UCLA Dept. of Meteorol. Tech. Rep. No. 8, 99pp. Semtner, A. J., Jr. and R. M. Chervin, 1988: A simulation of the global ocean circulation with resolved eddies, J. Geophys. Res., 93, 15,502-15,522. Semtner, A.J., Jr. and R. M. Chervin, 1992: Ocean general circulation from a global eddy-resolving model. J. Geophys. Res., 97, 5493-5550. Smith, R. D., J. K. Dukowicz, and R. C. Malone, 1992: Parallel ocean general circulation modeling, Physica D Amsterdam, 60, 38-61. Spall, M.A., 1990: Circulation in the Canary Basin: A model/data analysis. J. Geophys. Res., 95C, 9611-9628. Spall, M.A., 1992: Rossby wave radiation in the Cape Verde frontal zone. J. Phys. Oceanogr., 22, 796-807. Stammer, D. and C.W. BSning, 1992: Mesoscale variability in the Atlantic Ocean from Geosat altimetry and WOCE high resolution numerical modeling. J. Phys. Oceanogr., 22, 732-752. Stevens, D.P. and P.D. Killworth, 1992: The distribution of kinetic energy in the Southern Ocean. A comparison between observations and an eddy resolving general circulation model. Phil. Trans. Roy. Soc. B., 338, 251-257. Toggweiler, J.P~, and B. Samuels, 1993: Is the magnitude of the deep outflow from the Atlantic Ocean actually governed by Southern Hemisphere winds? The Global Carbon Cycle, M. Heimann, Ed., Springer-Verlag, 303-331. Toggweiler, J.P~, K. Dixon, and K. Bryan, 1989a: Simulations of radiocarbon in a coarse-resolution World Ocean model. 1. Steady state prebomb distributions. J. Geophys. Res., 94, 8217-8242. Toggweiler, J.P~, K. Dixon, and K. Bryan, 1989b: Simulations of radiocarbon in a coarse-resolution World Ocean model. 2. Distributions of bomb-produced Carbon 14. J. Geophys. Res., 94, 8243-8264. Trenberth, K. and A. Solomon, 1994: The global heat balance: heat transports in the atmosphere and ocean. Climate D~amics, 10, 107-134. Weaver, A,J, and E.S. Sarachik, 1991a: The role of mixed boundary conditions in numerical models of the ocean's climate. J. Phys. Oceanocyr., 21, 1470-1493. Weaver, A.J., and E.S. Sarachik, 1991b: Evidence for decadal variability in an ocean general circulation model: An advective mechanism, Atrnos.-Ocean, 29, 197231.

56 Weaver, A.J., E.S. Sarachik, and J. Marotzke, 1991: Internal low-frequency variability of the ocean's thermohaline circulation, Nature, 353, 836-838. Weaver, A.J., J. Marotzke, P.F. Cummins, and E.S. S a r a , 1993: Stability and variability of the thermohaline circulation, J. Phys. Oceanogr., 23, 39-60. Webb, D.J., 1995: The vertical advection of momentum in Bryan-Cox-Semtner ocean general circulation models. J. Phys. Oceanogr., submitted. Zhang, S., R.J. Greatbatch, and C.A. Lin, 1993: A reexamination of the polar halocline catastrophe and implications for coupled ocean-atmosphere modeling, J. Phys. Oceanogr., 23, 287-299.

Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

57

Oceanographic data for parameter estimation Nelson G. Hogg D e p a r t m e n t of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, 02543, United States of America

Abstract

Assimilation of ocean d a t a into numerical models represents an intimate coupling of numerical ocean modeling with ocean d a t a collection. Knowledge of climatologies, error sources and property covariances is essential and, used appropriately, represents a major step forward in being able to use inherently sparse d a t a technologies to u n d e r s t a n d b e t t e r the large scale ocean dynamics. In this chapter we will discuss several aspects of ocean d a t a t h a t are related to the assimilation problem. In particular, m e a s u r e m e n t techniques and their errors are discussed; assemblage of d a t a into climatologies is described; and a form for the d a t a covariance matrix is presented based on m o o r e d and satellite observations in the North Atlantic.

I. INTRODUCTION There are many motivations for incorporating d a t a into numerical models of the ocean. Some researchers employ this tool in much the same way as do meteorologists: as a means of using as much knowledge as possible of the present ocean state to predict the future with some quantifiable skill (e.g. the prediction of the E1 Nifio p h e n o m e n o n as in Chapter 4 or the evolution of the Gulf Stream as in Chapter 5). Others wish to use models and d a t a to establish the present state of the slowly evolving ocean climate or general circulation (e.g. Chapters 3 and 6). Still others see d a t a assimilation as a sophisticated means of optimal interpolation a technique for determining a time dependent ocean state which is dynamically consistent with some approximation to the equations of motion and statistically consistent with observations - - so as to u n d e r s t a n d b e t t e r ocean processes (e.g. Chapter 6). There are, as well, many approaches to the process of incorporating d a t a into models but each has two aspects in common, at least as far as the demands on the d a t a go. One is the need for an initial ocean state to be used as a starting point for predictive studies, the climatological mean for "optimal interpolation" or a best first guess for inverse modelling. The other is the property covariance matrix explicitly included in the cost function which is to be minimized in all three approaches. In this chapter we

58

Table 2.1 Measurement errors for some oceanographic instruments. These should be t r e a t e d as representative r a t h e r t h a n exact. Actual errors depend on i n s t r u m e n t preparation and model. Instrument CTD

Current Meter

Acoustic Tomography

ADCP Satellite Altimetry ALACE drifters RAFOS drifters Surface drifters

Variable

Measurement Error

Reference

temperature salinity pressure temperature speed direction temperature velocity relative vorticity velocity sea surface height 26 day mean 1 day mean 1 day mean

+0.002~ +0.002 psu +2.5 db +0.01~ 4-1 cm/s +2 ~ +0.1~ 4-4 cm/s 4-10-6/s 4-1 cm/s 4-5 cm 4-1 mm/s 4-1 cm/s 4-1 cm/s

Fofonoff et al. (1974)

McCullough (1975)

Chester (1993)

Dickey et al. (1996) Fu et al. (1994) Davis et al. (1991) Owens (pers. comm.) Niiler et al. (1995)

will describe what is known and available for each in turn and restrict the discussion to in-situ ocean data: meteorological forcing will be omitted, because of ignorance on the part of the author, and satellite d a t a will be treated more completely in C h a p t e r 2.3.

2. M E A S U R E M E N T S

AND

THEIR

ERRORS

Ocean d a t a are available from many instrument sources. Table 2.1 lists the most common tools presently in use and gives rough estimates of their associated measurement errors. For assimilation purposes there are several error sources. Firstly there is the m e a s u r e m e n t error derived from the instrument noise and the limited capabilities of the sensor, as listed in Table 2.1. Secondly, there is ocean noise related to unresolved small scale processes. For eddy-resolving models this noise would arise from internal waves and other submesoscale ageostrophic processes. For non-eddy-resolving models, the eddies themselves contribute to the noise. Although instrument m e a s u r e m e n t errors can be quantified with some accuracy, the ocean noise is highly variable, especially in space, and for some observation methods (particularly those capable of spatial or temporal averaging) depends on the design of the d a t a gathering system. For eddy-resolving models much of the geophysical noise variance is derived from the internal wave field and can be e s t i m a t e d from the Garrett and Munk (1972, 1975) empirical model which, however, ignores possibly significant contributions from subgridscale ageostrophic processes whose dynamics is not t h a t of internal waves.

59

decade- .

Satellite Altimetry .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

i

=

i

............ cu

year

-

.'.t Met. :

I i

Moo. . .e.ng. s. ......

Acoustic r o m o g r a p h y

.

o

i

. ,

m i

|

i

i

i e

e i

CO

month

E

:

9 .

!

day

hour

o

I

10

I

100

Space Scale

I

1000

~

I

lO, O00km

Figure 2.1. Space and time scales accessible by various measurement techniques.

Observational methods can be subdivided in several ways. A particularly useful one for our purposes is to consider the length and time scales accessible by each system (Figure 2.1). Shipboard hydrographic stations using CTDs (Conductivity and Temperature vs. Depth) are typically completed in several hours and grouped into a section across an ocean basin in one month, the so-called "synoptic" time scale. Repeats of these measurements have occurred on the decadal time scale but it is difficult or impossible to determine the effects of aliasing of higher frequencies. Ship-based surveying is presently the only available method for obtaining information on many of the dissolved compounds used as tracers in oceanography, such as the nutrients (phosphate, silicate, nitrate), the man-made chlorofluoromethanes, and many others. Many of these are crucial to the development of models which include biological processes and the carbon cycle. Moored arrays can cover regions of several hundred kilometers in scale and make nearly continuous measurements for severn years, and newer ADCPs (Acoustic Doppler Current Meters) measure velocity versus time and depth for ranges up to several hundred meters in the vertical. Hydrographic and moored measurements are complimentary, one

60 covering the spatial dimension and the other the temporal, and they have formed the backbone of traditional observational work. Newer methods are allowing more complete coverage of both dimensions. Neutrally buoyant floats and surface drifters, carried more or less with the flow, can cover large parts of ocean basins. However, large numbers of floats are required to give useful estimates of long term mean velocities away from the strong western b o u n d a r y currents (Davis, 1991). Acoustic tomography has the potential to provide more accurate estimates of velocity and thermal structure over larger scales than either current meters or shipboard measurements because of its inherent ability to average over small spatial scales. However, just as it cannot be done for moored current meters it is not feasible to consider the deployment of an eddy-resolving tomographic array on the basin scale. But, if one wishes to assimilate data into a non-eddy-resolving model, acoustic t o m o g r a p h y is an ideal tool as both temperature and velocity information can be obtained which is averaged over the eddy scale and large portions of an ocean basin can be covered with a practical number of instruments. (See Chapter 2.4 for an in-depth discussion of the capabilities of acoustic tomography.) Finally, the satellite altimeter has a unique capability of measuring the sea surface elevation over the whole globe although the repeat of the orbit in time and space makes it a marginal mesoscale tool. For example, the Topex/Poseidon altimeter now flying has a repeat time of about 10 days and distance between tracks at the equator of about 315 km (however, the along-track resolution is much finer at about 6 km) and has a hoped-for life of 5 years with follow-on systems being planned. Satellites are presently limited to acquiring information about the surface of the ocean although this information, when incorporated into a numerical ocean model, is a strong constraint on interior motions (e.g. Capotondi et al., 1995a, 1995b). Their capabilities are treated more fully in Chapter 2.3. For purposes of ocean prediction, acquisition of data in real time is required. At present satellite systems are the most reliable source of such information although surface drifters can provide some sparsely sampled meteorological and upper ocean information. Special purpose data collection activities have been a t t e m p t e d for model verification purposes (see Chapter 5.3). Telemetry of moored data is under active development but the cost is large and it is unlikely that such systems will be deployed in sufficient numbers to have a significant impact on basin scale modeling, although they are being used successfully in the T O G A - T A O (Tropical Ocean-Global A t m o s p h e r e Tropical Atmosphere Ocean) Array to return upper ocean and atmospheric temperatures in real time from the western Pacific (McPhaden, 1993).

3. C L I M A T O L O G I E S Water property information has been collected for well over a century and, by now, vast amounts of data exist and are archived in data centers round the world such as NODC (the National Ocean Data Center in Washington, D.C.). D a t a quality and coverage are uneven. Quality has improved but remains sensitive to operator practices

61

and experience. Coverage is particularly good in the North Atlantic but improving in other ocean basins. Since about 1980 there have been a number of efforts to assemble the archived data and construct climatologies of water properties, temperature and salinity in particular. These climatologies can be used in several ways. In a "relaxation" mode, forcing terms proportional to the difference between the model state and the climatological state are added to the right-hand-sides of the equations to ensure that the model never deviates too far from presumed reality (the so-called "robust diagnostic" method introduced by Sarmiento and Bryan, 1982). Models, not initialized at rest, also need a realistic starting point and climatologies can provide this. However, they are usually constructed by averaging data into boxes and, therefore, represent spatially smoothed versions of the oceans. For the gyre interiors this may not represent much of a problem but for frontal zones and western boundary currents the resulting smoothed out structures are unrealistic and depend somewhat on the averaging process (see below). As additional data accumulate the smoothing scale can be reduced if one assumes that there is negligible long term change. The time dependence of ocean currents introduces a natural smoothing process into the Eulerian mean. For example, the meandering of the Gulf Stream over several degrees of latitude will blur the time-averaged fields and the resulting Eulerian mean, no m a t t e r how finely resolved in the horizontal, will bear no resemblance to any instantaneous configuration of the Stream. This may be the appropriate climatology for inverse computations of the mean Eulerian circulation but could be inappropriate for initializing predictive models: hence the development of "feature" models of the region which include idealized building blocks (the jet, rings and recirculations) located by use of satellite imagery and available in-situ data (e.g. Chapter 5.3). Without an accurate parameterization of unresolved eddy fluxes attempts to use such smoothed climatologies to estimate property fluxes are doomed to failure (Marotzke and Wunsch, 1993): although smoothing has little effect on integrated mass fluxes the inherent reduction in velocities biases heat and freshwater fluxes downward (see also Rintoul and Wunsch, 1991; Martel and Wunsch, 1993). Most of the available climatologies deal with water properties alone. Direct velocity measurements remain too few to provide more than a few regional pictures of the mean circulation. We will first discuss those climatologies based on hydrographic data (Table 3.1) and end with a brief review of the available velocity information. The most widely used ocean climatology is probably that due to Levitus (1982), a global atlas of spatially smoothed mean and seasonal fields of potential temperature, salinity and dissolved oxygen on one-degree squares at 33 standard depth levels. Derived quantities such as Brunt-Vs163163frequency and oxygen saturation are also computed. An iterative scheme with an effective smoothing radius of about 550 km is used to produce the gridded fields. Some seasonal information is provided in the upper 250 m. The "Levitus Atlas" has been updated recently and given the formal name "World Ocean Atlas 1994" (Conkright et al., 1994; Levitus and Boyer, 1994a, 1994b; Levitus et al., 1994a). The analysis procedure is similar to the earlier atlas with almost the

52 Table 3.1 Numerical climatologies based on hydrographic measurements Source

Region

Levels

Smoothing

Comments

Levitus (1982) World Ocean Atlas 1 9 9 4 Bauer and Robinson (1985) Lozier et al. (1995) Fukumori et aJ. (1991) Olbers et al. (1992) Gouretski & Jancke (1995)

Global Global N. Hemisphere N. Atlantic N. Atlantic S. Atlantic S. Ocean

33 33 33 12 7 42 42

,,, 1100 km ,,~ 1100 km > 1~ >__1~ multiscale 450 x 350 450 x 350

6, S, 02 T, S, 02, nutrients T, S 8, S, 02, isopycnic 8, S, 02, nutrients 6, S, 02 6, S, 02

same effective smoothing scales. Major improvements are derived from the addition of considerably more data as more have been collected and some have been recovered through the activities of a "data archaeology and rescue" project (Levitus et al., 1994b). Throughout the water column the number of observations has more than tripled (for temperature and salinity, somewhat less for oxygen) with a proportionately higher increase in the deep water (e.g. 2981 temperature observations at 5500 m in 1994 versus only 138 in 1982). In contrast with the earlier version in-situ temperature is used rather than potential temperature. Major enhancements include the addition of the nutrients phosphate, nitrate and silicate to the analysis suite, and monthly analyses in addition to the seasonal and annual ones. The potential user should be aware that, for the first release of this atlas, the deeper portions of many CTD profiles were truncated. A corrected release is being prepared at this writing. Bauer and Robinson (1985) have prepared an electronic atlas, at 1~ resolution, for the northern hemisphere (north of 5~ using edited data from a variety of sources. Their atlas also includes a global climatology for the ocean mixed layer down to 150 m depth. The other available climatologies are regional rather than global. This has the advantage that they can be better tailored to the available information in a particular area: the smoothing scales employed in the "World Ocean Atlas 1994" were determined as a global compromise between data-rich and data-poor areas. Lozier et al. (1995) use the data from NODC for the period 1904 to 1990 to construct a 1-degree climatology for the North Atlantic, the best measured of all the ocean basins. Some 144,000 hydrographic stations were available. In a significant departure from the Levitus scheme Lozier et al. (op cit) choose to average, within the 1-degree squares, on potential density surfaces, rather than isobars. They show that both time dependence and spatial averaging produce pools of anomalous water properties in frontal regions where isopycnals slope steeply and the property-property relationship is not linear. This effect is most apparent in the Gulf Stream region (Figure 3.1) where the 0-S relationship is bowed upward in the thermocline. A comparison between isopycnal and isobaric averaging in the al000 = 31.85 surface (right panels in Figure 3.1) shows the existence of a pool of anomalously warm (and, consequently, salty) water beneath the Stream axis in

63

Figure 3.1. A comparison between the Lozier et al. (1995, upper panels) and Levitus (1982, lower panels) climatologies for the North Atlantic. Shown are the pressure (left panels) and potential temperature (right panels) of the potential density surface al000 = 31.85. (Courtesy R. G. Curry, personal communication, 1995.)

54 the isobaric case (lower right panel, Figure 3.1). In addition to salinity and potential temperature Lozier et al. (op cit) construct fields for dissolved oxygen. Another approach for the North Atlantic has been provided by Fukumori et al. (1991) who incorporate CTD-based hydrographic data taken during the 1980-1985 p e r i o d - a quasi-snapshot of the ocean at that time. These data have been objectively mapped on seven isobaric surfaces from the surface to 4500 db with a multiscale covariance function (the sum of three Gaussian functions of spatial lag with decay scales of 1000, 500 and 250 km). As well as potential temperature, salinity and oxygen, the standard nutrients are also provided. Atlases with very similar design philosophy have been prepared for the Southern Ocean (Olbers et al., 1992) and the South Atlantic (Gouretski and Jancke, 1995). Both use all data available up to the early 1990s, excluding new data from the World Ocean Circulation Experiment (WOCE), and both use a gridding algorithm based on the objective analysis technique with identical parameters (Table 3.1). Current meter arrays have been too sparsely deployed over the years to permit anything but fragmentary pictures of the ocean general circulation (e.g. Dickson et al., 1985; Hogg et al., 1986). However, they have been set in strategic areas such as the Florida Straits, the Drake Passage and a number of deep passages so as to provide useful constraints on numerical models. Neutrally buoyant floats are most suitable for the purpose of obtaining the large scale flow and these are presently being deployed in large numbers over the globe as part of WOCE. The closest we have to a useful climatology at present is the synthesis provided by Owens (1991), a compilation of SOFAR (sound fixing and ranging) float velocities averaged in 1~ squares for the western North Atlantic at the depths of 700 m, 1500 m and 2000 m. Both Martel and Wunsch (1993) and Mercier et al. (1993) have made use of the Owens (1991) climatology in their inversions and both find that solutions consistent with hydrography can be obtained, although in the former this is the 5-year "quasi-synoptic" climatology and in the latter it is derived from data collected over a 30-year period. Electronic atlases for other regions of the world's oceans, suitable for use in data assimilating numerical models, are lacking at this time. We can anticipate that the vast amount of new data, both hydrographic and velocity, provided by W O C E will permit the construction of new climatologies for the Pacific and Indian Oceans and more accurate ones for the North and South Atlantics.

4. D A T A C O V A R I A N C E S Most assimilation techniques employ a procedure in which a "cost function" is either implicitly or explicitly minimized. In its simplest form the cost function is the sum of squared differences between observations and model variables weighted by the observation error covariance. Although this error covariance is often taken as "white" such that the off-diagonal terms vanish, this may not always be appropriate, especially in the eddy-resolving context. In addition, the parameters being calculated in the model are

65

Figure 4.1. Eddy kinetic energy for the western North Atlantic as measured by moored current meters near 4000 m depth (numbers) overlayed on the surface eddy kinetic energy determined by surface drifters from Richardson (1983). (From Schmitz, 1984).

not necessarily those being observed and the model grid points do not necessarily coincide with observation locations. Hence the spatial covariance structure of the property fields is required. The diagonal of this spatial covariance function contains the property variances: velocity, temperature and sea surface height being the typical ones, and these can be estimated by a variety of means. Enough isolated moorings have been deployed to give a coarse picture of velocity and temperature variances over much of the globe with the best coverage again being in the North Atlantic, and this information has been compiled and summarized by Dickson (1983, 1989). The best spatial detail is provided in the deep water near 4000 m depth of the western North Atlantic where Schmitz (1984) has compared spot values of eddy kinetic energy from moorings with the distribution of surface eddy kinetic energy estimated from surface drifter data by Richardson (1983) (Figure 4.1). Typical of western boundary regions the eddy energy increases dramatically toward the axis of the Gulf Stream. It also increases downstream of Cape Hatteras, where the Stream has more freedom to meander, before decreasing

65

Figure 4.2. The global distribution of sea surface height variance as measured by the Topex/Poseidon altimeter. (Courtesy L.-L. Fu, personal communication, 1995).

again. Halkin and Rossby (1985) and Hogg (1994) have shown that as much as 2/3 of this increased energy results just from the simple meandering of a frozen jet structure past the mooring site. Global maps of eddy kinetic energy have been produced using reports of merchant ship drift (Wyrtki et al., 1976) and, more recently, velocities derived geostrophically from the various satellite altimeter measurements of sea surface height. The sea surface height, itself, reveals the system of western boundary currents in the global ocean (Figure 4.2). Using the methods and assumptions outlined below additional information on surface velocity field covariances can be estimated. Numerous surface drifters are being released in support of W O C E and TOGA as well, and these will add to the surface velocity data base and provide direct estimates of the near surface ageostrophic flow. Covariance information at spatial lags is also required. For eddy-resolving models we can anticipate that there will be significant covariances at lags corresponding to the mesoscale, typically of order one hundred kilometers. In the ocean water column such information is obtained practically only by moored current meter arrays. The associated expenses are such that just a few arrays, with sufficient density to be useful for these purposes, have been deployed. Recent moored observations from SYNOP (the

67 Synoptic Ocean Prediction Experiment) and older ones from the LDE (Local Dynamics Experiment) permit some estimation of the covariances in the Gulf Stream region and these will be discussed below. The recent Geosat and the on-going Topex/Poseidon altimeter missions now allow some estimation on a global basis although this is limited to properties related to sea surface height. In the absence of any concrete information on the nature of the property-property covariances researchers usually take the simplest approach and assume homogeneity, isotropy and even that the off-diagonal elements are zero. If we assume that the ocean dynamics is geostrophic, then all covariances are mutually related (Bretherton et al., 1976). In particular, defining the streamfunction spatial covariance to be: F ( x l , Yl, Zl, X2, Y2, Z2) ~- (l~(Xl, Yl, Zl )r

Y2, Z2))

(4.1)

then the velocity-temperature covariances are obtained by cross-differentiation, e.g.

02F (U(Xl' Yl' Z1) ?)(x2' Y2' Z2)) "-- --(~Yl (OX2

(4.2) and

(U(Xl, Yl, Zl) T(x2, y2, z2)) --

b2F -c~y 1 (~z2

with analogous formulae for the other combinations. On the larger, non-eddy-resolving scale, little information about spatial covariances is available from in-situ data. Correlations at this scale, observed by point measuring systems such as moorings, are dominated by the eddy signal which now would be considered noise. Integrating methods, such as acoustic tomography, are more suited to gathering information on spatially smoothed fields of velocity and thermal structure and, of course, the satellite altimeter fields can be smoothed to permit calculations of larger scale covariances. The Mid-Ocean Dynamics Experiment (MODE) in the early 1970s was the first serious attempt to resolve spatially the mesoscale eddy field but its usefulness was hampered by the technical inability to maintain instrumentation for sufficient durations to estimate meaningfully velocity and temperature covariances. Since that time mooring technology has advanced and it has become routine to deploy moorings for two years or more. In particular, several arrays have been maintained in and near the Gulf Stream (Figure 4.3). Analysis of velocity and temperature from the SYNOP East array led Hogg (1993) to propose the following statistical model for the streamfunction covariance field: F (x~, x ~ ) - r 1 6 2

2) (r

2) f(r, zi,z2)

(4.3)

with r - r - x,)2 + (y2 - yl)2. The form suggested by equation (4.3) is relatively simple: it is horizontally homogeneous and isotropic in terms of the correlation function but not for the streamfunction covariance, itself. For the velocity-temperature covariances, the derivatives of the nonhomogeneous variance terms imply that they also are neither homogeneous nor isotropic even when normalized (see Eq. 4.2). The simplest form for the correlation function, f(r, Zl, z2), that is consistent with the SYNOP data is:

68

Figure 4.3. Eddy-resolving moored arrays that have been maintained in the North Atlantic for a sufficient duration to permit estimation of spatially lagged covariances. The LDE was part of Polymode. ABCE stands for the Abyssal Circulation Experiment.

f(,.,z,,z~)

= h(,-,z~ + z~.)g(z~ - z~)

where h(r, za + z 2 ) - e -c~(za + z2)r2

(4.4a) (4.4b)

is the term determining the horizontal dependence. The vertical dependence is contained in both a(zl + z2) and 9(z2- zl), the latter being an even function which is parabolic at the origin from where its value decreases slowly from unity to about 0.95 at lags of 3.5 km the largest available from the current meter array. Implicit in this formulation and the large vertical correlation is that motions near the Gulf Stream are "weakly depthdependent" (Schmitz, 1980) and much of the temperature variance arises from simple advection of a nearly frozen structure by Gulf Stream meanders (Hogg, 1991). For the purposes of this article the statistical model of (4.3) and (4.4) has been applied to three other moored arrays to investigate its generality, albeit still within a region strongly influenced by the Gulf Stream. When fit to data, the model functions need further specification. We have chosen to model the streamfunction variance field as the exponential of a quadratic function of horizontal position, different for each array. The temperature field, being related to the vertical derivative of strearnfunction, also requires parameterization and we chose a hyperbolic tangent function of meridional direction (multiplied by the exponential factor) to model the frontal structure of the

69 Table 4.1 E s t i m a t e d covariance function parameters with their 95% confidence limits

Depth Range 400-600 m

1000-1500 m 3500-4000 m

Array SYNOP ABCE SYNOP LDE SYNOP ABCE SYNOP SYNOP ABCE SYNOP

a(104 km-2)

Central East Central East Central East

0.60 4- 0.06 0.65 4- 0.18 0.61 4- 0.06 0.30 4- 0.09 0.52 4- 0.04 0.48 4- 0.08 0.47 4- 0.11 1.20 4- 0.20 0.46 4- 0.09 0.46 4- 0.05

-g"(0) (km-~)

E~

0.21 4- 0.18 1.16 4- 0.08 -0.31 4- 0.94 0.62 4- 0.16 0.76 4- 0.39 0.58 4- 0.14 0.45 4- 0.24 0.32 + 0.05 0.22 4- 0.07 1.11 4- 0.07 0.003 4- 0.03 0.16 4- 0.20 0.008 4- 0.006 0.00024- 406 0.03 4- 0.017 1.49 4- 0.26 0.00034- 0.001 0.40 4- 0.10 0.00054- 0.0002 0.464- 0.10

E, 0.71 4- 0.20 0.27 4- 0.55 0.0 4- 10009 0.26 4- 0.08 0.37 4- 0.18 0.07 4- 0.06 0.054- 0.06 0.17 + 0.05 0.024- 0.005 0.024- 0.004

Stream. Finally, constant factors were included to account for the small scale noise in the system, one for the velocity components and one for temperature. For exaxnple, the m e a s u r e d velocity component variances were assumed to be related to the true variances by: =

(1 +

and the noise variance factor, Ev, was determined by the fit. We have modelled covariances only on the horizontal plane: Hogg (1993) gives results for a full three-dimensional fit which do not differ significantly from what will be given here. T h e end result is that 14 parameters are determined using a nonlinear fitting procedure. The Kolmogorov-Smirnov test for goodness of fit (e.g., Press et al., 1992) rejects three of the 10 cases at the 95% confidence level (the 10 cases result from three d e p t h levels at each of the sites except the LDE which had sufficient i n s t r u m e n t density only at the upper level). The largest d e p a r t u r e from the model is at the b o t t o m level of the S Y N O P Central location where it is suspected that the presence of a strong b o t t o m slope polarizes the deep motions thereby breaking the isotropic assumption. Of the 14 parameters, the ones describing the spatial form of the variances and the t e m p e r a t u r e structure will vary from location to location as the field is nonhomogeneous. However, we might expect those parameters describing the correlation function and the noise fields to be less position sensitive. Table 4.1 fists these parameters and their errors. The two noise parameters are the most variable and are generally highest at the S Y N O P Central site and more similar elsewhere. The correlation spatial decay p a r a m e t e r , a, is quite uniform across all arrays and all depths with a value of about 0.5 (100 km) -2 yielding a decay scale of 140 km for the Gaussian. T h e one outlier is the deep S Y N O P Central site where the model is not an adequate representation of the covariances, because of the proximity of the Continental Rise as was previously mentioned. It is also clear from Table 4.1 that the 9"(0) parameter, the curvature of the vertical correlation function at zero lag, is inconsistently estimated by upper and deep ocean

70

1.0 SECTION

.8

'A': 0~

50 ~ N, 3 0 ~

~ W

.6 "

.4

_o

.2

O ._

t,,_ t,...

O

o

O

-.2

-.4 -.6

0

160

'

2t30

'

3(~0

'

4(30

'

500

Figure 4.4. Spatially lagged autocorrelation functions for sea surface height as determined from the Geosat altimeter for different 10 ~ boxes. The numbers refer to different 10 ~ latitude bands starting at 0~ ~ (From Stammer and B6ning, 1992).

measurements. The shallow levels all suggest values in the neighborhood of 0.4 km -2 while the deeper ones arc much smaller. This suggests that the parameterization of the vertical structure of the covariance function is inadequate, although when the statistical model is applied to all depths simultaneously, it does produce a more consistent suite of parameters (Hogg, 1993). Current meter moorings are practically the only tool which can give a reasonably complete description of the structure of the error covariances over the full depth of the ocean. Analysis of satellite altimeter data is allowing us now to extend that view across the surface of the globe. Stammer and B6ning (1992) have analyzed the Geosat altimeter data for the North and South Atlantic Oceans. To calculate the spatial covariance of sea surface height they have computed wavenumber spectra for along-track data grouped into 10 ~ by 10 ~ longitude-latitude squares, averaged these spectra over all tracks within the squares and all repeats of the tracks, and then performed the inverse Fourier transform to obtain the spatial covariance function. Examples, normalized to unity at the origin, for various regions are shown in Figure 4.4. A characteristic of all regions is decay of the correlation function over a scale of 100 km which then crosses through zero in the 100-200 km range. The position of the zero crossing point mainly depends on latitude (Figure 4.5) in a way that is consistent with the dependence of the Rossby deformation radius, Ri, on latitude. Stammer and B6ning (op cit) give the following empirical relationship between the distance to the first zero crossing, L0, and Ri: L0 -- (79.2 + 2.2 Ri) km

71

280 0

"4

240E tn

0 uJ

O 0

200-

L0:79.2.2.18

0 0 0

Ri (r:0.91]

160-

O

120-

o 10" S - 10' II 910' S/II - 60" S/II

80

0

I

I 100

I

I 200

I

Ri (kin)

Figure 4.5. The distance to the first zero crossing in the autocorrelation function versus the Rossby radius of the first baroclinic mode. (From Stammer and BSning, 1992).

valid for the region between 60 ~ S and 60 ~ N outside the tropical band from 10 ~ S to 10 ~ N, a result which indicates that the scale parameters in (4.3) and (4.4), particularly a, should be considered slowly varying functions of the environment. By doing their analysis on both ascending and descending tracks, and finding no significant differences, Stammer and BSning (op tit) conclude that there is no measurable anisotropy to the calculated height covariances. If we treat sea surface height as proportional to the streamfunction at the surface, a quasi-geostrophic approximation, then the existence of the zero crossing in the spatial covariance function is at odds with the Gaussian form suggested by the current meter moorings, as discussed above. We have used more elaborate models for the spatial covariance, which would include the possibility of a zero crossing, but have been unable to find any form which goes significantly negative. A possible explanation for this is the following. As explained previously the satellite analysis has been done by computing wavenumber spectra on track segments that are 10 ~ long. After averaging these over all the repeats of that segment the averaged spectrum is inverse Fourier transformed to give the spatial covariance function. This spatial averaging can span the gyre interior, the recirculation gyres and the Gulf Stream in the western North Atlantic. Well away from the Stream more linear, wave-like dynamics should apply to the mesoscale eddy field and imprint a more periodic signature on the covariance function. Motions near the Stream are larger amplitude, more turbulent and have less dynamical basis for periodicity. An alternative explanation is that the Stammer and BSning procedure, through removing means and trends from each track (over the 10 ~ analysis scale), is filtering out significant low wavenumber temporal (as well as spatial) variance. Using data, corrected for tidal aliasing and seasonal steric changes, from one Topex/Poseidon track which cuts

72

Topex/Poseidon track 202 40

35 Z L.. (D

-~ 30

25

25

30

latitude (~

35

40

Figure 4.6: A correlation matrix of sea surface height measured along Topex/Poseidon descending track no. 202 which crosses 20~ at about 55~ and 40~ at about 68~ Filled areas are between - 0 . 2 and zero. The heavier line accentuates the +0.2 contour. Taking each repeat of the track to be an independent measurement yields an uncertainty in correlation coefficient of +0.2 at the 95% confidence level.

across the western North Atlantic we have computed the correlation matrix (Figure 4.6) using the more direct approach of calculating correlations based on the time series from different locations. Although there is some evidence for weak negative lobes to the north, correlation scales become abruptly broader below about 30~ much larger than suggested by the Stammer and BSning analysis. At low latitudes, apparently, there are significant covariances beyond the eddy scale, a result which needs further quantification and study, but which implies that off-diagonal terms in the covariance matrix could be important even in non-eddy-resolving models.

73 5. S U M M A R Y

During this century a vast amount of information has been collected from the ocean and assembled in forms which are useful for assimilating into numerical models. In particular, there exist a number of climatologies of water properties, both global and regional. Because there still exist regions of the ocean in which few or no observations have been made, these climatologies are usually quite highly smoothed although there do exist some better resolved ones for the North Atlantic. Smoothing blurs sharp frontal features in the ocean. Provided the smoothing scale is no greater than the smoothing imparted to mean statistics by the natural time variability (meandering jets, eddies) of the ocean, the resulting climatology should be adequate for assimilation efforts aimed at estimating the time mean circulation, provided that the associated effects of eddies are properly parameterized. Otherwise these smoothed fields will underestimate fluxes of temperature, salt and other properties. Least well known of the ocean properties needed for data assimilation is the data covariance matrix at nonzero spatial lags. Calculations based on a small number of eddy-resolving arrays from the northwest Atlantic suggest that covariance function for streamfunction decays approximately in a Gaussian fashion with a decay scale of about 140 km. Analyses of satellite-derived sea surface height suggest similar scales but that the covariance function has a zero crossing and negative values at lags greater than a distance of order 100 km which depends on the local radius of deformation for the first baroclinic Rossby wave, although this result appears to depend crucially on analysis technique. A preliminary analysis of sea surface height data from the Topex/Poseidon altimeter indicates that correlation scales are very long in the western North Atlantic below 30~ such that, even in the non-eddy-resolving context, there could be significant contributions from the off-diagonal terms in the data covariance matrix. In addition, the geostrophic constraint, through Eqs. 4.2, imply that there can be significant covarying relationships between different water properties at zero spatial lag and these should be accounted for in the formulation of the cost function.

6. A C K N O W L E D G M E N T S This work has been supported by the Office of Naval Research (grant N00014-90-J1465) and the National Science Foundation (grant OCE 90-04396) for which the author is grateful. Comments from two reviewers helped to improve the text significantly. Tom Shay kindly provided the basis for Figure 4.1.

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75 Garrett, C. J. R. and W. H. Munk, 1972. Space-time scales of internal waves. Geophys. Fluid Dyn., 3, 225-264. Garrett, C. J. R. and W. H. Munk, 1975. Space-time scales of internal waves: A progress report. J. Geophys. Res., 80, 291-297. Gouretski, V. and K. Jancke, 1995. A consistent pre-WOCE hydrographic data set for the South Atlantic: Station data and gridded fields. WOCE Hydrographic Programme Special Analysis Centre Technical Report No. 1, WOCE Report No. 127/95, Bundesamt ffir Seeschiffahrt und Hydrographie, Hamburg, Germany, 81 pp. Halkin, D., and T. Rossby, 1985. The structure and transport of the Gulf Stream at 73~ J. Phys. Oceanogr., 15, 1439-1452. Hogg, N. G., 1991. Mooring motion corrections revisited. J. Atmos. Oc. Tech., 8(2), 289-295. Hogg, N. G., 1993. Toward parameterization of the eddy field near the Gulf Stream. Deep Sea Res. I, 40(11/12), 2359-2376. Hogg, N. G., 1994. Observations of Gulf Stream meander induced disturbances. J. Phys. Oceanogr., 24(12), 2534-2545. Hogg, N. G., R. S. Pickart, R. M. Hendry, and W. J. Smethie, Jr., 1986. The Northern Recirculation Gyre of the Gulf Stream. Deep-Sea Res., 33 (9), 1139-1165. Levitus, S., 1982. Climatological atlas of the world ocean. NOAA Prof. Paper 13, U.S. Dept. of Commerce, Washington, D.C., 173 pp. Levitus, S. and T. P. Boyer, 1994a. World Ocean Atlas 1994, Volume 2: Oxygen. NOAA Atlas NESDIS 2, U.S. Dept. of Commerce, Washington, D.C., 186 pp. Levitus, S. and T. P. Boyer, 1994b. World Ocean Atlas 1994, Volume 4: Temperature. NOAA Atlas NESDIS ~, U.S. Dept. of Commerce, Washington, D.C., 117 pp. Levitus, S., R. Burgett and T. P. Boyer, 1994a. World Ocean Atlas 1994, Volume 3: Salinity. NOAA Atlas NESDIS 3, U.S. Dept. of Commerce, Washington, D.C., 99 pp. Levitus, S., R. D. Gelfeld, T. Boyer and D. Johnson, 1994b. Results of the NODC and IOC oceanographic data archaeology and rescue projects: Report 1. Key to Oceanographic Records Documentation No. 19, National Environmental Satellite, Data, and Information Service, National Oceanic and Atmospheric Administration, U.S. Dept. of Commerce, Washington, D.C., 73 pp. Lozier, M. S., W. B. Owens and R. G. Curry, 1995. The climatology of the North Atlantic. Progr. Oceanogr., in press. Marotzke, J., and C. Wunsch, 1993. Finding the steady state of a general circulation model through data assimilation: Application to the North Atlantic Ocean. J. Geophys. Res., 98(Cll), 20,149-20,167. Martel, F., and C. Wunsch, 1993. The North Atlantic circulation in the early 1980s an estimate from inversion of a finite difference model. J. Phys. Oceanogr., 23, 898-924. McCullough, J., 1975. Vector-averaging current meter speed calibration and recording technique. Woods Hole Oceanog. Inst. Tech. Rept., WHOI-75-44, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, 35 pp.

76 McPhaden, M. J., 1993. TOGA-TAO and the 1991-93 E1 Nifio-Southern Oscillation event. Oceanography, 6, 36-44. Mercier, H., M. Ollitrault, and P. Y. LeTraon, 1993. An inverse model of the North Atlantic general circulation using Lagrangian float data. J. Phys. Oceanogr., 23, 689-715. Niiler, P. P., A. K. Sybrandy, K. Bi, P. M. Poulain and D. Bitterman, 1995. Measurements of the water-following capability of holey sock and TRISTAR drifters. Deep-Sea Res., in press. Olbers, D., V. Gouretsky, G. Seit3 and J. SchrSter, 1992. Hydrographic Atlas of the Southern Ocean. Alfred Wegener Institute, Bremerhaven, Germany, xvii pp. 482 plates. Owens, W. B., 1991. A statistical description of the mean circulation and eddy variability in the Northwestern Atlantic using SOFAR floats. Progr. Oceanogr., 28, 257-303. Press, W. H., S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, 1992. Numerical Recipes in C: The Art of Scientific Computing. Second edition, Cambridge University Press, Cambridge, England, U.K., 994 + xxvi pp. Richardson, P. L., 1983. Eddy kinetic energy in the North Atlantic from surface drifters. J. Geophys. Res., 88(C7), 4355-4367. Rintoul, S. R., and C. Wunsch, 1991. Mass, heat, oxygen and nutrient fluxes in the North Atlantic Ocean. Deep-Sea Res., 38A, suppl., $355-$377. Sarmiento, J. L., and K. Bryan, 1982. An ocean transport model for the North Atlantic. J. Geophys. Res., 87, 394-408. Schmitz, W. J., Jr., 1980. Weakly depth-dependent segments of the North Atlantic circulation. J. Mar. Res., 38(1), 111-133. Schmitz, W. J., Jr., 1984. Abyssal eddy kinetic energy in the North Atlantic. J. Mar. Res., 42(3), 509-536. Stammer, D. and C. W. BSning, 1992. Mesoscale variability in the Atlantic Ocean from Geosat altimetry and WOCE high-resolution numerical modeling. J. Phys. Oceanogr., 22(7), 732-752. World Ocean Atlas 1994, NOAA Atlas NESDIS 1-4, U.S. Dept. of Commerce, Washington, D.C., 9 CD-ROMs. Wyrtki, K., L. Magaaxd, and J. Hager, 1976. Eddy energy in the oceans. J. Geophys. Res., 81, 2641-2646.

Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

77

A Case Study of the Effects of Errors in Satellite Altimetry on Data Assimilation Lee-Lueng Fu and Ichiro Fukumori J e t Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

Abstract Satellite altimetry provides synoptic observation of sea surface elevation t h a t manifests the ocean circulation through the entire w a t e r column. Assimilation of a l t i m e t r y d a t a thus provides a powerful tool for using an ocean model to e s t i m a t e the three-dimensional state of the ocean and its temporal variation. The u n c e r t a i n t y of the e s t i m a t e r e s u l t i n g from errors in satellite a l t i m e t r y is i n v e s t i g a t e d , in particular, the effects of the orbit and tide errors. Covariance e s t i m a t e s for these errors are t a k e n from the specifications of the TOPEX/POSEIDON mission, a stateof-the-art satellite altimetry mission. A shallow-water model of the tropical Pacific is used to carry out a case study. It is d e m o n s t r a t e d t h a t the e s t i m a t i o n errors become smaller as more information is used in the e s t i m a t i o n technique. An approximate K a l m a n filter t h a t m a k e s use of the past assimilated information performs the best. An optimal interpolation scheme t h a t does not take a d v a n t a g e of the history of assimilated information leads to inferior results. A direct inversion without using the model dynamics is the worst. In all three cases, the orbit error carries more impact t h a n the tide error as a consequence of the differences in their covariance functions.

1. I N T R O D U C T I O N It is known from direct observations of ocean currents, t h a t the fluid flow is t u r b u l e n t on an enormous range of spatial and temporal scales (ranging from 10 to 10,000 k m and days to years and possibly longer). As with any t u r b u l e n t flow, u n d e r s t a n d i n g of the ocean circulation can only be obtained if the system is sampled and t h e n described on all relevant scales. Attempts to forecast the ocean circulation so as to m a k e inferences about future climate are thus dependent upon having adequate observations - a formidable r e q u i r e m e n t for such a large fluid system. The lack of well-sampled observations has indeed been a major obstacle for the development and application of data assimilation techniques in oceanography. The only known practical approach to observing the global ocean with useful space and time resolution is from orbiting satellites. However, spaceborne observations are

78 restricted to the surface properties of the ocean (e.g., infrared sensors detect the "skin t e m p e r a t u r e " of the sea surface), generally producing m e a s u r e m e n t s of only limited use for m a k i n g inferences about the state of the ocean at depths. T h e r e is only one observable from space t h a t is directly linked to the circulation as a whole: the surface p r e s s u r e field, manifested as the sea surface elevation. If the ocean were at rest, the sea surface would coincide w i t h a g r a v i t a t i o n a l equipotential surface (the specific surface is d e s i g n a t e d the "geoid"). With the removal of such high frequency p h e n o m e n a as tidal variations, the elevation of the sea surface relative to the geoid is the ocean d y n a m i c topography, w h i c h is a m a n i f e s t a t i o n of the m o v e m e n t of the entire oceanic w a t e r column. Sufficiently accurate m e a s u r e m e n t s of the sea surface elevation t h u s provide very powerful constraints upon the large scale circulation and its variability. Via d a t a assimilation techniques, the information of the dynamic topography can be p r o p a g a t e d to other dynamic and t h e r m o d y n a m i c variables at all depths. Space m e a s u r e m e n t s of the sea surface elevation are based upon r a d a r a l t i m e t e r s (Wunsch and Gaposchkin, 1980; Stewart, 1985; Fu et al., 1988). T O P E X / P O S E I D O N is the first satellite altimetry system specifically designed for studying the circulation of the global oceans (Fu et al., 1994). L a u n c h e d on A u g u s t 10, 1992, the spacecraft has been operating in an orbit which repeats its u n d e r l y i n g g r o u n d - t r a c k every 10 days. Results to date show t h a t the mission is producing observations of the global sea surface elevation with an unprecedented accuracy b e t t e r t h a n 5 cm everywhere. Designed for a lifetime of 3-5 years, the satellite is providing oceanographers with their first t r u l y global observation system t h a t is able to m e a s u r e the sea surface elevation with sufficient accuracy and sampling to address its large-scale variabilities in relation to the ocean circulation. Assimilation of the T O P E X / P O S E I D O N a l t i m e t e r d a t a into an ocean circulation model would provide a dynamically consistent e s t i m a t e of the three dimensional state of the ocean and its time evolution with improved r e a l i s m (Blayo et al., 1994). We w a n t to emphasize the goal of establishing an "estimate" of the ocean. An estimate is useful only if its error is also e s t i m a t e d and provided. However, error e s t i m a t i o n can often be as computationally challenging as the s t a t e e s t i m a t i o n itself, if not more so (Thacker, 1989; Marotzke and Wunsch, 1993). The K a l m a n filter is a wellestablished technique t h a t provides a formal error estimate as p a r t of the calculation. The price paid is an enormous computational burden required to u p d a t e the error e s t i m a t e sequentially, m a k i n g its application to ocean general circulation models impractical. Various approximations to the technique have been developed recently to m a k e t h e calculation more feasible ( F u k u m o r i et al., 1993; F u k u m o r i a n d Malanotte-Rizzoli, 1995). In the p r e s e n t study we apply an approximate K a l m a n filter, as well as a couple of simpler methods, to a shallow-water model of the tropical Pacific Ocean to examine the effects of m e a s u r e m e n t errors in satellite a l t i m e t r y on the e s t i m a t i o n of ocean variables. This is a case study in which we examine the effects of two major errors in satellite altimetry, the orbit and tide errors, based on the T O P E X / P O S E I D O N results. The m a i n purpose is to d e m o n s t r a t e the methodology of e s t i m a t i n g the errors in ocean state estimation based on data assimilation, as well as the dependence of the errors on the sophistication of the assimilation scheme.

79 2. T H E K A L M A N F I L T E R The technique of the K a l m a n filter is well-documented in the literature (e.g., Ghil and Malanotte-Rizzoli, 1991, and references therein). Formally, it is an optimal s e q u e n t i a l l i n e a r filter t h a t m i n i m i z e s t h e e s t i m a t i o n e r r o r b a s e d on b o t h observations a n d model physics. The relation between the observables and all the physical variables in the model are explicitly accounted for in the formulation of the filter based on dynamics and statistics. This relation is carried forward in time and provides the basis for the optimal estimate. Applications of the K a l m a n filter to ocean models have a relatively short history, m a i n l y due to the prohibitive computational r e q u i r e m e n t resulting from the large dimension of the state vector of ocean models. Mathematically, the formulation of the K a l m a n filter can be w r i t t e n as follows: x(t) = x(t,-) + K(t) ( y(t)- H(t) x(t,-) )

(1)

where x is the state vector to be estimated and t denotes time. The m i n u s sign indicates an estimate before assimilation of d a t a at time t. Vector y r e p r e s e n t s observations, while H is a matrix such t h a t Hx is the model's estimation of y. At a given time, x(t,-) is forecasted from x(t-1) by the model as x(t,-) = A(t)x(t-1) + w(t-1)

(2)

where A is the model's state transition m a t r i x and w is external forcing. The weighting m a t r i x for the observation relative to the model forecast in (1), K, is the K a l m a n gain given by K(t) = P(t,-) H(t) v (H(t) P(t,-) H(t) v + R(t)) -1,

(3)

where P(t,-) is the error covariance for x(t,-), R(t) the error covariance for the observation, y(t), and H I the transpose of H. The error covariance for the filtered x(t), denoted by P(t), is given by P(t) = P(t,-) - K(t) H(t) P(t,-)

(4)

At a given tinle, P(t,-) is related to P(t-1) by P(t,-) = A(t) P(t-1) A(t) T + Q(t)

(5)

where Q is the error covariance due to the model error (or the process noise). The most time-consuming step in implementing the K a l m a n filter is (5), involving the update at each time step of a matrix of the dimension of the state vector, which is generally g r a t e r t h a n 100,000 for a general circulation model. Two key approximations to the K a l m a n filter have been developed recently. (See Malanotte-Rizzoli et al. (1995) for a review.) F u k u m o r i et al (1993) takes a d v a n t a g e of the fact t h a t P(t,-) oi~en approaches a steady-state, P(-) relatively fast. (The time

80

index will be dropped for asymptotic limits.) The asymptotic error covariance matrix can be calculated once and for all using (3)-(5) (called the Riccati equation). The result can then be used to form a Kalman gain, saving the time-consuming u p d a t e (equation (5)) of the error covariance at each time step (also see Fu et al., 1993). However, for most ocean general circulation models, even in a regional set up, the large dimension of the state vector still makes the calculation of such asymptotic Kalman gains beyond the capability of most modern computers. F u k u m o r i and Malanotte-Rizzoli (1995) m a d e a second a p p r o x i m a t i o n by extracting only the large-scale information in the observation for assimilation by the model. This was accomplished by estimating the state error covariance of the large-scales by transforming the model state into one of a reduced dimension. The Kalman gain for the reduced state can be formally derived and applied to the original model at the expense of not a s s i m i l a t i n g the small-scale i n f o r m a t i o n in the observation. A combination of these two approximations has made the K a l m a n filter feasible for even a global general circulation model. A demonstration of the approach was made by Fukumori (1995) using a shallow-water model of the tropical Pacific Ocean. Using the machinery of Fukumori (1995), we have investigated the effects of the m e a s u r e m e n t errors in the TOPEX/POSEIDON sea level observations on the estimation of oceanic variables. Before discussing the results, the characteristics of the TOPEX/POSEIDON m e a s u r e m e n t performance is briefly reviewed in the next section. 3. T O P E X / P O S E I D O N SEA L E V E L M E A S U R E M E N T The overall r m s accuracy of the sea level m e a s u r e m e n t made by T O P E X / POSEIDON is estimated to be about 5 cm (Fu et al., 1994), which is dominated by its time-varying component. This error is roughly equally partitioned between the altimetry error and the radial orbit error. The rms magnitude is 3.2 cm for the former and 3.5 cm for the latter. The time-invariant component of the orbit error is about 2 cm (Christensen et al., 1994), leaving 3 cm (rms) for the time-varying component. In addition to the m e a s u r e m e n t error, the residual tidal effects in sea level aider correction using tidal models amount to another 3-4 cm (rms) uncertainty for studying the low-frequency ocean current fluctuations (e.g., Schrama and Ray, 1994; Le Provost et al., 1995). The largest error in determining the absolute dynamic topography is t h a t of the geoid model. The uncertainty in the geoid increases with decreasing spatial scales. The error in the state-of-the-art geoid models has a magnitude t h a t exceeds the oceanic signals at wavelengths shorter t h a n 2000 km (Nerem et al., 1994). At wavelengths longer t h a n 2000 k m , the cumulative error of the best geoid model to date still has a magnitude of 10 cm. Therefore, most of the quantitative applications of satellite altimetry are still limited to the study of the time- dependent ocean circulation. To examine the effects of the various errors on the estimation of oceanic variables, one needs to have the knowledge of the error covariance functions. In the present

81

s t u d y we focus on two components of the t i m e - d e p e n d e n t errors: the tide a n d the orbit. Their error covariance functions can be e s t i m a t e d from t h e i r well-known characteristics. The errors in the altimeter range m e a s u r e m e n t is more complicated, because it is composed of several factors such as the r a d a r t r a n s m i s s i o n media, seas t a t e effects, a n d i n s t r u m e n t errors. The error covariance functions of these factors are more difficult to estimate. The purpose of the p a p e r is to d e m o n s t r a t e the methodology for e x a m i n i n g the properties of e s t i m a t i o n errors, r a t h e r t h a n a n e x h a u s t i v e s t u d y of all the error sources in satellite altimetry. 3.1 The orbit error

The u n c e r t a i n t y of the radial position of the satellite h a s characteristic scales on the order of the circumference of the E a r t h along the satellite's flight path. Shown in Figure I is a periodogram of the T O P E X / P O S E I D O N radial orbit error (Marshall

2.0

I

I

:

1.6

/.%

r

u

.2

I.r

0~) I--

.J I1. !-

~0.8

0.4

0.0 0.0

1 .0 FREQUENCY

2.0 (CYCLES/REVOLUTION)

3.0

Figure 1. Periodogram of the T O P E X / P O S E I D O N radial orbit error.

4.0

82

et al., 1995). Distinct peaks in the neighborhood of i cycle/revolution and 2 cycles/ revolution are clearly shown. These peaks can be explained in t e r m s of orbit dynamics and estimation procedures (Tapley et al., 1994; Marshall et al., 1995). Over a three-day interval, which is the time step for the data assimilation to be discussed later, an estimate for the correlation function for the orbit error is shown in Figure 2, which displays the value of the autocorrelation in the tropical Pacific Ocean (the model domain) for a given location indicated by the white circle. The calculation was made by simulating the orbit error using the spectrum of Figure 1 with random phases for each 10-day cycle. The rms amplitude of the error was n o r m a l i z e d to 3 cm, the t i m e - v a r y i n g c o m p o n e n t of the o r b i t error. T h e autocovariance was then calculated by using the simulated orbit errors from 74 cycles. Due to the orbit e r r o r ' s large scales along the s a t e l l i t e t r a c k , the autocorrelation is high along the track of the specified observation location and gradually decreases with increasing lag in the along-orbit distance. The p a t t e r n s in Figure 2 reflect basically the decrease of the autocorrelation with increasing lag. The rms amplitude of the error, however, is uniformly 3 cm everywhere. The inverse of the bandwidth of the spectral peak near i cycle/revolution is an estimate of the decorrelation time scale of the orbit error, which is about 3 days. 3.2 T h e t i d e e r r o r

The orbit of TOPEX/POSEIDON was designed to sample the ocean tides in a way t h a t most of the aliased tidal periods would be removed from major n a t u r a l periods such as the annual and the semi-annual ones. Consequently, the ocean

Figure 2. Spatial correlation of the orbit error with its value at a given point indicated by a white circle. The magnitude is indicated by the width of the lines along the satellite track. Positive (negative) values are in black (gray). The m a g n i t u d e of unity corresponds to the width of the line at the white circle.

83

Figure 3. Same as Figure 2 but for the tide error. tide models constructed from the TOPEX/POSEIDON d a t a are highly accurate, with an rms error estimated to be 3-4 cm (Le Provost et al., 1995). A s s u m i n g t h a t the spatial and temporal characteristics of the errors of the tide models are similar to those of the tides themselves, we used the tide model ofMa et al. (1994) to simulate the tide errors with an rms amplitude of 3 cm. An estimate of the autocorrelation function of the tide error based on simulations over 74 cycles was obtained (Figure 3). Because the tides are sampled at various phases over a 3-day period, the spatial s t r u c t u r e of the autocorrelation is quite complicated. A major difference between the tide error and the orbit error is the presence in the former (Figure 3) of relatively large negative values at large lags. This difference is responsible for the different effects of the two errors discussed later. Also note t h a t the r m s a m p l i t u d e of the simulated tidal error varies in space in proportion to the tidal a m p l i t u d e itself. 4. T H E M O D E L A N D A S S I M I L A T I O N S C H E M E The dynamic system we use in this study is t h a t of F u k u m o r i (1995). The model is a wind-driven, linear, reduced-gravity, shallow-water model of the tropical Pacific Ocean, with p a r a m e t e r s chosen to simulate the response of the first baroclinic mode to wind forcing. The model domain extends zonally across the Pacific basin, but limited m e r i d i o n a l l y w i t h i n 30 ~ from the equator, w i t h zonal a n d m e r i d i o n a l resolutions of 2 ~ and 1 ~ respectively. The dimension of the state vector of the model is about 12,000, m a k i n g direct application of the K a l m a n filter impractical. A reduced state was constructed on the grid shown in Figure 4, which has a zonal resolution of 7.5 ~ and a v a r y i n g meridional resolution from 3 ~ at the equator to 4 ~ at the boundaries. The dimension of the reduced state is 831. Transforming the state on the coarse grid to the original

84

Figure 4. The model domain and the TOPEX/POSEIDON ground tracks. Dots are the locations of the coarse grid on which the reduced model state is defined, at a nominal resolution of 10 ~ in longitude and 5 ~ in latitude. The gray border denotes the extent of the model domain. Thick solid lines are the satellite ground tracks for a particular 3-day period. The covariance of the estimation error is evaluated on these tracks. 2 ~ x 1~ grid is performed by objective mapping (Bretherton et al., 1976), using a Gaussian correlation function with zonal and meridional correlation distances of 7.8 ~ and 4 ~, respectively. The dynamic equations for the reduced state on this coarse grid are obtained by combining the model with an interpolation operator (objective mapping) between this coarse grid and the model grid plus its inverse transform. An asymptotic limit of this coarse state's error covariance will be obtained by solving the Riccati equation with a time-invariant observation pattern (Figure 4). TOPEX/POSEIDON's 10day orbit has a 3-day subcycle, such as the one shown in Figure 4, in which the satellite covers the entire globe nearly uniformly. The error covariance will be evaluated assuming t h a t the observation pattern is the same every 3 days and t h a t the m e a s u r e m e n t s of the oceanic signal (but not the m e a s u r e m e n t errors) are instantaneous. That is, we will ignore the effects on the error estimates of the different sampling patterns of the subcycles and the relatively small temporal c h a n g e s of t h e o c e a n d u r i n g t h e 3 - d a y i n t e r v a l . T h e s e are r e a s o n a b l e approximations, because the oceanographic variability typically has time-scales longer t h a n 3-days and that each subcycle samples the model domain nearly equally. Furthermore, to avoid effects of the artificial boundaries at 30~ and 30 ~ S, the data assimilation is limited within 20 degrees from the equator.

5. S T A T E E R R O R E S T I M A T I O N

The objective of the study is to evaluate the error covariance P in relation to the observational errors, represented by R. As in Fukumori et al. (1993), the doubling algorithm (Anderson and Moore, 1979) is used to solve for an asymptotic P(-) using

85

(3) -(5). The process noise, Q, required in this c o m p u t a t i o n is the s a m e as F u k u m o r i (1995), a n d is modeled in the form of s t a t i o n a r y wind error w i t h G a u s s i a n s p a t i a l covariance among the pseudo-stress components. Correlation distances were a s s u m e d to be 10 degrees zonally and 2 degrees meridionally. Wind speed error was a s s u m e d to be 2.2 m/s a n d the d r a g coefficient formulation of Kondo (1975) w a s used. F u r t h e r m o r e , wind errors were a s s u m e d to be completely correlated over 3days (assimilation cycle) b u t i n d e p e n d e n t from one t h r e e - d a y period to t h e next, while the errors of the meridional and zonal stresses were a s s u m e d to be uncorrelated w i t h each other. By combining (3) a n d (4), one obtains (using the a s y m p t o t i c variables), P = P(-) - P(-) H T (H P(-) H I + R) -1 H P(-) which can be r e w r i t t e n p = [p(_)-i + H wR-1 HI-1

(6)

See, for example, Gelb (1974) for the derivation of (6). In w h a t follows, the r e s u l t s of e v a l u a t i n g the dependence of P on R a n d P(-) are discussed. 5.1 D i r e c t i n v e r s i o n

with no data assimilation

As a bench m a r k , it is instructive to e x a m i n e the error of the model s t a t e w i t h o u t any d a t a assimilation; namely, the e s t i m a t i o n error for the coarse state, x, from a direct inversion of y = H x, with the left h a n d side being the observations. This is equivalent to a s s u m i n g P(-) = infinity in Equation (6). The resulting error covariance can t h e n be w r i t t e n P= (H T R "1 H) 1

(7)

To i l l u s t r a t e the i m p a c t of the orbit and tide errors on the s t a t e e s t i m a t i o n , we need a reference case to m a k e comparisons with. This reference case is chosen to be one in which the m e a s u r e m e n t error is a white noise, t a k e n to be the n o m i n a l a l t i m e t e r i n s t r u m e n t noise, whose r m s m a g n i t u d e is on the order of I cm after a 5point a v e r a g i n g along each satellite track. Errors (i.e., square root of the diagonal e l e m e n t s of P) for the sea level e s t i m a t e at the model grids are shown in Figure 5. The spatial s t r u c t u r e for the w h i t e noise case (the r e f e r e n c e error, F i g u r e 5 c) s i m p l y reflects t h e d i s t r i b u t i o n of t h e observational grid (i.e., the satellite tracks) u n d e r l y i n g the H m a t r i x (see F i g u r e 4). After adding additional errors (the orbit error or the tide error), the ratio of the r e s u l t i n g error to the reference error is shown in Figure 5 a (white noise plus the tide error) and Figure 5 b (white noise plus the orbit error). As noted in Section 3.1, the r m s m a g n i t u d e of the orbit error is uniform on the observation grids. The spatial p a t t e r n of the effect of the orbit error is dictated by the distance from the observation grids; the shorter the distance the larger the effect. On the other h a n d , the r m s m a g n i t u d e of the tide error varies in space w i t h its effect controlled by the s p a t i a l v a r i a b i l i t y of the error m a g n i t u d e itself. A s e c o n d a r y influence of the

85

Figure 5. The error in the estimate of sea level made by the direct inversion. The result from the case in which the d a t a error is white noise only (1 cm rms) is shown in (c) (unit in cm). The impact of adding additional errors is shown as the ratio of the r e s u l t i n g error to t h a t shown in (c) for: (a) the tide error plus the white noise, and (b) the orbit error plus the white noise.

87 observation grids is still noticeable though. Note t h a t the tide error has less i m p a c t t h a n the orbit error (the scale of the gray shade is different between Figures 5a a n d 5b). The rms error estimates are given in Table 1, along w i t h other cases discussed below. Dimensionally, the rms error estimates for the direct inversion are 0.78, 1.38, a n d 0.71 cm, for F i g u r e s 5a, 5b, a n d 5c, r e s p e c t i v e l y . The a p p a r e n t inconsequential n a t u r e of the tide errors relative to the orbit error is due to the difference in the s t r u c t u r e s of the error covariance. The tide error covariance h a s large negative side-lobes because of the periodic n a t u r e of the tides, w h e r e a s the m a g n i t u d e of the side lobes of the orbit error covariance is m u c h smaller. The large negative side-lobes make the tide error more benign t h a n the orbit error. This is because the H m a t r i x is a mapping operator composed of mostly positive off-diagonal elements, which will result in a smaller error w h e n the R m a t r i x has large negative off-diagonal elements as opposed to positive ones. A simple example helps illustrate the situation. Consider two d a t a points and one model point such that, R= (-1
H/I/ Then, (H T R 1 H) -1 = (1+r)/2, which is an increasing function of r, v a r y i n g from 0 to 1 as r varies from -1 to 1.

5.2 A s s i m i l a t i o n by a n a p p r o x i m a t e K a l m a n f i l t e r By assimilating the altimeter d a t a with a model, one is able to e s t i m a t e not only sea level, which are directly observed by the altimeter, but also velocities as well as other model variables based on the model dynamics and t h e r m o d y n a m i c s . The error in the e s t i m a t e d state of the ocean is determined by the assimilation scheme t h a t relates the observation to the state estimate. The error covariance for the state estimate resulting from the approximate K a l m a n filter described in Section 2 is computed according to Equation (6) with P(-) evaluated as the asymptotic solution to the associated Riccati equation. The resulting errors for the sea level and the two horizontal velocities are shown in Figure 6 with the d a t a error being the white noise. The rms error of the sea level estimate in the model domain has been reduced from 0.71 to 0.25 cm. There is little dependence of the error on the distribution of the observations, primarily due to the propagation of information in space from past m e a s u r e m e n t s by the model dynamics. The impact of the tide error and the orbit error becomes much smaller t h a n in the case of the direct inversion (Table 1), and their spatial structures (not shown) are similar to the case of white noise (Figure 6). The rms sea level errors are 0.26 and 0.31 cm for the tide error and the orbit error, respectively. The rms errors for the zonal and meridional velocity e s t i m a t e s for the case of white-noise observation error are 1.63 and 1.26 chris, respectively, and are largest n e a r the equator. The impact of the tide and the orbit errors on the velocities is also small (Table 1).

$$

F i g u r e 6. E r r o r (rms) in the e s t i m a t e d sea level, t h e zonal a n d m e r i d i o n a l velocities from t h e K a l m a n filter w i t h 1 cm w h i t e noise d a t a error.

It is i n s t r u c t i v e to c o m p a r e t h e s e e r r o r e s t i m a t e s to t h o s e of t h e m o d e l w h e n r u n by itself, n a m e l y e r r o r s of a simulation. S i m u l a t i o n e r r o r can be e s t i m a t e d by solving t h e Riccati e q u a t i o n (i.e., e q u a t i o n s (3)-(5)) w i t h no a s s i m i l a t i o n (H=0). T h e r e s u l t s a r e s h o w n in F i g u r e 7. T h e e s t i m a t e of s i m u l a t i o n e r r o r s e r v e s as a r e f e r e n c e as to

89 Table 1. R o o t - m e a n - s q u a r e error e s t i m a t e s for the various s t a t e variables r e s u l t i n g from the various e s t i m a t i o n methods. Values (in cgs units) are for the sea level (h), the zonal velocity (u), and the meridional velocity (v), as a function of the type of d a t a error (rows) a n d the e s t i m a t i o n m e t h o d (columns). D a t a Error

Direct Inversion

O p t i m a l Interpolation

K a l m a n Filter

h

h

u

v

h

u

v

white noise only

0.71

0.25

1.63

1.26

0.35

3.87

4.02

tide + white noise

0.78

0.26

1.66

1.28

0.40

3.88

4.03

orbit + white noise

1.38

0.31

1.71

1.31

0.60

3.92

4.04

how accurate the model is and how assimilation reduces the model's uncertainties. The velocity errors are largest n e a r the equator as in the case of the assimilation, reflecting the dynamics of the equatorial wave guide. There are significant errors in the model's extratropical sea level even though there are no pronounced velocity errors. The relative lack of extratropical velocity error is due to the dominance of available potential energy in the extratropical p l a n e t a r y waves, which are the p r i n c i p a l c a r r i e r s of the model e r r o r a w a y from the equator. The w e s t w a r d intensification of the extratropical sea level error is due to the w e s t w a r d propagation of the Rossby waves, w h e r e a s the e a s t w a r d intensification along the e q u a t o r is due to the dominance of the e a s t w a r d propagation of the Kelvin waves. The lack of such s t r u c t u r e in the sea level error for the K a l m a n filter (Figure 6) is caused by the correction by the d a t a assimilation, which has effectively removed the model error. However, the velocity errors from the K a l m a n filter, despite t h e i r m u c h s m a l l e r magnitudes, have the same spatial structures as the velocity errors of the simulation. This is because the velocity is not directly observed; its information is derived from the sea level via the model. The K a l m a n filter is t h u s less effective in reducing the velocity errors. For comparison to Table 1, the estimated r m s errors of the simulation are 6.91 and 4.80 cm/s for the zonal and meridional velocities, respectively, and 2.93 cm for the sea level.

5.3 Assimilation by optimal interpolation "Optimal interpolation" (OI) is used here to refer to the class of assimilation m e t h o d s based on approach similar to the K a l m a n filter, namely, m a k i n g e s t i m a t e s by optimally combining model forecast with observations (i.e., E q u a t i o n 6), b u t the forecast error covariance P(-) is based on empirical e s t i m a t i o n i n s t e a d of being a solution to the associated Riccati equation (e.g., Ezer and Mellor, 1994; White et al., 1990). A reasonable candidate for P(-) is the error of the model simulation, which m i g h t be e s t i m a t e d by computing the difference between model s i m u l a t i o n and

90

Figure 7. E r r o r (rms) in the sea level, the zonal and meridional velocities s i m u l a t e d by the model w i t h o u t d a t a assimilation.

observation. Alternatively, s i m u l a t i o n error can be e s t i m a t e d from the Riccati equation as was done for Figure 7. Using the e s t i m a t e d s i m u l a t i o n error based on the l a t t e r approach as P(-), the error variances resulting from E q u a t i o n (6) are shown in Figure 8. The basic spatial p a t t e r n s are similar to the r e s u l t s of the

9!

Figure 8. Same as in Figure 6 except for the optimal interpolation.

approximate K a l m a n filter, but the magnitudes of the errors are much larger (Table 1). However, the sea level error variances resulting from the OI are smaller t h a n those from the direct inversion. The decrease in estimation error with the increase in the use of the knowledge of dynamics and statistics is clearly demonstrated.

92 6. C O N C L U S I O N S A N D D I S C U S S I O N S The effects of the errors in satellite a l t i m e t r y on e s t i m a t i n g large-scale sea level a n d o c e a n c u r r e n t v e l o c i t i e s a r e e v a l u a t e d w i t h s e v e r a l t e c h n i q u e s . I t is d e m o n s t r a t e d t h a t the estimation errors become s m a l l e r as more information is used in t h e e s t i m a t i o n technique. The examples e x a m i n e d here, in the order of increasing use of information, are a direct inversion u s i n g no prior information, optimal i n t e r p o l a t i o n (OI), and an a p p r o x i m a t e K a l m a n filter. The difference between the OI and K a l m a n filter is in the specification of the prior error covariance matrix. The K a l m a n filter explicitly accounts for the effects of p a s t a s s i m i l a t e d information w h e r e a s OI does not. For example, the r m s e s t i m a t i o n error for sea level, based on a m e a s u r e m e n t error consisting of 1 cm white noise plus 3 cm orbit error (as in the case of TOPEX]POSEIDON), is 1.38 cm, 0.60 cm, a n d 0.31 cm, for the direct inversion, the OI, and the approximate K a l m a n filter, respectively. L a r g e r d a t a errors generally result in larger e s t i m a t i o n errors. However, the e s t i m a t i o n errors are less sensitive to the tide errors t h a n the orbit errors in all three cases considered. The difference in the effects of the two error sources is due to the d i s p a r a t e s t r u c t u r e of their covariances. While the orbit errors are n e a r l y u n c o r r e l a t e d from t r a c k to track, the tide errors are strongly correlated w i t h large negative covariances. Because the columns of the observation m a t r i x (H) which is being inverted are spatially smooth, the complex covariances of the tide errors are largely in their null space, m a k i n g the e s t i m a t i o n error less sensitive to the tide errors. The differences in sea level error among the three d a t a error scenarios become s m a l l e r as the e s t i m a t i o n m e t h o d becomes more accurate. For e x a m p l e , the differences in the r m s sea level errors between the cases of the orbit error and the reference white noise are 0.67, 0.25, and 0.06 cm for the direct inversion, the OI, and the K a l m a n filter, respectively (Table 1). As more prior information is used, differences in d a t a errors become less i m p o r t a n t for sea level estimates. In contrast, the sensitivity of the velocity e s t i m a t e s to d a t a characteristics is s o m e w h a t stronger for the K a l m a n filter t h a n the OI. This is because the velocity errors in the case of OI are largely dictated by the simulation error w i t h o u t being effected by the d a t a assimilation. On the other hand, the K a l m a n filter t r a n s m i t s the i n f o r m a t i o n contained in the observed sea level to the velocity e s t i m a t e s , m a k i n g t h e m more sensitive to the d a t a characteristics. The r e s u l t s of the study point to the importance of using advanced a s s i m i l a t i o n m e t h o d s and of u n d e r s t a n d i n g the error characteristics of the d a t a themselves. The covariance of the e s t i m a t i o n error h a s nontrivial s t r u c t u r e s t h a t are n e i t h e r homogeneous nor isotropic and differ a m o n g the various conducted experiments. Figure 9 shows an example of the error covariance for the K a l m a n filter e s t i m a t e s . W h a t is shown is a column of P, r e p r e s e n t i n g the covariance of various variables with the sea level at a grid node at (142 ~ W, 10 ~ N). The large covariance of the sea level w i t h the zonal velocity at the equator for the orbit error case is due to the large meridional correlation of the d a t a error and the model's large variance associated

93

Figure 9. E x a m p l e s of the error covariance among the various state variables e s t i m a t e d from the K a l m a n filter. Each panel shows the covariance of a given variable with the sea level at (142~ W, 10 ~N), representing a column of the covariance matrix. Lei~ (right) panels are for the tide (orbit) error plus the white noise d a t a error. Covariances for the case of white noise only are similar to the case of the tide error (not shown). Contour intervals are 0.02 (cm2/s), 0.01 (cm2/s), 0.01 (cm 2) for zonal velocity (u), meridional velocity (v), and sea level (h), respectively. Positive (negative) contours are in solid (dashed) lines.

with the equatorial wave guide (Figure 7). In comparison, the OI's covariance estimate shown in Figure 10 lacks the zonal and meridional a s y m m e t r y of the K a l m a n filter covariance. The spatial extent of the covariance is generally larger for the OI t h a n for the Kalman filter, and is the largest for the direct inversion (Figure 11). Although the results are sensible and instructive, we note some of the limitations in the present error estimate. The most significant simplification we have m a d e is the a s s u m p t i o n t h a t the tide errors are uncorrelated beyond 3 days. In actuality, the tide errors are narrow band processes with theoretically infinite correlation time scales, m a k i n g the error calculation for the K a l m a n filter an u n d e r e s t i m a t e . Temporally correlated m e a s u r e m e n t errors can be handled by modifying the K a l m a n filtering algorithm (Gelb, 1974). Another simplification we made is to estimate errors of only the coarse state, n a m e l y the large scales. The relative effects of the tide errors and the orbit errors could be different for the short scales not resolved by the present formulation. For example, the orbit errors have mostly large scales, whereas the tide errors have shorter scale variations. Therefore, the short scales m a y be more sensitive to the errors in the tidal corrections t h a n the errors in the orbit.

94

Figure 10. Error covariance from the OI. The panels are the velocity covariances with the sea level at (142 ~W, 10 ~N) assuming the orbit error plus the white noise data error. Contour intervals are 0.06 (cm2/s) and 0.03 (cm2/s) for the zonal (u) and the meridional (v) velocities, respectively. Positive (negative) contours are in solid (dashed) lines.

Figure 11. Sea level covariance from the direct inversion with the orbit error plus the white noise data error. Covariance is with the sea level at (142~ 10 ~ N). Contour interval is 0.19 (cm2). Positive (negative) contours are in solid (dashed) lines.

ACKNOWLEDGEMENTS

The authors are indebted to Andrew Marshall of NASA's Goddard Space Flight Center for providing the TOPEX/POSEIDON orbit error spectrum. Greg Pihos performed invaluable computer programming assistance. The research described in the paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under contract with National Aeronautics and Space Administration. Support from the TOPEX/POSEIDON Project funded under the NASA TOPEX/ POSEIDON Announcement of Opportunity is acknowledged (LLF).

95 REFERENCES

Anderson, B.D.O, and J.N. Moore, 1979. Optimal Filtering, Prentice-Hall, 357 pp. Bretherton, F.P., R.E. Davis, and C.B. Fandry, 1976. A technique for objective analysis and design of oceanographic experiments applied to MODE-73, Deep-

Sea Res., 23,559-582. Blayo, E., J. Verron, and J.M. Molines, 1994. Assimilation of TOPEX/POSEIDON altimeter data into a circulation model of the North Atlantic, J. Geophys. Res., 99, 24691-24705. Christensen, E.J., B.J. Haines, K.C. McColl, and R.S. Nerem, 1994. Observations of geographically correlated orbit errors for TOPEX/POSEIDON using the Global Positioning System, Geophys. Res. Lett., 21 (19), 2175-2178. Ezer, T. and G.L. Mellor, 1994. Continuous assimilation of Geosat altimeter data into a three-dimensional primitive equation Gulf S t r e a m model. J. Phys. Oceanogr., 24, 832-847. Fu, L.-L., D.B. Chelton, and V. Zlotnicki, 1988. Satellite altimetry: observing ocean variability from space. Oceanography Magazine, 1(2), 4-11. Fu, L.-L., E. J. Christensen, C.A. Yamarone, M. Lefebvre, Y. Menard, M. Dorrer, and P. Escudier, 1994. TOPEX/POSEIDON Mission Overview, J. Geophys. Res., 99, 24369-24381. Fu, L.-L., I. Fukumori and R. N. Miller, 1993. Fitting dynamic models to the Geosat sea level observations in the Tropical Pacific Ocean. Part II: A linear, wind-driven model, J. Phys. Oceanogr., 23, 2162-2181. Fukumori, I., J. Benveniste, C. Wunsch and D. B. Haidvogel, 1993. Assimilation of sea surface topography into an ocean circulation model using a steady-state smoother, J. Phys. Oceanogr., 23, 1831-1855. Fukumori, I., and P. Malanotte-Rizzoli, 1995. An approximate Kalman filter for ocean data assimilation; An example with an idealized Gulf Stream model, J. Geophys. Res., 100, 6777-6793. Fukumori, I., 1995. Assimilation of TOPEX sea level measurements with a reducedgravity shallow water model of the tropical Pacific Ocean, J. Geophys. Res., 100, in press. Gelb, A., 1974. Applied Optimal Estimation, M.I.T. Press, Cambridge, MA 374 pp. Ghil, M., and P. Malanotte-Rizzoli, 1991. Data assimilation in meteorology and oceanography, Adv. Geophys., 33, 141-266. Kondo, J., 1975. Air-sea bulk transfer coefficients in diabatic conditions, BoundaryLayer Meteorology, 9, 91-112. Le Provost, C., A.F. Bennett, and D.E. Cartwright, 1995. Ocean tides for and from TOPEX/POSEIDON, Science, 267,639-642. Ma, X.C., C.K. Shum, R.J. Eanes, and B.D. Tapley, 1994. Determination of ocean tides from the first year of TOPEX/POSEIDON altimeter measurements, J.

Geophys. Res., 99,24809-24820. Malanotte-Rizzoli, P., I. Fukumori, and R. E. Young, 1995. A methodology for the construction of a hierarchy of Kalman filters/smoothers for nonlinear primitive equation models, (this book).

95 Marotzke, J., and C. Wunsch, 1993. Finding the steady state of a general circulation model through data assimilation: application to the North Atlantic Ocean, J. Geophys. Res., 98, 20149-20167. Marshall, J.A., N.P. Zelensky, S.M. Klosko, S.B. Luthcke, K.E. Rachlin, and R.G. Williamson, 1995. The temporal and spatial characteristics of TOPEX/POSEIDON radial orbit error, J. Geophys. Res., 100, in press. Nerem, R. S., et al., 1994. Gravity model development for TOPEX/POSEIDON: Joint Gravity Models 1 and 2., J. Geophys. Res., 99, 24421-24447. Schrama, E. J. O., and R. D. Ray, 1994. A preliminary tidal analysis of TOPEX/ POSEIDON altimetry, J. Geophys. Res., 99, 24799-24808. Stewart, R.H., 1985. Methods of Satellite Oceanography. University of California Press, Berkeley, Ca. Tapley, B.D., et al. (14 coauthors), 1994. Precision orbit determination for TOPEX/ POSEIDON, J. Geophys. Res., 99, 24383-24404. Thacker, W.C., 1989. The role of the H e s s i a n m a t r i x in fitting models to measurements, J. Geophys. Res., 94, 6177-6196. White, W.B., C.-K. Tai and W.R. Holland, 1990. Continuous Assimilation of Geosat Altimetric Sea Level Observations into a Numerical Synoptic Ocean Model of the California Current. J. Geophys. Res., 95(C3), 3127-3148. Wunsch, C. and E.M. Gaposchkin, 1980. On Using Satellite Altimetry to Determine the General Circulation of the Oceans with Application to Geoid Improvement. Rev. Geophys. Space Phys., 18(4), 725-745.

Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

Ocean acoustic tomography-

97

Integral data and ocean models

Bruce D. Cornuelle and Peter F. Worcester Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California 92093, U.S.A.

Abstract

Tomographic data differ from most other oceanographic data because their sampling and information content are localized better in spectral space than in physical space. Approximate data assimilation methods optimized for localized physical space measurements can lose much of the non-local tomographic information, degrading the potential performance of the tomographic data in fixing the model state. In addition, methods which require inverting the tomographic data to a physical space grid before insertion into the model as point measurements incur special problems because of the non-local nature of the errors. We give a simple 1-D example to illustrate the problems that can arise. Fortunately, methods that directly assimilate integral measurements and preserve all of the information in the tomographic data, such as the Kalman filter, are becoming more practical as computer speed grows.

1. I N T R O D U C T I O N Ocean acoustic tomography is a remote sensing technique that exploits the transparency of the ocean to low frequency sound and the sensitivity of acoustic propagation to the ocean sound speed (temperature) and current fields (Munk and Wunsch, 1979). The travel times of acoustic pulses or the group delays of acoustic normal modes are interpreted to provide information about the intervening ocean. Tomographic data differ from more familiar data types, such as moored temperature and current measurements or hydrographic data, in that the data are integrals of the temperature and current fields along the acoustic paths, rather than values at a single location. Sound speed perturbations are at least an order of magnitude greater than current speeds, dominating travel time perturbations. Measurements of travel times between multiple points in the ocean can therefore be used to estimate the ocean sound speed field. The effects of sound speed perturbations and currents can be separated using acoustic transceivers, rather than separate sources and receivers. The difference in travel times of acoustic pulses traveling in opposite directions is to first order proportional only to the integral of current along the path. The sum of the travel times is proportional only to the integral of the sound speed field. Reciprocal transmissions can therefore be used to measure both sound speed and current. Linear inverse methods are commonly used to estimate the sound speed and current fields from tomographic data. These methods are well-developed for data collected more

98 or less simultaneously, but much less effort has been devoted to the methods needed to combine data collected at different times. Howe et al. (1987) applied a Kalman filter to combine data collected at different times from a single source-receiver pair, with the assumption that the ocean tends to climatology with a 10-day time constant (i.e., assuming that the perturbation state vector decays to zero with a 10-day time constant). Spiesberger and Metzger (1991) employed a generalized objective mapping approach in which a temporal covariance function was specified, in addition to the usual spatial covariance function, to analyze travel-time time series from a different experiment. Neither of these approaches takes advantage of our knowledge of ocean dynamics to constrain how the temperature and current fields evolve. Chiu and Desaubies (1987) included dynamics in the analysis of data from the 1981 Tomography Demonstration Experiment (Ocean Tomography Group, 1982; Cornuelle et al., 1985) in perhaps the simplest possible way, assuming that the ocean perturbations were made up of linear Rossby waves and using the data to estimate time-independent amplitudes and phases of selected Rossby wave components in a global (non-linear) least squares fit. Cornuelle (1990) used a Kalman filter with a linear ocean model to combine data obtained at different times in a simulated moving ship tomography experiment. More generally, however, fully non-linear ocean general circulation models (GCMs) are required to describe realistically the evolution of the ocean. Schrbter and Wunsch (1986) described one approach to forcing a steady ocean GCM to consistency with a comparatively sparse set of tomographic measurements, but did not consider the time-dependent problem. Munk et al. (1995) and Wunsch (1990) discuss the general problem of combining tomographic data with timedependent ocean models to obtain the best possible estimate of the state of the ocean. Sheinbaum (1989) performed simulations in which tomographic measurements were combined with a simple, time-dependent, one-layer ocean model. Fukumori and MalanotteRizzoli (1995) constructed an approximate Kalman filter for a nonlinear, primitive equation model of the Gulf Stream, and examined the assimilation of various pseudomeasurements, including tomographic observations. The same fundamental methods used for other oceanographic data are in principal also applicable to tomographic data. There are practical difficulties due to the large number of state variables in ocean GCMs, however. The computer time and memory required to implement exact assimilation methods have led to the development of various approximate methods for assimilating point measurements into ocean GCMs. Unfortunately, approximate methods suitable for point measurements often are not suitable for tomographic data, due to the quite different ways in which point sensors and tomographic measurements sample the ocean. Sequential methods that do not retain the full uncertainty covariance, for example, lose significant information from step to step when used with tomographic data. In this paper, it is not our goal to review the inverse methods required to combine tomographic measurements with ocean models, for which we refer the reader to Munk et al. (1995). Rather, we review the sampling properties of tomographic measurements, and present a cautionary note as to what can go wrong when approximate methods developed for point measurements are blindly applied to tomographic data.

99 2. S A M P L I N G P R O P E R T I E S OF A C O U S T I C RAYS Most tomographic applications outside the earth sciences are characterized using spectral methods, such as the projection-slice theorem central to the reconstruction in medical t0mography. The irregularity of the ray paths in geophysical and ocean tomography destroys the simplest spectral relationships, but the non-local character of the data remains.

2.1. Vertical slice: Range-independent A measurement of acoustic ray travel time is sensitive to the integral of sound speed and tangential current along the entire ray path, although the weighting is not uniform in range, because ray travel time is most sensitive to propagation speed perturbations at the upper and lower ray turning points. This depth dependence can be seen most easily in the range-independent case (Munk and Wunsch, 1982; Cornuelle et al., 1993). To first order, ray travel time perturbations are weighted averages of the sound-speed perturbations integrated along the unperturbed ray paths, F i , ds Ari = - ~ Cg(x,z)

A C (x ,z )

['i

where ocean currents have been neglected. For the range-independent case, in which Co(z ) and AC (z) are independent of x, this can be converted to an integral over z AT i = _ S

dz , C(2ql-(C0/C)

A C (z ) 2

using Snell's law, C(z)/cos 0 = C. (~ is the sound speed at the ray turning point and e is the angle relative to horizontal. The function

1 C~2 (z)N{ 1-(C0(z ) / 4 ) 2 gives the weighting with which the AC(z) contribute to A T i. The (integrable) singularity at the ray turning point depths, ~,, where C0(~,)= t~, clearly shows that ray travel times are most sensitive to the ocean at ray turning point depths. There is an ambiguity in that singularities occur at both the upper and lower turning points. As an example, Cornuelle et al. (1993) computed the travel time perturbations due to a sound speed perturbation with an amplitude o f - 1 m/s at 100 m dePth, linearly decreasing to zero at 90 and 110 m. The travel time anomaly, computed by subtracting the travel time in the unperturbed ocean from the travel time in the perturbed ocean, is sharply peaked for rays that turn between 90 and 110 m (Fig. 1). The anomaly is zero for rays with upper turning depths below 110 m because they do not sample the perturbed region. Rays that have upper turning depths above 90 m have non-zero anomalies, because they traverse the perturbed region, but the anomalies are relatively small because the ray weighting function falls off rapidly with distance from the turning point.

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2.2. Vertical slice:

Range-dependent

A ray trapped in the sound channel, with upper and lower turning points at regular intervals, samples the ocean periodically in space, so that its travel time is sensitive to some spatial frequencies, but unaffected by others. Chiu et al. (1987) found from simulations, and Howe et al. (1987) showed experimentally, that some range-dependent information on the sound speed field in a vertical slice can be extracted from data obtained with a single acoustic transceiver pair. Cornuelle and Howe (1987) subsequently showed that features which match the ray periodicity (i.e., have the same wavelength) generate large signatures in travel time. The ray paths are somewhat distorted sinusoids in midlatitudes, and so contain higher harmonics. Scales short compared to a double loop length that match these harmonics also generate significant signatures in travel time. This result can be qualitatively understood in the spatial domain (Worcester et al., 1991). Scales short compared to the double loop lengths of the rays affect only the few rays that pass through the feature, leaving the other rays unaffected. This differing sensitivity of the rays gives the fundamental information required to invert for scales short compared to double loop

101 lengths. Features which have wavelengths substantially longer than the ray double loop length (about 50 km in midlatitudes) affect all rays similarly and are indistinguishable from a change in the mean over the entire range between the source and receiver. Travel times are therefore sensitive to ocean features with wavelengths comparable to and shorter than a ray double loop length, as well as to the range-average. This behavior is most easily understood in the wave number domain, by choosing the model parameters used to represent the ocean sound speed perturbation field to be the complex Fourier coefficients in a spectral expansion. For any specific geometry, the sensitivity of the travel time inverse to various wave numbers can be quantified by plotting the diagonal of the resolution matrix. The resolution matrix gives a particular solution to any linear inverse problem as a weighted average of the true solution (e.g., Aki and Richards, 1980). For the specific case of two moorings separated by 600 km, with a source and five widely separated receivers on each mooring, the diagonal of the resolution matrix for spectral model parameters clearly shows the sensitivity of tomographic measurements to the mean and to harmonics with scales comparable to and shorter than a ray double loop length (Fig. 2). There are obvious spectral gaps for wave numbers between the mean and first harmonics of the ray paths, and again between the first and second harmonics. Travel times are not sensitive to wave numbers in these regions. The harmonics extend over bands of wave numbers because the eigenrays connecting the source and receiver have a range of double loop lengths.

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102

2.3. Horizontal slice As was the case for the vertical slice, the key to understanding the horizontal sampling properties of acoustic travel times is to consider the wave number domain, rather than physical space (Cornuelle et al., 1989). For simplicity, consider a two-dimensional ocean consisting of a horizontal slice. Rays are then straight lines connecting sources and receivers (neglecting horizontal refraction, which is usually small due to the small horizontal gradients in the ocean). The sound speed perturbation field A C ( x , y ) can be expanded in truncated Fourier series in x and y, giving the spectral representation: 2rti AC ( x , y ) - ~_~F, Pktexp---~- (kx + ly), k,l = O, +1 ..... +_N k

l

where L is the size of the domain and k,l are the wave numbers in x and y, respectively. With this representation of the ocean, the inverse problem is to determine the complex Fourier coefficients Pkl from the travel time data. Consider a scenario in which two ships start in the left and right bottom corners of a 1 Mm square and steam northward in parallel, transmitting from west to east every 71 km for a total of 15 transmissions (Fig. 3a). The inversion of the 15 travel times leads to an estimate which consists entirely of eastwest contours. All of the ray paths give zonal averages, with no information on the longitudinal dependence of the sound speed field. Similarly, transmissions between east-to-west moving ships give meridional averages, and the resulting estimate consists entirely of noah-south contours (Fig. 3b). To interpret these results in wave number space, note that with the field expressed as above, L

AC(x,y)dx -0

for k r

0 East-west transmissions therefore only give information on the parameters Pot, as can be seen in the expected predicted variance plot in wave number space of Fig. 3a. Similarly, with north-south transmissions only the parameters Pk0 are determined (Fig. 3b). Combining east-west and north-south transmissions determines both Pot and Pk0. Not surprisingly, this is still inadequate to generate realistic maps because most of the parameters remain unknown (Fig. 3c). More complex geometries with scans at 45 ~ give a distinct improvement by determining the parameters for which k = l, but at the cost of excessive ship time (Fig. 3d). In all cases, the division between well-determined and poorly-determined parameters is simplest in spectral space. To generate accurate maps of the ocean mesoscale field requires ray paths at many different angles to determine all of the wave number components. This requirement must be independently satisfied in all regions with dimensions comparable to the ocean decorrelation scale. Because this is impossible to achieve using two ships in any reasonable time period, a combination of moored and ship-suspended instruments was found to be required to achieve residual sound speed variances of a few per cent. These results are a direct consequence of the projection-slice theorem (Kak and Slaney, 1988).

103

Figure 3. Ship-to-ship tomography. The top center panel is the "true ocean", constructed assuming a horizontally homogeneous and isotropic wave number spectrum with random phases. Energy decreases monotonically with increasing scalar wave number, giving an approximately Gaussian covariance with l/e decay scale of 120 km. (a) W-~E transmissions between two northward traveling ships (left panel). Inversion of the travel time perturbations produce east-west contours in AC (middle) with only a faint relation to the "true ocean". Expected predicted variances in wave number space (right) are 0% (no skill) except for ( k , l ) = ( 0 , 1 ) , ( 0 , 2 ) , ' ' ' ,(0,7), which accounts for o 2 - 16% of the AC variance. (b) S-~N transmissions between two eastward traveling ships. (c) Combined W-~E and S-~N transmissions, accounting for 32% of the AC variance and giving a slight pattern resemblance to the true ocean. (d) Combined W-~E, S-~N, SW-+NE, and SE-~NW transmissions, accounting for 67% of the variance and giving some resemblance to the true ocean. (Adapted from Cornuelle et al., 1989.)

104 3. I N T E G R A L VS. P O I N T DATA: I N F O R M A T I O N C O N T E N T IN A 1-D EXAMPLE Because of the sampling properties of the tomographic measurements, when travel time data are used to estimate the sound speed or current in a volume of ocean, the uncertainty in the solution is generally local in wave number space, rather than in physical space. This is in contrast to the uncertainty in estimates made from independent point measurements, which tend to have errors localized in physical space. The non-locality can be characterized in least-squares estimation by examining the output model parameter uncertainty ('error') covariance. For model parameters localized in physical space (such as boxes or finite elements), significant off-diagonal terms in the output uncertainty covariance matrix P (see Appendix) show that the uncertainty at one point is related to the uncertainty at another point. These correlations arise in estimates made from tomographic data; the error at one point along a ray path tends to be anti-correlated with the error at other points on the ray path, because the sum of the points is known. (Off-diagonal correlations can arise in satellite altimetry measurements as well, because the orbit may contaminate many measurements along the ground track with approximately the same error.) It is common practice in objective mapping to display only the diagonal of the physical space uncertainty covariances, which is what is plotted in the 'error map' used in many papers (Bretherton, Davis, and Fandry, 1973). Assimilation methods that insert values at points in the model, such as 'nudging' (e.g., Malanotte-Rizzoli and Holland, 1986), sometimes use the local error bars from the objective map to adjust the strength of the data constraint. We will show below that although neglecting the off-diagonal components of the uncertainty covariance of the estimates is benign for point measurements with local covariances, it can be dangerous when the off-diagonal terms are significant, as when tomographic data are used. Doing the data insertion by some form of sequential optimal interpolation (OI), approximating the Kalman filter, is less dangerous, but the treatment of the model parameter uncertainty covariance between steps can still both destroy information and give misleading results when used with non-local data. In order to highlight the contrast between tomographic and point measurements, we have chosen a very simple model problem for pedagogical clarity. We use a I-D, periodic realm of 20 piecewise-linear finite elements, each with identical widths and randomly chosen temperatures, which are assumed to be exactly convertible to sound speeds. The unknowns (model parameters) are the temperatures at each of the points (and by interpolation, the temperature of the entire interval). The model parameters are assumed to be independent, identical, normally distributed random variables with zero mean and independent, equal (unit) variances. The initial model parameter uncertainty covariance matrix (P) is therefore diagonal with 1.0 on the diagonal. The 'error map' for this a priori state is a uniform, unit error variance in physical space, which is conveniently the same as the diagonal of the matrix plotted as a function of the location index. If we measure the value of one of these finite elements, say element 5, by sampling it at its center without noise, the output uncertainty variance is now zero for the sampled point (Fig. 4). If we instead measure the integral across all the points (as a representation of a tomographic measurement), the uncertainty variance is instead reduced only slightly at all the points (Fig. 4). This obscures the fact that something very specific has been learned, just as specific as for the point measurement. In fact, the sum of uncertainty variances (the trace of the covariance) has been reduced by the same amount (1 unit) in each case, to 19 from 20.

105 The full uncertainty covariance matrix gives complete information about the character of the remaining uncertainty. The first and fifth columns of the covariance are plotted in Fig. 5 for both point and averaged measurements. (This is the covariance between the error at points 1 or 5 and the error at all the rest of physical space.) The errors at the points missed by the point measurement show no correlation with the errors at their neighbors, while the fifth point (the site of the measurement) shows no error. In contrast, the average shows identical behavior at all points (Fig. 5). The variance is reduced slightly, as shown in Fig. 4, but there is a uniform negative correlation between the error at any point and the error at all other points, representing the knowledge of the average. That is, if the value at the one of the points is higher than estimated, the values at all the other points will tend to be lower than estimated, in order to keep the average the same.

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106 The similarity in the amount of information available from either type of measurement is most easily seen by decomposing the output error covariance P into eigenvalues and eigenvectors. ^

= UAU T where U is the matrix with the eigenvectors of P as its columns, and A is the diagonal matrix of eigenvalues. In both cases, there is one zero eigenvalue, representing the component of the model that is known exactly, and 19 eigenvectors with eigenvalue 1. The null space in each case is degenerate, and can be represented by many different sets of basis functions, but the eigenvector that is known is either a spike at the measured point or a uniform level across all points (Fig. 6). In either case, only 19 unknowns are left to be determined. Representing the estimate generated from the average as 20 point measurements with equal values and independent error bars with values as in Fig. 4, amounts to dropping the off-diagonal terms in the error covariance. Although the diagonalized covariance has the same trace as the exact covariance, and thus has a similar amount of information as measured using the trace, the eigenvectors of the diagonalized covariance are 20 unknown functions, each with variance 19/20 of the original. This is a considerably different state of knowledge than in the original error covariance, because 20 unknowns remain to be determined, and the slight reduction in their uncertainty is not very useful.

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107 4. I N T E G R A L VS. P O I N T DATA: EXAMPLE

E S T I M A T I O N IN A T I M E - D E P E N D E N T 1-D

Suppose now that the measurements are repeated, and that we wish to combine all the measurements to obtain an improved estimate of the field, assuming that we have a model for the dynamical evolution of the field. The time-dependent least-squares estimation problem can either be solved sequentially (the Kalman filter), or globally, giving identical results when fully optimal methods are used. In the examples to follow, the estimates have been described as the result of sequential estimation, since that is similar to many approximate schemes. A primitive dynamical example that includes advection demonstrates the effect of error propagation in a data assimilation scheme using non-local data. The 20-point domain was retained from the earlier example, but is now assumed to be periodic, with periodicity 20, so that the twenty-first point is the first point. The dynamics were uniform, constant advection of temperature as a passive tracer: every time step, the field shifted one place to the left, so 21 time steps completely rotate the domain back to the starting point. The a priori information state was again independent, unit variances for each point. It is obvious that 20 sequential, perfect point measurements, one per time step at a single location, would completely determine the field if the dynamics were modeled perfectly. On the other hand, 20 sequential, perfect 20-point averages would only determine the average of the domain, doing no better than a single perfect measurement of the average. If the measurements have error bars, then repeated averaged measurements increase the precision of the estimate of the average, but give no new information about the shorter scale structures, since every measurement is the same. Instead of 20-point averages, the tomographic data were therefore taken to be 5-point averages, which do not trivially repeat in the periodic domain and which yield some information on smaller scale structures. To avoid singularities due to perfect measurements, both point measurements and averages were assumed to be contaminated by noise. The point measurements were assumed to have an uncertainty variance of 0.1, representing measurement error. Because each tomographic measurement averages 5 points, the uncertainty variance was set at 0.02, one-fifth of the error variance assumed for the point measurements, keeping the signal-to-noise variance ratio (SNR) equal between the averaged and point measurements. (This is a relatively arbitrary choice, but is simplest for comparisons, because a single measurement of either type produces the same decrease in model uncertainty.) We used a Kalman filter (see Appendix) to combine the measurements, and compared the performance after all the measurements had been used. In this example, the tomographic measurements show significantly larger point error bars than the point measurements, even though the estimate used optimal error propagation (Fig. 7). Columns 1 and 5 of the output error covariance matrix (Fig. 8) show the contrast in structure between the point measurements and the tomography. The eigenvalue spectra (Fig. 9) show that the averaged data have a varying information content in spectral space. The 20-point mean is well determined (the smallest eigenvalue). The eigenfunctions corresponding to the next two smallest eigenvalues resemble sine and cosine functions with one cycle in the 20-point domain (Fig. 10). The eigenfunctions corresponding to the next two smallest eigenvalues similarly resemble sine and cosine functions with two cycles in the 20-point domain (Fig. 10). The 4 vectors in the null space, which each average to zero in any 5-point domain

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and are thus completely undetermined by the tomography, appear to be more complex, but correspond to aliased samples from sine and cosine functions that have either one or two cycles over the 5-point domain (Fig. 10). In spectral terms, the averaged measurements selectively determine the large-scale components better than the short-scale components, while the point measurements are equally sensitive to all scales. For geophysical systems with red signal spectra, this suggests that the averaged measurements may perform better than in this simulation, which assumes a white spectrum. Even in this simple example, the large-scale components of the field (i.e., the five eigenvectors shown in Fig. 10 associated with the five smallest eigenvalues in Fig. 9) are better determined by the tomographic measurements than by the point measurements. This is not entirely obvious from Fig. 9, because the eigenvectors of the 20 point measurements are 20 delta functions that are localized in physical space, while the eigenvectors of the tomographic measurements are localized in spectral space, and so the eigenvalues (variances) plotted in the figure do not correspond to similar eigenvectors for the two cases.

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Eigenvector 1 for the 20 5-point averages is related to the mean, for example, while eigenvector 1 for the 20 point measurements is a delta function at location 1. The variance of the mean deduced from the 20 point measurements is simply 1/20 of the 20 identical variances (eigenvalues) of the 20 eigenvectors. Because the eigenvectors are normalized to have unit length, the elements of eigenvector 1 for the 20 5-point averages all have magnitude 1/n/~--~, as can be seen in Fig. 10. The variance of the mean is then 1/20 of the variance (eigenvalue) of the eigenvector. The ratios of the eigenvalues in Fig. 9 therefore accurately reflect the ratios of the error variances of the estimates made using tomographic measurements and point measurements, showing that the large-scale components are better determined by the tomographic data. (Another way of looking at this is to note that because this simple example is homogeneous, sines and cosines are also eigenvectors of the point measurement covariance.)

110 A somewhat different question is how well the tomographic and point measurements resolve the detailed spatial structure of the field, rather than just the large-scale c o m ponents. The tomographic measurements were seen in Fig. 7 to have significantly larger point error bars than the point measurements for the case with 20 measurements. In that case, however, the tomographic measurements had more null space vectors than the point measurements. To give the tomographic and regular measurements similar numbers of null space vectors, we repeated the simulations using only 16 data in each case. The null space for the point measurements has four elements, representing the four points not measured, while the null space for the tomography remains the same. The trace of the output error covariances (the total uncertainty variance after the inverses) were 9.2 and 5.5 for the averaged and point measurements, respectively. The point measurements thus do better in resolving the detailed spatial structure of the field than the averaged measurements, when the unknown field has a white spectrum and when both types of data have equal SNR. The averaged measurements are most sensitive to the larger scales, as discussed above, and have to be differenced in order to resolve finer scales. The difference of two large numbers is easily contaminated by random noise. If the calculations are repeated giving the tomography data 0.1 of their original variances, (tomographic SNR = 10 time point measurement SNR), the trace of the output error covariance for the tomography is now 5.1, so the greater measurement precision has greatly improved the ability of the tomographic data to resolve the detailed spatial structure. This result has been shown before in a number of places, (e.g., Cornuelle et al., 1985), but rarely in such a simple example. The performance of averaged measurements is equal to that of point measurements (as measured by the trace) as long as the averaged measurements do not overlap. The redundant data generated by overlapping averages reduce the calculated performance, just as repeated point sampling in the same place would.

4.1. Approximate sequential methods with advection The issue that remains is to compare measurements fed in sequentially without keeping the off-diagonal model error covariance elements. We again use the example of 20 sequential measurements over 20 time steps, but approximating the forecast of the model parameter uncertainty covariance matrix (Appendix, equation A7). This is meant to model sequential optimal interpolation methods, where only a simplified version of the model parameter uncertainty is propagated between steps. If only the diagonal of the covariance is kept, then the total expected error for the tomographic measurements changes only slightly, increasing by about 3% compared to the exact (full covariance) result, while the total error for the point measurements is unchanged. The diagonal-only Kalman filter is still optimal for the point measurements because the covariance is completely local, and simple advection does not produce off-diagonal terms during the evolution of the model. Because the total expected error changes only slightly for the tomography, it is tempting to assume that the loss of off-diagonal terms has only slightly degraded the estimates. Unfortunately, a look at the eigenvalue spectra from a sequential, diagonal-only estimation with 20 tomographic or point measurements (Fig. 11) shows that for the tomography, the approximate sequential interpolation arrived at a vastly different (and incorrect) state of information than the exact Kalman filter. The spectrum of eigenvalues for the tomography no longer shows the large-scale components as being best determined, and the model state apparently includes information about all components (no zero eigenvalues, so no

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null space). This is in contrast to the point measurements, whose eigenvalue spectrum is unchanged. Because of the approximate error propagation, the error covariance is no longer a good figure of merit, and the true performance of the simplified method can best be evaluated by Monte Carlo methods, simulating an ensemble of true fields and looking at the error in the reconstruction. We only wish to point out that the diagonal-only method remains optimal for the point measurements, while becoming severely suboptimal for the tomographic measurements, but in a subtle way that could easily be overlooked. This contrast is heightened by the the trivial dynamics chosen for the simulations. Realistic dynamics, such as quasi-geostrophic flow in three dimensions, generally creates non-local covariances, even from point sampling, so that the sequential optimal interpolation would degrade the point measurements somewhat. On the other hand, for short time scales and normal advection velocities, the point measurement information will remain much more local than tomographic information, and so is more compatible with local approximations. Conversely, if the dynamical model is built in spectral space, so the horizontal basis functions are sines and cosines, then the tomographic data is much more local than point measurements, which are sensitive to all scales. Most modern data assimilation methods do not completely ignore off-diagonal terms in the model parameter uncertainty covariance matrix, however, even for point measurements. It is therefore natural to ask how well other possible approximations to the uncertainty covariance matrix perform. Perhaps the simplest class of approximations are ones in which varying numbers of diagonal bands of off-diagonal elements are retained, while the remaining elements are set to zero. Plotting the eigenvalue spectra as a function of the number of bands retained (Fig. 12) shows that retaining one off-diagonal band, in addition to the diagonal elements, results in the reduction of a single eigenvalue, corresponding to the mean. Little further change in the spectra is evident as additional off-diagonal bands are retained, until 15 off-diagonal bands are included. At that point the spectra begin to

112 resemble the spectrum obtained when the full matrix is used. For the simple example considered here, retaining additional off-diagonal bands of the uncertainty covariance matrix is therefore not a particularly effective approximation, as nearly the complete uncertainty covariance matrix needs to be retained before the results are similar to those obtained using the full matrix. The decomposition of the error covariance into eigenvectors suggests a more natural approximation for sequential assimilation, however, in which only the components of the model error covariance with large eigenvalues are propagated by the model. In the case of a single measurement, the savings are small, because 19 out of 20 vectors need to be propagated, but with more complete observations, the savings could be larger.

4.2. Separating the inverse from the assimilation Even the approximate method used in the previous example kept the inverse as part of the update of the model. Some older assimilation methods invert the measurements and then blend in the results as pseudo-point measurements with error bars. This approach is impossible when using the averaged measurements, because the uncertainty of the output estimate is not local, and so the pointwise error bars cannot express the infinite (but correlated) uncertainty imposed on the solution by the elements of the null space. Even if the data are inverted outside the model, it is necessary to use the model state as the reference; otherwise the inversion procedure will tend to pull the model toward whatever reference state is used. This problem of infinities is avoided in exact sequential optimal estimation and the Kalman filter, because the data are merged into the model directly, inverting for corrections to the current best forecast of the model parameters, and the a priori error bars describe the model's current state of knowledge.

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113 5. D I S C U S S I O N These simple examples were constructed to emphasize the spectral nature of tomographic measurements, in contrast to the local nature of point measurements. This is closely related to the projection-slice theorem (Kak and Slaney, 1988), but the irregularity of the ray paths in ocean acoustic tomography destroys the simplest spectral relationships, concealing the spectral nature of the sampling. The example reported in Figs. 9 and 10 shows that the error covariance matrix of the averaged measurements has sines and cosines as eigenvectors, while the error covariance matrix of the point measurements is diagonal with delta functions as one set of eigenvectors. For an unknown field with a white spectrum, and data with equal signal-to-noise ratios, non-overlapping averaged measurements increase our knowledge of the unknown field by the same amount as the same number of point measurements, but the spectral content of that knowledge is very different. Because the averaged measurements determine the lower wave numbers better than the higher wave numbers, they have advantages if the spectrum of the unknown field is red. Determining high wave number information from the averaged measurements is more difficult, unless the measurement errors are sufficiently small to make differencing of the integral measurements practical. The relative utility of tomographic measurements and point measurements thus depends strongly on the goal of the measurement program. The non-local nature of the averaged measurements also makes it difficult to use approximations to the Kalman filter in dynamical models with local parameterization. Conversely, the averaged measurements can be used efficiently by an approximate Kalman filter based on spectral functions. Acknowledgments. This work was supported by the Office of Naval Research (ONR Contracts N00014-93-1-0461 and N00014-94-1-0573) and by the Strategic Environmental Research and Development Program through the Advanced Research Projects Agency (ARPA Grant MDA972-93-1-0003).

APPENDIX The form of least-squares estimation used here assumes that the expected value of the model parameter vector has been removed, so <m> = 0, and that an initial guess exists for the covariance of the uncertainty around the expected value, <mmT> = P. The data are related to the model parameter vector by a linear relation, d = Gm + n

(AI)

where n is the random noise contaminating the measurements. Any known expected value of the noise is assumed to have been removed, so - 0 , and the noise is assumed to have covariance = N and to be uncorrelated with the model parameters. This relation can be inverted to obtain an estimate of the model parameters, rh = PG T (GPG T + N)-ld and the expected uncertainty in this estimate is

(A2)

114 = P-

p G T ( G P G r + N)-lGp

(A3)

If dynamics are available to forecast the model parameter vector between time steps, so that mt+l - A m / + q

(A4)

where A is the transition matrix, and q is the uncertainty in the forecast due to errors in the dynamics (with zero mean and uncertainty covariance Q = ). The Kalman filter performs a sequential cycle, correcting the starting guess by inverting the differences between the observations and the predicted data, ~rlt - mt + Pt GT (GPt Pt

--

GT

+

N)-l(dt - Gmt)

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(A5a) (A5b)

and forecasting the estimate and covariance to the start of the next step: mt +1 - Atilt Pt+z = APt AT + Q

(A7)

This cycle then repeats.

REFERENCES

Aki, K., and P. Richards, 1980. Quantitative Seismology, Theory and Methods. 2 Vols. W.H. Freeman and Co. Bretherton, F. P., R. E. Davis, and C. B. Fandry, 1976. A technique for objective analysis and design of oceanographic experiments applied to MODE-73. Deep Sea Res., 23, 559-582. Chiu, C.-S., and Y. Desaubies, 1987. A planetary wave analysis using the acoustic and conventional arrays in the 1981 Ocean Tomography Experiment. J. Phys. Oceanogr., 17, 1270-1287. Chiu, C.-S., J. F. Lynch, and O. M. Johannessen, 1987. Tomographic resolution of mesoscale eddies in the marginal ice zone: A preliminary study. J. Geophys. Res., 92, 6886- 6902. Cornuelle, B.D., 1990. Practical aspects of ocean acoustic tomography. In: Oceano-

graphic and geophysical tomography: Proc. 50th Les Houches Ecole d'Ete de Physique Theorique and NATO ASI, Y. Desaubies, A. Tarantola, and J. Zinn-Justin, eds., Elsevier Science Publishers, 441-463. Cornuelle, B.D., and B.M. Howe, 1987. High spatial resolution in vertical slice ocean acoustic tomography. J. Geophys. Res., 92, 11,680-11,692. Cornuelle, B.D., W.H. Munk, and P.F. Worcester, 1989. Ocean acoustic tomography from ships. J. Geophys. Res., 94, 6232-6250. Cornuelle, B.D., P.F. Worcester, J.A. Hildebrand, W.S. Hodgkiss Jr., T.F. Duda, J. Boyd, B.M. Howe, J.A. Mercer and R.C. Spindel, 1993. Ocean acoustic tomography at 1000kin range using wavefronts measured with a large-aperture vertical array. J. Geophys. Res., 98, 16,365-16,377.

115 Cornuelle, B.D., C. Wunsch, D. Behringer, T.G. Birdsall, M.G. Brown, R. Heinmiller, R.A. Knox, K. Metzger, W.H. Munk, J.L. Spiesberger, R.C. Spindel, D.C. Webb and P.F. Worcester, 1985. Tomographic maps of the ocean mesoscale, 1: Pure acoustics. J. Phys. Oceanogr., 15, 133-152. Fukumori, I., and P. Malanotte-Rizzoli, 1995. An approximate Kalman filter for ocean data assimilation: An example with an idealized Gulf Stream model. J. Geophys. Res., 100, 6777-6793. Howe, B.M., P.F. Worcester and R.C. Spindel, 1987. Ocean acoustic tomography: Mesoscale velocity. J. Geophys. Res., 92, 3785-3805. Kak, A.C., and M. Slaney, 1988. Principles of Computerized Tomographic Imaging. IEEE Press, New York. Malanotte-Rizzoli, P., and W.R. Holland, 1986. Data constraints applied to models of the ocean general circulation, Part I: the steady case. J. Phys. Oceanogr., 16, 1665-1687. Munk, W., P.F. Worcester, and C. Wunsch, 1995. Ocean Acoustic Tomography. Cambridge Univ. Press, Cambridge. Munk, W., and C. Wunsch, 1979. Ocean acoustic tomography: A scheme for large scale monitoring. Deep-Sea Res., 26, 123-161. Munk, W., and C. Wunsch, 1982. Up/down resolution in ocean acoustic tomography. Deep-Sea Res., 29, 1415-1436. Ocean Tomography Group, 1982. A demonstration of ocean acoustic tomography. Nature, 299, 121-125. SchrlSter, J., and C. Wunsch, 1986. Solution of nonlinear finite difference ocean models by optimization methods with sensitivity and observational strategy analysis. J. Phys. Oceanogr., 16, 1855-1874. Sheinbaum, J., 1989. Assimilation of Oceanographic Data in Numerical Models. Ph.D. Thesis, Univ. of Oxford, Oxford, England, 156 pp. Spiesberger, J.L., and K. Metzger Jr., 1991. Basin-scale tomography: A new tool for studying weather and climate. J. Geophys. Res., 96, 4869-4889. Worcester, P.F., B.D. Cornuelle, and R.C. Spindel, 1991. A review of ocean acoustic tomography: 1987-1990. Reviews of Geophysics, Supplement, U.S. National Report to the International Union of Geodesy and Geophysics 1987-1990, 557-570. Wunsch, C., 1990. Using data with models: Ill-posed problems. In: Oceanographic and

geophysical tomography: Proc. 50th Les Houches Ecole d'Ete de Physique Theorique and NATO ASI, Y. Desaubies, A. Tarantola, and J. Zinn-Justin, eds., Elsevier Science Publishers, 203-248.

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Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

119

Combining Data and a Global Primitive Equation Ocean General Circulation Model Using the Adjoint Method Z. Sirkes a, E. Tziperman b and W. C. Thacker c aCenter for Marine Sciences, The University of Southern Mississippi, Stennis Space Center, MS 39529-5005 bEnvironmental Sciences, The Weizmann Institute of Science, Rehovot 76100, Israel CAtlantic Oceanographic and Meteorological Laboratory, Miami FL 33149 USA

Abstract

A Primitive Equation Ocean General Circulation Model (PE OGCM) in aglobal configuration similar to that used in coupled ocean-atmosphere models is fitted to climatological data using the adjoint method. The ultimate objective is the use of data assimilation for the improvement of the ocean component of coupled models, and for the calculation of initial conditions for initializing coupled model integrations. It is argued that oceanic models that are used for coupled climate studies are an especially appropriate target for data assimilation using the adjoint method. It is demonstrated that a successful assimilating of data into a fully complex PE OGCM critically depends on a very careful choice of the surface boundary condition formulation, on the optimization problem formulation, and on the initial guess for the optimization solution. The use of restoring rather than fixed surface-flux boundary conditions for the temperature seems to result in significantly improved model results as compared with previous studies using fixed surface-flux boundary conditions. The convergence of the optimization seems very sensitive to the cost formulation in a PE model, and a successful cost formulation is discussed and demonstrated. Finally, the use of simple, suboptimal, assimilation schemes for obtaining an initial guess for the adjoint optimization is advocated and demonstrated.

Introduction Oceanographic data assimilation is a rapidly evolving field with very diverse objectives and hence many different possible methodologies to address these objectives. Two of the main purposes of combining ocean models and data are the improvement of ocean models, and the calculation of an optimal estimate of the oceanic state, based on both model dynamics and the available data (Malanotte-Rizzoli and Tziperman, Chapter 1 of this book). These two objectives are very general, and apply to a wide spectrum of

120

models, from high resolution to coarse, and a variety of uses can be found for the optimal ocean state estimated by data assimilation or inverse studies. One class of ocean models for which these two objectives are especially relevant and important consists of the ocean models used in coupled ocean-atmosphere model studies. Model improvement in this context refers to the need to improve these ocean models, including their sub-grid scale parameterizations, their poorly known internal parameters such as various eddy coefficients, the surface boundary forcing fields which are often known with large uncertainty, etc. Data assimilation may be used to find those model parameters that result in a better fit of the model results to observations, and therefore in an improved performance of the model when run within a coupled ocean-atmosphere model. The state estimation problem in this context refers to the need to find "optimal" initial conditions for coupled model climate simulations. Such initial conditions, based on both the model dynamics and the oceanic observations, would hopefully result in better climate forecasts. The combination of OGCMs and oceanographic data for the above purposes can be formulated as an optimization problem. Such an optimization would search for a set of model parameters and for an optimal ocean state which together satisfy the model equations and fit the available data as well as possible. This is done by formulating a cost function to be minimized, which measures the degree to which the model equations are satisfied, as well as the distance to the data. The minimization of this cost function is a most complex nonlinear optimization problem, requiring very efficient methodologies. A common solution for such large scale optimization problems is to use gradient-based iterative algorithms such as the conjugate gradient (c-g) algorithm. The minimization is carried out in a huge parameter space comprising of all model parameters and of the 3D model initial conditions for the temperature, salinity and velocities. The efficient estimation of the gradient of the cost function with respect to these many parameters is a crucial part of the methodology. This is done using a numerical model based on the adjoint equations of the original model equations. Thus this optimization approach is often referred to as the "adjoint method" (e.g. [1]-[4]). The adjoint method is very efficient compared to other ways of estimating the gradient of the cost function, but is still computationally intensive. Given the power of todays computers, the adjoint method is therefore adequate primarily for medium to coarse resolution models. Due to the very high computational cost of coupled models, they are also presently limited to a fairly coarse resolution. Clearly the data assimilation problems related to coupled models are therefore an excellent match to the capabilities of the adjoint method. Moreover, it may be expected that as available computers become more powerful and allow higher resolution coupled ocean-atmosphere models, the new computational resources will also enable the use of such higher resolution models with the adjoint method. We would like to present here a step towards the ultimate goal of using the adjoint method with the ocean component of coupled ocean-atmosphere models. We still cannot claim to having improved the model or having produced optimal initial conditions, but hopefully have made some progress. Inverting a three dimensional GCM (that is, assimilating data into a three dimensional GCM using an optimization approach) is basically a very technical problem, yet we will demonstrate here that a successful application of

121 the adjoint method to this problem requires a very good understanding of both the ocean circulation dynamics and of the technical issues involved. In fact, we try to emphasize here precisely those issues that require the understanding of the dynamics in order to formulate and successfully solve the inverse problem of combining ocean GCMs and data. The use of a fairly coarse resolution model here implies, of course, that we do not attempt here to produce a highly realistic simulation of the oceanic state. Rather, the above objectives are all related to the ultimate improvement of coupled ocean-atmosphere model simulations whose main tool is similar coarse-resolution models. Although the objective of combining 3D ocean climate models with data is of obvious interest, it is surprising to realize that there have only been very few efforts so far trying to apply the adjoint method to full complexity 3D ocean models. Tziperman et al. [5, 6] have examined the methodology using simulated data and then real North Atlantic data; Marotzke [7], and Marotzke and Wunsch [8] (hence after MW93) have considerably improved on the methodology and analyzed a North Atlantic model; Bergamasco et al. [9] used the adjoint method in the Mediterranean Sea with a full P E model, and Thacker and Raghunath [10] have examined some of the technical challenges involved in inverting a P E model. This relatively small number of studies has a simple reason: the technical difficulties in constructing an adjoint model of a full GCM are almost overwhelming. Fortunately, this difficult task was successfully tackled by Long, Huang and Thacker [11], who have generously made the results of their efforts available to others and the present study is a direct outcome of their efforts. (The adjoint code of [11] was modified here to be consistent with the global configuration and eddy parameterizations used in this study, so that the adjoint code used here is the precise adjoint of our finite difference global model). All of the above works use the the model equations as "hard" constraints. This implies that errors in the model equations are not considered explicitly. It is worthwhile noting that adjoint models can also be used for different data assimilation approaches than used here [12, 13]. Within the framework of using climate models with the adjoint method, this study has three specific objectives. First, we would like to investigate the issue of model formulation for such optimization problems, and in particular the surface boundary condition formulation. There are two commonly used surface boundary condition formulations. One is fixed-flux conditions, in which the heat flux is specified independently of the model SST. The second is restoring boundary conditions in which the heat flux is calculated by restoring the model SST to a specified temperature distribution (possibly the observed SST). Previous applications of the adjoint method to 3D GCMs used the fixed-flux formulation in an effort to calculate the surface fluxes that results in a good fit to the temperature observations. However, the optimal solution was characterized by large discrepancies, of up to 6 degrees, with the observed SST [6, 8]. Tziperman et al. [6] suggested that this discrepancy is the result of using flux boundary conditions, rather than restoring conditions that are normally used in ocean modeling. MW93 [8] suggested that this discrepancy might be a result of the use of a steady model which lacks the large seasonal signal in the SST, and that this problem might be resolved using a seasonal model. We explain and demonstrate below that using restoring boundary conditions, is better motivated physically as well as seems to eliminate the large SST discrepancies observed in previous optimization studies (section 4.2).

122

Our second objective is to examine various possibilities for the formulation of a cost function measuring the success of the optimization problem and their influence on the success of the optimization. Finally, we shall discuss and demonstrate methods for increasing the efficiency of the adjoint method by initializing the gradient based optimization with solutions obtained using simpler, sub-optimal, assimilation methodologies. Ocean models presently used in coupled ocean-atmosphere studies are coarse, noneddy-resolving, yet usually include the seasonal cycle. Faithful to our philosophy of trying to use the same models for data assimilation studies we should have used a seasonal model, and indeed work is underway to do just that. In this present work, however, we have made several steps forward going from basin to global scale, and from a simplified 3D GCM to a full P E model. These steps turned out to involve a sufficient number of new challenges, so we have decided to maintain the steady state assumption, and progress to a global PE seasonal model only at a following stage. We expect that the lessons learned from the steady state problem will be very useful at the next stage, as time dependent, presumably seasonal, models are inverted. In the following sections we describe the model and data used in this study (section 2), discuss in detail the formulation of the optimization problem (section 3). We then present the results of the model runs carried out here (section 4), and finally discuss the lessons to be learned for future work and conclude in section 5.

Model and data Ultimately, our objective is to use data to improve ocean models used in climate simulations; therefore the model used for the optimization study needs to be the same model that can be run independently in a simulation mode. This determines many of our choices concerning the model and surface boundary condition formulation. We use the GFDL PE model, derived from the model of Bryan [14], with later modifications by Semtner [15] and Cox [16], in a coarse resolution global configuration similar to that of Bryan and Lewis [17], with the main difference being that the Arctic ocean is not included in our model. The model's geometry and resolution are also similar to those presently used by coupled ocean-atmosphere models. The model's geometry is shown in Fig. la. The model has 12 vertical levels, with the eddy mixing coefficients for the temperature and salinity varying with depth according to the scheme proposed by Bryan and Lewis [17]. The mixing coefficients for the temperature and salinity are given by Ag(k) = rH(k)2 • 107cm2/sec in the horizontal direction, and Ay(k) = ry(k) • 0.305cm2/sec in the vertical direction, where rH(k) and ry(k) are given in Table 1. The momentum mixing coefficients are 25 • l0 s, and 50 cm2/sec in the horizontal and vertical directions correspondingly. The choice of surface boundary condition formulation turns out to be a crucial factor in the optimization problem we have set out to solve here. We explain and demonstrate below that using restoring boundary conditions, rather than the fixed-flux formulation used previously is better motivated physically as well as eliminates the large SST discrepancy observed in previous optimization studies (section 4.2). Under restoring boundary conditions the model is driven with an implied air-sea heat flux H ssT that is calculated

123 Table 1 Model levels and horizontal and vertical mixin$ coefficients. level depth horizontal vertical (k) (m) mixing factor (rH) mixing factor ( r v ) 1 25.45 1.0000 1.000 2 85.10 0.8923 1.003 3 169.50 0.7794 1.007 4 295.25 0.6620 1.015 5 482.80 0.5475 1.028 6 754.60 0.4482 1.053 7 1130.65 0.3733 1.109 8 1622.40 0.3218 1.288 9 2228.35 0.2853 2.904 10 2934.75 0.2553 4.048 11 3720.90 0.2274 4.193 12 4565.55 0.2000 4.244

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Figure 1: The steady state model solution for the surface temperature obtained without the use of interior data: (a) Model geometry and the sea surface temperature at steady state. Contour intervals are 2.5~ Negative areas are dotted. (b) Total meridional heat flux for the global ocean (solid), for the Atlantic ocean (dash), overturning circulation contribution to the meridional heat flux (short-dash) and gyre contribution of the meridional heat flux (dot). (c) North Atlantic meridional stream function. (d) Temperature section through the North Atlantic model sector solution.

124 at time step n from the model upper level temperature, Tinj,k=l, and the temperature data at this depth, Tdj,k=l, (where the indices i, j denote horizontal grid point location, and k vertical level) as follows [_[S ST,n

llij

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[E-P]i j

' = 7SAzl(Sdj,k=

n

1 -- S i , j , k = l ) / S o ,

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where So is a constant reference salinity used to convert the virtual salt flux to an implied fresh water flux. In our runs, where Azl = 50m, we use ,),T __ 1/30days-1 and 7s = 1/120days -1. Following Hirst and Cai [21], we restore our model surface temperature and salinity to (-1.9 ~ C, 34.84ppt) in the North Atlantic portion of our model, at two grid points only, located at (68.9N; 7.5W and 11.25W) using restoring coefficients that are 10 times larger than those used elsewhere. This results in an improved simulation of the NADW formation and spreading. Finally, as the Mediterranean Sea is not included in our model, a sponge layer is used at two grid points near the Mediterranean outflow region, in which model temperature and salinity are restored to the Levitus data at all depths. The steady state model results obtained by integrating the model for about 1500 years (without data assimilation) are shown in Fig. 1 Depicted are the surface temperature field (Fig. la), the global and North Atlantic meridional heat flux (Fig. lb, see [17, 18] for the meridional heat flux decomposition used here) the North Atlantic overturning circulation (Fig. lc) and a temperaturc section through the North Atlantic ocean (Fig. ld). Note that the overturning circulation is about 16Sv at 30N, close to the commonly assumed value of about 18 Sv there. This is due to the strong restoring at the two northern surface grid points mentioned above, without which the overturning at 30N reduces by about

25%. The "data" used in this study are the annually averaged temperature and salinity analysis of Levitus [22]; the annually averaged climatologies of heat flux from Esbensen and Kushnir [20], of fresh water flux ([E-P]) from Baumgartner and Reichel [23] and of winds from Hellerman and Rosenstein [24]. All of these are, in fact, gridded analyses rather than raw data. While it is clearly more convenient to use such analyses, future applications of the adjoint method may use the raw data instead. The use of the raw observations, together with detailed error information, may result in more reliable results and better error statistics for the model solution than is possible here.

3

Optimization Problem

One of the main lessons that have been learned over the past few years while trying to combine 3D ocean models and data, is that the correct formulation of the inverse problem is of crucial importance to the success of the optimization. Much thought and understanding of the dynamics should enter the process of posing the optimization problem. This

125 process includes the choice of a cost function that measures the optimization success and that needs to be minimized, the specification of the initial guess for the optimization solution from which the iterative minimization should begin, and the choice of control variables which are varied in the optimization. We now examine each of these steps in some detail. The results of an optimization formulated according to the ideas presented in this section are shown and discussed in section 4.

3.1

Cost Function

Once the data and model have been specified, the next stage in the formulation of the inverse problem is to specify a measure for the success of the optimization, i.e., the cost function to be minimized. The cost function measures both the fit of the model results to the data, and the degree to which the dynamical constraints are satisfied. A given dynamical constraint can be formulated in many different ways. It has been shown for simpler GCMs that the ability of the optimization to minimize the cost function critically depends on the precise form of the cost function [7]. We find that a Primitive Equations model is even more sensitive to the precise cost formulation. Let us consider the various dynamical and data constraints and the possibilities of specifying them within a cost function to be minimized. Begin from the dynamical constraints, which in our case are the requirement for the solution to be as close as possible to a steady state of the model equations. This condition may be obtained by minimizing a measure of the deviation of the model from a steady state solution. Tziperman and Thacker [4] and then Tziperman et al [5, 6] have suggested to minimize the finite difference form of (OT/Ot) 2, obtained by stepping the model from the initial conditions T ~ ~ a single time step to T ~ 1, and minimizing the sum of terms such as ( T ~ 1 - T ~ ~ 2. This seems reasonable, and worked for a QG model [4], yet encountered major difficulties when applied to a 3D model [5, 6]. Marotzke [7], in an important contribution, suggested to use instead (T[~ N - TD~~ 2, such that the model integration time N A t corresponds to the time scale of physically relevant processes in the model (e.g. O(10 years) for a problem involving the upper ocean, longer time scales for the deeper ocean, etc). Marotzke's suggestion resulted in most significantly improved convergence of the optimization, ~s presented in both Marotzke [7] and MW93 [8]. A useful perspective for evaluating the usefulness of a given formulation of the dynamical constraints in the cost function is the conditioning of the resulting optimization problem. The cost surface in parameter space near the cost minimum is of a bowl shape. The bowl may be nearly flat in some directions and very steep in others. If such a discrepancy occurs, the optimization is said to be ill conditioned [25]. An ill conditioned optimization may stall and not progress towards the minimum even after many iterations of the minimization algorithm. If the steepness of the cost surface is nearly even in all directions, the optimization is said to be well conditioned, and the solution is found within a few iterations. The conditioning issue was discussed in detail in Tziperman et al. [6], where the analysis pointed out to some possible ways of improving the conditioning using various formulations for the cost function. The conditioning of the steady penalties of temperature and salinity for the PE model used here is examined in section 3.1.1. For a primitive equation model such as used here, there are additional considerations con-

126

cerning the form of the dynamical constraints for the velocity field which turns out to be most crucial for the success of the optimization, and these are discussed in section 3.1.2. Finally, the cost formulation for the penalties requiring the model heat flux (and fresh water flux) to be close to the observations is discussed in section 3.1.3. 3.1.1

D y n a m i c a l c o n s t r a i n t s for t e m p e r a t u r e

and salinity

In order to evaluate the conditioning of the dynamical constraints, we have plotted them together with the data penalties along a somewhat arbitrary section between two points in parameter space. The two points correspond to two choices for the 3D temperature, salinity, velocity and stream function initial conditions. The two points were obtained by running a few iterations of the optimization algorithm once from the steady state solution and once from a robust diagnostic solution ([27]; see below for details). The plotted cost function is of the form j ( T o SO u o,

,r

=

W k (Tij_~

_ T~j~o)2 + W T ( T ~ k _ T ~ O ) 2

. .

(3) where T O = T n=~ is the initial condition for temperature, and similarly for S ~ u ~ v ~ ~0. The precise choice of the weights is discussed below. Let the two points in parameter space be Xl, x2. Then the various terms of the cost function were evaluated and plotted along the straight line in parameter space connecting these two points at x = x l + r(x2 - xz), with r varying from r = -0.6 to r = 1.6 at intervals of Ar = 0.1, using an integration time of N A t = 2 years. The results are shown in Fig. 2. ! 5O 45 - ~ 40 i 35 30 i I

25

20 15

!

|

Tdata Sdata

Tsleady

Ssleacly

'il,

Total

,i I

10 5 0

-0.5

0

0.5 Range

1

1.5

Figure 2" Cost function along a section in parameter space. Shown are the steady temperature penalties (short-dash); steady salinity penalties (dot); data temperature penalties (solid); data salinity penalties (dash) and the total cost (dash-dot). The data penalties along the section are clearly simple parabolas. The dynamical constraints for the temperature and salinity, however, have a very nonlinear character, reflecting the nonlinearity of the model equations used to obtain T n=N from T n=~ These terms of the cost function are nearly flat between the two points (r = 0 and r = 1), and then rise very rapidly outside of the interval. In particular, going from the minimum point

127 at r ,,~ 0, corresponding to the optimization started at the robust diagnostic solution, to r = 1, the data penalties increase significantly, indicating a very significant change in the temperature and salinity fields (Fig. 2). Yet the steady penalties hardly change. This seems to indicate a possible ill conditioning of the dynamical constraints, so that they are not well constraining the optimization which would feel mostly the variation of the data penalties along this section. As these dynamical constraints were evaluated using a 2 year integration time, they are presumably much better conditioned than using a single time step or other short integration time. It seems likely, however, that a more thoughtful formulation of the steady penalties may result in an even better conditioned form of the dynamical constraints. While there is probably room for improvement in the cost formulation, we wish to emphasize that an optimization problem formulated using a cost function similar to the above is, in fact, successfully solved below (section 4). 3.1.2

D y n a m i c a l c o n s t r a i n t s for velocities a n d b a r o t r o p i c s t r e a m f u n c t i o n

Under the primitive equation approximation, there are .5 prognostic fields: temperature, salinity, two horizontal baroclinic velocities and the barotropic stream function. In principle, each of these needs to be required to be at a steady state if such a model solution is desired. We have attempted to do this by adding to the cost function 3 terms such as

....

=

E

ijk

_-):

+

_

r

(4)

ij

Several optimizations were performed using this formulation, starting from the data, from the steady state or from a robust diagnostic solution (see next section). In all cases, the optimization efficiently reduced the steady penalties for the velocities and stream function using minute changes to the temperature and salinity, leaving the steady and data penalties for the temperature and salinity nearly unchanged. This could, of course, be due to a poor choice of the cost weights, although we feel that we have come up with a reasonable choice for them (see Table 2 and discussion below). Note that given the density stratification, the velocity field in a rotating fluid must adjust to the density stratification within a few pendulum days. Therefore, there seems to be no point in penalizing the velocity field separately from the temperature and salinity fields. Once the temperature and salinity penalties are minimized by the optimization, the velocity field just adjusts to the optimal stratification. Indeed, removing the velocity and stream function penalties resulted in an immediate improvement of the convergence of the optimization, and the steady velocity penalties are therefore not used in this study. It is interesting to note that this problem did not arise in previous studies such as [5][8], because they were all using a simpler GCM in which the momentum equations were diagnostic, and therefore did not require separate steady velocity penalties. The issue of dynamical constraints for the velocity field in a P E model is one of the new insights we seem to have gained by going to a full PE model in the present study.

128 3.1.3

C o n s t r a i n t s for surface flux data

In all previous applications of the adjoint method to a 3D GCM, the model was formulated using fixed surface-flux boundary conditions for the temperature and salinity. Then an optimal flux which minimizes the cost function was sought using the optimization algorithm. This involved penalizing the deviations of the optimal heat flux, H, from the heat-flux data, H d as follows:

]

=

( i,j - Hi,j) 2 t3

(5)

Note that the cost function in this case is an explicit function of the heat flux H which is used as a control variable to be directly calculated in the optimization. In previous applications of the adjoint method, this formulation resulted in very large discrepancies between the model surface temperature and the observed one, in spite of the data penalties in the cost function. In this study, we wish to examine the suggestion of Tziperman et al. [6] that restoring conditions may resolve the problem of large SST discrepancies, by using a cost function of the form

J(ssr) = E [w" (M~ ~ , ~ : ~

- H+

t3

where H ssT'n=~ is the restoring conditions heat flux (1) at the beginning of the run, and the control variable is the surface temperature, rather than the flux itself. Let us now write the complete cost function (selected parts of this cost function are used in the optimization presented below):

t3

+

..

+

.~ [WT(T~d k -- T i ~ ~

+

~(r~

w~ (%~

~ - ri~~ ~ + ,,~,~,j~

-

W~(S~k-S,~,~

- % 7 0 ) ~ + w ~ (~,j~

21

(7)

- ~,j~ )~

..

..

+

,_E-P

s'n=~ -

i,j) 2

.

:2

The data weights for the temperature, salinity and velocities are the inverse square error in the temperature data as estimated in Table 2, normalized by the number of model's grid points, M. The steady penalties require that the drift in temperature (or salinity) during a period of 15 years is equal to the assumed data error. The integration time of 2 years used to evaluate the steady penalties dictates the following choice for the steady penalties [6, 7]:

129 --T 1 (2yearsxek(T)) Wk = M 15 years

-2 (8)

The steady penalties for the velocities and stream function are similarly calculated from and e(r given in Table 2. The errors in the flux data were assumed to be 50Watts/m 2 for the climatological heat flux and 50cm/yr for the evaporation minus precipitation data [18]. The above choice of weights implied uncorrelated error statistics. For correlated errors, non diagonal weight matrices must be used. The errors in oceanic observations are not only correlated, but the correlation distances are, in fact, variable. This necessitates the use of non diagonal, inhomogeneous and non-isotropic error statistics. The use of horizontally uniform diagonal weights here is due to both the simplicity of this formulation and to the lack of reliable information about error statistics in oceanic observations.

ek(U)

Table 2 Error estimates used to calculate the cost function weights. level ek(T) ek(S) ek(U) (~ (ppt) (cm/sec) 1 2.000 0.2500 5.000 2 1.858 0.2323 4.677 3 1.675 0.2095 4.258 4 1.436 0.1796 3.712 5 1.142 0.1429 3.041 6 0.8218 0.1029 2.309 7 0.5249 0.06580 1.630 8 0.2976 0.03742 1.111 9 0.1555 0.01967 0.7866 10 0.08189 0.01048 0.6185 11 0.04942 0.006425 0.5444 12 0.03676 0.004844 0.5154 With the above choice for the cost weights, a given constraint can be said to be consistent with the assumed error level if the corresponding term in the cost function is less than one. Larger value of the temperature data penalties, for example, would indicate that the solution is not consistent with the requirement that the solution is near the Levitus analysis. A large steady penalty contribution indicates that the solution is not consistent with the steady state model equations. An optimal solution should have all terms, representing dynamical constraints as as well as data constraints, smaller than one.

3.2

Initial guess

The minimization of a cost function based on the equations of a complex OGCM as constraints is a highly nonlinear optimization problem. If started too far from the absolute minimum of the cost function, the gradient based optimization could lead to a local minimum of the cost function which does not represent the optimal combination of dynamics

130 and data. Tziperman et al. [6] found evidence for such local minima and MW93 [8] also found that when starting their optimization directly from the data it seemed to converge to a different solution than the one they felt reflects the optimal state. It is clearly important, therefore, to initialize the optimization with a good initial guess for the optimization solution. This can reduce the possibility of falling into a local minimum, as well as save much of the effort of minimizing the cost function through the expensive conjugate gradient iterations. The initial guess for the optimization solution can be obtained by using simpler assimilation methods that are not optimal in the least square sense, yet have been shown to produce a very good approximation for the optimal solution. Let us briefly consider two such methods and demonstrate them using the present global model. Suppose that our cost function consists of steady and data penalties for the temperature,

: z

[

-

-

i,j,k

(the steady penalty here is simply the square of the steady state model equations). Because each term in the cost function is weighted by its expected error, we expect that at the optimal solution the total contribution of the steady penalties over the entire model domain should be roughly of the same order as that of the data penalties [6]. Assuming (with no rigorous justification) that this global condition may be applied locally, we have

(uvr- K v:r- I

vrzz) 2

[WijT ] (7~_ T)2,

(10)

which is exactly the robust diagnostic equation [27] for the temperature OT Ot at a steady state, with the restoring coefficient set to [6] 1

= [w,j ~ T

(12)

In order to demonstrate the efficiency of the robust diagnostics approach, when used in the above fashion, to produce a good guess of the optimal solution, we show in Table 3 the cost parts obtained from the points in parameter space representing the Levitus data [entry (a)], the steady state model solution [entry (b)], and the robust diagnostic solution [entry (c)]. As may be expected, the point representing the Levitus data is characterized by large steady penalties and zero value for the data penalties; the steady state has vanishingly small values for the steady penalties but relatively large values for the data penalties, indicating that the steady state is not consistent with the data. Finally, the robust diagnostic solution has a well balanced distribution of steady and data penalties such that they are all small, and has therefore produced a near-optimal solution of our inverse problem, as anticipated in the above discussion.

131 Table 3 S u m m a r y of model runs and assimilations used in this study. Run (a) (b) (c) (d) (e) (f)

data 0.00 / 19.41 / 0.31 / 0.62 / 0.31 / 0.31 /

steady 9.18 / 0.01 / 0.51 / 1.11 / 0.51 / 0.32 /

T/S

0.00 61.3 0.32 0.74 0.34 0.32

Cost Parts steady u,v/r data H / [ E - P ] 8.98 12.00/3221. 0.15 / 1.87 0.02 0.00 / 0.00 0.25 / 1.92 0.49 0.06 / 1.47 0.15 / 1.81 1.24 0.08 / 1.67 0.00 / 0.00 0.50 0.06 / 1.48 0.10 / 0.66 0.42 *0.03/'2.17 "0.15/'1.81

Comments

T/S

data steady state robust (rest. b.c) robust (flux b.c) extended robust optimization

Terms marked by "*" were not part of the cost function used in the optimization and are only given for comparison with the other runs. A second example of using a simple assimilation technique to obtain a good approximation of a complex optimization problem involves the optimal combination of heat-flux data and SST data [18]. Given the SST data, an estimated implied heat-flux field H s s T may be obtained using the restoring conditions formulation (1). Given also a climatological flux estimate, H d, we can formulate an optimization problem in order to calculate an optimal heat flux H which is based on both estimates H a and H s s T . The appropriate cost function is of the form:

J(S T,H)

Z

-

(HU-

ij

]

+ WH(H:d,y -- H i , j ) 2 .

(13)

To obtain an approximate solution to the optimization problem posed by the above cost function, we simply write the model heat flux at every time step as a weighted average of the implied fluxes obtained from the restoring boundary conditions, and the climatological flux data H ~ = aTHd

+ (1 -- a T ) H ssT'~,

(14)

Integrating the model to a steady state using this heat flux, we obtain a solution for H which serves as the approximated solution to the above optimization problem. To derive an expression for c~T, we again use the expectation that at the minimum of the cost function, the different cost terms have roughly the same magnitude, ~ ..- ] [ w S S T ( H i S j j

sT -- H i , j ) 2

g -- Hi,j)2 ] . ~ /~j .. [ w H ( H ',3

(15)

*3

assuming this holds locally and taking the square root, we have W S S T ~/2

-

i,,)-

(16)

A final manipulation of (16) brings us to the form postulated before in (14) and the relation between the weights in the cost function (13) and the coefficient c~7 is found to be [18]

132

olz : [1 + IWssT/WH] -1

(17)

The runs in Table 3 demonstrate how the above scheme, which we term "extended robust" serves to minimize the heat-flux penalties in the cost function. The heat flux and fresh water flux penalties in entries (a-c) in Table 3, reflecting the data, steady state and robust diagnostics, are relatively large. Entry (e) represents the solution obtained using the robust diagnostics scheme (11) in the ocean interior plus the extended robust diagnostics scheme (14) at the surface. The extended robust scheme can be seen to be very efficient in reducing the value of the flux terms in the cost function, demonstrating again that simpler assimilation methods, when used wisely, can most efficiently calculate a near-optimal solution of most complex nonlinear optimization problems. Both of the simple assimilation schemes used above can be shown to be equivalent to a corresponding optimization problem and give the same results under certain simplifying assumptions such as linearity, a single time step in evaluating the steady penalties etc. Thus the success of the simpler methods is not surprising. It is important to note however, that these simpler methods cannot replace the optimization approach for its ultimate objectives of parameter estimation and 4D data assimilation, both of which are still not tackled here.

3.3

C o n t r o l V a r i a b l e s for a P E o p t i m i z a t i o n

A primitive equation ocean model such as we use here requires the specification of temperature, salinity, horizontal baroclinic velocity field and the barotropic stream function as initial conditions. This multiplicity of initial conditions that must be calculated by the optimization algorithm poses two potential difficulties. First, the parameter space is significantly larger due to the addition of the baroclinic velocities and stream function as control variables. In general, the larger the parameter space, the more iterations are required to locate the cost minimum. Second, the additional control variables are very different from the temperature and salinity initial conditions, and thus pose new conditioning problems. Some of the complexities of using the baroclinic velocities and barotropic stream function as control variables, and the resulting ill conditioning were carefully examined by Thacker and Raghunath [10]. These potential difficulties with the velocity initial conditions lead Tziperman et al. [5, 6] to develop and use a model with diagnostic momentum equations for which only temperature and salinity initial conditions needed to be specified. However, in the present work we are faced with an optimization based on a full P E model, with more than double the number of initial conditions (per a given model resolution) than in Tziperman et al. [5, 6]. As before, we can use our knowledge of the physics to formulate the optimization problem in a way that is more likely to result in an efficient solution. It is known, and this fact has been used above to formulate the steady cost penalties, that given the density stratification, the velocity field in a rotating fluid must adjust to the density stratification within a few pendulum days. It seems most reasonable, therefore, that one would not need to calculate initial conditions for the velocities, and restrict the optimization problem to finding only the optimal temperature and salinity. The optimal velocity field will be found by the model after a very short initial adjustment period that should not have a

133 significant effect on the cost function that is based on the difference in temperature and salinity over an integration period of years. Every several iterations, the initial conditions for u, v, ~p may be updated by integrating the model for a few days starting from the last initial conditions for the temperature and salinity calculated by the c-g optimization and saving the results for the adjusted velocities and stream function and other models variables to be used as the new starting point for the optimization. Because of the short integration period, the temperature and salinity hardly change from their value calculated by the optimization. This procedure should result in a better conditioning of the optimization problem due to the significantly reduced number of control variables. In Fig. 3 we show the reduction of the cost function for the optimization (run (f) in Table 3) that was started from a robust diagnostic solution. The optimization procedure was able to reduce the value of the cost, but eventually stalled after about 17 iterations. It seems that the optimization has converged to a local or global minimum solution; however after restarting the optimization with only T,S as control variables, additional progress was obtained, indicating that the stalling was more likely due to ill conditioning. Note that if the solution found at iteration 17 (Fig. 3) was indeed a minimum solution in the full parameter space spanned by T, S, u, v, ~p, then it is also a minimum in the subspace of T, S, and no further progress should have been obtained. 1.65

i

!

i 5

i 10

i

!

i

i 15

20

25

1.60 1.55

2

1.50 1.45 1.40 1.35

Iteration

30

Figure 3" Cost value as function of iteration number for the optimization (run (f) in Table 3) beginning from the extended robust diagnostic solution (run (e) in Table 3). Another issue related to the choice of control variables for the optimization is that of preconditioning. Preconditioning refers to a transformation of the control variables in order to improve the conditioning of the optimization. The control variables may be measured in various units and have very different typical numerical magnitudes. This may result in a badly conditioned optimization and therefore in the optimization stalling and not progressing towards the minimum of the cost function. The simplest remedy is to scale the control variables so that they all have similar numerical ranges. This may be improved upon by scaling the variables by the diagonal of the Hessian matrix if it can be estimated. The control variables may also be sealed by a non-diagonal transformation if a reasonably efficient transformation is available (see, e.g., [2.5, 26, 10]). Although somewhat neglected in the discussion here, the issue of preconditioning is a most important one.

134

4

Results

So far we have discussed in detail the issues of correctly formulating the optimization problem, and trying to guarantee its successful solution by starting from a good initial approximation of the optimization solution. We now wish to describe the results of a few model runs in some more details. We begin in section 4.1 by describing and analyzing the solution of optimization (f) in Table 3. We then analyze model solutions obtained under restoring conditions and under flux conditions in section 4.2. 4.1

The

optimization

solution

One of the advantages of nonlinear optimization is that it can be used to re-map the data in a way that is consistent with the model equations. Fig. 4 shows the horizontal temperature and salinity fields at model levels 2 and 7, as obtained from the optimization (run (f) in Table 3), as well as the Levitus data at the same levels. The data residuals at levels 2 and 7 for the temperature and salinity (Fig. 5a) are quite small over most of the ocean volume, as indicated by the fact that the global measure of the data penalties (see Table 3) is less than one for both the temperature and the salinity. But there are some regions, most notably the western boundary regions in the North Atlantic and North Pacific, as well as the equatorial Pacific region, in which the deviations from the data are systematic and larger than the errors specified by the cost function weights (Table 2). In these regions, the optimization has clearly modified some features of the Levitus analysis quite substantially [See for example (Fig. 4) the temperature field in the tropical Pacific at level 2, or the smoother salinity contours created by the optimization at level 7]. In some cases the changes made by the optimization could be considered improvements, in others they are certainly a reflection of model deficiencies. Considering the coarse model we use here, we do not wish to claim to have improved on the Levitus analysis. But the temperature and salinity distributions we find are clearly more consistent with the model dynamics and therefore more appropriate for starting a coupled model integration using an ocean model similar to ours than is the original Levitus analysis. The steady residuals at levels 2 and 7 for the temperature and salinity are shown in Fig. 5b. The quantity plotted is the temperature after two year integration from the optimal state, minus the optimal state, multiplied by 7.5, to get the extrapolated drift expected in a 15 year period, as it appears in the cost function. The projected temperature drift is quite small at level 2, except in the Pacific sector of the southern ocean, where a strong convection creates some numerical noise of no physical significance. At level 7 one notices systematic warming in the north west Atlantic, probably due to the inability of the model to create the NADW at the right level and to have it spread southward correctly. In the north east Atlantic, the cooling trend is related to the Mediterranean tongue outflow that while simulated fairly reasonably thanks to the Mediterranean sponge layer, is still not sufficiently consistent with the data in that region. The steady salinity residuals reflect basically the same model problems indicated by their temperature counterparts. It is important to understand that while the optimization results suffer some obvious deficiencies as indicated above, they still provide a significant improvement over both the steady state model solution obtained without data assimilation and the Levitus analysis.

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This is seen from Table 3 which shows that the cost values for the Levitus data and the steady state solution are significantly larger than for the optimal solution. Fig. 5c shows the steady temperature residuals at level 7 estimated for the Levitus analysis as well as the temperature data residuals at level 7 estimated at the steady state model solution. Clearly both the data and the steady state are not optimal in the sense that they minimize one type of cost terms (data or steady penalties), but on the expense of a large increase in the other cost terms. The North Atlantic overturning circulation for the optimal solution is shown in Fig. 6a. The overturning circulation at 30N is only 10Sv instead of the expected 16-20Sv. This feature of the solution cannot be considered an improvement over the prognostic run of Fig. 1. Fig. 6b shows the meridional heat flux for the optimal solution. Again, no significant improvement is obtained over the prognostic model solution of Fig. 1, and the northward heat flux carried by the North Atlantic ocean at 25N is still significantly less than the expected 1PW (1015 watts). These limitations of the meridional circulation

138 and meridional heat flux for the optimal solution are not surprising, considering the model performance in these areas. It seems that the only appropriate solution is to improve the prognostic model, perhaps by using isopycnal mixing or another eddy mixing parameterization [19]. As is quite clear from Table 3, most of the cost reduction as compared to the Levitus analysis or steady state solution has been obtained during the robust diagnostics initialization run [entry (c) in Table 3. Still, the cost reduction during the optimization itself is not negligible (Fig. 3), in particular for the steady penalties. Fig. 7 shows the steady residuals at the end of the robust diagnostic solution. The general picture is of fairly significantly reduced steady residuals in the optimization as compared to the robust diagnostics (compare Fig. 5b.1 and Fig. 7a). The reduction is spread over the entire domain, showing again the effectiveness of the optimization. A similar comparison of the salinity steady residuals (not shown) shows a similar reduction. A comparison of the distribution of the data residuals does not show a significant difference between the robust diagnostics solution and the optimization, as may be expected from the results in Table 3.

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Figure 7: (a) Steady temperature residuals for robust diagnostic solution (run (c)in Table 3) at level 2; (b) same, at level 7. Contour intervals are 1.0~ and 0.25~ respectively. Negative areas are dotted. Note that during the optimization, the steady residuals were reduced significantly more than the data residuals [compare entries (c) and ( f ) i n Table 3]. This indicates that a relatively small change in the temperature and salinity fields can induce a larger change in the steady penalties. This asymmetry between the steady and data penalties is again a possible indication that the cost function is not sufficiently well conditioned. Perhaps a better cost formulation may be able to better balance steady and data penalties. Interestingly, the asymmetry observed here between the data and steady penalties is of an opposite nature to that seen in the cost section of Fig. 2, where a small change to the steady penalties (between r = 0 and r = 1) involved a large change in the data penalties. Clearly the highly nonlinear structure of the steady penalties, reflecting the nonlinear model equations, accounts for these complex behaviors of the steady penalties.

4.2

Restoring

vs fixed-flux surface boundary

conditions

Let us now consider the issue of fixed-flux vs restoring surface boundary conditions in inverse problems based on an ocean GCM. Tziperman et al. [5] have shown that when

139 using fixed flux boundary conditions with the flux as a control variable, small errors may be amplified by the optimization in areas of deep convection resulting in huge errors in the calculated heat flux. Furthermore, Tziperman et al. [6] found a very large discrepancy between their optimization solution for the SST and the data, and suggested that this is due to the use of flux rather than restoring conditions. Marotzke and Wunsch [8] encountered a similar large discrepancy in SST which they interpreted as a drift towards winter conditions and felt that this is the result of the absence of seasonal cycle in their model. We would like to suggest here that these large SST discrepancies may be eliminated by the use of restoring boundary conditions. We further argue that such a boundary condition formulation is more physically motivated as well as more successful from a practical point of view. Based on the success of the robust diagnostic approach in obtaining a near-optimal solution to the least square optimization problems, we shall base our discussion on the two robust diagnostic solutions represented by entries (d) and (e)in Table 3. Run (d) uses flux boundary conditions with the surface fluxes of heat and fresh water specified to be the climatological data sets described in section 2, while run (e) uses restoring boundary conditions and combines the climatological flux data and the restoring to the observed SST using the extended robust diagnostics approach [18] described in section 3.2.

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Figure 8: SST for (a) robust diagnostics run using flux conditions with the climatological heat and fresh water flux data [entry ( d ) i n Table 3), (b) an extended robust run using restoring surface boundary conditions [entry (e) in the Table 3]. Contour intervals are 2.5~ Negative areas are dotted. Fig. 8 shows the SST for both runs. The surface temperature field for run (d), using flux boundary conditions with climatological flux data, is very far from the observed field. Note that the temperature and salinity at all levels are still restored in this run to the Levitus data by the robust diagnostic term in the model equations. The restoring time, however, is 15 years, rather than 30 to 120 days normally used for the surface fields under restoring conditions. The structure of the temperature field is consistent with a contraction of the large scale shape of the thermocline in the north-south direction, as seen in a much more pronounced form in ocean model runs under flux conditions without restoring at the interior. The mid-latitude regions and poleward are colder than the Levitus datal while the tropical regions are warmer. The large discrepancy in SST is reminiscent of the results of Tziperman et al. [6] and MW93 [8]. In our run (d), the

140 entire North Pacific ocean north of about 20N is significantly colder than the data, giving an impression that it tends towards a winter temperature distribution. We note, however, that the restoring conditions run of entry (e) produces a very reasonable fit to the Levitus SST, while also being able to reduce the distance to the observed climatological fluxes (see heat-flux penalty terms for this run in Table 3). Moreover, both the data and steady penalties for the temperature and salinity under the flux conditions are significantly larger. It seems, therefore, that inverse models should use restoring conditions even when trying to estimate the optimal air-sea flux. The enforcement of the flux data can be done by including it in the cost function as in (6). Such a formulation seems capable of producing a reasonable compromise of heat-flux data, SST and interior temperature. I

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Figure 9: Heat-flux residuals for the extended robust diagnostics run of Fig. 8b. Contour intervals are 50Watts/m 2. Negative areas are dotted. Fig. 9 shows the heat-flux data residuals for run (e), that is, the optimal heat flux of run (e) minus the climatological data of Esbensen and Kushnir [20]. There are clearly large systematic deviations from the heat-flux data in many areas such as the North Atlantic, equatorial Pacific and Indian Ocean. Large systematic heat-flux residuals in MW93 have lead the authors to suggest that the optimization's solution tends towards winter conditions with strong cooling over their entire basin. It seems to us that such large heat-flux residuals may, in fact, be related to the inability of the model to correctly simulate the North Atlantic meridional circulation [19], and therefore the meridional heat flux. Such a poor simulation of the meridional heat flux is directly linked to poor simulation of the air-sea fluxes [18], and hence the large heat-flux residuals seen in Fig. 9, and possibly also in Mwg3. The meridional heat flux for runs (d) and (e) is shown in Fig. 10. The run under flux conditions has a somewhat enhanced northward flux both in the northern hemisphere of the global ocean and in the North Atlantic ocean. But the price paid for this enhancement in terms of deviation from the temperature data is clearly too large. The large SST discrepancy indicate that the model cannot be forced to simulate the correct air-sea fluxes, possibly because of its inability to produce the correct overturning circulation. Runs (d) and (e) are, of course, not optimizations but solutions of a robust diagnostic model which was previously shown to closely simulate the optimal solution of a corresponding optimization. We have recently repeated the above analysis for two optimizations using restoring and flux boundary conditions correspondingly. The results

141

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Figure 10" Meridional heat flux for both runs of Fig. 8: (a) run using fixed surface-flux boundary conditions (run ( d ) i n Table 3); (b) run using restoring boundary conditions (run ( e ) i n T~b]e .3). Se~ caption of Fig. lb. fully support the above conclusions reached for runs (d) and (e) and will be published elsewhere with a fuller analysis of the boundary condition formulation problem for inverse studies. To estimate the effect of the missing seasonal cycle in our model, a comparison can be made between our results and those of Bryan and Lewis [17] who have used a very similar model, under seasonal forcing. Comparing, for example, the annually averaged meridional heat flux in their model, the to the steady state meridional heat flux in ours (Fig. 1), we see that there is not much of a difference. The addition of seasonality does not necessarily improve the simulation of the meridional heat flux (and therefore of the implied surface flux). Let us summarize the issue of boundary condition formulation, both from the point of view of the physics, and from a practical point of view. First, from the point of view of the physics of air-sea interaction, we note that flux boundary conditions imply modeling the atmosphere by assuming that the air-sea heat flux does not depend on the SST, and ignoring the obviously important feedback between SST and air-sea heat flux. In restoring conditions, this feedback is crudely included, as the restoring to the SST observations is somewhat reminiscent of the restoring of SST to the lower atmospheric t e m p e r a t u r e which occurs in the actual coupled system. Ultimately, one may want to use more elaborate parameterizations of the air-sea heat flux as function of the SST, and perhaps use as control variables the restoring times, or the atmospheric t e m p e r a t u r e which may appear in these parameterizations. This seems to make much more physical sense than formulations in which the flux is calculated directly, ignoring the SST-flux feedback. Second, from a more practical point of view, we note that ocean GCMs poorly simulate the observed ocean t e m p e r a t u r e when driven by specified surface heat fluxes. Similarly, when run under restoring conditions to observed SST and surface salinity, ocean models produce very poor estimate of the surface fluxes of heat and fresh water, and therefore of the meridional fluxes of heat and fresh water [18]. It seems, therefore, that ocean models

142 can presently produce either surface heat fluxes that are consistent with observations, or an interior solution that is consistent with observations, but not both. Now, in coupled model studies, it is presently more crucial for the ocean model to get the SST right than the heat flux, as the latter is corrected for using the artificially added flux correction. This dictates a choice of surface boundary condition formulation that is different from what was used in previous applications of the adjoint method to similar models, namely a restoring boundary condition rather than flux boundary condition. Restoring boundary conditions formulation is also consistent with our wish to use a model that can be run in a simulation mode, as this requirement cannot be met using fixed-flux surface boundary conditions.

5

Conclusions

This study attempted to combine a global primitive equation ocean model with climatological data for temperature, salinity and surface fluxes, using the adjoint method of data assimilation. This is a step towards using the adjoint method with the ocean component of coupled ocean-atmosphere models in the hope of achieving two ultimate goals. The first goal is the improvement of ocean climate models by estimating, for example, internal model eddy parameters and parameterizations so that the model simulations are closer to the observed ocean even when they are run without data assimilation. The second ultimate goal is the calculation of an ocean state based on the available data and the model equations to be used as initial conditions for coupled ocean-atmosphere climate simulations. While we have not achieved these goals as yet, we believe an important progress was made. Let us briefly summarize the main lessons we have learned here. Because our goal is to work with ocean models that can also be run without data assimilation, we have taken the approach that the model used for the inverse calculation must be formulated such that it can run independently in a simulation mode. A particular consequence of this approach has been the use of restoring rather than fixed-flux surface boundary conditions. We argued that the restoring boundary conditions are both better physically motivated and more successful from a practical point of view in producing a goo d inverse solution. Large discrepancies between the optimization solution for the SST and the data have been encountered in previous inversions using the adjoint method [6, 8]. These inversions used a fixed-flux surface boundary condition formulation. Marotzke and Wunsch have suggested that the SST discrepancy in their model was due to a drift towards colder surface temperatures ("winter conditions") which results from the lack of a seasonal cycle in their steady model. We have shown that the SST discrepancy can be eliminated in our model by using restoring surface boundary conditions, while still incorporating the available climatological flux data into the optimization. A comparison of our model results with a seasonal version of the present model run by Bryan and Lewis [17] seems to indicate that the annually averaged results of the seasonal model are quite close to the results of the model when driven with annually averaged steady forcing. We suspect, therefore, that at least for the present model, the use of flux conditions, rather than the lack of a seasonal cycle may be the cause of the SST discrepancies.

143 We have run our model in a simulation mode and have tuned the model as well as we can to produce the best possible results. Still, it is evident that according to our measure of success (the value of the cost function), the steady results with no data. assimilation were grossly inconsistent with the Levitus data. Similarly, the Levitus analysis was found to be just as inconsistent with the requirement that it satisfies the steady state model equations. However, our optimization approach provided a better solution than both the steady state model solution obtained with no data assimilation and the original climatological data sets. This solution was much more consistent with both the data and steady constraints, and therefore significantly more optimal in the least square sense. The ultimate test of this optimality would be, of course, to use such a state as initial conditions for a climate simulation. Several important lessons have been learned here concerning the cost function formulation for PE models. We have demonstrated that there are still conditioning difficulties for the dynamical constraints with the presently used cost formulations, and that there seems to be room for improvement in this area. This issue requires much further research. A second conclusion we have come to concerning the cost formulation for PE models is that it does not seem necessary to include explicit steady penalties for the baroclinic velocities and barotropic stream function. Because the velocity field very rapidly adjusts to the stratification in rotating fluids, it seems sufficient to penalize the deviations of the temperature and salinity from a steady state. Similarly, we expect that future studies using seasonal models with an optimization approach, should enforce the dynamical constraints requiring the model solution to be seasonal on the temperature and salinity fields and not on the velocity field. The fast adjustment of the velocity field to the stratification has also led us to suggest that one can do well by using only the temperature and salinity as control variables to be calculated by the optimization, allowing the model to adjust the velocity field. Such a procedure, outlined in more detailed in the previous sections, may improve the conditioning of optimization problems based on PE ocean models. A most successful part of this study has been the use of simple assimilation method to obtain good approximations to the optimization problem. These approximations are then used to initialize the optimization, significantly reducing the minimization effort in the optimization itself. More importantly, they reduce the possibility of encountering local minima that will prevent the c-g optimization from finding the global minimum representing the desired optimal solution. We have demonstrated how dynamical constraints can be combined with data constraints using the simple robust diagnostic approach [27] to obtain a near-optimal solution. We have also shown how surface flux data may be combined with the dynamical constraints and the surface temperature and salinity data using the extended robust diagnostic approach of Tziperman and Bryan [18], again resulting in a near-optimal solution. Such simple assimilation approaches can help but not replace the optimization approach of the adjoint method, because they cannot be used to estimate parameters such as eddy coefficients etc, a goal for which the adjoint method itself is well suited. We feel that the technical aspects of inverting complex PE ocean models treated here, as well as the more general issues we dealt with, should be useful to future studies directed at using data assimilation with ocean climate models. There is a clear and urgent necessity of improving ocean models used for climate studies, and of using these models to estimate

144

the ocean state as well as is allowed by the available data. We have argued here that the adjoint method is a most appropriate tool for obtaining these goals, and we feel that they should and can be achieved in the near future.

6

Acknowledgments

This study would not be possible without the efforts of Long, Huang and Thacker who have developed the adjoint code for the GFDL PE model. We are grateful to them for allowing us the use of this code. We wish to thank Jochem Marotzke for sharing with us his code improvements allowing the use of multiple time steps for the cost evaluation. Thanks also to Carl Wunsch and to two anonymous reviewers for useful comments on an earlier draft. Computer time for this study was partially provided by a grant from NCSA Illinois. Partial support for ET was provided by grant 89-00408 from the United States- Israel Binational Science Foundation. Partial support for ZS and computer time were provided by grant N00014-93-1-0831 from the Office of Naval Research - Navy Ocean Modeling Program.

References [1] F. Le Dimet and O. Talagrand, Tellus, 38A (1986) 97. [2] W. C. Thacker and R. B. Long, J. Geophys. Res., 93 C2 (1988) 1227. [31 C. Wunsch, J. Ceophys. Res., 93 C7 (1988) 8099. [4] E. Tziperman and W. C. Thacker, J. Phys. Oceanogr., 19 (1989) 1471. [5] E. Tziperman, W. C. Thacker, R. B. Long and Show-Ming Huang, J. Phys. Oceanogr., 22 (1992) 1434. [6] E. Tziperman, W. C. Thacker, R. B. Long, Show-Ming Huang and S. Rintoul, J. Phys. Oceanogr., 22 (1992) 1458.

[7]

J. Marotzke, J. Phys. Oceanogr., 22 (1992) 1.5.56.

[8] J. Marotzke and C. Wunsch, J. Ceophys. Res., 98 (1993) 20149. [9] A. Bergamasco, P. Malanotte-Rizzoli, W. C. Thacker and R. B. Long, Deep-Sea Res.. II, 40 (1993) 1269. [10] W. C. Thacker and R. Raghunath, J. Geophys. Res., 99 (1993) 10131. [11] R. B. Long, S.-M. Huang and W. C. Thacker, The finite-difference equations defining the GFDL-GCM and its adjoint, Unpublished report, Atlantic Oceanographic and Meteorological Laboratory, Miami, Florida, 1989 [12] Bennett, A. F., and P. C. McIntosh, 1982. Open ocean modeling as an inverse problem: tidal theory. J. Phys. Ocean., 12, 1004-1018.

145 [13] Bennett, A. F. Inverse methods in physical oceanography, Cambridge Monographs, Cambridge University Press, 346 pp, 1992. [14] K. Bryan, J. Computat. Phys., 4 (1969) 347. [15] A. J. Semtner, An oceanic general circulation model with bottom topography. UCLA dept of Meteorology Tech. Rep. No. 9, 1974. [16] M. D. Cox, A primitive equation 3 dimensional model of the ocean, GFDL ocean group technical report No 1. Princeton University. [17] K. Bryan and L. J. Lewis, J. Geophys. Res., 84 (1979) 2503. [18] E. Tziperman and K. Bryan, J. Geophys. Res., 98 C12 (1993) 22629. [19] C. W. Boening, F. Bryan, W. R. Holland and J. C. McWilliams, An overlooked problem in model simulations of the thermohaline circulation and heat transport in the Atlantic ocean, Manuscript, 1994. [20] S. K. Esbensen and Y. Kushnir, The heat budget of the global ocean: an atlas based on estimates from surface marine observations, Oregon State Univ., Climate Research Institute Report No. 29. [21] A. C. Hirst and W. Cai, J. Phys. Oceanogr., 24 (1994) 1256. [22] S. Levitus, Climatological atlas of the world ocean, NOAA Tech. pap., 3, 1982. [23] A. Baumgartner and E. Reichel, The world water balance, Elsevier, NY, 1975. [24] Hellerman and Rosenstein, J. Phys. Oceanogr.13 (1983) 1093. [25] P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Springer-Verlag, Heidelberg, 1981. [26] Tziperman, E., W. C. Thacker and K. Bryan, 1992: Computing the steady state oceanic circulation using an optimization approach. Dynamics of Atmospheres and Oceans. Vol 16 No 5 pp. 379-404. [27] J. L. Sarmiento and K. Bryan, J. Geophys. Res., 87 (1982).

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Modern Approaches to Data Assimilation in Ocean Modeling

edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

147

D a t a A s s i m i l a t i o n M e t h o d s for O c e a n T i d e s Gary D. Egbert and Andrew F. Bennett College of Oceanic and Atmospheric Sciences, Oregon State University, Ocean. Admin. Bldg. 104, Corvallis, OR 97331, USA D e d i c a t e d to G e o r g e W . P l a t z m a n For his c o n t r i b u t i o n s to o u r u n d e r s t a n d i n g of o c e a n tides Abstract Using a common notational framework, we compare a range of assimilation schemes for combining dynamical and observational information into a single tidal solution. We consider approaches which make only indirect use of dynamics (fitting modes, objective analysis), approaches which incorporate dynamics directly but in an ad hoc manner (nudging), and formal generalized inverse approaches based on minimizing an explicitly defined penalty functional. We show that solutions for each of these schemes can be expressed in terms of a set of basis functions which is characteristic of the particular assimilation method. Understanding the similarities and differences between the approaches can be greatly facilitated by examination of these bases. We show in particular that representers, which arise naturally in the generalized inverse approach, can be interpreted as dynamically consistent, spatially inhomogeneous tidal error covariances. We contrast these to simpler, spatially homogeneous covariances used in objective analysis, to Green's functions (which arise in nudging), and to various dynamical modes (normal modes, Proudman functions) which have been used in tidal modeling. As an illustration of these ideas we consider a hypothetical tidal inverse problem for the Gulf of Mexico.

1. I N T R O D U C T I O N In a sense, the assimilation of data into dynamical models for ocean tides has a long history. The tides are certainly the most predictable of oceanic phenomena, and as a result accurate tidal harmonic constants have been known at isolated coastal and island locations for over a century. Similarly, the basic equations of tidal dynamics are comparatively simple, having been understood since the time of Laplace. Over the years a voluminous literature has evolved demonstrating a qualitative consistency of observable tidal data and theory. Since the development of the digital computer more quantitative calculations have been pursued, with the goal of extending our precise knowledge of coastal tides across the global ocean. Some efforts in this direction have involved direct solution of the governing dynamical equations, with no reference to the available data. (e.g., Accad and Pekeris, 1978). However, it was recognized early on that data from coastal and island tide gauges could be used to help constrain the tidal solutions (e.g., see Hendershott and Munk (1970) and references therein). These early efforts, in which coastal data were imposed as elevation boundary conditions, could be viewed as the first tentative applications of data assimilation methods in tidal modeling.

148 In the past few decades advances in tidal data acquisition technology have allowed for direct measurement of tides in the open ocean, altering the requirements, and the possibilities, for tidal data assimilation. First, with the availability of pelagic tidal observations, more general approaches to data assimilation, which allow in a natural way for observations away from boundaries, were required. The "hydrodynamic interpolation" scheme of Schwiderski (1978) was the first serious effort in this direction. Bennett and McIntosh (1982) and McIntosh and Bennett (1984) pursued a more formal approach, in perhaps the first papers to treat the tidal data assimilation problem explicitly. More recently, high precision satellite altimetry data have made direct measurement of open ocean tides on a global scale possible (e.g. Cartwright and Ray, 1990). Superficially, this development would appear to lessen the need for dynamical information in tidal modeling. Quite accurate estimates of open ocean tides can be obtained from the data alone (e.g., Schrama and Ray, 1994). However, accurate observation of variations in the small amplitude, climatologically interesting signals that altimetric missions are designed to detect requires essentially complete removal of the tides. Current accuracy requirements for tidal models are thus very stringent (Koblinsky et al., 1992), and the need to separate tides from other, general much lower amplitude oceanic motions has come to the fore. Since the dynamical information can be of great help in this critical task, the need for data assimilation methods in tidal modeling remains undiminished. In this chapter we offer a broad, but necessarily incomplete, overview of modern approaches to tidal data assimilation. The common thread running through these various approaches is the inclusion of both observational data, and dynamical information into a single tidal solution. The approaches described here vary considerably in the way that the dynamical information is incorporated. We consider approaches that make only indirect use of the dynamics (e.g., fitting tidal data with Proudman functions, objective analysis), heuristic approaches which directly incorporate dynamics (e.g., nudging), and formal generalized inverse approaches which directly incorporate dynamics by minimizing an explicitly defined penalty functional. While our overview is broad, it is not necessarily completely balanced. We devote much of our discussion to the generalized inverse method, focusing in particular on the representer approach described in detail in Egbert, Bennett and Foreman (1994; hereinafter referred to as EBF; see also Bennett (1992) for a more thorough and general discussion of the representer approach). This chapter is organized into two main sections. In the first, we summarize a variety of tidal assimilation methods within a common mathematical framework. For this purpose we adopt a strictly discrete viewpoint, as appropriate for a numerical implementation of the assimilation methods. The resulting matrix equations are readily comparable, thus clarifying similarities and differences between possible approaches. In the second section we compare and contrast various basis functions which arise naturally in conjunction with the assimilation solutions. Our goals in this section are two. First, we seek to explain to the reader the physical nature and interpretation of the representers. Second, by comparing representers to more familiar basis functions which arise in various other approaches to tidal inversion (Green's functions, Proudman functions and normal modes) we hope to further clarify relationships among methods. As a concrete, but hypothetical, example we consider inversion for tides in a semi-enclosed basin, the Gulf of Mexico.

149

2. A S U M M A R Y

OF ASSIMILATION METHODS

2.1. G e n e r a l F r a m e w o r k We adopt a general but somewhat simplified framework for the tidal inverse problem. As we discuss the work of others within this framework, we focus on the basic ideas, omitting many (often technically important) details. For the hydrodynamic equations we use the general notation (1)

S u = fo.

Here u represents the tidal fields, fo the astronomical forcing, and S the dynamical equations plus boundary conditions. Unless explicitly stated to the contrary, we take u = (U, V, h) where U and V are volume transports, and h the elevation of the ocean surface. As in EBF, we assume linearized, time separated shallow water dynamics with simple parameterizations for dissipation, for tidal loading and for ocean self attraction. To keep things simple we consider only a single tidal constituent at frequency w. While a full treatment of the tidal inverse problem should consider multiple constituents simultaneously (this is especially necessary for direct inversion of time domain data; see EBF), the essential ideas can be understood without reference to this complication. In this case (1) represents a coupled elliptic system of linear first order partial differential equations S~u~ - (So +

iwI)u~ =

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(2)

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Here J" is the coriolis parameter, H is water depth, fl is a scalar correction for tidal loading, ~ is a linear drag coefficient, and g is gravitational acceleration. Boundary conditions are: no flow across the coast, and specified elevations on open boundaries. In the time domain the dynamical equations take the form (0, + S o ) u = fo.

(4)

Most all of our general discussion is independent of these details of dynamics and boundary conditions. As the general notation of (1) is meant to suggest, all of the following will be applicable, with minor modifications, to other linear formulations of the tidal equations (e.g., a second order elliptic equation for h, obtained by eliminating U and V (e.g., Le Provost and Poncet, 1978) or formulation in terms of potentials and stream functions (e.g., Platzman, 1978), and to other sorts of boundary conditions (e.g., radiation conditions on open boundaries). Note also that some of the assimilation methods discussed here can be extended to allow for non-linearities (e.g., see Bennett et al. (1995) for one example). We shall not treat these variants explicitly here, and unless it is necessary to be explicit, we will use the generic notation introduced in (1). In this paper we work only with the spatially discretized form of the dynamical equations. Thus, u and fo are taken to be complex N-dimensiona,1 vectors, which specify the tidal fields and forcing on a discrete numerical grid (e.g., a "C" grid, with a total of N

150

lattice points). As long as the dynamics are linear, S is just an N x N matrix. With a rational choice of dynamics and boundary conditions, the tidal equations will have a unique solution, so we shall always assume that S is invertible. There are several justifications for our focusing here on the discrete version of the tidal inverse problem. This approach allows us to focus more on basic ideas, by simplifying notation and reducing the need to refer to technical mathematical details. In fact, most all of the mathematics in the following is reduced to straightforward manipulation of a small number of matrices. Similarities and differences between the various approaches are clearly exhibited in the resulting matrix equations. The discrete approach also allows us to be more explicit about certain computational aspects of the assimilation methods. Since in practice all data assimilation methods require heavy computation, these issues are of great importance. Of course there are also some disadvantages to a purely discrete description of the assimilation problem. In particular the hydrodynamical equations and boundary conditions are most simply and clearly expressed in continuous form. Furthermore, the "technical mathematical details" are not always insignificant. For instance, we would very much like the results of our assimilation method to be relatively insensitive to details of our numerical implementations (e.g., grid resolution). As we show in the next section, this will not be true for all approaches. These difficulties are in fact best understood with reference to the full continuous formulation of the inverse problem. We shall discuss this issue in some detail in the next section. We focus on the special but important case where the data consist of estimates of harmonic constants for tidal elevation at discrete points, such as would be obtained from a standard tide gauge. In keeping with the spatial discretization assumed for the dynamics, error free data from a tide gauge located at grid point nk can be expressed as dk = A~u,

(5)

where Ak is the N-dimensional vector with component nk equal to one, and all other components equal to zero, and the superscript asterisk represents the conjugate transpose. Other forms of tidal data, such as velocities obtained from deep-sea moorings (Luyten and Stommel, 1991), or from reciprocal acoustic tomography (e.g., Dushaw et al., 1995) can be represented in the same general form d = O ' u , where O represents the discrete approximation to the appropriate spatial averaging kernel. Again, we will focus on the special case (5) above. For a set of K such data we write d = L*u where L = ( A , . . . / ~ g ) .

(6)

Equations (1) and (6) formally represent our tidal information: dynamics and data, respectively. In general, these two equations will be inconsistent. There will be no u which satisfies both, so we must compromise between (1) and (6). The various assimilation methods reviewed here achieve this compromise in several ways. These methods can be roughly categorized by two factors: the degree to which the dynamical equations (1) are used directly in the assimilation, and the degree to which the tradeoff between (1) and (6) is made explicit.

151 2.2. M o d a l B a s i s F u n c t i o n s One approach which has received a fair amount of attention recently is to use the dynamics to define a "natural" set of basis functions ym, so that the spatial variations of the tides can be adequately described by a low order expansion of the form M

u-

~

bmym.

(7)

m--1

Provided M < K the coefficients in (7) can be found by standard least squares procedures, i.e., by minimizing the data misfit sum of squares M

l i d - L" ~ ymbmll2 = l i d - Ybll 2.

(8)

rn--1

In (8) Y is a K x M matrix, with entry Ykm representing mode Ym evaluated at the kth data location. Note that in practice we must have M << K for stability of the solution to (8). See Sanchez et al. (1992) for a discussion of this issue. The justification for the modal basis function approach is perhaps clearest when the ym are taken to be an appropriate subset of the normal modes for the dynamical system (4). Assuming linear dynamics, and using standard Sturm-Liouville theory, we can express the forced solution to (2) as oo

(1 -

-

(9)

rn=l

Here C~ gives the amplitude and phase of the astronomical forcing, Ym and Cm are the normal modes and associated frequencies, while fm is a scalar "shape factor" determined from the spatial variations of Ym and the forcing fo~; see Platzman (1991). Note that in practice normal mode calculations have assumed dissipationless dynamics (Platzman, 1972; 1978; 1984; Rao and Scwhab, 1976). In such a case, the modal frequencies Cm are real, and the modes Ym are free oscillations, i.e., unforced solutions to (4). The normal mode expansion of (9) emphasizes those Ym for which w ~ era. Such modes thus seem a priori likely to be a useful for fitting spatial variations of the tides at such a frequency w. This approach was taken by Woodworth and Cartwright (1986), who used the 14 normal modes with natural frequencies closest to M2 (as computed by Platzman et al. [1982]) to fit SEASAT altimetry data over the globe. Although the accuracy of the resulting model did not represent a significant improvement over existing open ocean tidal solutions, major tidal features were recovered from a relatively small amount of very noisy data, demonstrating the value of even this rather indirect use of dynamical information. Computation of a large number of normal modes is a challenging computational task (Platzman, 1978). Recent efforts to use modal expansions for tidal modeling have thus adopted a somewhat simpler approach, based on fitting Proudman functions (PFs; e.g., Sanchez et al., 1985; 1992; Sanchez and Cartwright, 1988). PFs are the gravitational normal modes for a fluid in a non-rotating basin (Rao and Schwab, 1976). That is, the PFs represent a subset of the free oscillations for (4) with dissipation and coriolis terms omitted from So. The remaining free oscillations for this simplified system are vorticity modes, which have no effect on surface elevations (in the case of PFs). Obviously these

152 vorticity modes would not be useful for fitting observations of tidal elevations, though they could be necessary for an adequate description of tidal currents. In practice PFs can be computed by finding the eigenfunctions for a real symmetric eigenvalue problem, or depending on the coordinate system, a generalized eigenvalue problem. Numerical evaluation of PFs for a given bathymetry is thus considerably simpler than computation of normal modes for the full dynamical system. In fact, PFs were initially developed as a first step toward calculation of normal modes (Proudman, 1918; Rao and Schwab 1976). Each PF individually satisfies mass conservation over the model area, and at least approximately correct boundary conditions. (The standard PF boundary condition is H i ) n h = 0 at the coast; on a rotating earth this is not equivalent to the usual no-flow boundary condition Un = 0, as the example of a Kelvin wave clearly shows). The normal modes are more closely related to the full dynamical equations (1), although the omission of dissipation (and tidal loading) implies some degree of dynamical approximation for these modes as well. Both sets of modal basis functions share an important property. Lengths characteristic of the normal modes ym scale as ~ / w , and are thus strongly dependent on bathymetry (H): shorter in shallow water, longer in deep. Length scales in tidal solutions fitted to such modes will also tend to be dynamically consistent with bathymetry. This represents a significant advantage of modal expansions over simpler approaches to smoothing or fitting of tidal data, such as fitting a low order spherical harmonic model. However, the modal basis function approach uses the dynamics only indirectly, ignoring potentially useful information implicit in (1) or (9). In particular, no use is made of our knowledge of the forcing (represented by C~ and fm in (9)). Furthermore, the tradeoff between fealty to the dynamics (1) and the data (6) cannot be made explicit in any compelling way.

2.3. Objective Analysis A rather different approach to tidal data assimilation which also makes only indirect use of the dynamics is the statistical interpolation approach proposed by Mazzega and Jourdin (1991) and Jourdin et al. (1991). Their approach is essentially a variant on classical methods for objective analysis (OA) of meteorological fields (Bretherton et al., 1976). The method is also closely related to the stochastic inversion approach advocated for geophysical data by Tarantola (1987). As we will show, the statistical approach is also formally equivalent to the representer approach discussed below. Here we summarize only the essential ideas, retaining the notation and simplifying assumptions introduced above. The reader is referred to the original works for details, and a proper treatment. The basic idea here is to assume that u is a realization of a stochastic process with known mean and covariance. The goal then is to use this statistical information to find the "best" estimate of u given a data vector of the form d = L*u + e.

(10)

In (10) e represents errors in the tidal data. We include in this term such things as instrumental noise or calibration errors, the effects of non-tidal ocean signals, and errors in analysis and reporting of tidal constants. These errors are treated as random variables (independent of the random tidal field u), and assumed to have zero mean: E[e] - 0, and covariance matrix E[ee*] = lee. With our discrete formulation, the covariance of u

153 is also expressed as a matrix E[(u-

u 0 ) ( u - u 0 ) * ] - ]E,

(11)

where u0 - E u is the mean or expected value of the tidal field. In the applications to tidal modeling by Mazzega and Jourdin (1991) and Jourdin et al. (1991), u0 was taken to be zero. Initially we adopt the same approximation here. There are several ways to proceed in deriving the unique optimal estimate fi, given the data d. One approach is to estimate the components of u as the linear combinations of the d a t a K

un -

~

cn~dk -- c~d

(12)

k--1

for which the expected estimation variance E[fin - un] 2

(13)

is minimized. T h a t is we seek the Gauss-Markov (linear unbiased minimum variance estimator; e.g., Ripeley (1981)) of u. Substituting (12) into (13), and varying the result with respect to the coefficients c~k, yields the standard result c~ : (L*]EuL + ]Ee)-IL*]EuA..

(14)

Putting (14) and (12) together, we have for the estimate of u: K

fi = (]EuL)(L*~uL + ~ e ) - l d = ( ~ u L ) b = ~

bkqk

(15)

k=l

where b = (L*~uL + ~ ) - l d , and qk is the kth column of the N x K matrix ~uL*. For tide gauge data, where L takes the form given in (6), qk is just the nkth column of ]Eu. Thus the optimal solution in this case is a linear combination of the columns of the covariance matrix ]Eu. For the more general case where u0 ~= 0, it is straightforward to show that the solution (15) should be modified to K

fi = u0 + (]EuL)b = u0 + ~

bkqk

(16)

k=l

where now, b = (L'Y~uL + ]E~)-'(d - L'u0).

(17)

Note that u0 could be the solution to the astronomically forced dynamical equations (1), or some other a p r i o r i "most likely" tidal model. An alternative derivation of the OA approach begins with the assumption of a joint Gaussian distribution for the independent random vectors u and e with covariances ]Eu, lee. fi is then calculated by maximizing the joint likelihood function with respect to

154 u. This leads to minimization of a quadratic form, which is essentially minus the loglikelihood:

(d- L*u)*]E.-'(d-L'u)+ (u- Uo)*~u-'(u- Uo).

(18)

K:(x, x') -- E a~Pt(x, x') = E a~Pl(cos A).

(19)

It is simple to verify that the minimizer of (18) is given by (16), or by (15) in the special case where u0 = 0. Although the two ways of posing the optimal interpolation problem yield the same final answer, the different formulations suggest different solution strategies. In particular, the system in (17) is K x K, while direct minimization of (18) with respect to the elements of the vector u leads to an N x N system of equations. Since typically K << N, the first solution strategy will generally be much more efficient. To apply this stochastic inversion approach, Mazzega and Jourdin (1991) and Jourdin et al. (1991) assumed a tidal covariance K: which was spatially homogeneous and isotropic over the globe. Such a covariance can be expressed in terms of spherical harmonics as (e.g., Yaglom, 1961) 1

1

In (19) Pl is the Legendre function of degree l, and A is the spherical angle between points x and x'. ]Eu is of course then just the discretized approximation of K:(x, x'). In order to estimate the spherical harmonic power spectral coefficients a~, Mazzega and Jourdin (1991) made a spherical harmonic analysis of an existing global tidal model (Schwiderski, 1980). Estimates of al were computed by averaging over all spherical harmonic coefficients of degree l, and then fit to a simple low order parametric model for the spherical power spectrum. By computing cross-spectra between tidal constituents, correlation between tidal constituents was introduced. The OA tidal solution can be interpreted as the minimizer of the penalty functional given in (18). This approach allows for a direct treatment of the tradeoff between fitting the data, and agreement with prior dynamical information (i.e., the prior model u0, and the spatially homogeneous covariance JEll). However, the restriction to spatially uniform tidal length scales (as implied by the homogeneous covariance) is rather severe, and certainly non-physical. In principal, this restriction could be relaxed. In fact, the representer approach discussed in 2.5 provides a straightforward way to develop a spatially inhomogeneous random tidal model with dynamically plausible covariances. 2.4. N u d g i n g An heuristic combination of the dynamics (1) and data (6) is given by the relaxation or "nudging" scheme Su = f o - A ( L * u - d),

(20)

where A is the N x K nudging matrix. Any of the techniques that could be used to solve (1) could be extended to solving (20). Implementation of (20) is particularly straightforward with a time stepping scheme. The simplicity of (20) as a data assimilation scheme has made it very popular in a broad range of oceanographic and meteorological applications. The "hydrodynamical interpolation" method of Schwiderski (1980) can be viewed as a special case of (20). Letting A = vL (where L is as given in (6)) (20) can be written Su = fo - ~[~ v(unk - dk)Ak. k

(21)

155

As is readily seen from (21), in the limit of large v the solution at the data sites is forced to agree with the data. In this case the data are essentially introduced as elevation "boundary conditions" at the tide gauge locations, with the dynamical equations used to interpolate between gauges. As v is decreased, the agreement of u with the data becomes only approximate. More generally, by using a full (non-diagonal) nudging matrix A, additional spatial smoothing of the data residuals L*u - d can be incorporated into the assimilation, allowing the data to have more remote influences. The importance of such smoothing will be illustrated in the examples of the next section. In contrast to modal expansions and OA, nudging makes direct use the dynamics. Indeed, data are incorporated in such a simple fashion that relatively more complex treatments of the dynamics should be feasible. For example, if (4) is solved by time stepping, inclusion of non-linearities in the tidal equations is straightforward (Kantha, 1995). The principal scientific difficulty with (20) lies in making a rational choice for the nudging matrix A. This matrix controls the trade off between the data and the dynamical constraints, and controls the smoothness of the solution. In practice A must be chosen by trial and error. The more complicated generalized inverse schemes of the next section are in effect rational procedures for constructing A from prior information. 2.5. T h e G e n e r a l i z e d Inverse Approach Equations (1) and (6) can be expressed together as

is u:

,-,

In general, no tidal state u will exactly satisfy the N + K equations in N unknowns represented by (22). The inconsistency arises both from observational errors plus nontidal oceanography aliased into the data d, and from the inevitably approximate nature of the (discretized) dynamical constraints of (1). The generalized inverse (GI) method amounts to constructing a generalized inverse for the singular matrix operator on the left hand side of equation (22) (Reid, 1968). Because no u exactly satisfies (22), we must approach this as a minimization problem, in which we try to satisfy both the data constraints, and the hydrodynamical constraints "well enough" We thus replace (22) with the requirement that our tidal estimate fi minimize the quadratic penalty functional fl[u] = (L*u - d)*~E~X(L*u - d) + (Su - fo)*lEfl(Su - fo).

(23)

Here IEe and ~Ef are the covariances for the data and dynamical errors, which are assumed uncorrelated. These covariances express our a priori beliefs about the magnitude and correlation structure of errors in the data and dynamical equations, and allow us to make precise the notion of satisfying (22) "well enough" Note that in our discrete formulation IEe and ]Er are K • K and N • N matrices, respectively. Note also that the classical "forward" model is equivalent to the assumption of infinite data error (or zero dynamical error, with finite data error). Once the covariances are specified, the tidal solution which minimizes (23) is completely determined, at least in theory. The science in tidal data assimilation thus lies in the specification of these error covariances. Of the assimilation methods considered here, only the GI approach makes these scientifically critical aspects of the problem fully explicit.

156 This represents a major advantage of the approach. While precise a priori specification of lee is difficult, if not impossible, some progress has been made on developing reasonably realistic models for dynamical error variances. Through an analysis of approximations in the discretized dynamical equations and boundary conditions, McIntosh and Bennett (1984) estimated appropriate scales for dynamical errors in a model for Bass strait. EBF extended this analysis to derive a spatially varying estimate of dynamical error magnitudes for a global scale model. For their dynamical error covariance, EBF assumed an isotropic correlation with a globally constant length scale of 5 degrees. However, the general class of covariances developed by EBF (based on solutions to a diffusion equation) could easily be extended to allow for anisotropy and/or inhomogeneous correlation length scales (by using an anisotropic/inhomogeneous diffusion coefficient in the covariance smoothing operator). Further progress on developing realistic statistical characterizations of dynamical residuals will be critical to future advances in tidal data assimilation. A similar statement of course holds with regard to the data error covariances. In tidal data assimilation the proper separation of tides from other oceanographic phenomena will depend on a correct characterization of these other motions. These issues are beyond the scope of this paper. For the examples considered here we adopt a simple diagonal form for lee, recognizing that more complicated error covariances may often be appropriate. In theory, minimization of fl is straightforward. Eq. (23) is just a large weighted least squares problem, for which direct solutions are readily available (e.g., Tarantola, 1987). However, in most practical circumstances the number of unknown parameters (i.e., the dimension of u) is too large to make such an approach feasible. For example in a global scale model with a reasonable number of tidal constituents (8) and moderate resolution (1 degree), N is of the order of 106. Alternative strategies must thus be pursued. One approach which has been tried is to use some sort of iterative descent method to minimize fl directly. As an example, Zahel (1991) used conjugate gradients to minimize a penalty functional of the form (23) iteratively, in an inversion of tide gauge data for global ocean tides. However, with this approach it is still necessary to invert the N x N dynamical error covariance matrix ]Er. Unless the form of ]Er is severely restricted to make this inversion relatively easy, iterative descent offers no advantage over more direct solution of the least squares normal equations. In the example already cited, Zahel (1991) assumed a diagonal form for ]Ef, a form which seems difficult to justify physically. Furthermore, as discussed in EBF (see also Bennett, 1990), rigorous mathematical justification for the GI method breaks down when a white-noise (5-correlated) dynamical error covariance is assumed. Even for the finite dimensional discretized inverse problem considered here, the practical consequences of this mathematical technicality can be significant. Inverse solutions computed for white noise dynamical error covariances can contain localized peaks or holes at data locations. Furthermore, these peaks or holes tend to grow in height or depth as grid resolutions are refined. Thus, solution properties can be strongly dependent on the numerical discretization. We provide evidence for this phenomenon in section 3. An alternative formulation of the problem, in terms of the representers of the data functionals, leads to a scheme which is practical for general dynamical error covariances ~Ef. This approach has seen extensive application in solid earth geophysics (e.g., Parker, 1994). A general discussion of applications to problems in oceanography is given in Bennett (1992). More specific applications to tidal problems can be found in EBF. The basic

157 ideas behind the method can be easily derived in terms of some elementary Hilbert space theory (e.g., see Bennett, 1992). Here we take a slightly simpler approach appropriate for our restriction to spatially discretized problems. First, note that the penalty functionals of (23) and (18) are virtually identical. In fact, the two penalty functionals arise from special cases of the same general inverse approach. The major difference lies in the direct inclusion of the dynamical equations in the misfit penalty functional of (23). However, if we make the identifications u0 = s - l f o

Eu = s - l ~ f ( S - 1 )

*,

(24)

then (23) reduces precisely to (18). The minimizer of ,7 is thus given by (16), with ]Eu and u0 as given in (24). We repeat this equation here, using slightly different notation: K !~1 -- U0 + E bkrk,

(25)

k-1

where rk = ]EuAk = s - l ~ f ( S - 1 ) ' m k ,

k = 1, K

(26)

are the so-called representers (Yosida, 1980) for the data functionals defined by Ak. These satisfy dk -- Ak'u

=

,

(27)

where the inner product (.,-) is defined for vectors Ul and u2 by

(ux, u~) = u~r~u-lu~ = u~S'r~-'Su~.

(28)

Recall that S includes boundary conditions. Note that (28) defines a natural inner product for our minimization problem: the dynamical error part of the penalty functional in (23) is just (u - u0, u - u0). Ak is the data functional which evaluates u at a single point; rk is the vector which "represents" this functional with respect to the natural inner product. By the "representer approach" to data assimilation we mean explicit calculation of these representers for the data functionals. At this point it is worth recalling what the steps in the matrix product in (24) and (26) really entail. Of course the actual calculation of S -1 is impractical, as well as unnecessary. In particular, the vector u0 - S-lfo, to which we refer as the prior model, is calculated by solving the astronomically-forced hydrodynamic equations (4). Various approaches are possible; in EBF the tidal equations were solved by time-stepping with a periodic forcing. Similarly, the representer calculation can be broken down into three steps. First, solve the adjoint hydrodynamic system with impulsive forcing at the sampling location: S*c~k = Ak.

(29)

(Note that the matrix S* is not necessarily the discretization of the adjoint dynamic system). Second, multiply the solution to (29) by the dynamical error covariance matrix, thereby computing Eras. Note that for the class of Gaussian covariances suggested in

158 EBF this multiplication can be accomplished by repeated local 'diffusing' of ak. Finally, solve the hydrodynamic system Srk

:

(30)

]~'~fC~k.

With the representers rk so calculated, the solution to the inverse problem can be completed by computing the representer coefficients bk, k = 1, K. These are found by solving the K • K system of equations (R + ]E~)b = (L*]EuL + ]E~)b = (d - L*uo).

(31)

In (31) R is the (Hermitian, non-negative definite) representer matrix with elements =

=

(32)

That is, the elements of R are obtained by evaluating the representers at the data locations. The great similarity between the GI and objective analysis (OA) approaches is of course no accident. The vectors qk in (16) are also representers for the data functionals Ak, but with respect to a different inner product. In both cases the inner product is defined by the quadratic form u~]Eu-lU2, where ~Eu specifies a spatial covariance for the tidal fields u (see (28)). However, the nature of this covariance is very different in the two cases. For the OA case, it is by choice homogeneous and isotropic. In contrast, the tidal field covariance ]Eu for the GI approach is inhomogeneous and anisotropic, with correlation length scales and relative amplitudes determined primarily by the bathymetry and the dynamical equations. We will discuss the nature of this inhomogeneous covariance further in section 3, where we turn to examples. We can also relate the GI approach to nudging. Applying S to both sides of (25) and using (24) and (26) we have Su = fo + ]Ef(S-1) *Lb-

(33)

From (31) we derive b = ]E; 1 (d - L* u).

(34)

Combining (33) and (34) we arrive at Su - ] E f ( S - 1 ) * L ] E ; I ( d - L ' u ) - A ( d -

L'u).

(35)

Thus, the G I approach can be viewed as providing a rational way to combine all of our information about dynamical errors (]El), data errors (Y]e), dynamics (S), and the sampling properties of the observing system (L) into a "nudging matrix" A.

2.6. An Indirect Representer Approach With the representer approach the principal computational task lies in solving (29) and (30). In principle, these equations must be solved once for each data location. For very large assimilation problems a full calculation of all representers at a reasonable grid resolution may be impractical. For example, for the global inversion of T O P E X / P O S E I D O N

159

altimetry crossover data described in EBF, there were over 6000 data locations. With the solution resolution of ~ 2/3 degree, and with the full treatment of cross-correlated multiple constituents used in EBF, computation of the full set of representers would require on the order of 1017 floating point operations. Storage of the representers would require 200 Gbytes. Even with today's massively parallel supercomputers such gargantuan calculations are impractical. EBF thus used a variety of strategies to reduce computational requirements to a manageable level. Here we extend and unify these ideas into a more general coherent scheme, which we will refer to as the indirect representer approach. See Bennett et al. (1995) for a recent application of these ideas to a non-linear data assimilation problem. For motivation, note that if we write out (16) explicitly for the GI approach we have fi = u0 + ~u = u0 + ]EuLb = u0 + S-1]Ef(S-1)*Lb.

(36)

Once the vector b is obtained (by solving (31)), 5u can be computed by solving (29) and (30), with the right hand side of (29) replaced by the vector Lb = ~k bkAk 9Note that this vector is just a discretized version of a comb of scaled delta functions at the observation locations. This "open loop trick" can be used to avoid saving the full representers, since solution of (31) requires only the representer matrix R. This simple idea also provides the basis for an alternative approach to minimization of J . Instead of directly minimizing (23) with an iterative descent method (as used by Zahel (1991)), we can use an iterative method (such as conjugate gradients) for solving the system (31), for the vector b (e.g., Press et al., 1986). To solve the equations in this way we must be able to calculate the matrix-vector product y = P x efficiently for an arbitrary vector x, without actually calculating the full matrix P _= R + ]E~. This can be accomplished using an analogue of the open loop trick. Since y = (R + ]E~)x =

L*S-1]Ef(S-1)*Lx + ]EeX,

(37)

y can be calculated by first doing the representer calculation of (29)-(30) forced by Lx, thereby computing v = S-1]Ef(S-t)*Lx; and then sampling v at the K data locations (i.e., computing L ' v ) and adding ]E~x to the result. Repeated application of this calculation scheme to solve iteratively the Hermitian positive definite system (31), via conjugate gradients or some other iterative solution method, forms the basis for the indirect representer approach. However, since the system (31) will typically be rather poorly conditioned, convergence of such an iterative solution scheme may be too slow to be useful. To improve convergence some sort of preconditioner is desirable. We seek a preconditioner P which is a reasonable approximation to the matrix P. P should be cheap to compute and store, compared to the full matrix P. It should also be reasonably easy to invert, so that we can efficiently precondition (31) by premultiplying both sides of this system of equations by p-1. As usual, there is something of a tradeoff here. A better approximation to P will lead to more rapid convergence of the iterative solver, but will be expensive to compute and apply as a preconditioner. The approaches discussed here lean very heavily toward computing a very good approximation to P. Further research is warranted on the appropriate balance to strike here. ^

100 There are two general strategies for approximating P. First, we can simplify the dynamical equations in order to reduce the computational burden in the representer calculation (29)-(30). The most obvious application of this idea is to do all representer calculations on a grid with coarser resolution than that desired for the full solution. That is, the elements of 15 could be computed at coarse resolution, and used as a preconditioner to solve (31) on a finer grid. This approach was used in EBF, and by Bennett et al. (1995) who applied this idea to a non-linear data assimilation problem. More generally, one could use relatively crude (but simple to implement) approximations in the dynamics for the representer calculation, and refine these only in the final iterative calculation. The second strategy, appropriate for very large data sets such as the T / P altimetry crossovers, is to calculate a judiciously chosen subset of the representers, and use these to compute an estimate of P. Consider the K x K representer matrix R. Each column of R corresponds to a representer for a single observation sampled at each of the K data locations. Thus if we calculate the first K ~ << K representers we can partition R as R - [Rll R21

R12 R2,]-[Rc

RNC]

where R c gives the columns of R corresponding to the K t calculated representers. Our goal is to approximate all of R as well as possible, given only R e . Let R - UAU* be the eigenvector decomposition of R. Although it is difficult to make rigorous generalizations, the eigenvalues of R are generally found to fall off rapidly (see the next section for an example). This means that the K columns of R can all be approximated quite well in a much lower dimensional subspace (e.g., the space spanned by the dominant eigenvectors). This is especially true in the case of interest here, where K is large (see EBF for examples). If the K t calculated representers are chosen as a reasonable sample of the observable data space, the columns of R N C should be well approximated as linear combinations of the columns of R e . Again, it is difficult to be rigorous about this, but intuitively we should choose data locations for the reduced representer calculation which are spread out in such a way so as to minimize sampling redundancy. In EBF this reduction in the representer calculation was effected by choosing an approximately uniformly spaced subset of ~ 1000 (out of ~ 6000) crossovers. In retrospect, this choice was less than optimal. In the deep ocean representers corresponding to observations separated by a few hundred kilometers (the typical spacing of TOPEX/Poseidon crossovers) are virtually identical, and thus redundant. However, this is not true over the continental shelves, where tidal wavelengths are much shorter, and nearby data thus much less redundant. A better strategy would thus be to calculate a non-uniform subset of representers - perhaps keeping the spacing of calculated representers proportional to x/~. To approximate R, we seek an estimate of R y e which is linear in R e R N C -- R c B +

E ..~ R c B .

(3s)

We would like to choose B so that the error in this approximation is small, e.g., by minimizing IIEII2. However, the only information we have about RNc is the upper block R12 = 1~1. This suggests that we should choose the coefficients B to minimize the part of the error E which we can calculate, that is, {IR12- R11BII2. This error can be reduced to

161 zero by choosing 13 = Ri-)P~I, so that our approximation to the full representer matrix, in terms of the calculated representers, is just l~-

[ Rll 1~1 R21 R21RIIP~I

(39)

"

The desired approximation to P is then I~ + ]E~. (Note the error in this approximation (R22 - R21RI-1P~1) is just the conditional covariance of tidal errors at the uncalculated sites, given the observations at the calculated sites). The K x K matrix 15 will in many cases be too large to directly invert or store. However, l~ = RcR11R~ - QQ* where Q = RcR111/2, is K x K' (K' << K). For the very important special case where lee = a21 the singular value decomposition (SVD) of Q can be adapted to develop an efficient preconditioner for the system (31).

3. B A S I S F U N C T I O N S

IN TIDAL DATA A S S I M I L A T I O N

As long as we restrict ourselves to linear hydrodynamics (and linear data functionals), all of the methods considered above yield solutions which are linear combinations of some set of fundamental basis functions. In some cases the use of these basis functions is explicit (e.g., fitting Proudman functions, or minimizing (23) with the representer approach). In other cases (e.g., nudging, minimizing (23) by iterative descent, OA), no direct reference to any sort of basis functions is made. Whether this aspect of the inverse solution is made explicit or not, consideration of these basis functions can provide significant insight into the similarities and differences between the various methods. In this section we compare and contrast representers (calculated under various assumptions), PFs, normal modes, and Green's Functions (impulse responses). To be concrete, we focus our discussion on tidal data assimilation in a single basin: the Gulf of Mexico (GoM; Figure 1). Bathymetry in the GoM is fairly simple, with depth contours roughly parallel to the coast. The central basin is deep and elongated along an east-west axis. Broad, shallow continental shelves occur along the northern and eastern coast, as well as north and west of the Yucatan peninsula. The GoM is connected to the Atlantic Ocean and Caribbean Sea through two open segments or ports: the Straits of Florida and the Yucatan channel, respectively. These ports have a significant impact on tidal dynamics in the Gulf. Grace (1932) found that co-oscillations with the Caribbean almost completely masked the "independent" part of the tide (see also Platzman, 1972). For the representer and impulse response calculations discussed here we apply the computational approach described in EBF to the simplified linear shallow water dynamics as summarized in the previous section. The dynamical and adjoint equations are solved on an Arakawa "C" grid by adopting a periodic forcing and time stepping forward (or backward, as appropriate) from uniform initial (or final) conditions. Complex harmonic amplitudes for each constituent are then obtained by harmonic analysis of the time dependent asymptotic solutions. For purposes of our discussion here, we consider only the /142 constituent by itself. For most of the calculations discussed here we have used a grid resolution of ~ 0.2 ~ We shall illustrate the effect of grid spacing by repeating selected calculations at higher resolution (~ 0.1~ The coarser grid resolution can be seen in the

162 surface plots of representers displayed in Figure 2.

30

2S "

"

(

20

260

265

270

275

280

Figure 1. Gulf of Mexico bathymetry with hypothetical tide gauge array. Locations of the observations (one degree apart in latitude and longitude) are marked by stars. The two triangle denote sampling locations for the representers displayed in Fig. 2. A representer is calculated for each of the 143 data locations. For examples of representers calculated via (29)-(30), we consider two principal cases: a white noise (i.e., di-correlated) dynamical error covariance (i.e., diagonal :Er), and a covariance with a finite correlation length scale (non diagonal ]Ef). For the later case, we use the formulation of EBF: dynamical error variances vary with location, while decorrelation length scales are taken to be constant (~ 1.5~ This length scale was chosen to maintain resolution over the shelves, but is probably too short for the deep central basin. A spatially varying decorrelation length scale would almost certainly be more reasonable here. Variances for the differential equations, and for the coastal boundary conditions, were calculated as described in EBF, using tidal elevations and currents from the TPXO.2 global inverse solution (Egbert et al., 1994) to estimate magnitudes for the various terms in the dynamical error budget. As shown in EBF, a dynamical error covariance calculated in this way is at least approximately self-consistent for the global ocean. Errors in the specified elevations on the open boundaries were assigned a standard deviation of 2 cm (for the M2 constituent), and a decorrelation length scale of 1.5 ~ The effect of varying the assumed uncertainty in boundary conditions will be discussed further below. For all of these calculations we consider a hypothetical set of 143 tide gauges, evenly spaced 1 degree apart over the GoM (see Figure 1). This sampling configuration is not

163 meant to be representative of a realistic array of tide gauges, which would inevitably be unevenly (and in places, sparsely) distributed. However, this idealized array is well suited to our present pedagogic purposes. Furthermore, as we shall show, representer calculations based on this uniform array provide considerable insight into the sampling properties of more realistic uneven arrays.

3.1 Representers and Covariances As noted above, the OA and GI approaches can be viewed as special cases of a more general scheme for optimal interpolation. In both cases our prior knowledge about the tides can ultimately be expressed in terms of the mean u0 and covariance ]Eu of a random tidal state u. In each case the "optimal" solution is then the Gauss-Markov smoother of the data, appropriate for the assumed mean and covariance. These solutions can also both be expressed as a sum of u0 and a linear combination of representers (see (16) and (25)). However, the representers qa for OA are very different in character from the representers rk for the GI approach. With the OA approach of Mazzega and Jourdin (1991) and Jourdin et al. (1991), a spatially homogeneous and isotropic covariance for the tidal fields is assumed. With the solution covariance specified by the kernel K: as in (19), the representer for a tide gauge at xk is then just )U(x, xk). In our discrete formulation, the OA representer qk would be this function of x approximated on the grid. Since by (19),)U(x, Xk) depends only on the radial distance between x and xk, the representers qk, k - 1, K are all just translated versions of a single, radially symmetric, generic representer. These representers will generally be peaked at the data locations, and fall off smoothly at a rate determined by the assumed decorrelation length scale. In practice, a simple analytical form is assumed for K:, and thus for the generic representer. For instance, in a global scale application of OA to GEOSAT altimetry data, Mazzega and Jourdin (1991) assumed a simple damped cosine form for the radial dependence of K:. Decorrelation length scales (,~ 15~ and typical variances were estimated from the model of Schwiderski (1980). In contrast, with the GI approach there is no simple analytical form for the representers. Both amplitudes and shapes of the GI representers rk can vary considerably depending on the sampling location. This is illustrated in Figure 2, where amplitudes and phases for two of the representers computed for our GoM example are plotted. Sampling locations for these representers are denoted by the triangles in Figure 1. The first location (Figures 2a and 2b) is near the open boundary of our model domain in approximately 100 meters of water, right at the edge of the shelf. The representer has a broad peak in amplitude near the sampling location, with secondary peaks in several shallow areas around the Gulf coast (Figure 2a). The second data location (Figures 2c and 2d) is in the center of the deep western part of the basin. In this case it is impossible to discern the sampling location from inspection of the representer amplitude (Figure 2c). There is no evidence of any peak in amplitude anywhere in the central Gulf. Rather, the representer has peaks in several places at the coast, generally in the same areas where the secondary peaks are seen in Figure 2a. The two representers displayed in Figure 2 are in a sense typical of the full set of 143 representers computed for the hypothetical sampling array of Figure 1. Representers for data locations over broad continental shelf areas are peaked at least to some degree near the sampling locations. Representers for observations in deep water, in

154 the middle and along the western edge of the Gulf, are all very similar to Figure 2b. This general pattern reflects the variation of coherent length scales (shorter in shallow water, longer in deep water) implicit in the dynamical equations.

Figure 2. Amplitude and phase for the two representers marked by triangles in Figure 1. Top panels give amplitude, bottom panels phases. For (a) and (b) the sampling location is in an area of 100 meter water depth, where the amplitude peaks in (a). View in (a) is from the northeast. For (c) and (d) the sampling location is in the center of the Gulf, and cannot be discerned from the amplitude plot (view from the southwest). Note the diffraction of waves across the shelves evident in the phase plots. The phase plots (Figures 2b and 2d) also clearly illustrate the dynamically appropriate nature of the representers. Rapid variations in phase occur only in shallow water. In the 100 meter depths on the broad shelf off the west coast of Florida phase isolines (in 30 degrees increments, corresponding to one hour for this semi-diurnal constituent), are roughly 100 km apart - as would be predicted from the dispersion relation for shallow

165 water waves (w ~ v / ~ a ) . Diffraction of the tidal waves, as they propagate around the basin and across the shelf breaks, is also clearly seen in these two representers. It is instructive to consider more directly some of the properties of the spatially inhomogeneous tidal covariance ]E~ - S-1]Ef(S-1) *. Note that we have not actually calculated all of ~ as this would require calculating representers for observations at every node in our discrete grid. However, by applying (39) we can use the much smaller set of 143 representers calculated for the uniform array of Figure 1 to compute at least a reasonable approximation to all of ]E~. In particular, the diagonal of Z~, which gives the variance of u as a function of position in the model domain, can be efficiently computed in terms of of the matrix Q = RcRI~/2 defined in section 2.6.

Figure 3. RMS tidal error (i.e., the square root of the diagonal of ]Eu) computed for two different assumed values of open boundary elevation error RMS: (a) 2 cm RMS, (b) 10 cm RMS. In Figure 3 we plot the standard deviation of u resulting from this calculation. This figure gives our estimate of the magnitude of errors in our prior, purely dynamical estimate of the 71//2 tidal constituent u0. For our assumed lee the expected standard deviation of errors in u0 is only 1-2 centimeters over most of the Gulf. Errors are expected to be appreciably larger (up to several tens of centimeters) only in shallow shelf areas - exactly those areas where the example representers of Figure 2 exhibit secondary peaks. This pattern of spatial heterogeneity in tidal error magnitudes is physically quite plausible. Virtually all sources of error in the dynamical equations are likely to be significantly larger in shallow water (EBF). Furthermore, tidal amplitudes are generally greatest in shallow water. The representers can also be interpreted as covariances. The representer rk (corresponding to an observation of a tidal constituent at node nk) gives the covariance of u at node nk with u at every other node in the numerical grid. Using the estimated variance of u (i.e., (]Eu)nn) we can rescale this covariance into a complex correlation

pk. = (rk)./((~.).~.~ (:~u)..) -'/2.

(40)

166

This allows us to eliminate the spatial heterogeneity of variances implicit in the representers, and focus more explicitly on the correlation structure in ]Eu - i.e., on the expected spatial structure in the discrepancy between the true tides and the prior solution ( u - u 0 ) . Results of this exercise, for the two representers of Figure 2, are given in Figure 4. The shallow water peaks in the representers are not evident in these plots (except, of course, for the peak at the shallow water sampling location in Figure 4a). This implies that tidal errors in these shallow coastal areas are weakly correlated with errors at other points in the basin (specifically, at the two sampling locations under consideration). In contrast, tidal errors in the center of the basin have extremely high correlations. Figure 2b shows correlations very close to one across the entire deep-water portion of the western Gulf. This would imply that assimilation of one very good tide gauge (e.g., at 24~ 267~ would suffice for accurate tidal prediction in the central basin. This high degree of correlation is consistent with the size of the basin, which is roughly an order of magnitude smaller than the deformation radius appropriate for the deep water in the center of the Gulf. Any perturbations to the tides resulting from errors in the dynamical equations will thus be nearly in phase everywhere in the central deep basin. Tidal errors in this area resulting from multiple, weakly correlated, dynamical errors scattered around the basin will thus still be highly coherent.

Figure 4. Correlation functions corresponding to the two representers of Figure 2. The plots give amplitudes of correlation between tidal errors u - u0 at the sampling location and other points in the basin. See discussion in text. So far we have emphasized the effect of the dynamics on the tidal error covariance ]Eu. This covariance is of course also strongly effected by our assumptions concerning errors in the dynamics (]Ef). As an example of this aspect of the problem, consider again Figure 3, where we plot RMS tidal errors for two different assumed open boundary condition error variances. In the case already discussed (Figure 3a), RMS elevation errors were taken to be 2 cm on the open boundaries. In Figure 3b the results are plotted for much larger

167 open boundary condition errors (10 cm RMS). These figures show that the magnitude of errors in the interior of the Gulf is very sensitive to the accuracy of the open boundary conditions. Indeed, with perfect elevations on the two boundary segments, RMS errors in the central Gulf (not shown) are well below 1 cm. This result is perhaps rather obvious - tides in a comparatively small semi-enclosed basin will be forced to a significant degree at the open boundary. Any errors in the boundary conditions will thus force errors in the interior. Note however, that RMS tidal errors over the shelves are relatively insensitive to the magnitude of open boundary condition errors. Tidal errors in these areas are dominated by more local errors in the dynamical equations. 3.2 S i n g u l a r i t y o f R e s p o n s e F u n c t i o n s

The representers discussed above were calculated for a full (non-diagonal) dynamical error covariance. In order to make efficient use of an iterative descent method for minimizing the penalty functional (23), Zahel (1991) assumed a simpler, diagonal or "white noise" form for the dynamical error covariance ~f. Although this algorithm does not involve representers, the unique solution will still take the form given in (25). We can thus understand something about the character of solutions computed in this manner from an examination of these "white noise representers" When the nudging approach is used with a simple "diagonal" nudging matrix (e.g., A - vL, as with Schwiderski (1980), or Kantha (1995)), the solution will take the form (for linear dynamics) fi = Uo + ~ v(dk - u , k ) S - ' A k ~ ckgk. k

(41)

k

In this case the solution must be a linear combination of u0 and the vectors gk, which give the dynamical response for delta function forcings at the data locations. Note that the gk are closely related to the Green's function for the linear hydrodynamic problem, G - (S-1) t. With our discrete matrix formulation, convolution with G is expressed by a matrix product such as Gfo, where G = S -1. It is readily verified that the vector gk is just the nkth column of G, and thus is just a discretized "slice" of the adjoint Green's function. Again, while the impulse responses are not generally calculated explicitly with the nudging approach, we can understand some aspects of the solution character by examination of these functions. In Figure 5 we compare three response functions calculated for the shallow water data location (a) in Figure 1: the representer calculated with the non-diagonal ]Ef described above (Figure 5a); the white noise representer (Figure 5b); and the impulse response. For the two representer calculations the open boundary elevation error variance was set to zero. This allows a more direct comparison to the impulse response, which was computed for homogeneous open boundary conditions for elevation. Also, the error scale for the white noise case was chosen so as to make RMS tidal errors comparable to that for the non-diagonal dynamical error case (plotted in Fig. 5a). The white noise representer (Figure 5b) is dominated by a significant spike at the observation location. Because the data will be implicitly fit to a linear combination of such functions, the tidal solution will inevitably have peaks and holes at some data locations. This problem is eliminated with the non-diagonal dynamical error covariance

168

Figure 5. Various response functions calculated for a shallow water tidal observation at 273~ 22~ (a) Representer calculated with a dynamical error decorrelation length of 1.5 ~ (b) Representer calculated with a 5-correlated dynamical error covariance. In (a) and (b) the open boundary elevation RMS has been set to zero. (c) Impulse response for a point forcing at the sampling location. The second column (d-f) give the same functions calculated on a grid with finer resolution (0.1 ~ vs. 0.2~ Note that the spikes in (b) and (c) grow in relative amplitude as resolution is refined.

169 (Figure 5a), which adds a smoothing step (i.e., convolution with lee) to the representer calculation. The impulse response (Figure 5c) is a pure spike - elevations away from the data location are essentially zero. As result, we would expect nudging with a diagonal A to yield even rougher solutions than would be obtained with uncorrelated dynamical errors. The comparatively greater smoothness of even the white noise representers is easy to understand. The white noise representers (computed with a spatially uniform dynamical error variance) are proportional (see 24) to a column of the matrix GG* essentially the square of the Green's function - while the impulse response is a column of G. Since the Green's function acts to smooth out the forcing, we should expect that applying G twice would result in greater smoothing. Note that the spikes in the white noise representer and impulse response are typical of all shallow water data locations. However, for deep water locations the situation is somewhat different. For example, for data location (b) in Figure 1, the white noise representer is similar to that calculated for the full non-diagonal covariance - there is no peak anywhere near the data location. In this case the dynamics alone impose a great deal of smoothness on the representer. The situation is similar for the impulse responses, although in this case small localized spikes are still present for at least some deep water locations. Throughout this paper we have maintained a discrete formulation for the tidal data assimilation problem. We now briefly consider effects of discretization and grid resolution. In Figure 5d-5f we plot the three response functions of Figures 5a-5c, computed now on a grid with twice the resolution (0.1 ~ as opposed to 0.2~ For the white noise representer and impulse response the spike at the data location grows in relative magnitude as the resolution is refined. In contrast, the representer calculated for the full non-diagonal lee changes very little. The changes in the impulse response (Figures 5c and 5e) are readily understood. The Green's function for the shallow water equations is singular on the diagonal (Bennett and McIntosh, 1982). With the discretized formulation, the matrix G is of course perfectly well defined, with the singularity replaced by a finite peak. However, as grid resolution is refined and the approximation of the Green's function is improved, this peak will become sharper. The relative growth of the spike in the white noise representer (Figures 5b and 5d) ultimately results from the same singularity. Strictly speaking, the white noise representers are not well defined in the continuum limit (see EBF for a brief discussion; Bennett 1990 for further details). This is because when a white noise covariance is used in the norm which defines the dynamical error penalty, the data functionals corresponding to height measurement at an isolated point are not continuous, or even bounded. What this means in practice is that there will be pathological tidal solutions u = u0 + 6u which exactly fit the data, and at the same time have arbitrarily small dynamical misfits (i.e., [l~u[I is arbitrarily small). These pathological solutions will consist of spikes at the data locations superposed on u0. As discretization is refined, the inverse solution computed under the assumption of 5-correlated dynamical errors will also begin to exhibit these pathologies. See Bennett (1990) for further examples of this phenomena. We stress that this behavior of the solution is implicit in the penalty functional, and does not depend on how the minimizer of (23) is found. One final comment on Figure 5 is in order. The scale used for the white noise representer is reduced by a factor of 4 when the grid is refined by a factor of 2. This complication arises

170 from the non-realizable nature of white noise processes. We have calculated rk for the two resolutions using (26), with ]Ef = a~I, using the same value of a in both cases. This is the approach that a naive direct minimization of (23) would suggest, but a simple dimensional analysis shows that this cannot be correct. With this assumption the dynamical error part of the penalty is just a~-211fll2, where f -- Shu. Since f is just the dynamical error sampled at the grid nodes, the standard Euclidean norm Ilfll will tend to grow in proportion to the total number of grid nodes. The key point is that the dynamical penalty term is s u p p o s e d to represent a discrete approximation of an integral over the basin. It thus should be of the form a ] 2 f * W f , where W is a diagonal matrix of weights (proportional to grid cell area) needed to represent the integral properly. The inconsistency of scales in Figures 5b and 5d would be resolved if the proper (grid dependent) form for the white noise covariance were used. The dependence of spike amplitudes on grid resolution can only be eliminated by assuming a finite correlation length scale for the dynamical errors.

3.3 R e p r e s e n t e r Array Modes, P r o u d m a n Functions, and N o r m a l M o d e s The GI tidal solution can be neatly expressed in terms of the eigenvector decomposition of R (e.g., see Bennett, 1992). We summarize the key ideas here. Diagonalizing the representer matrix as R - VAV*, we have, for ~e = a2I u-

u0 + ~--~.(Ak/()~k + a2))d~r k

= u0 + ~ W k d ~ r ' k ,

(42)

k

In (42) Ak is the kth largest eigenvalue of R, and r'k -

~

Vk, krk,

d'k -- ~

k'

Y j k ( d k , -- L*uo).

(43)

k'

The linear combinations of representers r'k are the a r r a y m o d e s . The corresponding components d~ of the transformed residual data vector are mutually uncorrelated, and have variances E[d'k*d'k] - Ak + a e2 9

(44)

In (44) /kk gives the component of residual data variance due to the dynamical errors - the signal that we are trying to recover - and a~ gives the variance due to data errors, The wk in (42) can be viewed as data weights which vary between zero and one, depending on a p r i o r i signal and noise variances. For our hypothetical GoM tide gauge array eigenvalues fall off rapidly (Figure 6a). The corresponding weights are plotted in Figure 6b for data error variances of ae2 - 1 c m 2 (dashed line) and a~2 = .01cm 2 (solid line). Array modes r'k corresponding to weights near zero can be omitted from the sum in (42) with little effect on the solution. Figure 6 implies that this truncation of the representer expansion could be fairly severe. This is particularly true for the realistic case of data with RMS errors of 1 cm, where the data weights drop to ~ 0.1 by mode 35. In Figure 7 we plot selected eigenvectors Vk of R for the M2 constituent. The complex elevation components of Vk are displayed in polar form on a map of data locations as "sticks" with length proportional to amplitude, and direction denoting phase. Note that vk = A~-IL*'rk.

(45)

171 That is, the eigenvectors are proportional to the array modes sampled at the data locations. With the dense and uniform set of data locations in our hypothetical array, these stick plots thus provide a good representation of the elevation components of the array modes r~,. Note, however, that with the dynamics formulated in terms of the variables (U, V, h), the representers and array modes have transport components as well as the plotted elevation components.

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Figure 6. (a) Eigenvalues Ak of the representer matrix R computed for the hypothetical 143 tide gauge array of Figure 1. (b) Data weights computed from the eigenvalues. The dominant eigenvector (Figure 7a) is a basin scale seiche-like mode, with phases reversed at either end of the Gulf. Elevations are in phase, and of approximately constant amplitude over a large area in the western gulf. Note that the area of reversed phase in the eastern gulf is much smaller, with generally lower amplitudes. This requires net flow in and out of the Gulf, suggesting a significant co-oscillation, or Helmholtz mode (Miles, 1971), component in the dominant eigenvector. Note that this array mode does not exhibit the large amplitudes in shallow areas seen in the example representers of Figure 2, or the plots of RMS tidal error in Figure 3. The effects of bathymetry on this mode is seen only in the relative phase lags over the shelves. Eigenvectors 2 and 6 (Figures 7b-7c) are typical of all of the array modes up to number 26, and most of the next 20 or so array modes as well. These modes are all well localized in one or more of the broad shelf areas surrounding the gulf- i.e., the areas of large RMS tidal error in Figure 3. Note that array mode 26 (not shown) looks very much like a higher order seiche mode. The plotted eigenvectors can also be interpreted as the complex conjugates of the coefficients for the transformed data d~ given in (42). For small eigenvalues Ak these coefficients specify linear combinations of tidal data which are a priori likely to be small. For example, eigenvector 90 (Figure 7d) corresponds to a linear combination of the tide gauges with an expected RMS of a few tenths of a millimeter. The elements of Vg0 (and thus the array mode rg0) vary rapidly across the interior of the Gulf. Such rapid variations of tidal elevations are not dynamically reasonable in the deep (> 3000 meters) water in

172 this area. Our prior information strongly suggests that the transformed data residual d~0 will be dominated by noise. This noise will be heavily damped in the GI estimate (42).

. . . . . .

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Figure 7. Selected eigenvectors vk of the representer matrix R, computed for the hypothetical 143 tide gauge array of Figure 1. The complex components of vk are plotted as "sticks" on a map of station locations. Length is proportional to amplitude, direction gives phase. Eigenvectors are ordered according to eigenvalue magnitude. Note that the eigenvectors are just the array modes sampled at the 143 data locations. These plots thus also effectively display the array modes. The dominant array modes are completely consistent with the examples of representers discussed in the previous section, and illustrated in Figures 2-4. Errors and approximations in the dynamical equations and boundary conditions lead to basin scale, highly coherent errors in tidal elevations in the interior. This part of the tidal error is well represented by a single mode, consistent with the very high correlations of Figure 4b. In contrast, errors in tidal predictions over the shelves are coherent over much smaller scales. In particular, the relatively large tidal errors in these areas are not coherent with the basin scale errors which dominate the first array mode. A comparatively large set of

173 localized basis function is thus required to adequately approximate the correction to the prior model (Su) in these areas. All of our prior information - the dynamical equations plus estimates of error covariances for data and dynamics - are used to determine the number and form of dynamically consistent basis functions which can be stably fit to the data.

Proudman function number 1

Proudman function number 2

Figure 8. Selected Proudman functions computed for the same model region used for the representer examples. Since the PFs are standing waves, they can be displayed as a real, signed scalar field; solid contours denote positive elevations, dashed contours negative. Note the general similarity of form to the array modes of Figure 7. It is natural to ask how the array modes vk compare to the dynamical modes ym (i.e., PFs and Normal modes) considered in section 2. First, we should stress that the array modes are forced solutions to the dynamical equations, while the dynamical modes are homogeneous solutions. The two sorts of modes do, however, share some important properties. A selected subset of PFs calculated for the GoM are plotted in Figure 8. Note

174 that the P Fs are standing waves and can be represented as real functions giving elevation at time t = 0. There are several general similarities between this figure and the previous one. The first PF (corresponding to the lowest eigenfrequency) is indeed the fundamental seiche mode, and thus bears some resemblance to the dominant array mode of Figure 7. As with the array modes, other low order PFs are dominated by shorter length scale high amplitude features over the broad continental shelves. Finally, higher order modes become increasingly rough, as for the array modes. There are however, some very significant differences between the PFs and the array modes. Perhaps the most significant of these can be traced to the manner in which the open boundaries on the eastern edge of the Gulf are treated. For the PF calculation these boundaries are taken to be closed, and the standard rigid boundary conditions (Onh = O) are imposed. This condition prohibits any flow in and out of the Gulf, thus eliminating the Helmholtz mode evident in the dominant M2 eigenvector of Figure 7a. The importance of the Helmholtz mode to tides in the GoM is illustrated by the normal mode calculations reported by Platzman (1972). The two slowest gravitational modes calculated by Platzman for a GoM with open ports are reproduced in Figure 9. The slowest, with a calculated period of 21.2 hours, is the Helmholtz mode. The next slowest, with a period of 6.68 hours, is essentially the fundamental seiche mode, which of course bears a close resemblance to the first PF (see Platzman, 1972). The period of the Helmholtz mode is such that we should expect a significant contribution to tides, especially in the diurnal band. Note that the dominant /1//2 array mode of Figure 7a also includes a significant Helmholtz mode component. Indeed, this array mode can be well approximated as a superposition of the two normal modes in Figure 9. This implies that the Helmholtz mode will be critical for fitting semi-diurnal constituents as well. It seems unlikely that tides in this semi-enclosed basin could be adequately approximated without some modifications of the usual PF basis. The simplest refinement of this sort would be to include the spatially constant (zero frequency) PF mode. One could also replace the usual PF boundary condition (HOnh = 0) by h = 0. These are in fact the boundary conditions used by Platzman (1972) in his normal mode calculations in the GoM. This approach would clearly allow the Helmoltz mode. However, with this alternative formulation fitting non-zero elevations on the open boundary would become a problem. It is interesting to note that the dominant array mode for the K1 constituent (not shown) is essentially identical to the slowest mode illustrated in Figure 9a. This is completely consistent with the near resonance of this mode at the K1 frequency. Note also that the importance of the Helmholtz mode in the representers depends strongly on the assumed open boundary elevation error variance. When elevation data on the boundaries are taken to be perfect, the Helmholtz mode virtually disappears from all array modes. In the Appendix we derive a normal mode expansion for the representer matrix R. This allows us to show that for a very contrived tidal inverse problem, array modes are equivalent to normal modes. The rather bizarre conditions required for this equivalence very nicely illustrate some of the significant differences between dynamical modes and array modes. First, the equivalence requires that data be sampled at all locations, and for all three components (U, V, h). Obviously, dynamical modes have nothing to do with the sampling configuration of any observing array. Array modes, in contrast, are optimized

175

to data coverage. This property of the array modes will be especially important when sampling is uneven. With modal basis functions it is difficult to fit small scale detail where justified by data coverage (and dynamics), without introducing spurious oscillations in areas of poor data coverage (see Sanchez et al., 1992, for a discussion of difficulties in fitting PFs to unevenly distributed tide gauges in the Mediterranean). To attain equivalence between normal modes and array modes, sampling must be perfectly uniform.

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275

280

Figure 9. The two slowest normal modes for the Gulf of Mexico. In this case connections to the Caribbean Sea through the Straits of Florida and the Yucatan Channel are open. (a) The slowest gravitational mode (period = 21.2 hours) is the Helmholtz mode. (b) The next mode (period - 6.68 hours) is essentially the fundamental seiche mode. After Platzman (1972). Equivalence also requires a very special form for :Er and ~e, which must be diagonal, with error variances (for both dynamical and data errors) proportional to water depth. All boundary condition variances must be zero. All of these conditions are extremely unphysical. We have already discussed difficulties that may arise from a diagonal Ze. Furthermore, as discussed in EBF, virtually all sources of dynamical error are likely to be large in shallow water, and much smaller in deep water, exactly the opposite of the condition required for equivalence. More generally, dynamical modes cannot be tuned to reflect our prior knowledge of dynamical error statistics. This important sort of information is incorporated in the representer array modes. In summary, representer array modes are an optimal set of basis functions for minimizing (23). These basis functions are tuned to the observing system, and to our prior assumptions about errors in dynamics and data. As the normal mode expansion given for R in the Appendix shows, dominant array modes for a tidal constituent of frequency are likely to resemble normal modes with nearby oscillation frequencies. This is just what we have seen in the GoM example, as illustrated by Figures 7 and 9. However, normal modes are not generally equivalent to array modes, and we should not expect a tidal solution computed by fitting a small number of normal modes to be equivalent to a solution computed from an equivalent number of array modes.

176 4. S U M M A R Y

AND CONCLUSIONS

The GI approach is the most complicated of the data assimilation schemes considered here. However, this approach alone makes explicit use of all available information: the data and the dynamics, plus estimates of the magnitude and structure of errors or inadequacies in both of these. The GI approach can be viewed, along with OA, as special cases of the classical optimal interpolation method. As such, both approaches make the tradeoff between the dynamical information and the data explicit and rational. However, with OA the dynamical information is included only in a very gross spatially averaged sense, through a homogeneous covariance. In contrast, the GI approach uses the dynamics directly to construct a plausible, dynamically consistent, inhomogeneous covariance. While the modal basis function and nudging approaches include dynamics more directly, the critical tradeoff between data and dynamics is implicit, at best. The GI solution can be expressed as a linear combination of the representers for the data functionals. With the representer approach these natural basis functions are computed explicitly. In fact, all of the other assimilation approaches considered here lead to solutions which are linear combinations of a particular set of basis functions. Whether computation of these basis functions is explicit or not, we can understand aspects of the solution character by examination of typical basis functions. Applying this idea to a synthetic example in the Gulf of Mexico, we have shown how two possible assimilation approaches (GI with a diagonal Ef, and nudging, with a diagonal nudging matrix), will result in "spikey" tidal solutions. For very large assimilation problems, a literal application of the representer approach may not be practical. We have proposed here a general scheme which uses an approximate or truncated representer calculation to construct a preconditioner for an iterative solution to the GI problem. This, general scheme, which is an area of active research, opens up the possibility of using the rather involved GI approach, with a full non-diagonal dynamical error covariance, on very large assimilation problems.

APPENDIX:

A NORMAL

MODE EXPANSION

F O R ]Eu

In this appendix we derive an expression for the tidal error covariance matrix JEll in terms of the normal modes of the tidal equations. We assume dissipationless dynamics and homogeneous boundary conditions, and use notation established in the main text. In particular, the tidal field u is represented in primitive form in terms of volume transports and elevation (U, V, h), and the dynamical equations are as given in (2), with ~ = 0. As above, we restrict ourselves to the discretized case. Let C = diag[(j3gH) 1/2, (/3gH) ~/2, 1]. Then a simple calculation shows that the N x N matrix i C - ~ S o C is Hermitian, and can thus be decomposed, in terms of a unitary matrix U and a diagonal matrix ~I,, as U~I'U*. Setting Y = C U , noting that Y-~ = U*C -~, and combining the resulting expression for So with (4), we have for the time domain dynamical equations:

(Or-

iYq~Y-~)u = fo.

(46)

From (46) it can be seen that the normal modes for the tidal equations are given by the

177 columns Ym of Y, with the eigenfrequencies given by the corresponding elements Cm of the diagonal matrix ~ (i.e., for every m, ymexp[iCmt] is a solution to (46) with fo - 0). Returning now to the frequency domain we have

S~ = i w I - i y ~ y - 1

= iy~y-1,

(47)

where ~ , ~ = (w-Cm)hnm. Away from resonance (i.e., w ~ Cm for all m), S~ is invertible. From (24) and (47)we thus have Eu = [ Y ~ - ~ Y - ~ ] E f [ Y ~ - ~ Y - ~ ] * = Y [ ~ - I U * ( C - ~ E f C - ~ ) U ~ - ~ ] Y *.

(48)

In the very special case where

Ef = a}C 2,

(49)

this reduces to Eu - a } [ Y ~ - 2 Y ' ] . = a~ ~-~(w - Cm )YmYm 2 *.

(50)

m

Eq. (50) demonstrates that normal modes near the driving frequency w will tend to make large contributions to the tidal error. If we now assume that observations are made at all nodes (U and V nodes, as well as h), then L = I and R = Eu. In this case the normal modes are very nearly the same as the array modes for R. This is not quite true, since Y is not a unitary matrix. Rather, Y satisfies the orthogonality condition Y * C - 2 Y -- I. If we further assume that the data errors are of the form :Ee = a eC 2 2 (i.e., observational error variances are also proportional to water depth), then this is the proper orthonormality condition to define the array modes (e.g., Parker, 1994). In this extremely contrived case, array modes reduce to normal modes, with the eigenvalues of R of the form a~(w- era) 2. The dominant array modes will be those for which Cm is closest to w. A c k n o w l e d g m e n t s : We thank Richard D. Ray for calculating Proudman Functions for the Gulf of Mexico, and for helpful discussions and comments on this manuscript. This work was supported in part by the Office of Naval Research under grant N00014-9410926.

REFERENCES

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178 Bennett, A.F., B.S. Chua, and L.M. Leslie (1995), Generalized inversion of a global numerical weather prediction model, submitted to Meteorology and Atmospheric Physics. Bretherton, F.P., Davis, R.E. and C.B. Fandry (1976), A technique for objective analysis and design of oceanographic experiments applied to Mode-73, Deep Sea Research, 23, 559-582. Cartwright, D. E. (1991), Detection of tides from artificial satellites, in Advances in Tidal Hydrodynamics 547-568, edited by B. Parker, John Wiley, New York. Cartwright, D. E., and R. D. Ray (1990), Oceanic tides from GEOSAT altimetry, J. Geophys. Res., 95 3069-3090. Dushaw, B.D., B.D. Cornuelle, P.F. Worcester, B.M. Howe, and D.S. Luther (1995), Barotropic and baroclinic tides in the central North Pacific Ocean determined from long-range reciprocal acoustic transmissions, J. Phys. Oceanogr., 25, 631-647. Egbert, G.D., A.F. Bennett, and M.G.G. Foreman (1994), TOPEX/POSEIDON tides estimated using a global inverse model, J. Geophys. Res., 99, 24,821-24,852. Grace, S.F. (1932), The principal diurnal constituent of tidal motion in the Gulf of Mexico, Mon. Notices Roy. Astron. Soc., Gophys. Suppl., 3, 70-83. Hendershott, M.C., and W.H. Munk (1970), Tides, Ann. Rev. Fluid Mech., 2, 205-224. Jourdin, F. O., Francis (1991), P. Vincent and P. Mazzega, Some results of heterogeneous data inversions for ocean tides, J. Geophys. Res., 96 20267-20,288. Kantha L.K. (1995), Barotropic Tides in the Global Oceans from a nonlinear tidal model assimilating altimetric tides, submitted to J. Geophys. Res. Koblinsky, C.J., P. Gaspar, and G. Lagerloef 1992), The Future of Space-borne Altimetry: Oceans and Climate Change Joint Oceanographic Institution Inc., Washington D.C. Le Provost, C., and A. Poncet (1978), Finite element method for spectral modelling of tides, Int. J. Num. Meth. Engng., 12 853-871. Luyten, J. R., and H. M. Stommel (1991), Comparison of M2 tidal currents observed by some deep moored current meters with those of Schwiderski and Laplace models, Deep Sea Res., 38, Suppl. 1 $573-$589. Mazzega P. and F.O. Jourdin (1991), Inverting SEASAT altimetry for tides in the Northeast Atlantic: Preliminary results, in Advances in Tidal Hydrodynamics 569-592, edited by B. Parker, John Wiley, New York. McIntosh, P. C., and A. F. Bennett (1984), Open ocean modeling as an inverse problem: M2 tides in Bass Strait, J. Phys. Oceanogr., 14 601-614. Miles, J.W. (1971), Resonant response of harbors: an equivalent-circuit analysis, J. Fluid Mech., 46, 241-245. Parker, R.L. (1994), Geophysical Inverse Theory, Princeton University Press, Princeton, N J, 386 pp. Platzman, G.W. (1972), Two-dimensional free oscillations in natural basins, Jour. of Phys. Ocean, 2, 117-138. Platzman, G.W. (1975), Normal modes of the Atlantic and Indian Oceans, Jour. of Phys. Ocean, 5, 201-221. Platzman, G.W. (1978), Normal modes of the world ocean. Part I. Design of a finiteelement barotropic model, Jour. of Phys. Ocean, 8, 323-343. Platzman, G.W. (1984), Normal Modes of the World Ocean. Part III: A procedure for tidal synthesis, Jour. of Phys. Ocean, 14, 1521-1531.

179 Platzman, G. W. (1991), Tidal evidence for ocean normal modes, in Advances in Tidal Hydrodynamics 13-26, edited by B. Parker, John Wiley, New York. Platzman, G.W., G.A. Curtis, K.S. Hansen, and R.D. Slater (1981), Normal modes of the world ocean. Part II: Description of modes in the period range 8 to 80 hours. Jour. of Phys. Ocean, 11,579-603. Press, W. H., B. P. Flanerrry, S.A. Teukolsky, W.T. Vetterling (1986), Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 818 pp. Proudman, J. (1918), On the dynamical equations of the tides, Proc. Lon. Math. Soc., Set. 2, 18, 1-68. Rao D.B., and D.J. Scwhab (1976), Two dimensional normal modes in arbitrary enclosed basins on a rotating earth: applications to Lakes Ontario and Superior, Phil. Trans. R. Soc. Lond., A, 281, 63-96. Reid, W. T. (1968), Generalized inverses of differential and integral operators, in T. L. Boullion and P. L. Odell, Eds, Theory and Applications of Generalized Inverses of Matrices Texas Technical College, Lubbock, 1-25. Ripley, B. (1981), Spatial Statistics John Wiley, New York. Sanchez, B.V., R.D. Ray, and D.E. Cartwright (1992), A Proudman-function expansion of the M2 tide in the Mediterranean Sea from satellite altimetry and coastal gauges, Oceanol. Acta, 15, 325-337. Sanchez, B.V., D.B. Rao, and P.G. Wolfson (1985), Objective analysis for tides in a closed basin. Mar. Geod., 9, 71-91. Sanchez, B.V., and D.E. Cartwright (1988), Tidal estimation in the Pacific with application to Seasat altimetry, Mar. Geod., 12, 81-115. Schrama, E. O. J., and R. D. Ray (1994), A preliminary tidal analysis of TOPEX/Poseidon altimetry, submitted to J. Geophys. Res. Schwiderski, E. W. (1978), Global ocean tides, Part I: a detailed hydrodynamical interpolation model, NSWC/DL TR-3866, Naval Surface Weapons Center, Dahlgren, VA. Schwiderski, E. W. (1980), Ocean tides, II, A hydrodynamic interpolation model, Mar. Geod., 3 219-255. Tarantola, A. (1987), Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation Elsever, Amsterdam, 613 pp. Woodworth, P.L., and D.E. Cartwright (1986), Extraction of the M2 ocean tide from SEASAT altimeter data, Geophys. J. Roy. Astron. Soc., 84,, 227-255. Yaglom, A.M. (1961), Second order homogeneous random fields, in Proceedings of the Forth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 593-622, University of California Press, Berkeley. Yosida, K. (1980), Functional Analysis 6th ed., Springer-Verlag, Berlin, 500 pp. Zahel, W. (1991), Modeling ocean tides with and without assimilating data, J. Geophys. Res., 96 20,379-20,391.

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Modern Approaches to Data Assimilation in Ocean Modeling

edited by P. Malanotte-Rizzoli 1996 Elsevier Science B.V.

181

Global Ocean Data Assimilation System A. Rosati, R. Gudgel and K. Miyakoda GFDL/NOAA, Princeton University, Princeton N J 08542, USA Abstract

A global oceanic four-dimensional data assimilation system has been developed for use in initializing coupled ocean-atmosphere general circulation models and also to study i n t e r a n n u a l variability. The data inserted into a high resolution global ocean model consists only of conventional sea surface t e m p e r a t u r e observations and vertical t e m p e r a t u r e profiles. The data are inserted continuously into the model by u p d a t i n g the model's t e m p e r a t u r e solution every timestep. This update is created using a statistical interpolation routine applied to all data in a 30-day window for three consecutive timesteps and then the correction is held constant for nine timesteps. Nut updating every timestep allows for a more computational efficient system without affecting the quality of the analysis. The data assimilation system was run over a ten year period from 1979-1988. The resulting analysis product was compared with independent analysis including model derived fields like velocity. The large scale features seem consistent with other products based on observations. Using the m e a n of the ten-year period as a climatology, the data assimilation system was compared with the Levitus climatological atlas. Looking at the sea surface t e m p e r a t u r e and the seasonal cycle, as represented by the mixed layer depth, the a g r e e m e n t is quite good, however, some systematic differences do emerge. Special attention is given to the tropical Pacific examining the E1 Nifio signature. Two other assimilation schemes based on using Newtonian nudging of SST, are compared to the full data assimilation system. The heat content variability in the data assimilation seemed faithful to the observations. Overall, the results are encouraging, d e m o n s t r a t i n g t h a t the data assimilation system seems to be able to capture m a n y of the large scale general circulation features t h a t are observed, both in a climatological sense and in the temporal variability.

1. I N T R O D U C T I O N As the availability of ocean data increases dramatically in quality and q u a n t i t y in the near future, and both ocean and atmosphere models improve, the predictability of the coupled ocean-atmosphere system becomes feasible. The long-term goal of our work is to provide a description,understanding, and prediction of the coupled system as complete and reliable as t h a t which now exists for the atmosphere alone. In w e a t h e r forecasts, it has been customary to use data assimilation methods for g e n e r a t i n g initial conditions. For forecasts with coupled air-sea general circulation models (GCM), it would also be reasonable to consider that the ocean and atmosphere data assimilation (DA) are the optimum methods for the production of initial conditions even though the variability of the coupled system is quite different from any simple function of its parts (Ghil and Malanotte-Rizzoli 1991). As the first step toward this goal, a scheme of the oceanographic DA was developed by Derber and Rosati (1989).

182

This paper is a report on the implementation of this technique to a long-time series of ocean data analysis. The period of the analysis is 10 years from J a n u a r y 1979 to December 1988. During the decade, various national and international projects of d a t a acquisition and processing were carried out, i.e., COADS, MOODS, TOGA, and FOCAL; the acronyms and the contents will be described later. These d a t a s e t s are utilized for this study. We will examine the utility of the DA system both as a climatology and for capturing temporal variation. Another objective of this paper is to compare Derber-Rosati's DA variational analysis to two surface assimilation schemes. The two surface assimilation schemes are based on nudging the ocean model surface t e m p e r a t u r e toward observed SSTs and forcing one with winds from an operational analysis (as in the DA) and the other from model generated winds from a coupled model. If using only surface d a t a proves comparable to the DA, in t e r m s of analysis quality, then assimilating only surface d a t a may be a viable option for the generation of initial conditions and the production of a climatic d a t a s e t of ocean analysis. The assimilating model along with the surface forcing and oceanic data are described in section 2. This system is essentially the same as the one in Derber and Rosati (1989) with a modification to the frequency of inserting the t e m p e r a t u r e correction t h a t results in a computational savings. As in the previous study, the ocean data has been limited to conventional surface t e m p e r a t u r e data and vertical t e m p e r a t u r e profiles. The following three sections (3,4,5) examine the results looking at a comparison to independent observed analysis, for i n s t a n t a n e o u s and decadal means, and finally the interannual variability in the tropical Pacific. For the tropical Pacific, the heat content from the two surface assimilation schemes is compared to the DA system. The s u m m a r y notes discuss the utility of the DA system.

2. DATA A S S I M I L A T I O N SYSTEMS 2.1. D A s y s t e m f o r a n o c e a n m o d e l

The ocean model configuration is the same as described in Rosati and Miyakoda(1988).The model equations were solved on a nearly global grid (excluding the Arctic Ocean) with realistic topography. In the horizontal, a staggered 1~ (longitude) x 1~ (latitude) grid was used except between 10~ -10~ where the northsouth resolution was increased to 1/3 ~ in order to resolve finer scale equatorial structures. Since this is an upper ocean model 10 of the 15 vertical levels are in the top 177m. The vertical extent is 3km. The model includes subgrid scale parameterizations, non-linear viscosity for horizontal mixing, and turbulence closure scheme for vertical mixing. The atmospheric forcing consists of the surface wind stress and surface h e a t flux.The surface wind, t e m p e r a t u r e , and moisture d a t a were obtained from the twice daily analysis of NMC (National Meteorological Center). The method for computing h e a t flux is described in detail in Rosati and Miyakoda. Note t h a t the quality of NMC d a t a during this 10 year period was not very good although it improved during the latter years. Also, since the NMC analysis scheme changed over this period m a n y times, the r e s u l t a n t d a t a s e t is not self-consistent. Two other possible wind d a t a sources were the ECMWF and FSU analyses. However, the ECMWF winds were available only after 1985. The FSU winds are not global and are monthly m e a n d a t a and we found t h a t the most effective way for the vertical mixing scheme was to include the high frequency wind forcing, so as to capture the work done by the wind. Therefore the NMC analysis seemed the best for this study. In this study, the oceanic data have been limited to conventional surface t e m p e r a t u r e data and vertical t e m p e r a t u r e profiles. The ocean surface t e m p e r a t u r e data for the entire ten years are taken from COADS (Comprehensive Ocean-

183 A t m o s p h e r e D a t a Set. See Woodruff et al. 1987). The SST d a t a were m o n t h l y a v e r a g e d d a t a w i t h i n 2 ~ x 2 ~ latitude-longitude quadrangles. The d a t a coverage is very s p a r s e over the S o u t h e r n Ocean and therefore the analysis in this region would t e n d to be less accurate. This situation could be alleviated by the inclusion of satellite data, however, blending of this field would require the u n d e r s t a n d i n g of the satellite errors, i.e. d a y vs. n i g h t bias a n d a t t e n u a t i o n correction Reynolds et al. (1989). Such an i n v e s t i g a t i o n is beyond the scope of this paper. The vertical t e m p e r a t u r e profiles are based on the subsurface w a t e r t e m p e r a t u r e s m e a s u r e d m a i n l y by m e r c h a n t , fishing and r e s e a r c h vessels (see White et al. 1985; Kessler 1989), and here, t a k e n from t h r e e sources of dataset. They are: NODC (the N a t i o n a l Oceanic D a t a Center) for 5 years from 1984 to 1988, U.S. Navy's MOODS (Master Oceanic Observation D a t a Set, see B a u e r 1985) for 6 y e a r s from 1979 to 1984, a n d TOGA (Tropical Ocean and Global A t m o s p h e r e Project) d a t a set for 4 y e a r s from 1985 to 1988. This is a typical example of the m o n t h l y coverage from the t h r e e d a t a sets. The XBT profiles normally extend to about a depth of 450 m. The observed d a t a were continuously inserted into the ocean model by applying a correction to the forecast t e m p e r a t u r e field at every model timestep. The spatial objective analysis technique is based on the statistical interpolation analysis scheme of G a n d i n (1963). The m e t h o d used by Derber-Rosati is based on the v a r i a t i o n a l principle (Sasaki 1958; Lorenc 1986), in which the ocean model solutions are used as the first guess for t e m p e r a t u r e and the final analysis are d e t e r m i n e d by the i n s e r t e d oceanographic d a t a in such a way t h a t a functional be statistically minimal. The functional consists of two terms, first the fit of the corrected t e m p e r a t u r e field to the guess field weighted by an e s t i m a t e of the first guess error covariance matrix, and secondly the fit of the corrected t e m p e r a t u r e field to the observations weighted by the observational error covariance matrix. The functional is m i n i m i z e d using a Conjugate G r a d i e n t algorithm. E r r o r estimates, which d e t e r m i n e the spatial s t r u c t u r e and a m p l i t u d e of the correction field, are specified for each observation and for the first guess. The observational error covariances are set equal to an e s t i m a t e of the observational error variance. This variance e s t i m a t e is t a k e n from the COADS e s t i m a t e for the SSTs. For the t e m p e r a t u r e profiles, the variance is set equal to (0.25~ 2. These error e s t i m a t e s are t h e n weighted by a time factor which increases linearly from zero to one and back to zero as the difference b e t w e e n the observation time a n d the model solution time goes from -15 days to zero to +15 days. The observations are given no weight when the time difference is g r e a t e r t h a n 15 days. The inclusion of the time factor allows the use of 30 days of observations in the analysis scheme, yet gives the observations closest to the p r e s e n t model t i m e s t e p more weight. This has the built in a s s u m p t i o n t h a t ones i n t e r e s t is in the low frequency p h e n o m e n a . Currently, the first guess error covariance m a t r i x is defined so t h a t the vertical correlations are ignored and the spatial correlations are a s s u m e d to be the s a m e for each model level. Away from the e q u a t o r the horizontal covariances are defined by a G a u s s i a n with an e-folding scale equal to the distance b e t w e e n two grid points. This decreases away from the e q u a t o r by the cosine of the latitude allowing for s m a l l e r scale features in the corrections at higher l a t i t u d e s . N e a r the equator, the eastwest distance e n t e r i n g into the calculation is decreased by a factor of 2.28. This results in an anisotropic covariance function n e a r the equator, with the function s t r e t c h e d parallel to the e q u a t o r by a factor of 2.28. This anisotropic covariance is included to account for the well-known longer east-west correlation scales n e a r the equator. U n f o r t u n a t e l y these statistics are not well known and thus, are now defined empirically. As these statistics become b e t t e r defined, the results should also improve. In addition, to m o m e n t u m flux and h e a t flux a salinity b o u n d a r y condition is specified at the surface (at z=0), i.e., p C p K s = l.t ( Sclim -- S )

(2.1)

184 where s is the model salinity at level one (2.5m), Sclim is the monthly climatological salinity taken from the atlas of Levitus (1982) and u is a restoring time scale, set at 30 days-*. Of course, this is an ad hoc a r r a n g e m e n t ; it is hoped t h a t in the future observed salinity data will be used. (Carton and Hackert 1990 included observed salinity d a t a in their data assimilation.) Assembling the observed data from within a 15 day interval to either side of the current timestep, the ocean data are injected into the model. The time step for integrating the t e m p e r a t u r e and salinity equations is At I = 2 hs, while the time step for the m o m e n t u m equation is At2 = 1 h. Thus the modeI runs continuously, while the observation data are injected into the model. Unfortunately the solving of the minimization of the functional by the Conjugate Gradient method is quite time-consuming, although this technique itself is extremely efficient. In order to save computer time, the assimilation procedure is skipped for certain time-steps. With this process, all oceanographic variables such as the ocean currents, t e m p e r a t u r e and salinity are determined consistently within the f r a m e w o r k of the ocean GCM and the associated boundary conditions.

2.2. S u r f a c e d a t a a s s i m i l a t i o n s As an alternative to the full DA, a simple method, i.e., Newtonian nudging of surface boundary conditions, is investigated. In our case, the surface nudging technique is applied in two ways. One way is to use the coupled atmosphere-ocean model, in which only the top level of the ocean model is nudged toward the observed SST. Specifically, the t e m p e r a t u r e equation for the upper-most layer of ocean model is modified by adding a Newtonian nudging term, i.e., ~T i)t

.

.

.

.

.

.

.

.

.

= -~, ( T -

Lbs)

(2.2)

where k is the Newtonian damping coefficient, (3 day) -1, and Wobs is the observed SST; Tobs in this study is the monthly m e a n SST field of Reynolds (1982). This simplified DA is originally used for assessing the "systematic bias" of SST in the coupled model. I n t e g r a t i n g the model equations for 10 years with eq. (2.2), the systematic bias or flux a d j u s t m e n t term is calculated based on the time averages of the t e r m on the r.h.s, of eq. (2.2) (Miyakoda et al. 1989). The idea is t h a t surface winds g e n e r a t e d by the atmosphere model, influenced from the observed SST field, would initialize subsurface fields t h a t would be in balance with the atmosphere model winds, and therefore the models would be in adjustment with one another and not shocked when coupled. This case is referred to as the SST nudging. The atmosphere model is a global spectral GCM (Gordon and Stern, 1982). The horizontal resolution is T30 which is the spectral t r i a n g u l a r truncation at zonal w a v e n u m b e r 30, corresponding to a Gaussian grid of approximatly 4.0 ~ longitude by 4.0 ~ latitude. The vertical resolution is 18 levels. The second way is to use the SST nudging term in eq.(2.2) but with the same wind field as used in the DA, taken from the NMC 12 hourly wind data. This case is referred to as the simulation. Therefore the simulation case uses the same surface forcing as the DA but does not include subsurface data assimilation. The essential differences between the two nudging cases is t h a t in the first case the ocean model is forced by the T30 atmospheric model winds, t h a t are consistent with observed SSTs and in the second case the ocean model is forced by NMC wind analysis as in the DA.

2.3. I n i t i a l i z i n g t h e a s s i m i l a t i o n s The main objective is to produce an ocean analysis using DA for a given ten year period. In parallel, auxiliary runs are made with the SST n u d ~ n g , and also the simulation, for about the same ten years. In order to integrate these assimilations,

185 initial conditions are required at t=0, i.e., 00GMT 1 J a n u a r y 1979. These conditions were generated by r u n n i n g the ocean GCM for more t h a n 9 years in the climatological mode, using climatological forcing, i.e., the wind stress of H e l l e r m a n and Rosenstein (1983), and nudging toward Levitus climatological SSTs. Nine years was sufficiently long for spinning up the ocean GCM, to attain quasi-equilibrium conditions for the upper ocean from the surface down to about 400 m depth. For the SST n u d ~ n g , additional initial conditions are required for the atmospheric model. These conditions are taken from the Level III NMC analysis. Some examples of the preliminary analysis based on this DA system were shown by Derber and Rosati (1989), however, this sequel paper examines the full decadal series of analysis, comparing to independent analyses. The results of the auxiliary runs will be shown by way of comparison.

3. ANALYSIS C O M P A R I S O N S 3.1. M e r i d i o n a l

cross sections

Figure 1 displays latitude-depth sections of t e m p e r a t u r e at about 155~ for September 1982. The lower panel is a plot of connected observed vertical t e m p e r a t u r e profiles m e a s u r e d by a commercial ship, crossing at 140~ 30~ (Japan) to 166~ 21~ (New Caledonia) and taking 10 days or so in August and September. The upper panel is the monthly mean of the DA results along 155~ for September. It should be noted t h a t this particular ship m e a s u r e m e n t was not included in the DA. The overall a g r e e m e n t between them is good, although within the upper 50 meters the DA appears warmer, and the isothermal layers at 30~ and 18~ in the DA do not exist in the ship track; the reason for this discrepancy may be due to the winter convective overturning and will be discussed later.

le~'l

s e e i e M l e l 1912

DATA ASSL~ILATION

foe

E

3oo

;~~/'

.

, ,/\

,,oo

1o

le~-e

o

Auo

io

- see 1~ml

2o

9'

li Vh,~

20"S

.

io

JO'N

oaseevATJON l i O ' ! L ~ -

......

o

ib

2"0

30"N

Figure 1. Temperature distributions along about 155~ longitude for September 1982. Upper: Data assimilation, and lower: direct plot of t e m p e r a t u r e profile by ship measurement. Contour interval is I~

186 Figure 2 is the meridional section of zonal current near the e q u a t o r at 95~ for November 1982. Different from Figure l's upper panel, Figure 2's u p p e r panel is a model derived variable, as opposed to a model variable t h a t has been corrected by observations. The lower panel is the geostrophic calculation based on the observed t e m p e r a t u r e distribution. The calculation at and near the equator is based on a method after Tsuchiya's (1955). The region of 95~ corresponds to NINO 3, i.e., the e a s t e r n equatorial Pacific, and the Equatorial U n d e r c u r r e n t (EUC) is normally coincident with the shallow thermocline depth in this region. The 1982-83 E1 Nifio has already commenced and we see the surfacing of the EUC in both the DA and the geostrophic analysis. This comparison demonstrates not only the model's capability to simulate the quasi-geostrophic n a t u r e of the temperature-velocity relationship but also t h a t the assimilated model t e m p e r a t u r e field m u s t have been similar to the observed one.

Figure 2. Isotachs of zonal current along 95~ l o n g i t u d e for November 1982. Upper: Data assimilation, and lower: the geostrophic calculation (after Lukas-personal commun.). Contour interval is 10 cm s -1. E a s t w a r d currents are stippled.

187

Figures 3a and 3b are latitude-depth diagrams of t e m p e r a t u r e and zonal current at 165OE, in the western Pacific, for J a n u a r y 1986. The observations were obtained by hydrographic casts and velocity profilers, mounted on a 20~ - 10~ cruise, under a special TOGA program of the ORSTOM Centre in Noum~a, New Caledonia (Delcroix et al. 1987). Comparison between the DA and observations in Figure 3a show r e m a r k a b l e similarity for all of the major features. Centered about the equator isotherm spreading associated with the EUC is observed. Below 200 m, the isotherms bend concave downward symmetrically about the equator, as required if the EUC is to be in geostrophic balance. The isotherms are convex upward between 140 and 200 m, but no evidence of equatorial upwelling is found above 140 m, which is contrary to the situation in the eastern (Figure 13) and central (Figure 11) Pacific. North of 4~ the rising and steepening of the thermocline correspond to the North Equatorial C o u n t e r c u r r e n t (NECC) extending to 9~ A pool of w a r m w a t e r with t e m p e r a t u r e s above 29~ is in the surface layer, within 17~ and 5~ asymmetric about the equator; with the w a r m e s t w a t e r (>30~ located between 10~ and 3~ over a depth range of 60 m. In the section on zonal current (Figure 3b), the latitudinal position and the depth of the EUC correspond well between the m e a s u r e m e n t (Vmax = 50 cm s -1 at a depth of 200 m) and the DA ( U r e a x = 40 c m s -1 at a depth of 220 m); the e a s t w a r d flowing NECC (9 ~ -- 4~ also agrees, but the intensities are different (Ureax = 60 cm s "1 vs. 30 cm s-l). Two branches of the westward flowing South Equatorial C u r r e n t (SEC)(4~ - 5~ and 11 ~ 15~ also correspond between the observation and the DA, though their intensities are different. The underestimation of the s t r e n g t h of the c u r r e n t vectors appears to be a deficiency of this DA. If there was sufficient confidence in the wind field, the vertical mixing scheme could be tuned to get the amplitude of the currents better.

Figure 3a. Isotherms along 165~ longitude for J a n u a r y 1986. Upper: Data assimilation, and lower: direct plots of t e m p e r a t u r e profile by ship m e a s u r e m e n t for 10 -- 20 J a n u a r y (Delcroix et al. 1987). Contour interval is I~

Figure 3b. Isotachs of zonal current along 165 ~ or 166~ longitude for U p p e r : Data January 1986. assimilation, and lower: the direct observation (Delcroix et a1.1987~. Contour interval is 10 cm s --. E a s t w a r d currents are stippled.

188

3.2. C o m p a r i s o n w i t h TAO d a t a Surface moorings deployed as part of the TOGA program have t a k e n m e a s u r e m e n t s of ocean t e m p e r a t u r e and currents (McPhaden and McCarty 1992, McCarty and McPhaden 1993). The data has been analyzed over multi-year monthly m e a n time series which allows us an opportunity to validate the DA. We have chosen three mooring sites for the comparison, all with long records. Two sites, 0~176 and 0~176 are situated in the equatorial Pacific cold tongue where SST anomalies associated with ENSO events tend to be largest, and where ocean dynamics are crucial to the development of ENSO variability. The data span the years 1980-1991 at 110~ and 1983-1991 at 140~ The third site, 0~176 is located in the western equatorial Pacific w a r m pool where SST are the highest in the world ocean. The d a t a span the years 1986-1992. Figures 4, 5, and 6 show the means, over the time span available for each site, for zonal current, meridional current, and t e m p e r a t u r e for both the moorings and the DA. These mooring data were not inserted into the DA during the time period for the comparison. The surrounding XBT profiles are providing the observational influence for the DA.

Figure 4. Contoured time series of zonal velocity (u), meridional velocity (v) and t e m p e r a t u r e (T) at 0 ~ 165~ Climatologies (left) and DA (right). Velocities are in cm s "1 and t e m p e r a t u r e is in ~ Dashed contours are for westward or southward flow. Shading highlights zonal velocities > 50 cm s 1, meridional velocities > 6 cm s -1, and t e m p e r a t u r e s between 15~176

At all three sites the t e m p e r a t u r e is well represented, although too w a r m in the upper 25 m. This is somewhat puzzling since the DA has COADS SST inserted and Figures 7 and 15 show the DA to be close to the NMC analysis. The seasonal cycle and

189 the extent of the thermocline are simulated well. The a n n u a l cycle in the currents, however, does not compare well, and the amplitude of the EUC is u n d e r e s t i m a t e d in the DA. At 0~176 the zonal component of the DA in the top 100 m is always westward, w h e r e a s the mooring shows w e s t w a r d flow for J a n - A p r and t h e n e a s t w a r d flow. This difference in c u r r e n t structure, even though the t e m p e r a t u r e field is well simulated, m a y be underscoring the problems of only inserting t e m p e r a t u r e d a t a along the equator, w h e r e geostrophy is not the major balance.

Figure 5. The same as in Figure 4 except t h a t at 0 ~ 140~ and t h a t shading highlights zonal velocities > 100 cm s -1, meridional velocities > 6 cm s -1, and t e m p e r a t u r e s between 15 ~ 20~

4. D E C A D A L M E A N S In this section we compare the ten year m e a n of the DA, which will be considered a climatology, with independent climate analysis. We define the m e a n and the deviation as below. mean

( ) =

/0yi )/oyia / ( )dr /

o

dt

(4.1)

o

and deviation

( )'=

( )-(

) (4.2)

190

Figure 6. The same as in Figure 4 except t h a t at 0 ~ ll0~ and t h a t shading highlights zonal velocities > 100 cm s 1, and t e m p e r a t u r e s between 15 ~ 20~

4.1. S S T Figure 7 is the annual mean SST, which is compared with that of NMC analysis, Reynolds (1988), Reynolds and Marsico (1993). The top panel is the ten year average from the DA, eq. (4.1), the middle panel is Reynolds climatology from 1970-86, and the bottom panel is the deviation, eq. (4.2). The largest differences (> I~ are found off Newfoundland, around Marvinas, over the Southern Ocean, and the extension of the Kuroshio current. In general, the large differences are predominantly negative, and they are located in the middle latitude, implying that the SSTs in the DA are lower t h a n in Reynolds. Although not shown, the largest negative differences are associated with the winter hemisphere. This may imply a bias in the surface heat flux forcing. Overall the a g r e e m e n t is quite reasonable however, the regions where ocean dynamics play a d o m i n a n t role the differences are largest. The differences are such t h a t the n o r t h e r n p a r t is dominantly negative and the southern part is positive, implying t h a t the isotherms are located relatively southward in the DA. In other words, the isotherms in the DA are concentrated along the Gulf Stream, while those in the NMC are overly smoothed. This feature is stronger in DJF than in J J A (not shown here). 4.2. C o m p a r i s o n w i t h Levitus c l i m a t o l o g y One interesting feature to consider is how well does the DA system compares with established climatological data sets such as Levitus. This method, therefore, may be considered as a precursor toward producing climatological data sets t h a t are

191 dynamically constrained as well as objectively analyzed. Toward this goal, a consistent forcing field, such as the CDAS product (Climate Diagnostic Analysis System-see Kalnay et al. 1993), would help to produce an oceanic analysis t h a t includes i n t e r a n n u a l variability and climate trends. For this study, we have a fixed ocean DA system, however, the atmospheric DA from NMC was subject to m a n y modifications during this time period and so we lack consistent forcing. Nevertheless, we will compare the DA ten year mean as if it were a climatological data set to the Levitus data set. Figure 8 is the annual mean isotherms at 160 m depth in the DA and in Levitus (1982). The a g r e e m e n t is surprisingly good. Although here, as in Figure 1, the DA is colder at depth in the high latitudes, indicating t h a t winter convective mixing is deeper, and also in the western tropical Pacific indicating t h a t the thermocline is shallower. The discrepancies larger t h a n 2~ exist near the date-line along the equator, and also

Figure 7. Maps of sea surface t e m p e r a t u r e , based on the decadal (1979-88) average of the data assimilation (top), the Reynolds (middle), and the difference (bottom). Contour interval is 2~ for the top and middle panels and 0.5~ for the bottom panel, the negative areas are stippled.

Figure 8. Temperature at 160 m depth. The data assimilation (top), Levitus map (middle), and the difference, i.e., the data assimilation minus Levitus (bottom). Contour interval is 2~ in the top and middle panels, and it is I~ in the bottom panel, where the negative areas are stippled.

192 in Southern Ocean, Newfoundland, and Marvinas. Figure 9 shows the seasonal cycle of the climatological thermocline depth in the equatorial Pacific (2~176 The timelongitude charts of the contours of 20~ depth are compared among the SST nudging (first strip from left), the simulation (second strip), the DA (third strip) and Levitus data (fourth strip, i.e., far right). The results of the DA and the nudging are the decadal averages, while the Levitus result is based on climatology. This figure reveals that the phase of the annual cycle, most pronounced in the east, agree between the DA and Levitus, to a reasonable extent; the minimum depth is during September-October, and the maximum depth is during May-June. There is, however, a distinct difference west of the date-line; the depth in the DA is shallower by about 10-20 m than that in Levitus, reflecting the temperature difference by 2~ in the bottom panel of Figure 8. On the other hand, the annual cycle of the temperature nudging (first strip) shows a deeper thermocline depth from those of the other analysis, particularly in the western part of the basin. Perhaps the largest difference of the SST nudging is the amplitude; the magnitude of the annual cycle in the SST nudging is very large, whereas that of the simulation is smaller.

Figure 9. Annual cycle of the depth of the 20~ isotherm along the equator. From left to right, SST nudging, simulation, data assimilation, and Levitus. Contour interval is 20 m.

One way to see how well the model simulates the seasonal cycle in the upper ocean is to examine mixed-layer depth (MLD). Figures 10a and 10b display the MLD in a time-longitude section along various latitude bands, for both Pacific and Atlantic basins. Here the DA is compared to the climatological MLD after Levitus. Overall, the agreement between the two analysis is good, however, the tropical bands do show significant differences. Within the tropical band, (2.5~176 the characteristic east-west slope of the thermocline, deep in the central and western part of the basins as compared to the very shallow mixed layer in the east may be seen. This basic equilibrium state is simulated

193

Figure 10a. Time-longitude charts of mixed layer depth for the data assimilation (left) and Levitus (right) for various and latitude bands, i.e., (top) middle latitude 20~176 (bottom) extratropics 10~176 Contour interval is 20m.

Figure 10b. As in Figure 10a but for latitude bands, i.e., (top) 2.5~176 and (bottom) at the equator 2.5~176

by the DA, also seen in the (2.5~176 band, although it tends to be not as deep as the Levitus climatology. The reason for this has more to do with the ocean model and forcing data t h a n the assimilation scheme. In the simulation run the model tends to diffuse the thermocline and not m a i n t a i n the deep mixed layer in the western Pacific

194 w a r m pool. One source of the problem may be t h a t the heat flux is too high giving rise to a buoyant layer. This is consistent with Figure 8. The model tendency toward a shallower equatorial thermocline in the western Pacific is discussed in Rosati and Miyakoda(1988). The seasonal cycle is evident for the 10~176 and 20~176 bands. The DA seems to be quite reasonable, with the phase and amplitude of the m a x i m u m and m i n i m u m in the correct place. The MLD, within the 20~176 band for the western region during the winter does appear to be slightly deeper, perhaps due to excessive convective overturning.

5. T E M P O R A L VARIATIONS IN T H E T R O P I C A L P A C I F I C From the standpoint of climate variability in the atmosphere-ocean coupled system, the influence of the tropical Pacific SST appears very profound. In particular, the phenomenon, called the ENSO (El Nifio / Southern Oscillation), has a large impact on the state of the global atmosphere and ocean for time scales from seasonal to i n t e r a n n u a l (Ropelewski and Halpert 1987). For this reason, it is worthwhile to pay special attention to the results of the DA in this area. The DA system is not only important to study the interannual variability as a source for verification but also to produce oceanic initial conditions for coupled model forecasts.

5.1. T i m e - d e p t h c h a r t s Figures 11 and 12 are the time-depth charts of t e m p e r a t u r e and u-component at the central equatorial Pacific, 159~ longitude, comparing the DA (middle panel)with the observations (bottom panel). The bottom panel is a direct plot of ship track m e a s u r e m e n t s over a 16-month time span, which is based upon absolute current profilers, under the program of Pacific Equatorial Ocean Dynamics (Firing et al. 1983), and is redrawn for the purpose of comparison in this paper. Note t h a t the top panel displays the results of the simulation, which uses the same model configuration, initial conditions, winds and heat flux as the DA but there is no insertion of subsurface data. One of the characteristic features in Figure 11 is that the thermocline in the middle and lower panels is clearly visible at a depth of about 150 m, however, the simulation has a thermocline that is too diffuse. As one may note from the observations, there was an increase in the t e m p e r a t u r e of the surface layers but not a deepening of the thermocline. Then in December 1982 the thermocline rose, associated with the development of the E1 Nifio. Both the simulation and the DA reproduce these changes, with the DA being more faithful to the data. The u-component in Figure 12 represents a unique situation, associated with the E1 Nifio. As Firing et al. mentioned, the EUC disappeared beginning in September 1982 continuing until early J a n u a r y 1983. The DA has some difficulty in reproducing the exact features of the zonal c u r r e n t (Figure 12). The reason for the discrepancy is not clear; it could be due to some deficiency in the ocean GCM or the poor quality of the wind stress data. Since the simulation also shows this discrepancy it is not due to the DA scheme. Figures 13 and 14 are cross sections of t e m p e r a t u r e and zonal velocity along the equator at 95~ for 1982 and 1983. Once again the upper figures are the model simulation, the middle figures are the data assimilation, and the lower figures are the observed values after Halpern (1987). The observations of Figure 13 show the complex vertical structure of the evolution of heat content during the 1982-83 E1 Nifio event at 95~ We see the thermocline steadily deepening during 1982. In April and May 1983, we see a rise in SST r a t h e r than the continued deepening of the thermocline, and finally, in J u n e 1983 the restoration to normal conditions (for a more extensive discussion see Philander and Siegel, 1984). Overall, the ability of the model to simulate these changes at the equator is demonstrated. Two serious flaws of the model simulation are the lack of a well-defined thermocline and the inability to r e t u r n to

195

Figure 11. Time-depth diagrams of t e m p e r a t u r e at the central equatorial 0~176 Pacific, i.e., 158~176 Simulation (/eft), data assimilation (middle) and observation (bottom). Contour interval is I~

Figure 12. The same as Figure 11 but for the zonal component of current, u. Contour interval is 20 cm s -1. The -l areas of u > 8 0 cm' s are shaded dark, and the areas of u < 0 are stippled.

normal conditions toward the end of 1983 (Rosati and Miyakoda, 1988; Derber and Rosati, 1989). Although these discrepancies may be related to errors in the wind field, we see t h a t the DA does maintain a tight thermocline gradient and does recover from the E1 Nifio after July 1983. In Figure 14 the observations show a deceleration of the EUC during 1982,the appearance of an eastward jet during April-June 1983, and finally, normal conditions. Once again the model simulates the gross feature. However, the major shortcomings are a considerable underestimation of the EUC speed and a poor simulation of the phase and r e t u r n to normal. The data assimilation did show good a g r e e m e n t with observations for the t e m p e r a t u r e field however the zonal current sustains a strong easterly jet and does not show a timely r e t u r n to normal conditions. P e r h a p s for the same reasons as mentioned for Figure 12. Figures 11-14 demonstrate the ability of the DA to simulate the changes in the vertical structure of the flow at locations with considerable differences in the variability. The extent to which errors in the surface boundary conditions, especially in the wind field, caused the discrepancies between the m e a s u r e m e n t s and the model and to a lesser extent the DA are not completely known and will be assessed as the atmospheric surface analyses improve. For example, the EUC is driven by the

196

Figure 13. The same as Figure 11, except for the eastern equatorial 0~176 The Pacific, i.e., 95~ areas of > 28~ are stippled.

Figure 14. The same as Figure 12, except the location, i.e., 95~ (see -! Figure 13). The areas of u < 0 cm s are shade~t, and the areas of u > 60 cm s are stippled.

eastward pressure force which is maintained by the westward trades. Differences between s t r e n g t h of the EUC in the DA and the observations may well be due to the inconsistency between the wind stress and the east-west slope of the density field inferred from the data.

5.2. Hovm611er d i a g r a m s Hovm611er diagrams in this paper are time-longitude plots of any variable. Figure 15 is the equatorial Pacific time-longitude diagram for the SST anomalies from 1981 to 1988. The left side is the result of the DA and the right side is the CAC (Climate Analysis Center, NMC) analysis (Kousky and L e e t m a a 1989). The latter is based on the Reynolds' SST analysis. The a g r e e m e n t between the two analyses, i.e., the left and the right, is reasonable. Two E1 Nifios (warm phase), i.e., 1982/83 and 1986/87, and two La Nifias (cold phase), i.e., 1984/85 and 1988 are clearly identified in the 140~176 sector. The largest differences are found in t h e western Pacific, and

197

Figure 15. Longitude-time diagrams of SST anomalies along the equator. The data assimilation (left), and the NMC analysis (right). Contour interval is l~

overall the m a x i m u m or the minima are more intense in the DA t h a n in the Reynolds. The seasonal variation of the depth of the 20~ isotherm along the equator was already shown in Figure 9. The 20 ~ isotherm runs in the middle of thermocline along the equatorial belt throughout the Pacific, Atlantic and Indian Oceans. Therefore, it has been customary to use the depth of 20~ isotherm as a m e a s u r e of the available heat content in the upper ocean. In the NMC analysis, this depth has been monitored operationally as an index of ENSO (see L e e t m a a and Ji 1989). Figure 16 is the time series from 1985 to 1988 of the depth of the 20~ isotherm based on the DA and also the DA as calculated by Leetmaa and Ji (1989). The two diagrams agree with each other reasonably well, and yet, one wonders why they are different, considering t h a t both analyses are based on the same model and also the same DA scheme. The explanation may be attributed to a number of factors: a) the forcing data, b) the ocean model configuration, c) quality control and quantity of data, d) statistics used in DA. Next we examine the relation between the heat content and 20~ isotherm. The h e a t content defined in this paper is the part contained in the upper ocean from the surface to 248 m depth (the l l t h level), and computed by

Heat Content =

o ~ pCpTdZ -248m

(5.1)

198

20~

Jan

DEPTHS

DATA ASSIMILATION

NMC

85 Jul

Jan

86 Jul

Jan

87 Jul

Jan

88 Jul

140~

160 180 160 140 120 100 80~

LONGITUDE

140~

160 180 160 140 120 100 80~

LONGITUDE

Figure 16. Longitude-time charts of the 20~ depth along the equatorial Pacific Ocean, based on the data assimilation (left) and on the NMC (right). Contour interval is 20 m. The thick contours are for 140 m. where p = 1.02g'cm -3 and c = 4.187 J(g.deg) -1. Figure 17 shows HovmSller diagrams of heat content, along ~he equator, for SST nudging at the left, simulation in the middle and DA at the right. First it should be pointed out t h a t the i n t e r a n n u a l variability, due to the rise and fall of the thermocline, has a similar signature in both the DA h e a t content as seen in Figure 17 (right side) and the DA depth of the 20~ isotherm as seen in Figure 16 (left side). The anomalies of these quantities also resemble each other very well (not shown here). This demonstrates t h a t the depth of the 20~ isotherm is a good substitute for the variability of heat content, within the Tropics. The two SST nudging cases, the left and the middle panels in Figure 17, do not show good a g r e e m e n t with each other. The SST nudging case shows a pronounced annual cycle in the eastern part of the basin. Although this is a feature of the SST, observed annual variations in thermocline depth are weak and therefore, the annual cycle in h e a t content for this case is unrealistic. In the western part of the basin, the SST nudging case contains much more heat t h a n the other two cases. W h a t this reveals is the sensitivity of the subsurface thermal field to the wind field. In the SST nudging case the winds that drove the ocean model were generated from the

199

Jan

SST NUDGING

HEAT CONTENTS SIMULATION

Jan

85 Jul

Jul

Jan

Jan

86 Jul

Jul

Jan

Jan

87 Jul

Jul

Jan

Jan

88 Jul

Jul

140E 160 180 160 140 120 100 80W

140E 160 180 160 140 120 100 80W

DATA ASSIMILATION

37

~8

140E 160 180 160 140 120 lO0 80W

LONGITUDE

Figure 17. Longitude-time d i a g r a m of h e a t content along the e q u a t o r in the Pacific Ocean, based on the SST n u d g i n g (left), s i m u l a t i o n (middle) and on the d a t a assimilation (right). Contour interval is 109 J m -1. The thick contours are for 352 10 9 J m- 1.

a t m o s p h e r i c model. E x a m i n a t i o n of the model winds showed s t r o n g e r t h a n observed westerlies in the w e s t e r n equatorial Pacific d u r i n g DJF. This r e s u l t e d in convergence at the e q u a t o r and caused downwelling which influxed a large a m o u n t of h e a t into the ocean. This was possible since the model surface t e m p e r a t u r e was being forced to observed SSTs and therefore there was a positive h e a t flux into the ocean. In the coupled system, this wind bias increases evaporation and reduces the h e a t flux into the ocean resulting in SSTs t h a t are too cold. W h e n the observed wind field (NMC analysis) is used the systematic westerly bias is removed and a more r e a s o n a b l e h e a t content is simulated, as shown in the simulation and DA cases. It was also found t h a t the a t m o s p h e r i c model h a d a bias toward Trade winds t h a t were too strong. Without the SST n u d g i n g t e r m (eq. 2.2) this would have resulted in increased upwelling and SSTs t h a t would be too cold in the central and e a s t e r n Pacific. However, with the SST n u d g i n g t e r m e n o r m o u s a m o u n t s of h e a t h a d to be fluxed into the ocean model, to m a i n t a i n the observed a n n u a l cycle of SST, due to the systematic error in the model wind field. This resulted in the pronounced a n n u a l cycle in the h e a t content. Looking at the h e a t content over the central Pacific, a m u c h t i g h t e r g r a d i e n t m a y be observed in the DA as opposed to the other two cases. Once again this shows t h a t the ocean model produces too diffuse a thermocline, and t h a t this bias m a y be alleviated by the addition of subsurface d a t a in the DA. It a p p e a r s t h a t for the p r e s e n t s y s t e m subsurface t h e r m a l d a t a are necessary and t h a t using only the wind fields, either from the a t m o s p h e r e model or analysis, is not a d e q u a t e to derive the ocean subsurface

200 t h e r m a l structure. The conclusions could alter with improved wind products. Figure 18 shows time-longitude diagrams of the heat content anomalies along the equator, in the Pacific, from 1981 to 1988. As in the previous figure, the right panel is the DA; the middle is the simulation; and the left SST nudging. Where the total heat content showed distinct differences between the three assimilations, the anomalies, about their individual means, show t h a t the i n t e r a n n u a l fluctuations of the thermocline are similar. The most pronounced feature in Figure 18 is the two distinct episodes of E1 Nifios and of La Nifias, and t h a t these positive and negative anomalies of heat content propagate eastward. Comparison with SST anomalies, Figure 15, shows the two E1 Nifio signatures are evident, however, the 1986/87 event is more distinct in the SST anomalies than in the heat content anomalies, and for the whole period, the eastward propagation is dominant in the heat content, as opposed to some element of westward propagation in the SST. This coherence between SST anomalies and heat content anomalies appears to be essential to ENSO predictability. It would seem of p a r a m o u n t importance that the assimilation scheme, used to produce initial conditions for ENSO forecasting, must contain an accurate r e p r e s e n t a t i o n of the phase and amplitude of these propagating heat content anomalies. Forecasts of Nino-3 SST anomalies using initial conditions generated from the three assimilation ocean model runs, showed t h a t forecasts based on the DA scheme d e m o n s t r a t e d the most skill.

Figure 18. Longitude-time diagrams of ocean heat content anomalies along the equator from the SST nudging (left), the simulation (middle), and the data assimilation (right). Contour interval is 109 J m -1. Regions larger than 10 X 109 J m -1 and less t h a n -10 X 109 J m -1 are stippled differently.

201 6. SUMMARY AND C O N C L U S I O N S A global oceanic data assimilation system has been developed. This data assimilation system was created primarily for the initialization of coupled oceanatmosphere models for use in producing seasonal forecasts.We examine the fields produced by the assimilation procedure, over the ten year period t h a t was run, both for their temporal variability and as a climatological data set. The ocean d a t a used in the DA system consisted only of conventional t e m p e r a t u r e observations. For surface observations, 2 ~ x 2 ~ COADS data (Woodruff et al. 1987) were used. Vertical t e m p e r a t u r e profiles were incorporated from NODC and the U.S. Navy's MOODS dataset. While the coverage of the sea-surface t e m p e r a t u r e d a t a was quite good during this period, the vertical t e m p e r a t u r e profile data contained large gaps, particularly, in the equatorial region and in the Southern Hemisphere. The assimilation procedure was developed using a modified version of a global high resolution numerical model developed by Rosati and Miyakoda. This model is based on the primitive equations with the atmospheric forcing provided from the 12 hourly atmospheric analysis of NMC. The data were inserted into the model using a continuous insertion technique. A t e m p e r a t u r e correction field was created and inserted into the model solution. Instead of creating a correction field every timestep, it was found t h a t quite a computational savings could be realized, without compromising quality, by calculating a new correction for three consecutive timesteps and then apply the last correction for nine timesteps. The t e m p e r a t u r e correction was created by applying a statistical objective analysis routine to the differences between the model solution and the data in a 30-day window around the analysis timestep. The results from the assimilation system applied over the decade from 1979 to 1988 are encouraging. The SST fields compare well to the operational analyses in terms of large scale features. For smaller scales, the analysis captures some features not contained in the operational analyses, but it probably contains too much noise. At subsurface levels, the model solution is made much more realistic by the inclusion of data. The decadal means compared well in the upper ocean with the established climatologies of NMC SST analysis and Levitus. Their a g r e e m e n t is of particular interest since the basis of the DA is a dynamical model whereas the other two use objective analysis. This bodes well for ocean DA as a source for future climatologies, since one could expect that as the numerical model improves so will the analysis. Another advantage would be that the DA would also contain m e a n information about the general circulation. The interannual variability was compared to a variety of analyses for the tropical Pacific and it was found t h a t the DA captured the main features of the t h e r m a l structure including the ENSO signature. Although the velocity structure did not agree as well with observations. The ability of the DA to realistically simulate the variability on seasonal to interannual time scales has led to using it to provide initial conditions and verification fields for the experimental seasonal forecasts of the coupled model. As a first step toward producing a self consistent data assimilation system for coupled models, whereby the atmosphere and ocean model are in balance with one another, a coupled model scheme was investigated. This procedure was based on using the coupled model using Newtonian nudging of SST. The surface nudging systems produced very different heat contents as compared to the DA system. It would appear, t h a t at least for the present model, the wind field is not enough to define the subsurface t h e r m a l structure correctly. Although the simulation was more realistic than the SST nudging case. The only difference between the two runs was the wind forcing, thus indicating, not only, the sensitivity to atmospheric forcing, but also, t h a t the coupled model winds are not very good. The assimilation procedure may be improved by expanding the d a t a b a s e and by improving the quality control. Since the period of this study, only a fraction of the

202 TOGA TAO data was available. The assimilation should be substantially improved, in the tropical Pacific, with the addition of this database. Also, the database may be expanded by using other observation types, such as satellite derived SSTs, altimetry data, and current measurements. Changes to the assimilation procedure itself can also enhance the results. The statistics currently in use in the statistical objective analysis scheme are somewhat ad hoc. The first-guess error statistics are obviously deficient since they only vary in the meridional direction. The ocean dynamics are very different in regions with strongly sloping bottom topography or along the equator. The model's error characteristics are quite different in these regions because of this and should be accounted for in the first guess error covariances. With improved knowledge of the model's and the data's error characteristics many of the errors in the present system may be reduced. Any improvements to the numerical model solution feeds back to improve the analysis produced by the assimilation system. These improvements may be made directly to the model dynamics or physics, or may enter through atmospheric forcing. At this point in time, the effects of inaccurate atmospheric forcing on the assimilation are not completely known. Given the complexity of the whole system, i.e. the data and its' associated errors and distribution; the ocean model with unknown impacts from resolution and physics parameterizations; the assimilation procedure w i t h the difficulty in specifying error statistics; and the extent to which errors in the atmospheric forcing cause problems; it is extremely difficult ascertain where things go wrong and why. It seems obvious, however, that one can still produce useful and reasonable solutions from the present DA system. The incorporation of all potential enhancements can only improve the results.

7. R E F E R E N C E S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Ghil, M. and P. Malanette-Rizzoli, Adv. Geophys., 33 (1991) 141. Derber, J. and A. Rosati, J. Phys. Oceanogr., 19 (1989) 1333. Rosati, A. and K. Miyakoda, J. Phys. Oceanogr., 18 (1988) 1601. Woodruff, S.D., R.J. Slutz, R.L. Jenne and P. M. Steuer, Bull. Amer. Meteor. Soc., 68 (1987) 1239. Reynolds,R.W., C.K. Folland, and D.E. Parker, Nature, 341 (1989) 728. White, W.B., G.A. Meyers, J.R. Donguy, and S.E. Pazan, J. Phys. Oceanogr., 15 (1985) 917. Kessler, W.S., Proceedings of the Western Pacific International Meeting and Workshop on TOGA COARE, held at Centre ORSTOM de Noum~a, New Caledonia, May 1989, ed. by J. Picaut, R. Lukas and T Delacroix 185. Bauer, R.A., FNOC (1985) 477 pp. [Available from Fleet Numerical Oceanographic Center, Monterey, California 93940]. Gandin, L.S., Gidrometeor, Isdat., Leningrad (1963) 242 pp. [English translation, Israeli Program for Scientific Translations, Jerusalem, 1966]. Sasaki, Y., J. Meteor. Soc. Japan, 36 (1958) 77. Lorenc, A., Quart. J. Roy. Meteor. Soc., 112 (1986) 1177. Levitus, S., Climatological Atlas of the World Ocean (1982). Carton, J.A. and E.C. Hackert., J. Phys. Oceanogr., 20 (1990) 1150. Reynolds, R. W., NOAA Tech. Rep. NWS 31, Washington, DC, (1982) 33 pp. Miyakoda, K., J. Sirutis, A. Rosati and R. Gudgel, Proceedings of Workshop on Japanese-Coupled Ocean Atmosphere Response Experiments, 23-24 October 1989, ed. by A. Sumi (1989) 93. Gordon, C.T. and W. F. Stern, Mon. Wea. Rev., 110 (1982) 625. Hellerman, S. and M. Rosenstein, J. Phys. Oceanogr., 13 (1983) 1093.

203

18 Tsuchiya, M., J. Oceanogr. SOC.Japan, 11 (1955) 1. 19 Delcroix, T., G. Eldin, and C. Henin, J. Phys. Oceanogr., 17 (1987) 2248. 20 McPhaden, M.J. and M.E. McCarty, NOAA Tech. Rep. PMEL-95, Seattle WA. (1992) 118 pp. 21 McCarty, M.E. and M.A. McPhaden, NOAA Tech. Rep. PMEL-98, Seattle WA, (1993) 64 pp. 22 Reynolds, R.W., J. Climate, 1 (1988) 75. 23 Reynolds, R.W. and D.C. Marsico, J. Climate, 6 (1993) 114. 24 Kalnay, et al. , MC Office Note 401 (1993). 25 Ropelewski, C. and M.Halpert, Mon. Wea. Rev., 114 (1987) 2352. 26 Firing, E., R. Lukas, J. Sades, and K. Wyrtki, Science, 222 (1983) 1121. 27 Halpern, D., J. Geophys. Res., 92(C8) (1987) 8197. 28 Philander, S.G.H., and A.D. Seigel, Coupled Ocean-Atmosphere Models. J.G.J. Nihoul, Ed., Elsevier Oceanography Series, No. 40 (1984) 517. 29 Kousky, V.E. and A. Leetmaa, J. Climate, 2 (1989) 254. 30 Leetmaa, A. and M. Ji, Dyn. Atmos. and Ocean, 13 (1989) 465.

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Tropical Ocean Applications

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Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All fights reserved.

207

Tropical data assimilation" Theoretical aspects Robert N. Miller a and Mark A. Cane b aCollege of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon, 97331-5503, USA bLamont-Doherty Earth Observatory of Columbia University, Palisades, New York, 109648OOO, USA Abstract

In this chapter, some of the theoretical issues underlying the application of optimized methods of data assimilation to the tropical oceans are discussed. By "optimized" methods of data assimilation, we mean methods which minimize some objective measure of error. Methods formulated in this way are cast in terms of statistical hypotheses, which can be tested by standard statistical methods. The efficacy of simple models of the tropical ocean has been a major advantage in the practice of data assimilation for this region. We discuss physical reasons for the effectiveness of these simple models, but also remind the reader that much of this apparent simplicity stems from the nature of the agenda in tropical oceanography. Since the focus in the community is on phenomena relevant to ocean-atmosphere interaction and climate prediction, the highest priority is large scale, low frequency low latitude motions. More complex models are necessary for reasonably accurate descriptions of the dynamics of the tropical ocean on shorter spatial or temporal scales, or more than about 10~ from the equator. We discuss some of the theory of the data assimilation methods as such, and conclude that the crucial research issues revolve around the prior error estimates that largely determine the product of any practical data assimilation method.

1. I N T R O D U C T I O N In the emerging field of ocean data assimilation, direct application of optimized methods has come furthest in application to the tropical ocean. By "optimized methods" we mean those based on minimization of some measure of error, subject to given assumptions. One important reason for this is purely practical: as data assimilation in numerical weather prediction is driven by the demand for operational weather forecasts, data assimilation in the tropical ocean (at least the Pacific) is driven by the demand for prediction and analysis of climate change, due to the well known if imperfectly understood influence of the tropical ocean on world climate. An equally important reason is the emphasis in the tropical oceanography community on the large spatial scale, long time scale, low latitude

208

phenomena which play an important role in seasonal to interannual climate variability. It is the problem of climate variability that motivates virtually all data assimilation efforts in the tropical oceans. To a greater extent than in other venues, researchers in tropical oceanography share a common goal. As is true elsewhere in the ocean, the tropics contain energetic and interesting mesoscale and fine scale motions, albeit with special regional forms (e.g., the well known Legeckis waves; see, e.g., Weisberg, 1987). The complex physics of these motions are often nonlinear and even chaotic, and dense observations in time and space are necessary to analyze them adequately. However, none of this seems to matter greatly in realizing the common goal, which concerns only the large scale, low frequency variations. On these scales, simple models have been found to capture much of the observed variability, making implementation of optimized data assimilation methods practical for real applications. The sparsity of the data makes the use of optimized methods particularly advantageous. Practical data assimilation systems are the results of compromises made under pressure of computing resource limitations. In the mid-latitudes and in numerical weather prediction, data assimilation systems have consisted of large resource-intensive models and highly simplified data assimilation schemes; see, e.g., Ghil and Malanotte-Rizzoli (1991) or Daley (1991). The purpose of these systems is often to analyze results of specific observation programs such as SYNOP in the gulfstream region. In the tropics, simple models capture much of the physical phenomena of interest, so advanced data assimilation schemes can be implemented within the limits of available computational resources, and the sparsity of data relative to the natural time and space scales makes the use of optimized schemes necessary in order to extract as much information as possible from the observations. In this chapter, we shall concentrate on optimized methods. Because the minimization problems by which optimized methods are defined are difficult to solve in practice even for simple models, much of the literature has been devoted to techniques for solving the minimization problem, but the quality and usefulness of the computed solution depends upon the form of the function (usually known as the "cost function") to be minimized. Cost functions are necessarily built around error estimates; it is the estimated errors in the model and the observations which ultimately determine the relative influences of observation and model output in the final product. In some cases the estimates may be implicit, but they nonetheless control the analysis. We shall therefore concentrate on the error estimates in real models of the tropical ocean. Section 2 contains a brief survey of optimized methods, with particular attention to the problems of the tropical ocean. In section 3 we present quantitative evidence for our still controversial assertion that the large spatial scale slow time scale motion in the tropical ocean is well described by simple models. We compare results from different models of the tropical Pacific, Atlantic and Indian oceans to find the simplest model of each ocean which captures the phenomena of greatest interest, and to provide a basis for estimating the errors in these models. Since statistical hypotheses about the model and observation errors are the defining quantities of data assimilation methods, we devote section 4 to error estimates and their consequences. Section 5 contains discussion and summary.

209 2. D A T A A S S I M I L A T I O N MODELS

IN THE

CONTEXT

OF TROPICAL

OCEAN

2.1. O p t i m i z e d M e t h o d s Optimized d a t a assimilation schemes are based on minimization of some measure of error, usually some weighted least squares formulation. Typically, one starts with a model in the form of an evolution equation:

(1)

ut = L u + F ( x , t ) + q

where u is the model state function, L is typically a partial differential operator and F is the forcing by wind and surface fluxes. The function q is often called the system noise; it represents the unknown error in either the model or the forcing. For the present purpose, we shall take L to be linear, and make appropriate note of the simplest generalizations to nonlinear models. We presumably have a vector of measurements d which is related to the state vector u by: d - Hu + e

(2)

where e is the measurement noise and H specifies the relation of the model state to the quantities being measured. In this general setting, H can be a differential or integral operator. It is i m p o r t a n t to note that the measurement noise contains neglected physics as well as instrument error. We may illustrate this with a simple example. Consider a linear long-wave model consisting of evolution equations for Kelvin and Rossby waves. Suppose the d a t a consist of ten day running means of sea level height anomalies from tide gauge stations. The actual error in measuring the sea level height anomaly is very small, but on ten day time scales, the sea level height is strongly influenced by physical effects such as wave setup due to local wind forcing that cannot be represented in terms of long waves. If this signal were assimilated, the effects would propagate as long waves, which would result in systematic errors in the large scale analysis. It is therefore necessary to formulate a prior estimate of the magnitude of the contribution of the long waves to the sea level height anomaly, and assign all other contributions to the measurement noise term. An equivalent view is that the observation is a measurement at a point influenced by all the physics and forcing at all scales, whereas the model is a model for only a subset of physics and scales. If one wants it all then the system noise q should be augmented to account for what the model misses. If, as assumed above, one wants only the larger space and time scales, then the measurement error e should be augmented to include the sampling error inherent in measuring at a point only. In most cases of practical interest, H can be expressed as an integral operator with typical form: Hu -

/0 /o

G(xl, x2, tl, t2)u(x2,

t2)dx2dt2

where f~ is the spatial domain, x~ and tl are the measurement locations and times and G could be some convolution kernel. In the case of a single point measurement,

210 G ( x l , x 2 , tl, t2) is proportional to 6(x~ - x2, t~ - t2), where 6 denotes the Dirac 6 function. Bennett (1990) described the construction of a kernel G which would yield Kelvin or Rossby wave amplitudes from values of the dynamic height anomaly observed along a ship track. D a t a assimilation proceeds by finding the state function u which minimizes a positive definite functional J, usually known as the cost function or objective function: J(u) "--"

2

+

~

'

1/o/o

-~9 ~s

(u(x~, o) -

,

,

,

uo(x~))rV(x~,x~)(u(x~,o)

- uo(x~))dxldX~

(3)

where u0(x) is the prior estimate of the initial condition, q is the estimate of the system noise and e is the estimate of the observation noise. W, w and V are the weights given to residuals relative to the analysis of the forecast, data and prior initial conditions. If W, w and V are chosen to be the inverses of the system noise covariance function, the measurement noise covariance and the error covariance function of the initial estimate u0, then the minimizer of J ( u ) will be the estimate which is best in the least-squares sense. If in addition the model and measurement functions are linear and the system noise, the measurement noise and the error in the estimate of the initial conditions are Gaussian, then the minimizer of J ( u ) will also be the maximum likelihood estimate of the system conditioned on the observations; see, e.g., chapter 5 of Jazwinski (1970). Of course, all of these desirable consequences follow from the assumptions that we have accurate knowledge of the statistics of the errors, which, in general, we do not. The cost function defined in (3) contains the assumption that the different noise sources are uncorrelated. If the model was formally or informally "tuned" against the same data, this is unlikely to be true. Moreover, the mismatch in scales between model and data noted above may tangle the measurement and model noise. Nonetheless, every ocean data assimilation scheme we know of contains this assumption. Relaxing this assumption would involve explicit estimates of the covariances of the measurement and system noises. Such estimates would be difficult in the extreme to construct and verify. In fact, even with this assumption, available data sets are barely sufficient to validate the crude error models in common use. The matrix valued function W is often simplified by the assumption of homogeneity, i.e., W ( X l , t l , X 2 , t2) : ~/'(Xl - x2, tl,t2). This is hard to justify anywhere in the ocean, since proximity to boundaries can be expected to affect model errors. It is even less justifiable in the tropics, where distance from the equator exerts such a strong influence on dynamics. It is customary to collapse first integral in (3) to an integral over space alone by assuming that the system noise is white in time, i.e., W ( x ~ , t l , x 2 , t2) = IYd(x~, x2)~(tx t2). This assumption is almost always made in practice, although it cannot be true unless the model has no systematic flaws. Yet most modelers would acknowledge that there are certain situations where their model will make the same sort of mistake for many consecutive time steps. Chan et al. (1995) is the only study we know of in which this

211 assumption is dropped, and their results show a marked improvement in the assimilation. In the case of a time series of observations, it is also customary to assume that the measurement errors are white in time. This assumption is certainly false in general. In our general treatment of the minimization problem, we retain the continuous form (1) of the model equations, rather than the discretized form used in practice so that we can apply the methods of the calculus of variations (see, e.g., Courant and Hilbert, 1953), which allow us to write, with some abuse of notation:

eTwG

OJ/Ou

=

-At - L*A

o J / O u ( x , o)

=

(u(x, o) - uo(x))

-

v -

o).

(4)

where L* is the adjoint of L. In the case of a nonlinear model, L* is replaced by the adjoint of the linearization of the model evolution operator about the current estimate of the state vector. Requiring the gradient of the cost function J to vanish results in: At + L*A A(x, 0)

=

=

--eTwG (u(x, 0) - U0(x))Tv

(5) (6)

These two equations, along with the final condition: A(x, T) = 0

(7)

and the definition of the adjoint variable A: ut- Lu-

F = W-1A T

(8)

make up the classical Euler-Lagrange equations. The most common means of solving the minimization problem are the adjoint methods, which by now have a considerable history in both the numerical weather prediction (e.g., Courtier and Talagrand, 1987) and oceanographic (e.g., Long and Thacker, 1989a,b; Tziperman and Thacker, 1989) communities. In most of the methods described in the literature, the actual computation is simplified considerably by setting q = 0, i.e., assuming that the model and forcing fields are perfect. These methods are known as strong constraint methods, after Sasaki (1970). This formulation can be recovered from the Euler-Lagrange equations (5) through (8) by taking the limit as W -1 --+ 0. Substitution of the definitions q = ut - L u - F and A -- q T w into the first integral in (3) shows that the adjoint variable becomes the Lagrange multiplier in the usual strong constraint method. The gradient of the cost function with respect to the initial state vector u(x, 0) is calculated from (4) given the initial value of A, which, in turn, is obtained by integration of (5) backward in time from (7). Given this gradient, the cost function is typically minimized by the application of a descent method. Conjugate gradient methods are commonly used for this purpose. Without further constraint, the conjugate gradient method is not efficient, since it must choose a search direction in state space from a number of candidate directions equal to the state dimension of the model. Since the number of independent degrees of freedom in the data is almost always much smaller than the number of possible search directions, the value of the cost function will not change as a result of searching parameter space in

212 most directions. In symbolic language, for most directions z, z - V J ~ 0. Bennett (1992) describes a formulation of the variational problem in which the solution can be shown to lie in a linear space of dimension no greater than the number of observations. The basis of this space consists of the representer functions. For details, see Bennett (1990, 1992). Since the final estimate of the system state in a strong constraint method is an exact solution to the model equation with q = 0, the only free parameters are the initial conditions, and physical parameters such as drag coefficients, wave speeds and biases. These may be estimated by adding additional terms to the cost function. One such example can be found in the work of Greiner and P&igaud (1994). They replaced e in the above cost function with d - H u - Ar where d was a sequence of thermocline displacement fields derived from altimetric data and A r was a function of space alone, to be estimated in the course of the assimilation process. For most applications, the tropical ocean is a forced dissipative system. In this, as in any such system, the influence of the initial conditions decays; the memory of the tropical ocean for its initial conditions is no longer than a year. Strong constraint simulations in which initial conditions are the only controls are therefore useful for assimilation runs of a year or less. A broader conceptual problem with strong constraint methods is the'well known fact that the error in actual forcing data sets such as wind stress is substantial. Thus even if one were to maintain that the numerical model were practically perfect, q is still significantly different from zero. The principal motivation for imposing q - 0 is computational simplification: without it the backward step (5) and the forward step (8) are fully coupled. (Bennett (1992) suggests an alternate approach to decoupling based on representers.) Also, there is something to be said in favor of obtaining an analysis that exactly satisfies the model equations. But there is less to be said for it in studying the upper layers of the tropical oceans, where misfits to the model equations may be "eliminated" a posteriori simply by absorbing them into the poorly known forcing fields, i.e., changing F to F + q. These criticisms notwithstanding, strong constraint adjoint methods have been applied with some success, most notably by Greiner and P&igaud (1994) in the Indian ocean and Sheinbaum and Anderson (1990a,b) and Smedstad and O'Brien (1991) in the Pacific. Sheinbaum and Anderson's experiments had a duration of six months; Greiner and P&igaud limited their experiments to a single year, and their results imply strongly that one year is the limit in that situation. Smedstad and O'Brien, noting explicitly the limited influence of initial conditions on the tropical ocean over long time periods, used the Kelvin wave speed, considered as a function of space and time, as a control. The second integral in (3) represents the square of the norm of the deviation of the initial field from the first guess. If this term is omitted, the resulting initial field will have spurious high wavenumber content; this was noted by Sheinbaum and Anderson (1990a, b). The origin of this roughness in the estimated initial field is now understood (Bennett and Miller, 1991). Small, high wavenumber perturbations in the initial field will have little effect on subsequent data misfits, especially in dissipative systems. Smooth and rough fields will therefore result in nearly identical values of the cost function, and the roughness will not be eliminated in the minimization process. Sheinbaum and Anderson were able to achieve smoother fields in one of three ways: by limiting the number of iterations in the optimization process; by including explicit

213 smoothing terms proportional to fa(Vu)2dx, and by including a term similar to the second integral in (3). Their interpretation of the phenomenon appears in Sheinbaum and Anderson (1990b). The review by A. J. Busalacchi in this volume shows the need for smoothness constraints to be a recurrent theme in the application of variational (adjoint) methods to tropical oceanography (cf. Long and Thacker (1989a,b) and Moore et al. (1987) in addition to Sheinbaum and Anderson (1990b)). In addition to dissatisfaction with the structure of the solutions obtained, Long and Thacker (1989b) found that allowing too much structure made the search for the optimal solution poorly conditioned. The need for smoothness constraints is related to the issue of regularity studied by Bennett and Budgell (1987). They showed that the Kalman filter will create unrealistic local features if the noise model has too much power at small scales. 2.2. T h e K a l m a n S m o o t h e r and t h e K a l m a n F i l t e r The Kalman smoother is a particular class of algorithms for minimizing the cost function given by (3) (cf., e.g., Jazwinski, 1970) in the general case in which the errors in the model, initial conditions and measurements are all non zero. Along with the solution of the minimization problem, it provides a theoretical estimate of the analysis error based on the prior error estimates W, V, and w. The Kalman filter can be derived as a recursive algorithm for minimizing the error (i.e., finding the best estimate of the state) at a time T based on all information at all times t _< T. The algorithm is recursive in that it produces the best estimate of the state at time T from the best estimate of the state at time T-1 and new information (measurements) at time T. As with the smoother, it produces a theoretical estimate of the analysis error. The Kalman filter is most commonly written in discretized form, since it is implemented in discrete form in practice. We have a forecast model"

Ukf+l- Lu~ + Fk

(9)

where L is the model evolution operator, discretized in time and space. The subscript k refers to quantities evaluated at the k th time step, i.e., t = tk and the superscripts f and a refer to the forecast and analyzed quantities respectively. Forecast quantities at t = tk contain the impact of all data assimilated at times up to but not including the present. Analyzed quantities are those which result from assimilation of data at the present time. The forecast error covariance Pk/ evolves according to:

Pk/+l -- LP~LT + Q

(10)

where Q = AtW -1 The assimilation step is given by:

U~+1 : Ukf+l -~- Kk+l(dk+l- Hk+lUk/+l)

(II)

where dk+l is the vector of observations at time tk+1 and Hk+l is the operator which transforms the state vector at time tk+l into the vector of observed quantities. Kk+1 is the Kalman gain: i k + , -- P~+IHLI(Hk+IPL+IHLI q- w-l) - '

(12)

214

and the new covariance matrix P~+I is given by: P;+I-

( I - gk+lHk+l)PIk+l.

(13)

The methods of successive corrections and optimal interpolation can be viewed as approximations to the Kalman filter. In devising a successive correction scheme, one begins by performing a forecast step (9). The updated field is then calculated iteratively according to: u(J+l) _ (j) K(J+I) ( d k + 1 - Hk+lUk+l) . (j) k+l = Uk+, + where 1Uk+ "(j) is the analysis following the jth correction, Uk+ 9(0)1 =Uk/+ 1 a n d K (j) i s a p r e d e termined sequence of gains. Successive gains g (j) usually have decreasing spatial scales. This was the method used by Moore et al. (1987). The procedure can be cast in the same form as (10) with K a polynomial function of the K(J)'s. Unlike the Kalman filter, K is fixed once and for all: it is not adjusted as the expected error in the first guess changes over time. Optimal interpolation, the data assimilation method most commonly used in operational numerical weather prediction, involves the use of an assumed form of the forecast error covariance P. The gain K is calculated from (12) each assimilation cycle. In both successive corrections and OI, the necessity of computing the explicit evolution of P from (10) is avoided. In OI as in the Kalman filter, the gain matrix Kk will change as the array of observations changes because this changes Hk. With the Kalman filter Kk might change even if Hk doesn't because P will change as the assimilation evolves. For constant Hk and a model which describes a forced dissipative system, P will eventually approach a constant vahle asymptotically, at which time computing it via (10) is no longer necessary. At this point the filter is very much like OI. One strategy for reducing the computational cost of the Kalman filter is to use the asymptotic P for all time (e.g., Fu et al., 1993; see below for additional discussion). However the optimized solution is calculated, explicit estimates of the error statistics W, w and V are required. Before turning explicitly to errors, we must discuss the models to which these methods will be applied.

3. E F F I C A C Y

OF SIMPLE MODELS

It is a fact that simple models are unnaturally effective in the tropical oceans. Using a wind driven one layer shallow water model, Busalacchi and O'Brien (1980, 1981) were able to simulate the major features of interannual variability in the Pacific. Using a wind driven two mode linear shallow water model restricted to long waves and low frequencies (Cane and Patton, 1984), Cane (1984) was able to simulate the major features of the seasonal cycle in the Pacific. The standard by which success of these and most other models has been judged is comparison to pressure anomaly data in the form of sea level height anomalies or dynamic height increments. Sea level data carry the E1 Nifio-Southern Oscillation (ENSO)

215

signal, and will therefore remain important for assimilation. Velocity and subsurface temperature data undoubtedly will become increasingly useful as more faithful models are developed, but at present the most valuable data sets are sea level data sets, including data from a variety of sources, from tide gauges to moored instruments to satellite altimeters. In seeking a natural explanation for the effectiveness of simple models, the most striking feature is the extraordinarily sharp and shallow tropical thermocline. It is always within 200 m of the surface, and temperature changes of 10~ within 50 to 100 m are typical. Thus the real tropical oceans approximate the theorists' two layer ocean, and behave accordingly. Motions in the upper l a y e r - the upper few hundred meters- may be calculated to reasonable accuracy by ignoring all motions in the abyssal l a y e r - the nearly 4 km below the thermocline. This simplifies tropical models and drastically reduces data requirements. Still, from any map of ocean surface currents it is obvious that the tropical current system is highly structured and strong, with speeds and mass transports of the same order of magnitude as the western boundary currents. Below the surface lies the equatorial undercurrent; the extensive literature devoted to this highly nonlinear feature attests to its fascination. Nonetheless, it seems that the upper layer integrated mass transports of these currents may be calculated from linear theory to acceptable accuracy. Recall that Sverdrup (1947) theory was first formulated as an explanation for the north equatorial countercurrent in the Pacific. It seems plausible to us that while nonlinear corrections may well be important for calculating the structure of currents, they are unimportant for the integrated transports of these currents. Off the equator, currents are not strong enough to violate time dependent Sverdrup theory (i.e., linear quasi-geostrophic theory), which may be viewed as a theory for the upper layer integrated transports. This theory can be applied right through the equator with the addition of one mode that is not quasigeostrophic, the equatorial Kelvin wave. In the literally hundreds of papers touching oil seasonal and longer time scale transports, including those employing the most complex ocean general circulation models (GCM's), explanations rarely go beyond linear low frequency dynamics that include only the equatorial Kelvin and Rossby waves. The equatorial undercurrent deserves further comment. In the Pacific it has been observed to attain speeds up to 140 cm/s and transports of 40 Sverdrups (1 Sverdrup = 106ma/sec), yet models that treat it poorly or not at all (either deliberately or inadvertently) still are perceived to be adequate for seasonal to interannual studies. With the exception of Pedlosky (1987), all theories treat it as the centerpiece of a special boundary layer circulation that is closely confined to the equator, having no connection to mass exchanges at higher latitudes. Net mass transports near the equator are constrained by the need to match the mass balances at higher latitudes; these are constrained by dynamics that assign no role to the undercurrent. The undercurrent flows eastward along the equator, perhaps entraining water meridionally, but surely losing mass to the surface layers. Its eastward transport is compensated by westward surface (and perhaps subsurface) flows on or close to the equator, and it is nearly spent by the time it arrives at the eastern boundary. The larger scale circulations do demand that mass be carried from west to east along the equator, but this task is assigned to equatorial Kelvin waves, not the nonlinear undercurrent.

216 The most compelling reason for data assimilation in the tropical oceans is the prediction of climate variations, especially those associated with the ENSO phenomenon. These are ocean-atmosphere interactions, and, in the end, only one oceanic variable matters: sea surface temperature (SST), which is all that the atmosphere sees of the ocean. SST is determined by horizontal advection in the surface mixed layer, upwelling, entrainment into the mixed layer, and surface heat fluxes. On annual time scales the last of these is the most important. It is also the one least affected by the quality of the ocean model since it is independent of all ocean properties except SST. Equatorial mixed layers have far less seasonal and other variability than those in higher latitudes, so relatively simple mixed layer physics formulations are sufficient. For example, a constant depth mixed layer yields decent simulations of annual (Seager et al., 1988) and interannual (Seager 1989) SST variations. The same formulation is used in the Zebiak and Cane (1987) coupled model. Simple models should be at a disadvantage in computing advective effects. Horizontal advection is of some importance, and vertical advection is thought to be an essential link in the ENSO process (e.g., Seager, 1989): variations in thermocline depth give rise to variations in the temperature of the water upwelled and then entrained into the surface layer. As this plays out in the eastern equatorial Pacific it generates the SST variations characteristic of the ENSO cycle. In principle, GCM's should do a better job of simulating this process, but at the present state of the modeling art they are hampered by an inability to simulate the mean state correctly. The simple models (e.g., Zebiak and Cane, 1987) specify this mean and compute only the perturbations, a much easier task. The judgment that simple models perform at state of the art levels reflects not only the state of models, but of data. Tropical forcing data - wind stress and heat flux - have large errors (e.g., Halpern and Harrison, 1982; Blumenthal and Cane, 1989; Busalacchi et al., 1993 and references therein). That makes it difficult to conclude rigorously that one ocean model is better than another, since so much of the failure to agree with oceanic observations could be due to errors in the forcing. Of course, the problem is compounded by the scarcity of data to verify against. Tile complex models may indeed perform better than the simple ones, but it is hard to tell. The sophisticated comparison of models of the tropical Atlantic performed by Frankignoul et al. (1995) is one of the few to establish that a complex model is significantly better than a simple one. In the Pacific, it is even reasonable to neglect the complex coastal geometry of South America in the east and Australia, New Guinea and Indonesia in the west, and model the basin as a rectangle. This may be a result of the sheer size of the Pacific, a.s much ms anything else: much of the basin is far from boundaries relative to the relevant scales of forcing and wave motion, but it is remarkable how well models with rectangular geometry perform at reproducing the sea level height anomalies at coastal tide gauge stations in South America and New Guinea. Most, but not all, of the simple models are formulated by applying separation of variables to the linearized primitive equations of motion on the equatorial 13-plane. Separation of the vertical dependence of the motion from the horizontal and temporal results in a system of discrete vertical modes, with amplitudes governed by a set of equations formally identical to the shallow water equations for each vertical mode. This necessarily implies that such models contain the a.ssumption that the wave speed Cm corresponding

217 to the m th vertical mode is constant for each m (but see, e.g., Smedstad and O'Brien, 1991), despite the fact that the thermocline in the tropical Pacific shoals toward the east to the extent t h a t the depth of the thermocline off the coast of Peru is only half that off New Guinea. This would imply a variation in the wave speeds of 40% over the Pacific basin. Wave speeds for models of the Pacific are typically calculated from data taken around 160~ (see, e.g., Cane, 1984), so they are representative of most of the interior of the basin. The tropical ocean is a forced-dissipative system on large scales, with most of the dissipation taking place near boundaries. Waves do not propagate undisturbed across the entire basin; for the most part they are forced by wind, which itself varies on large scales. If we figure that the error in the wave speed is 20%, and the wave propagates over half the basin, then we would expect the phase error of the model Kelvin wave in the Pacific to be perhaps 10 days, figuring 3 months to cross the entire basin. To demonstrate the effectiveness of simple models, we can compare results from different models of the tropical Pacific, Atlantic and Indian oceans. Table 1 contains a summary of the performance of three models of the variation of the dynamic topography of the tropical Pacific for the Geosat period, 1986-1988: a simple linear long wave model, based on direct calculation of Kelvin and Rossby wave amplitudes (Fu et al., 1993); a coarse-resolution finite difference model based on a linear long wave shallow water model (Miller et al., 1993, 1995), and a GCM based on the primitive equations (Chao et al., 1993). All of these models were driven with surface wind stresses computed from the monthly pseudostress data from Florida State University (Stricherz et al., 1992; hereafter "FSU"). Results shown in Table 1 are comparable for the three models. Because the time series are so short it is not possible to establish that the differences between models are significant. Errors in wind forcing of about 2 m/sec will result in errors in the sea level height and dynamic height which are as large as the differences between observations and the models driven by available wind analyses, and much larger than the effects of neglected nonlinearity or coarse resolution. Miller (1990) performed simulation experiments in which a simple model was driven by two different wind data sets: the FSU winds, and the FSU winds to which a white sequence of perturbation fields with given spatial covariance was added. The differences between the sea level height anomalies produced from these two simulations were comparable to the differences between the actual observations and the output of the model driven by the FSU winds. Harrison et al. (1989) performed experiments with a detailed nonlinear model of the tropical Pacific and five different wind products, and found differences among the resulting dynamic height fields that are comparable in magnitude to the error estimates calculated by Miller and Cane (1989). Consistent results were also obtained from comparisons between different wind products and model responses to those wind products by Busalacchi et al. (1990, 1993). When the data from the T O G A Atmosphere-Ocean (TAO) array are fully analyzed and when the anticipated scatterometer data become widely available, the errors in the wind fields may be reduced to the point that some other source of error becomes the dominant one, but for studies of the tropical Pacific ocean prior to at least 1990 or so, error in the wind dominates all other sources of error, for the purpose of estimation of the seasonal to interannual variation of dynamic topography. Direct model comparisons for the Atlantic and Indian oceans are not so easily culled

218 Table 1: Comparative performance of three different models of sea surface height variability during the Geosat period 1986-88: FFM, the simple Kelvin-Rossby wave model used by Fu, Fukumori and Miller (1993); MBH, the coarse-grid version of the shallow water model of Cane and Patton (1984), as implemented by Miller, Busalacchi and Hackert (1995); CHP, the primitive equation model of Philander et al. (1987), as implemented by Chao, Halpern and P~rigaud (1993). Correlation coefficients and RMS differences between model output and data from island tide gauge stations are presented. Temporal means are subtracted from both model output and data. RMS differences are presented in cm.

Station

FFM

MBH

Corr RMS Diff Corr RMS Diff Rabaul 0.61 * 0.84 6.59 Nauru 0.26 * -0.18 8.37 Ponape 0.62 * N/A N/A Christmas 0.68 * 0.80 7.82 Santa Cruz 0.83 * 0.80 4.55 Callao N/A N/A 0.49 6.29 Kapingamarangi 0.67 * 0.78 5.59 Tarawa 0.43 * -0.25 8.09 Canton 0.67 * 0.38 9.31 Fanning N/A N/A 0.71 6.88 Truk 0.72 * 0.71 7.90 Kwajalein 0.75 * 0.64 5.89 Yap N/A N/A 0.79 10.17 Honiara 0.80 * 0.92 6.14 Majuro 0.57 * N/A N/A a,,N/A,, indicates that no results were presented for that site. * Fu et al. did not report RMS differences.

CHP Corr N/A a N/A 0.77 0.86 0.91 N/A N/A N/A 0.39 N/A 0.44 0.73 0.72 N/A 0.62

RMS Diff N/A N/A 7.0 6.5 3.5 N/A N/A N/A 6.5 N/A 11.2 6.0 8.2 N/A 6.8

from the literature. There is nothing in either of these oceans which corresponds to the Pacific tide gauge network, and the corresponding analysis of the relation of the tide gauge data with other dynamical quantities (e.g., Rebert et al., 1985). As in the Pacific, linear models reproduce much of the variability of the dynamic topography in the tropical Atlantic on seasonal and longer time scales (Busalacchi and Picaut, 1983; du Penhoat and Treguier, 1985; du Penhoat and Gouriou, 1987; Reverdin and du Penhoat, 1987). Models of the Atlantic have been verified against the Seasonal Response of the Equatorial Atlantic/Fran(;ais Oc6an Climat dans l'Atlantique (SEQUAL/FOCAL) data set, as well as some which have been compared to inverted echo sounder (IES) data and to altimetric data from Geosat. Du Penhoat and Gouriou compared tile results of the linear model of Cane and Patton

219 (1984) driven by two different wind data sets with SEQUAL/FOCAL observations, and, in their conclusions, attributed most of the discrepancies to errors in the wind forcing data. Longitude-time plots of model dynamic height along the equator for 1982 through 1984 from the experiments performed by du Penhoat and Gouriou are shown in figure 1. Panels a and b of figure 1 correspond to the responses of the model to two different wind data sets. One, "SPB", was derived by objective analysis of ship observations, averaged monthly on 5~ x 2~ grid boxes; see Servain et al. (1985) and Picaut et al. (1985). The other, "FSIIB," was derived from a combination of winds from the ECMWF forecast model and the SPB monthly mean winds.

9 8 4

9 8

o

85--

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2 50*W

a

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J

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Figure 1: Time longitude plot of surface dynamic height relative to 500m along the equator, as simulated by a linear model driven by two different wind data sets. a. SPB forcing, b. FSIIB forcing. Reproduced from Figure 4 of du Penhoat and Gouriou (1987) with permission of the American Geophysical Union. The results of the linear study are comparable to those from the nonlinear model study of Reverdin et al. (1991), who used a full primitive equation model with realistic coastlines in a region from 65~ to 15~ and 40~ to 50~ Figure 2 shows longitudetime plots from that study of dynamic height along the equator. Panel a is derived from objective maps of the SEQUAL/FOCAL data, and panel b is the corresponding plot from the simulation. The weakening of the zonal slope in the eastern part of the basin in early 1984, which appears in both the analysis (panel a) and the nonlinear simulation (panel b) is also present in both panels of figure 1, but the recovery of the slope in the east, along with its decay in the west is better represented in panel lb than in la. The weakening of the slope in the western part of the basin in the spring of 1984 also appears in the analysis, the nonlinear simulation and both linear simulations.

220

Figure 2: Time-longitude plot of surface dynamic height relative to 400m along the equator, a: objective analysis of observed data b: result of nonlinear simulation. Redrawn from figure 23 (panels a and b) of Reverdin et al. (1991) with permission of Elsevier Science Ltd. Figures 3 and 4 show computed and observed zonal dynamic height slope. Raw dynamic height differences calculated from FOCAL data are shown as crosses in figure 3. Results of an attempt to correct for asynopticity of the FOCAL data by temporal interpolation are shown as asterisks. This comparison highlights one of the practical difficulties in comparing model results with observations. The comparison between figure 3 and the corresponding result from the GCM calculation of Reverdin et al. (1987), shown here as figure 4, shows one systematic difference between the GCM and the simple linear model. The linear model overestimates the slope from mid 1983 to early 1984, and shows a distinct phase lag of about 2 months, while the GCM result tracks the slope reasonably faithfully, and without the spurious phase shift. This may be due in part to inadequate wind data, but du Penhoat and Gouriou point out that the inclusion of nonlinearity in the model should result in more realistic dynamic height gradients. Linear models are less successful in reproducing the dynamic topography of the tropical Indian ocean. A direct comparison between single layer reduced gravity linear and nonlinear models of the tropical Indian ocean was presented by P~rigaud and Delecluse (1989). In that study, the authors were primarily interested in the northwest region of the Indian ocean where the "Great Whirl" appears in the summer. There is probably a greater body of literature devoted to this prominent feature than to any other aspect of the Indian Ocean circulation. Both linear and nonlinear models produced the large scale features of the dynamic topography reasonably well, but it is not surprising that the Great Whirl was much better represented in the nonlinear than in the linear simula-

221

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.

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Figure 3: Dynamic height differences between 29~ and 6~ solid line: FSIIB simulation. dotted line: tide gauges, crosses: FOCAL cruises, asterisks: interpolated values. See text. Reproduced from Figure 7 of du Penhoat and Gouriou (1987) with permission of the American Geophysical Union. tions. They also found that the transient behavior of the simulations depended strongly on the position of the Great Whirl; this sensitivity is one indication of the importance of nonlinearity. In that study, the linear model showed much greater sensitivity to changes in the wind forcing than did the nonlinear model. Evidently, and plausibly, internal dynamics are more important relative to fluctuations in wind forcing for the Great Whirl than for the basin scale adjustments that were the basis of the cited studies of the tropical Pacific and Atlantic oceans. Much of the strong annual and interannual variability of the tropical Indian ocean occurs in the band of latitudes from 10~ to 20~ Interaction between Rossby waves and variations in the thermocline depth is significant in that region, more so than it is nearer to the equator. In the equatorial waveguide, assimilation experiments with a model similar to that of Miller and Cane achieved some degree of success (P6rigaud and Fu, 1990), but this has not been the focus of attention in research on the tropical Indian ocean. At least in part because the focus of interest in the Indian ocean has been the region outside the waveguide, work on assimilation has proceeded with nonlinear models, albeit very simple ones: the model used by Greiner and P6rigaud (1994) for their assimilation study was a very coarsely resolved single layer reduced gravity model. In summary, we note that in the Pacific, the difference between results from detailed nonlinear models and simple linear models can be explained by uncertainty in the forcing data. Since the quantities of interest are determined by basin scale adjustment processes, it is likely that the simple models capture much of the variation observed in nature. The

223 Greiner and P~rigaud (1994), in their adjoint assimilation of Geosat data in a model of the tropical Indian ocean also used a diagonal matrix for V, but they did not use a multiple of the identity. Instead they derived a spatially varying estimate of the error variance from a difference between two altimeter products. They used this same form of the error weighting for both V and w. It is important to note that w, the inverse of the observation error covariance matrix, is the coefficient of the inhomogeneous term in the adjoint equation (5), and therefore influences the solution explicitly. Leetmaa and Ji (1989) and Carton and Hackert (1990) used optimal interpolation to perform assimilation in the Pacific and Atlantic oceans respectively with detailed fully nonlinear primitive equation models for which full implementation of optimized data assimilation methods is impractical at this time. The reader should recall at this point that OI is based on an assumed form of the matrix P, as opposed to V, w or W; refer t o t h e previous section for definitions. Carton and Hackert implemented their scheme by producing gridded fields of estimated forecast errors based on objective mapping of the model residuals, i.e, the quantities dk+l Hk+lUk+ f 1 in our notation. The forecast error covariance Pk/ is then proportional to the correlation function used in the objective map. Carton and Hackert used a spatially homogeneous correlation function for temperature measurements given by: -

At) - (1

+

+

AR-

XR /3)

[(Ax/170) 2 +

-

(l

tl/40)]

(Ay/45)2] I/2

where x and y are in km and time is measured in days. These correlation scales are much shorter than any that have been used and/or validated with the simple models such as those used by Miller and Cane (1989), Miller et al. (1995) or Gourdeau et al. (1992, 1995), and will give rise to a much rougher field. The temperature correlation, and hence the density correlation is not differentiable at the origin. This could cause trouble at small grid spacings, since the derivatives of the density error are related through geostrophy to the velocity error. Hao and Ghil (1994) used optimal interpolation in their series of Observing System Simulation Experiments (OSSE). They used a linear model and an assumed error covariance of Gaussian form with a zonal decorrelation length of 10 ~ and a meridional decorrelation length of 1.6 ~ similar to the scales used by Leetmaa and Ji (1989). These results are reported in greater detail in the following chapters by A. J. Busalacchi and by A. Leetmaa and M. Ji. The forecast error covariances in any model of the tropical ocean are almost certainly not homogeneous due to the influence of boundaries and of the equatorial waveguide. This is evident in Figure 5, which shows the estimated root mean square (RMS) error in the forecast sea level height of a linear shallow water model with two baroclinic modes. This was assumed to be a pure forecast experiment with no data assimilation. This figure was constructed by integrating (10) for seven years, and extracting the variance of sea level error at all grid points from the expected error covariance matrix of the state vector; the figure shows the square roots of the error variance, i.e., the expected RMS errors. The system noise covariance matrix Q was derived from the assumption that the forcing errors were white in time and spatially homogeneous, with a spatial covariance function

225 EQ

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160W 140W Lon~liat)Ude

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100W

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EQ

(D "10

10S

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20S EQ

10S

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Lon~ic~ude Figure 6: Maps of estimated RMS errors in sea level height anomaly for three assimilation experiments with a linear shallow water model, a: assimilation of sea level anomaly data from selected tide gauges. Locations of station from which data were assimilated are shown as filled circles. Open circles represent stations from which data were held back for comparison, b: assimilation of dynamic height anomaly data derived from XBT casts taken along indicated ship tracks, c: assimilation of sea level anomaly data and dynamic height anomaly data. Reproduced from figure 16 of Miller et al. (1995) with permission of the American Geophysical Union.

226 measurement errors, i.e., 3 cm. Outside of the waveguide, improvement of the analysis by assimilation is mostly local. For measurements near the equator, influence at large distances compared to the equatorial deformation radius of about 3 ~ results from propagation of equatorially trapped waves. The influence of these measurements is therefore greater in the zonal than in the meridional direction. Figure 7 shows the influence of data from the tide gauge at Nauru on the analysis based on a model which consists of equatorial wave dynamics (Miller and Cane, 1989). The map shown is actually a representation of the matrix GKeN, where G is the matrix which maps the state vector onto a gridded map of sea level height anomalies, K is the equilibrium Kalman gain matrix for the Miller and Cane model for the given error estimate, and e g is the unit vector with a 1 in the component that corresponds in the observation vector to sea level anomaly at Nauru. When d a t a are assimilated at Nauru, a pattern of sea level anomalies proportional to the map in figure 7 is added to the forecast sea level anomaly. The constant of proportionality is the difference between the forecast and observed values at Nauru. This is the content of equation (11). In this figure, we see that the influence decays to zero within 4o-5 ~ of the equator, while the influence extends over much of the equatorial basin in the zonal direction. Fu et al. (1993) performed an experiment in which (10) was integrated to equilibrium, and the result used in an OI scheme, which performed as well as the Kalman filter for assimilation of the Geosat data. This is the only example we know of in which an inhomogeneous form of the forecast error covariance matrix was used in an OI scheme in modeling of the tropical ocean. We know of no specific comparisons between OI schemes with homogeneous and inhomogeneous error covariance matrices, so the actual impact of the erroneous assumption of spatial homogeneity of the forecast error covariance matrix P remains to be determined. IInplementation of weak constraint inverse methods (i.e., minimization of (3)) requires prior estimates of the model and forcing errors, in the form of explicit estimates of W or its inverse Q. Miller and Cane (1989) in their Kahnan filter study derived Q from the assumption that the system noise was dominated by the response to errors in the wind forcing. As noted above, the wind errors were assumed to be spatially homogeneous and Gaussian in form. Decorrelation scales were estimated by comparing statistics of misfits between d a t a and unfiltered model output with prior estimates of those statistics derived from integration of (10) with different candidate values of Q. The best fit was achieved with zonal and meridional error decorrelation scales of 10 ~ and 2 ~ respectively. Similar error models were used by Gourdeau et al. (1992, 1995) and by Fu et al. (1993). We remind the reader that the dynamical model used by Miller and Cane was only valid in a narrow band of latitudes near the equator. Miller et al. (1995) found that application of that same error model to a more general dynamical model resulted in unrealistic overestimates of the sea level height anomaly errors at stations at north and south latitudes of 9 ~ and poleward. At these latitudes the coriolis parameter f is large enough t h a t the effects of wind stress curl are significant, and it is easily shown that the variance of the error in the wind stress curl is inversely proportional to the square of the meridional decorrelation length of the forcing error. Satisfactory overall performance was obtained with a meridional decorrelation length of 4 ~ Bennett (1990) performed an OSSE with simulated XBT d a t a taken along shipping

227

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%

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%

i: C h r i s t m a s Jurawu 9 1 Nauru '~'~---J

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Figure 7: Contour map of the influence of data from the tide gauge station at Nauru. In this experiment, data were also assimilated at Rabaul, J arvis, Christmas, Santa Cruz and Callao. Taken from Figure 8 of Miller and Cane (1989), by permission of the American Meteorological Society. routes. The zonal and meridional error covariance scales Lx and L v were chosen to be Lx -- 32 ~ and L v = 8 ~ much longer than those used in other studies. Bennett performed some tests of sensitivity to these scales, and found that the conditioning of the inverse problem was not very sensitive to decreases in decorrelation scales from the ones he used. Data assimilation experiments clearly have shown an ability to improve analyses. However, comparisons of computational results to real data, as well as comparisons of prior estimates of data misfits with actual data misfits indicate that our implementations are far from optimal. Several deficiencies in proposed error models can be identified. It is highly unlikely that errors in wind forcing fields taken from ship observations are statistically homogeneous, if only because ships tend to travel along well-defined routes, leaving large data voids. Comparisons of ship-derived wind fields with wind fields derived from remote sensing data show systematic spatial patterns (see, e.g., Busalacchi et al., 1993). Miller et al. (1995) relaxed the assumption of spatial homogeneity of wind error statistics, but found little sensitivity beyond some improvement in the analysis near the coastlines. Relaxing the assumption that the noise from various sources is white may be more fruitful. Like the homogeneity assumption, the whiteness assumption is made more for convenience than from evidence. Taft and Kessler (1991) estimated spatial and temporal correlations of errors in dynamic height calculations that result from the use of a mean T - S relation to calculate density from temperature measurements. The errors have a temporal autocorrelation of about 0.6 at a lag of one month. The zero-crossing of the autocorrelation is between three and four months.

4.2. Error E s t i m a t e s , P o s t e r i o r The final judgment of the accuracy and reliability of any data assimilation calculation

228 must come from the posterior error estimates, i.e., the testing of the statistical hypotheses underlying the method. At its optimum value, the cost function (3) is a random variable with a )/2 distribution on a number of degrees of freedom equal to the number of independent measurements (see Bennett, 1992). Similarly, in a properly tuned Kalman filter, the innovation sequence, i.e., the time series dk--Hkuk/, will be white. This has been presented in the numerical weather prediction literature by Dee et al. (1985) and by Daley (1992), following the work of Kailath (1968). Intuitively, a white sequence contains no information on which to base skillful predictions of the future; if the innovation sequence is not white, then there is information remaining which is not being extracted. In the optimal case, the random variables (dk+l -- Hk+lUfk+l)T(Hk+ 1P/,+IHT+I+w f -1)-1 (dk+l - Hk+ lUk+l) f will be X 2 distributed. In theory, one should be able to perform X2 tests on the results of a data assimilation experiment in order to establish confidence limits on the hypotheses that the model errors, data misfits and initial errors are random variables with the specified covariances. This ideal has not yet been achieved in any practical data assimilation experiment with real data from the tropical ocean, but it is reasonable to hope for such tests in the near future. Statistical tests of this sort are especially valuable in cases such as the present ones in which d a t a are sparse and often inadequate for the simple exercise of holding some back for verification. This is true in particular of assimilation of satellite altimetric data, whose error characteristics are not well known. In the Atlantic and Indian oceans, surface verification of analyses of satellite data is particularly hard to come by. Greiner and P6rigaud (1994) found that their final cost function from their strong constraint assimilation experiment with Geosat data was consistent with prior estimates of errors in the Geosat sea levels. Miller and Cane (1989) and Miller et al. (1995) found the error estimates to bc reasonable for the most part, but some systematic differences between actual and estimated model-data misfits remained. Systematic underestimates of variances of model data misfits in XBT-derived dynamic height comparisons were particularly troublesome. The error estimates produced by the Kalman filter follow directly from prior estimates of statistics of model, measurement and initial errors. Since direct independent information about these errors is limited, it is more likely that the assimilation results themselves will be used to refine error models, as in the work of Chan et al. (1995). Chan et al. (1995) analyzed the innovation sequence from Miller and Cane's (1989) experiment, and found that they were well fitted by autoregressive models with onemonth time lags. The monthly innovation sequences themselves were modeled by univariate first-order autoregressive processes; no improvement was realized by fitting a multivariate autoregressive model to the innovation sequence. From the point of view of implementation, this amounts to adding a new state variable for each site at which tide gauge d a t a are assimilated. The resulting innovation sequences are white; in cases in which d a t a are held back from the assimilation process for verification, the sequence of residuals at stations which are not assimilated are also white. This one-month autoregressive structure may arise from autocovariance in the model error, the observation error or both. Daley (1992) suggested a way in which the lagged innovation covariances could be used to distinguish between the effects of system noise and observation noise, but a dense d a t a set is required for this determination.

229 While mathematical rigor is certainly a desirable goal, we must approach it with the caution that we invariably work with small samples. Monte-Carlo experiments (Miller, 1990) have shown that even if all of the relevant statistical quantities are known exactly, the variances and covariances that will be calculated from the actual data misfits may be far from their ensemble means. The smallness of the samples we work with limits our ability to test our error models. More importantly, it diminishes the likelihood that even a perfect, true error model will yield the best possible results: the most effective error model would be the one that exactly represents the error structure during the relatively short assimilation period. Two or three decades of monthly data is a sample length of only a few hundred. It is likely that our view of the series of data misfits as a representative sample drawn at random from an infinite underlying population with specified statistics may not be the most useful one for design of more detailed error specifications for more detailed models.

5. D I S C U S S I O N It is a boon to data assimilation studies that simple models are extremely effective in the tropics. We noted that their success is not a consequence of tropical ocean circulations being exceptionally simple. Indeed, the phenomenology of the tropics is no less rich than elsewhere in the world ocean. There are energetic motions at a vast range of time and space scales, and a strong interplay between dynamics and thermodynamics. The simplifications are enabled by the research agenda in this region. The highest priority is seasonal to interannual variability, especially aspects of ocean - atmosphere interaction important for climate prediction. Tropical oceanography is fortunate in its problem: what matters most is captured by relatively simple models. However, we noted that this should not be regarded as an eternal truth. The poor quality of presently available forcing data and the paucity of oceanic data to verify against make it difficult to demonstrate the superiority of more complex models. As the data improve, standards will be raised and the inadequacies of the simple models will be uncovered. In the meantime we can exploit their speed and simplicity to learn more about data assimilation in an important realistic context. Little is known and little has been done on the subject of assimilation of data into the coupled models of the tropical ocean and atmosphere which are used for short to intermediate term climate prediction. This is clearly the next frontier and the most obvious application in tropical oceanic and atmospheric science of assimilation of data in ocean models. As noted in the introduction to this chapter, SST is the most important oceanic variable for the purposes of initialization of coupled models. The purpose of data assimilation in models which include SST, at least in the context of examination of the conditions in the present and recent past is not to produce the best possible SST analysis. Processed data sets which include remotely sensed data can now produce maps of SST with sufficient accuracy for present purposes (Reynolds, 1988, Reynolds and Smith, 1994). Therefore, assimilation of SST is not so much to improve the SST analysis as the hope of improving the analyses of other quantities. Unfortunately, given the present primitive state of affairs, the model SST which gives rise to the

230 best predictions may be systematically different from the observed SST. Insertion of the observed SST into a coupled model will give rise to a dynamic adjustment process which may not be a faithful representation of the evolution of the real ocean and atmosphere. This is the problem of initialization, well known in the numerical weather prediction community; see, e.g., Daley (1991). This topic was explored in the context of tropical ocean models by Moore and several others (see Moore, 1990 and references therein). A summary of that work appears in this volume in the chapter by A. Busalacchi. More recently, Chen et al. (1995) applied a very simple assimilation technique to assimilate observed winds into the coupled model of Zebiak and Cane (1987). Earlier forecasts with the Zebiak and Cane model had been initialized by decoupling the model, spinning up the ocean model with observed winds, and then, at the start of a prediction experiment, coupling the model atmosphere to the ocean model state which resulted from the spin-up calculations. Improved ENSO model predictions result from the new initialization process. Chen et al. argue persuasively that the improvement results from improved initialization. A number of highly sophisticated optimized data assimilation methods have been applied to the tropical oceans, including adjoint methods and the Kalman filter. All of them may be regarded as different computational schemes for carrying out the same least squares minimization to find the best estimate of the state of the ocean. But this is not quite true, because each has a preferred set of simplifications to the prior error estimates that determine the outcome of the procedure. For example the adjoint methods most often assume the model is error free, which is computationally convenient, but hard to defend. It is more reasonable to assume that errors in the model, the observations, and thc initial conditions are all significant. We must then confront the fact that we have very little idea of what these errors are. Without good prior error estimates, there is no guarantee that the "optimized" methods will yield the best possible answers. It may be necessary to adopt methods that adapt to the structure of the posterior errors. Chan et al. (1995) is the sole example we are aware of to take such an approach in the context of tropical oceanography. The key to improved tropical ocean data assimilation is hidden in the noise. The task ahead is to find the way to it ....

6. R E F E R E N C E S Bennett, A. F., Inverse methods for assessing ship-of-opportunity networks and estimating circulation and winds from tropical expendable bathythermograph data. J. Geophys. Res., 95, 16,111-16,148, 1990. Bennett, A. F., Inverse methods in physical oceanography, Cambridge University Press, New York, 346 pp., 1992. Bennett, A. F. and W. P. Budgell, Ocean data assimilation and the Kalman filter: spatial regularity. J. Phys. Oceanogr., 17, 1583-1601, 1987. Bennett, A. F. and R. N. Miller, Weighting initial conditions in variational assimilation schemes. Mon. Wea. Rev., 119, 1098-1102, 1991.

231 Blumenthal, M. B. and M. A. Cane, Accounting for parameter uncertainties in model verification: an illustration with tropical sea surface temperature. J. Phys. Oceanogr., 19, 815-830, 1989. Busalacchi, A. J. and J. J. O'Brien, The seasonal variability in a model of the tropical Pacific. J. Phys. Oceanogr., 10, 1929-1951, 1980. Busalacchi, A. J. and J. J. O'Brien, Interannual variability of the equatorial Pacific in the 1960s. J. Geophys. Res., 86, 10,901-10,907, 1981. Busalacchi, A. J. and J. Picaut, Seasonal variability from a model of the tropical Atlantic ocean. J. Phys. Oceanogr., 13, 1564-1587 1983. Busalacchi, A. J., M. J. McPhaden, J. Picaut and S. R. Springer, Sensitivity of winddriven tropical Pacific Ocean simulations on seasonal and interannual times scales. J. Mar. Syst., 1, 119-154, 1990. Busalacchi, A. J., R. M. Atlas and E. C. Hackert, Comparison of special sensor microwave imager vector wind stress with model-derived and subjective products for the tropical Pacific. J. Geophys. Res., 98, 6961-6977, 1993. Cane, M. A., Modeling sea level during E1 Nifio. J. Phys. Oceanogr., 1~, 1864-1874, 1984. Cane, M. A. and R. J. Patton, A numerical model for low- frequency equatorial dynamics. J. Phys. Oceanogr., 1~, 1853-1863, 1984. Carton, J. A. and E. C. Hackert, Data assimilation applied to the temperature and circulation in the tropical Atlantic, 1983-84. J. Phys. Oceanogr., 20, 1150-1165, 1990. Chan, N-H, J. B. Kadane, R. N. Miller and W. Palma, Predictions of tropical sea level anomaly by an improved Kalman filter. J. Phys. Oceanogr., subjudice, 1995. Chao, Y., D. Halpern and C. P~rigaud, Sea surface height variability during 1986-1988 in the tropical Pacific ocean. J. Geophys. Res., 98, 6947-6959, 1993. Chen, D., S. E. Zebiak, A. J. Busalacchi and M. A. Cane, An improved procedure for E1 Nifio forecasting. Preprint, 16 pp., 1995. Courant, R. and D. Hilbert, Methods of Mathematical Physics, Vol. 1, Wiley Interscience, 560 pp., 1953. Courtier, P. and O. Talagrand, Variational assimilation of meteorological observations with the adjoint vorticity equation. II: Numerical results. Quart. J. Roy. Met. Soc., 113, 1329-1347, 1987. Daley, R., Atmospheric Data Assimilation, Cambridge University Press, 1991. Daley, R., The lagged innovation covariance: A performance diagnostic for atmospheric data assimilation. Mon. Wea. Rev., 120, 178-196, 1992. Dee, D. P., S. E. Cohn and M. Ghil, An efficient algorithm for estimating noise covariance in distributed systems. IEEE Trans. Autom. Control, A C-30, 1057-1065, 1985. Du Penhoat, Y. and A-M Treguier, The seasonal linear response of the tropical Atlantic. J. Phys. Oceanogr., 15, 316-329, 1985. Du Penhoat, Y. and Y. Gouriou, Hindcasts of equatorial sea surface dynamic height in the Atlantic in 1982-1984. J. Geophys. Res., 92, 3729-3740, 1987. Frankignoul, C., S. Fevrier, N. Sennechael, J. Verbeek and P. Braconnot, An intercomparison between four tropical ocean models. Part 1: Thermocline variability. Tellus, in press, 1995. Fu, L-L, I. Fukumori and R. N. Miller, Fitting dynamic models to the Geosat sea level observations in the tropical Pacific ocean. Part II: A linear, wind-driven model. J. Phys. Oceanogr., 23, 2162-2181, 1993. Ghil, M. and P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography. Adv. Geophys., 33, 141-266, 1991. Gourdeau, L., S. Arnault, Y. Menard and J. Merle, Geosat sea-level assimilation in a tropical Atlantic model using Kalman filter. Oceanologica Acta, 15, 567-574, 1992. Gourdeau, L., J. F. Minster and M. C. Gennero, Sea level anomalies in the tropical Atlantic from Geosat data assimilated in a linear model, 1986-88, J. Geophys. Res., subjudice, 1995.

232 Greiner, E. and C. P~rigaud, Assimilation of Geosat altimetric data in a nonlinear reduced-gravity model of the Indian ocean. Part I: adjoint approach and model-data consistency. J. Phys. Oceanogr., 2~, 1783-1804, 1994. Halpern, D. and D.E. Harrison, Intercomparison of tropical Pacific mean November 1979 surface wind fields. Report 82-1, Department of Meteorology and Physical Oceanography, Massachusetts Institute of Technology, 40 pp., 1982. Hao, Z. and M. Ghil, Data assimilation in a simple tropical ocean model with wind stress errors. J. Phys. Oceanogr., 2~, 2111-2128, 1994. Harrison, D. E., W. S. Kessler and B. J. Giese, Ocean circulation and model hindcasts of the 1982-83 E1 Nifio: Thermal variability along the ship-of-opportunity tracks. J. Phys. Oceanogr., 19, 397-418, 1989. Jazwinski, A. H., Stochastic Processes and Filtering Theory, Academic Press, NY, 376 pp., 1970. Kailath, T., An innovations control approach to least square estimation- Part I: Linear filtering in additive white noise. IEEE Trans. Autom. Control, 13, 646-655, 1968. Leetmaa, A. and M. Ji., Operational hindcasting of the tropical Pacific. Dyn. Atmos. Oceans, 13, 465-490, 1989. Long, R. B. and W. C. Thacker, Data assimilation into a numerical equatorial ocean model. I. The model and the assimilation algorithm. Dyn. Atmos. Oceans., 13, 379-412, 1989a. Long, R. B. and W. C. Thacker, Data assimilation into a numerical equatorial ocean model. II. Assimilation experiments. Dyn. Atmos. Oceans., 13, 413-440, 1989b. Miller, R. N., Tropical data assimilation experiments with simulated data: the impact of the Tropical Ocean and Global Atmosphere Thermal Array for The Ocean. J. Geophys. Res., 95, 11,461-11,482, 1990. Miller, R. N. and M. A. Cane, A Kalman filter analysis of sea level height in the tropical Pacific. J. Phys. Oceanogr., 19, 773-790, 1989. Miller, R. N., A. J. Busalacchi and E. C. Hackert, Comparison of geosat analysis with results of data assimilation for the period November, 1986 to September, 1989. In abstract volume, "Satellite Altimetry and the Oceans," 29 November- 3 December, Toulouse, France. CNES, 1993. Miller, R. N., A. J. Busalacchi and E. C. Hackert, Sea Surface Topography Fields of the Tropical Pacific from Data Assimilation. J. Geophys. Res., 100, 13,389-13,425, 1995. Moore, A. M., Linear equatorial wave mode initialization in a model of the tropical Pacific ocean: an initialization scheme for tropical ocean models. J. Phys. Oceanogr., 20, 423-445, 1990. Moore, A. M., N. S. Cooper and D. L. T. Anderson, Initialization and data assimilation in models of the Indian Ocean. J. Phys. Oceanogr., 17, 1965-1977, 1987. Pedlosky, J., An inertial theory of the equatorial undercurrent. J. Phys. Oceanogr., 17, 1978-1985, 1987. Pe!rigaud, C. and P. Delecluse, Simulations of dynamic topography in the northwest Indian ocean with input of Seasat altimeter and scatterometer data. Ocean Air Interact., 1, 289-309, 1989. P~rigaud, C. and L-L Fu, Indian ocean sea level variations optimally estimated from Geosat and shallow-water simulations. Proc. Int. Syrup. on Assimilation of Observations in Meteorology and Oceanography, World Meteorological Organization, Clermont-Ferrand, France, 510-514. 1990. Philander, S. G. H., W. Hurlin and A. D. Seigel, A model of the seasonal cycle in the tropical Pacific ocean. J. Phys. Oceanogr., 17, 1986-2002, 1987. Picaut, J., J. Servain, P. Lecomte, M. Seva, S. Lukas and G. Rougier, Climatic Atlas of

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233 Rebert, J. P., J. R. Donguy, G. Eldin and K. Wyrtki, Relations between sea level, thermocline depth, heat content and dynamic height in the tropical Pacific ocean. J. Geophys. Res., 90, 11,719-11,725, 1985. Reverdin, G. and Y. du Penhoat, Modeled surface dynamic height in 1964-1984: an effort to assess how well the low frequencies in the equatorial Atlantic were sampled in 1982-1984. J. Geophys. Res., 92,, 1899-1913, 1987. Reverdin, G., P. Del~cluse, C. L~vy, P. Andrich, A. Morli~re and J. M. Verstraete, The near surface tropical Atlantic in 1982-84: results from a numerical simulation and a data analysis. Prog. Oceanogr., 27,, 273-340, 1991. Reynolds, R. W., A real-time global sea surface temperature analysis. J. Clim., 1, 75-86, 1988. Reynolds, R. W. and T. M. Smith, Improved global sea surface temperature analyses using optimum interpolation. J. Clim., 2~, 929-948, 1994. Sasaki, Y., Some basic formalisms in numerical variational analysis. Mon. Wea. Rev., 98, 875-933, 1970. Seager, R., A simple model of the climatology and variability of the low level wind field in the tropics. J. Climate, ~, 164-179, 1989. Seager, R., S. E. Zebiak and M. A. Cane, A model of the tropical Pacific sea surface temperature climatology. J. Geophys. Res., 93, 11,587-11,601, 1988. Servain, J., J. Picaut and A. Busalacchi, Interannual and seasonal variability of the tropical Atlantic Ocean depicted by sixteen years of sea surface temperature and wind stress, in Coupled Ocean-Atmosphere models, 16th Liege Colloquium on Ocean Hydrodynamics, edited by J. C. Nihoul, pp. 211-237, Elsevier, New York, 1985. Sheinbaum, J. and D. L. T. Anderson, Variational assimilation of XBT data, Part I. J. Phys. Oceanogr., 20,, 672-688, 1990a. Sheinbaum, J. and D. L. T. Anderson, Variational assimilation of XBT data, Part II. J. Phys. Oceanogr., 20,, 689-704, 1990b. Smedstad, O. M. and J. J. O'Brien, Variational data assimilation and parameter estimation in an equatorial Pacific ocean model. Prog. Oceanogr., 26, 179-241, 1991. Stricherz, J. N., J. J. O'Brien, and D. M. Legler, Atlas of Florida State University Tropical Pacific Winds for TOGA, Mesoscale Air-Sea Interaction Group Technical Report, Florida State University, Tallahassee, 261 pp., 1992. Sverdrup, H. U., Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Nat. Acad. Sci. Wash., 33, 318326, 1947. Taft, B. A. and W. S. Kessler, Variations of zonal currents in the central tropical Pacific during 1970-1987: sea level and dynamic height measurements. J. Geophys. Res., 96, 12,599-12,618, 1991. Tziperman, E. and W. C. Thacker, An optimal-control/adjoint equations approach to studying the oceanic general circulation. J. Phys. Oceanogr., 19, 1471-1485, 1989. Weisberg, R. H., Observations pertinent to instability waves in the equatorial oceans, in Further Progress in Equatorial Oceanography, edited by E. J. Katz and J. Witte, pp. 335-350, Nova University Press, Ft. Lauderdale, FL, 1987. Zebiak, S. E. and M. A. Cane, A model E1 Nifio-Southern Oscillation. Mon. Wea. Rev., 115, 2262-2278, 1987.

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Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 1996 Elsevier Science B.V.

DATA ASSIMILATION CIRCULATION STUDIES

IN

SUPPORT

235

OF

TROPICAL

OCEAN

Antonio J. Busalacchi Laboratory for Hydrospheric Processes, NASA Goddard Space Flight Center, Greenbelt, Maryland, 20771

Abstract

The assimilation of data into tropical ocean models is an active area of research for a variety of reasons. The deterministic nature of the tropical ocean circulation, the rapid time scale at low latitudes, the two-layer approximation, the non-local impact of assimilated data, the increased quantity of in situ data obtained as part of the TOGA Program, and the important role played by the tropics in short-term climate prediction have stimulated data assimilation in tropical oceanography. This paper describes the extent of the approaches in data assimilation supporting tropical ocean circulation studies. Data assimilation efforts in the tropics encompass initialization experiments, observing system simulation experiments, estimation of model parameters, and real-time analyses for the tropical Pacific Ocean. The data that are most frequently assimilated are observations of the vertical structure of temperature and sea level height. Observations of sea level and thermocline depth are normally assimilated into reduced-gravity models with the more advanced assimilation schemes such as the Kalman filter and the adjoint method. The effect of the data usually increases the amplitude of the variability in the model. Four-dimensional temperature observations are used to constrain primitive equation, general circulation models via optimal interpolation and successive correction methods. The principal influence of the assimilated data is to eliminate systematic biases in the model temperature fields.

1. I N T R O D U C T I O N The tropical oceans are a laboratory for the design, implementation, and testing of a number of approaches to four-dimensional ocean data assimilation. The dominant physics at low latitudes, together with the role of the tropical oceans in seasonal to interannual climate prediction, contribute to make tropical ocean data assimilation an attractive avenue for research. In the previous chapter, Robert Miller and Mark Cane laid down the theoretical foundation for ocean data assimilation in the tropics. The chapter that follows by Ants Leetmaa and Ming Ji describes the importance of providing the best possible set of oceanic initial conditions to coupled ocean-atmosphere forecasts of short-term climate phenomena such as E1 Nino. The present chapter traces the evolution of the burgeoning activities in data assimilation for the tropical oceans.

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Considerable progress in tropical ocean data assimilation has been enabled for a variety of reasons, foremost of which may be that the physics of the low-latitude oceans is relatively straightforward. The fact that the Coriolis term goes to zero at the equator renders this region unique, but it also has important implications for data assimilation. To first order, the large-scale ocean circulation in the tropics is wind-driven and linear. Thus, the ocean dynamics at low latitudes comprise a fairly deterministic system. Stochastic variability, such as that often associated with mesoscale eddies, is of lesser importance. Temporally, the time scale in the tropics is fast. The equatorial wave guide serves as an efficient conduit to transmit rapidly (O-months) information from one end of a tropical ocean basin to the other. Horizontally, the variability in the tropical oceans is predominantly zonal. Zonally propagating equatorial waves significantly influence the redistribution of mass and heat. Vertically, most of the variability is in the upper portion of the water column and can be described by a few baroclinic modes. Therefore a tropical ocean is often characterized as a two-layer system where deep subthermocline fluctuations are neglected within the context of the reducedgravity formulation. The implications for tropical ocean data assimilation of these time-space attributes are that ocean observations ingested into one region of a model domain can have a far ranging impact over a short period of time. Moreover, observations of surface quantities such as sea level are easily projected onto the subsurface structure because of the limited degrees of freedom in the vertical. There is ample evidence to suggest that given accurate initial conditions and wind forcing, the dynamic response of the tropical ocean circulation (e.g., vertically integrated quantities such as sea level, dynamic height, and heat content) can be simulated with a fair degree of certainty. Unfortunately, more often than not, the initial conditions and the momentum fluxes can not be considered as being given with a high degree of confidence. Even if they could be, information on sea level and upper ocean heat content provides a rather limited view of the state of the tropical ocean. In terms of the importance to the coupled ocean-atmosphere problem, sea surface temperature (SST) is a more essential variable. However, the thermodynamic controls on SST are clearly nonlinear and depend critically on poorly known surface heat fluxes. Consequently, if one wishes to obtain an accurate, or at least, the best possible depiction of the state of a tropical ocean, assimilation of ocean observations into a numerical ocean model offers a means to counterbalance the deficiencies in initial conditions, surface forcing, and model physics. In situ and remotely sensed observations of the tropical oceans have provided ample opportunities to do just that. Another related factor that has spurred progress in tropical ocean data assimilation has been the recently completed Tropical Ocean Global Atmosphere (TOGA) program. Much of our recent knowledge of the variability of the tropical Pacific Ocean is derived from the vast amount of observations collected during the TOGA program. In contrast to other regions of the world ocean where data assimilation serves to synthesize various and disparate ocean observations, in the tropics, ocean data assimilation is also needed to provide the best set of initial conditions to coupled ocean-atmosphere models used for seasonal-to-interannual climate prediction. In a sense this is analogous to the demand that was created for atmospheric observations by the advent of numerical weather prediction. The 10-year TOGA program began in 1985 with three main objectives (WCRP, 1985): 1. to gain a description o f the tropical oceans and the global atmosphere as a time-dependent system in order to determine the extent to which this system is

237

predictable on time scales of months to years and to understand the processes underlying its predictability 2. to study the feasibility of modeling the coupled-atmosphere system for the purposes of predicting its variations on time scales of months to years, and 3. to provide the scientific background for designing an observing and data transmission system for operational prediction if this capability is demonstrated by coupled ocean-atmosphere models. As a result of the progress that was made in support of these objectives, the TOGA program generated a need for real-time monitoring of the tropical Pacific Ocean and the subsequent ocean initialization of coupled forecasts. This has quite naturally brought the modelling and observational components of the program together. Ocean data assimilation constitutes the link that binds the two components together. At the conclusion of TOGA, one of the principal legacies of this program will have been the design and implementation of a Tropical Pacific Ocean Observing System (Figure 1); elements of which are being transitioned to operational status to ensure the routine monitoring of many of the key variables needed in support of E1 Nino/Southern Oscillation (ENSO) prediction (National Research Council, 1994). In order of importance, the critical oceanographic variables for the TOGA problem are sea surface wind stress, sea-surface temperature (SST), upper-ocean thermal structure, sea level, and current velocity. Although TOGA was designed to consider all three tropical oceans, in reality most of the attention was focused on these observables in the tropical Pacific basin because this was the domain of the most intense ocean-atmosphere coupling that constitutes the ENSO phenomenon. Indeed, most of the tropical ocean data assimilation studies to date have focused on the tropical Pacific Ocean. The backbone for the Tropical Pacific Ocean Observing System was the TOGA Tropical Atmosphere Ocean (TAO) array. This array of approximately 70 moored surface buoys spanning the equatorial Pacific between 8~ and 8~ is used to monitor, in near real time, surface wind velocity, SST, surface air temperature, humidity, and subsurface temperature at 10 levels to 500 m (McPhaden, 1993). In addition, upper-ocean currents are measured at five of the moorings along the equator. The TAO array has also proved to be a very valuable "platform of opportunity" for a limited number of sensors to measure precipitation, short-wave radiation, and salinity. A variety of other platforms and measurements were used to complement the TAO array. For example, volunteer observing ships were used to obtain standard surface meteorological observations. A particularly valuable enhancement to the measurements obtained from these merchant ships was the observation of upper-ocean thermal structure from expendable bathythermographs (XBT) deployed along three main clusters of ship transects in the western, central, and eastern tropical Pacific (Meyers et al., 1991). Routine XBT surveys were also performed in the tropical Atlantic and Indian Oceans. Surface drifting buoys have been a very effective means for obtaining basin-scale coverage of SST, near surface currents, and sea level pressure. Lastly, a global network of island and coastal tide gauges was used to monitor the large-scale fluctuations in sea level. Taken together, the TAO array, the XBT network, the surface drifters, and the tide gauge network provide a rich data set for tropical ocean assimilation studies.

238

Figure 1. Schematic of the in situ Tropical Pacific Ocean Observing System developed as part of the Tropical Ocean Global Atmosphere (TOGA) program. Island and coastal tide gauge stations providing sea level measurements are indicated by circles. Moored buoys of the Tropical Atmosphere Ocean (TAO) Array are indicated by diamonds (wind and thermistor chain moorings) and squares (current meter moorings, also with winds and temperatures). Drifting buoys providing SST and surface current estimates are indicated by arrows. Expendable bathythermograph (XBT) measurements providing upper ocean thermal profiles along volunteer observing ship (VOS) lines are indicated by the shaded transects. Most data from this array are telemetered to shore in real-time via satellite. (Courtesy of Michael J. McPhaden, NOAA/Pacific Marine Environmental Laboratory). In addition to the in situ observations of the ocean, the T O G A program benefited from remotely sensed ocean observations from several satellite platforms. Of all the space-based observations of the oceans, SST is the one measurement that has reached operational status. Infrared sensors, e.g., the Advanced Very High Resolution Radiometer ( A V H R R ) , on polar orbiting and geostationary meteorological satellites have provided routine observations of SST for over fifteen years. When blended with in situ SST observations from the drifting buoys (to remove retrieval biases) this data set (Reynolds, 1988; Reynolds and Smith, 1994) has been used routinely to monitor the development of interannual SST anomalies in the tropical Pacific, and also to constrain the simulated SST in numerical ocean simulations. With the launch of the Geosat altimeter in 1985, nearly uninterrupted coverage of variations in the sea surface topography has been obtained on the mesoscale to the basin-scale. The launches of

239 the ERS-1 and TOPEX/Poseidon altimeters have continued this class of observations into the 1990's with an increase in measurement precision and overall accuracy of the sea level retrievals. These measurements have been shown to be particularly effective at monitoring the basin-scale variability of equatorial wave dynamics (Miller et al., 1988; Delcroix et al., 1991; Busalacchi et al., 1994). In view of the two-layer approximation that is often used in tropical oceanography, these global observations of sea surface topography offer considerable potential for use in data assimilation applications for all three tropical oceans. Now that there are extensive in situ and space-based observing systems in place providing comprehensive information for parts of the tropics, the incorporation of these data into assimilation schemes requires that there be a rigorous estimation of the error characteristics of these observations. As cited below, some progress in this regard has been made for surface and subsurface thermal measurements. Yet, little is known about the error covariance structure for fields of data such as sea level or surface fluxes of momentum and heat. This is a critical issue to be confronted when merging data with models, and is usually dealt with by making certain generalizations. Often the spatial and temporal scales of the measurement errors are used interchangeably with those for the signal itself because so little is known about the error structures. The past several years have seen a wide range of applications of data assimilation methodology in tropical oceanography beyond providing initial conditions to coupled oceanatmosphere models or providing a statistically optimal blend of observations and ocean model solutions. Ocean data assimilation approaches have been used to assess the space/time impact of various data types, to perform observing system simulation experiments (OSSE), to estimate model parameters, to highlight problem areas in model domains, and to identify model physics in need of improvement. The range of ocean models used to perform tropical ocean data assimilation experiments extends from reduced-gravity and multi-baroclinic mode linear models to fully nonlinear, primitive equation, ocean general circulation models (GCM). This range of models and the nature of the problems in the tropics have fostered a corresponding range of data assimilation approaches, from relatively simple direct insertion and optimal interpolation (OI) schemes to the more advanced Kalman filter and adjoint methods. Most of the studies to date have concentrated on the assimilation methodology and techniques and less on the application of these techniques in support of process oriented research. More often than not, the more sophisticated data assimilation techniques have been implemented into the simpler models and vice versa. In fact, some of the first applications of the Kalman filter and the variational method have been with tropical ocean observations. A detailed description of these assimilation methods is provided in the previous chapter by Miller and Cane. This chapter is meant to focus on model-based data assimilation in the tropics. It will not address objective analyses that are also used to produce oceanic data fields (e.g., Smith, 1991; Meyers et al., 1991; Smith et al., 1991). This should not be construed as a commentary on the importance of such efforts. To the contrary, studies of this kind have and will continue to stand on their own merits. In the past, these analyses of in situ thermal data have proven to be very useful in determining the decorrelation scales of ocean variables and the inherent errors in their measurement. It is expected that such information will continue to be essential input to any four-dimensional data assimilation scheme. Similarly, this chapter will not consider data assimilation studies for coupled models or for producing surface fluxes. Global

240 ocean assimilation studies that may have relevance to the tropics will not be discussed here as they are addressed elsewhere in this book. The next section describes the tropical ocean data assimilation studies that can be categorized as "identical twin" experiments in which synthetic data or model-based "observations" are assimilated. This will be followed by a presentation of the advances that have taken place with the assimilation of real ocean observations, both in situ and space based.

2. A S S I M I L A T I O N O F S Y N T H E T I C D A T A Some of the initial attempts at tropical ocean assimilation did not even use real data. Instead, this class of studies would subsample or degrade fields of model variables and reinsert this information back into the model as if it was observed data. These experiments were performed to gain a better appreciation for what the impact of real data might be, and to determine what observations should be assimilated. For example, experiments were performed with these so-called synthetic data to assess the relative importance of assimilating sea level and/or subsurface thermal structure observations versus current velocity measurements, to estimate the impact of neglecting salinity, and to evaluate planned observing systems. The first efforts in this regard relied on simple approaches to data assimilation. Initialization experiments were designed to determine the time span over which initial conditions continued to affect a model solution, and subsequently, the interval on which it would be necessary to update a model with observations. Philander et al. (1987) studied the effect of different initial conditions in a multi-year GCM simulation of the tropical Pacific Ocean that had been used previously to study the 1982-1983 El Nino (Philander and Seigel, 1985). This interannual simulation forced by winds from the National Meteorological Center (NMC) was used as a control run. In one experiment the model was stopped on March 1, 1984, and all model fields except the forcing were replaced with climatological values. The model was restarted and compared with the control run that did not have the restart. In contrast to the control run, the initial conditions from the restart were not in balance with the wind forcing and an equatorial adjustment ensued (Figure 2). The time scale of the readjustment is of the order of one year. Along the equator this is essentially the time it takes the zonal pressure gradient to readjust to the zonal wind stress. The pressure gradient is brought back into equilibrium by equatorially trapped waves excited as part of the adjustment process. Kelvin waves can be seen to take approximately three months to propagate eastward across the basin and Rossby waves take an additional nine months to propagate west across the basin. In a second restart experiment the current fields were set to zero, but the thermal structure from the control run was retained. In this instance the initial temperature gradients were correct. Because the available potential energy of the system is greater than the kinetic energy, or equivalently, the radius of deformation is greater than the horizontal scale of currents, specifying information on the density field is an efficient means of initialization. Off the equator, the current field undergoes a rapid (O-days) geostrophic adjustment. On the equator, where direct wind forcing is important in determining the surface equatorial currents, the eastern end of the equatorial waveguide takes several months to readjust because of the accumulated influence of the Kelvin waves.

241

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Figure 2. The difference in surface zonal velocity (in 10 cm s-~ intervals) and 55 m temperature (in I~ intervals) between the control run and the restart experiment, along the equator. The difference is negative in shaded areas (from Philander et al., 1987). The specific effects of initializing equatorial waves were treated in greater detail by Anderson and Moore (1989). They demonstrated that updating the mass (or height) field but not the velocity field, and vice versa, produces a Kelvin wave amplitude one-half the true solution when both height and velocity information are available. That is, along the equator, the mass and velocity fields have equal importance for initializing the amplitude of the Kelvin waves. This is because the total energy for the Kelvin wave is split equally between kinetic and potential energy. For the Rossby waves, the kinetic energy is high near the equator and potential energy is dominant away from the equator. Thus the mass field is more important for initializing Rossby waves away from the equator and velocity information is important near the equator. In contrast to the Kelvin waves, the importance of the velocity data for Rossby wave initialization near the equator decreases as the dissipation of the system increases. Initialization studies have not been limited to the tropical Pacific Ocean. Moore et al. (1987) used a G C M and a linear reduced-gravity model to investigate the sensitivity to initializing an Indian Ocean model. They asked the questions: 9 What is the memory time of the ocean (for how long in the past do you need to know the forcing)? 9 Which measurements contain most information and which are redundant? 9 What resolution in the observations is adequate? The "truth" for these numerical experiments was a 110 day integration of an ocean GCM forced by 5-day mean winds for June through September 1979. The main area of interest was the region of the Somali Current. The impact of repeated initialization or updates to the model fields was studied in four case studies. In each case study the model fields were modified on day 50 by setting the velocity field to zero and setting the temperatures back to an initial state of uniform vertical stratification. Three of the experiments were then updated with model data from the truth run inserted every 10 days for the remaining 60 days as follows:

242 1) control run, day 50 velocity field set to zero, day 50 temperatures reset to initial state of uniform vertical stratification no updating 2) same restart conditions as (1), all gridpoint temperatures updated from truth run every 10 days 3) same restart conditions as (1), all gridpoint velocities updated from truth run every 10 days 4) same as (2), wind forcing provided from a different source but for the same time period. Updating with the temperature fields (Cases 2 and 4) significantly reduced both temperature and velocity errors within a few months as compared to the control run that was not updated (Figure 3). The results from Case 4 imply that even if there are errors in the wind field, updating the temperature field can help rectify problems caused by inadequacies in the wind forcing. Updating the velocity field was found to reduce the initial errors somewhat, but was not as effective as updating the temperature. 1. O0

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These results suggested that subsurface thermal structure measurements from an Indian Ocean TOGA XBT network could help constrain a model simulation. However updating the model solution at each and every model gridpoint is not representative of the sparse sampling in the real ocean. In a follow-on assimilation experiment, the temperature structure from the truth run was sampled along the Indian Ocean XBT tracks and, after the model fields were reinitialized on day 50, these synthetic data were blended back into the model using a successive correction method (Bergthorsson and Doos, 1955). The effect of the first assimilation cycle was to reduce the initial error similar to the update experiments, but after this first cycle, the subsequent assimilation did not continue to reduce the error from the control experiment. It was determined that errors could not be reduced further because the sampling and assimilation scheme could not resolve small-scale structures in the temperature

243 field. This was not the case when a linear reduced-gravity model was used in assimilation experiments of the height field. Since the reduced-gravity simulations did not have the same degree of small-scale structure, the initialization error was able to be reduced beyond the first cycle of the assimilation and continued to be low through the full course of the assimilation. The work of Moore et al. (1987) was complemented by Cooper (1988) with a similar set of experiments investigating the impact of updating the salinity field in the Indian Ocean. In these studies the density in the GCM was a function of temperature and salinity. The model was updated using the full density field and again using only the temperature or salinity fields. The initialization error was reduced only in the experiment where the full density field was updated. If either temperature or salinity was updated, but not both, the imbalance with the density field created errors worse than if no updates were made. The relative importance of height field and velocity assimilation has also been studied against the backdrop of different errors in idealized wind forcing. Hao and Ghil (1994) used a reduced-gravity model and an assimilation approach based on optimal interpolation to study the reduction in simulation errors caused by timing errors, systematic errors, and stochastic errors in the wind forcing. A timing error can be considered to be an error in the initial state. Consistent with the previous initialization studies, the timing error is easily corrected by assimilating height field observations. A systematic error in the forcing is more problematic. Every time data is assimilated, the model variables are drawn to the observations and away from a balance with the erroneous forcing. As the integration proceeds away from the assimilation time, the model fields go back to being in balance with the erroneous forcing. The oscillation that is set up is best constrained by assimilating height and velocity observations. When the equatorial wind stress has a random error, assimilating height and velocity is not very effective underneath the forcing. However downstream from the forcing, the errors in the solution are reduced by the information propagated by equatorial waves. If the stochastic forcing is in the west, equatorial Kelvin waves excited as part of the assimilation adjustment reduce the errors in the east. When the forcing is in the east, Rossby waves are important factors in reducing the simulation error in the west. Some of the beginning work with more advanced techniques in ocean data assimilation also relied on synthetic data. Long and Thacker (1989a, b) constructed the adjoint for a linear, continuously stratified, equatorial ocean model similar to that of McCreary (1981). This strong constraint variational approach was used to find the optimal estimate for model initial conditions when simulated sea level, as might be obtained from a satellite altimeter, and/or subsurface density observations are assimilated. The forward model equations and the inverse adjoint equations were iterated as part of a conjugate gradient descent method to minimize the cost function or misfit between the synthetic observations and the expansion coefficients (modal amplitudes) for the model velocity and pressure fields. Experiments were performed with an equatorial ocean basin 120 ~ wide that had been spun-up for five years with a constant zonal wind stress. The last 54 days of the solution were used as a control run. The initial conditions were set to zero for the assimilation experiments. Studies were performed as a function of varying degrees of freedom in the model and different sampling strategies for the observations. When the model was configured with only one baroclinic mode, essentially a reduced-gravity model, and simulated sea level information was assimilated, the adjoint method converged rapidly and the initial error was virtually eliminated. Two measures quantifying the effect of the assimilation are given in Figure 4. One measure

244

tracks the residual root mean square (rms) error for the model expansion coefficients presented as a ratio of the rms error between the control run and the assimilation run at each iteration relative to the initial rms error for the zero initial conditions. For the case of a single baroclinic mode the remaining rms error was 0.1% after 64 iterations. The second parameter indicates the fractional reduction in the cost function given as a ratio of the cost function at each iteration relative to the cost function for the zero initial conditions. When the model system consists of two vertical modes the adjoint is not as effective at reducing the error. As a result of this increase in the number of degrees of freedom in the model the residual rms error was 22% after the same number of iterations (Figure 4). A third model experiment increased the number of vertical modes to four, but kept the total degrees of freedom in the model constant by reducing the number of meridional modes in the solution. In this instance the residual rms error increased to 3 8 % . A s the vertical r e s o l u t i o n o f the

model increases, the optimization of the adjoint becomes more ill conditioned. There is insufficient information to constrain the projection of the sea level observations onto the subsurface structure of the model. Therefore, without additional a priori or observational information on the vertical structure, the value o f the sea l e v e l o b s e r v a t i o n s b e c o m e s limited.

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245

In a second series of experiments, the sea level observations were supplemented with simulated density observations as might be estimated from XBT observations. The model was configured with four vertical modes. Density information was assimilated at 50 m intervals between 50 and 500 m, and along meridional sections spaced 6 ~ apart in the zonal direction. This additional information on the subsurface thermal structure dropped the error from the 38% of the previous experiment to less than 1% (Figure 4). However the 6 ~ zonal sampling is rather dense compared with the real ocean. If the sampling spacing is increased to 18~ the error increases to 14%, but is still considerably less than the case without any density information. As the zonal spacing of the density observation increases, the ability to resolve small-scale features decreases, similar to the experience of Moore et al. (1987). Inspection of the individual solutions indicated that the coarse density sampling was introducing grid-point oscillations into the assimilated fields. In a final set of experiments, spurious short-wavelength structure was suppressed by adding a curvature penalty component to the cost function. Second derivatives of the assimilation fields in the zonal direction were penalized. With the addition of this smoothness constraint and the 18~ zonal spacing for the density observations, the rms error is reduced to 4% (Figure 4). Thus the addition of the curvature penalty permitted the model initial conditions to be reconstructed at nearly the same level of accuracy as before, but with onethird the amount of density observations. Most assimilation studies have been designed to accommodate Eulerian observations. Kamachi and O'Brien (1995) developed an adjoint method to assimilate the Lagrangian trajectories of synthetic drifting buoys into a nonlinear reduced-gravity model. In this study, the control parameter was the upper-layer thickness or height field for a tropical Pacific Ocean model. The cost function for the drifter trajectories was minimized to obtain the optimal spatial structure for the model height field. The model was spun up for 20 years forced by the seasonal climatology of the Florida State University (FSU) wind stress product (Goldenberg and O'Brien, 1981). The 21st year was the control experiment and 40 simulated buoy trajectories were extracted from the simulation. The assimilation experiments proceeded by changing the amplitude of the seasonal forcing by 5 %, 10%, and 20%. The objective of these experiments was to determine if the synthetic buoy trajectories could help recover the true model interface depth for the assimilation experiments with amplified forcing. The assimilation procedure, over a span of three months, was shown to recover the true height field near the equator in the vicinity of the buoy trajectories. Off the equator, in the vicinity of the North Equatorial Current, where the current flow and buoy trajectories were in the same direction as Rossby wave propagation, there was insufficient independent information to compute the gradient of the cost function. Advanced assimilation techniques have also been used to assess actual observational arrays as part of observing system simulation experiments. Miller (1990) used a Kalman filter to evaluate the importance of tide gauge observations and TOGA TAO estimates of dynamic height for assimilation studies of sea level in a linear model of the equatorial Pacific model. The ocean model was the linear wave model of Cane (1984) consisting of two vertical modes. The meridional structure was decomposed into five Rossby modes. A reference solution was obtained by forcing the model with the FSU winds for the years 1978-1983 together with a random component to the forcing. This random error in the forcing had a covariance structure corresponding to approximately a 2 m s-~ rms error and decorrelation lengths of 10~ zonally

246 and 2 ~ meridionally. Synthetic observations were extracted from this solution at tide gauge locations and some positions for the early deployments of TOGA TAO moorings. These data were assimilated into a model simulation that was forced by the FSU winds but without the random perturbations. The assimilated fields were then compared with simulated observations of the reference solution at island and TAO locations that were withheld from the assimilation. In one experiment synthetic data were extracted at locations corresponding to the sea level stations at Rabaul, Nauru, Jarvis, Christmas, Santa Cruz, and Callao. This is best described as two stations each in the western, central, and eastern Pacific. The western and central Pacific islands of Kapingamarangi, Tarawa, Canton, and Fanning were treated as validation sites. In the second experiment the tide gauge retrievals were augmented with synthetic observations at TAO mooring locations. Sea level from the reference solution was considered to be equivalent to dynamic height sampled at 12 TOGA TAO locations: equatorial moorings at 165~ 170~ 140~ 125~ 110~ and seven additional offequatorial moorings along 140~ and 110~ The locations of four proposed TAO mooring deployments were used as verification sites. A third experiment considered the subsurface information the TAO moorings could provide. In this experiment the individual contributions to the sea level height from two baroclinic modes were used rather than a total sea level measure. One of the benefits of the Kalman filter approach is that the scheme provides theoretical error estimates that quantify the uncertainty in the analyses. The assimilation experiments from Miller t1990) can be summarized by a contour map of the expected rms error in the sea level height anomaly (Figure 5). These error estimates are obtained from the error covariance matrix after it had achieved a steady state. The unfiltered simulation, i.e., no data assimilation, is estimated to have errors of 6 to 10 cm. Assimilating information at the six sea level stations reduces the error in the vicinity of the stations to less than 3 cm. The error within the equatorial waveguide is also reduced somewhat. The addition of the TAO observations has a significant impact. The influence of the synthetic TAO observations reduces the error to 2-4 cm across the full width of the waveguide. The presence of the simulated TAO observations had the greatest impact across the 8,000 km gap between the Jarvis and Christmas pair of islands in the central equatorial Pacific and Santa Cruz in the east. The separate assimilation of sea level contributions from the two baroclinic modes did not reduce the error beyond that achieved previously. In addition to the island tide gauges and TOGA TAO moorings, the TOGA XBT network is an important part of the Tropical Pacific Ocean Observing System. Bennett (1990) used a weak constraint variational approach to assess the simulated sampling of the TOGA Ship of Opportunity XBT array in the tropical Pacific Ocean. In this inverse method a finite number of representers or influence functions are sought to minimize the cost function. A linear, reduced-gravity model was used as a weak constraint to smooth synthetic XBT observations. The model was forced by idealized winds consisting of energy at 0.25, 0.5, and 1.0 cycles per year. The observational array consisted of thirteen ship tracks from the XBT network between 20~ and 20~ The cost function was the misfit between the model simulation and the synthetic observations projected onto the first six Hermite functions in latitude. The representer matrix provides an assessment of the efficiency of the observational array before data are actually collected. Its structure indicates where the data have an

247

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248 influence, and its conditioning number is a measure of the redundancy of the array. Three ship tracks that were farther than 11~ from the equator in the northwestern tropical Pacific were deemed insignificant within the context of this study. Information from these ship tracks was not important because the tracks lie north of the turning latitudes for the Hermite functions used. Four additional tracks in the central and eastern Pacific were eliminated because the observations that would be obtained along these tracks were redundant with observations from nearby tracks. In the end, due to the large zonal scales of the fields being considered from the linear model, only every other ship track was needed to form an efficient array. In summary, the information obtained from these assimilation studies with synthetic data has helped to guide future assimilation experiments that use genuine ocean observations. As more and more real data have come in from the observing systems, there has been less of a need for the assimilation studies with synthetic data. Yet, these initial efforts at tropical ocean data assimilation have served to foreshadow some of the results obtained with real data. These studies demonstrated how equatorial wave processes rapidly transmit assimilated information east along the equator and more slowly west off the equator. Assimilating observations of the subsurface density or thermal structure have been shown to be more important than velocity measurements for the usual situation in the tropics where the potential energy is greater than the kinetic energy. Sea level from island tide gauges and satellite altimeters is easily assimilated into linear reduced gravity-models. However, when the assimilation model is configured with more than just a few degrees of freedom in the vertical, supplemental information on the subsurface vertical structure, such as provided by XBT observations, is needed to use the sea level information effectively. In a similar manner, horizontal sampling strategies that have been successfully implemented in linear models may prove inappropriate in GCMs where small-scale structure, not present in the simple models, can introduce noise into the assimilated fields.

3. A S S I M I L A T I O N OBSERVATIONS

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OCEAN

The assimilation of real data into tropical ocean models can be placed into two general categories based on the type of ocean model being applied. On one hand there are the studies that use a reduced-gravity model or at most a linear model with a few baroclinic modes. In these studies some measure or proxy for the height field is assimilated. This includes observations from island tide gauges, XBTs, and the GEOSAT altimeter. Because of the simplicity of the model, these data often tend to be assimilated with the more advanced assimilation methods such as the Kalman filter and the adjoint method. In the second category, an ocean GCM is used to assimilate SST, subsurface thermal structure from XBTs and TAO moorings, and sea level from tide gauges. Only the simpler data assimilation approaches such as optimal interpolation and the successive correction method have been used to date in ocean GCMs. 3.1. R E D U C E D - G R A V I T Y M O D E L A P P L I C A T I O N S In a series of papers by Moore and Anderson (1989) and Moore (1989, 1990) the successive correction method has been used to assimilate XBT information into a linear,

249 reduced-gravity model for the tropical Pacific Ocean. The depth of the 16~ isotherm, as measured by the XBTs, was used as a proxy for the thermocline or pycnocline depth in the model. The period of study was June 1979 to December 1983, and the model was forced by the FSU winds. Comparisons between the model solutions and the XBT data prior to assimilation indicated that the mean depth of the model thermocline was similar to the observed depth of the 16~ isotherm in the western and central Pacific. In the eastern Pacific, however, the model thermocline was too deep by a factor of two. In terms of the variability about the mean, the displacements of the model thermocline were generally smaller than that observed. The XBT information was assimilated into the model on a monthly basis. In the western Pacific the impact was minimal because the model first guess there was reasonably correct. In other regions where the model was not as good initially, the data assimilation changed the depth of the model thermocline by as much as 50 m. For example, in the eastern equatorial Pacific the rms error was cut in half at the time that the model was being updated. In between updates, the model error rose quite rapidly. This was because the eastern equatorial Pacific was a region in the model with an erroneously weak zonal pressure gradient. This would imply that the model wind forcing was too weak or that the model stratification (also speed of the first baroclinic mode) was too large. The influence of the data was to raise the model thermocline in the east causing it to be out of balance with the model wind forcing. In between updates the pressure gradient would rapidly relax. This is in contrast to the situation off the equator where slower time scales dominate. Away from the equator in regions such the North Equatorial Countercurrent Trough, the data have a sustained impact on the depth of the model thermocline over long zonal scales. As demonstrated in some of the earlier initialization studies with synthetic data, on the equator, Kelvin waves set the time scale on which the information from the assimilation leaves the insertion region. Away from the equator, the slower moving Rossby modes, excited as part of the assimilation, are able to influence the model over a longer period of time. These same space-time scale arguments contribute to there being a more severe initialization shock along the equator, especially for observations assimilated in the westem Pacific at the origin of the Kelvin wave characteristic. Attempts to filter out the Kelvin modes with a normal mode initialization scheme inspired by that used in numerical weather prediction offered mixed results. While spurious Kelvin modes that contribute to initialization shock can be eliminated, there is also a serious risk of filtering out information on real Kelvin waves contained in the data. Most of the other assimilation studies in this category of simple models have been performed with the more advanced assimilation methods. Prior to the Kalman filter study of Miller (1990) that considered synthetic tide gauge and TOGA TAO data, the work of Miller and Cane (1989) was one of the initial efforts in all of oceanography to apply the Kalman filter to real observations. Unfiltered solutions for equatorial Pacific sea level were obtained by using the FSU winds to force the same linear wave model, consisting of two baroclinic modes, for 1978 through 1983. For the most part, this hindcast of sea level underestimated the amplitude of sea level change during the 1982-1983 El Nino. The purpose of the assimilation study was to produce monthly maps of sea level anomalies for the equatorial Pacific, and, as part of the process, assess the impact of assimilating tide gauge data. The system noise field for the Kalman filter assumed that the main source of error was the wind stress error and it was assumed to be statistically

250 homogeneous. Sea level information was assimilated at six locations. Four additional stations were withheld from the assimilation and used as validation points. The overall effect of the assimilation was to increase the amplitude of the sea level anomalies and to add detailed structure that was not contained in the unfiltered solutions. The estimated rms error for the unfiltered model sea level was about 5 cm within a few degrees of the equator. The assimilation reduced this error by 1 cm. At first, this might not appear to be a very significant result. Nonetheless, it is noteworthy that, independent of the absolute reduction of the error, the assimilation at only six locations was able to reduce the error by 20% across the full 17,000 km width of the equatorial Pacific Ocean. This work was recently extended in several ways by Miller et al. (1995). In this followup study the linear wave model was replaced by the grid-point model of Cane and Patton (1984). This permitted the latitudinal range of interest to be extended from about + 6 ~ from the equator to approximately +15 ~ A relatively coarse resolution of 5 ~ in longitude and 2 ~ in latitude was used to keep the size of the state space manageable. The latitudinal extension of the domain permitted sea level data from a total of eight stations to be assimilated, and data from seven more stations to be withheld for validation. Another extension to the previous work was the assimilation of dynamic height calculated from XBT observations along shipping tracks in the western, central, and eastern tropical Pacific. A third enhancement was the inclusion into the Kalman filter of a nonhomogeneous error model for the wind forcing. Four experiments were performed: the unfiltered sea level simulation, assimilation of the tide gauge observations, assimilation of the XBT dynamic heights, and the assimilation of the tide gauge sea levels and the XBT dynamic heights. A statistical objective analysis was also performed on the in situ sea level and XBT dynamic heights for comparison purposes. The objective analysis clearly demonstrated that the amplitude of the unfiltered sea level solution was deficient. Spatially, however, the objective analysis suffered from having unrealistic structure within data void regions away from islands and between the ship tracks. The data assimilation experiments remedied both of these shortcomings. In all three assimilation trials the total variance of the sea level was enhanced considerably relative to the total variance of the observations. Figure 6 depicts the error reduction induced by assimilating the two data types separately and together. In the central and western tropical Pacific significant error reduction occurred off the equator and to the west, i.e., downstream for Rossby wave propagation, of islands and to a lesser extent XBT assimilation locations. This was verified by the point comparisons at the withheld stations of Yap, Truk, and Honiara. At other locations such as Nauru and Fanning the improvement was less significant. In the east, the Santa Cruz and Callao sea level stations, as well as the eastern portion of the XBT track served to constrain the eastern end of the equatorial waveguide and the coastal sea level. A common assumption in assimilation studies like this is that the system noise is homogeneous. In this study a nonhomogeneous noise model was constructed based on the uncertainty in different wind products available to force the model. In the end, though, the use of an nonhomogeneous error model did not have a significant bearing on the results. One of the drawbacks of using a Kalman filter is the computational expense involved in updating the error covariance matrix at each assimilation interval. This can render a number of applications for the Kalman filter impractical since the size of this matrix is the square of the state space for the model Variables. It was this computational limitation that required Miller et al. (1995) to control the size of the state space with a coarse 5~ x 2 ~ resolution for

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252 their ocean model. In a related study by Cane et al. (1995), a procedure was developed to reduce the size of the state space for the Kalman filter without having to sacrifice resolution or complexity in the model. The philosophy of the method is to reduce the number of degrees of freedom in the error covariance matrices used to compute the Kalman gain. This is accomplished at every assimilation update by projecting the full state space of the model onto multivariate empirical orthogonal functions (MEOF) from a control run of the model. The error covariance structure is then determined using a truncated number of the lead MEOFs. Besides the efficiency offered by this approach, the truncation also limits the amount of smallscale structure in the covariance of the system noise. This reduced state space Kalman filter was applied to a similar version of the tropical Pacific ocean model used by Miller et al. (1995). This time the model resolution was 2 ~ zonally and 0.5 ~ meridionally. Observations from the tropical Pacific sea level network were assimilated at 34 tide gauge locations between 29~ and 29~ for the period 1975 to 1986. The results of the reduced state space Kalman filter compared quite favorably with the full state Kalman filter. A total of 296 MEOFs described 100% of the model variance. However, the best results of the reduced-state filter were obtained using only 17 MEOFs. In the unfiltered control run these EOFs accounted for 80% of the total variance. In the filter runs, these structures maximized the information extracted from the 34 gauges as determined by correlations and rms differences with withheld data. Adding more EOFs ended up increasing the small-scale structure of the solution without any apparent increase in quality. The reduction of the state space for the Kalman filter allowed literally hundreds of assimilation experiments to be performed as part of this work. Experiments with the same set of tide gauges and time period as Miller et al. (1995), indicated that the same results or better could be achieved using the reduced state space Kalman filter with the high resolution ocean model versus the full Kalman filter with the coarse resolution model. If only a few tide gauges or XBT tracks could have a positive impact on large-scale tropical ocean simulations, the prospect of having basin-scale coverage of sea level, such as that afforded by a satellite altimeter, is an even greater impetus for tropical ocean data assimilation. Once again some of the first assimilation studies of satellite altimeter data in the tropics relied on the application of the Kalman filter. In a pair of papers by Fu et al. (1991, 1993) Geosat altimeter data were assimilated into two different linear ocean models. In the first study, the model was composed of four equatorial wave modes: the Kelvin wave, the mixed Rossby-gravity wave, and the first and second meridional mode Rossby waves. The model did not include the effects of any wind forcing or ocean boundaries. GEOSAT data from November 1986 to March 1988 were projected onto the meridional structure of these four wave modes. The zonal structure was decomposed into 10 sinusoidal zonal wavelengths. Only the first baroclinic mode was included in the vertical. The Kalman filter was used to update the amplitude and phase of the wave modes. Of the 40 possible wave components, it was found that the best description of the Geosat sea level observations was obtained with only the two longest Kelvin waves and the two longest, first symmetric mode, Rossby waves. Incorporating the remaining 36 wave modes tended to propagate more noise than signal, and therefore degraded the assimilated solution. The total variance of the Geosat data, 75 cm 2, was partitioned as follows. The altimeter measurement error variance was estimated to be 28 cm 2 with the remaining 47 cm 2 being the estimated oceanic signal variance. The assimilated data were able to account for 11 cm 2 or 23% of the signal variance. This relatively low

253

fraction of the total variance was attributed to the simplified model physics, e.g., no wind forcing or boundary effects. In the second paper by Fu et al. (1993) these limitations were addressed by using the same model Miller and Cane (1989) used to assimilate the island sea level observations. Now wind forcing, boundary effects, richer meridional structure, and two vertical modes were included. Figure 7 shows the cross correlation between the unfiltered model sea level versus Geosat and tide gauge data for November 1986 through November 1988. Generally, the point correlations between the unfiltered model and the tide gauges were higher than between the model and the Geosat data. Negative correlations with the Geosat data were present along the eastern boundary and in the northeastern portion of the basin. The corresponding correlations for the assimilated solution are also presented in Figure 7. While it should not be a surprise that the correlations with Geosat are improved by assimilating Geosat into the ocean model, it is worth noting that the correlations were improved at all tide gauges except the one station (Santa Cruz) in the east. This overall improvement was consistent with the finding that 68% of the signal variance was now being accounted for using the better ocean model versus the 23% that was obtained previously in Fu et al. (1991). In the eastern region the unfiltered model disagreed with the altimeter data and the assimilation of the altimeter data degraded the comparison with the local tide gauge, suggesting that the altimeter data may contain large errors in this region. Additional experiments indicated that the same quantitative results could be obtained using a sub-optimal Kalman filter in which a steady-state limit to the error covariance matrix was reached early in the filtering procedure. The error covariance matrix was then kept constant for the remainder of the assimilation. The use of this steady-state filter approach had the advantage of reducing the computational requirements by a factor of 25 while yielding essentially the same results as before. The steady-state filter accounted for 62% of the signal variance. The use of a steady-state Kalman smoother that incorporated both past and future data at update intervals did not have a significant impact. For this experiment 66% of the signal variance was accounted for by the smoothed solution. This indicated that including future data slightly improved the results over the sub-optimal Kalman filter, but the recovered variance was still slightly less than the 68% recovered with the full Kalman filter. The Kalman filter has also been used by Gourdeau et al. (1992, 1995~ to assimilate Geosat observations into a tropical Atlantic Ocean model. In these studies the altimeter data were assimilated into a reduced-gravity model chosen to have a phase speed corresponding to the second baroclinic mode. The monthly mean wind stress of Servain and Lukas (1990) was used to force the model. The initial study dealt with seven months of altimeter data. In the latter study, Geosat data were assimilated from November 1986 to November 1988. Similar to Fu et al. (1993), a steady-state Kalman filter was used to reduce the computational overhead. Validation of the filtered solutions is a problem in the tropical Atlantic because there are few open-ocean tide gauge locations. In this study, a comparison could only be made at the island of Principe, just north of the equator in the Gulf of Guinea. At this location the cross correlation between an objective analysis of the altimeter data and the tide gauge data was 0.59. Assimilation of the altimeter data into the linear model raised the correlation with the station data to 0.70. An interesting finding in this work was that even though a second baroclinic phase speed of 1.32 m s-~ was chosen for the model, the assimilation of the altimeter data inserted information in January 1988 with amplitude of 2-4 cm that was seen to

254

propagate eastward along the equator at a first-mode wave speed of 2.2 m s -~. The filtered solution was also used to identify a biennial signal in the large-scale sea level that may be associated with zonal wind stress changes in the western equatorial region.

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255 The use of the adjoint method with real data in simple tropical ocean models was pursued by Sheinbaum and Anderson (1990a, b) as an extension to the successive correction studies of Moore and Anderson (1989) and Moore (1989, 1990). Once again XBT observations of the depth of the 16~ isotherm were used to correct the interface depth of a linear, reduced-gravity model. The cost function to be minimized was the difference between the observed and modeled height field. The adjoint was used to determine the optimal initial condition for h that gave the best space-time fit to the XBT data for January to June 1980. The adjoint was forced by the misfit to the data at 10 day intervals. This assimilation scheme was shown to improve the initial thermocline depth in the southem and eastern tropical Pacific. In general, this improved fit to the data was retained for the entire six month period except in the eastern equatorial Pacific. In this region the misfit was degraded to the level of the wind-forced control run without assimilation. This was the same region where the successive correction method exhibited spurious Kelvin waves as a result of an imbalance between the pressure gradient of the assimilated field and the overlying wind forcing. With the use of the adjoint method, spurious waves are not excited because no discrepancies between the observations and the assimilated fields are introduced. Instead, the adjoint is unable to fit all the data equally well in space and time. This was not a problem caused by data coverage in the east, but rather an inconsistency between the model, forcing, and the observations. Attempts to reduce the misfit in the east by increasing the magnitude of the forcing or decreasing the stratification proved problematic. While this could improve the situation in the east, it also served to increase the misfit in other portions of the domain. The adjoint was also noted to introduce small-scale structure into the initial conditions when drawing toward noisy data. Additional experiments were performed to smooth this structure by either adding a penalty term to the cost function, weighting the model first guess (i.e., prior estimate) which was essentially large scale, or reducing the number of iterations of the optimization algorithm. While these procedures gave smoother initial conditions, this information was lost several months earlier in the eastern equatorial Pacific than in the original assimilation experiment without any smoothing. Smedstad and O'Brien (1991) used the adjoint method to optimize the phase speed, c, used in a linear, reduced-gravity model for the tropical Pacific Ocean. The number of degrees of freedom for the adjoint was limited by only seeking solutions where the phase speed was allowed to vary as a function of longitude. The ocean model was spun up using the FSU winds for 1972 through 1983. A series of identical twin experiments was performed at first. These included experiments with synthetic observations being assimilated everywhere in order to estimate a constant phase speed and next a zonally varying phase speed. This was followed by a third experiment where synthetic sea level observations were assimilated at three locations and used to estimate a zonally varying phase speed. All three experiments showed that the optimization recovered the correct phase speed. Two contrasting experiments were performed with real data. In the first experiment tide gauge observations were assimilated at Santa Cruz, Truk, and Jarvis islands for 1979. This year was chosen because normal conditions were prevalent in the tropical Pacific. The initial guess for the square of the phase speed was a constant 6.0 m 2 s 2. Figure 8 shows the zonal structure of c 2 after one, three, and six iterations of the optimization algorithm. The phase speed was high in the west and decreased steadily to about 160~ and then was basically constant to the east. This was consistent with what is expected from a two-layer system with a deep thermal structure in the

256 west and a shallower thermocline in the east as observed. This use of the phase speed as a control parameter had a direct impact on the quality of the solutions. Cross correlations with tide gauge data increased at all three stations used in the assimilation, but more importantly, they were increased at two locations, Guam and Nauru islands, that were not part of the assimilation. In the second experiment, the same procedure was repeated for June 1982 through May 1983 at the height of the E1 Nino. This time the phase speed increased in value from west to east up to 160~ before leveling off and decreasing slightly (Figure 8). This is also consistent with the observed shoaling of the thermocline in the west and deepened thermocline in the east during this extreme event. Correlations with observed sea level were again improved.

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257 The adjoint method has also been applied to remotely sensed sea level data. Greiner and Perigaud (1994) have used the adjoint of a nonlinear reduced-gravity model for the Indian Ocean to assimilate Geosat altimeter data. Because of uncertainties in the geoid, it is impossible to reference the Geosat data to an absolute level. The purpose of this study was to demonstrate that altimeter data could be used to improve the mean spatial structure of the thermocline depth in a nonlinear ocean model, and thereby provide a reference level for the altimeter data. Two main experiments were performed. The first experiment optimized the initial conditions for the thermocline and a second experiment optimized both the initial thermocline depth and the mean thermocline depth (reference surface) across a one-year assimilation period. The first guess for the reference surface was obtained by averaging the thermocline depth from a wind-forced control run of the model for 1987-1988. The Geosat data were assimilated every 10 days beginning November 1986. Additional experiments were used to optimize the drag coefficient, diffusion coefficient, and density ratio used in the model. A principal result of the data assimilation was to change the mean depth of the thermocline on the order of 10-30 m. Figure 9 shows the initial reference level from the windforced simulation and the changes to this resulting from the Geosat assimilation. The largest modifications are found in the south and east near the South Equatorial Current and the Indonesian Throughflow. These changes to the mean topography of the thermocline suggest that the model may need to account for the mean mass transports from the Pacific Ocean, into the Indian Ocean, and out to the Atlantic Ocean. Smaller-scale regional changes to the thermocline depth occurred in the Arabian Sea and Bay of Bengal implying changes to their internal circulations. The assimilation of the observed data increased the amplitude of the temporal variability of the thermocline, consistent with many previous assimilation studies. Similar to the situation in the tropical Atlantic Ocean, independent validation data are hard to come by. In this work comparisons could only be made relative to the Geosat data prior to assimilation. The Geosat sea-level variations alone, converted into thermocline variations, had an rms variation of 21 m. The average rms variability of the thermocline depth increased from 11 m for the wind-forced model without assimilation to 15 m for the first experiment and to 16 m for the second experiment. The mean correlation with Geosat increased from 0.40 without assimilation to 0.62 for the two runs with assimilation. The rms difference with Geosat decreased from 20 m to 18 m for the first experiment and to 16 m for the second experiment. These assimilation studies have shown that observations of the oceanic height field, be they sea level, dynamic topography, or thermocline depth, can be readily assimilated into simple tropical ocean models. In view of the large length scales present, significant error reduction can be obtained with relatively sparse sampling. The major effect of the data assimilation in many of these studies was to increase the amplitude of the variability. Data assimilation was also useful at improving model parameters and pointing to potential problems in both the models and the data. Good examples of this were the results from several assimilation studies in the eastern equatorial Pacific. Assimilation of Geosat data indicated potential problems with the altimeter retrievals in the eastern Pacific. Assimilation of XBT observations indicated problems with the balance between the zonal pressure gradient and zonal wind forcing.

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259 simulating the 1982-1983 E1 Nino (Philander and Seigel, 1984), and in view of the forthcoming data to be obtained by TOGA, the same primitive equation model was ported from the Geophysical Fluid Dynamics Laboratory (GFDL) to NMC in support of a real-time ocean analysis capability (Leetmaa and Ji, 1989). The model was configured with a horizontal resolution of 1~ zonally and 1/3 ~ meridionally within 10~ of the equator and 27 levels in the vertical. Initially, the model was forced with an analysis of ship-wind observations. Since October 1987 the model has been forced with NMC analyses of the surface wind field. An optimal interpolation scheme was employed to update the thermal field of the model. The purpose of this effort was to provide the best possible description of the state of the tropical Pacific via routine nowcasts that incorporated the in situ observations of the TOGA program. In this regard, the model was serving as a tool to synthesize the observations and produce a four-dimensional description of the tropical Pacific Ocean that could not have been obtained by using separately either the model or observations. The continuous updates or constraints placed on the model by the data helped to counteract the inadequacies in the forcing functions and to identify regions of the model that were consistently deficient. The analyses that resulted were well suited to ocean process studies of the dynamical and thermodynamical variations in the tropical Pacific on seasonal to interannual time scales. Furthermore, the real-time nature of this capability provided the basis for using these fields as initial conditions to coupled oceanatmosphere climate forecasts. The importance of this latter point is elaborated on in greater detail by Leetmaa and Ji in the following chapter. Based on the results of the initialization and identical twin studies, together with the quantity of data available, only temperature observations are assimilated. A blended SST product (Reynolds, 1988, Reynolds and Smith, 1994) is used to determine the mixed layer temperatures in the model. The incorporation of these SST observations is used to counter the uncertainties in surface heat flux estimates. At depth, XBT observations, and more recently, TOGA TAO temperature measurements are assimilated monthly. At the mid-point of each month, the upper ocean (450 m) temperatures from the model are differenced with the observations obtained within + 15 days. The OI scheme of Derber and Rosati (1989) is used to update the temperature structure of the model using the difference field between modeled and observed temperatures. The decorrelation scales for the analysis are 10~ zonally and 2 ~ meridionally. The model is then integrated forward for one month and the assimilation procedure is repeated. The major impact of the assimilation is to correct bias problems in the temperature field, both SST and the depth of the thermocline. Along the equator, the model SST simulation without assimilation is too cold in the east and too warm in the west. The depth of the 15~ isotherm is also too deep on the equator in the east, suggesting there are problems with the vertical stratification in the model. Much of the same low-frequency temperature variability is simulated without assimilation, but with incorrect vertical temperature structure. Comparisons with data from the first few TOGA TAO moorings that did not make it into the assimilation (Hayes et al., 1989) demonstrate that these errors are reduced as a result of the assimilation. The observations at the 110~ mooring on the equator are presented in Figure 10. Most of the cold bias in SST has been eliminated. Similarly, the depth offset for the 15~ isotherm was reduced. It is also worth noting that the assimilation of the temperature data is seen to impart some improvement to the zonal velocity at this location. Halpem and Ji (1993) suggest that additional improvements along the equator might be possible if the one month

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Figure 10. Low-pass-filtered time series of 0~ 110~ of SST, depth of 20~ (Z20) and 15~ (Zis) isotherms, and zonal (U) and meridional (V) velocity at 10-m and 80-m depths. The heavy solid line shows the observations, the dashed line is the model simulation without data assimilation, and the light solid line is the model simulation with data assimilation (from Hayes et al., 1989). Temperature biases also exist away from the equator. Hayes et al. (1989) show that without assimilation a cold SST offset is present south of the equator. However, when the SST observations are assimilated the SST is not corrected adequately, but rather becomes too warm. In between update intervals, the model SST increases by 1-2 ~ beyond SST verification observations. This would imply there is a problem with the heat budget in the region and heat is not being removed as efficiently as it needs to be. The work of Ji and Smith (1995) has shown that the assimilation procedure corrects large-scale errors in the depth of the thermocline away from the equator. Figure 11 illustrates the annual mean impact of the data

261

Figure 11. 1 l-year mean for the depth of the 20 ~ isotherm for a simulation forced by Hellerman and Rosenstein climatology (x0.9) plus interannual anomalies from the FSU wind data (upper), same as above with data assimilation of temperature observations (middle), and the difference field (lower). The units are in m. For the difference field, areas where the H20 difference is greater than 20 m are in dark shading, areas where the H20 difference is lower than -40 m are in light shading; for the mean fields, areas where H2o is between 100 m and 140 m are in light shading, areas where the H20 is greater than 240 m are in dark shading (from Ji and Smith, 1995).

262 assimilation in a series of 11-year hindcast experiments forced by different wind products. In this example the model was forced with the seasonal wind forcing of the Hellerman and Rosenstein (1983) climatology plus the interannual wind anomalies from FSU for February 1982 to December 1993. The effect of the data assimilation is to deepen the mean depth of the 20 ~ isotherm by 30-50 m in the southeast and to raise the thermocline by 10-20 m in the vicinity of the North Equatorial Countercurrent Trough. Either there is a problem with the climatological wind stress curl forcing in these regions or there is a problem with the model physics. Additional experiments with lesser quality winds indicated that the assimilation of temperature observations was necessary for compensating for some of the errors in the surface forcing, but not sufficient. A better ocean analysis was obtained by using both better wind forcing and data assimilation. Based on the success of assimilating ocean data into the NMC GCM, assimilation schemes have been developed for other GCMs of the tropical Pacific and Atlantic Oceans. The successive correction method has been used by Fischer and Latif (1995) to continuously update a primitive equation model for the tropical Pacific. In a series of assimilation experiments with SST, island sea level, and subsurface thermal observations, the rms differences between the model fields and observations were reduced by factors of at least two to three. In their first experiment, SST observations were assimilated at every time step with decorrelation scales of 2000 km zonally and 200 km meridionally and with a sliding time window of 15 days. The model was forced by the FSU winds from January 1966 through December 1986. The surface heat flux was parameterized with a Newtonian damping to a prescribed climatological air temperature. The mean rms difference between the model SST without assimilation and the observed SST varied, as a function of time, between 1.1 ~ and 1.7~ The assimilation of the observations brought the rms difference down to 0.4 ~ to 0.6~ The SST assimilation had the effect of heating the surface layer by greater than 0.5~ along the equator to remedy anomalously cold model SST (Figure 12). The insertion of SST observations into the upper level of the model had a related cooling effect at depth. At 200 m, temperatures were cooler by 0.5~ as a result of a shoaling of the undercurrent brought on by changes to vertical mixing processes. The SST assimilation contributed to a more stable stratification of the upper reaches of the water column. This led to a reduction in the Richardson number dependent mixing. Subsequently, the equatorial undercurrent became stronger and rose in the depth. The temperatures in the deeper layers decreased as a result. In a second experiment, sea level observations from 22 tide gauges in the western tropical Pacific were assimilated for January 1975 to December 1989. Rather than directly assimilating sea surface height, the sea level observations were projected onto the vertical thermal structure using an empirically derived fit between model sea level and model subsurface temperatures. Decorrelation lengths of 1000 km zonally and 100 km meridionally were used for the assimilation. Once again a 15 day data window was used. Prior to this modification of the subsurface thermal structure, the rms difference between the modeled and observed sea level anomalies was between 2 and 8 cm. The assimilation reduced the rms differences down to about 1 cm. This indicates that the assimilation scheme was sucessful at incorporating the observational information, but this should not be confused as being indicative of the actual measurement error of the sea level observations themselves which is significantly greater than 1 cm. Empirical orthogonal functions of the temperature changes in the equatorial (x-z) plane indicated the assimilation had the greatest impact on the subsurface

263

thermal variability at a depth of 70 m in the western one-third of the basin. This is not too surprising since most of the sea level stations were in the west as well. However, significant changes in the subsurface temperature variability were also induced in the eastern equatorial Pacific as a consequence of assimilating sea level information in the west.

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Figure 12. Difference of the long-term mean temperature between SST assimilation experiment and control run for September at the equator (from Fischer and Latif, 1995). The last assimilation experiment considered subsurface temperature observations. The period of interest was from January 1979 to December 1988. Subsurface temperature observations between 10 m and 300 m were assimilated using length scales of 1000 km zonally and 200 km meridionally. A wider data window of 30 days was used for these XBT observations. The mean rms error for the three-dimensional temperatures without assimilation was between 0.5 ~ and 0.9 ~ C. The assimilation of the same observations reduced the error to 0.1 ~ Restart experiments indicated that this information was retained by the model for about two months at the equator and for about three to four months outside +5 ~. Data assimilation studies with GCMs for the tropical Atlantic Ocean have focused on incorporating XBT and hydrocast temperature observations from the Seasonal Response of the Equatorial/Francais Ocean et Climat dans l'Atlantique Equatorial ( S E Q U A U F O C A L ) Experiment of 1983-1984. Morliere et al. (1989) assimilated 3610 temperature profiles from 1984 into an ocean GCM forced by the wind analysis of Servain et al. (1987). Once a month a successive correction method was used to map the difference field between the modeled and observed three-dimensional temperatures. The radii of influence for the temperature observations were chosen to be 800 km zonally and 300 km meridionally. As opposed to other assimilation schemes where the model temperatures would be updated directly at this point, in this approach the model was restarted at the previous month with a heating term proportional to the model-data misfit added to the temperature equation. The modified model

264 was integrated forward to the analysis time and the corrected model state was then used as the initial conditions going forward into the next month. On the basin scale, the effect of the assimilation was to raise the mean depth of the model thermocline by approximately 25 m. Along the equator, the assimilation raised the thermocline even more and strengthened the zonal pressure gradient. The magnitude of the zonal equatorial currents also improved as a result of the temperature assimilation. For example, the magnitude of the equatorial undercurrent near the mid-point of the basin for the model run without assimilation was approximately 40-50 c m s "1 in June 1984. The constraints imposed on the meridional temperature gradients by the assimilation increased the undercurrent magnitude to more than 60 cm sl. Direct current measurements made as part of FOCAL in July 1984 observed undercurrent speeds up to 80 cm s ~. In a separate study with the SEQUAL/FOCAL observations, Carton and Hackert (1990) developed an assimilation scheme for the GFDL model applied to the tropical Atlantic Ocean. Optimal interpolation was used to minimize the mean-square error for analyzed temperatures similar to the implementation of this model by the NMC for the tropical Pacific Ocean. E-folding scales of 340 km zonally, 180 km meridionally, and 40 days in time were used to analyze the temperature residuals. The model was forced by winds from the EurOpean Centre for Medium Range Weather Forecasts and the temperature fields were updated once a month. Similar to the experience with this model in the tropical Pacific, the principal effect of the data assimilation was to reduce systematic errors in the temperature field indicative of problems in modelling the mean stratification. Whereas the model of Morliere et al. (1989) had an anomalously deep thermocline, in this application of the GFDL model the thermocline was generally too shallow. Along the equator, the thermocline in a run without assimilation was too shallow by about 20 m, except in the far west. The SST in the Gulf of Guinea was also too warm by l~ The shallow thermocline and warm SST were characteristic of an anomalously strong stratification in the east. Data assimilation helped to correct the equatorial SST, but not the shallow thermocline in the east. Temporally, the seasonal amplitude of the zonal pressure gradient was too weak without assimilation. The seasonal variability improved due to the assimilation, but unwanted month-to-month noise was an apparent byproduct of the updating process. Away from the equator a shallow thermocline bias was also present. The model temperatures without assimilation were systematically too cool at the depth of the thermocline (Figure 13). Similar to the results of Ji and Smith (1995) with the Pacific version of this model, large errors exist in the southem hemisphere because the thermocline is too shallow. However in contrast to the Pacific Ocean simulations, in the northern hemisphere the depth of the countercurrent trough in the Atlantic is too shallow not too deep. A sizable fraction of this error was reduced when the assimilated temperatures deepened the model thermocline. As was the situation along the equator, the temporal variability of the gradients across this meridional trough-ridge structure was weak compared with the observations. At some longitudes, the data assimilation helped to increase the amplitude of the seasonal variability and at other longitudes the assimilation contaminated the north-south fluctuations with monthly noise. Although the impact on changes in the meridional temperature gradients was ambiguous, comparisons with direct measurements of the zonal equatorial currents indicated that assimilating temperature observations had improved the current simulations; this had been

265

the experience in the tropical Pacific (Hayes et al., 1989) and in the other tropical Atlantic study of Morliere et al. (1989). o

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4. SUMMARY A number of factors combine to make the tropical oceans a fertile area for ocean data assimilation. The deterministic nature of the low-latitude ocean, the development of an ocean observing system under the auspices of the TOGA Program, and the role of the tropical oceans in short-term climate prediction, have all been a catalyst for data assimilation studies in the tropical Pacific, Atlantic, and Indian Oceans. The concentration of TOGA observations in the tropical Pacific Ocean has meant that most tropical ocean assimilation studies have been in this basin. Applications of data assimilation methods to the tropics have resulted in analyzed fields of ocean properties, initialization studies, evaluation of observational array designs, estimation of model parameters, identification of problems in model physics, and identification

266 of problems in the data. Some of the first applications of the Kalman filter and adjoint methods to in situ ocean data have been in the tropics. Usually, these have been investigations assimilating data into a reduced-gravity or multi-mode linear model. Most assimilation efforts with general circulation models have relied on optimal interpolation or successive correction methods. Initialization studies with synthetic data demonstrated that assimilating height or subsurface temperature observations were more effective than velocity observations because the potential energy is normally greater than kinetic energy for tropical oceans. This fostered a number of assimilation experiments with in situ data, since observations of subsurface thermal structure and sea level are considerably more numerous than velocity observations. In reduced-gravity models, the effect of assimilating height field information was to increase the amplitude of the variability in the model. Often, assimilated data highlighted potential problems between the simulation of the zonal pressure gradient and the overlying zonal wind stress in the eastern equatorial Pacific. Studies also showed that the value of assimilating sea level observations, e.g., from tide gauges or from a radar altimeter, was limited when the model had more than a few degrees of freedom in the vertical, unless additional information on the subsurface vertical structure was included. In GCMs, the most important influence of the data assimilation was to correct bias or systematic errors in the surface and subsurface temperature fields of the model. Once these problems with the thermal stratification of the models had been remedied, the assimilation of temperature observations had the added benefit of improving the simulation of zonal equatorial currents. It is anticipated that this confrontation between observations and models brought about by data assimilation will lead to model improvements in an iterative manner. In the future as the mixing parameterizations, mixed layer physics, and surface fluxes used in these models improves, the impact of data assimilation should become relatively more important for time dependent phenomena and less so for correcting systematic biases. Assimilation methods have used both synthetic and in situ observations to assess components of the tropical Pacific Ocean Observing System as it began to evolve and be implemented. Observing system simulation experiments have considered the island tide gauge network, the XBT network, and some of the initial deployments of the TOGA TAO array. Now that this ocean observing system is completely deployed, it is an opportune time to evaluate the system as a whole. Such an assessment of the observing system will identify possible redundancies, and also should consider the potential of observations not presently being assimilated and observational components in need of enhancement. One example of this are the current measurements as a function of depth that are available at five points along the equator in the Pacific Ocean. Although subsurface temperature observations may be more important on the basin-scale, the equator is a region of energetic zonal currents that are critical to the zonal advection of temperature. It remains to be seen if these observations add value to assimilated data sets for the tropical Pacific Ocean. Moreover, the availability of routine space-based observations from radar altimeters requires that the relative importance and merit be established for in situ versus remotely-sensed observations of the tropical oceans. It is only a matter of time before the initial assimilation studies with Geosat altimeter data are followed by experiments assimilating TOPEX/Poseidon data into not only reduced-gravity models, but GCMs as well. Similarly, satellite scatterometer and passive microwave measurements of the surface wind field are beginning to become routine. In particular, the

267 whole subject of surface fluxes is in need of attention. A prime reason why data are assimilated into tropical ocean models is the uncertainty in surface fluxes of momentum and heat. Future assimilation studies will likely address this problem and treat the surface forcing as a control variable. Characterization of the errors in the forcing, errors in the observations, and errors in the models are all at a rather rudimentary level. Numerous assumptions and simplifications are made when constructing present error covariance structures. If the merging of models with data is to be truly optimal, there will be an ever increasing need for improved error estimates as more and more data from the observing systems in the tropics become assimilated. This will include a closer examination of what constitutes noise as opposed to real small-scale structure supported by the observations. Access to more data should also permit data to be withheld in support of more rigorous evaluations of the impact of assimilated data. A case in point are the theoretical estimates of the analysis errors provided by the Kalman filter, and the need to verify these estimates against actual data. With a few exceptions, most of the data assimilation efforts for the tropical oceans have demonstrated the potential of a particular assimilation method. However, the relative importance of one scheme versus another for practical applications has not been addressed. For example, do the benefits of implementing a Kalman filter or adjoint method outweigh the computational expense involved? In other words, for the present implementation of these methods in reduced-gravity models, what improvements are made to the model height fields above and beyond those with a straightforward use of optimal interpolation? Do the f'mite degrees of freedom contained in today's limited ocean data sets justify the vast state spaces required to solve the Kalman filter or the adjoint? Is it practical and advisable to implement versions of the Kalman filter and the adjoint in tropical ocean models more complex than linear, shallow-water models? Alternatively, can more information be extracted when assimilating today's ocean data into a GCM if assimilation techniques other than optimal interpolation or successive corrections are used? As more of these questions are answered data assimilation will become less of an end onto itself and more of a tool in support of larger process and phenomenological studies. A unique advantage of tropical ocean data assimilation is that some of these questions will be answered in the context of the impact on initial conditions for coupled ocean-atmosphere model forecasts. Prediction skill will be a very powerful metric. Routine short-term climate predictions will create a sustained demand for the data and provide a means to quantify the impact of a particular data type and assimilation methodology.

Acknowledgments The author wishes to thank Mark Cane, Martin Fischer, Zheng Hao, Bob Miller, and Michele Rienecker for their helpful comments.

5. REFERENCES Anderson, D. L. T., and A. M. Moore, Initialization of equatrial waves in ocean models, J. Phys. Oceanogr., 19, 116-121, 1989.

268 Bennett, A. F., Inverse methods for assessing ship-of-opportunity networks and estimating circulation and winds from tropical expendable bathythermograph data, J. Geophys. Res., 95, 16,111-16,148, 1990. Bergthorsson, P., and B. R. Doos, Numerical weather map analysis, Tellus, 7, 329-340, 1955. Busalacchi, A. J., M. J. McPhaden, and J. Picaut, Variability in equatorial Pacific sea surface topography during the verification phase of the TOPEX/POSEIDON mission, J. Geophys. Res., 99, 24,725-24,738, 1994. Cane, M. A., Modeling sea level during E1Nino, J. Phys. Oceanogr., 14, 1864-1874, 1984. Cane, M. A., and R. J. Patton, A numerical model for low-frequency equatorial dynamics, J. Phys. Oceanogr., 14, 1853-1863, 1984. Cane, M. A., A. Kaplan, R. N. Miller, B. Tang, E. C. Hackert, and A. J. Busalacchi, Mapping tropical Pacific sea level: data assimilation via a reduced state space Kalman filter, J. Geophys. Res., submitted, 1995. Carton, J. A., and E. C. Hackert, Data assimilation applied to the temperature and circulation in the tropical Atlantic, 1983-84, J. Phys. Oceanogr., 20, 1150-1165, 1990. Cooper, N. S., The effect of salinity on tropical ocean models, J. Phys. Oceanogr., 18, 697707, 1988. Delcroix, T., J. Picaut, and G. Eldin, Equatorial Kelvin and Rossby waves evidenced in the Pacific Ocean through Geosat sea level and current anomalies, J. Geophys. Res., 96, 32463262, 1991. Derber, J., and A. Rosati, A global oceanic data assimilation system, J. Phys. Oceanogr., 19, 1333-1347, 1989. Fischer, and M. Latif, Assimilation of temperature and sea level observations into a primitive equation model of the tropical Pacific, J. Mar. Systems, 6, 31-46, 1995. Fu, L.-L., J. Vazquez, C. Perigaud, Fitting dynamic models to the Geosat sea level observations in the tropical Pacific Ocean. Part I: A free wave model, J. Phys. Oceanogr., 21,798-809, 1991. Fu, L.-L., I. Fukumori, and R. N., Miller, Fitting dynamic models to the Geosat sea level observations in the tropical Pacific Ocean. Part II: A linear, wind-driven model, J. Phys. Oceanogr., 23, 2162-2181, 1993. Goldenberg, S. B., and J. J. O'Brien, Time and space variability of tropical Pacific wind stress, Mon. Wea. Rev., 109, 1190-1207, 1981. Gourdeau, L., S. Arnault, Y. Meynard, and J. Merle, Geosat sea-level assimilation in a tropical Atlantic model using Kalman filter, Oceanologica Acta, 15, 567-574, 1992. Gourdeau, L., J. F. Minster, and M. C. Gennero, Sea level anomalies in the tropical Atlantic from Geosat data assimilated in a linear model, 1986-88, J. Geophys. Res., 100, submitted, 1995. Greiner, E., and C. Perigaud, Assimilation of Geosat altimetric data in a nonlinear reducedgravity model of the Indian Ocean. Part 1: Adjoint approach and model-data consistency, J. Phys. Oceanogr., 24, 1783-1804, 1994. Halpern, D., and M. Ji, An evaluation of the National Meteorolgical Center weekly hindcast of upper-ocean temperature along the eastern Pacific equator in January 1992, J. Clim., 6, 1221-1226, 1993. Hao, Z., and M. Ghil, Data assimilation in a simple tropical ocean model with wind stress errors, J. Phys. Oceanogr., 24, 2111-2128, 1994.

269 Hayes, S. P., M. J. McPhaden, and A. Leetmaa, Observational verification of a quasi real time simulation of the tropical Pacific Ocean, J. Geophys. Res., 94, 2147-2157, 1989. Hellerman, S., and M. Rosenstein, Normal monthly wind stress over the world ocean with error estimates, J. Phys. Oceanogr., 13, 1093-1104, 1983. Ji, M., and T. M. Smith, Ocean model response to temperature data assimilation and varying surface wind stress: Intercomparison and implications for climate forecast, Mon. Wea. Rev., 123, in press, 1995. Kamachi, M., and J. J. O'Brien, Continuous data assimilation of drifting buoy trajectory into an equatorial Pacific Ocean model, J. Mar. Systems, 6, 159-178, 1995. Leetmaa, A., and M. Ji, Operational hindcasting of the tropical Pacific, Dyn. Atmos. Oceans, 13, 465-490, 1989. Long, R. B., and W. C. Thacker, Data assimilation into a numerical equatorial ocean model. I. The model and the assimilation algorithm, Dyn. Atmos. Oceans, 13, 379-412, 1989a. Long, R. B., and W. C. Thacker, Data assimilation into a numerical equatrial ocean model. II. Assimilation experiments, Dyn. Atmos. Oceans, 13, 413-440, 1989b. McCreary, J. P., A linear stratified ocean model of the equatorial undercurrent. Philos. Trans. R. Soc. London, 298, 603-635, 1981. McPhaden, M. J., TOGA-TAO and the 1991-93 ENSO event, Oceanography, 6, 36-44, 1993. Meyers, G., H. Phillips, N. Smith, and J. Sprintall, Space and time scales for optimal interpolation of temperature - Tropical Pacific Ocean, Progr. Oceanogr., 28, 189-218, 1991. Miller, L. R., R. E. Cheney, and B. C.Douglas, Geosat altimeter observations of Kelvin waves and the 1986-87 E1 Nino, Science, 239, 52-54, 1988. Miller, R. N., Tropical data assimilation experiments with simulated data: The impact of the Tropical Ocean and Global Atmosphere Thermal Array for the Ocean, J. Geophys. Res., 95, 11,461-11,482, 1990. Miller, R. N., and M. A. Cane, A Kalman filter analysis of sea level height in the tropical Pacific, J. Phys. Oceanogr., 19, 773-790, 1989. Miller, R. N., A. J. Busalacchi, and E. C. Hackert, Sea surface topography fields of the tropical Pacific from data assimilation, J. Geophys. Res., 100, in press, 1995. Moore, A. M., Aspects of geostrophic adjustment during tropical ocean data assimilation, J. Phys. Oceanogr., 19, 435-46 1, 1989. Moore, A. M., Linear equatorial wave mode initialization in a model of the tropical Pacific Ocean: An initialization scheme for tropical ocean models, J. Phys. Oceanogr., 20, 423-445, 1990. Moore, A. M., N. S. Cooper, and D. L. T. Anderson, Initialization and data assimilation in models of the Indian Ocean, J. Phys. Oceanogr., 17, 1965-1977, 1987. Moore, A. M., and D. L. T. Anderson, The assimilation of XBT data into a layer model of the tropical Pacific Ocean, Dyn. Atmos. Oceans, 13, 441-464, 1989. Morliere, A., G. Reverdin, and J. Merle, Assimilation of temperature profiles in a general circulation model of the tropical Atlantic, J. Phys. Oceanogr., 19, 1892-1899, 1989. National Research Council, Ocean-atmosphere observations supporting short-term climate predictions, National Academy Press, Washington, D. C., 51 pp., 1994. Philander, S. G. H., and A. D. Seigel, Simulation of El Nino of 1982-1983, Coupled OceanAtmosphere Models, J. Nihoul, Ed., Elsevier, 517-541, 1985.

270 Philander, S. G. H., W. J. Hurlin, and R. C. Pacanowski, Initial conditions for a general circulation model of tropical oceans, J. Phys. Oceanogr., 17, 147-157, 1987. Reynolds, R. W., A real-time global sea surface temperature analysis, J. Clim., 1, 75-86, 1988. Reynolds, R. W., and T. M. Smith, Improved global sea surface temperature analyses using optimum interpolation, J. Clim, 24, 929-948, 1994. Servain, J., M. Seva, S. Lukas, and G. Rougier, Climatic atlas of the tropical Atlantic wind stress and sea surface temperature: 1980-1984, Ocean-Air Interactions, 1, 109-182., 1987. Servain, J., and S. Lukas, Climatic atlas of the tropical Atlantic wind stress and sea surface temperature: 1985-1989, Centre ORSTOM de Brest, IFREMER, Plouzane, France, 1990. Sheinbaum, J., and D. L. T. Anderson, Variational assimilation of XBT data. Part I, J. Phys. Oceanogr., 20, 672-688, 1990a. Sheinbaum, J., and D. L. T. Anderson, Variational assimilation of XBT data. Part II: Sensitivity studies and use of smoothing constraints, J. Phys. Oceanogr., 20, 689-704, 1990b. Smedstad, O. M., and J. J. O'Brien, Variational data assimilation and parameter estimation in an equatorial Pacific Ocean model, Progr. Oceanogr., 26, 179-241, 1991. Smith, N. R., Objective quality control and performance diagnostics of an oceanic subsurface thermal analysis scheme, J. Geophys. Res., 96, 3279-3287, 1991. Smith, N., J. Blomley, and G. Meyers, A univariate statistical interpolation scheme for subsurface thermal analyses in the tropical oceans, Progr. Oceanogr., 28, 219-256, 1991. World Climate Research Programme, Scientific plan for the Tropical Ocean and Global Atmosphere Programme, WCRP Publication Series, No. 3/WMO TD No. 64, World Meteorological Association, Geneva, 146+xxvii pp, 1985.

Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

271

Ocean Data Assimilation as a Component of a Climate Forecast System Ants Leetmaa and Ming Ji National Centers for Environmental Prediction, NWS/NOAA, 5200 Auth Road, Camp Springs, Md. 20746 U.S.A. Abstract The E1 Nifio Southern Oscillation (ENSO) phenomena is the major source ofint e r a n n u a l climatic variability in the Tropics. It results primarily from ocean-atmosphere interactions in the tropical Pacific. Over the past decade as p a r t of the Tropical Oceans Global Atmosphere Experiment (TOGA) considerable progress was made in implementing observing systems to document this variability, developing a hierarchy of models, statistical as well as dynamical, to study its physics, and to implement routine experimental forecasts for aspects of ENSO related variability. During the past decade at the National Meteorological Center (NMC), presently the National Centers for Environmental Prediction (NCEP), a unified system for seasonal climate prediction was developed. This consisted of the routine assimilation of the in situ thermal data sets collected by TOGA into an ocean general circulation model to provide analyses for real time climate diagnostics and to provide ocean initial conditions for forecasts and a coupled ocean-atmosphere general circulation forecast model. Conceptually similar systems are currently being implemented at a n u m b e r of other Centers internationally. A basic requirement for climate diagnostics and prediction is the best definition of the state of the ocean. In the Tropics where the ocean is strongly and directly forced, a model simulation forced with observed stress fields combined with in situ observations through data assimilation, can give a good estimate. These modelbased analyses can provide the basis for diagnostic studies, verification of model simulations and forecasts, and the initial conditions for the forecasts. Comparisons of simulations using existing wind stress products and models to analyses produced using data assimilation show large differences indicating t h a t models and stress fields can still be improved. Without data assimilation model simulations contain significant errors both in their mean spatial structure and also in their low frequency variability. The thermocline topography in the mean is too weak, especially south of the equator where the subtropical gyre is not well defined. Experiments with several different wind products suggest t h a t this is more a result of model r a t h e r t h a n forcing field errors. Simulations without data assimilation are also unable to capture the full amplitude and structure of the low frequency variations associated with E1 Nifo. Data assimilation can overcome m a n y of these deficiencies. Even with assimilation, incremental improvements in analysis accuracy are further achieved

272 when better wind forcing is used. However, large corrections can also alter strict dynamical balances. One impact of this is t h a t the near equatorial currents in the western Pacific in NCEP's model-based analyses appear unrealistic. An improved estimation of the low frequency variability of the ocean should lead to higher skill levels in forecasts. This appears to be the case but the results are seasonally dependent. Forecasts initiated from late spring to fall for two versions of the N C E P forecast model show improved skill when data assimilation is used to derive the initial conditions. However, little positive impact is found for forecasts initiated in the winter months. If data assimilation is needed to correct for large errors, then in the forecast mode, where assimilation is not possible, the corrections, especially to the m e a n field, can lead to large systematic forecast errors. F u t u r e skill improvements will result from improvements in the forcing fields and ocean model used in the initialization, and improvements to the coupled forecast model. Indications from experiments at NCEP are that the largest impact on forecast skill is from improvements in the coupled ocean-atmosphere model used in the forecasts. The central role of data assimilation is in producing the best analyses t h a t can be used for improving the ocean models and forcing fields. 1. Introduction

The largest climatic variability on interannual time scale is the E1Ni~o-Southern Oscillation (ENSO) phenomena. Bjerknes (1969) was the first to suggest t h a t ENSO is the result of coupled interactions between the tropical Pacific Ocean and the global atmosphere. The memory of this component of the climate system resides in the ocean because of its greater heat capacity and longer adjustment time relative to the atmosphere (Wyrtki 1975, 1985). Therefore, understanding and documenting the physical mechanisms of ENSO and its global impact depends crucially on our ability to observe and analyze states of the tropical ocean, especially the tropical Pacific. The optimal combination of data, forcing fields and ocean model, simulations using modern data assimilation techniques, plays a central role in this. The tropical ocean's state is primarily determined by the history of the surface wind stress and heat and fresh water flux forcing by the atmosphere. Thus an ocean general circulation model(OGCM) forced with accurate fluxes gives a good estimate of the ocean circulation in the tropics in regions where internal instabilities are not dominant. The forcing fields can be obtained from analyses based on surface marine observations such as those produced at Florida State University (FSU, Goldenberg and O'Brien, 1981) or derived from operational analyses produced from atmospheric global forecast systems at operational centers such as the N C E P or the European Center for Medium Range Weather Forecasting (ECMWF). Unfortunately both ocean models and flux fields need improvement. Therefore assimilation of observed in situ data into an OGCM simulation can be an effective method for improving estimates of the state of the ocean.

273 The observing system for in situ measurements, especially for subsurface variables, of the ocean has been slow to develop. Over the past decade as a result of the Tropical Ocean Global Atmosphere (TOGA) experiment and World Ocean Circulation Experiment (WOCE), sampling of the subsurface thermal structure by expendable b a t h y t h e r m o g r a p h s (XBT) has become more systematic (Fig. 1). However, because the sampling is primarily confined to major shipping lanes, the coverage is spatially sparse and temporally sporadic, tks a result of TOGA an extensive a r r a y of moored buoys, TOGA-TAO (Hayes et al. 1991; McPhaden 1993), now regularly reports surface marine information as well as subsurface temperature information using satellite communications (Fig. 1). At the sea surface the XBT and TAO measure-

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ments are supplemented by a network of tide gauges for sea level and a global surface drifter program to provide climate quality sea surface t e m p e r a t u r e (SST) and near surface current information. Global coverage for sea surface t e m p e r a t u r e and relative sea level is now routinely available from satellites. The routine availability of the newer data sets, especially those from TAO and the TOPEX altimeter, give oceanographers an unprecedented opportunity to study the ocean and develop models for prediction. Ocean data assimilation provides an objective way to combine these data sets into consistent analyses t h a t can be used to study the ocean, improve models, and initialize coupled forecast systems. Ocean data assimilation has become an active area of research. Some examples of the growing body of literature in this area are: Miller (1985,1990); Miller and Cane (1989); Moore (1989,1990); Moore, et al. (1987); Moore and Anderson (1989); Sheinbaum and Anderson (1990a,b); Leetmaa and Ji (1989); Derber and Rosati (1989); and Hao and Ghil (1994). A comprehensive review of these and m a n y additional works in support of tropical ocean circulation studies are found in the previous Chapter of this volume by Antonio Busalacchi.

274 In addition to research efforts on ocean data assimilation for ocean circulation studies, dynamical model-based ocean data assimilation system have been developed to produce ocean initial conditions for coupled ocean-atmosphere forecast models for E1 Nifio predictions. Ocean analysis systems, based on the variational ocean data assimilation system of Derber and Rosati (1989), have been implemented at the NCEP (Ji et al. 1995) and the Geophysical Fluid Dynamical Laboratory (GFDL, Rosati et al. 1995). These systems assimilate in situ temperature data from XBTs and TAO buoys into ocean general circulation model to produce initial conditions for coupled ocean-atmosphere general circulation forecast models. Kleeman et al. (1995) used an adjoint method to assimilate both subsurface t e m p e r a t u r e and observed surface wind information into an intermediate complexity coupled model to achieve improved ocean initialization and improved forecast skill. Chen et al. (1995) developed an ocean initialization procedure to assimilate the FSU surface wind analyses into an intermediate coupled ocean-atmosphere model developed by Zebiak and Cane (1987). This resulted in enhanced low frequency climate signals in the coupled model and improved balance between the ocean initial conditions and the coupled model. The E1Nifio prediction skill of the Zebiak and Cane model utilizing this initialization procedure is significantly increased. These studies show t h a t various ocean data assimilation techniques can, by improving the accuracy of ocean initial conditions and improving balance between ocean initial conditions and coupled models, lead to improved ENSO forecasts. In this Chapter, we focus on the ocean data assimilation system at NCEP as a component of an end to end climate forecast system (Ji et al. 1994a) for ENSO and the i n t e r a n n u a l climate anomalies (height, surface temperature and precipitation) associated with ENSO. In this system, the ocean data assimilation system functions primarily to produce real time and retrospective ocean analyses for documenting and studying seasonal to interannual climate variability of the oceans (discussed in Section 3) and to produce high quality ocean initial conditions for coupled ocean-atmosphere forecast models. The NCEP's ocean analysis system will be briefly described in Section 2, details of the overall ocean data assimilation system can be found in Ji et al. (1995). The discussion in Section 4 examines the impact of the assimilation on correcting for stress errors and deficiencies in the ocean model. Difference in mean thermal structures between purely wind forced ocean simulations and analyses which are improved by data assimilation are examined in this section. As will be shown in Section 5, the improved ocean analyses lead to improvements in the forecast skill for SST variations in the tropical Pacific related to ENSO. However, despite these skill improvements, complications can arise t h a t impact the skill at longer lead times. One problem is the imbalance between the mean state of the analyses t h a t are used as the initial condition for the forecasts and the mean state of the coupled forecast system itself. This difference in the mean states is essentially the correction the in situ data makes to compensate for errors in the forcing field and deficiencies in the ocean model. This will also be discussed in Section 5. A s u m m a r y is given in Section 6.

275

2. An ocean analysis system The NCEP's climate forecast system consists of three major components: an ocean d a t a assimilation system which produces ocean analyses and initial conditions for coupled model; a coupled ocean-atmosphere general circulation model (CGCM) which is used for forecasting i n t e r a n n u a l SST variations in the tropical Pacific; and a climate atmospheric general circulation model (AGCM) which is used to produce ensemble seasonal climate forecasts for the tropics and extra-tropics using the CGCM predicted SSTs as the lower boundary condition for the atmosphere. The ocean analysis system consists of an ocean general circulation model r u n jointly with a four dimensional variational data assimilation system developed by Derber and Rosati (1989). The ocean model was developed at the GFDL (Bryan 1969; Cox 1984; Philander et al. 1987). In the Pacific domain, it extends from 45~ to 55~ and 120~ to 70~ The bottom topography is variable and there are 28 model levels in the vertical. The zonal grid spacing is 1.5 degrees. The meridional grid spacing is 1/3 degree within 10 degrees of the equator and gradually increases outside this zone. Poleward of 20~ and 20~ the meridional resolution is one degree. The time step for the model integration is one hour. A Richardson n u m b e r dependent formulation for vertical mixing is used in the upper ocean (Pacanowski and Philander 1981). The variational data assimilation method computes a horizontal t e m p e r a t u r e correction field obtained by solving the optimal interpolation objective analysis equation using an equivalent variational formulation (e.g., Lorenc 1986). This is done by minimizing an objective function given by I = 1TrE-,T

+ I(D(T ) _ To)rF-,(D(T)

_

To)

(1)

where T represents the correction to the first guess field, E is an estimate of the first guess error covariance matrix, To represents difference between the observations and the first guess field, D is an operator representing interpolation of first guess from model grid to observation locations and F is an estimate of observations error covariance matrix. The first term on the right hand side in (1) is a measure of the fit to the first guess weighted by the inverse of the first guess error covariance m a t r i x (E) and the second term is a measure of the fit to the data weighted by the inverse of the observational error covariance matrix (F). Assimilation is done continuously during model integration. Observed SST data from satellite, moored and drifting buoys, XBTs and volunteer observing vessels (VOS), between one week before and one week after the model integration time, and all subsurface thermal data from XBTs and moored buoys collected two weeks before and two weeks after this time are used. By limiting ones attention to low frequency phenomena, observations can be kept in the analysis system for several weeks, hence effectively increasing their influence. The model t e m p e r a t u r e field computed at the previous time step serves as first guess field. Error estimates are assigned to each observation and to the first guess. The observational error covariances are defined for each observation type and are weighted based on the distance in

276 time from the observation time to the model integration time. This weight reaches a m a x i m u m of one when the model and observation times are the same and reaches zero when the time difference is greater t h a n one week for surface d a t a and two weeks for subsurface data. The horizontal first guess error covariances are defined by a Gaussian function with e-folding scale of four degrees at the equator. This scale decreases away from the equator by the cosine of the latitude resulting in smaller scale horizontal correlation functions away from the equator. A weak coupling of the horizontal to the vertical analyses is included through a 1 - 2 - 1 smoothing of first g u e s s - d a t a differences (To) on adjacent vertical levels. F u r t h e r improvements to the system by producing and including more realistic vertical and horizontal error covariance functions based on structures of observed ocean anomalies are underway. When realistic vertical covariance functions are included, the coupling in the vertical dimension through the vertical error covariance m a t r i x requires the entire three-dimensional problem be solved at once. This system is used for producing analyses for climate monitoring and the initialization of a coupled forecast system for ENSO. Weekly mean analyses are produced each week in n e a r r e a l - t i m e for the Pacific and Atlantic basins. In addition, retrospective analyses have been performed for both basins for the period 1982-1994. These document the climatic variability t h a t has taken place during the last thirteen years, and in the Pacific provided the initial conditions required for the development of a coupled ocean-atmosphere forecast system for ENSO.

3. Ocean analyses for documenting and understanding interannual climate variability A significant advantage t h a t the model based analyses provide is a description of the large scale structure associated with the interannual variability t h a t is difficult to obtain from the observations by themselves. U n d e r s t a n d i n g of spatial structure and temporal evolution of ENSO variability is essential for developing and improving coupled ocean-atmosphere forecast models. To document the large scale, i n t e r a n n u a l climate variability in the Pacific was an l l - y e a r retrospective Pacific Ocean analysis for the period 1982 to 1993 produced using the ocean d a t a assimilation system of Ji et al. (1995). This data set is denoted as RA3. The forcing for this consisted of the m e a n annual cycle surface stresses of Hellerman and Rosenstein (1983, H&R hereafter) and the FSU monthly stress anomalies which were obtained by removing the 1965-1985 m e a n annual cycle from the FSU stress analyses. The n e a r equatorial region in the Pacific is where the strongest coupling takes place between the ocean and atmosphere during ENSO. In this region the depth variations of the 20~ isotherm (H2o) are frequently t a k e n as a surrogate for variations in the depth of the main thermocline or upper ocean heat content. The time history of this shows strong i n t e r a n n u a l fluctuations (right panel; Figure 2). Largest amplitudes are located in the central and eastern Pacific. E a s t w a r d propagation is suggested for the major signals. These anomalies represent the changes in the e a s t - w e s t slope of the equatorial thermocline associated with ENSO episodes. Dur-

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ANOMALOUS Hm ALONG THE EQUATOR

Fig. 2 Anomalous depth of 2OoC isotherm along the equator for the HCMP simulatio (middle) and the FtA3 analysis (right). Contour interval is 10 m. Dark (light) shading than 20 m (-20 m).

278

ing this time there were three w a r m episodes, 1982/83, 1986/87, and 1991/92, and two cold episodes, 1984/85 and 1988/89. During the time t h a t the anomalies are largest along the eastern boundary, anomalies of opposite sign are already present in the western part of the basin. These frequently lead to the transition to an event of the opposite sign; although t h a t was not the case in 1992/93. This m o d e l - b a s e d reanalysis (RA3) is a good rendition of the ocean as compared to i n d e p e n d e n t tide gauges (Ji and Smith, 1995). The other two panels in Figuire 2 show estimates of H2o variations for this same period derived from model simulations, i.e. no data assimilation, using two different wind products. The center panel (HFSU) uses the wind anomalies from FSU, i.e. the same anomalies used in RA3; the left panel (HCMP) uses anomalies derived from the climate AGCM which was forced with the observed SSTs for this time period. It is clear t h a t neither of these simulations captures the amplitude or s t r u c t u r e of the variability as shown in the right panel. Both underestimate the relaxation (enhancement) of the e a s t - w e s t slope of the thermocline associated with w a r m (cold) episodes. This is especially the case for the simulation shown in the left panel where in addition to amplitude discrepancies there are differences even in the sign of the anomalies at times (1987 for example). Since the right panel used the same forcing field as the center one, it is clear t h a t data assimilation contributes considerably to improving the description of the ocean for this region. One advantage of the model-based analyses is t h a t they provide basin scale, continuous in time descriptions of the variability. These form the basis for ENSO related climate diagnostic studies at NCEP. For example, a convenient way to look at the temporal and spatial variations of several fields is to use combined empirical orthogonal function (EOF) analysis (Nigam and Shen 1993). (For the analyses to be discussed here, EOF analysis of the component fields yields essentially the same description of the variability.) The OGCM computes surface pressure variations on its rigid lid; in a straight forward m a n n e r these can be related to sea level variations. Since the most significant variability on interannual time scale is related to ENSO, it is of some interest to see aspects of coherent sea level and SST variations on interannual time scale associated with ENSO variability. (The variability along the equator, as represented by variations in H20, was shown in Fig. 2). The leading two combined EOFs of sea level and SST anomalies are shown in Fig. 3 and Fig. 4. The sea level anomalies are derived from RA3 analyses and the SST fields are from the blended analyses produced at NCEP (Reynolds and Marsico, 1993). The leading EOF mode of sea level and SST anomalies carries approximately 18% of the variance. Its time series (Fig. 3) clearly shows t h a t its positive peaks correspond to the ENSO w a r m episodes of 1982/83, 1986/87, 1991/92, and the negative peak to the cold episodes of 1988/89. The SST anomaly p a t t e r n is t h a t which is present during the height of the event. Largest positive anomalies are located in the equatorial eastern Pacific surrounded by a horseshoe shaped p a t t e r n of negative anomalies in the western and offequatorial region. For a w a r m episode, the sea level signal shows a relaxation of the e a s t - w e s t pressure gradient within about 10 ~ of the equator, i.e. positive anomalies in the east and negative in the west. The positive signal extends poleward along the eastern boundary of the basin indicating poleward

279 propagation as expected. The negative anomalies in the w e s t e r n Pacific have off equatorial m i n i m a which are suggestive of s t r u c t u r e s associated w i t h Rossby waves.

Fig. 3 The leading mode from the combined EOF analysis for SST (top) and Sea level (SL, middle) anomalies. The time series for the leading mode (solid) and the second mode (dash) are shown in the lower panel. Contour interval for SST (SL) is I~ (5 cm). In addition, contours of-0.5~ and 0.5~ are depicted for SST. Shaded areas are where SST (SL) anomalies are above I~ (10 cm).

The time series of the second combined EOF mode suggests both i n t e r a n n u a l and i n t r a d e c a d a l variations. The i n t e r a n n u a l variations indicate a w a r m i n g (cooling) and sea level increase (decrease) in the vicinity of the dateline before m a t u r e w a r m (cold) episodes. For SST the p r e d o m i n a n t spatial s t r u c t u r e is of an e a s t - w e s t v a r i a t i o n w h e r e a s for sea level it has a strong meridional component. The intradecadal v a r i a t i o n indicates t h a t since about 1990 the region out by the dateline has been persistently w a r m e r t h a n normal and shows positive sea level anomalies. Although these time series end in 1993, such anomalous conditions have persisted t h r o u g h mid-1995. The presence of SST and sea level anomalies extending well outside the n e a r equatorial region indicates t h a t off equatorial wind forcing probably is involved. An e x a m i n a t i o n of the wind field for the period 1990-95 shows a general reduction of the winds from 20~ to 20~ (Ji et al. 1996). An E O F analysis of the spatial and temporal differences between ocean analyses and wind forced simulation (HFSU) results (Ji and L e e t m a a 1996) shows that, once a m e a n difference has been removed, the leading two difference E O F p a t t e r n s have large scale resemblance to the first two sea level EOFs as shown in Fig. 3 and 4. Without d a t a assimilation the amplitudes of these d o m i n a n t modes of variability

280 are u n d e r e s t i m a t e d . The m e a n difference field is discussed in the next Section. Fig. 4 The same as in Fig. 3, except for the second combined EOF mode. Contour interval for SST (top) is 0.25~ Contour interval for Sea Level (SL) is (5 cm). The time series for the second mode (solid) and the first mode (dash) are shown in the lower panel. Shaded areas are where SST (SL) are below -0.5~ (-10 cm).

4. Improvements of analyses by subsurface temperature data assimilation The t r u t h of the m o d e l - b a s e d analyses discussed in the previous section depends strongly on the quality of the wind forcing and how good the ocean model is. The quality of the analyses can be assessed by comparisons of the analyses and purely wind forced model simulations with observations not used in the analyses. The impact of the quality of the winds can be examined by using several different wind products. Available for such comparisons are two 11-year retrospective Pacific Ocean analyses for the period 1982 to 1993 produced using the ocean d a t a assimilation system of Ji et al. (1995), i. e. the RA3 and a similar analysis denoted as RA2. Both d a t a sets assimilated the same observed subsurface t e m p e r a t u r e d a t a from XBTs and the TAO buoys but the anomalous stress forcing for RA2 was obtained from an ensemble average of l l - y r (1982-1993) AGCM simulations forced with observed SSTs. The AGCM is a climate version of the NCEP's operational m e d i u m range forecast model (MRF, K a n a m i t s u 1989) with modified physical p a r a m e t e r i z a tions in convection, cloudiness and vertical diffusion for improved climate simulation results. Detail of these modifications are described in Ji et al. (1994b). The fields from both RA2 and RA3 compare well against independent mooring and tide gauge sea level observations (Ji and S m i t h 1995).

281 In addition, two simulations using no assimilation were produced, i.e. H F S U and HCMP. The designation of nomenclature for these d a t a sets is for consistency with those used in Ji et al. (1995) and Ji and S m i t h (1995). H F S U used the same stress forcing as in RA3, HCMP used the same anomalous stress forcing as in RA2. The left panel in Fig. 2 shows the evolution of the H20 anomalies along the e q u a t o r produced from HCMP. Compare to the H F S U (middle) and the RA3 (right) results in the same figure, it is obvious t h a t the stress anomalies used in H C M P are of lower quality t h a n those used in HFSU. This can be more directly verified by comparisons to i n d e p e n d e n t in situ data. Tide gauge d a t a is convenient for this because they are not used in the assimilation and long continuous record are available (Wyrtki 1979). 30 15 0 -15

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Shown in Fig. 5 are comparisons of the model simulated and observed sea level anomalies (only for HCMP and RA3). Recall t h a t the RA3 analysis used the F S U stress anomalies, and t h a t the RA2 analysis results from the same ocean d a t a used in the assimilation but with the HCMP stress anomalies. RA2 and RA3 reproduce the observed anomalous sea level variations associated with the ENSO episodes almost equally well. HCMP, which was produced w i t h o u t

282

d a t a assimilation, compares poorly to the observations, p a r t i c u l a r l y in the e a s t e r n e q u a t o r i a l Pacific ( S a n t a Cruz, 90~ The observed response to the E N S O at this island s t a t i o n is completely missing in the H C M P simulation. However, u s i n g the s a m e wi n d s and the s a m e model but with the addition of the a s s imila tion of observed subsurface t e m p e r a t u r e data, RA2 is able to reproduce the observed a n o m a lous sea level response at all island stations t h r o u g h o u t the Pacific basin including the S a n t a Cruz with an average r m s error of about 5.5 cm over these four tide ga uge stations down from about 8.2 cm before. This indicates t h a t d a t a a s s i m i l a t i o n can c o m p e n s a t e for poor stress product. F u r t h e r m o r e , RA3, which used a b e t t e r s t r e s s a n o m a l y product t h a n RA2, shows f u r t h e r i m p r o v e m e n t in the accuracy of the ocean analyses. The error statistics (Fig. 5) quantify this result; at all four tide gauge stations, RA3 h a s lower analysis error t h a n RA2. Note t h a t the r m s errors from H F S U , which uses no d a t a assimilation, are comparable or slightly b e t t e r to those from RA2. F u r t h e r m o r e , the addition of d a t a using the s a m e forcing (i.e. RA3) continues to improve the analyses. The basin wide average r m s errors over about 35 tide gauge stations for the 1983-1993 period are 7.2, 6.2, 6.0 a nd 5.5 cm for the HCMP, RA2, H F S U a n d RA3, respectively. These comparisons indicates t h a t d a t a a s s i m i l a t i o n can do a lot to improve a simulation using a poor wind stress product, a nd t h a t for b e t t e r w i n d products the d a t a assimilation still m a k e s i n c r e m e n t a l i m p r o v e m e n t s . The accuracy of ocean analyses at a limited n u m b e r of in situ observation locations such as at island tide gauge stations is one metric for the quality of analyses, Fig. 6 The 11-yr mean H2o field for the HFSU simulation (a) and the RA3 analysis (b). The difference field (HFSU-RA3) is shown in ( c ) . The units are in m. For the difference field, areas where the H20 difference is greater than 20 m are in dark shading, areas where the H20 difference is lower than -30 m are in light shading; for the mean fields, areas where H2o is between 100 m and 140 m are in light shading, areas where the H20 is greater than 240 m are in dark shading.

283

i.e. the m e a s u r e of fit to observations. As will be shown in the next section, skill of forecasts which use the analyses as initial conditions is an a l t e r n a t i v e metric for analyses which are produced for the purpose of initialization of forecast model. A major contribution of d a t a assimilation is to correct for errors in the m e a n field. The m e a n and the difference of the depth of 20~ isotherm field b e t w e e n RA3 and H F S U is shown in Fig. 6. Large differences are p r e s e n t in the subtropical gyre in the s o u t h e r n tropical Pacific and in the region of the thermocline ridge n e a r 10~ These large differences, of the order of 50 m south of the equator, are indicative of serious errors either in the forcing fields or in the model physics or possibly a combination of both. In order to get a better sense as to the origin of this error, several e x p e r i m e n t s were conducted using different stress fields. In addition to the F S U stress analyses used in H F S U simulation, wind stress forcing fields were available from the operational atmospheric analyses produced at N C E P and from an analyses derived from remotely sensed winds from the E u r o p e a n Remote Sensing ( E R S - 1 ) satellite. Since the E R S - 1 based product was only available since 1992, differences between simulations using d a t a assimilation and those w i t h o u t using d a t a assimilation are shown only for the past two and a h a l f years (Fig. 7). These m e a n differFig. 7 Mean differences in the depth of 20~ isotherm between model simulations and data assimilation using the wind stress forcing of HFSU, NCEP and ERS-1 wind stress products. The comparison is for the two and a half year period from January 1992 to June 1994. The units are in m. Areas where the H20 difference is greater (less) than 20 (-30) m are in light (dark) shading.

ence fields are very similar to those shown in the lower panel of Fig. 6 for each of these stress fields. Without d a t a assimilation the subtropical gyres are too w e a k in both hemispheres, especially the southern one, and the ridge at 10~ is also too

284 weak, i.e. the thermocline at this location is not shallow enough. The suggestion from these experiments is t h a t the cause of the problem is likely with the model since each stress field exhibits the same general error. The w e a k e r thermocline topography in H F S U results also in differences in the m e a n c u r r e n t structures. Obvious feature are a w e a k e r South E q u a t o r i a l C u r r e n t (SEC) and N o r t h Equatorial Counter C u r r e n t (NECC) (Fig. 8). However, all of the impacts of assimilation are not necessarily positive. The strong w e s t w a r d flow j u s t south of the equator in RA3 in the far w e s t e r n Pacific does not a p p e a r to be in the observations. It seems to appear because the d a t a assimilation is p u t t i n g in a large correction field j u s t south of the equator. The resulting velocity field is not in balance with the local wind forcing.

Fig. 8 l l - y r mean difference in the currents between HFSU and RA3 averaged over the top 100 m. The contours represent the mean zonal current difference field; the arrows represent the mean difference of total currents. Light (dark) shading is for areas where the zonal current difference is greater (less) than 5 (-10) cm s -1.

5. Impact of subsurface temperature data assimilation on El Niflo prediction The skill of E1Nifio forecasts out to a year or so depends on the quality of the ocean initialization. Two techniques are currently used to initialize coupled forecast models. The first uses the past history of the tropical winds, typically stress fields based on the FSU p s e u d o - s t r e s s product are used, to spin up the ocean component before the forecast is started. The second uses the past history of winds and d a t a assimilation to improve the specification of the ocean fields. The time h i s t o r y of the initial conditions for the N C E P ocean model obtained in these two ways, i.e. H F S U and RA3, was shown in Fig. 2. Since the FSU stress analysis is based on surface marine d a t a which poorly sample much of the tropical Pacific, one can anticipate t h a t the l a t t e r technique provides for a better initialization. (Fig. 2 shows t h a t t h e r e certainly was a big difference between the r e s u l t a n t fields.) To d e m o n s t r a t e t h a t this leads to improved forecast skill, sets of forecast experiments was conducted using these two sets of initial conditions.

285 The coupled forecast model used in these experiments, denoted as CMP6, is described in Ji et al. (1994b). It consists of the GFDL OGCM configured for the Pacific a n d a global a t m o s p h e r i c GCM which is the modified climate M R F model previously described, with a spectral resolution ofT40 and 18 vertical levels. The coupled model uses a n o m a l y coupling for stress and short wave flux and full coupling for sensible, l a t e n t and longwave h e a t flux components. The H&R stress climatology is used for m e a n stress forcing. The short wave flux climatology is obtained based on the bulk formula of Reed (1977) and the cloud climatology e s t i m a t e d from the i n t e r n a t i o n a l satellite cloud climatology project (ISCCP). Stress and short wave flux anomalies from the AGCM are combined with the prescribed stress and short wave flux climatologies to force the ocean model. However, the total SST fields from the ocean model are used to force the atmospheric model. The AGCM climatologies of stress a n d short wave flux were obtained from an ensemble of decadal AGCM simulations forced with observed m o n t h l y SST. These simulations also produced atmospheric initial conditions for coupled model. No observed d a t a were a s s i m i l a t e d into the atmospheric initial conditions. The ocean initial conditions used for these forecast e x p e r i m e n t s were produced from the H F S U simulation and the RA2 analyses previously described. (These exp e r i m e n t s were done before RA3 was available). Results of forecasts initiated I

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Fig. 9 The ACC and RMS errors for Nifio-3 SST anomalies between observations and forecasts. Upper: For the RA2 (solid) and HFSU (dashed) forecasts using the CMP6 model. The forecasts were initiated monthly from May through November for 1984 to 1992. Lower: For the RA3 forecasts using the CMP10 model (solid) and the HFSU forecasts using the CMP6 model (dashed). The forecasts were initiated in the northern winter season (December-February) for 1983/84 to 1992/93. (see Section 5).

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286 monthly from May through November using ocean initial conditions from RA2 and H F S U are shown in solid and s h o r t - d a s h e d curves in the upper panels of Fig. 9. The temporal anomaly correlation coefficients (ACC) for area averaged Nifio-3 (150~ 5~176 SST anomalies between forecasts and observations (Reynolds and Marsico 1993) as a function of forecast lead time are shown in the u p p e r left panel, the root m e a n square (RMS) errors for the forecasts are shown in the upp e r - r i g h t panel. Although simple in concept, ACC and RMS errors are a common way to estimate forecast skill (Latif et al. 1994). Recall t h a t in comparisons to tide gauges RA2 and H F S U produced sea level analyses of comparable accuracy, however, comparisons of forecast skills as shown in the upper panels of Fig. 9, the RA2 forecasts clearly outperform the H F S U forecasts. Since the main difference between these two sets of forecasts lies in the ocean initial conditions, these results suggest t h a t assimilation of subsurface temperature data significantly improves forecasts initiated from late spring to late fall with lead time up to one year. As suggested by Fig. 2, this improvement in forecast skill results from an improved definition of the ocean fields in the central and western equatorial Pacific. However this is not so clear for forecasts initiated in the winter (December-February). Using the same coupled model, forecasts initiated in the winter using RA2 initial conditions showed lower skill t h a n those using H F S U initial conditions (not shown). The exact reason for this is not clear. Shown in the lower panels of the Fig. 9 are ACC and RMS errors for forecasts initiated in the winter using an improved version of N C E P coupled model (Ji et al. 1996), denoted as CMP10, and oceanic initial conditions produced with and without data assimilation. Results from these forecasts suggest t h a t for forecasts initiated in the winter, data assimilation has only a very small positive impact on forecast skill for lead time of up to two seasons. For the third season, the positive impact is more significant. 1.2 0.8 0.6

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Fig. 10 ACC and RMS errors for Nifio-3 SST anomalies for RA2 forecasts using the CMP6 model (solid) and for RA3 forecasts using the CMP10 model (dot-dashed). Forecasts were initiated~monthly during the common period from October 1983 through May 1993. A limited n u m b e r of forecasts have also been carried out using RA3 initial conditions and the CMP6 model. Since RA3 was a slightly more accurate analysis t h a n RA2, one would expect slightly better results from these forecasts. Comparisons of

287 RA2 a n d RA3 forecasts (not shown) for common periods showed t h a t t h e r e is essentially no difference in forecast skills b e t w e e n them. However, forecast e x p e r i m e n t s for s t a r t s in all seasons using RA3 initial conditions and the CMP10 model showed significant i m p r o v e m e n t in forecast ofNi~o-3 SST anomalies over those u s i n g the RA2 initial conditions and CMP6 model (Fig. 10). A limited n u m b e r of forecasts for common periods using the CMP10 model and RA2 and RA3 initial conditions also h a d comparable skill (not shown), hence the skill difference shown by the solid and dot--dashed curves in Fig. 10 is not likely due to use of more accurate oceanic initial conditions (RA3) but more likely due to i m p r o v e m e n t in the forecast model. The m o s t significant difference between the CMP10 and CMP6 versions is in the coupling physics. The CMP10 model incorporates a model statistic o u t p u t (MOS) correction procedure to correct stress anomalies produced from the AGCM before t h e y are used to force the ocean model. Recall t h a t w h e n the AGCM was r u n w i t h observed SSTs, the stress anomalies obtained w i t h o u t this MOS correction, w h e n used to force the ocean model, produced a very poor simulation (cf. Fig. 2, left panel). These results suggest t h a t forecast model deficiencies can limit the potential imp r o v e m e n t s in forecast skill resulting from better analyses. Once a certain skill threshold is reached, i n c r e m e n t a l i m p r o v e m e n t s in forecast models a n d assimilation techniques, done jointly, are needed in order to f u r t h e r improve forecast skill. The previous examples d e m o n s t r a t e d t h a t assimilation of subsurface t e m p e r a t u r e d a t a can improve short t e r m coupled forecast skill. However, d a t a assimilation Fig. 11 Mean forecast errors for SST, zonal stress (~x) and depth of 20~ isotherm (H2o) for forecasts with lead time of 7 month from the RA2 forecast experiment. Contour intervals for SST, H2o and ~x are 0.5~ 10 m, and 0.1 N m -2, respectively. The arrows in the middle panel are the total stress errors.

288 plays a p a r t at p r e s e n t in introducing systematic errors into the forecasts (Mo, et al. 1994), and hence limits the skill at longer lead times. The m e a n difference in the thermocline depth between analyzed (RA3) and model s i m u l a t e d (HFSU) ocean fields was discussed in Section 4 (cf. Fig. 6c). This difference r e p r e s e n t s an imbalance in the m e a n t e m p e r a t u r e state between oceanic initial conditions produced w i t h d a t a assimilation and conditions during a coupled forecast. The former would have a m e a n thermocline structure similar to Fig. 6b; the latter, if it is the N C E P coupled model which uses the same OGCM and the H&R stress climatology as the m e a n stress forcing, would have a model climatology similar to Fig. 6a. Therefore, a forecast initiated from an analyzed ocean state will drift from the analyzed thermal state produced with data assimilation to the coupled model's equilibrium state d u r i n g the forecast. An indication of the initial circulation changes associated w i t h this was shown in Fig. 8. The a d j u s t m e n t results in a w e a k e n i n g of the SEC and an erroneous advection of w a r m w a t e r from the w e s t e r n Pacific w a r m pool e a s t w a r d into the equatorial Pacific. In a coupled model, this tendency is amplified by coupled interactions and mimics conditions leading to w a r m ENSO events. Shown in Fig. 11 are the m e a n forecast errors for SST, zonal stress and the 20~ i s o t h e r m depth field for forecasts with a lead time of 7 months. These are averages over 122 forecasts s t a r t i n g on the first of each m o n t h from October 1983 to December 1993 regardless of forecast s t a r t i n g m o n t h and forecast target month. The m e a n error in the H20 field is very similar to the m e a n H2o difference field shown in Fig. 6c, indicating t h a t during these forecasts, the H20 field is drifting towards the coupled model's ocean climatology. The m e a n SST error signal is a w a r m i n g in the c e n t r a l - e a s t e r n equatorial Pacific with a m a x i m u m centered n e a r 130~ 5~ The m e a n zonal stress error (Zx) is westerly in the equatorial Pacific and is collocated with the m e a n w a r m SST error. This suggests an E N S O - l i k e response of the winds to the w a r m i n g of SST which in t u r n accelerates the w a r m i n g of SST in the equatorial central to e a s t e r n Pacific.

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The combined E O F analysis for m e a n errors of SST, H20 and ~x show t h a t they grow simultaneously and rapidly (Fig. 12), and reach s a t u r a t i o n after about three

289 seasons of coupled integration. The spatial loading p a t t e r n s of these variables are very similar to the m e a n errors shown in Fig. 11. It is evident t h a t the initial growth of the systematic forecast errors is caused by the relaxation of the m e a n thermocline structure in the ocean, and enhanced by the E N S O - l i k e positive feedback between the SST error and the stress error. Results from the H F S U forecast experiments, which do not have this difference in climatologies between the initial conditions and the forecast model, support this conclusion. The m e a n SST errors from these forecasts were projected onto the leading combined m e a n error EOFs of the RA2 forecast experiment. The m e a n SST error was significantly smaller and grew at a much slower rate (Fig. 12, heavy curve). This comparison indicates t h a t when forecasts are initiated from ocean initial conditions which have a m e a n thermocline state similar to the coupled model, the growth rate of the m e a n SST error is much less t h a n those from forecasts which the oceanic initial conditions are not balanced with the coupled model. A reduction of the s y s t e m a t ic error shown in Fig. 6c is required in order to take full advantage of the ocean d a t a assimilation for initialization.

6. Discussion and Summary Ocean d a t a assimilation is required, at least for the system at NCEP, because significant errors r e m a i n in the forcing fields and in the ocean models. Its usefulness was shown in the earlier sections by the improvements in the analyses for the thermal and sea level fields and in the increase in skill in the forecasts. Despite these early successes, further improvements can and need to be made. These, however, should be considered in the context of improvements in the overall system which includes the forcing fields, the ocean models, and the coupled forecast model. The metric for w h a t constitutes analysis improvement will u l t i m a t e l y depend on the purpose for which the analyses are to be used (and is not always obvious). Comparisons of RA3, RA2, and HFSU to island tide gauge stations indicate t h a t RA3 has the lowest errors, followed by H F S U and RA2. Yet when fields from these are used in forecasts for s u m m e r and fall starts, RA3 and RA2 produce forecasts of comparable skill, which are significantly better t h a n those from H F S U produced initial conditions. On the other hand this was not the case for winter s t a r t s with the CMP6 model, where H F S U produced better forecasts t h a n either RA2 or RA3. Even with the CMP10 model, forecasts produced with RA3 initial conditions are only marginally better t h a n those produced with H F S U initial conditions. Although the reasons for this are not clear, these results point to the fact t h a t an overall evaluation using another consideration, i.e. a complex forecast system, in addition to the analysis itself can lead to unexpected results. C u r r e n t l y at N C E P much of the impact of data assimilation is to improve the m e a n oceanic field. From the analysis point of view this is a positive contribution. However, this is another example when the best possible analyses, where the figure of m e r i t is j u s t fit to the data, do not necessarily lead to the best forecasts. The difference between the m e a n states with assimilation and in the forecast mode (where as-

290 similation is not used), leads to an a d j u s t m e n t during the forecasts which contributes to the growth of strong systematic forecast errors. It is likely t h a t forecast errors would be reduced if this m e a n error could be eliminated. This will require improvem e n t s in the ocean model r a t h e r t h a n in the assimilation technique. Since the assimilation system at N C E P presently only a s s i m i l a t e s t h e r m a l data, the velocity s t r u c t u r e r e m a i n s unconstrained. Unexpected and erroneous effects can arise since the t h e r m a l fields have been modified but the stress field rem a i n s the same. This a p p e a r e d to be the case in the n e a r equatorial region in the far w e s t e r n Pacific where RA3 has a strong w e s t w a r d flow (on the order of 40 cm s -i) in the m e a n . Comparisons of the RA3 surface currents to those e s t i m a t e d from drifting buoys suggests t h a t the RA3 currents in the n e a r equatorial w e s t e r n Pacific are erroneous. One possible reason for this t h a t is being explored is t h a t the erroneous signals r e s u l t from Rossby waves g e n e r a t e d by corrections to the t h e r m a l field f u r t h e r to the e a s t r e s u l t i n g from stress or model physics errors. Even a more sophisticated, m u l t i v a r i e n t assimilation scheme, which assimilates c u r r e n t s and t e m p e r a t u r e , despite locally producing a more consistent analysis, probably would obscure b u t not completely eliminate this basic problem. W h a t is needed is a systematic improvem e n t of the models and the forcing fields. In this process assimilation is a tool, not j u s t an end in itself. Despite this broader issue, d a t a assimilation has proven to be a useful tool at NCEP. It is capable of improving the simulations t h a t result from poor forcing fields as d e m o n s t r a t e d by the experiments shown in Section 4 (cf. Fig. 5). Comparisons of the operational N C E P winds and the F S U p s e u d o - s t r e s s analyses to directly observed winds from the TAO a r r a y suggest t h a t f u r t h e r i m p r o v e m e n t s can be m a d e to stress forcing fields. Use of these would possibly f u r t h e r reduce the r m s errors, which for RA3 were 3 to 6 cm, in comparison to the tide gauges. However, for the p a s t two and a h a l f years, r m s errors at some tide gauge stations, resulting from use of the N C E P stresses, have been of the order of 2 cm. One suspects t h a t a p l a t e a u is being reached. With the large a m o u n t of TAO and XBT d a t a t h a t is available, the d a t a is constraining the analysis and i m p r o v e m e n t s to the stress fields will be less obvious. Hence "less assimilation" m a y be necessary in order to see the benefits of improved forcing fields and models more clearly. C u r r e n t ocean d a t a assimilation practices have increased the skill of the forecasts. One would anticipate t h a t more accurate initial conditions resulting from still improved winds with d a t a assimilation will produce still increasingly b e t t e r forecasts. In fact in going from the RA2 initial conditions to those of RA3 using the s a m e coupled model (CMP6), no i m p r o v e m e n t s in forecast skill were achieved. However in going from the CMP6 to CMP10 model, significant i m p r o v e m e n t in forecasting Nifio-3 SST anomalies was achieved. This suggests t h a t more significant increases in forecast skill will come from improving the models and in developing coupled assimilation s y s t e m specifically designed to capture the predictable components of the low frequency variations of the coupled climate system.

291 References

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Regional Applications

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Modern Approaches to Data Assimilation in Ocean Modeling

edited by P. Malanotte-Rizzoli 9 1996 Elsevier Science B.V. All rights reserved.

297

A M e t h o d o l o g y for the C o n s t r u c t i o n of a H i e r a r c h y of K a l m a n Filters For Nonlinear Primitive Equation Models Paola Malanotte-Rizzoli a, Ichiro Fukumorib and Roberta E. Young c aDepartment of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology Cambridge, MA 02139 bJet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109 CDepartment of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract

In this paper we present a methodology for the construction of a hierarchy of Kalman filters for nonlinear primitive equation (PE) models. We capitalize on the approximations introduced in Fukumori and Malanotte-Rizzoli (1995) that make the filter feasible and efficient for a PE model of the Gulf Stream jet in an idealized zonal channel configuration. These approximations involve a) a method that reduces the model's effective state dimensions; b) use of the error's asymptotic steady-state limit; c) a time-invariant linearization of the dynamic model used only for the time integration of the state error covariance matrix. Here we eliminate the time-invariance with two different procedures. In the first one we allow for full time evolution of the error covariance but preserve the time-invariant linearization of model dynamics around a unique mean state. The second procedure allows for the covariance time-evolution by time evolving the linearization of the model dynamics around successive 10 day intervals. The error covariance matrix evaluated asymptotically for each 10 day interval is then updated accordingly. We use again the idealized zonal channel configuration for the evolution of an unstable, highly nonlinear jet. The assimilated dataset consists of velocity pseudo-observations taken at two identical arrays of 13 moorings each designed to encompass the region of growing, finite amplitude meanders mimicking the Gulf Stream behavior. We summarize our results as follows. The first conclusion concerns the importance of specifying full covariances instead of the usual assumption of white noise. Estimates based upon the linearized dynamics of the model provide quite successful assimilation results even in the steady-state asymptotic limit. This result strongly suggests that accurate specification of process noise maybe the most critical issue for Kalman filtering. The second conclusion is that nonlinearities in the model may be more important than the covariance time evolution per se when based on a time-invariant linearization. Allowing for time variation of the covariance through the second procedure outlined above

298 produces better assimilation estimates of the model variables. Thus a procedure similar to, but simpler than, extended Kalman filtering would be affordable and efficient while allowing to take into account important nonlinearities through successive linearizations around different mean states. 1. INTRODUCTION The primary objective of physical oceanography has always been to estimate accurately the state of the ocean on different time and space scales. This goal, formidable per se, was not even deemed to be attainable until recent years because of the discontinuity between observations and their analysis on one side and theoretical modeling on the other. Traditionally, in fact the two fields have evolved and grown rather independently from each other. Both have important and somewhat complementary limitations. Oceanographic observations are by and large far from synoptic. Even the most recently available altimetric dataset of TOPEX/POSEIDON, while providing a global map of the ocean surface topography every 10 days, leaves completely unmonitored the mostly unknown oceanic deep and abyssal layers. Apart from the accuracy of oceanographic observations, their sparcity in space and time makes the ocean state estimated from these observations yery poorly resolved either in space or in time or both. And, even if synoptic observations were available, they would not be sufficient. As the ocean is mostly a forced system, synoptic observations of heat and momentum fluxes would still be required. These fundamental deficiencies are not overcome even by the most sophisticated methods of inductive analysis of direct observations based upon inverse modeling (Wunsch, 1978; Wunsch and Grant, 1982). Theoretical modeling on the other hand is based on first principles and does not suffer from the above limitations of the observations. It is capable of resolving almost the entire spectrum of time/space scales of oceanic motions on the basin-wide and global scale. However, even the most sophisticated numerical models based on fully nonlinear, timedependent primitive equation dynamics would require computer power presently inconceivable to resolve all motions down to the fine turbulent scales where dissipation plays the major role. Thus, the finest scales not resolved by the model must be parameterized. This sub-grid scale parameterization leads to the most serious deficiency of all the prognostic simulations of the ocean state. Moreover, an intrinsic unpredictability problem lies at the core of the nonlinear primitive equations of motion. A numerical model, even when initialized with a realistic oceanic state, will lose the memory of this initial state after a finite time - the predictability time - and evolve in a fully unrealistic way, diverging exponentially from the observational state. These fundamental deficiencies of observations and models can be overcome through the approach of data assimilation, a recent field of investigation that has come to the forefront of research in the last decade. Data assimilation combines models and data through methodologies that allow the data to constrain the model evolution to follow closely the oceanic observational state while the model acts as a dynamical interpolator/extrapolator to the space and time scales not resolved by the data. Methodologies have been transplanted to oceanography from different fields, such as meteorology and engineering control theory. One of these optimal methods having the greatest potential for oceanographic applications is the Kalman filter (Tarantola, 1987; Miller, R.N., 1987; Ghil and Malanotte-Rizzoli, 1991). The Kalman filter is based on formal estimation theory that minimizes the data-model misfit under the constraint of the model dynamics. It has advantages, as well as disadvantages, when compared with similar optimal techniques such as the adjoint method. The adjoint method is rather simple to implement in fully nonlinear numerical models, but

299 requires a complex procedure for the evaluation of model errors, i.e., the evaluation of the Hessian matrix of the cost function, which is often impossible if not computationally unaffordable for the most realistic ocean models. In the Kalman filter on the other hand, the formulation of the formal error estimate and of its time evolution is rather straightforward and rigorous for linear models. However, it is not rigorous for nonlinear models. Its formidable computer requirements, analogous to those necessary for the evaluation of the Hessian in the adjoint, have so far made the oceanographic applications of the filter limited to simple models with approximate physics and dynamics and/or poor space resolution. The computational load of Kalman filtering resides in the evaluation of the time evolution of the state error covariance matrix. In fact, the amount of storage and number of computations required in such a calculation is proportional to the square and cube of the state's dimension, respectively. Methods that approximate this error evaluation in order to make Kalman filtering more feasible have been recently introduced by Fukumori et al. (1993), and Fukumori and Malanotte-Rizzoli (1995, hereafter FR95). Fukumori et al. (1993) describe an asymptotic approximation of the error evolution for the filter that significantly reduces the storage and computational requirements. FR95 further propose utilizing transformations that approximate the model state with one having fewer degrees of freedom, thus effectively reducing the size of the estimation problem and hence the number of computations involved. FR95 is the first study in which the usefulness of such approximations is explored in a highly nonlinear, primitive equation model in a series of twin experiments of an idealized Gulf Stream jet. In this paper we extend the study of FR95 by presenting a methodology for the construction of a hierarchy of Kalman filters that can be used for complex, nonlinear models. We extend and implement the asymptotic filter of FR95 by eliminating the timeinvariant assumption in two different manners. We then gauge the impact of these two implementations with respect to the static asymptotic filter by comparing the relative reductions in the state's true error in the context of twin experiments analogous to those of FR95. The paper is organized as follows. In section 2 we briefly review the foundations of the filter, its difficulties, and the three approximations introduced in FR95. In section 3 we present the results of FR95 relevant for the present work. In section 4 we introduce two different implementations to eliminate the time invariance of the filter and we discuss and compare the results of the twin experiments carried out with each of them. Finally in section 5 we summarize our conclusions and suggest future applications of the hierarchy of filters developed here for more realistic model configurations and related numerical calculations. 2. A P P R O X I M A T E KALMAN FILTER 2.1 The Kaiman Filter and its Difficulties The Kalman filter is a recursive least-squares inversion of observations for model variables, using a dynamic model as a constraint. Operationally, the filter performs a weighted average of model estimates and data, where the weights are based on the relative accuracies of the two. The result is an improved estimate of model variables, where the improvement is achieved in a statistical sense; the result has the least expected error given the measurements and the model, along with their error statistics. The filter also provides an estimate of the model error covariance matrix and of its time-evolution. For reference, the algorithm for the filter is briefly reviewed below. Let there be, at time t a model estimate x(t,-) where the minus sign denotes a model prediction, and a set of observations y(t), with corresponding independent error covariance matrix estimates P(t,-)

300 and R(t), respectively. Estimation theory (Gelb, 1974) states that the optimal combination of the model x and observation y is given by (regardless of the model being linear or nonlinear), x(t) = x(t,-) + K(t)[y(t)-H(t)x(t,-)]

(1)

where the weight K(t) (Kalman gain) is, K(t) = P(t,-)HT[HP(t,-)HT+R]-I

(2)

Bold lower and upper case characters denote vectors and matrices, respectively. H(t) is a matrix such that H(t)x(t) is the model's theoretical estimate of what is observed. The error of the improved estimate x(t) is given by P(t) as follows: P(t) = P(t,-)-K(t)H(t)P(t,-)

(3)

The Kalman gain K(t) can also be written in terms of this P(t) as K(t) - P(t)HT(t)R - l(t)

(4)

which is a useful representation when applying the approximations later. The improved estimate x(t) (called the analysis) is then time-stepped by the model until another set of measurements are available (time t+l) and the assimilation is repeated. For simplicity, let the dynamic model be linear so that the evolution equation can be written as" x(t+l,-) = A(t)x(t)+u(t)

(5)

where u(t) denotes forcing and boundary conditions and A is termed the state transition matrix. Then the error of x(t+ 1,-) can be estimated by P(t+ 1,-) = A(t)P(t)A(t)T+Q(t)

(6)

where Q(t) is the error covariance of the model when time stepped from time t to t+l, which includes errors of u(t). Again, it was assumed that this process noise is uncorrelated in time and with the observation errors R. Equations (3) and (6) describe the error's time evolution and together form the so-called Riccati equation. Correlated process noise and/or observation errors can be treated by a slight modification of the algorithm (Gelb, 1974). For a recent review of the mathematical foundations of Kalman filters and of its applications to meteorology and oceanography, see Ghil and Malanotte-Rizzoli (1991). Although theoretically straightforward, difficulties arise in applying the Kalman filter to oceanic data assimilation. The difficulties are of two-fold nature. First, the filter is rigorous for linear models but is approximate for nonlinear ones. This is because time evolution of the error covariance matrix for nonlinear systems involves higher order statistical moments and cannot be written in the closed form of the second moment as in Eq. (6). The Extended Kalman Filter (EKF) approximates this error evaluation by a piecewise linearization thus circumventing this difficulty. Second, and even more important from the practical point of view, is the enormous computational requirements in integrating in time the state's error covariance matrix P, which is used to generate the Kalman gain matrix K that performs the least-square averaging (Eq. (1)). The error covariance matrix evolves in time according to the model dynamics (Eq. (6)), just as the model state itself does. Integrating each column of the

301 matrix is computationally equivalent to integrating the full model equations. Thus, integration of P requires the size-of-the-model times larger computational resources than the numerical prognostic calculations. For a model like SPEM in the FR95 configuration, with over 170,000 variables, this integration is computationally unfeasible, both for storage and for required CPU time (for a full discussion of these difficulties, see FR95). FR95 bypassed these difficulties by performing a series of approximations that make the filter feasible, affordable and efficient. The approximations involve a) a method that reduces the model's effective state dimension; b) use of the error's asymptotic steady-state limit (Fukumori et al., 1993) and c) a time-invariant linearization of the dynamic model (but only for the time integration of the state error covariance matrix).

2.2 Reduced State Approximation The grid size of a model and the resulting dimensionality of the model state are often dictated by numerical accuracy and stability. On the other hand, most energetic scales are typically much larger than the smallest grid spacing, and available observations are often sparse. Then, extraction and assimilation of the large-scale information content of the measurements may be the most effective approximation in terms of the amount of improvement made in the estimate for the computations involved. This can be achieved by approximating the model error covariance matrix with one of fewer degrees of freedom that resolve the covariances of the large-scale (Fukumori and Rizzoli, 1995). For example, suppose there exists some approximtion, x'(t), of the original model state (x(t)) with a smaller dimension, x(t)-x = Bx'(t)

(7)

The approximation is define_d, without loss of generality, around some prescribed timeinvariant reference state, x. Matrix B is the transformation matrix defining the approximation. Given such transformation, we can approximate the error covariance of x, i.e., P, by the error of x ~ i.e., P', by P(t) = BP'(t)B T (8) m

which can be substituted into the Kalman gain (Eq. (2)) for assimilation. Note that, since x is a prescribed reference, it has no error and is statistically inconsequential. Given the smaller dimension, derivation of the statistical properties of the 'coarse' state x~ will be computationally less demanding than that for the original model, x(t). The dynamical equations for x'(t) used to evaluate P~ may be constructed by directly combining the transformation B with the original model for x(t). Eq. (8) is an approximation and the exact relationship involves the null space of the transformation and can be written as, x(t)-x = Bx'(t)+c(t)

(9)

where c is a vector in the null space of the columns of B. Defining the pseudo inverse of B as B-~, Eq. (9) may be inverted as, D

B-l(x(t)-x) = x'(t)

(10)

Let the dynamic model for x(t) be written as x(t+ 1) = F[x(t)] Then substituting Eqs. (9, 10) into Eq. (11) yields

(11)

302 x'(t+l) = B-1F[x+Bx'(t)+c(t)l - B -1 x

(12)

Finally, Eq. (12) is approximated into a closed set of equations for x' by assuming that the null space c(t) is dynamically uncoupled from the reduced-state x'(t) and then treat it as a statistically independent noise q(t); x'(t+l) = B-1F[x+Bx'(t)]+q(t) - B -1 x (13) The observation equation, which theoretically relates the model state to the model equivalent of the observations and is used in evaluating P', may be approximated likewise in terms of x'. All statistical quantities of the reduced-order model may now be estimated based on Eq. (13), using the standard Kalman filter equations (Riccati equation). The statistical properties of the original model will be approximated according to Eq. (8) and in turn substituted into the filter for assimilation. Such state dimension reduction greatly reduces the computational requirements of Kalman filtering, because the storage and matrix operations involved in Kalman filtering are proportional to the square and cube of the model dimension, respectively. 2.3 Asymptotic Approximation

Although the state approximation outlined above results in a substantial computational savings in performing Kalman filtering, further computational reduction can be achieved by employing an asymptotic approximation of the model error covariance matrix P' (or P). When data are regularly assimilated, estimation errors often approach a steady-state limit. Using such limit throughout the assimilation eliminates the need for storage and for continuous integration of the error covariance matrix. Two issues arise concerning such an approximation. The first is the existence of such a limit, and the second is its derivation. Under certain conditions, the Riccati equation is proven to converge exponentially fast to a unique limit. This can be demonstrated for timeinvariant linear systems, in which all system matrices (the state transition matrix, observation matrix, and their error covariances) are time invariant, and both the unstable and neutral modes of the modelare observable and controllable (Goodwin and Sin, 1984). Observability is the ability to determine the model state from data in the absence of data errors and model errors, and controllability is the ability to drive the model to an arbitrary state by the model errors (Gelb, 1974). In many situations, strict convergence does not occur, because of time-varying models (including nonlinear ones) and/or aperiodic observations. However, experience shows that asymptotic errors derived based on approximating such systems as time-invariant can still be effective when used in Kalman filtering of time-varying systems. Although existence of an asymptotic limit may be known, integration of the Riccati equation to this limit is not trivial. A direct solution of the Riccati equation produces a nonlinear matrix equation whose general solution is not known. Heemink (1987) uses a so-called Chandrasekhar-type algorithm in solving for the asymptotic limit. Anderson and Moore (1979), Stengel (1986) and other textbooks describe this and other algorithms to derive the steady-state solution of the Riccati equation. One method, which was used by Fukumori et al. (1993) and FR95, is the "doubling algorithm", a recursive method that allows one to integrate P in increasing time steps of powers of two when the system (Eqs. 2,3,6) is time-invariant. Thus, the doubling algorithm approximates the time-evolving system with a time-invariant one. For reference, the doubling recursion may be written as (Anderson and Moore, 1979),

303 q~(t+ 1) = O(t)[I+~P(t)0(t)]-l
(14a)

~P(t+ 1) = ~P(t)+O(t)[I+~P(t)0(t)] - 1W(t)oT

(14b)

0(t+ 1) = 0(t)+oT(t)0(t)[l+~P(t)0(t)]-lo(t)

(14c)

where the recursion is started from O(1) = A T ~P(1) = HTR-1H

(14d)

0(1) = Q and error at time 2 t is given by p(2t, -) = 0(t) This recursion as written above assumes P(0) = 0, but such assumption is not critical for the asymptotic solution because it is unique. The doubling iteration is repeated until the error covariance, 0, converges.

2.4 Time-lnvariant Linearization Approximation For time-varying systems, an approximate asymptotic limit of P' can still be computed based on a time-invariant approximation of the model dynamics. Non-stationary observations, in which the observation pattern or what variables are measured vary in time, can be approximated as a stationary observation matrix for the purpose of deriving an approximate error. The simplest approximation for nonlinear models is a time-invariant linearization, and the corresponding state transition matrix A' can be computed numerically for u s e i n the doubling algorithm. For example, linearizing the reduced state model around xx'(t+ 1) = B- 1F(x +Bx')-B- 1 x = B-1F(x)+B -1 d---~-FBx'-B-1x dx = B-1F(x)+A'x' - B-1 x

(15a)

Then, the i-th column of A', ai, may be obtained by ai = A'ei = B- 1 F( x +Bei)-B- 1F ( x )

(15b)

where ei is the i-th column of the identity matrix spanning the reduced state. The two terms on the right hand side of the last equation can be evaluated numerically using the model. Other system matrices may be constructed easily likewise. Such numerical evaluation allows derivation of system matrices for evaluating the evolution of P' straightforward even for models with complex numerical code, and can be used to obtain relevant matrices for linear models as well.

2.5 Summary of the Reduced-Dimension, Static, Linearized Kalman Filter Using the three approximations outlined above, the model's state error covariance matrix P(t) may be approximated by

304 P(t) = BP'B T

(16)

where P' is the asymptotic limit of the error of the reduced state based on a time-invariant approximation of the dynamic system. Substitution of this approximation into the Kalman gain matrix yields the reduced-dimension, static, linearized Kalman filter. K(t) = P(t)HT(t)R-I(t) = BP'BTHT(t)R - l(t)

(17)

Although such approximations were effective in FR95, some of them might be relaxed with the associated computational increase still being manageable. In particular, the prognostic dynamical experiments of FR95 were fully nonlinear, whereas the error estimates were evaluated by approximating the model with a time-invariant linearization. Among the various limitations of the suboptimal filter developed in FR95, perhaps the most important ones are related to the complete time-invariance of the assimilating filter. This involves fil'st the use of a time-invariant transition matrix A, i.e., a time-invariant linearization of the dynamical equations to evaluate the time evolution of the error covariance matrix. 3. THE MODEL CONFIGURATION AND THE C O N T R O L E X P E R I M E N T The numerical model is a nonlinear primitive equation model of an east-west jet that simulates an idealized Gulf Stream. The numerical code is based on the Semi-Spectral Primitive Equation Model (Haidvogel, et al., 1991). The model domain shown in Fig. 1 is a zonal channel, 1875 km long and 1400 km wide, with a flat bottom at a constant depth of

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305 4000 m. The total model grid is 129 x 97 horizontally, with a horizontal resolution o f - 1 4 km. Five collocation points are used in the vertical, at 0-600-2000-3400-4000 m depths. The state reduction for the approximate filter (transformation B in eq. 9) is achieved by using the barotropic and first baroclinic model amplitudes as state elements in the vertical. Horizontally, these model amplitudes are defined on a coarse 20 x 14 grid and using bicubic spline interpolation as for B. The twin experiment approach consists of the following steps. First, a control simulation is carded out by running the model for 360 days from an initial condition given by a geostrophically and hydrostatically balanced zonal jet. The jet is unstable; meanders grow to finite amplitude, elongate and bend giving birth to warm and cold rings. Day 60 to 120 of the control run are the "true" ocean from which pseudo-observations are taken at two day intervals at the two arrays of Fig. 1. Day 180 to 240 of the control run are the "false" ocean whose initial condition (day 180 of the control run) is statistically uncorrelated from the time history of the true ocean. (The decorrelation time of the control run is - 1 2 days). The pseudo-observations that are taken from the true ocean are then assimilated into the false ocean, thus providing the "model" ocean, i.e., the simulation that starts from the initial condition of the false ocean (day 180 of the control), but is modified by the assimilation. If the assimilation is successful, the model ocean will gradually diverge from its initial state, the false ocean, to converge to the behavior of the true ocean. Fig. 2 schematizes the twin experiment approach. Fig. 3 shows an example of the evolution of the surface density anomaly in the true ocean (left panels) and the false ocean (fight panels). These two 60-day intervals of the control run will be used for comparison purposes in the assimilation experiments of this work.

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Figure 3. Time evolution of surface density anomaly for the "true" ocean and "false" ocean. A meander, A at day 82, grows to finite amplitude (B), breaks on day 106 (C) and disintegrates by day 118, leaving a warm (D) and cold (E) anomaly north and south of the jet. In contrast, the false ocean generates a southward meander (F) which eventually breaks, leaving a cold eddy (G) to the south. The false ocean also has a strong eddy (H) north of the jet. Contour interval is 10-5 g/cm3.

307 A series of assimilation experiments were carried out in FR95. Specifically, the assimilated measurements at the two arrays of Fig. 1 were a) velocity data only at all 5 collocation points in the vertical; b) density data only at all 5 collocation points in the vertical; c) total transport data only (streamfunction). In this paper we carry out assimilation experiments only with velocity measurements, thus only the velocity results of FR95 are summarized here. The sea surface density field of the model ocean for the velocity data assimilation of FR95 is shown in Fig. 4 and should be compared with Fig. 3. At day 202 the model ocean in Fig. 4 is still very similar to day 202 of the false ocean (Fig. 3 fight panel). As time evolves, however, the model ocean converges to the true one. The surface density field at day 238 in Fig. 4 is totally different from day 238 of the false ocean (Fig. 3 fight panel), and reproduces all the features observed in day 118 of the true ocean (Fig. 3 left panel). The velocity assimilation resolves extremely well the major meanders of the true ocean, labeled A,B,C,D,E in Fig. 3, while at the same time suppressing the southward meander of the false ocean, labeled F in Fig. 3, day 202. A quantitative measure of the assimilation success is given by the root-mean-square (rms) difference between the assimilated estimate and the true ocean at a particular depth and time" I N~ (fassim. - ftrue) 2 eassim. =

~/

(18)

where N is the total number of grid points and f is one of the state variables (velocities, density, streamfunction). In FR95 the assimilation rms eassim.was normalized by the rmserror of the false ocean with respect to the true one, efalse, at that particular time. This normalization removed the ambiguity that a decrease in eassim, might be due to a decrease of the false ocean error, i.e., to a false ocean becoming more similar to the true one, and not to the actual effect of the assimilation itself. In this work however we want to assess the success of the assimilations carded out with the new approximations discussed in the following sections with respect to the FR95 results. The appropriate norm for eassim, is therefore the error of the FR95 velocity assimilation experiment itself with respect to the true ocean, i.e., ~ (fFR95 - ftrue EFR95 =

N

(19) N The normalized error eassim./eFR95 in fact quantifies any improvement (or worsening) of the present assimilations over the results of FR95: a ratio less than 1 defining an improved assimilation and a ratio greater than 1 a degraded one. We need first to "calibrate" the present results against the false ocean which represents the worst case of departure from the true ocean. Thus we evaluate the rms-error of the false ocean. I ~ ( f f a l s e - ftrue )2 efalse =

N

and normalize it with eFR95 for consistency of comparisons.

(20)

308

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(d)

t = 238

Figure 4. Sea surface density anomaly of the "model" ocean with velocity data assimilation from FR95. Contour interval is 10-5 g/cm3.

309 Fig. 5 shows the ratio Cfalse/eFR95 for the barotropic streamfunction (upper panel) and for the u-velocity component (thin line-lower panel), and v-velocity component (thick linelower panel), at the mid-depth collocation point at 2000 m. At the initial times, the ratio is very near 1, as the initial state for the assimilation experiments is the false ocean itself. In the time evolution, however, the assimilation of FR95 successfully converges to the true ocean as shown in Figs. 3 and 4, while the false ocean diverges from it. By day 30 the ratio has overshot the value of 2, and remains around a 2-average value until day 60. Fig. 5 constitutes the reference for comparison with the assimilations of the next section 4.

4. Implementations of the FR95 filter We now present a methodology for the construction of a hierarchy of Kalman filters based upon implementing the FR95 filter by eliminating the time-invariance with two different procedures. In the first one we remove the steady-state approximation of the error covariance matrix P by letting it evolve in time but keeping a time-invariant transition matrix A, i.e., the linearization of the dynamic model for the error integration is the same for all times and carried out around the time average over 360 days of the control run. In the second procedure we let the error covariance P evolve in time by updating the transition matrix A every ten days, i.e., by re-linearizing the model dynamics every 10 days around the time-average of the prior 10 days, and then evaluating the error covariances P as the asymptotic limits of the successive Riccati equations.

4.1 Time-varying P but static A Here we use the first procedure and investigate the importance of employing temporally varying error covariance matrices. The temporal evolution of P is evaluated by integrating in time the Riccati equation of the reduced state using, however, the same time-invariant transition matrix A' linearized as in FR95 around the time-average of the control run. Three different initial error matrices P'(0) were investigated. Specifically, they are: (i) a diagonal matrix with variance equal to 10 times the variability of the prognostic simulation around its mean; (ii) the asymptotic P' when no velolcity data are assimilated; (iii) twice the initial P'(0) of (ii). Fig. 6 shows the ratio eassim./EFR95 in the assimilation of case (i), respectively for the barotropic streamfunction (upper panel) and the u-velocity component (thin line-lower panel) and v-velocity component (thick line-lower panel) at the mid-depth collocation point. Obviously the assimilation result is much better than the false ocean behavior quantified in Fig. 5. However, after an initial decrease, all the ratios shoot to a value greater than 1, to decrease to an average value of roughly 1 in the final period of the assimilation. The overall average ratio is -1.2 for the streamfunction and .--1.1 for the velocity components, indicating that case (i) filtered estimates are rather degraded with respect to the FR95 case of static asymptotic covariance. Figs. 7 and 8 show the same ratios, barotropic streamfunction (upper panel) and u-velocity (thin line-lower panel), and v-velocity (thick line-lower panel) respectively for case (ii) (Fig. 7) and case (iii) (Fig. 8). Case (ii) in Fig. 7 appears to be the best, with (u,v) Eassim./eFR95 ratios basically constant at the value of 1 and the average streamfunction ratio also 1. Even though the ratios for case (iii) shown in Fig. 8 have broader fluctuations around 1, the overall average value is still roughly 1, thus showing that the two experiments (ii) and (iii) are equally successful in providing convergence to the true ocean. However, the ratios' behavior also shows that no improvement is actually achieved over FR95.

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The fact that assimilation (i) is the least successful clearly demonstrates the importance of specifying full covariances instead of assuming uncorrelated errors for assimilation, as it is usually done. Using as initial condition for P' the asymptotic error covariance based on the approximate model dynamics A' obviously allows for a more realistic error evolution. All the experiments have greater accuracies than FR95 at the beginning of the assimilation, as shown by the ratios becoming less than 1, because FR95 asymptotic error underestimates model uncertainties during the initial stage. However, in experiments (ii) and (iii) the differences become rather small at later times, especially between case (ii) and FR95 (compare Figs. 7 and 8). For this particular model configuration, the similarities between cases (ii) and (iii) with FR95 clearly indicate that no substantial improvement in the estimates is achieved by actually integrating in time the Riccati equation for the error covariance matrix P.

4.2 Time-varying P by time-varying A With the second procedure we investigate the significance of the model nonlinearities on the evolution of the error covariance by employing a time-varying transition matrix A' in the Riccati equation, a procedure similar to the extended Kalman filter. However, to keep the computational costs manageable two simplifications are performed. First, the model dynamics are linearized every 10 days around the 10 day time-mean. Second, the error covariance is updated every 10 days by the doubling algorithm solution while leaving it constant between updates. The numerical evaluation of the state transition matrix is the largest computational process and is done according to Eq. (15b). The results of the velocity data assimilation according to this second procedure are presented in Fig. 9, showing the ratios Eassim./EFR95 again for the barotropic streamfunction (upper panel) and the u-velocity (thin line-lower panel), and v-velocity, (thick line-lower panel). Now a slight but definite improvement is achieved over FR95 results. The streamfunction ratio decreases and remains below 1 during the last 40 days of the assimilation, with an overall average value o f - 0 . 9 . The ratios for the (u,v) velocities are basically 1 until the last 20 days where a decreasing trend is achieved, especially for the v-velocity. Even though the actual improvement is slight, this second implementation based upon successive linearizations of the model dynamics appears to be the most promising for application to more realistic, and more nonlinear, model realizations, as will be discussed in the next section 5. 5. CONCLUSIONS In this paper we present a methodology for the construction of a hierarchy of Kalman filters for complex nonlinear models that, starting from the use of the static, asymptotic error covariance matrix and time-invariant linearization of model dynamics of FR95, partially removes the time-invariance by adopting two different procedures. The first one is to allow for full time evolution of the error covariance but preserve the time-invariant linearization of model dynamics around a unique mean state, section 4.1. The second procedure allows for the covariance time-evolution by time evolving the linearization of the model dynamics around successive 10 day intervals, thus leading to 6 successive Riccati equations over 60 days of assimilation. The error covariance matrix is then evaluated asymptotically from the Riccati equation for each 10 day interval and updated accordingly, section 4.2. The assimilated dataset consists of velocity pseudo-observations taken at two identical arrays of 13 moorings each located in a zonal channel and designed to encompass the region of an unstable jet having meanders which grow to finite amplitude, steepen and bend in a fashion reminiscent of the Gulf Stream. The configuration idealizes the Gulf Stream

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316 system and its processes. The model used is the fully nonlinear primitive equation SPEM model. The twin experiment approach is used in all the assimilations whose success is quantified by monitoring the time decrease of a normalized rms error. The following conclusions can be reached by the comparison of the different assimilation experiments. All the results of the time-varying assimilations were extremely similar in success to those of FR95. Specifically, section 4.1 cases (ii) and (iii) were almost identical to FR95 while the experiment of section 4.2 was actually slightly better. Section 4.1 case (i) on the other hand was clearly worse. It must be pointed out that there are significant differences in the computational requirements of the different experiments. The filter derivation of FR95 required approximately 2.6 CPU hours on a Cray YMP. In comparison, the three experiments of section 4.1 required 2.9 hours each and the experiment of section 4.2 required 16.1 hours. The numerical evaluation of the state transition matrix is the dominant computational cost in all the experiments, requiring by itself 2.2 CPU hours for evaluation. For the experiment described in section 4.2, the state transition matrix is evaluated six times, amounting to 13 hours. The first conclusion that is clearly demonstrated is the importance of specifying full covariances instead of the usual assumption of white noise. Even though one of the frequently cited advantages of Kalman filtering is the evaluation of temporally evolving error estimates, the examples discussed in section 4 indicate that a more crucial aspect of Kalman filtering is the estimation of the covariance. Estimates based upon the approximate (linearized) dynamics of the model provide quite successful assimilation results even in the steady-state asymptotic limit. Adding the time-evolution of the error covariance itself when starting from the static limit as the initial condition does not produce any improvement in the assimilations. This result strongly suggests that accurate specification of process noise maybe the most critical issue for Kalman filtering. While the process noise for the present idealized experiments was due to the simplifications of the reduced state, in practice it would include also errors of the model physics. A second conclusion which may be inferred, even though on less firm ground, is that nonlinearities in the model are more important than the covariance time evolution per se when based on a time-invariant linearization. Allowing for time variation of the covariance by updating the linearized transition matrix (experiment in section 4.2) does in fact produce slightly better assimilation estimates of the model variables. If this is indeed the case, then procedure 4.2, similar to but simpler than extended Kalman filtering, would be computationally affordable and efficient while allowing it to take into account important nonlinearities through successive linearizations around different mean states. The timeinterval over which a mean state is evaluated should be chosen according to the time-scale of growth and decay of the important nonlinear processes characterizing the system being studied. There may be in fact a dynamical reason explaining the similarity of results of assimilation for the experiment in section 4.2 with those of FR95. The 360 day control run used in FR95 as well as in the present paper shows a jet which, even though unstable and meandering to finite amplitude, does not present the richnesss of nonlinear events characterizing the Gulf Stream system. The overall system comprising the jet, meanders and mesoscale eddy field evolves rather slowly. This can be seen from Fig. 3 presenting the "true" and "false" oceans, i.e., the two different 60 day intervals of the control run. As a consequence, the 10-day mean states around which the successive linearizations are performed are rather quiescent. The model jet is evolving very gradually and does not show the dramatic changes that are often present in actual biweekly realizations of the Gulf Stream system based on the best available observations for that region (Malanotte-Rizzoli and Young, 1995). As a result, the 6 asymptotic covariances evaluated from the 6 successive Riccati equations thus obtained may have only very slight differences and not be very different altogether from the asymptotic covariance obtained in the unique linearization around the overall mean state. The plausibility of this explanation for the

317 similarities of the assimilation of section 4 and FR95 can be confirmed only by transferring the methodology and hierarchy of filters developed here to simulations of a much more realistic Gulf Stream system. Such an application is currently under development.

Acknowledgements This research was carried out with the support of the Office of Naval Research, Grant No. N00014-95-1-0226 (P. Malanotte-Rizzoli and R.E. Young), and in part by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (I. Fukumori).

REFERENCES Anderson, D.L.T. and A.M. Moore, 1989: Initialization of equatorial waves in ocean models, J. Phys. Oceanogr., 19, 116-121. Fukumori, I., J. Benveniste, C. Wunsch, and D.B. Haidvogel, 1993: Assimilation of sea surface topography into an ocean circulation model using a steady-state smoother, J. Phys. Oceanogr., 23, 1831-1855. Fukumori, I. and P. Malanotte-Rizzoli, 1995: An approximate Kalman filter for ocean data assimilation; an example with an idealized Gulf Stream model, J. Geophys. Res., 100, 6777-6793. Gelb, A., 1974: Applied Optimal Estimation, MIT Press, Cambridge, MA, 374 pp.. Ghil, M. and P. Malanotte-Rizzoli, 1991: Data assimilation in meteorology and oceanography, Advances in Geophysics, 33, 141-266. Goodwin, G.C. and K.S. Sin, 1984: Adaptive Filtering Prediction and Control, PrenticeHall, Inc., Englewood Cliffs, NJ, 540 pp. Haidvogel, D.B., J. Wilkin and R. Young, 1991: A semi-spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear coordinates, J. Comp. Phys., 94, 151-185. Heemink, A.W., 1987: Two-dimensional shallow water flow identification, Appl. Math. Model, 12, 109-118. Miller, R.N., 1987: Theory and practice of data assimilation for oceanography, Rep. Meteorol. Oceanogr. No. 26, Harvard University, Cambridge, MA. Malanotte-Rizzoli, P. and R.E. Young, 1995: Assimilation of global versus local datasets into a regional model of the Gulf Stream system, Part I: Data effectiveness, submitted to

J. Geophys. Res. Stengel, R.F., 1986: Stochastic Optimal Control: Theory and Application, Wiley and Sons, 638 pp. Tarantola, A., 1987: Inverse Problem Theory, Methods for data fitting and model parameter estimation, Elsevier, Amsterdam. Wunsch, C., 1978: The North Atlantic general circulation west of 50~ determined by inverse methods, Rev. Geophys. Space Phys., 16, 583-620. Wunsch, C.I. and B. Grant, 1982: Towards the general circulation of the North Atlantic ocean, Prog. Oceanogr., 11, 1-59.

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Modern Approaches to Data Assimilation in Ocean Modeling edited by P. Malanotte-Rizzoli 1996 Elsevier Science B.V.

319

Data assimilation in a North Pacific Ocean monitoring and prediction system M. R. Camesa, D. N. Foxa, R. C. Rhodesa and O. M. Smedstadb aNaval Research Laboratory, Stennis Space Center, MS 39529, USA bPlanning Systems, Inc., Slidell, LA 70458, USA

Abstract The Naval Research Laboratory recently began experimenting with a Pacific Ocean nowcast/forecast system. It is being developed to eventually run operationally at the Fleet Numerical Meteorology and Oceanography Center (FNMOC) to provide real-time nowcasts and forecasts of the ocean's temperature, salinity, sound speed and surface currents for Naval operations. This paper describes the models and assimilation schemes used and presents some early results from a pseudo-operational three-month series of test runs. Operationally available real-time data were used, and evaluations were performed using in situ observations which were not assimilated.

1. INTRODUCTION As ocean models, computers, ocean observations, and assimilation techniques have evolved and improved, the possibility of skillful real-time ocean prediction has become more realistic. The needs of the Navy and of the ocean-going industry for knowledge of the ocean environment are well known, ranging from underwater acoustic sensing and detection to ship routing. To fill these needs, several ocean models are currently run operationally by the Navy at the Fleet Numerical Meteorology and Oceanography Center (FNMOC) and the Naval Oceanographic Office (NOO). The FNMOC models, reviewed in Clancy (1992), run on global and regional scales, and analyze and predict ocean thermal structure and circulation, sea ice, and sea state. Notable models include the Thermal Ocean Prediction System (TOPS), which is an upper ocean mixed layer model, and the Polar Ice Prediction System model (Preller, 1992). The TOPS model also provides sea surface temperatures as boundary conditions for the Navy Operational Global Atmospheric Prediction System (NOGAPS) (Hogan and Rosmond, 1991). Models run at NOO include regional implementations of the Princeton primitive equation model for the Persian Gulf (Horton et al., 1992), the Red Sea, and the Mediterranean Sea. The subject of this paper is a new ocean modeling system being developed for operational use at FNMOC which is eddy-resolving, covers most of the Pacific basin, and assimilates altimeter-derived sea surface heights (SSH) as well as other satellite and in situ observations in near-real time. Its predecessor is the NOGUFS 2.0 system (Fox et al., 1992, 1993). NOGUFS 2.0 was designed specifically to forecast the position of the surface thermal front of the Gulf Stream. NOGUFS 2.0 uses a two-layer, primitive equation, eddy-resolving model

320 (Thompson and Schmitz, 1989). Each forecast is initialized from a surface height field derived from the Optimum Thermal Interpolation System (OTIS) (Cummings and Ignazewski, 1991) feature model based on the observed position of the surface thermal front of the Gulf Stream. No subsequent data assimilation is performed after initialization of the forecast. The initial field for the lower layer is derived from the surface layer values using statistical relationships derived from multi-year runs of the model (Hurlburt et al., 1990). This system provides skill of 40% better than persistence for a two week forecast when initialized with high quality research-grade data sets (such as those produced for the DAMEE Gulf Stream data assimilation and modeling experiments (Perkins, 1993)), and 20% better than persistence when initialized with frontal positions supplied from real-time daily operational analyses by the NOO (labelled as OPCHECK in Figure 1). Persistence here means use of the initial field as the forecast (Figure 1). It is important to note that this system is entirely a cold-start system and no data assimilation of any kind is done during each 2-week forecast. To examine the impact of this lack of continuous assimilation on the forecast skill, perfect sea surface height was taken from a twin model run and provided to the system as though it had been generated by OTIS. The skill in this case (labelled MODEL in Figure 1) was no better than that produced with the research-grade datasets, implying that the cold-start system had reached its limit of forecast capability. The limiting factor appeared to be the cold-start initialization. Other factors, such as the artificial boundaries of the regional model and the effect of using only two layers could not be evaluated in the identical-twin experiments, but likely further reduced skill in real forecasts. The need for further improvements in skill, geographic coverage, and in the type of output fields led to development of the Pacific Basin system described in this paper. The new Pacific basin system consists of several linked components. The two numerical models of the system are an eddy-resolving six-layer primitive equation hydrodynamic model coveting the Pacific Ocean north of 20 ~S, which is forced by surface wind stresses, and an upper ocean mixed layer model (TOPS), coupled loosely to the hydrodynamic model, and forced by surface wind stresses and heat fluxes. The data assimilation procedures for the hydrodynamic model include preparation of the sea surface height field by regional rubbersheeting to match features seen in IR imagery, optimum interpolation of altimeter surface height anomalies, statistical inference of subsurface model fields, and nudging. The coupling to the thermodynamic model allows indirect assimilation of the altimeter heights into the TOPS model fields and is accomplished by generation of synthetic temperature and salinity profiles from historical data as a function of surface observations and hydrodynamic model parameters. Assimilation of synthetic and observed data into TOPS model fields is done by optimum interpolation. Each component is described in the following sections.

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The diagram in Figure 2 outlines the analysis and forecast scheme used in the Pacific basin system. The upper block of the diagram shows the time sequences from three consecutive analysis/forecast cycles, labelled cycles i-l, i, and i+l. A new cycle starts every two days, but the start of each cycle begins two days before the analysis date. The lower diagrams show the procedures executed during cycle i for the hydrodynamic and thermal portions of the system. The hydrodynamic block begins by modification of the two-day forecast SSH field from cycle i-1 by rubber sheeting (described below) in selected regions to match positions of distinct feature such as fronts and eddies observed in the IR images from the analysis date. It is also modified to match front and eddy positions along ERS-1 and TOPEX~OSEIDON altimeter SSH tracks and dynamic heights estimated from recent XBTs. In the next step, this rubber-

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sheeted SSH field is used as the first guess field for optimum interpolation assimilation of altimeter SSH measurements made during the previous two days. The covariance of the errors of this first-guess field is assumed to have a Gaussian form with e-folding length scales of 150 km and 100 km in the east-west and north-south directions, respectively, a time scale of one day, and the ratio of the expected variances of the errors of the SSH measurements divided by the errors of the first-guess field (the rubber-sheeted two-day forecast) is set to 0.5. A SSH anomaly grid is computed as the difference between the analyzed grid and the two day forecast SSH field from cycle i-1. Next, pressure anomalies in each model layer are derived from the computed SSH anomaly using the statistical inference technique, described in section 2.1. Geostrophic velocity anomalies are also computed at latitudes greater than 5 degrees from the equator. At this point, the hydrodynamic model is restarted using the unaltered model fields from the analysis day of the previous cycle (two days before the analysis date of cycle i). The pressure and velocity anomalies are nudged in for two days. A two day forecast is then performed to provide fields for the next cycle, and the forecast is then continued for at least two weeks. However, when the system is run in real time, wind stresses to force the model are available from the global meteorological models for only three days. After the initial cycle, each subsequent two-day cycle of the thermal block consists of a data assimilation step and then a two-day forecast step to bring the fields up to the beginning of the

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323 next cycle. The assimilation combines both observed and synthetic temperature and salinity data with the previous forecast fields. The sequence of operations in the assimilation step are shown in the bottom block of Figure 2. First, satellite MCSSTs and XBT profiles taken during the previous two-day period are checked for quality and deleted if bad. Next, an analysis of sea surface temperature (SST) is performed by optimum interpolation. The SST forecast up to the analysis day of cycle i is used as the first-guess field, and MCSST observations and climatological SSTs from the Generalized Digital Environmental Model (GDEM) climatology ffeague et al., 1990) are the data. The GDEM surface "observations" cause a slow drift back to climatology in regions lacking observations for a long period of time. The expected errors of the GDEM SSTs are assumed to be spatially correlated, minimizing the effect of redundancy (having a value at every grid position). The covariance is set somewhat arbitrarily in this experiment, but will later be modified based on test results. Next, the vertically-averaged potential density (at zero pressure), Y., is computed at each grid position by averaging the model density from the surface to 1000 m from the cycle i hydrodynamic model analysis-day results. A density correction is then applied as explained below. Synthetic profiles of temperature and salinity are computed at every model grid position as a function of the vertically-averaged density and the analyzed SST. The synthetic profiles are then modified to match the near-surface vertical gradients of temperature and salinity from first-guess fields (previously forecast up to the present cycle's analysis date). The final analyzed grids of temperature and salinity are computed by optimum interpolation using the previous forecast as the first guess and the synthetic and observed profiles as data. Errors of the synthetic profiles are assumed to be spatially (horizontally) corr