Electromagnetic Sounding of the Earth's Interior, Volume 40 (Methods in Geochemistry and Geophysics)

ELECTROMAGNETIC SOUNDING OF THE EARTH’S INTERIOR METHODS IN GEOCHEMISTRY AND GEOPHYSICS (Volumes 1–28 are out of prin...

Author: Viacheslav V. Spichak

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ELECTROMAGNETIC SOUNDING OF THE EARTH’S INTERIOR

METHODS IN GEOCHEMISTRY AND GEOPHYSICS (Volumes 1–28 are out of print) 29.

V.P. Dimri – Deconvolution and Inverse Theory – Application to Geophysical Problems

30.

K.-M Strack – Exploration with Deep Transient Electromagnetics

31.

M.S. Zhdanov and G.V. Keller – The Geoelectrical Methods in Geophysical Exploration

32.

A.A. Kaufman and A.L. Levshin – Acoustic and Elastic Wave Fields in Geophysics, I

33.

A.A. Kaufman and P.A. Eaton – The Theory of Inductive Prospecting

34.

A.A. Kaufman and P. Hoekstra – Electromagnetic Soundings

35.

M.S. Zhdanov and P.E. Wannamaker – Three-Dimensional Electromagnetics

36.

M.S. Zhdanov – Geophysical Inverse Theory and Regularization Problems

37.

A.A. Kaufman, A.L. Levshin and K.L. Larner – Acoustic and Elastic Wave Fields in Geophysics, II

38.

A.A. Kaufman and Yu. A. Dashevsky – Principles of Induction Logging

39.

A.A. Kaufman and A.L. Levshin – Acoustic and Elastic Wave Fields in Geophysics, III

40.

V.V. Spichak – Electromagnetic Sounding of the Earth’s Interior

Methods in Geochemistry and Geophysics, 40

ELECTROMAGNETIC SOUNDING OF THE EARTH’S INTERIOR

Edited by

Viacheslav V. Spichak Geoelectromagnetic Research Center IPE RAS Troitsk, Moscow Region, Russia

Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2007 Copyright r 2007 Elsevier B.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 without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-52938-1 ISBN-10: 0-444-52938-1 ISSN: 0076-6895 For information on all Elsevier publications visit our website at books.elsevier.com

Printed and bound in The Netherlands 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

Contents Preface

Part I:

xiii

EM Sounding Methods

Chapter 1 Global 3-D EM Induction in the Solid Earth and the Oceans A.V. Kuvshinov 1.1. Forward Problem Formulation 1.2. Basic 3-D Earth Conductivity Model 1.3. Ocean Effect in S q Variations 1.4. Ocean Effect of Geomagnetic Storms 1.5. Magnetic Fields due to Ocean Tides 1.6. Magnetic Fields due to Ocean Circulation 1.7. Mapping Conductivity Anomalies in the Earth’s Mantle from Space 1.8. Conclusions References

4 5 7 9 13 16 18 21 21

Chapter 2 Magnetovariational Method in Deep Geoelectrics M.N. Berdichevsky, V.I. Dmitriev, N.S. Golubtsova, N.A. Mershchikova and P.Yu. Pushkarev 2.1. Introduction 2.2. On Integrated Interpretation of MV and MT Data 2.3. Model Experiments 2.4. MV–MT Study of the Cascadian Subduction Zone (EMSLAB Experiment) References

27 30 33 38 51

Chapter 3 Shallow Investigations by TEM-FAST Technique: Methodology and Examples P.O. Barsukov, E.B. Fainberg and E.O. Khabensky 3.1. 3.2. 3.3.

Introduction Advantages of TEM in Shallow Depth Studies On the TEM-FAST Technology

55 56 57

vi

Contents

3.4. Transformation of E(t) Data into r(h) 3.5. One-Dimensional Inversion and TEM-FAST’S Resolution 3.6. Joint Inversion of TEM and DC Soundings 3.7. Side Effects in TEM Sounding 3.7.1. Superparamagnetic Effect in TEM 3.7.2. Effect of Induced Polarization 3.7.3. Antenna Polarization Effect (APE) References

59 61 63 67 68 71 73 76

Chapter 4 Seismoelectric Methods of Earth Study B.S. Svetov 4.1. 4.2.

Seismoelectric Effect of the First Kind Seismoelectric Effect of the Second Kind: Historical Outline and Elements of Theory 4.3. Physical Interpretation of Seismoelectric Phenomena 4.4. Modeling of Seismoelectric Fields 4.5. Laboratory Studies of Seismoelectric Effects on Rock Samples 4.6. Experimental Field and Borehole Seismoelectric Studies References

Part II:

79 80 83 86 90 96 100

Forward Modeling and Inversion Techniques

Chapter 5 3-D EM Forward Modeling Using Balance Technique V.V. Spichak and M.S. Zhdanov 5.1.

Modern Approaches to the Forward Problem Solution 5.1.1. Methods of Integral Equations 5.1.1.1. The method of volume integral equations (VIE) 5.1.1.2. The method of surface integral equations 5.1.2. Methods of Differential Equations 5.1.2.1. The FD technique 5.1.2.2. The FE technique 5.1.3. Mixed Approaches 5.1.4. Analog (Physical) Modeling Approaches 5.2. Balance Method of EM Fields Computation in Models with Arbitrary Conductivity Distribution 5.2.1. Statement of the Problem 5.2.2. Calculation of the Electric Field 5.2.2.1. Equations and boundary conditions 5.2.2.2. Discretization scheme 5.2.3. Calculation of the Magnetic Field

106 106 107 108 109 109 111 112 114 116 116 117 117 118 120

Contents

5.2.4. Controlling the Accuracy of the Results 5.2.4.1. Criteria for accuracy 5.2.4.2. Comparison with high-frequency asymptotic solution 5.2.4.3. Comparison with results obtained by other techniques 5.3. Method of the EM Field Computation in Axially Symmetric Media 5.3.1. Problem Statement 5.3.2. Basic Equations 5.3.3. Boundary Conditions 5.3.4. Discrete Equations and their Numerical Solution 5.3.4.1. Discrete equations 5.3.4.2. Basis functions 5.3.4.3. Numerical solution of discrete equations 5.3.5. Code Testing References

vii 121 121 122 123 124 124 125 127 127 127 129 129 131 134

Chapter 6 3-D EM Forward Modeling Using Integral Equations D.B. Avdeev 6.1. 6.2.

Introduction Volume Integral Equation Method 6.2.1. Traditional IE Method 6.2.1.1. Comparison with other methods 6.2.1.2. Straightforward solution 6.2.1.3. Neumann series 6.2.2. Modiﬁed Iterative Dissipative Method 6.2.2.1. Krylov subspace interation 6.3. Model Examples 6.3.1. Induction Logging Problem 6.3.2. Airborne EM Example 6.4. Conclusion References

143 144 145 146 146 146 147 149 151 151 152 152 152

Chapter 7 Inverse Problems in Modern Magnetotellurics V.I. Dmitriev and M.N. Berdichevsky 7.1.

Three 7.1.1. 7.1.2. 7.1.3. 7.2. Three 7.2.1. 7.2.2. 7.2.3.

Features of Multi-Dimensional Inverse Problem Normal Background On Detailness of Multi-Dimensional Inversion On Redundancy of Observation Data Questions of Hadamard On the Existence of a Solution to the Inverse Problem On the Uniqueness of the Solution to the Inverse Problem On the Instability of the Inverse Problem

159 159 162 162 163 164 164 172

viii

Contents

7.3.

MT and MV Inversions in the Light of Tikhonov’s Theory of Ill-Posed Problems 7.3.1. Conditionally Well-Posed Formulation of Inverse Problem 7.3.2. Optimization Method 7.3.3. Regularization Method References

175 175 177 178 183

Chapter 8 Joint Robust Inversion of Magnetotelluric and Magnetovariational Data Iv.M. Varentsov 8.1.

Adaptive Parameterization of a Geoelectric Model 8.1.1. A Background Structure and Windows to Scan Anomalies 8.1.2. A Priori Model Structure and Constrains 8.1.3. Window with Correlated Resistivities of Inversion Cells 8.1.4. Window with Finite Functions 8.2. Inverted and Modeling Data 8.3. Inversion as a Minimization Problem 8.3.1. Minimizing Functional 8.3.2. Robust Misﬁt Metric 8.3.3. Cycles of Tikhonov’s Minimization 8.3.4. Newtonian Minimization Techniques 8.3.5. Solution of Linear Newtonian System and Choice of Scalar Newtonian Step 8.3.6. Multi-Level Adaptive Stabilization 8.3.7. Post-Inversion Analysis 8.4. Study of Inversion Algorithms using Synthetic Datasets 8.4.1. Comparison of Three Model Parameterization Schemes in 2-D Inversion 8.4.2. 2-D Inversion with Numerous Finite Functions 8.4.3. 3-D Inversion Example 8.4.4. Resolution of a System of Local Conductors using the CR-Parameterization 8.4.5. Reduction of Strong Data Noise and Static Shifts 8.5. Conclusions References

186 186 186 187 187 188 190 190 191 192 192 194 195 196 197 197 205 209 211 213 215 216

Chapter 9 Neural Network Reconstruction of Macro-Parameters of 3-D Geoelectric Structures V.V. Spichak 9.1. 9.2. 9.3.

BackPropagation Technique Creation of Teaching and Testing Data Pools Effect of the EM Data Transformations on the Quality of the Parameters’ Recognition 9.3.1. Types of the Activation Function at Hidden and Output Layers 9.3.2. Number of the Neurons in a Hidden Layer

220 223 224 225 227

Contents

9.3.3. 9.3.4. 9.4. Effect 9.5. Effect 9.5.1. 9.5.2.

Effect of an Extra Hidden Layer Threshold Level of the Input Data Type of the Volume and Structure of the Training Data Pool Effect of Size Effect of Structure 9.5.2.1. Random selection of synthetic data sample 9.5.2.2. Gaps in the training data base 9.5.2.3. ‘‘No target’’ case 9.6. Extrapolation Ability of ANN 9.7. Noise Treatment 9.8. Case History: ANN Reconstruction of the Minou Fault Parameters 9.8.1. Geological and Geophysical Setting 9.8.2. CSAMT Data Acquisition and Processing 9.8.3. 3-D Imaging Minou Fault Zone using 1-D and 2-D Inversion 9.8.3.1. Synthesis of Bostick transforms 9.8.3.2. 2-D inversion results 9.8.4. ANN Reconstruction of the Minou Geoelectric Structure 9.8.4.1. ANN recognition in terms of macro-parameters 9.8.4.2. Testing ANN inversion results 9.8.5. Discussion and Conclusions References

ix 229 229 229 232 232 233 234 234 236 238 239 242 242 243 245 245 246 247 250 251 252 253

Part III: Data Processing, Analysis, Modeling and Interpretation Chapter 10 Arrays of Simultaneous Electromagnetic Soundings: Design, Data Processing and Analysis Iv.M. Varentsov 10.1. Simultaneous Systems for Natural EM Fields Observation 10.2. Multi-Site Schemes for Estimation of Transfer Operators 10.3. Temporal Stability of Transfer Operators 10.4. Methods for the Analysis and Interpretation of Simultaneous EM Data 10.5. Conclusions References

259 262 264 266 270 271

Chapter 11 Magnetotelluric Field Transformations and their Application in Interpretation V.V. Spichak 11.1. 11.2.

Linear Relations between MT Field Components Point Transforms of MT Data

276 277

x

Contents

11.2.1. Impedance Transforms 11.2.2. Apparent Resistivity Type Transforms 11.2.3. Induction and Perturbation Vectors 11.3. Examples of the Use of MT Field Point Transforms for the Interpretation 11.3.1. Dimensionality Indicators 11.3.2. Local and Regional Anomalies 11.3.3. Constructing Resistivity Images in the Absence of Prior Information 11.4. Integral Transforms 11.4.1. Division of the MT Field into Parts 11.4.2. Transformation of the Field Components into Each Other 11.4.3. Synthesis of Synchronous MT ﬁeld from Impedances and Induction Vectors 11.4.3.1 Magnetic ﬁeld synthesis from know impedance 11.4.3.2 Magnetic ﬁeld synthesis from known tipper References

277 285 287 291 291 292 293 298 298 302 303 304 304 305

Chapter 12 Modeling of Magnetotelluric Fields in 3-D Media V.V. Spichak 12.1.

A Feasibility Study of MT Method Application in Hydrocarbon Exploration 12.1.1. Statement of the Problem 12.1.2. Numerical Modeling 12.2. Testing Hypotheses of the Geoelectric Structure of the Transcaucasian Region from MT Data 12.2.1. Geological and Geophysical Characteristics of the Region 12.2.2. Alternative Conductivity Models 12.2.3. Numerical Modeling of Magnetotelluric Fields 12.2.4. Conclusions 12.3. MT Imaging Internal Structure of Volcanoes 12.3.1. Simpliﬁed Model of the Volcano 12.3.2. Synthetic MT Pseudosections 12.3.3. Methodology of Interpretation of the MT Data Measured over the Relief Surface 12.4. Simulation of MT Monitoring of the Magma Chamber Conductivity 12.4.1. Geoelectric Model of a Central Type Volcano 12.4.2. Detection of the Magma Chamber by MT Data 12.4.3. Estimation of MT Data Resolving Power with Respect to the Conductivity Variations in the Magma Chamber 12.4.4. ‘‘Guidelines’’ for MT Monitoring Electric Conductivity in a Magma Chamber 12.5. Simulation of MT Monitoring the Ground Water Salinity 12.5.1. Statement of the Problem 12.5.1.1 The data 12.5.1.2 Prior information 12.5.2. Modeling of the Salt Water Intrusion Zone Mapping by Audio-MT Data References

314 315 315 321 321 324 325 330 331 331 332 335 338 338 338 340 343 344 344 346 346 347 348

Contents

xi

Chapter 13 Regional Magnetotelluric Explorations in Russia V.P. Bubnov, A.G. Yakovlev, E.D. Aleksanova, D.V. Yakovlev, M.N. Berdichevsky and P.Yu. Pushkarev 13.1. 13.2. 13.3. 13.4.

Introduction Observation Technology MT-Data Processing, Analysis and Interpretation Case Histories 13.4.1. East-European Craton 13.4.2. Caucasus, the Urals, Siberia, and North East Russia 13.5. Conclusion References

351 353 354 356 356 360 366 366

Chapter 14 EM Studies at Seas and Oceans N.A. Palshin 14.1.

Conductivity Structure of Sea and Ocean Floor 14.1.1. Background Conductivity Structure of the Ocean Crust and Upper Mantle 14.1.2. Principle Objectives of Marine EM Studies 14.2. Instrumentation for Marine EM Studies 14.2.1. Seaﬂoor Controlled Source Frequency and Transient EM Sounding 14.2.2. Measurements of Variations of Natural EM Fields on the Seaﬂoor 14.3. Some Results of EM Sounding in Seas and Oceans 14.3.1. Studies of Gas Hydrates in Seabed Sediments of Continental Slopes 14.3.2. Studies of Buried Salt Dome-like Structures 14.3.3. The Reykjanes Axial Melt Experiment: Structural Synthesis from Electromagnetics and Seismics (RAMESSES Project) 14.3.4. Seaﬂoor MT Soundings of the Eastern-Paciﬁc Rise at 91500 N 14.3.5. Mantle Electromagnetic and Tomography Experiment 14.4. Deep Seaﬂoor EM Studies in the Northwestern Paciﬁc References

370

377 378 379 380 382

Subject Index

385

370 370 371 371 373 375 376 377

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Preface This book is prepared on the basis of underlying lectures given in the First Russian School-Seminar on electromagnetic (EM) soundings of the Earth held in Moscow on 15th November, 2003. In recent years, considerable progress has been made both in the design of EM equipment and in the development of methods for processing, analysis and interpretation of electromagnetic data. Therefore, the aim of the School-Seminar was to acquaint the scientists and technologists engaged in this ﬁeld of science with the latest achievements in theory, techniques and practical applications of the methods of electromagnetic sounding. The program of the School included scientiﬁc contributions from leading Russian scientists involved in active research in this ﬁeld. It should be mentioned that the suggested selection of lectures is not a textbook on electric prospecting, but rather a live pattern of actual works at the leading edge of geoelectrics. Only a few monographs more or less related to topics discussed here appeared (see, e.g., 3-D Electromagnetics (Eds. Oristaglio & Spies, 1999), Geophysical Inverse Theory and Regularization Problems (Zhdanov, 2002), Practical Magnetotellurics (Simpson and Bahr, 2005)). The present book updates the above monographs, in particular, as regards the MT, magnetovariational and seismo-electrical (SE) methods and the practice of 3-D interpretation as well. In this book, the reader will ﬁnd both theoretical and methodological ﬁndings as well as examples of application of recently developed algorithms and software in solving practical problems. The book comprises 14 chapters. In Part I, EM sounding methods of the Earth are considered on a global, regional and local scale. Chapter 1 addresses the problems of 3-D induction due to ionospheric and magnetospheric currents, ocean tides and global ocean circulation. Chapter 2 tells about a ‘‘new breath’’ of the magnetovariational method that has been opened owing to the recent results obtained by the authors. In essence, a revision of the traditional approach to regional (mainly magnetotelluric) soundings is proposed with an emphasis on the use of magnetovariational data. The transient electromagnetic method as applied to solving the problems of shallow geoelectrics is considered in Chapter 3. Inﬂuences of the superparamagnetic effect, induced polarization and the ‘‘antenna effect’’ on the resolving power of the method are analyzed. Numerous examples of application of the method to the solution of practical problems are given. A special section of Part I is devoted to seismoelectric methods, a rapidly growing ﬁeld of interest nowadays. In this section, theoretical grounds for seismoelectrical

xiv

Preface

studies on both the borehole and ﬁeld setup are considered. Experimental corroboration is given to the developed theory adequate to the observed phenomena and to that ensuing from the theory conclusions regarding information capabilities of SE method of geophysical prospecting . In Part II, modern methods for solving forward and inverse problems of geoelectrics are analyzed. Particular attention is paid to contemporary approaches to EM data modeling and interpretation in the class of three-dimensional models. In Chapter 5, up-to-date approaches to the forward problem solution in three-dimensional media are considered. Main emphasis is laid on the balance method application aimed at obtaining efﬁcient discrete schemes. This approach is illustrated by an example employing axial-symmetric models of the medium that are particularly helpful in methodical studies. In Chapter 6, solution of the forward problem with the method of integral equations is discussed. Here, major attention is given to the modiﬁcation of this method known as a modiﬁed iterative-dissipative method. Generalizations of this method on media with displacement currents and anisotropic conductivities are considered, and examples of its application in solving the induction logging and airborne EM problems are given. The subsequent chapters of Part II are devoted to the solution of inverse problems. In Chapter 7, speciﬁc features of the solution of multidimensional inverse problems are considered. Much attention is given to the basic issues of existence, uniqueness and stability of the inverse problem solution of magnetotelluric and magnetovariational soundings in the light of Tikhonov’s theory of the ill-posed problems solution. Joint inversion of magnetotelluric and magnetovariational data in piecewisecontinuous media is addressed in Chapter 8. General approach to the solution of non-linear inverse problems in geoelectrics is discussed within the framework of the trial-and-error method for a broad class of piecewise-continuous models. In this approach, model parameterization schemes combine well with the traditions of Tikhonov’s regularization, and the ideas of robust estimation impart a new quality to the methods of non-linear Newtonian minimization. The consideration is focused on the synthesis and adaptive implementation of the new and traditional approaches with the purpose of attaining the effective trade-off between the accuracy and stability of a solution. Finally, in the last chapter of this part, a non-traditional approach to the EM data inversion based on the pattern recognition techniques is considered. A neural network algorithm of data interpretation that enables interpretation of incomplete, inhomogeneous and strongly noised data when traditional interpretation methods do not work is presented. It is shown that, following the proposed scheme of information processing, one can perform the inversion of electromagnetic data in the chosen class of three-dimensional model media. An example is given for a threedimensional dike macro-parameters determination from the magnetotelluric data measured in Minou fault zone (Kyushu, Japan).

Preface

xv

Part III of the book is allotted to the results of regional EM (mainly magnetotelluric) on-land and sea soundings. In Chapter 10, methods for analyzing synchronous data of electromagnetic soundings of the Earth with natural ﬁelds, multipoint systems of their relevant transfer operators, and speciﬁc methods for data analysis and interpretation are considered. By the examples of international experiments such as BEAR and TESZPomerania it is shown that synchronous EM soundings of the Earth crust open new resources of noise-suppressing data processing and provide additional interpretation possibilities. Considered in Chapter 11 are various transformations of the magnetotelluric ﬁeld used in data interpretation. Here, much attention is given to integral transformations of the ﬁeld that are not often used in practice. In Chapter 12 by means of three-dimensional numerical modeling, methodological questions concerning such applications of magnetotellurics as hydrocarbon detection, magma chamber mapping and others are analyzed. Cited in Chapter 13 are the results of magnetotelluric sounding carried out at a network of regional proﬁles used to study the geodynamic conditions of the region, estimation of the extent and development of dangerous geological processes, and mineragenic zoning of territories. It is shown that electrical prospecting efﬁciently complements the seismic data by the information about physical properties of rocks speaking about characteristic of their lithology, ﬂuid saturation, rheological state and so on. Various methods of electric prospecting (MTS, frequency sounding, transient domain electromagnetic method) are considered, and their efﬁciency in solving the above-listed tasks is analyzed. Examples of interpretation of magnetotelluric sounding data obtained at the East European Platform are given. Finally, in the last chapter of this part the principal methods applied in sea and ocean ﬂoor studies are considered: ﬂoor-bottom frequency and magnetotelluric soundings. The former one is used mainly for investigation of the upper layers of the ocean crust, while the latter one is employed for deep studies of both the oceanic crust and the upper mantle. Main targets of EM studies in seas and oceans are gaseous-hydrate and/or permafrost layers in bottom sediments of continental slopes, salt domes buried in sedimentary rocks, and deep conductors in the oceanic crust and upper mantle, particularly in the regions of lithospheric plate junction (rift and subduction zones). Results of basic research of the deep structure of oceanic crust and upper mantle obtained with EM methods are presented. In conclusion, I would like to note that an efﬁcient solution to the problems of electromagnetic sounding of the Earth is not only of purely scientiﬁc interest; it is of obvious practical importance as well, since it enables speaking about a construction of a closed technological cycle incorporating the systems for measurements, processing, analysis and three-dimensional interpretation of electromagnetic data. Certainly, the whole list of topical problems of geoelectrics is not exhaustingly covered by the issues touched in the book. At the same time, the profound consideration of the latter will hopefully give an impetus to the advancement in ement this branch of geophysics.

xvi

Preface

I am pleased to express my thanks to Alexandra Goidina and Marina Nazarenko who assumed not an easy charge to prepare the manuscript for publication. Viacheslav V. Spichak February 2006

Part I: EM Sounding Methods

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Chapter 1 Global 3-D EM Induction in the Solid Earth and the Oceans A.V. Kuvshinov1,2 1

Danish National Space Center, Juliane Maries Vej 30, 2100 Copenhagen, Denmark 2 Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, RAS, Troitsk, Moscow region, Russia

There has been an increasing interest in global electromagnetic (EM) induction studies during the past years, mainly because of two reasons. Firstly, due to the tremendous growth of the amount of global geomagnetic data that has happened recently, coming mainly from satellite measurements. Indeed, 20 years after the Magsat satellite mission (1979–1980), the satellites Oersted (launched, February 1999), CHAMP (launched, July 2000) and Oersted-2/SAC-C (launched, November 2000) measure the vector and scalar magnetic ﬁelds from their low altitudes (400–800 km), circular polar orbits with unprecedented accuracy (cf. Neubert et al., 2001; Reigber et al., 2002). Moreover, the geomagnetic low-orbiting (450–550 km of altitude) three-satellite constellation mission Swarm is scheduled to be launched by the European Space Agency in 2010 (cf. Friis-Christensen et al., 2006). In contrast to land-based data from geomagnetic observatories, which are sparse and irregularly distributed (with only few in oceanic regions), satellite-borne measurements provide an excellent spatio-temporal coverage with high-precision data of uniform quality. In addition to satellite measurements, variations of voltage difference measured in transoceanic submarine telecommunication cables have been introduced recently for deep EM studies in oceanic regions (e.g. Lanzerotti et al., 1992; Vanyan et al., 1995; Utada et al., 2003). By combining land- and ocean-based observations with satellite-borne measurements, we have an intriguing chance to approach the solution of the most challenging problem of deep EM studies: the

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40001-9

4

A.V. Kuvshinov/Global 3-D EM Induction in the Solid Earth and the Oceans

recovery of three-dimensional (3-D) variations of electrical conductivity in the Earth’s mantle, beneath the continents as well as the oceans. Since conductivity reﬂects the connectivity of constituents as graphite, ﬂuids, partial melt and volatiles (all of which may have profound effect on rheology), 3-D conductivity images provide information about mantle convection and tectonic activity in the Earth. The second reason for the renewed interest in global EM induction studies is again due to satellite geomagnetic investigations. It is known that the primary objectives of satellite geomagnetic missions are the studies of core dynamics, geodynamo processes, core–mantle interaction as well as mapping of the lithospheric magnetization and its geologic interpretation. For these studies, it is essential that the magnetic ﬁeld models be contaminated as little as possible by ﬁelds originating, say, from the ionosphere and magnetosphere and their Earth-induced counterparts. However, accurate extraction of the contribution of these sources from the satellite signals is a nontrivial problem of modern geomagnetic ﬁeld modeling. So far, the conducting Earth considered even in the most sophisticated modeling schemes (e.g. Sabaka et al., 2004) is assumed to be spherically symmetric (1-D). As a consequence, the EM effects and/or signals originating in the oceans are ignored. But oceans play a special role in the EM induction problem due to their relatively high conductance of extremely large lateral variability. Electric currents that generate secondary magnetic ﬁelds are induced in the oceans by two types of sources: by time-varying external magnetic ﬁelds, and by motion of the ocean water through the Earth’s main magnetic ﬁeld. Signiﬁcant progress in the accurate and detailed prediction of the magnetic ﬁelds induced by these sources has been achieved during the last years, utilizing a realistic 3-D conductivity model of the oceans, crust and mantle. In addition to these improvements in the prediction of 3-D induction effects, much attention has been paid to identify magnetic signals of oceanic origin in observatory and satellite magnetic data. In this chapter, recent results of the 3-D model studies that aim at quantitative estimating magnetic signals (at ground and satellite altitude) induced by a variety of realistic sources are presented. In particular, the 3-D induction due to ionospheric currents, magnetospheric currents, ocean tides and global ocean circulation is considered. Finally, a scheme how to process satellite geomagnetic data in order to detect possible mantle inhomogeneities is discussed.

1.1. FORWARD PROBLEM FORMULATION The results that will be presented rely on a solution of the 3-D forward problem, which is the accurate and detailed prediction of EM ﬁelds induced by a given timevarying source in a given spherical 3-D conductivity model of the Earth. For this problem, the EM ﬁelds in the frequency domain obey Maxwell equations r H ¼ sE þ jext

ð1:1Þ

r E ¼ iom0 H

ð1:2Þ

A.V. Kuvshinov/Global 3-D EM Induction in the Solid Earth and the Oceans

5

where m0 is the magnetic permeability of free space, jext the extraneous current, o ¼ 2p=T angular frequency, T the period and s the 3-D conductivity distribution in the models, which consist of a number of anomalies of conductivity s3D ðr; W; jÞ, embedded in a host section of conductivity sb ðrÞ. Here r, W, and j are the distance from the Earth’s centre, colatitude and longitude, respectively. In Equations (1.1) and (1.2) we assume that time dependency is eiwt, and ignore the displacement currents (which is a valid approximation for considered periods of a few hours and larger). During the last decades, a number of algorithms have been developed to solve Maxwell equations numerically (in spherical geometry) – either in the frequency or in the time domain (cf. Fainberg et al., 1990a,b; Tarits, 1994; Everett and Schultz, 1996; Weiss and Everett, 1998; Martinec, 1999; Uyeshima and Schultz, 2000; Hamano, 2002; Koyama et al., 2002; Yoshimura and Oshiman, 2002; Tyler et al., 2004; Velimsky and Martinec, 2005). For the model studies of this chapter we used our own numerical solution (Kuvshinov et al., 2002a, 2005), which is based on a volume integral equation approach. This approach combines the modiﬁed iterative dissipative method (MIDM) (Singer, 1995) with the conjugate gradient iterations. In accordance with MIDM, Maxwell Equations (1.1) and (1.2) are reduced to a special scattering equation (cf. Pankratov et al., 1997; Avdeev et al., 2002), which is then solved by the generalized bi-conjugate gradient method (Zhang, 1997). Once the scattering equation is solved (and thus the electric ﬁeld at depths occupied by the 3-D anomalies is determined), the electric, E, and magnetic, H, ﬁelds at the observation points r 2 V obs are calculated as Z Z e e 0 ext 0 0 ^ EðrÞ ¼ G b ðr; r Þj ðr Þ dv þ G^ b ðr; r0 Þjq ðr0 Þ dv0 ð1:3Þ V ext

Z HðrÞ ¼ ext

V e G^ b

V mod

Z

h G^ b ðr; r0 Þjext ðr0 Þ dv0 þ V

h G^ b ðr; r0 Þjq ðr0 Þ dv0

ð1:4Þ

mod

h where j ¼ ðs sb ÞE, and G^ b the respective ‘‘electric’’ and ‘‘magnetic’’ Green tensors of the host radially symmetric section, r ¼ ðr; W; jÞ, r0 ¼ ðr0 ; W0 ; j0 Þ and V ext and V mod the spherical layers which comprise the extraneous current jext and the 3-D anomalies, respectively. The explicit expressions to calculate the elements of e h Green tensors G^ b and G^ b are presented in the Appendix of Kuvshinov et al. (2002a). More details of the approach can be found in Chapter 6 of this book. q

1.2. BASIC 3-D EARTH CONDUCTIVITY MODEL Global EM induction simulations require a model of the electric conductivity of the Earth’s interior. The basic 3-D model that will be used in the following sections consists of a thin spherical shell of conductance SðW; jÞ at the Earth’s surface and a radially symmetric spherical conductivity sðrÞ underneath. A realistic model of the

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shell conductance SðW; jÞ is obtained by considering contributions both from sea water and from sediments. The conductance of the oceans has been derived from the global 50 50 NOAA ETOPO map of bathymetry/topography, multiplying the water depth by the sea water conductivity. Note that sea water conductivity, sw , varies between 3 and 5 S/m, depending on salinity, temperature and pressure. In principle, the World Ocean Atlas (WOA, www.nodc.noaa.gov) and ocean circulation models (e.g. ECCO, www.ecco-group.org) provide the spatial distribution of these parameters. Based on WOA, the global conductance map of oceanic regions has been recently updated (Manoj et al., 2006), revealing many regions (for example, Mediterranean and Black Seas, polar and equatorial regions) where the new conductance values differ noticeably (thousands of Simens) from those based only on bathymetry and a constant value of sw : However, for the model studies presented here a mean value of sw ¼ 3.2 S/m is used; the errors introduced by deviations from that value are believed to be smaller than those due to insufﬁcient knowledge of mantle conductivity. The conductance of the sediments (for oceanic and continental regions) has been derived from the global sediment thicknesses given by the 11 11 map of Laske and Masters (1997) by using a heuristic procedure similar to that of Everett et al. (2003). In general, the sediments contribute with up to 10% to the total surface conductance. However, in areas such as the Gulf of Mexico, Arctic Ocean, Black and Caspian Seas, the conductance of the accumulated sediments is comparable to that of the sea water. Fig. 1.1 shows the conductance of this surface shell. It is seen that conductance varies from fractions of Siemens inland up to tens of thousand Siemens in the oceans. The underlying conductivity sðrÞ is compiled from the four-layer model of Schmucker (1985a) for

Fig. 1.1. Conductance of the surface shell describing oceans and sediments. Also shown are the locations of the observatories used for the source determination (small red dots; see Section 1.4 for details) and of those used in Fig. 1.7 (large red dots). (Reproduced from Olsen and Kuvshinov, 2004.)

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7

depth greater than 100 km (0.014 S/m between 100 and 500 km, 0.062 S/m between 500 and 750 km, and 2.4 S/m deeper than 750 km), whereas for the upper 100 km we take 3 104 S/m. A mesh of 11 11 spatial resolution was used for most of the simulations presented here.

1.3. OCEAN EFFECT IN S q VARIATIONS The ﬁrst example deals with a quantitative estimation of the time–space distribution of Sq geomagnetic daily variations in the presence of the oceans. As primary (inducing) source, we take realistic equivalent Sq current system for equinoctial conditions (21 March 2001), based on the Comprehensive Model of Sabaka et al. (2004). The Sq currents, generated by the motion of ionosphere conducting matter in ambient magnetic ﬁeld of the Earth form two large loops on the day-lit side of the Earth, which are symmetric to the dip equator during equinoxes. Viewed from the Sun, the Earth rotates under these ionospheric current loops. Note that numerical estimating the effect of distribution of resistive continents and conductive oceans on EM induction in the Earth induced by Sq currents was the topic of numerous studies (e.g. Ashour, 1965; Bullard and Parker, 1970; Hewson-Brown and Kendall, 1978; Beamish et al., 1980; Hobbs, 1981; Fainberg et al., 1990b; Takeda, 1991). A detailed and systematic study of the coastline effect in Sq variations on surface observatories (and a review of previous work on the subject) was done by Kuvshinov et al. (1999). Tarits and Grammatica (2000) and Grammatica and Tarits (2002) qualitatively estimated the inﬂuence of near-surface heterogeneities in Sq ﬁelds at satellite altitudes. Fig. 1.2 shows global maps of the total (upper panels) and anomalous (lower panels) vertical magnetic ﬁelds, Z ¼ Br , simulated at sea level for two instants (07:00, left panels and 19:00, right panels) of universal time (UT). Here we determine as anomalous effect the difference between the results with and without nonuniform oceans. Hereinafter we present notably vertical component of magnetic ﬁeld, since this component is to a largest extent inﬂuenced by the induction. As expected, this difference is largest in the oceans and near the coasts and reaches 12 nT in amplitude, which is approximately half of the maximum of the total signal. Fig. 1.3 presents in a similar way the anomalous Z at CHAMP altitude (h ¼ 400 km). At CHAMP altitude the ocean effect appears to be smoother and smaller in amplitude but is still about 6 nT on average. It is of interest to compare the model predictions and observations. Fig. 1.4 shows the modelled and observed (average of the ﬁve quietest days of September 1964) Sq variations in Z at the Japanese observatory Kakioka (KAK). A scheme how to deduce an external Sq source ﬁeld from observations has been presented by Kuvshinov et al. (1999). The results are shown in local time, T ¼ t þ j (where t is UT), since daily variations are basically functions of local time. Coloured lines show the model results, with red lines for the nonuniform oceans model and with blue lines for the model without oceans. It is clearly seen that anomalous behaviour of Z at this coastal observatory can be identiﬁed as the inﬂuence of oceans.

8 A.V. Kuvshinov/Global 3-D EM Induction in the Solid Earth and the Oceans Fig. 1.2. The total (upper panels) and anomalous (lower panels) Z (in nT) at sea level for 07:00 UT (left panels) and 19:00 UT (right panels) on 21 March 2000.

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Fig. 1.3. The anomalous Z (in nT) at CHAMP altitude for 07:00 UT (left panel) and 19:00 UT (right panel) on 21 March 2000.

Fig. 1.4. The modelled and observed (as an average of ﬁve quietest days in September 1964) Sq variations in Z at Japanese observatory Kakioka (KAK). Coloured lines are for model results. Red lines for nonuniform oceans model and blue lines the model without oceans. (After Kuvshinov et al., 1999.)

1.4. OCEAN EFFECT OF GEOMAGNETIC STORMS To the ﬁrst order, geomagnetic storms can be described by an intensiﬁcation of the (westward directed) magnetospheric ring current (e.g. Rostoker et al., 1997). Its time change induces a corresponding internal current system of reversed sign, and hence the major part of induced currents during storms is eastward directed. While a signiﬁcant part of the induced currents ﬂow in the open oceans (and in the underlying mantle), coastlines will force them to deviate from the west–east geometry dictated by the external (inducing) currents. This leads to current channelling,

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which is especially pronounced at the edges of continental barriers like southern Africa. A number of numerical modelings (Kuvshinov et al., 1990; Takeda, 1993; Tarits, 1994; Weiss and Everett, 1998) of the ocean effect using realistic conductivity models have been performed during the past years to estimate this effect in Cresponses. The common understanding was that the ocean effect in the responses becomes negligible for periods greater than a few days. Kuvshinov et al. (2002b) reconsidered the ocean effect by making detailed and systematic studies in the period range from 1 to 64 days with subsequent comparison of modelled and observed C-responses at a number of coastal observatories. They concluded that, for all considered observatories, sea water is a major contributor to the anomalous behaviour of C-responses in the period range up to 20 days. Recently, Olsen and Kuvshinov (2004) presented an approach for modeling the ocean effect of geomagnetic storms in the time domain. Their results for several major storms show much better agreement between the observed and the simulated magnetic vertical component at coastal sites if the oceans are considered. Using model studies, Kuvshinov and Olsen (2005a) demonstrated that induction in oceans considerably inﬂuences the magnetic ﬁeld even at satellite altitudes. Fig. 1.5 shows global maps of the total (upper panels) and anomalous (lower panels) vertical magnetic ﬁeld at sea level for two UT instants (3:00, left panels and 6:00, right panels) of the main phase of the storm of 5–6 November 2001. The timevarying magnetospheric source is determined from Oersted and CHAMP satellite data (cf. Kuvshinov and Olsen, 2005c). For this example, the source geometry was approximated by the ﬁrst zonal harmonic P01 ¼ cos Wd in geomagnetic coordinates (where Wd is geomagnetic colatitude); this geometry is clearly seen in the total Z that is shown in the upper panels of Fig. 1.5. To obtain the time series of the induced ﬁeld for a given 3-D conductivity model of the Earth, a time-domain scheme (cf. Olsen and Kuvshinov, 2004; Kuvshinov and Olsen, 2005a) was applied which relies on a Fourier transformation of the inducing ﬁeld, and a frequency domain forward modeling. Fig. 1.6 presents the anomalous effect in Z at CHAMP altitude. The ocean effect, here mainly manifested itself as a sharp ﬁeld increase near the coasts, reaches tens of nT during the main phase of the storm both at sea level (80 nT maximum amplitude) and at CHAMP altitude (30 nT maximum amplitude). Fig. 1.7 demonstrates the ocean effect in more detail. The left panels of the Fig. 1.7 present the time series of observed and modelled Z at selected coastal observatories during the storm of 13–14 July, 2000. The source geometry for these simulations was derived from an hour-by-hour spherical harmonic analysis of worldwide distributed observatory hourly mean values (the observatory distribution is shown in Fig. 1.1). The time series of the 15 expansion coefﬁcients m of the external potential, qm n ðtÞ, sn ðtÞ, for n; m 3 are thus determined (cf. Equation (1.6)). The results of the 3-D model calculations (with nonuniform oceans) are shown in red, those of the 1-D model (without oceans) in blue and the black lines present the observed data. The green lines show the values based on the Dst-index. It is seen that there are considerable differences between the 1-D and the 3-D results. The largest difference is found at the South African observatory Hermanus (HER); the peak of

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11

Fig. 1.5. Total (upper panels) and anomalous (lower panels) Z (in nT) at sea level for two UT instants (3:00 and 6:00) of the main phase of the storm of 5–6 November 2001.

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Fig. 1.6. Anomalous Z (in nT) at CHAMP altitude for two UT instants (3:00 and 6:00) of the main phase of the storm of 5–6 November 2001.

Fig. 1.7. Left panel: time series of observed and modelled Z(in nT) at selected observatories. 3-D model results are shown in red, 1-D results in blue and observed ﬁelds in black. The green lines present values based on Dst index. t ¼ 0 corresponds to 13 July 2000, 00:00 UT. Right panel: time series of observed and modelled Z (nT) at Hermanus observatory for selected geomagnetic storms. The numbers at the left present rms deviations (see text). (After Olsen and Kuvshinov, 2004.)

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13

Z during the maximum of the storm is 220 nT for the 3-D case, which is much closer to the observed value (250 nT) than the 1-D result (50 nT). Japanese observatories (KAK, KNY, HTY, KNZ) also show a clear ocean effect due to their proximity to a deep-sea trench. There is also in this case better agreement between the observations and the model results if the oceans are considered. For comparison, the time series of Z at four inland observatories (ASP, TAM, SUA and KSH) are also shown in the ﬁgure. As expected for sites faraway from the coast, 1-D and 3-D results are rather similar. It is also seen that in addition to the induction effects the contributions from higher harmonics (besides the dominant P01 source) are important (cf. the results are based on Dst-index, which relies on the P01 assumption). Strictly speaking the anomalous induction near coastlines has at least two possible contributions: the ocean effect, and the conductivity discontinuities in the crust and upper mantle speciﬁcally associated with continent–ocean boundaries (for instance, subduction slabs). However, simulations using conductivity models with and without laterally inhomogeneous lithosphere and upper mantle at the continent–ocean transition indicate that the ocean effect is dominating (cf. Kuvshinov et al., 2005). Finally, the right panel of Fig. 1.7 presents the results at observatory Hermanus for major geomagnetic storms. Again, only the 3-D results reproduce the observations. The superiority of the 3-D results is also evident when comparing the root mean-square deviation between the observed and predicted Z. For example, for the 14 July storm the differences between observations and predictions based on 1-D and the 3-D models are 63 and 17 nT, respectively (cf. numbers at the left side of Fig. 1.7).

1.5. MAGNETIC FIELDS DUE TO OCEAN TIDES Another source of magnetic signals originating in the oceans is the motionally induced currents. As the electrically conducting water in the oceans moves in the ambient magnetic ﬁeld of the Earth, it induces secondary electric and magnetic ﬁelds. In the last few years much attention has been given to the periodic magnetic signals caused by lunar tidal ocean ﬂows. For example, Tyler et al. (2003) demonstrated that the magnetic ﬁelds generated by the lunar semidiurnal M2 (of 12.42h period) ocean ﬂow can be clearly identiﬁed in magnetic satellite observations. They compared their numerical simulations of magnetic ﬁelds due to the M2 tide with CHAMP observations and found quite close agreement between the observations and predictions. Their conductivity model consists of a surface thin shell and an insulating mantle underneath. The discrepancy between observations and predictions have been addressed to the absence of a coupling between the surface shell and the mantle. Maus and Kuvshinov (2004) and Kuvshinov and Olsen (2005b) performed model studies and derived the magnetic signals of various tidal constituents in the presence of a conducting mantle. Note that for these ‘‘tidal’’ simulations the extraneous current jext in Equation (1.1) degenerates to the sheet

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Fig. 1.8. Depth-integrated velocities (in m2/s), U, of M2 tide. Right and left panels are real and imaginary parts of U, respectively. Maximum arrow length is 240 m2/s.

Fig. 1.9. Brm component of the main magnetic ﬁeld (in nT).

current density, Jext t , which is calculated as m Jext t ¼ sw ðU er Br Þ

ð1:5Þ

where U is the depth-integrated tidal velocity (transport) (see Fig. 1.8), taken from the TPXO6.1 global tidal model of Erofeeva and Egbert (2002), er is the outward unit vector and Bm r is the radial component of the main magnetic ﬁeld (see Fig. 1.9) derived from the model of Olsen (2002).

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Fig. 1.10. Z (in nT) due to M2 tide at sea level (upper panels) and at CHAMP altitude (lower panels). Right and left panels are real and imaginary parts of Z, respectively. (After Kuvshinov and Olsen, 2005b.)

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Fig. 1.11. Z (in nT) due to O1 tide at CHAMP altitude. Right and left panels are real and imaginary parts of Z, respectively. (After Kuvshinov and Olsen, 2005b.)

Fig. 1.10 presents global maps of the predicted amplitude of the vertical component of the magnetic ﬁelds due to M2 tide at sea level (upper panels) and at CHAMP altitude (lower panels). Fig. 1.11 shows O1 tide (25.82 h period) magnetic signal at CHAMP altitude. These two tides are chosen among others, since they are known as those mainly produced by the oceans. In accordance with the geometry of the exciting current (which, in particular, is governed by the vertical component of the main magnetic ﬁeld; cf. Fig. 1.9 and Equation (1.5)), the signals are negligible at the dip equator and increase towards the magnetic poles. Also, the maxima of the magnetic ﬁeld amplitudes follow those of the depth-integrated velocities, as expected. The largest amplitudes of the M2 tidal magnetic ﬁeld (about 5 nT at sea level) occur in the Indian Ocean, the western part of the South Paciﬁc Ocean, in the North Paciﬁc Ocean and in the North Atlantic Ocean. The magnitude at CHAMP altitude is decreased (down to 2 nT) and smoother compared to that at sea level. The magnetic signals of the O1 tide are at least three times smaller compared to M2 and have quite different geometry. The largest amplitudes are observed in the North Paciﬁc Ocean and in the region between Australia and Antarctica. The simulations of magnetic signals due to other tidal constituents (S2, N2, K2, K1, P1, Q1; not shown here) have revealed that their strength is also much smaller compared to M2. However, as demonstrated by Maus et al. (2006), accounting for the predicted results of all tidal constituents suppresses tidal noise in CHAMP magnetic observations. Finally, it is relevant to mention that, recently Kuvshinov et al. (2006b) performed simulations of tidal electric ﬁelds and demonstrated that the predictions are in good agreement with the observed electric tidal signals from sites in northern Germany and tidal voltage signals from the northern Paciﬁc Ocean cables.

1.6. MAGNETIC FIELDS DUE TO OCEAN CIRCULATION One more source of detectable magnetic signals above the Earth from the moving sea water is the global ocean circulation. Ocean circulation is driven by winds on the surface and density differences due to varying water temperature and salinity. Attempts to estimate ocean-induced electric or/and magnetic ﬁelds with realistic

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Fig. 1.12. Depth-integrated velocity (in m2/s) from ECCO ocean circulation model. Maximum arrow length is 490 m2/s. (After Manoj et al., 2006.)

ocean circulation models have been made in a number of papers (e.g. Stephenson and Bryan, 1992; Flosadottir et al., 1997; Tyler et al., 1997; Palshin et al., 1999; Vivier et al., 2004; Manoj et al., 2006). Note that there are evident differences in the source compared with tides. Firstly, we examine now the steady ﬂow (rather than periodic tides), and second, the ﬂow has a completely different velocity distribution. Fig. 1.12 presents the average depth-integrated velocity compiled by Manoj et al. (2006) from the ECCO ocean circulation model. The main feature here is a prominent Antarctic circumpolar current (ACC). Fig. 1.13 shows the dominant ocean-induced magnetic signal – vertical component – at sea level (left) and CHAMP altitude (right). The simulations are predominantly inﬂuenced by the ACC; the eastward ﬂowing ACC results in two prominent anomalies to the east and west of the southern geomagnetic pole (located in the South Australian Ocean). Both at sea level and CHAMP altitude the predictions show a relatively signiﬁcant contribution of the ocean circulation generated by magnetic ﬁelds signals with amplitude range of 6 and 2 nT, respectively. Note, however, that in contrast to tidal signals, the time-independent magnetic signals of the steady ﬂow is extremely difﬁcult to distinguish from the crustal magnetic ﬁeld (which in many regions has comparable amplitudes at satellite altitudes). But in analogy with the correction for tidal signals (cf. Maus et al., 2006), it is reasonable to correct/improve crustal ﬁeld anomaly maps by subtracting predicted signals due to realistic ocean circulation models from the magnetic satellite observations.

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Fig. 1.13. Z (in nT) due to global ocean circulation at sea level (left panel) at CHAMP altitude (right panel). (After Manoj et al., 2006.)

1.7. MAPPING CONDUCTIVITY ANOMALIES IN THE EARTH’S MANTLE FROM SPACE As mentioned in introduction, satellite-borne measurements provide a powerful source for improving our knowledge about 3-D variations of the electrical conductivity in the Earth’s mantle due to their good spatio-temporal coverage. However, low-orbit satellites move typically with a speed of 7–8 km/s and thus measure a mixture of temporal and spatial changes of the magnetic ﬁeld. This makes satellite data analysis more challenging compared to ground-based data. In spite of this problem several successful attempts have been made to derive the global conductivity-depth (1-D) distribution from magnetic satellite measurements (cf. Didwall, 1984; Oraevsky et al., 1993; Olsen, 1999; Olsen et al., 2002; Constable and Constable, 2004; Velimsky et al., 2006). Conversely, until now 3-D induction studies with satellite data are mostly conﬁned to simulating magnetic effects of conductivity anomalies at satellite altitudes (cf. Kuvshinov et al., 1998; Olsen, 1999; Tarits and Grammatica, 2000; Grammatica and Tarits, 2002; Everett et al., 2003; Velimsky et al., 2003; McCreadie and Martinec, 2005; Velimsky and Everett, 2005; Kuvshinov and Olsen, 2005a). Recently, Kuvshinov et al. (2006a) made a ﬁrst attempt to demonstrate that deep 3-D anomalies can be successfully mapped from space. Their analysis deals with a recovery of global maps of C-responses by a processing of realistic signals simulated in the frame of a closed-loop simulation of the Swarm multisatellite mission (cf. Olsen et al., 2006). The authors demonstrated that there exists principal possibility to detect 3-D mantle regional conductivity anomalies from satellite geomagnetic data. Shortly, a scheme of the recovery of global maps of C-responses can be explained as follows. Let magnetic signals due to magnetospheric sources, ðsÞ ðsÞ ðsÞ ðsÞ ðBrðsÞ ; BW ; BjðsÞ Þ at time instant ti and position ðri ; Wi ; ji Þ, be provided by a constellation of satellites. Here ti ¼ iDD ; i ¼ 1; 2; :::; N D ; s ¼ 1; 2; :::; N S , where DD is sampling interval, and N D and N S the number of samples and satellites,

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respectively. Assuming that no electric currents exist at satellite altitude, the magnetic ﬁeld can be derived from a scalar magnetic potential, B ¼ grad V , that is approximated by the spherical harmonic expansion r n i m Pm ðqm ðtÞ cos mj þ s ðtÞ sin mjÞ n n n ðcos WÞ a n¼1 m¼0 Ni X k akþ1 X l l ðgk ðtÞ cos lj þ hk ðtÞ sin ljÞ þa Plk ðcos WÞ r k¼1 l¼0

V ¼a

N X n h X

ð1:6Þ

l with a as the mean Earth’s radius, and Pm n , Pk as associated Legendre functions. This allows reconstructing time series (with some sampling interval, DC ) of the external and induced expansion coefﬁcients from magnetic signals using a leastsquare approach. Note that in 1-D Earth’s conductivity models external coefﬁcients induce internal coefﬁcient of the same degree n and order m. In the general case of a 3-D conductivity model, the external coefﬁcients produce a whole spectrum of internal coefﬁcients. Thus in the frame of 3-D conductivity models we have to l m l consider N i N . Once the external, qm n ðtÞ; sn ðtÞ and internal, gk ðtÞ; hk ðtÞ coefﬁcients have been determined, time series of Br ðtÞ and rH BH ðtÞ (with the same sampling interval DC ) are reconstructed on a regular grid at the surface of the Earth by spherical harmonic synthesis. Signal processing of Br and rH BH allows for an estimation of Cðo; r; W; jÞ using equation

Br ðo; r; W; jÞ ð1:7Þ rH BH ðo; r; W; jÞ n o @Bj @ðBW sin WÞ 1 þ (cf. Schmucker, 1985b) with rH BH ðo; r; W; jÞ ¼ r sin being the @j W @W horizontal divergence of the horizontal component, BH . For the validation of the approach, three years of realistic synthetic data at simulated orbits of the forthcoming Swarm constellation of three satellites have been used. We used a conductivity model that consists of a thin surface layer of realistic conductance and a 3-D mantle that incorporates, in particular, a hypothetic deep regional conductor of 1 S/m located between 400 and 700 km depths beneath the Paciﬁc Ocean plate (see Fig. 1.14). The regional conductor is embedded in a radially symmetric section consisting of a relatively resistive 400-km-thick layer of 0.004 S/m, a 300-km-thick transition layer of 0.04 S/m and an inner uniform sphere of 2 S/m. Fig. 1.15 compares global maps (on a mesh of 5 5 ) of real parts of the m reference (also called ‘‘true’’) C-responses (left) from a given time series qm n ðtÞ, sn ðtÞ l l and gk ðtÞ, hk ðtÞ (obtained in a course of Swarm synthetic data simulations) and Cresponses recovered directly from these data (right). The maps are presented for a period of 7.8 days. We used synthetic data sampled at every 1 min.The sampling interval of the resulting coefﬁcients, the length of the time series, and the number of internal coefﬁcients were chosen to be 12 h, three years (1999–2002) and N i ðN i þ 1Þ ¼ 99(N i ¼ 9). The anomalous behaviour of C-responses near the magnetic equator is due to the fact that Br as well as rH BH are close to zero here Cðo; r; W; jÞ ¼

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Fig. 1.14. Conductivity distribution (S/m) at depths from 400 km down to 700 km in the 3-D model considered in Section 1.9. (After Kuvshinov et al., 2006a.)

Fig. 1.15. Real parts of C-responses at period of 7.8 days, which were estimated from a given time series of inducing and induced coefﬁcients (left panel) and from Swarm synthetic satellite data (right panel). The geometry of deep-seated anomaly is shown by the red line. (After Kuvshinov et al., 2006a.)

(since source geometry is dominated by P01 in geomagnetic coordinates), which makes the estimation of C-responses unstable in this region. By comparing the results one can conclude that the recovered C-responses are consistent in geometry and amplitudes with the true C-responses. The key is the availability of simultaneous observations of the magnetic ﬁeld variation at different local times, i.e. a

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spatio-temporal coverage sufﬁcient to reproduce the magnetic ﬁelds induced by magnetospheric source. Finally, note that to make synthetic data as realistic as possible a rather sophisticated model of magnetospheric source is used. It is derived from an hour-by-hour spherical harmonic analysis of worldwide distributed observatory hourly mean values of the years 1997–2002 and includes the time series of m expansion coefﬁcients of the external potential, qm n ðtÞ, sn ðtÞ, with n; m 3.

1.8. CONCLUSIONS The model studies demonstrate that the 3-D induction (ocean) effect and motionally induced signals from the oceans contribute signiﬁcantly to the near-Earth magnetic ﬁeld. These effect and signals can be predicted with required accuracy and detail (both at ground and at satellite altitudes) by using 3-D numerical solution based on integral equation approach. It is remarkable that the predictions agree well with the observations. It is believed that incorporating the 3-D EM induction into the geomagnetic ﬁeld modeling schemes could improve the resulting models. Besides the model studies an approach to detect large-scale conductivity anomalies deeply embedded in the mantle by analysis of magnetic signals from low-Earth-orbiting satellites is discussed. The approach deals with recovery of long-period C-responses on a regular spatial grid. It is demonstrated that the global maps of C-responses can be successfully recovered from multi-satellite magnetic data. For this demonstration synthetic magnetic signals from a given realistic magnetospheric source and realistic 3-D conductivity model are used. Eventually, the C-responses recovered on a regularly spaced grid at a set of periods could serve as input for a rigourous 3-D inversion, yet to be developed. The ﬁnal remark is that the C-response approach described in the last section of the chapter is only one of the several possible ways to tackle the 3-D satellite induction problem. The complicated spatio-temporal characteristics of satellite data may favour the application of time-domain techniques (cf. Everett and Martinec, 2003; Martinec and McCreadie, 2004; Kuvshinov and Olsen, 2005d; Velimsky et al., 2006). Acknowledgments Most of the results presented in this review have been obtained in a close collaboration with Nils Olsen and Chandrasekharan Manoj. Author appreciates very much their contribution. This work has been supported in part by European Space Agency through contract No. 17263/03/NL/CB and by Russian Foundation for Basic Research under grant no. 06-05-64329-a.

REFERENCES Ashour, A.A., 1965. Electromagnetic induction in ﬁnite thin sheets. Quart. J. Mech. Appl. Math., 18: 73–86.

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Avdeev, D.B., Kuvshinov, A.V., Pankratov, O.V. and Newman, G.A., 2002. Threedimensional induction logging problems, Part I: An integral equation solution and model comparisons. Geophysics, 67: 413–426. Beamish, D., Hewson-Browne, R.C., Kendall, P.C., Malin, S.R.C. and Quinney, D.A., 1980. Induction in arbitrarily shaped oceans IV: Sq for a simple oceans. Geophys. J. R. Astron. Soc., 60: 435–443. Bullard, E.C. and Parker, R.L., 1970. Electromagnetic induction in the oceans. In: J. Maxwell (Ed.), The Sea – Ideas and Observations on Progress in the Study of the Seas, Vol. 4, Wiley, New York, pp. 695–730. Constable, S. and Constable, C., 2004. Observing geomagnetic induction in magnetic satellite measurements and associated implications for mantle conductivity. Geochem. Geophys. Geosystems, 5: doi:10.1029/2003GC000,634. Didwall, E.M., 1984. The electrical conductivity of the upper mantle as estimated from satellite magnetic ﬁeld data. J. Geophys. Res., 89: 537–542. Erofeeva, S. and Egbert, G., 2002. Efﬁcient inverse modeling of barotropic ocean tides. J. Oceanic Atmosph. Technol., 19: 183–204. Everett, M.E., Constable, S. and Constable, C.G., 2003. Effects of near-surface conductance on global satellite induction responses. Geophys. J. Int., 153: 277–286. Everett, M.E. and Martinec, Z., 2003. Spatiotemporal response of a conducting sphere under simulated geomagnetic storm conditions. Phys. Earth Planet Int., 138: 163–181. Everett, M.E. and Schultz, A., 1996. Geomagnetic induction in a heterogeneous sphere: Azimutally symmetric test computations and the response of an undulating 660-km discontinuity. J. Geophys. Res., 101: 2765–2783. Fainberg, E.B., Kuvshinov, A.V. and Singer, B.Sh., 1990a. Electromagnetic induction in a spherical earth with non-uniform oceans and continents in electric contact with the underlying medium – I. Theory, method and example. Geophys. J. Int., 102: 273–281. Fainberg, E.B., Kuvshinov, A.V. and Singer, B.Sh., 1990b. Electromagnetic induction in a spherical earth with non-uniform oceans and continents in electric contact with the underlying medium – II. Bimodal global geomagnetic sounding of the lithosphere. Geophys. J. Int., 102: 283–286. Flosadottir, A.H., Larsen, J.C. and Smith, J.T., 1997. Motional induction in North Atlantic circulation models. J. Geophys. Res., 102: 10353–10372. Friis-Christensen, E., Lu¨hr, H. and Hulot, G., 2006. Swarm: A constellation to study the Earth’s magnetic ﬁeld. Earth Planets Space, 58: 351–359. Grammatica, N. and Tarits, P., 2002. Contribution at satellite altitude of electromagnetically induced anomalies arising from a three-dimensional heterogeneously conducting Earth, using Sq as an inducing source ﬁeld. Geophys. J. Int., 151: 913–923. Hamano, Y., 2002. A new time-domain approach for the electromagnetic induction problem in a three-dimensional heterogeneous earth. Geophys. J. Int., 150: 753–769. Hewson-Brown, R.C. and Kendall, P.C., 1978. Some new ideas on induction in inﬁnitely-conducting oceans of arbitrary shapes. Geophys. J. R. Astron. Soc., 53: 431–444.

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Hobbs, B.A., 1981. A comparison of Sq analyses with model calculations. Geophys. J. R. Astron. Soc., 66: 435–447. Koyama, T., Shimizu, H. and Utada, H., 2002. Possible effects of lateral heterogeneity in the D’’ layer on electromagnetic variations of core origin. Phys. Earth Planet Int., 129: 99–116. Kuvshinov, A., Junge A. and Utada, H., 2006b. 3-D modelling the electric ﬁeld due to ocean tidal ﬂow and comparison with observations. Geophys. Res. Lett., 33: doi: 10.1029/2005GL025043. Kuvshinov, A. and Olsen, N., 2005a. Modelling the ocean effect of geomagnetic storms at ground and satellite altitude. In: Ch. Reigber, H. Luhr, P. Schwintzer and J. Wickert (Eds), Earth Observation with CHAMP. Results from Three Years in Orbit, Springer, Berlin, pp. 353–358. Kuvshinov, A. and Olsen, N., 2005b. 3-D modelling of the magnetic ﬁelds due to ocean tidal ﬂow. In: Ch. Reigber, H. Luhr, P. Schwintzer and J. Wickert (Eds), Earth Observation with CHAMP. Results from Three Years in Orbit, Springer, Berlin, pp. 359–366. Kuvshinov, A. and Olsen, N., 2005c. Satellite Induction Studies in the Presence of Induction in the Oceans, and Accounting for an Asymmetric Magnetospheric Ring Current. 10th IAGA Scientiﬁc Assembly, Toulouse, France. Kuvshinov, A. and Olsen, N., 2005d. Mantle conductivity obtained by 3-D inversion of magnetic satellte data – an approach and its validation. Geophys. Res. Abstr., 7: 08607. Kuvshinov, A., Sabaka, T. and Olsen, N., 2006a. 3-D electromagnetic induction studies using the Swarm constellation: Mapping conductivity anomalies in the Earth’s mantle. Earth Planets Space, 58: 417–429. Kuvshinov, A.V., Avdeev, D.B. and Pankratov, O.V., 1998. On deep sounding of a nonhomogeneous earth using satellite magnetic measurements. Phys. Solid Earth, 34: 326–331. Kuvshinov, A.V., Avdeev, D.B. and Pankratov, O.V., 1999. Global induction by Sq and Dst sources in the presence of oceans: Bimodal solutions for nonuniform spherical surface shells above radially symmetric Earth models in comparison to observations. Geophys. J. Int., 137: 630–650. Kuvshinov, A.V., Avdeev, D.B., Pankratov, O.V., Golyshev, S.A. and Olsen, N., 2002a. Modelling Electromagnetic Fields in 3-D Spherical Earth Using Fast Integral Equation Approach: Three-Dimensional Electromagnetics. Elsevier, Holland, pp. 43–54. Kuvshinov, A.V., Olsen, N., Avdeev D.B. and Pankratov, O.V., 2002b. Electromagnetic induction in the oceans and the anomalous behaviour of coastal C-responses for periods up to 20 days. Geophys. Res. Lett., 29(12): doi:10.1029/2001GL014409. Kuvshinov, A.V., Pankratov, O.V. and Singer, B.Sh., 1990. The effect of the oceans and sedimentary cover on global magnetovariational ﬁeld distribution. Pure Appl. Geophys., 134: 533–540. Kuvshinov, A.V., Utada, H., Avdeev, D.B. and Koyama, T., 2005. 3-D modelling and analysis of Dst C-responses in the North Paciﬁc Ocean region, revisited. Geophys. J. Int., 160: 505–526.

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Lanzerotti, L.J., Sayres, C.H., Medford, L.V., Kraus, J.S. and Maclennan, C.J., 1992. Earth potential over 4000 km between Hawaii and California. Geophys. Res. Lett., 19: 1177–1180. Laske, G. and Masters, G., 1997. A global digital map of sediment thickness. EOS Trans., AGU, 78: F483. Manoj, C., Kuvshinov, A.V., Maus, S. and Luhr, H., 2006. Ocean circulation generated magnetic signals. Earth Planets Space, 58: 429–439. Martinec, Z., 1999. Spectral-ﬁnite element approach to three-dimensional electromagnetic induction in a spherical Earth. Geophys. J. Int., 136: 229–250. Martinec, Z. and McCreadie, H., 2004. Electromagnetic induction modelling based on satellite magnetic vector data. Geophys. J. Int., 155: 33–34. Maus, S. and Kuvshinov, A., 2004. Ocean tidal signals in observatory and satellite magnetic measurements. Geophys. Res. Lett., 31: doi:10.1029/ 2004GL000634. Maus, S.M., Rother, M., Hemant, K., Stolle, C., Luhr, H., Kuvshinov, A. and Olsen, N., 2006. Earth’s lithospheric magnetic ﬁeld determined to spherical harmonic degree 90 from CHAMP satellite measurements. Geophys. J. Int., 164: 319–330. McCreadie, H. and Martinec, Z., 2005. Geomagnetic induction modeling based on CHAMP magnetic vector data. In: Ch. Reigber, H. Luhr, P. Schwintzer and J. Wickert (Eds), Earth Observation with CHAMP. Results from Three Years in Orbit, Springer, Berlin, pp. 335–341. Neubert, T., Mandea, M., Hulot, G., von Frese, R., Primdahl, F., Jørgensen, J.L., Friis-Christensen, E., Stauning, P., Olsen, N. and Risbo, T., 2001. Ørsted satellite captures high-precision geomagnetic ﬁeld data. EOS, 82(7): 81, 87, and 88. Olsen, N., 1999. Induction studies with satellite data. Surv. Geophys., 20: 39–340. Olsen, N., 2002. A model of the geomagnetic ﬁeld and its secular variation for epoch 2000 estimated from Orsted data. Geophys. J. Int., 149: 454–462. Olsen, N., Haagmans, R., Sabaka, T., Kuvshinov, A., Maus, S., Purucker, M., Rotter, M., Lesur, V. and Mandea, M., 2006. Swarm end-to-end mission simulator study: Separation of the various contributions to Earth’s magnetic ﬁeld using synthetic data. Earth Planets Space, 58: 359–371. Olsen, N. and Kuvshinov, A., 2004. Modelling the ocean effect of geomagnetic storms. Earth Planets Space, 56: 525–530. Olsen, N., Vennerstrom, S. and Friis-Christensen, E., 2002. Monitoring magnetospheric contributions using ground-based and satellite magnetic data. In: Ch. Reigber, H. Luhr and P. Schwintzer (Eds), First CHAMP Mission Results for Gravity, Magnetic and Atmospheric Studies, Springer, Berlin, pp. 245–250. Oraevsky, V.N., Rotanova, N.M., Bondar, T.N., Abramova, D.Yu. and Semenov, V.Yu., 1993. On the radial geoelectrical structure of the mid-mantle from magnetovariational sounding using Magsat data. J. Geomagn. Geoelectr., 45: 1415–1423. Palshin, N., Vanyan, L., Yegorov, I. and Lebedev, K., 1999. Electric ﬁeld induced by the global ocean circulation. Phys. Solid Earth, 35: 1028–1035.

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Pankratov, O.V., Kuvshinov, A.V. and Avdeev, D.B., 1997. High-performance three-dimensional electromagnetic modeling using modiﬁed Neumann series. Anisotropic case. J. Geomagn. Geoelectr., 49: 1541–1548. Reigber, C., Luhr, H. and Schwintzer, P., 2002. CHAMP mission status. Adv. Space Res., 30: 129–134. Rostoker, G., Friedrich, E. and Dobbs, M., 1997. Physics of magnetic storms. Geophys. Monogr. Se., 98, 149–160, AGU, Washington, DC. Sabaka, T., Olsen, N. and Purucker, M., 2004. Extending comprehensive models of the Earth’s magnetic ﬁeld with Orsted and CHAMP data. Geophys. J. Int., 159: 521–547. Schmucker, U., 1985a. Electrical properties of the Earth’s interior. Landolt-Bornstein, New Series, 5/2b, Springer, Berlin, pp. 370–395. Schmucker, U., 1985b. Magnetic and electric ﬁelds due to electromagnetic induction by external sources. Landolt-Bornstein, New Series, 5/2b, Springer, Berlin, pp. 100–125. Singer, B.Sh., 1995. Method for solution of Maxwell equations in non-uniform media. Geophys. J. Int., 120: 590–598. Stephenson, D. and Bryan, K., 1992. Large-scale electric and magnetic ﬁelds generated by the oceans. J. Geophys. Res., 97: 15467–15480. Takeda, M., 1991. Electric currents in the ocean induced by the geomagnetic Sq ﬁeld and their effect on the estimation of mantle conductivity. Geophys. J. Int., 104: 381–385. Takeda, M., 1993. Electric currents in the ocean induced by model Dst ﬁeld and their effects on the estimation of mantle conductivity. Geophys. J. Int., 114: 289–292. Tarits, P., 1994. Electromagnetic studies of global geodynamic processes. Surv. Geophy., 15: 209–238. Tarits, P. and Grammatica, N., 2000. Electromagnetic induction effects by the solar quiet magnetic ﬁeld at satellite altitude. Geophys. Res. Lett., 27: 4009–4012. Tyler, R., Maus, S. and Luhr, H., 2003. Satellite observations of magnetic ﬁelds due to ocean tidal ﬂow. Science, 299: 239–240. Tyler, R., Mysak, L.A. and Oberhuber, J., 1997. Electromagnetic ﬁelds generated by a 3-D global ocean circulation. J. Geophys. Res., 102: 5531–5551. Tyler, R.S., Vivier, F. and Li, S., 2004. Three-dimensional modelling of ocean electrodynamic using gauged potentials. Geophys. J. Int., 158: 847–887. Utada, H., Koyama, T., Shimizu, H. and Chave, A., 2003. A semi-global reference model for electrical conductivity in the mid-mantle beneath the north Paciﬁc region. Geophys. Res. Lett., 30(4): doi: 10.1029/2002GL016092. Uyeshima, M. and Schultz, A., 2000. Geoelectromagnetic induction in a heterogeneous sphere: A new 3-D forward solver using a staggered-grid integral formulation. Geophys. J. Int., 140: 636–650. Vanyan, L.L., Palshin, N.A. and Repin, I.A., 1995. Deep magnetotelluric sounding with the use of the Australia–New Zealand cable 2. Interpretation: Phy. Solid Earth, 31: 417–421.

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Velimsky, J. and Everett, M.E., 2005. Electromagnetic induction by Sq ionospheric currents in a heterogeneous Earth: Modelling using ground-based and satellite measurements. In: Ch. Reigber, H. Luhr, P. Schwintzer and J. Wickert (Eds), Earth Observation with CHAMP. Results from Three Years in Orbit, Springer, Berlin, pp. 341–347. Velimsky, J., Everett, M.E. and Martinec, Z., 2003. The transient Dst electromagnetic induction signal at satellite altitudes for a realistic 3-D electrical conductivity in the crust and mantle. Geophys. Res. Lett., 30: doi:10.1029/ 2002GL016,671. Velimsky, J. and Martinec, Z., 2005. Time-domain, spherical harmonic-ﬁnite element approach to transient three-dimensional geomagnetic induction in a spherical heterogeneous Earth. Geophys. J. Int., 161: 81–101. Velimsky, J., Martinec, Z. and Everett, M.E., 2006. Electrical conductivity in the Earth’s inferred from CHAMP satellite measurements – I. Data processing and 1-D inversion. Geophys. J. Int., 166: 529–542. Vivier, F., Maier-Reimer, E. and Tyler, R.H., 2004. Simulations of magnetic ﬁelds generated by the Antarctic Circumpolar Current at satellite altitude: Can geomagnetic measurements be used to monitor the ﬂow? Geophys. Res. Lett., 31: doi:10.1029/2004GL019804. Weiss, C.J. and Everett, M.E., 1998. Geomagnetic induction in a heterogeneous sphere: Fully three-dimensional test computations and the response of a realistic distribution of oceans and continents. Geophys. J. Int., 135: 650–662. Yoshimura, R. and Oshiman, N., 2002. Edge-based ﬁnite element approach to the simulation of geoelectromagnetic induction in a 3-D sphere. Geophys. Res. Lett., 29(2): doi:10.1029/2001GL014121. Zhang, S.-L., 1997. GPBi-CG: Generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems. SIAM J. Sci. Comput., 18: 537–551.

Chapter 2 Magnetovariational Method in Deep Geoelectrics M.N. Berdichevsky, V.I. Dmitriev, N.S. Golubtsova, N.A. Mershchikova and P.Yu. Pushkarev Moscow State University, Geological Department, Russia

2.1. INTRODUCTION Deep geoelectrics studies of the Earth’s crust and upper mantle include two methods: (1) the magnetotelluric (MT) method using the electric and magnetic ﬁelds and (2) the magnetovariational (MV) method using only the magnetic ﬁeld. Following a common practice, a leading part belongs to the MT method with impedance tensor Z^ and apparent resistivity ra (vertical stratiﬁcation of the medium, geoelectric zoning, mapping of underground topography, detection of conductive zones in the Earth crust and upper mantle, recognition of deep faults), ^ whereas the MV method with tipper vector W and horizontal magnetic tensor M helps in tracing of horizontal conductivity contrasts, localization of geoelectric structures, determination of their strike. Such a partition of MT and MV methods is reﬂected even in the MT nomenclature: if the MT studies are referred to as MT soundings, the MV studies are considered as MV profiling (Rokityansky, 1982). The MT–MV geoelectric complex is widely and rather successfully used throughout the world. It provides an unique information on the Earth’s interior (porosity, permeability, graphitization, sulﬁdizing, dehydration, melting, ﬂuid regime, ground-water mineralization, rheological characteristics, thermodynamic, and geodynamic processes). Corresponding author: e-mail: [email protected]

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40002-0

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The weak point of deep geoelectrics with MT priority is that inhomogeneities in the uppermost layers may severely distort the electric ﬁeld and consequently the impedance tensor along with the apparent resistivity. The distortions are of galvanic nature – they extend over the whole range of low frequencies causing static (‘‘conformal’’) shifts of the low-frequency branches of apparent resistivity curves. The near-surface inhomogeneities affect the apparent resistivities, no matter how low the frequency is. They spoil the information on the deep conductivity. There is a plethora of techniques for correcting these distortions. But all these techniques are fraught with information losses or even with subjective (sometimes erroneous) decisions resulting in false structures. We can considerably improve the MT–MV complex by realizing to the full extent the potentialities of the MV method. The generally recognized advantage of MV method is that with lowering frequencies the induced currents penetrate deeper and deeper into the Earth, so that their magnetic ﬁeld and consequently the tipper and magnetic tensor are less and less distorted by subsurface inhomogeneities and convey more and more information about buried inhomogeneities. This remarkable property of the magnetic ﬁeld gives us the chance to protect the deep geoelectric studies from the static-shift problem (no electric ﬁeld is measured). But excluding the electric ﬁeld, we face the problem of informativeness of the MV method. It is commonly supposed that ‘‘MV studies determine only horizontal conductivity gradients, while the vertical conductivity distribution is not resolved’’ (Simpson and Bahr, 2005). Is it true? The fallacy of this statement is clearly seen from Fig. 2.1,

0

Fig. 2.1. Illustrating the resolution of MT and MV soundings. Model parameters: r01 ¼ 100 O m; r1 ¼ 10 O m; w ¼ 8 km; h1 ¼ 1 km; r2 ¼ 10; 000 O m; h2 ¼ 24; 49; 99; 149 km; r3 ¼ 1 O m. Curve parameter: h ¼ h1+h2.

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which shows a two-dimensional model with an inclusion of higher conductivity in the upper layer resting on the resistive strata and conductive basement. The half width of the inclusion is 8 km.A depth to the conductive basement ranges from 25 to 150 km.Let us compare the longitudinal apparent resistivity curves rxy , (site O1 , y ¼ 29 km), with the real-tipper curves measured outside the inclusion ReW zy ¼ Re H a H y , measured z . at the same site O1 , and with the magnetictensor curves M yy 21 ¼ H ay H ny , measured inside the inclusion (site O2 ; y ¼ 0Þ: ReW zy and In the model under consideration the bell-shaped MV curves M yy 21, derived from the ratio between the vertical component of the anomalous magnetic ﬁeld to the horizontal component of the magnetic ﬁeld and from the ratio between the horizontal component of the anomalous magnetic ﬁeld to the horizontal component of the normal magnetic ﬁeld, resolve the vertical conductivity distribution no worse than the customary MT curves rxy . Generalizing these indications, we can say that the MV method reveals not only horizontal variations in the Earth’s conductivity but the vertical variations as well. Moreover, we can appeal to the uniqueness theorem proved by Dmitriev for 2-D tipper and 2-D horizontal magnetic tensor and state that the 2-D piecewise analytical distribution of conductivity is uniquely deﬁned by exact values of the tipper or the horizontal magnetic tensor given over all points of inﬁnitely long transverse proﬁle in the entire range of frequencies from 0 to N (Berdichevsky et al., 2003; Dmitriev and Berdichevsky, Chapter 7, this volume). The physical meaning of this unexpected result is rather simple. Naturally, the MV studies of horizontally homogeneous media with zero MV anomalies make no sense. But in the case of the horizontally inhomogeneous medium, the MV studies can be considered as an ordinary frequency soundings using the magnetic ﬁeld of excess currents distributed within a local horizontal inhomogeneity, which plays a role of the buried source. So, we have every reason to revise the traditional MT–MV complex and consider a new MV–MT complex, within which the MV method, as being tolerant to subsurface distortions, plays a leading part and gives a sound geoelectric basis for MT-detailed speciﬁcation. This approach goes back to the MT experiments that were performed in 1988–1990 in the Kirghiz Tien Shan mountains by geophysical teams of the Institute of High Temperatures, Russian Academy of Sciences (Trapeznikov et al., 1997; Berdichevsky and Dmitriev, 2002). These measurements were carried out at a proﬁle characterized by strong local and regional distortions of apparent resistivities that dramatically complicated the interpretation of resulting data. The situation has normalized only with MV soundings. Fig. 2.2 shows the real tippers, Re W zy , and the geoelectric model ﬁtting these observation data. The model contains an inhomogeneous crustal conductive layer (a depth interval of 25–55 km) and vertical conductive zones conﬁned to the known faults, the Nikolaev line (NL) and the Atbashi–Inylchik faults (AIF). The ﬁgure also presents the model reconstructed from seismic tomography data. The geoelectric model agrees remarkably well with the seismic model: low resistivities correlate with lower velocities. This correlation conﬁrms the validity of geoelectric reconstructions based on MV data. We see that MV soundings not only outline crustal conductive zones but also stratify the lithosphere.

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Fig. 2.2. Magnetovariational sounding in the Kyrgyz Tien Shan Mountains. (A) Plots of the real tipper along a proﬁle crossing the Kyrgyz Tien Shan. (B) The resistivity section from MV data (Trapeznikov et al., 1997): NL, Nikolaev line; AIF, Atbashi-Inylchik fault. The resistivity values in O m are given within blocks; the lower-resistivity crustal zone is shaded. (C) The velocity section from seismic tomography data (Roecker et al., 1993). Values of P wave velocities in km/s are given within blocks; the low-velocity crustal zone is shaded.

The advancement of the MV–MT complex with MV priority is facilitated by the emergence of programs combining MV and MT automatized inversions (Siripunvaraporn and Egbert, 2000; Nowozynski and Pushkarev, 2001; Varentsov, 2002). Our paper is devoted to strategy of integrated MV and MT inversions. We consider general questions of MV–MT complex, describe model experiments on synthetic data and present a new model of the Cascadian subduction zone constructed with MV priority.

2.2. ON INTEGRATED INTERPRETATION OF MV AND MT DATA The inverse problem of MV and MT soundings is unstable. An arbitrarily small error in the measurement data can give rise to an arbitrarily large error in the conductivity distribution. Such a problem is meaningful if we use a prior information and limit the parameters to be found so that an approximate solution of the

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inverse problem is sought within a compact set of plausible solutions forming an interpretation model. An interpretation model should reﬂect current notions and hypotheses as to the sediments, crust, and upper mantle. It can either smooth or emphasize geoelectric contrasts and incorporate inhomogeneous layers and local inclusions of higher or lower electric conductivity. An approximate solution of an inverse problem constrained by the interpretation model is chosen by criteria ensuring the agreement of the solution with the available a priori information and observations. The number of such criteria is deﬁned by the number of response functions in use (real and imaginary or amplitude and phase functions). If a few response functions are used in the inversion, the problem is referred to as multicriterion. The 2-D integrated interpretation of MV and MT data belongs to the class of multicriterion problems. The electric conductivity of the Earth can be determined from the TE mode with the response functions Re Wzy, Im Wzy, rk and jk (real and imaginary tippers, longitudinal apparent resistivities, and phases of longitudinal impedances) and from the TM mode with the response functions r? and j? (transverse apparent resistivities and phases of transverse impedances). These functions differ in sensitivity to target geoelectric structures and in stability with respect to subsurface distortions (Berdichevsky and Dmitriev, 2002). The TE mode is more sensitive to deep conducting structures and less sensitive to the resistance of the lithosphere, whereas the TM-mode is less sensitive to deep conducting structures and more sensitive to the resistance of the lithosphere. Also note that apparent resistivities over the entire range of low frequencies can be subject to strong static distortions due to local 3-D subsurface inhomogeneities (geoelectric noise), whereas low-frequency tippers and impedance phases are free from these distortions. An algorithm of the 2-D bimodal inversion should implement such a procedure that the used characteristics would support and complement each other: gaps arising in the inversion of one response function should be ﬁlled through the inversion of another. In inverting various characteristics, one should give priority to the most reliable elements of the model and suppress the least reliable ones. The following two approaches are possible in solving multicriterion inverse problems: (1) parallel (joint) inversion of all characteristics used and (2) successive (partial) inversions of each of the characteristics. The parallel inversion summarizes all inversion criteria related to various response functions. In the 2-D problem it reduces to the minimization of the Tikhonov’s functional ( ) M X 2 gm F m ðy; oÞ I m ðsÞ þ aOðsÞ ð2:1Þ inf p

m¼1

where the following notation is used: p, vector of the sought-for parameters; Fm, response function in use; y, coordinate of the observation point; o, frequency; Im, operator determining Fm from the known distribution of the conductivity s; gm, signiﬁcance coefﬁcient of the model misﬁt (deviation of Im from Fm); O, criterion of solution selection (stabilizer) adjusting the solution to a priori information;

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a, regularization parameter (the signiﬁcance coefﬁcient of the prior information); M, number of the response functions used. At ﬁrst glance, the parallel inversion seems to be the most effective because it incorporates all the speciﬁc features of the multicriterion problem together and signiﬁcantly simpliﬁes the work of the geophysicist. However, this approach is open to criticism. If various characteristics Fm have the same sensitivity to all parameters p(p1, p2 , y , ps) of the geoelectric structure and the same immunity to subsurface distortions, their parallel inversion is not very advantageous because only a single one, the most reliably determined, characteristic is sufﬁcient for a comprehensive inversion. The use of several characteristics Fm makes the inversion more informative if they differ signiﬁcantly in their sensitivity to various parameters of the geoelectric structure and in their immunity to distortions. However, in this case their joint inversion can become inconsistent, because they put different constraints on the geoelectric structure and are related to different criteria of model misﬁts and solution selection. Clearly they can interfere with one another. True enough, it is possible that in some cases a fortunate choice of weights allows one to construct a self-consistent model with a small overall misﬁt. However, the adequate selection of such weights is itself a complex problem that often cannot be solved as yet. Apparently, the SPI method (successive partial inversions) is the best approach to the solution of a multicriterion inverse problem. Let a response function Fm be the most sensitive to the vector of parameters p(m). Then, the partial m-th inversion of the multicriterion 2-D problem consists in the minimization of the following Tikhonov’s functional on the set of the parameters p(m), with other parameters being ﬁxed: n o 2 inf F m ðy; oÞ I m ðsÞ þ aOðsÞ

ð2:2Þ

pðmÞ

The successive application of the functions Fm, m ¼ 1, 2, y , M reduces the solution of the multicriterion problem to a succession of partial inversions. Each partial inversion is intended for the solution of a speciﬁc problem and can be restricted to speciﬁc structures. A decrease in the number of parameters minimizing the Tikhonov’s functional signiﬁcantly enhances the stability of the problem. Partial inversions comprehensively incorporate speciﬁc features of the response functions used, their informativeness, and their conﬁdence intervals. They allow the information exchange between various functions, enable a convenient interactive dialog, and are easily tested. We believe that this direction of research is most promising for further development of methods designed for the integrated interpretation of MV and MT data. The method of partial inversions is corroborated by results of studies carried out in various geological provinces (Trapeznikov et al., 1997; Berdichevsky et al., 1998, 1999; Pous et al., 2001; Vanyan et al., 2002). Below, we brieﬂy describe some model experiments elucidating the potentials of SPI with MV priority.

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2.3. MODEL EXPERIMENTS Fig. 2.3 displays a 2-D model schematically illustrating geoelectric structure of the Kyrgyz Tien Shan (Trapeznikov et al., 1997). This model, referred below as the TS model, includes (1) inhomogeneous sediments, (2) a inhomogeneous resisting crust, (3) a deep crustal layer with a resistivity increasing monotonically from 10 O m in the south to 300 O m in the north, (4) three conducting zones A–C branching from the crustal conducting layer, and (5) a poorly conducting mantle underlain by a conducting asthenosphere at a depth of 150 km.The model is excited by a vertically incident plane wave. The forward problem was solved with the use of the ﬁnite element method (Wannamaker et al., 1987). Gaussian white noise was added to the response functions: it had 5% standard deviations for longitudinal and transverse apparent resistivities rk and r? , 2.51 for phases of longitudinal and transverse impedances jk and j? , and 5% for real and imaginary parts of the tipper ReW zy and ImW zy . To simulate the static shift caused by small 3-D near-surface inhomogeneities, the apparent resistivities were multiplied by random real numbers uniformly distributed in the interval from 0.5 to 2. The integrated interpretation of the synthetic data obtained in the TS model was performed by the method of partial inversions. The construction of the interpretation model is the most important step of interpretation (Berdichevsky and Dmitriev, 2002). The interpretation model should meet the following two requirements: it should be informative (i.e., reﬂect the target

Fig. 2.3. The TS model. The resistivity values in O m are shown within blocks; blocks of lower crustal resistivities are shaded.

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layers and structures) and it should be simple (i.e., be determined by a small number of parameters ensuring the stability of the inverse problem). It is evident that these requirements are antagonistic: a more informative model is more complex. Therefore, an optimal model, both simple and informative enough, should be chosen. This is a key point of the interpretation, deﬁning the strategy and even, to an extent, the solution of the inverse problem. The choice of the interpretation model is restrained by a priori information, qualitative estimates, and reasonable hypotheses on the structure of the medium under study. Constructing the interpretation model for inversions of the synthetic data in the TS model, we assumed that the following a priori information on the studied medium was available: (1) the sedimentary cover is inhomogeneous, with an average thickness of 1 km, (2) the consolidated crust is inhomogeneous and can contain local conducting zones, its resistivity can experience regional variations, and an inhomogeneous conducting layer corresponding to the seismic waveguide can exist in its lower part at depth of 35–50 km, (3) the upper mantle consists of homogeneous layers, and its resistivity at depths below 200 km can amount to 20 O m, and (4) the area under observation is framed by asymmetric media which slowly vary with distance. To detail these assumptions, we inverted the tippers using a smoothing program capable of identifying and localizing crustal conductors. We applied the REBOCC code (Siripunvaraporn and Egbert, 2000) and use a homogeneous half-space with a resistivity of 100 O m as an initial approximation. Fig. 2.4 presents this trial model, resulting from the inversion of Re W zy and Im W zy . The model yields clear evidence of three local crustal conducting zones A–C (ro30 O m branching from the crustal conducting layer) but fails to stratify the crust and upper mantle.

Fig. 2.4. The trial model: Inversion of Re W zy and Im W zy using the REBOCC program; A–C are conducting zones in the crust (cf. Fig. 2.3).

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Fig. 2.5. The interpretation block model; starting values of resistivities in O m are shown within blocks.

The prior information complemented with data on local crustal conductors provides a reasonable basis for the construction of a block interpretation model. This model, presented in Fig. 2.5, consists of 70 blocks of a ﬁxed geometry and different starting resistivities. The partition density depends on the position and size of tentative structures and is highest within the sedimentary cover, local crustal conductors, and crustal conducting layer. Partial inversions of the synthetic data were performed in the class of block structures with the use of the II2DC program (Varentsov, 2002) in the following succession: (1) Re W zy and Im W zy inversion, (2) jk inversion, and (3) r? and j? inversion. All the inversions were carried out automatically. We consider each inversion separately. 1. Inversion of Re W zy and Im W zy . The starting model is shown in Fig. 2.5. The tipper inversion results in the TP model (Fig. 2.6), which agrees well with the initial TS model. The divergence between the tippers calculated from both models is generally no higher than 5–7% within the period range from 1 to 10,000 s. Using the MV data alone, we successfully reconstructed the most signiﬁcant elements of the initial model, including the inhomogeneous sedimentary cover; the local crustal conductors A–C; and the inhomogeneous crustal conducting layer whose resistivity varies from 234 O m in the north to 16 O m in the south (from 300 to 10 O m in the initial model). Also resolved was the contrast between the nonconductive and conductive mantle (1667 O m/109 O m in the TP model against 1000 O m/10 O m in the initial TS model). We see that the MV response functions measured on a 200 km proﬁle allowed us not only to detect the local conducting zones but also to determine the stratiﬁcation of the medium (with an accuracy sufﬁcient for obtaining gross petrophysical estimates). 2. Inversion of jk . At this step, without going beyond the TE mode, we can control the tipper inversion and gain additional constraints on the stratiﬁcation of the medium. A difﬁculty consists in the fact that the rk -curves of the longitudinal

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Fig. 2.6. The TP model: Inversion of Re W zy and Im W zy using the II2DC program; the resistivity values in O m are shown within blocks; blocks of lower crustal resistivities are shaded (cf. Fig. 2.3).

Fig. 2.7. The TE model: Inversion of jk using the II2DC program; the resistivity values in O m are shown within blocks; blocks of lower crustal resistivities are shaded (cf. Fig. 2.3).

apparent resistivity are distorted by subsurface 3-D inhomogeneities that create geoelectric noise. We avoided this difﬁculty by conﬁning ourselves to the inversion of the undistorted jk curves. If rk and jk are interrelated through dispersion relations, the disregard of the rk curves does not lead to a loss of information. We interpreted the jk curves using the TP model, obtained from the tipper inversion, as a starting model. Inversion of jk resulted in the TE model, shown in Fig. 2.7. The divergences between the phases from the TE model and initial TS model do not exceed 2.51. Comparing the TE and TP models, we see that the phase inversion agrees reasonably well with the tipper inversion. Two points are of particular interest: (1) the edge resistivities of the inhomogeneous crustal layer (343 and 10 O m) became closer to their true values (300 and 10 O m), and (2) the contrast between the

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nonconductive and conductive mantle became sharper (3801 O m/15 O m in the TE model against 1000 O m/10 O m in the initial TS model). Thus, the phase inversion visibly improved the accuracy of the medium stratiﬁcation. 3. Inversion of r? and j?. This inversion is sensitive to galvanic effects. It is focused on estimating the resistivity of the upper highly resistive crust. The TE model, obtained from the jk inversion, was used as a starting model. Here, we ﬁxed all resistivities except for blocks that contact the sedimentary cover. The inversion of r? and j? yielded the TM model, shown in Fig. 2.8. It conﬁrms the galvanic connection between the conductive zone B and sediments, and reveals the asymmetry of the highly resistive upper crust whose resistivity changes from 283,000 O m in the north to 13 000 O m in the south (in the initial TS model, from 100 000 O m in the north to 10 000 O m in the south). The TM model is the ﬁnal model obtained from the successively applied automatic partial inversions. Its agreement with the initial TS model is evident. All of the major TS structures are well resolved in the TM model. Misﬁts between these models do not exceed 5–7% in tippers and 2.51 in phases. For comparison, Fig. 2.9 presents the PI model, obtained by the parallel (joint) inversion of all response functions (Re W zy , Im W zy , jk , r? and j?) used in constructing the TM model (the starting model was the same as in the tipper inversion). In the PI model: (1) resistivity contrasts in the sedimentary cover are signiﬁcantly smoothed, (2) the resistivity contrast in the upper, highly resistive crust is also signiﬁcantly smoothed, (3) the conducting zones A and C are resolved with some degree of certainty, but the central through-the-crust conducting zone B is completely destroyed, (4) the contrast between the two edge resistivities in the crustal conducting layer is much lower, and (5) the monotonic decrease of the mantle resistivity is disturbed (a poorly conducting layer appears in the conducting mantle). We see that the parallel inversion of all response function used impairs the interpretation result.

Fig. 2.8. The TM model: Inversion of r? and j? using the II2DC program; the resistivity values in O m are shown within blocks; blocks of lower crustal resistivities are shaded (cf. Fig. 2.3).

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Fig. 2.9. The PI model: Parallel inversion of Re W zy , Im W zy , jk , r? and j? using the II2DC program; the resistivity values in O m are shown within blocks; blocks of lower crustal resistivities are shaded (cf. Fig. 2.3).

Of course, the parallel inversion is the simplest approach to a multicriterion problem, and apparently this is the reason why it is popular among geophysicists fascinated by the possibility of automatic inversions eliminating the necessity of comprehensive analysis. The transition to the technique of SPI undoubtedly complicates the work, and this is a possible reason for the objections raised in the discussions. However, our experiments on the integrated interpretation of MV and MT data indicate that the game, albeit more difﬁcult, is worth the candle.

2.4. MV–MT STUDY OF THE CASCADIAN SUBDUCTION ZONE (EMSLAB EXPERIMENT) The above scheme of SPI of MV and MT data was applied in constructing the geoelectric model for the Cascadian subduction zone (Wannamaker et al., 1989a; Vanyan et al., 2002). We used data obtained in 1986–1988 by geophysicists from the United States, Canada, and Mexico on the Paciﬁc North American coast within the framework of the experiment ‘‘electromagnetic study of the lithosphere and beneath’’ (EMSLAB). Fig. 2.10 presents a predictive petrological and geothermal model of the Cascadian subduction zone along an E–W proﬁle generalizing modern ideas and hypotheses on the structure of the region and its ﬂuid regime (Romanyuk et al., 2001). The subducting Juan de Fuca plate originates at an offshore spreading ridge (about 500 km from the coast). In the eastward direction, the proﬁle crosses (1) an abyssal basin with a sedimentary cover 1–2 km thick and a pillow lava layer 1.5–2 km thick; (2) the Coast Range, formed by volcanic-sedimentary rocks; (3) the Willamette River valley, ﬁlled with a thick sequence of sediments and basaltic intrusions; (4) the Western (older) and Eastern (younger) Cascade ranges, consisting of volcanic and

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39 Fig. 2.10. Predictive geothermal and petrological CASCADIA model constructed along an E–W proﬁle across central Oregon (Romanyuk et al., 2001).

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volcanic-sedimentary rocks typical of a recent active volcanic arc; and (5) the Deschutes Plateau, covered with lavas. The abyssal basin is characterized by a typical oceanic section with the asthenosphere at a depth of about 40 km (the 9001C isotherm). The continental crust above the subducting slab has lower temperatures. A subvertical zone of higher temperatures reaching the melting point of wet peridotite (9001C) has been localized beneath the high Cascades. The release of ﬂuids from the upper part of the slab appears to be due to a few mechanisms. First, at depth to about 30 km, free water is released from micropores and microfractures under the action of the increasing lithostatic pressure. Dehydration of minerals such as talc, serpentine, and chlorite starts at depths of 30–50 km, where the temperature exceeds 4001C. Finally, the basalt–eclogite transition can start at depth greater than 75 km, and exsolution of amphibolites can take place at depths of more than 90 km.All these processes are accompanied by the release of ﬂuids. Supposedly, ﬂuids released at small depths migrate through the contact zone between the oceanic and continental plates. At greater depths, ﬂuids can be absorbed by mantle peridotites (serpentinization). They disturb the equilibrium state of material and cause ‘‘wet’’ melting. The melts migrate upward toward the Earth’s surface, producing a volcanic arc. Two 2-D geoelectric models of the Cascadian subduction zone constructed along the Lincoln line (an E–W proﬁle in the middle part of Oregon) have been discussed in the literature: EMSLAB-I (Wannamaker et al., 1989b) and EMSLABII (Varentsov et al., 1996). The EMSLAB-I model, shown in Fig. 2.11, was constructed by a trial-and-error method with a strong priority given to the TM mode (the latter, in the opinion of the authors of this model, is least subjected to 3-D distortions). The EMSLAB-I model minimizes the misﬁts of the curves r? and j? and ignores the curves rk and jk . Its main elements are (1) the upper conductive part of the plate, sinking at a low angle beneath the Coast Range, (2) a sub-horizontal conducting layer in the middle continental crust broadening in the area of the High Cascades, and (3) a welldeveloped conductive asthenosphere beneath the ocean. The problem of the junction between the slab and the crustal conductor remains open in this model. The continental asthenosphere is ignored, although the shape of the experimental curves rjj and jjj suggested a low resistivity of the upper mantle. The absence of catching the eye divergences between the model values of Re Wzy and Im Wzy and the experimental data is considered by the authors as evidence of the reliability of the model. The EMSLAB-I model is vulnerable to criticism. A cold continental mantle contradicts current geodynamic ideas of the Cascadian subduction zone (compare the EMSLAB-I model with the predictive CASCADIA model shown in Fig. 2.10). Analysis of the EMSLAB-I model has shown that the TM mode is weakly sensitive to variations in the electrical conductivity of the mantle and that the bimodal inversion alone, using both the TE and TM modes, can be effective in studying the asthenosphere (Vanyan et al., 2002). Experiments on the bimodal interpretation of MT and MV data obtained in the Cascadian subduction zone resulted in the 2-D EMSLAB-II model (Fig. 2.12). It

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Fig. 2.11. Geoelectric 2-D model EMSLAB-I of the Cascadian subduction zone (Wannamaker et al., 1989b); CB, Cascadia basin; NB, Newport basin; CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

Fig. 2.12. Geoelectric 2-D model EMSLAB-II of the Cascadian subduction zone (Varentsov et al., 1996); CB, Cascadia basin; NB, Newport basin; CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

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was constructed with the automatic inversion program INV2D-FG, optimizing resistivities on 20 blocks of a ﬁxed geometry (Varentsov et al., 1996). An algorithm of parallel weighted inversion was applied to j?, Re W zy , and Im W zy (maximum weight), jk and j? (normal weight), and rk (minimum weight). The EMSLAB-II model has much in common with EMSLAB-I. Both have the same oceanic asthenosphere, subducting slab, and crustal conducting layer. However, the EMSLAB-II subducting plate joins the crustal conductor, and a conducting asthenosphere is present in the continental mantle. Thus, the geoelectric data have revealed partial melting in the continental mantle. The main drawback of the EMSLAB-II model is its sketchiness due to the limited possibilities of the INV2D-FG program. Presently, the INV2D-FG program has given way to more efﬁcient software tools designed for the automatic 2-D inversion of MV and MT data. These are the smoothing program REBOCC (Siripunvaraporn and Egbert, 2000) the programs IGF-MT2D (Nowozynski and Pushkarev, 2001) and II2DC (Varentsov, 2002). These programs enable the optimization of models containing 512 and more blocks of a ﬁxed geometry and provide new possibilities for interpreting the EMSLAB experimental data (Vanyan et al., 2002). Three-dimensional model estimates obtained for the Paciﬁc coast of North America and analysis of experimental data, induction arrows, and polar diagrams show that the regional structure along the Lincoln line is favorable for the 2-D interpretation of MV and MT data. The interpretation consisted of three stages. At the ﬁrst stage, the 1-D inversion of short-period MT curves (T ¼ 0,01–100 s) was performed and an approximate geoelectric section of the continental volcanicsedimentary cover was constructed to a depth of 3.5 km.This section agrees with the near-surface portion of the EMSLAB-I model (Wannamaker et al., 1989b). At the second stage, the REBOCC program was applied for the 2-D smoothed trial inversion. With the complicated conditions of the Cascadian subduction zone, the parallel inversion of the TE and TM modes yielded whimsical alternation of low- and high-resistivity spots with a poor minimization of the misﬁt. It is difﬁcult to recognize real structures of the subduction zone in these spots. The most interesting result was obtained from the partial inversion of Re W zy , Im W zy , and jk (Fig. 2.13). Here, the western and eastern conducting zones are separated by a Tshaped region of higher resistivity that can be associated with the subducting slab. An oceanic asthenosphere whose top can be ﬁxed at a depth of about 30 km is recognizable in the western conducting zone. The eastern conducting zone coincides with the crust-mantle zone of wet melting in the predictive CASCADIA model shown in Fig. 2.10. It is noteworthy that the upper boundary of the eastern conductor closely resembles the topography of the crustal conducting zone in the EMSLAB-I and EMSLAB-II models shown in Fig. 2.11. At the third, ﬁnal stage, the method of SPI was applied and a new 2-D geoelectric model of the Cascadian subduction zone was constructed (Vanyan et al., 2002). This model was called EMSLAB-III. It was constructed with the II2DC program (Varentsov, 2002) minimizing the model misﬁt in the class of media with a

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Fig. 2.13. The trial model of the Cascadian subduction zone; 2-D inversion of Re W zy , Im W zy , jk with the use of the smoothing REBOCC program; CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

ﬁxed geometry of blocks. The interpretation was conducted in the regime of testing hypotheses. We consider three hypothetic models of the Cascadian subduction zone: (1) CASCADIA predictive model, (2) EMSLAB-I model, and (3) EMSLABII model. The interpretation model is shown in Fig. 2.14. The ocean ﬂoor topography and thickness of the seaﬂoor and shelf sediments were taken according to bathymetric and sedimentary thickness maps. The resistivities of the water, sediments, and oceanic crust are 0,3, 2, and 10,000 O m, respectively. The depth to the oceanic mantle and its resistivities were chosen in accordance with the CASCADIA, EMSLAB-I, and EMSLAB-II models. The slab surface was determined from seismic and seismic tomography data. The structure of the volcanic-sedimentary cover was speciﬁed from the 1-D inversion of short-period MT curves. The crust and mantle of the continent were divided into homogeneous blocks. The division density and block geometry were chosen so that they admit a free choice of crust and mantle structures within the framework of the three hypotheses considered. A hypothesis best ﬁtting the observed data can be chosen automatically in the process of misﬁt minimization. The continental crust and mantle have a resistivity of 1000 O m in the START model constructed on the basis of the interpretation model. Below, we consider the SPI. 1. Inversion of Re Wzy and Im Wzy. The START model was taken as the starting one. The TP model resulting from the inversion is shown in Fig. 2.15. The tipper misﬁt (the RMS deviation of model tippers from observed values) in this model is

44 M.N. Berdichevsky et al./Magnetovariational Method in Deep Geoelectrics Fig. 2.14. The interpretation block model; resistivity values of the initial START model are shown within blocks; CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

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Fig. 2.15. The TP model of the Cascadian subduction zone; 2-D inversion of Re W zy and Im W zy with the use of the block program II2DC (resistivity values in O m are shown within blocks); CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

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5–10 times smaller than the tipper amplitude (the difference between the maximum and minimum tipper values), which is an evidence of good agreement between the model and observations. A remarkable feature of the TP model is the conducting continental asthenosphere and the vertical low-resistivity zone that branches off the asthenosphere and crosses the continental crust in the High Cascade region. This feature distinguishes the TP model from EMSLAB-I and EMSLAB-II models and makes it similar to the predictive CASCADIA model, where a vertical high-temperature zone of wet and dry melting is evidently characterized by low resistivities. 2. Inversion of uJ. At this stage, we controlled the tipper inversion. To avoid difﬁculties associated with subsurface distortions of the curves rjj , we conﬁned ourselves to the inversion of the curves jk , which satisfy the dispersion relations. The TP model, obtained from the tipper inversion, was used as a starting model. The inversion of longitudinal phases yielded the TE model, shown in Fig. 2.16. The phase misﬁt (the RMS deviation of model phases from observed values) in this model is 5–10 times smaller than the phase amplitude (the difference between the maximum and minimum phase values), indicating good agreement of the model with observations. As distinct from the TP model, the TE continental crust includes a better delineated conducting layer (r ¼ 14–46 O m) in a depth interval of 35–45 km, whereas the subvertical conducting zone (r ¼ 12–46 O m) in a depth interval of 45–110 km, bounded by layers with resistivities of 147–1260 O m to the west and 215–612 O m to the east, is localized with a higher contrast. The TE model can be considered as an update of the TP model. 3. Inversion of q? and u?. At this stage, we inverted the TM mode, which is less sensitive to conducting zones in the crust and mantle but is more effective in resolving the structure of the junction zone between the slab and crustal conducting layer. Besides, it provides more reliable estimates of the resistivity in the upper consolidated crust. In inverting the TM mode, the TE model, obtained from the inversion of phases jk , was taken as a starting model. The inversion of transverse apparent resistivities and phases of the transverse impedance yielded the TM model shown in Fig. 2.17. In this model, the misﬁts of transverse apparent resistivities at most points vary within 6–12%, and the phase misﬁts are 7–10 times smaller than the phase amplitude (the difference between the maximum and minimum phase values). The TM model inherits the main features of the starting TE model (albeit with some deviations). The following implications of the TM model are noteworthy. First, no well-conducting junction is present between the conducting slab and the crustal conducting layer. Second, the upper consolidated crust of the continent has a resistivity of about 2000 O m, indicating that it is fractured. Synthesis. At this stage, we analyzed the TP, TE, and TM models and constructed the generalized EMSLAB-III model, smoothing insigniﬁcant details and enlarging blocks. All changes were made interactively with the calculation of local misﬁts and the correction of boundaries and resistivities. The resulting model shown in Fig. 2.18 provides a coherent geoelectric image of the subduction zone. The extent of its agreement with observed data is seen from Fig. 2.19, where the model curves r?, rjj , j?, jk , Re W zy , and Im W zy are compared with the observed curves (the static distortion in the observed rjj curves was removed by a vertical shift of

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Fig. 2.16. The TE model of the Cascadian subduction zone; 2-D inversion of jk with the use of the block program II2DC (resistivity values in O m are shown within blocks); CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

48 M.N. Berdichevsky et al./Magnetovariational Method in Deep Geoelectrics Fig. 2.17. The TM model of the Cascadian subduction zone; 2-D inversion of r? and j? with the use of the block program II2DC (resistivity values in O m are shown within blocks); CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

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Fig. 2.18. The ﬁnal EMSLAB-III model (resistivity values in O m are shown within blocks); ICV, soundings on the ocean ﬂoor; 1C15, soundings on the continent; CR, Coast Range; WV, Willamette Valley; WC, Western Cascades; HC, High Cascades; DP, Deschutes Plateau.

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Fig. 2.19. Comparison of the observed MT and MV curves with the curves calculated from the EMSLAB-III model: (1) - observations, (2) - EMSLAB-III model.

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their low-frequency branches). The model curves agree well with the observed curves at the majority of sites. In its oceanic part, the EMSLAB-III model is close to EMSLAB-I and EMSLAB-II models and exhibits a thick oceanic asthenosphere in a depth interval of 37.5–110 km. The structure of the continental part of EMSLAB-III is distinguished by the following signiﬁcant elements: (1) a crustal conducting layer (r ¼ 20 O m, a depth interval of 25–40 km) and a conducting asthenosphere (r ¼ 30 O m, a depth interval of 100–155 km) are distinctly resolved, (2) crustal and asthenospheric conductors are connected by a column-like conducting body (r ¼ 20–30 O m) crossing the lithosphere and reaching depths of about 7 km in the volcanic zone of the High Cascades, (3) a subducting slab, in a depth interval of 4–40 km contains a thin inclined conductor (r ¼ 20 O m) separated from the crustal conducting layer by a higher-resistivity zone (r ¼ 60 O m); apparently, the crustal conducting layer has a deep origin. The reliability of these elements is supported by the fact that the elimination of any of them noticeably increases the model misﬁts. These features of the continental section make the EMSLAB-III and predictive CASCADIA models similar. The ﬂuid regime of the subduction can be clearly observed here. The subducting slab entraps ﬂuid-saturated low-resistivity rocks of the ocean ﬂoor. As the slab moves down, the released free water migrates through the shear zone (the contact zone between the subducting oceanic and stable continental plates). The dehydration (the release of bound water) developing in the slab at depths of 30–40 km supplies ﬂuids to the mantle and causes the wet melting of asthenospheric material. The low-resistivity melts move upward through the lithosphere and form a volcanic arc. The heating of the lithosphere activates dehydration in the lower crust, producing the crustal conducting layer. Thus, using the MV–MT complex with MV priority, we managed to construct a meaningful geodynamic model of the Cascadian subduction and successfully complete the EMSLAB experiment. It seems that the development of the MV method should be regarded as a promising task of modern geophysics.

Acknowledgements We are grateful to P. Weidelt and U. Schmucker for discussions stimulating this work. The work was supported by the Russian Foundation for Basic Research, projects 05-01-00244 and 05-05-65082.

REFERENCES Berdichevsky, M.N. and Dmitriev, V.I., 2002. Magnetotellurics in the Context of the Theory of Ill-Posed Problems. SEG monograph, Tulsa, 215pp.

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Berdichevsky, M.N., Dmitriev, V.I., Golubtsova, N.S., Mershchikova, N.A. and Pushkarev, P.Yu., 2003. Magnetovariational sounding: New possibilities. Izvestiya, Phys. Solid Earth, 39, 9: 701–727. Berdichevsky, M.N., Dmitriev, V.I. and Pozdnjakova, E.E., 1998. On two-dimensional interpretation of magnetotelluric soundings. Geophys. J. Int., 133: 585–606. Berdichevsky, M.N., Vanyan, L.L. and Koshurnikov, A.V., 1999. Magnetotelluric sounding in the Baikal Rift Zone. Izvestiya, Phys. Solid Earth, 35, 10: 793–814. Nowozynski, K. and Pushkarev, P.Yu., 2001. The efﬁciency analysis of programs for two-dimensional inversion of magnetotelluric data. Izvestiya, Phys. Solid Earth, 37: 503–516. Pous, J., Queralt, P. and Marcuello, A., 2001. Magnetotelluric signature of the Western Cantabrian Mountains: Geophys. Res. Lett., 28, 9: 1795–1798. Roecker, S.W., Sabitova, T.M., Vinnik, L.P., Burmakov, Y.A., Golovanov, M.I., Mamatkhanova, R. and Munirova, L., 1993. Three-dimensional elastic wave velocity structure of Western and Central Tien Shan. J. Geophys. Res., 98, B9: 15579–15795. Rokityansky, I.I., 1982. Geoelectromagnetic Investigations of the Earth’s Crust and Mantle. Springer, Berlin, 381pp. Romanyuk, T.V., Mooney, W.D. and Blakely, R.J., 2001. A tectonic-geophysical model of the Cascadian subduction zone in North America. Geotektonika, 3: 88–110. Simpson, F. and Bahr, K., 2005. Practical Magnetotellurics. Cambridge University Press, Cambridge. Siripunvaraporn, W. and Egbert, G., 2000. An efﬁcient data subspace inversion method for 2-D magnetotelluric data. Geophysics, 65: 791–803. Trapeznikov, Ju.A., Andreeva, E.V., Batalev, V.Ju., Berdichevsky, M.N., Vanyan, L.L., Volykhin, A.M., Golubtsova, N.S. and Rybin, A.K., 1997. Magnetotelluric soundings in the mountains of the Kirghyz Tien-Shan. Izvestiya, Phys. Solid Earth, 1: 3–20. Vanyan, L.L., Berdichevsky, M.N., Pushkarev, P.Yu. and Romanyuk, T.V., 2002. A geoelectric model of the Cascadia subduction zone. Izvestiya, Phys. Solid Earth, 38, 10: 816–845. Varentsov, I.M., 2002. A general approach to the magnetotelluric data inversion in a piecewise-continuous medium. Izvestiya, Phys. Solid Earth, 38, 11: 913–934. Varentsov, I.M., Golubev, N.G., Gordienko, V.V. and Sokolova, E.Yu., 1996. Study of the deep geoelectric structure along the Lincoln Line (EMSLAB Experiment). Izvestiya, Phys. Solid Earth, 4: 124–144. Wannamaker, P.E., Booker, J.R., Filloux, J.H., Jones, A.G., Jiracek, G.R., Chave, A.D., Tarits, P., Waff, H.S., Young, C.T., Stodt, J.A., Martinez, M., Law, L.K., Yukutake, T., Segava, J.S., White, A. and Green, A.W., 1989a. Magnetotelluric observations across the Juan de Fuca subduction system in the EMSLAB project. J. Geophys. Res., 94, B10: 14111–14125.

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Wannamaker, P.E., Booker, J.R., Jones, A.G., Chave, A.D., Filloux, J.H., Waff, H.S. and Law, L.K., 1989b. Resistivity cross-section through the Juan de Fuca subduction system and its tectonic implication. J. Geophys. Res., 94, B10: 14127–14144. Wannamaker, P.E., Stodt, J.A. and Rijo, L., 1987. A stable ﬁnite element solution for two-dimensional magnetotelluric modeling. Geophys. J. R. Astron. Soc., 88: 277–296.

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Chapter 3 Shallow Investigations by TEM-FAST Technique: Methodology and Examples P.O. Barsukov, E.B. Fainberg and E.O. Khabensky Geoelectromagnetic Research Center IPE RAS, Troitsk, Moscow Region, Russia

3.1. INTRODUCTION In recent years, geophysical methods became widely used by geologists, engineers, hydrogeologists, and other specialists in geotechnical, hydrogeological, and archaeological studies, in for environment monitoring and other purposes. High ranking among these studies is shallow depth investigation. Generally speaking, contemporary geoelectrics provides tools for studying the electric conductivity distribution from the Earth’s surface down to the depth of hundreds of kilometers. Within the depth interval down to 10 m, Ground penetration radar (GPR) works quite well; the direct current (DC) methods of electrical prospecting (VES) are effective in the depth range of 30–50 m. The depth interval from 500 m to a few hundred kilometers is studied mostly by magnetotelluric methods. The depth of interest for our study lies within an interval between 5–10 m and a few hundred meters. This is the region where human activity is found, and where an emphasis of electric prospecting is laid. Electric prospecting methods are usually divided into those employing either the DC or alternating current (AC). The advantage of DC methods is their high sensitivity to local geoelectric inhomogeneities, especially high-resistivity ones. However, this advantage becomes a shortcoming if the survey is intended to study layers and objects overlaid by near-surface inhomogeneities and highly resistive layers. In such cases the efﬁciency of DC methods drops, and analysis and interpretation of Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40003-2

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the obtained results become considerably more difﬁcult. Methods based on the use of AC relates to the electromagnetic class of methods.

3.2. ADVANTAGES OF TEM IN SHALLOW DEPTH STUDIES Among the AC methods, the most popular ones are magnetotelluric sounding (MTS), frequency sounding (FS), and the transient electromagnetic method (TEM) and its modiﬁcations. MTS methods employs ﬁelds of natural origin, and they are indispensable in studying sections at depths from 500-600 meters to several hundred kilometers. FS methods are useful in relatively shallow depth studies. MTS and FS methods are based on the determination of electric conductivity functions in a frequency domain. TEM methods work in the time domain, which provides a higher resolution compared to the frequency sounding methods, although the EM ﬁelds in these methods are deﬁned by the diffusion equations. To compare the resolution of MTS and TEM methods, Fig. 3.1 presents the apparent resistivity curves calculated for the same multilayer section (parameters of the section are given in the right-hand part of the ﬁgure). As seen from the ﬁgure, TEM curves are more sensitive to the parameters of the section, and approach their asymptotes earlier. The enhanced resolving power of TEM is due to the speciﬁcity of the method – signal measurements are carried out at those moments when the current in the transmitter’s antenna is switched off and there is no primary ﬁeld. In the practice of electric prospecting, various TEM conﬁgurations are used. They differ in the shapes and relative position of transmitting and receiving antennae, pulse shapes, etc. In cases when electric dipole are used as a transmitting (TR) or receiving (R) antenna, the inﬂuence of local inhomogeneities in electric conductivity that are ever persistent in the Earth results in the occurrence of a

Fig. 3.1. Comparison of the resolving power of MTS and TEM methods.

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galvanic ﬁeld mode with characteristic slow spatial decay. This mode can considerably overstructure the ﬁeld pattern in both space and time, and complicate the interpretation of measured results. Therefore, the facilities with antennae conﬁgured as transmitting and receiving loops are preferable at surveying of conductive structures.

3.3. ON THE TEM-FAST TECHNOLOGY In the TEM-FAST technology, a single square or rectangular loop is used as the TR and R antennae (the so-called single-loop TEM conﬁguration). This enables three objectives to be reached at the same time: ﬁrst, time variations of the magnetic ﬂux through the loop are measured, and the effects of subsurface irregularities in the antenna vicinity are integrated maximally; second, efﬁciency of the ﬁeld experiment grows signiﬁcantly since a second loop placement or induction receiver installation is no longer necessary; and third, the measured transient process shows a series of unique helpful peculiarities used in performing the transformations. The most important characteristic of any TEM equipment is the duration of the self-transient process, that is, of the process dependent on the transmitter–antenna–receiver system properties. It is obvious that proper quality TEM signal measurements are feasible only within the time interval where the self-process of the system is missing. The shorter the self-process is, the earlier are the time delays when measurements become feasible, and the minimum depth of study grows less. The voltage and the current in a loop at small times depend on the loop size, its resistance, capacitance, and inductance, and show quite a complicated behavior. Figs. 3.2, 3.3 give examples of the normalized voltage and current behavior in the TEM-FAST 48HPC antenna. Signal measurement and processing is carried out via a portable computer that monitors the equipment condition, operation process, and data quality, and, in

Fig. 3.2. Voltage in a loop after switching-off.

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Fig. 3.3. Current in a loop after switching-off.

Fig. 3.4. Control and analysis of EM noise.

accordance with a special code, carries out signal stacking to provide the best signal-to-noise ratio. Simultaneously with measurements of the medium response at each time lag, electromagnetic noise is also measured (see Fig. 3.4). Information about the noise parameters is important in data averaging (ﬁltering) and in the inverse problem solution. Reconstruction of the section from the apparent resistivity curves (which are a usual form of the ﬁeld data representation) is carried out by either the transformation of the curves (in this case, a gradient section is calculated), or by solving the inverse problem in a class of layered media. Although both approaches have their own inherent signiﬁcance and can be applied independently, sometimes it is helpful to use the transformation results in constructing the initial model for inverse problem solution in a class of layered media.

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3.4. TRANSFORMATION OF E(t) DATA INTO q(h) A common practice in electric prospecting is to use, as the interpretation parameter, an apparent resistivity ra(t) of a homogeneous medium, the response of which at a given moment coincides with the signal measured in experiment. Traditionally, ra(t) is calculated by asymptotic formula for late times in near zone of transient ﬁeld when the condition t/(m0R2/ra(t))b1 is satisﬁed: ra ðtÞ ¼

pﬃﬃﬃ 5=2 4 2=3 p m R 20 t5=2 EðtÞ=I

ð3:1Þ

Here E(t)/I is the measured value of normalized voltage at antenna terminals, and R ¼ L/p1/2 the effective radius of a single-turn square antenna with a side L. However, we can introduce an apparent resistivity calculated from the complete formula for the ﬁeld valid at any stage of the transient process. The voltage in a single-loop antennae for a homogeneous half-space equals to: # Z 1 t=t " p ﬃﬃﬃﬃﬃﬃ ﬃ e 1 1m pﬃﬃﬃﬃﬃﬃﬃﬃﬃ et=t erfcð t=tÞ J 21 ðlrÞdl t ¼ 2 ð3:2Þ EðtÞ=I ¼ mpR2 t r l pt=t 0 Apparent resistivity rf(t) can be obtained by solving the Equation (3.2) for q, on the left-hand side of which there is the measured voltage. Apparent resistivity calculated from asymptotic formula for the near-zone (3.1) and from complete formula coincide at late times and essentially differ at earlier times where t/(m0R2/ra(t))o1 (Fig. 3.5). Note that rf(t), in contrast to ra(t), at all stages of transient process, does not distort the shape of apparent resistivity (the same as qk in VES). Transformation E(t)/I-r(h) in TEM-FAST follows the ideas of well-known Niblett, Bostick, Molochnov, Le-Vieta transformations of MT data (Berdichevsky and Dmitriev, 1972)

Fig. 3.5. Apparent resistivities calculated by asymptotic ra(t) and completed rf(t) transient ﬁeld formula.

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but has some peculiarities permitting adjustment of the r(h) resolution depending on the type and contrast of variations in the geoelectric section under study. For the E(t)/I-r(h) transformation, two functions are employed: rf(t) , the apparent resistivity calculated by complete formula, and its time derivative dr/dt ¼ r0 f(t). The procedures of rf(t) and r0 f(t) calculation (solution of equation (3.2)) relate to the class of unstable problems and require a special algorithm for smoothing the initial data E(t)/I. The smoothing algorithm is described in detail in Svetov and Barsukov (1984) and based on the transient process E(t) representation as a superposition of exponent functions: Z 1 EðtÞ ¼ EðsÞest ds ð3:3Þ 0

where the real function E(s) of a real exponent s is called an exponential spectrum. The use of additional information about the spectrum E(s) behavior essentially increases the stability of a solution. For example, in a single-loop TEM conﬁguration, irrespective of the given distribution pattern of frequency-independent electric conductivity in the medium, the exponential spectrum is positive for any values of s, from which it follows that the transient signals E(t) themselves and all their time derivatives have a property of ‘perfect monotony’ (Gubatenko and Tikshaev, 1979; Weidelt, 1982). Z 1 dn EðtÞ dn EðtÞ n ¼ ð1Þ sn EðsÞest ds ð1Þn 40; n 0 u t40 ð3:4Þ n dt dtn 0 This means that the transient process cannot change its polarity, nor can its time derivatives change their sign. But if the signal measured in a single-loop antenna decreases non-monotonically while changing its polarity, this unambiguously speaks for the frequency dependency of the studied medium (induced polarization, or IP-effect). Thus, ﬁrst the experimental data are smoothed (the approximation by superposition of exponent functions in accordance to Equation (3.3), and then the apparent resistivity is calculated by the formula rf(t). After this, the correcting coefﬁcient is calculated: kðtÞ ¼

1 ð1 vÞ

3=2

;

v¼

t drf ðtÞ dlnrf ðtÞ ¼ ; lnt rf ðtÞ dt

absðvÞo1

ð3:5Þ

and then r(h) and effective depth of sounding h for each time t can be found as rðhÞ ¼ kðtÞrf ðtÞ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tbðresÞ h¼ m0

ð3:6Þ

ð3:7Þ

Function b(res) has the dimension of O m and can vary from the value rf of ‘untransformed’ resistivity to the value of rh of (3.6). Variations in b(res) signiﬁcantly change the shape of the curve rh ¼ r(h) and the resolving power of transformations. For low-contrast media b-rh, for the higher-

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Fig. 3.6. Example of ra(t)-r(h) transformation. Interlayer spacing for left, middle and right panel is 3, 10, and 30 m, respectively.

contrast ones b-rf, and for intermediate contrast media b ¼ (rh rm)1/2 lnðrh Þ lnðrm Þ res 0 res 10 10 The relation between rh and rm in b(res) is controlled by a speciﬁc parameter resolution (res) – ‘resolving power of the transformation’ that can be changed either manually or automatically via the transformation program. Reconstruction of r(h) section is carried out automatically without involving any additional information. Since it is not necessary to specify the number of layers in r(h) calculation, the reconstructed sections, in spite of their being actually the pseudo-sections, in many cases reproduce the resistivity distribution with depth more adequately than the data inversion in the class of layered media. Fig. 3.6 illustrates the changes in the resolving power of transformations with the spacing between two thin high-conductive clay layers (r ¼ 3 O m) imbedded in quite high-resistive rocks (r ¼ 70 O m). Piecewise-homogeneous plot is the model, and a smooth curve is the transformation. At a spacing h ¼ 3 m the layers are indistinguishable; resolution increases with the increasing spacing of layers. lnðbðresÞÞ ¼ lnðrm Þ þ

3.5. ONE-DIMENSIONAL INVERSION AND TEM-FAST’S RESOLUTION As has been already mentioned, a single-loop conﬁguration ensures minimum inﬂuence of side irregularities; therefore, as is evident from practice, in the overwhelming majority of cases the data interpretation within 1-D model class proves quite sufﬁcient to provide satisfactory results.

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Values of resistivity and layer thickness are calculated from the minimum of the misﬁt functional of calculated and experimental data. Here, the measurement accuracy and a priori information are taken into account, as well as the medium polarization effects, if necessary. The choice of the initial model is based on the available information about the section or, as shown in Fig. 3.7, on the pseudosection r(h). Quite naturally, a question arises – how widely can a 1-D class of models be applied in the practice of TEM sounding with TEM-FAST technologies? To give an answer to this question, a series of experiments was carried out with 3-D models simulating situations most frequently encountered in practice: fault, horst, graben, inclined layer, and others. The models were made from thin aluminum foil of known thickness and conductivity. Measurements were carried out by a TEMFAST device with antennae 10 10 cm (scaled as 1:400), and then the results were recalculated to real spatial scales. An example of simulation results is shown in Fig. 3.8. The section contains two local vertical objects and several horizontal and inclined conductive layers – bold lines. As seen from the ﬁgure, the pseudo-section properly reproduces the upper edge of the conductors and their horizontal extension, and the relief does not cause any signiﬁcant distortions. Local structures are not ‘‘visible’’ already at distances as small as those comparable with the size of the antenna itself. In 3-D studies, a necessity emerges of automatic inversion of rather large volumes of TEM-FAST data (a few thousand transient sounding points). A special

Fig. 3.7. Determination of the initial model in 1-D inversion. Horizontal markers in the curve correspond to the depths where the second depth derivative of q(h) is equal to zero (knee points).

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Fig. 3.8. TEM-FAST 1D inversion of results of physical modeling.

Fig. 3.9. 1-D inversion of a real TEM-FAST data (3500 soundings with 100 100 m antennae, Russia).

algorithm is provided to solve this problem. This algorithm stabilizes the inversion procedure by using, as an initial model at each point, the inversion result at neighboring points of the studied area. In addition, to obtain geologically adequate results of the inversion it is necessary to specify the limit values of resistivities and layer thicknesses with all possible accuracy. An example of automatic inversion of TEM data measured during kimberlite prospecting in Russia is shown in Fig. 3.9. Inversion was carried out for a case of a ﬁve-layer section with resistivities ranging from 10 to 300 Ohm.m. Blue color maps the ancient river bed. The whole data array processing was run automatically and took about 12 h on 3.3 GHz PC.

3.6. JOINT INVERSION OF TEM AND DC SOUNDINGS As has already been noted, DC soundings are very sensitive to resistive layers and structures imbedded in a section, and insensitive to the conductive layers.

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Conversly, TEM is sensitive to conductive layers and insensitive to the resistive ones. The resolution of both methods can be improved by joint inversion of both DC and TEM data in a 1-D class of layered sections providing construction of a single model satisfying all experimental data. An essential property of the method is interactive ﬁtting of parameters of the section. The problems arising at interpretation of TEM sounding data are well known: weak sensitivity of the method to the poor conducting layers and rock blocks; loss of information about subsurface layers in the hole zone of early TEM times. DC soundings meet some problems as well, namely: shielding effects present even in thin high-resistance layers limit the depth and resolution of soundings; great extent of equivalence of the models obtained at the data inversion; ineffective ratio between sounding depth and the size of electrode remote. However, combined use of both methods is capable of enriching the advantages of each separate method and decreasing their imperfections substantially. The developed tool for TEM and DC data inversion is based on the analysis and joint inversion of both TEM and DC data in the class of layered sections. Joint inversion implies ﬁnding the minimum misﬁt functional O ¼ arDC RDC þ ð1 aÞrTEM RTEM where rDC and rTEM are experimental values of apparent resistivity for DC and TEM methods, RDC X rR 2 r R ¼ expð3D=rÞ r and RTEM are the model values corresponding to 1-D-layered model of the section similar for both methods,||?||– the norm which is determined by the experimental data on apparent resistivity and their errors, 0pap1 is the factor deﬁning the contribution of each method in the functional O. In case of TEM, the errors determining the norm||?||are calculated directly during ﬁeld measurements, and for DC – from the results of measurements with ﬁxed center of soundings and various orientation of AMNB line, D is measurement error for time t (TEM) or remote AB (DC), summation at calculation of functional O is carried out over all times t and offsets AB. The robust technology is applied for suppressing the ‘‘heavy tails’’ of measurements in order to decrease the weight of poor-quality measurements in the functional O. In the minimization process the factor a changes smoothly from 1 to 1/2 (or from 0 to 1/2), which allows avoiding local minima of the functional. In the beginning, a model adequate to the data provided by either a single-TEM or a single DC method is used as the initial model; then the model parameters gradually and interactively change the data on the basis of both methods. This strategy is an essential element of the process of the problem solution. 2-D and 3-D geological structures are thus represented as the geoelectric images constructed like tomograms on the base of local 1-D TEM and DC inversion adjusting the data of proﬁle

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or array measurements. Examples of such analysis and inversion are shown in Figs. 3.10, 3.11. Fig. 3.10 displays the measured data and the results of an individual and joint inversion, and Fig. 3.11 presents the corresponding sections for: (a) separate DC (a ¼ 1), (Schlumberger conﬁguration MN ¼ 2 m), (b) separate TEM (a ¼ 0), (one-loop 50 50 m conﬁguration ), (c) TEM and DC data together, (a ¼ 1/2), (d) geological section corresponding to the well located in the center of TEM and DC measurements. In separate inversions the misﬁt (mean square deviation) between the model and experimental data is minimal and falls within the conﬁdence limits of the error. In the case of joint inversion, this misﬁt is certainly higher; however, it is still within the conﬁdence interval. As one can see, in spite of the model and experimental curves of apparent resistivity being quite close in the case of separate inversions, the obtained sections are rather far from the real geological data. In joint inversion the result is quite adequate to the real geological section. An example of practical applications of the above-described technology of joint inversion in hydrogeological prospecting is presented by Fig. 3.12. A 5 km-long proﬁle goes along the Nile (Egypt); the purpose of this survey was a water springs prospecting. The geological structure is characterized by interbedding of limestone (head water horizon) and clay (conﬁning layer). A good agreement of sounding and well data is seen in this ﬁgure.

Fig. 3.10. Experimental TEM and DC data (Russian platform) with the modeling curve of apparent resistivity, calculated on the basis of joint inversion.

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Fig. 3.11. Three variants of DC+TEM-FAST inversion (Russian platform).

Fig. 3.12. Joint TEM and DC sounding and inversion (Nile, Egypt).

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Fig. 3.13. Study of the fracture zone in French Alps.

Several other examples illustrating an application of TEM-FAST technology for hydrogeological research are shown below. Figure 3.13 illustrates high resolution of TEM sounding in solving the problems of ground waters leaking. Such problems arise in regions of water storage reservoirs used for city water supply (Grenoble) or energy production. At data analysis and interpretation the above-described technology of data transformation was applied. In spite of the dam, the water leaks from upper reservoir to the lower one. A considerable part of the Netherlands is separated from the North Sea by dikes. From time to time the dikes are destroyed by pouring rains and sea pressure; therefore the monitoring of the dams’ state is of great importance. As seen from Fig. 3.14, TEM-FAST provides a good opportunity to monitor the dikes’ condition. The apparent resistivity curves are shown in the upper part of the ﬁgure. They give an impression on the quality of the measured data in some points and crosssection in these points respectively. There are two dams seperated by a distance of 1200 m. The main part of the water is held by ﬁrst dam; some part of the water reaches the second dam and is accumulated under it. Sometimes a problem of waste burial arises, and then it becomes necessary to check the integrity of this or that horizon. An example of a solution to such a problem is shown in Fig. 3.15. A break is distinctly seen close to the point 250. The water leaks down through this break.

3.7. SIDE EFFECTS IN TEM SOUNDING In practical TEM-FAST applications, as well as in other TEM methods, three physical phenomena take place that complicate the process of ﬁeld diffusion and

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Fig. 3.14. Geoelectric section across a protecting dike (the Netherlands).

Fig. 3.15. Mapping of fracturing in a highly resistive horizon (Krasnoyarsk, Russia).

may affect considerably the effectiveness of geological interpretation of the measured results: superparamagnetic (SPM) effect, induced polarization (IP) effect, antenna polarization (APE) effect. Depending on the problem at hand, these effects can be treated as ‘‘harmful’’ or as ‘‘useful,’’ that is carrying additional information about the medium under study. 3.7.1. Superparamagnetic effect in TEM SPM, or the effect of magnetic viscosity, has been studied by many researchers (Neel, 1950; Averianov, 1965; Nagata, 1961; Lee, 1981; Barsukov and Fainberg, 2001).

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The SPM effect in rocks is related to processes of orientation/disorientation of magnetic moments of very ﬁne (of the order of an Angstrom) grains of magnetic minerals at initial moments of off/on switching of the exciting magnetic ﬁeld. TEM-FAST studies of the SPM effect in different regions of the world show that the most intensive SPM effects are observed in regions of effusive and volcanogenic-sedimentary rock development, the most superparamagnetic formations being the subsurface clays covering the parent rock masses. SPM effects are encountered in permafrost conditions and are usually localized at the boundaries of the frozen rock thaw. Considerable SPM effects are observed in glaciers. SPM is generated, as a rule, by magnetite and maghemite particles (r ¼ 109–107 m). SPM effects are usually understood as a frequency dispersion of magnetic susceptibility of rocks (on the analogy with frequency dispersion of conductivity or IP phenomena). To identify SPM in the measured data, a function tE(t) is provided in the TEMFAST system interface. Later stages of the transient process containing an SPM component, when multiplied by t, give a function almost constant in time. In practice, due to various reasons, against the Neel theory, the observed SPM procwhere 0.2odo0.2. For coaxial circular antenna loesses decrease as E1/t1+d , cated above a superparamagnetic half-space at height h, SPM transient processes are described by formula EðtÞ=I ¼ m0 wSPM ðtÞ F ðR; r; hÞ Geometric function F is equal to (R,r,h) ¼ 1/2 (Rr)1/2 Q1/2(x), where Q1/2(x) is the Legendre function of the order of 1/2 with argument x ¼ (4 h2+r2+R2)/2rR. In case of coincident antenna conﬁguration at R ¼ r and small h/R oo 1, geometric function is proportional to the antenna perimeter F(R,R,h) ¼ 1/2 R ln(R/h), and at R ¼ r and h ¼ 0 F is equal to the antenna inductivity L. At roR and h ¼ 0 function F is equal to mutual inductivity of antennae M. With a decrease of the receiving antenna size r, function F drops abruptly within an interval 1>r/R>0.9, and further on approaches its asymptote proportional to the area of the receiving antenna F(R,r,h)r2. Precise calculations of transients for horizontally layered conductive and superparamagnetic medium show that the interaction between induction currents in the medium and SPM effects is negligible, that is, these effects can be assumed additive. In order to reduce the SPM effects inﬂuence on the sounding results, three approaches are possible: to lower mutual inductance of antennae M, to increase the size of coincident antenna facility, and to raise the coincident antenna above the medium. Three characteristic features of SPM distortions are as follows: late stages of the process are proportional to E(t)1/t; resistivity r(t) drops rapidly down to ‘‘unlikely low’’ values; curves r(t) at later times for different in size antennae show steeply falling parallel branches. To avoid problems associated with the SPM effect, one has to check the measured data for a distortion and to change the antenna conﬁguration if necessary.

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At the same time, SPM can also be used in analyzing the magnetic properties of rocks and in the prospecting of ore deposits accompanied by SPM anomalies in their overlaying soils. This possibility arises from the SPM effect properties. Chunhan et al. (1997) have experimentally studied the gas bubbles, with nitrogen, oxygen, argon, and methane being the major components within them. These volatiles are generated mainly by the gas breath of the mantle. The gaseous agents are continually moving from the depth to the surface, inﬂuenced by many factors, mainly the pressure. Studies of the gas aureole showed that the gas ﬂows can transport the metallic particles in a vertical direction both at high and at atmospheric temperatures. Analytical techniques of electron microscopy made it possible to ﬁnd that the gas emanating from the earth contains Si, Al, K, Na, S, and also Fe, Mg, Ti, V, Zn, Au, As, Ba, Ca, etc. Just these elements are responsible for the SPM effect. Some examples of successful analysis of emanating gas ﬂuxes as applied to ore prospecting in fault zones were described in (Kristiansson et al., 1990). An example of such SPM application in gold prospecting, based on the analysis of logarithmic relaxation velocity (d) measured at soil patterns sampled above the gold ﬁeld, is shown in Fig. 3.16 (Barsukov and Fainberg, 2001). As one can see from the ﬁgure, both conventional parameters magnetic susceptibility wmagn and wSPM do not indicate existence of any anomaly; at the same time parameter d shows clearly local anomaly.

Fig. 3.16. SPM effect over the gold placer. Soil samples are taken from South Ural.

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71

Fig. 3.17. Comparison of traditional (TEM and magnetic survey) and alternative (SPM) research methods.

It should be mentioned that the SPM effect does not allow determination of the occurrence depth of the object in such a way as is done in induction prospecting, but it exceeds the latter in horizontal resolution. An other example of TEM-FAST research with SPM application is shown in Fig. 3.17. This ﬁgure displays the results of searching for a large bomb, mine and shell arsenal left in adits since the Second World War (Sevastopol, Ukraine). The adits lie at a depth more than 25 m and are overlain by a thick layer of highly conductive clay. As seen from the ﬁgure, the magnetic survey (Fig. 3.17, left panel) revealed chaotic distribution of magnetic anomalies. Usual induction sounding does not feel metal objects due to the shielding effect of the clay. At the same time, the SPM effect rather well outlines the scheme of ammunition disposition that agrees with the results of speleological studies of the adit passages. 3.7.2. Effect of induced polarization The importance of studying the IP effect results from a wide prevalence of this effect in various geological media and its essential inﬂuence on the results of practically all EM studies. Often it is very hard to diagnose the IP effects from TEM ﬁeld data. The exceptions are cases of very intense manifestations of the polarization at low levels in induction processes when the observed signals multiply invert their sign. Cases occur when the entire observed signal from 4 ms to 4–10 ms has a negative sign. Much more often the polarization process distorts the observed signal

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but without changing the sign. In the interpretation in these cases, sometimes artifacts of low-conductance layers are revealed. In many cases IP effect is adequately described by the well-known Cole–Cole formula (Cole and Cole, 1941; Kamenetsky, 1997), but in a series of cases this formula is insufﬁcient and it is necessary to involve other considerations, e.g., those by Debye, Davidson and others (Pelton, Sill and Smith, 1983). As the analysis with TEM-FAST showed, the geoelectric situations most unfavorable for interpretation where IP effects show are as follows: thin conductive horizon of subsurface clay deposits with ro20–40 O m laying at rather high-resistive rock masses with r>300–500 O m.; at a noticeable polarization of this layer the later stages of transient process are distorted by IP effects; glaciers and frozen rocks; subsurface deposits highly polluted by industrial-waste products (including pollution by oil); weathering cores in crystal rocks and fault zones. An example of IP-effect measured in Egypt close to the fault at the Giza plateau is shown in Figs. 3.18, 3.19 (these measurements have been fulﬁlled jointly with NRIAG team). The studies started initially as purely geological and lead to an unexpected result: between the famous objects the Sphinx, the Cheops Pyramid and the Chephren pyramid, a region with abnormally high electric conductivity has been found. Conventionally, this region was believed to be associated with cavities in limestone formations of the Giza plateau that contain mineralized water and have rather low resistivity. In the year 2000, deep underground galleries were discovered at this site. Intensive IP effects were revealed near the Chephren pyramid where,

Fig. 3.18. Manifestation of IP effect during the study of the Giza plateau geological structure (Egypt, 1994).

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73

Fig. 3.19. IP effects in the fault zone at the Great Egyptian Pyramids plateau (Gizza, 1994).

according to the TEM-FAST measurements, the local fault does exist. This effect is shown in details in Fig. 3.19. The nature of the local zone of intense polarization effect at the foot of Chephren pyramid still remains a mystery. Analysis of numerous ﬁeld experiments where polarization effect was observed and numerical simulation showed that TEM is distorted, as a rule, by highly conducting subsurface rock horizons with ro 20–40 O, occurrence depth not exceeding 1–2 m with quite a weak background polarizability Z1–3%, polarization effects of rocks with r>100 appear only at considerably high polarizability Z>10%, time constants tip lie within a very wide range of 1–1000 ms. For surface IP effect, usually c1 (Eip(t)-exp(t/t) ) and distortions are observed within a narrow time interval, deep IP effect is characterized by c1/31/2 and distorts all the later stages of the signal; with IP present simultaneously in subsurface and deep horizons of a rock mass, at once two negative minima are observed at earlier and later times. It is important to note that since the IP effect speaks for the presence of polarizable bodies or ﬂuids, it can be used in the prospecting, tracing, and monitoring these objects.

3.7.3. Antenna polarization effect (APE) The antenna polarization effect has been detected in the TEM data by many researchers (Bishop and Reid, 2003) but only recently was an explanation to this effect was found. In practice, transient characteristics (or their time derivatives)

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Fig. 3.20. Negative in-loop TEM decays (site A is on outcropping granite and the overburden thickens progressively toward site C).

often change the polarity, which is due to the frequency dependence of electric conductivity of rocks (Fig. 3.20). In induction electric prospecting these effects are called ‘‘negative anomalies’’. Theoretically, it is not difﬁcult to calculate transient characteristics of horizontally layered media with conductivities described by complex frequency-dependent function (e.g., Cole–Cole formula). Experimental data comparison is calculated in such a way that transient characteristics show that discrepancies are often observed that do not ﬁt theoretical models of polarizable media; for example, the value of negative anomalies depends on the resistivity of the antenna wire. These discrepancies between the experiment and theory cannot be explained by horizontal inhomogeneity of the medium or imperfection of the modeling technique. It is obvious that the interpretation algorithms in such cases face insuperable difﬁculties. (Numerous ﬁeld experiments and theoretical analysis of this effect based on the antenna system consideration as a line with distributed parameters allowed to make the following conclusions.) Distributed antenna capacity and ever-persisting resistance in real devices employed in the study of geological media with frequency-dependent dielectric permeability produce the antenna polarization effect (APE). This effect shows at later stages of transient process and is detected as a slow decaying with time process 1/t(0.3–0.7) with its phase opposite to the phase of induction transient characteristics. APE depends on the dispersion parameters of the medium and, with a ﬁxed current, is proportional to the squared resistivity of the antenna wires. Distributed resistivity of antenna circuits where the currents run even in case of symmetric

P.O. Barsukov et al./Shallow Investigations By TEM-FAST Technique

Fig. 3.21. Dependency of the signal in coincident antenna on the wire resistance.

Fig. 3.22. Dependency of the signal in coincident antenna on the ‘‘ground-wire’’ capacitance.

75

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rotation of the antenna-medium system produces asymmetric electric ﬁelds with radial and vertical components. APE is detected in both the coincident receiving–transmitting and the spaced antenna, and should be taken into account in the ﬁeld data interpretation. The results of ﬁeld experiments that prove the dependence of the antenna polarization effect on the resistivity and capacity of wires of which the antenna are shown in Fig. 3.21, 3.22.

REFERENCES Averianov, V.S., 1965. Role of magnetic crystallographic anisotropy in the process of viscous magnetization of ferrites (in Russian). Izvestia, Fizika Zemli, 7: 82–89. Barsukov, P. and Fainberg, E., 2001. Superparamagnetic effect over gold and nickel deposits. Eur. J. Environ. Eng. Geophys., 6: 61–72. Berdichevsky, M.N. and Dmitriev, V.I., 1972. Magnetotelluric Sounding of Horizontally-uniform media (in Russian). Nedra, Moscow, 250pp. Bishop, J. and Reid, J., 2003. Some positive thoughts about negative TEM responses. ASEG 16th Geophysical Conference and Exhibition, Adelaide. Chunhan, T., Juchu, L. and Liangquan, G., 1997. Nano-scale particles of ascending gas ﬂows in the crust and geogas prospecting: Engineering and environmental geophysics for the 21st century. Proceedings of the International Symposium. Sichuan Publishing house of Science and Technology, Chengdu, China, pp. 337–342. Cole, K.S. and Cole, R.H., 1941. Dispersion and absorbtion in dielectrics. I. Alternating current ﬁeld J. Chem. Phys., 9: 341–351. Gubatenko, V.P. and Tikshaev, V.V., 1979. On changes in the sign of electromotive force of induction in the transient electromagnetic ﬁeld method (in Russian). Izvestia, Fizika Zemli, 3: 95–99. Kamenetsky, F.M., 1997. Electromagnetic geophysical Studies with the Method of Transient Processes (in Russian). GEOS, Moscow, 162pp. Kristiansson, K., Malmquist, L. and Persson, W., 1990. Geogas prospecting: a new tool in the search for concealed mineralization. Endeavour, New Series, 14, 1: 407–416. Lee, T., 1981. Transient electromagnetic response of a polarized ground. Geophysics, 46: 1037–1041. Nagata, T., 1961. Rock Magnetism. Plenum Press, New York, 350pp. Neel, L., 1950. Theorie du trainage magnetique des substances massives dans le domaine le Rayleigh. J. Phys. rad., 2: 49. Pelton, W., Sill, W. and Smith, B., 1983. Interpretation of complex resistivity and dielectric data, I. Geophys. trans., 29, 4: 297–330. Pelton, W., Sill, W. and Smith, B., 1983. Interpretation of complex resistivity and dielectric data, II. Geophys. trans., 30, 1: 11–45.

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Svetov, B.S. and Barsukov, P.O., 1984. Transformation of quasi-stationary geoelectric transient processes into equivalent wave processes (in Russian). Izvestia, Phys. Earth, 8: 29–37. Weidelt, P., 1982. Response characteristic of coincident loop transient electromagnetic system. Geophysics, 47: 1325–1330.

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Chapter 4 Seismoelectric Methods of Earth Study B.S. Svetov Geoelectromagnetic Research Center IPE RAS, Troitsk, Moscow Region, Russia

It is well known that the Earth, affected by some physical ﬁelds, experiences diverse complex variations. In geophysical respect, these changes can appear in two forms: (1) as changes in physical parameters of a geological medium (effect of the ﬁrst kind), and (2) as an emergence of physical–chemical processes in the medium that, in turn, gives rise to various physical ﬁelds (effect of the second kind). Currently, we are very far from comprehensive understanding of these phenomena, and very often we conﬁne ourselves to their phenomenological description based on the theory of physical ﬁelds in continuous, stationary and passive media. In this approach, the complex multi-phase rock structure is ignored and, as a result, a possibility is lost to obtain information about petrophysical parameters of rocks (porosity, ﬂuid permeability, ﬂuid saturation and others). Just the same, the energy state of a geological medium is also neglected in geophysical prospecting.

4.1. SEISMOELECTRIC EFFECT OF THE FIRST KIND Let us dwell on seismoelectric effect (SE) of the ﬁrst kind implying the elastic (seismic) ﬁeld inﬂuence on a geological medium and the resulting change in its conductivity (Ivanov, 1949). In real conditions, the geological medium is energetically unstable. This instability develops over a wide range of spatial scales – from the scale of porous and polyphase rocks having complex structure to scales of regional geological structures. Therefore, even weak seismic ﬁeld impacts on the medium can result in its signiﬁcant changes (in particular, changes in its electric resistivity). Let us show a result of an experiment carried out by geophysicists from Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40004-4

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Saratov, USSR, in Near-Caspian depression – a region of widely developed saltdome and fault tectonics (Ozerkov et al., 1998). Fig. 4.1 displays cross sections of apparent resistivity obtained by transient electromagnetic method: (a) before the vibrational impact on the medium, (b) a few minutes after a 3-min operation of seismic vibrator, (c) 24 h after the vibrator operation, and (d) 17 days after the operation. From the dynamics of these cross sections it is clearly seen how sharply the geoelectrical cross section is changing immediately after the vibrational impact, and how slowly and incompletely it is relaxing to its initial state. It is worth noting that such a pattern is observed only within extremely stressed regions; in other experiments the changes of geoelectrical sections were less evident, although still observed sometimes. Similar changes are encountered after and sometimes before earthquakes and are used for earthquake prediction. SE of the ﬁrst kind caused by controlled seismic impact can be applied in engineering geology as a marker of unstable zones unsuitable for building. Seismic impact is sometimes used to provoke weak and to prevent strong earthquakes. An obstacle for a widespread use of SE of the ﬁrst kind is its purely empirical foundations and difﬁculties in creation of a rather rigorous theory.

4.2. SEISMOELECTRIC EFFECT OF THE SECOND KIND: HISTORICAL OUTLINE AND ELEMENTS OF THEORY Situation with SE of second kind is more favorable. During recent decades, certain progress in understanding this phenomenon began to show. Let us consider this effect in more detail. Classical seismic prospecting and acoustic logging were and, up to now, are theoretically based mainly on the equation of elastic waves propagation in continuous media – the Lame equation. In a frequency domain (eiwt) and for isotropic media this can be written in the form mr r u ðl þ 2mÞrðr uÞ o2 u ¼ 0

ð4:1Þ

Here u is a vector of medium displacement, l ¼ K2/3m and m the Lame parameters (m the shear modulus), K the bulk modulus. Solution of this equation in a homogeneous medium is a sum of two elastic waves: longitudinal (potential) and transverse (vortex) ones. In the 1940s, just after the discovery of SE by A.G. Ivanov (1940) Ya.I. Frenkel (1944) gave a ﬁrst theoretical description to this phenomenon, concurrently laying the foundations to the theory of elastic waves propagation in a porous two-phase medium. Later on, M. Biot (1956) extended this theory and formulated his widely known equations (the Biot equations) describing elastic waves propagation in a two-phase porous ﬂuid- or gas-saturated medium that is a more adequate model of a rock. Written in the form suggested in (D. Schmitt et al., 1988), these equations for an isotropic medium look like mr r us þ Prðr us Þ þ Qrðr uf Þ þ o2 ðg11 us þ g12 uf Þ ¼ 0

ð4:2Þ

Qrðr us Þ þ Rrðr uf Þ þ o2 ðg12 us þ g22 uf Þ ¼ 0

ð4:3Þ

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Fig. 4.1. Seismoelectric effect of the ﬁrst kind. Sections of apparent resistivity: (a) before the vibrational impact, (b) 5 min after the impact, (c) 24 h later, (d) 17 days later. N, numbers of the sounding points.

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Here us and uf are displacements of a solid and ﬂuid phases in a medium, ð1 fÞð1 f wÞ þ fwK s K f 4 P ¼ Ks þ m, 3 1 f w þ fK s =K f ð1 f wÞfK s Q¼ , 1 f w þ fK s =K f R¼

f2 K s 1 f w þ fK s =K f

where Ks, Kf, Km are bulk moduli of a solid and ﬂuid phases and dry rock skeleton, m the rock skeleton shear modulus, w ¼ Km/Ks, f the porosity of the medium, g11 ¼ r11+ib/o, g12 ¼ r12ib/o, g22 ¼ r22+ib/o, b(o) ¼ f2H1 (o), r22 ¼ f2H2 (o)/o, r12(o) ¼ frfr22(o), r11(o) ¼ (1f)rsr12(o), H1(o)iH2(o) ¼ Z/k(o), n o1 1=2 io=ob is frequency-dependent permeakðoÞ ¼ k0 1 io=ob M B =2 bility of a porous medium, M B 2 ð1; 2Þ is a constant depending on a pore shape, ob ¼ f=a1 k0 Z=rf is the Biot critical frequency at which the diffusive motion of a porous ﬂuid changes into the wave motion, rf,rs are densities of a ﬂuid and solid phases, a1 ¼ 1–8 is pore tortuosity, Z the ﬂuid viscosity, k0 the permeability of a medium in a stationary ﬁeld. The Biot equation’s solution in a homogeneous medium is a sum of three waves: one transverse and two longitudinal (‘‘slow’’ and ‘‘fast’’). Dynamical and kinematical characteristics of these waves depend, as it follows from the equations, not only on the elasticity moduli and densities of a liquid and solid phases but also on pertophysical parameters of the medium and, ﬁrst of all, on its porosity and permeability. This has opened new informational possibilities for seismic prospecting and, particularly, for acoustic logging. From obvious physical considerations and, in particular, from the Biot equations it follows that the pore ﬂuid moves not in synchronism with a solid phase, but lags behind it. This results in the emergence of a ﬂuid ﬂow relative to a solid rock skeleton, the ﬂow density being w ¼ iofðus uf Þ ¼

kðoÞ 2 ðo rf us rpÞ Z

ð4:4Þ

Bracketed on the right-hand part of this expression is a force that generates the ﬂow and comprises inertial force and pore pressure gradient rp. Selective ion adsorption from a pore solution by a solid rock skeleton results in opposite charging of its liquid and solid phases, therefore their relative motion produces an extraneous electric current with density jext. This current generates an electromagnetic (EM) ﬁeld. The above is a physical substance of the second kind SE of electrokinetic nature. The value of extraneous current can be expressed in terms of surface-charge density on a solid phase boundary Q, or through zeta-potential z of a pore solution. As it has been already mentioned, the ﬁrst mathematical description of SE in rocks is given in the work by Ya. Frenkel (1944). Fifty years later, S. Pride (1994) gave a more rigorous description to this phenomenon using a self-consistent system of

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83

equations comprising the Biot and Maxwell equations. Subsequently, Svetov and Gubatenko (1999) and Svetov (2000) proposed a simpler formulation of this problem allowing for a negligibly weak EM ﬁeld back inﬂuence on the elastic ﬁeld and reducing to a successive solution of the Biot (4.2, 4.3) and Maxwell equations: ~ oÞE þ j ext ; r H ¼ sðr;

r E ¼ iomH

ð4:5Þ

~ oÞ ¼ s io is a complex electric conductivity of a medium. The extraHere sðr; neous current representation plays an interlinking part between the Biot and Maxwell equations. This representation follows from Onsager thermodynamical relations. Let us write it out in two forms obtained by S. Pride and B. Svetov-V. Gubatenko for a case when the thickness of a diffusional part of a double-layer in a pore solution is much smaller than the pore size: h i ZfLðoÞ ð4:6Þ ðus uf Þ ¼ ioZðus uf Þ j ext ¼ LðoÞ o2 rf us rp ¼ io kðoÞ j ext ¼ ioQðoÞyðus uf Þ ¼ io

QðoÞyðo2 rf us rpÞ

¼ ioZ1 ðus uf Þ ð4:6aÞ iorf 1=2 , Z ¼ ZfL(o)/k(o), Z1 ¼ Q(o)y, Here LðoÞ ¼ f=a1 f z=Z 1 io=ob 2=M B Q(o) is the frequency-dependent surface-charge density on the solid and liquid phase interface, y the speciﬁc surface of the pore space. Quantity Z has a sense of a frequency-dependent electrokinetic coefﬁcient since it deﬁnes the extraneous electric current generated by the pore ﬂuid ﬂow. This quantity contains basic information about the medium provided by the EM ﬁeld of electrokinetic origin. For the sake of objectivity it should be mentioned that, as laboratory experiments show (Ageeva et al., 1999), the both of the above expressions give rather crude approximation of processes taking place in the real rocks. Basic importance of these expressions consists in general description of electrokinetic coefﬁcient dependence pattern on pertophysical parameters of a geological medium. Zf kðoÞ

4.3. PHYSICAL INTERPRETATION OF SEISMOELECTRIC PHENOMENA To get a better understanding of what the EM ﬁelds develop in a porous ﬂuidsaturated medium (the ‘‘Biot medium’’), let us consider the simplest case of plane elastic waves propagation in such a medium. The medium is assumed homogeneous, then, dependent on the speciﬁc exciter type, fast (velocity a1) and slow (velocity a2) longitudinal Biot waves, or transverse wave (velocity b) or both wave types are generated in the medium. Assume these waves to propagate in the 0z axis direction and to have solid phase displacement amplitudes Ap1 ; Ap2 ; As at Z ¼ 0. Then rock skeleton displacements for longitudinal and transverse waves can be written as p

p

upz ¼ Ap1 eik 1 z þ Ap2 eik 2 z

usx ¼ As eik

Sz

ð4:7Þ

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These waves give rise to longitudinal and transverse extraneous currents moving together with the generative waves: p

p

p j ext ¼ Ap1 Zð1 x1 Þeik 1 z þ Ap2 Zð1 x2 Þeik 2 z 1z

s

s s ik z j ext 1x ¼ A Zð1 gÞe

ð4:8Þ

Here kp1 ¼ o=a1 ; kp2 ¼ o=a2 ; ks ¼ o=b are wave numbers of corresponding elastic waves (Re k40), and x1;2 ¼ P a21;2 g11 =a21;2 g12 Q ¼ Q a21;2 g12 =a21;2 g22 R and g ¼ g12 =g22 are the constants deﬁning the ratio of liquid-to solid-phase displacements in the medium that depend on the parameters of the medium (D. Schmitt, 1988). Longitudinal extraneous currents also give rise to longitudinal electric ﬁeld, not accompanied by a magnetic ﬁeld and equal to: p p 1 E pz ¼ ½Ap1 Zð1 x1 Þeik 1 z þ Ap2 Zð1 x2 Þeik 2 z ð4:9aÞ s Transverse extraneous current generates transverse electric and magnetic ﬁelds:

E sx ¼

s 1 k2e iks s iks z s A Zð1 gÞe ; H ¼ As Zð1 gÞeik z 2 y 2 s2 s k2e ks ke k

ð4:9bÞ

Here ke ¼ ðiomsÞ1=2 is a wave number of EM ﬁeld. All these ﬁelds propagate in a medium with the same velocities as the corresponding elastic waves do, and are non-zero only in that just place crossed by the elastic perturbation at a given moment. This circumstance laid the grounds to call such electric and magnetic waves ‘‘frozen-in’’ (Svetov, 2000). Note that the frozen-in transverse electric ﬁelds (Equation (4.9b)) due to k e =ks 1 (over a frequency range of interest) are much smaller than the longitudinal ones (Equation (4.9a)). Now assume longitudinal (Equation (4.9a)) or transverse (Equation (4.9b)) elastic waves orthogonally incident on a plane interface z ¼ 0 of two half-spaces different in their elastic, pertophysical or electric properties. The waves are partially reﬂected, partially penetrate into the lower half-space without generation of converted waves at orthogonal incidence. In this situation, longitudinal extraneous electric ﬁelds will hold the structure of (4.9a), being different in the upper and lower half-spaces only on account of the difference between electrokinetic coefﬁcients and conductivities. The case is another with transverse EM ﬁelds. Here, solutions of homogeneous Maxwell equations became non-zero, and general solution to these equations, e.g. in a lower half-space, takes the form E sx ¼

s 1 k2e As Zð1 gÞeik z þ E 0x eike z s k 2 k s2

e

H sy ¼

iks k2e

k

s

s2

As Zð1 gÞeik z þ H 0y eike z

ð4:10Þ

Second summands in Equation (4.10) describe usual EM waves propagating independently on elastic waves at a velocity many times higher than seismic waves. In seismoelectrics such waves can be called ‘‘fast’’ EM waves. The values E 0x and H 0y are found from the continuity conditions for the tangential components of electric

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85

and magnetic ﬁelds at the half-spaces interface. Fast EM waves depend complexly on the contrast in physical and petrophysical characteristics of the medium on either side of the interface. In a highly conducting medium, they decay with the distance to the interface. On the whole, from a geophysical point of view, EM ﬁeld of electrokinetic origin is much more informative than the parental seismic one. This ﬁeld includes four wave types differing in their kinematic (velocities) and dynamic characteristics: three waves frozen in their corresponding seismic waves and one fast EM wave. All these waves are differently, and more strongly than the seismic ones, dependent on the pertophysical properties of the medium. If the elastic and electric ﬁelds are measured simultaneously as it is usually done in seismoelectrics, a possibility arises to ﬁnd, directly from measurements, the so-called seismoelectric transfer functions W ðwÞ that are complex ratios, in a frequency domain, of electric ﬁeld strength to the medium displacement (or the velocity of this displacement) or to the well pressure. Owing to their relative nature they do not depend on the intensity of the excited elastic ﬁeld and its spectral composition, and therefore these functions are easier to use in the determination of the parameters of the medium than the direct measurements of the ﬁeld strengths. Functions W ðwÞ differ for different wave types. For frozen-in longitudinal and transverse waves the functions W(w) ¼ E/u equal to 1 W p1 ¼ ð1 x1 ÞZ; s

1 W p2 ¼ ð1 x2 ÞZ; s

Ws ¼

1 k2e ð1 gÞZ ð4:11Þ s k2e ks2

Fig. 4.2. Frequency dependences of real (a) and imagery, (b) parts of SE transfer function (p, porosity; pm, permeability).

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They depend on both the petrophysical parameters of the medium and on its elastic and electric properties. Of particular importance seems to be the theoretically predicted possibility of direct determination of rock porosity and permeability from the transfer functions. This possibility shows in different dependences of the real and imaginary parts of SE transfer functions on these parameters. Figure 4.2 displays frequency characteristics of normalized by conductivity real and imagery parts of SE transfer functions for a longitudinal wave. The real part is additionally normalized by o2 , and the imagery by o3 . At not very high frequencies ðRe k40Þ; the real part is proportional to porosity but is practically permeability-independent; quite a contrary, the imaginary part only weakly depends on porosity but is proportional to permeability. Such diverse and rich geophysical information that can be potentially yielded by SE methods destines the prospects of their application in ﬁeld studies and, particularly, in logging. Note the different character of information provided by the frozen-in and fast EM ﬁelds. The frozen-in wave measured by ﬁeld receivers describes the structure of the medium only within the vicinity of the receiving site. Within the frequency range of ﬁeld seismoelectrics (60–100 Hz) the size of this area is a few meters to a few dozens of meters, and at SE logging frequencies (2000–10 000 Hz) it amounts to a few meters. Hence, these waves can be used only in shallow (engineer and hydrogeological) ﬁeld studies and logging. Information about the deeper Earth layers required, e.g., in solving the problems of oil–gas geology, can be obtained only from fast EM waves. However, here the wave absorption along the path from the reﬂecting boundary to the ﬁeld receivers should be taken into account.

4.4. MODELING OF SEISMOELECTRIC FIELDS The recent decade was the time of intensive development of theoretical and methodological grounds to ﬁeld and, particularly, borehole seismoelectrics. Many scientists are involved in this research (S. Pride, M. Haartsen, B. Svetov, V. Gubatenko, P. Aleksandrov, B. Plyuschenkov, M. Markov, V. Verzhbitsky and others). In their works the seismoelectric 1-D problems for horizontally and radially stratiﬁed media are solved, corresponding software is built and the solutions are analyzed. Development of the software for 2-D problems solution (vertical well + horizontally stratiﬁed medium outside) got started. Experimental ﬁeld and borehole studies are performed and continue to be carried out. Seismoelectric phenomena are studied by rock sample testing. Here below we shall dwell mainly on the results obtained in the Geoelectromagnetic Research Center IPE RAS. In analytical solution of SE 1-D problems in horizontally and radially stratiﬁed media the Biot and Maxwell equations for components of the ﬁeld or potential are easily scalarized by corresponding integral transforms and reduce to the known ordinary differential equations (the Bessel equations). At interfaces of media with different parameters, the displacement, stress tensor and EM ﬁeld components

87

B.S. Svetov/Seismoelectric Methods of Earth Study Table 4.1. Conjugation conlitions. B–B usr ¼ usr f ufr usr ¼ f ufr usr p¼p cBrr ¼ cBrr cBrz ¼ cBrz uss ¼ uss

W–B (1)

W–B (2)

L–B

ur ¼ usr ð1 fÞ þ fufr

ubr ¼ usr ð1 fÞ þ ufr f

ur ¼ usr ð1 fÞ þ ufr f 0 ¼ ufr usr

upz 0 ¼ cBrr þ p p ¼ cBrr 0 ¼ cBrz

¼

p Ap1 eik1 z

p ¼ cBrr 0 ¼ cBrz

þ

p Ap2 eik2 z

crr ¼ cBrr crz ¼ cBrz uz ¼ usz

satisfy the necessary conjugation conditions. For the Biot equations in a radially stratiﬁed medium the conjugation condition are tabulated in Table 4.1. In this table, the columns present the conjugation conditions at interfaces between two Biot media (B–B), between a ﬂuid and a Biot medium with permeable (W–B1) and impermeable (W–B2) interfaces, and between continuous solid medium and a Biot medium (L–B). In the table, an equality is established of the displace^ ¼ fpI^ þ ð1 fÞc^ ment and total stress tensor components in a Biot medium c B s ^ (I^ is the unit tensor, cs the stress tensor of a solid phase) on either side of the contact media interface. Similar equations can be written for horizontal layers interfaces. Conjugation conditions for EM ﬁeld consist in the equality of the tangential components of electric and magnetic ﬁeld. The scientists of the Geoelectromagnetic Research Center developed the software necessary for solving such problems and calculated the SE ﬁelds for a series of typical model media. Shown in Figs. 4.3a, b, are the calculation results for the pressure ﬁeld and vertical component of the electric ﬁeld in a well enclosed by porous ﬂuid-saturated medium, excited by a radial elastic force transmitter. Base excitation frequency is 10 000 Hz.The abscissa axis is the time of ﬁeld observation; each wave trace is indexed by the distance to the elastic ﬁeld source. Fields at each trace are normalized by their maximum values. In the pressure plots (Fig. 4.3a), the longitudinal, transverse and surface (propagating along the drilling mud–porous medium interface) waves successively arriving at the receiver are seen. From these plots, one can ﬁnd the velocities of these waves and their spatial damping, and that is all what the acoustic logging gives. Using these data geophysicists, to this or that degree of conﬁdence, can separate the geological section in lithology and determine some petrophysical parameters important for gas–oil disposal estimation, e.g. porosity. Fig. 4.3b portrays the wave traces of the electric ﬁeld. Besides the same types of electric waves frozen in the acoustic ﬁelds, also fast EM waves are seen in the plots, instantly and practically simultaneously arriving at electric ﬁeld receivers spaced by different distances from the source of excitation. These waves originate at a borehole wall (media interface) once it has reached by an acoustic wave. All the geophysical information contained in the acoustic logging persists in the frozen-in electric waves, but this is complemented by new independent data yielded from SE transfer functions and fast waves. Shown in Figs. 4.4a, b, are the results of calculating the vertical components of a solid-phase displacement in a water-saturated medium and electric ﬁeld strength for

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Fig. 4.3. Calculated acoustograms (well pressure) (a) and electrograms (vertical component of electric ﬁeld strength), (b) for radially layered medium (well + porous ﬂuid-saturated medium).

an orthogonal intersection of a plane interface of two half-spaces by 2-m-long SE logging facility (the well inﬂuence is neglected in calculations). In the right-hand part of this and a series of next ﬁgures, the main parameters of the section are given: T layer, location of the section surface; Vp,VS, velocities of longitudinal and transverse seismic waves; ps, solid phase density; mo, porosity; Ks,

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Fig. 4.4. Calculated seismograms (vertical medium displacement) (a) and electrograms, (b) at orthogonal intersection of a plane interface of two halfspaces by SE logging facility.

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solid phase bulk modulus; Vf, elastic waves velocity in a ﬂuid phase of the medium; pf, ﬂuid phase density; ef =e, relative dielectric permittivity of the ﬂuid phase; k, ﬂuid permeability; nu, ﬂuid viscosity; x, pore tortuosity, the value of the M B constant, dzeta, zeta-potential; sigma, conductivity of the medium. The half-spaces have similar elastic and electrical parameters and differ only in their ﬂuid permeability and porosity. The interface is located at z ¼ 0, and the observation point of calculated signals is related to the ﬁeld receivers. This interface is almost invisible in plots of elastic displacement (on different sides of the interface the wave arrival time changes). In the wave images of electric ﬁeld, this boundary is clearly manifested in the emergence of reﬂected waves as the ﬁeld receivers pass through the interface. Thus, almost invisible in elastic waves, the petrophysical parameters interface is quite distinct in the SE ﬁeld. Note that the transverse frozen-in waves are practically not apparent in the electric ﬁeld plots. Similar situation is observed also when the same logging facility intersects a 1-m thick layer that differs from the embedding medium only in porosity and permeability (Figs. 4.5a, b) (the layer boundaries are at z ¼ 70–5 m). On the basis of the carried out calculations, the SE logging sensitivity to petrophysical parameters of the medium was analyzed and compared to that of acoustic logging. Figs. 4.6a, b show the porosity and permeability dependences of pressure P observed in a well (acoustic logging), electric ﬁeld and absolute values of SE transfer functions for frozen-in fast longitudinal Biot waves. It can be seen from the images that the electric ﬁelds and SE transfer functions show substantially higher sensitivity to the petrophysical parameters of the medium. The plots shown above illustrate capabilities of SE logging. Let us present an example of SE ﬁeld calculation in the context of ﬁeld seismoelectrics. Traces of vertical component of the displacement and radial component of the electric ﬁeld excited in a two-layer medium by a pulse of vertical force are depicted in Figs. 4.7 a, b in a similar form. The section is chosen to reproduce the wave pattern observed at one of the segments of experimental proﬁle. The traces are indexed by the distance to the exciter in meters. Besides the frozen-in longitudinal and surface waves, also rapidly damped fast EM waves are seen in the electric ﬁeld plots at small distances from the exciter. Currently, numerical methods for SE problems solution in 2- and 3-D media based on the use of integral and integral-differential equations are under development in the GEMRC.

4.5. LABORATORY STUDIES OF SEISMOELECTRIC EFFECTS ON ROCK SAMPLES The above stated theory of SE phenomena in a porous ﬂuid-saturated medium and that resulting in geophysical conclusions need an experimental validation and veriﬁcation. First of all, it is necessary to make sure in the theoretical description adequacy to the real geophysical processes in rocks. This can be done on the basis of laboratory studies of SE effect on rock samples (Ageeva et al., 1999). With this

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Fig. 4.5. Calculated seismograms (vertical medium displacement) (a) and electrograms, (b) at orthogonal intersection of a 1-m-thick layer by SE logging facility.

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Fig. 4.6. Well pressure (a) and SE transfer functions, (b) dependencies on the porosity and permeability of the medium.

purpose, a special facility has been designed for studying elastic and electric ﬁelds excited in rock cores by the piezoelectric transmitter of longitudinal waves. Using this equipment, a large set of terrigene and carbonate rocks differing in their porosity and permeability at different levels of water saturation and mineralization of pore solution were tested. In accordance with electrokinetic theory, the intensity of

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Fig. 4.7. Example of calculated seismograms (vertical-medium displacement) (a) and electrograms (horizontal electric-ﬁeld strength), (b) at different distances to the exciter above a two-layer medium.

SE observed in limestones and sandstones drops with the decrease in ﬂuid saturation of the pattern (Fig. 4.8). This effect is almost missing in dry rock samples. SE transfer functions dependences on petrophysical parameters of the patterns, such as porosity and permeability, were studied. Fig. 4.9 shows the obtained

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Fig. 4.8. SE transfer function dependency on ﬂuid saturation for (a) limestones and (b) sandstones.

porosity dependences of SE transfer function normalized by the pattern conductivity W i ¼ W =s for terrigene rocks. General character of the dependency and the order of magnitudes of the transfer function agree with those theoretically predicted, but a wide spread of the data is apparent (the upper panel). To some extent, this can be explained by imperfection of measurement technique, but there are also another, more fundamental reasons for this. If the whole set of patterns is divided into limestone and sandstone groups, the data scatter reduces and correlation coefﬁcient of linear regression increases (the two bottom panels). This is quite understandable: be the SE theory and its underlying model of the medium ever so perfect, they are incapable of allowing for the whole variety of pore space shapes of rocks and its gas–ﬂuid saturation as well as such important factors as clay contents

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Fig. 4.9. SE transfer function dependency on porosity for (a) sedimentary rocks, (b) limestones, (c) sandstones.

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in rocks, roughness of the solid phase surface and many others. All this necessarily predetermines the spread of SE transfer function values over a set of various patterns not grouped on the basis of any criterion or even united by common linotype. With the reduction of the pattern collection according to some additional features (e.g. geographical or stratigraphical), the resulting regression dependencies become more compact and coefﬁcients of correlation between W i and the studied petrophysical parameters increase. The laboratory studies carried out lead to a conclusion that theoretical expressions for transfer functions like those of Equation (4.11) are capable of characterizing only a few, although important, features of a real relationship between SE effect and petrophysical parameters of the medium. For experimental data interpretation, a parallel arrangement of laboratory studies on the rock samples of the studied object is necessary.

4.6. EXPERIMENTAL FIELD AND BOREHOLE SEISMOELECTRIC STUDIES In order to study the practical possibility of SE measurements in natural conditions and to corroborate their geophysical informativeness following from the theory, necessary equipment has been designed, and ﬁeld and well measurements were carried out. Fig. 4.10 displays the results of SE logging (Svetov et al., 2001; Svetov et al., 2004) of a well located within an edge zone of the Near-Caspian depression. Measurements were performed at a ﬁxed frequency of 9.5 kHz, the length of the sonde was 0.5 m. Elastic ﬁeld was excited by magnetostriction transmitter. Shown in the Fig. 4.10a are the recorded curves of pressure and electric ﬁeld strength measured directly in the well. Electric ﬁeld signal amounted to hundreds of microvolts and was many times as high as the noise. Within a metal-cased borehole section (down to the depth of 410 m) the signal dropped sharply. Within an open section (Figs. 4.10b, c) the electric ﬁeld is differential; its variations correlate with changes in the acoustic ﬁeld intensity and represent the speciﬁc features of the geological section known from coring and other logging methods. In particular, the enhanced electric signals (and SE transfer function) are associated with intervals of ﬁnest-pored carbonate rocks. Reliability of the obtained results is conﬁrmed by numerous repeated measurements. Detailed ﬁeld SE studies were carried out at the geophysical test area of the Moscow State University close to v. Aleksandrovka of Kalouga region (Ugra river ﬂood-plain) (Svetov et al., 2001; Svetov et al., 2004). The upper part of the geological section of interest consists of morainic deposits interstratiﬁed with layers of limestones, sands and clays. For ﬁeld SE studies, an eight-channel SE equipment has been designed (four seismic and four electromagnetic channels). Elastic oscillations were excited by the sledge blows. Detection of seismic signals was carried out using seismic detectors SV-20. Electric signals were measured at 1–2 m long grounded lines MN. Measurements of seismic and electric ﬁelds at each point of the proﬁle were carried out at progressively increasing separation of the blow site from the ﬁxed

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Fig. 4.10. SE logging results for a borehole in Saratov region: directly measured pressure P and electric ﬁeld E (a) same data averaged by low-frequency ﬁltering (b) and SE transfer function E/P (c).

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Fig. 4.11. Experimental seismograms (a) and electrograms, (b) obtained at one of the proﬁle points in Aleksandrovka at different distances to the blow point.

measurement point. Fig. 4.11 shows an example of a seismogram of the elastic ﬁeld vertical component measured at a certain point (a) and an electrogram oriented in the proﬁle direction of the horizontal component of the electric ﬁeld (b). Separate signal traces are indexed by the distance to the blow point in meters. In the righthand parts of all plots, the maximum values are written of seismic or electric signals

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(in m/s and V/m, correspondingly) that deﬁne vertical scales at each of signal traces. From the ﬁgure, a conclusion can be drawn about a complex character of seismic (and the more so as to electric) ﬁeld peculiar to complexly structured near-surface zone. At small time intervals, the refracted waves are recorded (seismic event G1) and at long time intervals intense low-frequency surface waves (seismic event G2). Intermediate time intervals are ﬁlled with mutually interfering waves of various types. Seismic waves propagation velocities are low, the values varying from 200–250 m/s (surface waves) to 400–500 m/s (refracted waves). Electric signals show a rather good correlation with seismic signals, which speaks for their frozenness into the elastic ﬁeld. Fast waves are distinguished only at small spacing r ¼ 4–12 m. Based on the results of the measured data processing (Svetov et al., 2004), the SE transfer function W ¼ E=vðVs=m2 Þ along the proﬁle was plotted (Fig. 4.12). The curve correlates with geological and geophysical data. In particular, the minimum in SE transfer function clearly marks the break in the clay water-resisting rock

Fig. 4.12. SE transfer function along the proﬁle in Aleksandrovka.

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accompanied by the decrease in rock humidity but not revealed by any other methods. Rather large amount of experimental ﬁeld SE studies are by now carried out with 24-channel SE equipment designed in the GEMRC IPE RAS (12 seismic and 12 electric channels). These works conﬁrmed, in general, the developed theory of SE phenomena and the prospects of seismoelectrics application in solving the problems of engineering, hydro- and mining geology. Resuming the studies carried out, one can say that their main result is the creation of theoretical basics for SE-prospecting studies both in borehole and ﬁeld modiﬁcations, and the experimental veriﬁcation of a certain adequacy of the developed theory to the really observed phenomena and the yielded from the theory conclusions about new informational possibilities of SE method of geophysical prospecting. To the moment, we believe the major applications of this method to be the logging studies of gas–oil and hydrogeological wells and the study of the upper part of geological section for the purposes of solving the shallow engineering, ecological and hydrogeological problems, although the use of the method in the oil–gas exploration is envisaged in the works by other Russian institute (VNIIGeophysica) with powerful vibrators. Acknowledgments The chapter is based on the results of joint work of the author with his collaborators P. Aleksandrov, V. Ageev, O. Ageeva, S. Karinskii, S. Kevorkyantz, Yu. Kuksa and with the professor of Saratov State Technical University V. Gubatenko to whom the author expresses his deep gratitude. The work was supported by RFBR grant No. 05-03-64467 and Shlumberger Corporation grant CRDF RGE1295.

REFERENCES Ageeva, O.A., Svetov, B.S., Sherman, G.H. and Shipulin, V.V., 1999. Seismoelectric effect of the second kind in rocks (laboratory studies) (in Russian). Novosibirsk. Geol. Geoﬁs., 40, 8: 1251–1257. Biot, M.A., 1956. Theory of propagation of elastic waves in a ﬂuidsaturated porous solids. J. Acoust. Soc. Am., 28: 168–186. Frenkel, Ya. I., 1944. To theory of seismic and seismoelectric phenomena in wet soil (in Russian). Izvestia, Geogr. Geoﬁs., 8, 4: 133–150. Ivanov, A., 1940. Seismoelectric effect of the second kind (in Russian). Izvestiya, Geogr. Geoﬁs., 5: 699–727. Ivanov, A., 1949. Seismoelectric effect of the ﬁrst kind in near-electrode regions (in Russian). Doklady Akad. Nauk SSSR, 68: 699–727. Ozerkov, E.L., Ageeva, O.A., Osipov, V.G., Svetov, B.S. and Tikshaev, V.V., 1998. On the vibroimpact inﬂuence on electric properties of geological medium (in Russian). Geoﬁsika, 3: 30–34. Pride, S.R., 1994. Governing equations for the coupled electromagnetics and acoustics of porous media. Phys. Rev., B, 50: 15678–15696.

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Schmitt, D.P., Bouchon, M. and Bonnet, G., 1988. Full-wave synhtetic acoustic log in radially semiinﬁnite saturated porous media. Geophysics., 53, 6: 807–823. Svetov, B.S., 2000. To theoretical substantiation of seismoelectric method of geophysical prospecting (in Russian). Geoﬁsika, 1: 28–39. Svetov, B.S. and Gubatenko, V.P., 1999. Electromagnetic ﬁeld of mechanoelectric origin in porous ﬂuid-saturated rocks: I. Statements of the problem (in Russian). Fis. Zemli, 10: 67–73. Svetov, B.S., Ageeva, O.A. and Lisitsyn, V.S., 2001. Logging studies of seismoelectric phenomena (in Russian). Geoﬁsika, 3: 44–48. Svetov, B.S., Ageev, V.V., Ageeva, O.A., Alexandrov, P.N. and Gubatenko, V.P., 2004. Seismoelectric methods of prospecting and logging (in Russian). Geoﬁsika, 1: 44–48. Svetov, B.S., Ageev, V.V., Alexandrov, P.N. and Ageeva, O.A., 2001. Some results of experimental ﬁeld seismoelectric studies (in Russian). Geoﬁsika, 6: 47–53.

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Part II: Forward Modeling and Inversion Techniques

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Chapter 5 3-D EM Forward Modeling Using Balance Technique V.V. Spichak1 and M.S. Zhdanov2 1

Geoelectromagnetic Research Center IPE RAS, Troitsk, Moscow Region, Russia 2 University of Utah, Salt Lake City, USA

In this chapter, the balance technique for the forward modeling of electromagnetic ﬁelds in 3-D media is being considered. Section 5.1 gives a brief review of numerical and analog approaches to the forward problem solution. Numerical methods, in turn, are subdivided into two large groups: integral equation methods and differential equation methods. A comparative analysis of these two groups of methods and hybrid approaches based on their combination is presented. Section 5.2 describes the main characteristic features of a balance technique for EM ﬁeld calculations in media with arbitrary 3-D distribution of the electric conductivity. At the same time, if the electric conductivity model shows a certain symmetry type, the above-mentioned purpose can be achieved using smaller computer resources. In particular, for 3-D models with a vertical axial symmetry the vector problem reduces to a scalar one. Section 5.3 addresses an algorithm for numerical calculation of quasi-stationary electromagnetic ﬁelds in a 3-D axially symmetric media based on a ﬁnite element modiﬁcation of the balance method. It is worth mentioning that adequate program realization of the developed methods is of great importance in getting accurate results. The codes based on the algorithms discussed in this chapter were tested in different ways. In Section 5.4 of this Chapter the solution accuracy is analyzed and the data of test calculations are presented.

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40005-6

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5.1. MODERN APPROACHES TO THE FORWARD PROBLEM SOLUTION To solve the forward problem (in a frequency domain), implies determining the electromagnetic ﬁeld (E,H) satisfying the Maxwell equations, ~ oÞE þ jext ; r H ¼ sðr;

r E ¼ iomH

ð5:1Þ

at a given frequency o, from the distribution of generalized complex electric con~ oÞ ¼ s io (m is the magnetic permeability, e the permittivity, and ductivity sðr; jext ðr; oÞ the extraneous current density) speciﬁed in a certain spatial region that includes an inhomogeneity. In this chapter, we assume for simplicity that j ext ¼ 0; m ¼ m0 ¼ 4p 107 H/m, where m0 is the free-space magnetic permeability, and neglect the displacement currents, i.e., s~ s. Various approaches to the solution of this problem exist, all of which fall under two groups comprising numerical and analog methods (see the reviews by Hohmann (1983, 1988), Varentsov (1983), Zhdanov and Spichak (1984), Cerv (1990), and Zhdanov et al. (1997)). Currently, numerical computer-aided modeling of electromagnetic ﬁelds has become a powerful and relatively easy-to-access tool for analyzing complicated situations. Whereas, formerly, the scope of geophysical consideration was conﬁned to simpliﬁed model media (cases of E- and H-polarization (see, e.g., Dmitriev, 1969; Jones and Pascoe, 1972; Varentsov and Golubev, 1982) and thin-sheet models (see, e.g.,Weaver, 1979; Debabov, 1980; McKirdy and Weaver, 1984; McKirdy et al., 1985; Singer and Fainberg, 1985), the advent of powerful fast computers made it possible to numericaly model electromagnetic ﬁelds excited by 3-D sources in a 2-D or three-dimensional medium or by 2-D sources in a three-dimensional medium. 5.1.1. Methods of integral equations The integral equation method for the numerical computation of electromagnetic ﬁelds was pioneered by Dmitriev (1969). The basic ideas of this method were developed by Raiche (1974), Hohmann (1975), Tabarovsky (1975), and Weidelt (1975). Subsequently, the integral equation method was successfully elaborated by Ting and Hohmann (1981), Wannamaker et al. (1984a,b), Hvozdara (1985), Dmitriev et al. (1985), Hvozdara et al. (1987), Khachaj (1988), Dmitriev and Pozdnyakova (1989), Wannamaker (1991), Xiong (1992), Pankratov et. al. (1995), Singer (1995), Singer and Fainberg (1995, 1997), Avdeev et al. (1997), Farquharson and Oldenburg (1999), Lee et al. (1999), Portniaguine et al. (1999), Singer et al. (1999), Xie and Li, (1999), Xiong et al. (1999a), Zhdanov et al. (2000), etc. (see also Chapter 6 and references therein). Parallel to the method of volume integral equations (VIE) that involves integration over the entire volume of the anomalous domain, a method of surface integral equations (SIE) implying integration only over the boundary of the domain, has been developed (see, e.g., Oshiro and Mitzner, 1967; Dmitriev and

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Zakharov, 1970; Tabarovsky, 1971; Smagin, 1980; Smagin and Tsetsokho, 1982; Liu and Lamontagne, 1999; (Xie and Li, 1999)). Let us brieﬂy consider these two approaches. 5.1.1.1. The method of volume integral equations (VIE) VIE technique is based on the numerical solution of the second-type Fredholm equation, which can be derived from the Maxwell equations using Green functions: ZZZ e EðrÞ ¼ Eb ðr0 Þ þ iom0 ðs sb ÞG^ ðr=r0 ÞEðr0 Þ dV 0 ð5:2Þ ^

where Eb is a background ﬁeld calculated by assuming s ¼ sb ; G e ðr=r0 Þ the electrictype Green tensor, r and r0 are coordinates of the observation and source points, respectively, and V the region in question. (Hereinafter, for the sake of deﬁniteness, only the equation for the electric ﬁeld E will be considered. Once the numerical solution to this equation is obtained, the magnetic ﬁeld H can be readily calculated from, e.g., the second Maxwell’s equation (see Section 5.2.3).) The pivotal idea of this approach is as follows. The anomalous region is divided by a spatial grid into cells. Within each cell the ﬁeld is assumed to be constant. Therefore, in the second term of (5.2), it can be taken outside the integral sign. To obtain a system of linear algebraic equations for the ﬁeld, one has to only calculate the respective tensor coefﬁcients within each unit cell. Let us discuss the main advantages and drawbacks of this method. The advantages comprise its greater physical transparency compared, e.g., with the methods of differential equations. In addition, as is apparent from (5.2), the second term on the right-hand side is non-zero only if sasb , i.e., integration must be carried out only over the anomalous region alone. Finally, the numerical approximation (5.2) does not involve the unstable procedure of numerical differentiation, which is typical of most of the differential equation methods. Signiﬁcant difﬁculties arise in the numerical approximation of the second term on the right-hand side of Equation (5.2). In particular, determination of the Green e tensor components G^ is a non-trivial computational problem; the solution involves the Hankel transforms (see eg., Hohmann, 1975; Weidelt, 1975; Farquharson and Oldenburg, 1999), the linear ﬁltration technique (Das and Verma, 1981; Verma and Das, 1982), and the method of path deformation in a complex plane of the integration variable (Tabarovsky, 1971; Dreizin et al., 1981). Computation of the tensor coefﬁcients can be as time consuming as solving the system of linear algebraic equations. Therefore, it is a ‘‘bottleneck’’ of the technique, and the success of the whole approach thus depends on how effectively this problem is solved. Another difﬁculty arises while solving the system of linear algebraic equations. Dense matrix of the system makes limited computer resources a critical factor. In turn, the limitation on the dimensionality of algebraic system ultimately entails limitations to the mathematical model: ‘‘we can only compute the ﬁelds for bodies that are not too large’’ (Dmitriev and Farzan, 1980). Xiong (1992) overcomes this drawback by partitioning the scattering matrix into many block submatrices and

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solving the whole system by a block iterative method. This reduces the computer memory requirements and the time of computation. (It is worth mentioning in this connection that a drastic reduction of the computer resources could be achieved by using the so-called ‘sparse matrix technique’ (Poggio and Miller, 1973) that implies ignoring the interaction among sufﬁciently remote areas of the modeling domain; the latter corresponds to zeroing those elements of the matrix that are small compared with the diagonal ones (Dmitriev and Pozdnyakova, 1989). Another simpliﬁcation (which also reduces the class of objects being modeled) is to solve the problem within the long-wave approximation. This enables one, ﬁrst, to simplify computation of the Green tensor (Hvozdara, 1981) and, second, to use in the numerical solution of (5.2) the Born approximation (Torres-Verdin, 1985; Torres-Verdin et al., 1992), localized non-linear approximations (Habashy et al., 1993; Torres-Verdin and Habashy, 1994), quasi-linear approximation and series (Zhdanov and Fang, 1996, 1997) and quasi-analytical series (Zhdanov et al., 2000). Finally, Portniaguine et al. (1999) suggested a way of reducing the time of computations by the use of ‘‘compression matrix’’ that converts the original dense matrix into a sparse one by constructing an interpolation pyramid in multiple dimensions. In evaluating the advantages and drawbacks of the VIE method on the whole, it should be noted that this method seems to be giving the most accurate results – in a comparatively short time – for anomalies that are small compared with the wavelength within the anomalous region, have a simple shape, and occur not too close to the point of ﬁeld calculation (although Wannamaker (1991) succeeded in overcoming the latter drawback). 5.1.1.2. The method of surface integral equations SIE technique makes use of electromagnetic ﬁeld representation as an integral over the surface of a domain by means of Stratton–Chu formulae (Dmitriev and Zakharov, 1970; Zhdanov and Spichak, 1983), method of auxiliary (ﬁctitious) sources (Tabarovsky, 1971), method of potentials (Smagin, 1980), or the method employing the Helmholtz scalar equations (Liu and Lamontagne, 1999). After passing the limit with the observation point tending to the boundary of the domain from inside and from outside in turn, one can, using the continuity of the tangential components of electromagnetic ﬁeld on the boundary, obtain the necessary equations only over the boundary of domain V with respect to unknown ﬁeld densities. Numerical solutions to these equations involves eg., the Krylov–Bogolyubov method, variational approaches, and Bubnov–Galerkin type methods. In particular, Smagin and Tsetsokho (1982) obtained the system of linear algebraic equations using the collocation method, and the densities sought for are approximated by smooth ﬁnite functions, which form a ﬁnite partition of unity over the surface of the domain. This technique has an obvious advantage over the integral equation method. Numerical solution to the equations written over the surface of the domain rather than over its volume involves a considerable reduction in the dimensionality of the

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system of linear algebraic equations. This is especially important when modeling large inhomogeneities, because the ratio of the dimensions of respective matrices decreases with an increase in the characteristic linear dimension L as 1/L. Despite this advantage, however, the use of SIE technique is limited to modeling homogeneous bodies with rotational symmetry, embedded in a homogeneous space (half-space) (Oshiro and Mitzner, 1967; Smagin and Tsetsokho, 1982; Liu and Lamontagne, 1999). This is due to the imperfection of the mathematical apparatus for modeling inhomogeneities with an arbitrary distribution of electric conductivity sa ¼ s(x,y,z) embedded in a horizontally layered section, and to the difﬁculties arising in the step of numerical approximation of singular integral equations and of solving the resulting system of linear algebraic equations with an ill-conditioned matrix. Therefore, integral equation methods are most helpful when modeling inhomogeneities of comparatively simple shape and small size (compared to the wavelength within them) embedded in a horizontally layered medium with a small number of layers. 5.1.2. Methods of differential equations Advances in geoelectrics call forth computing 3-D electromagnetic ﬁelds in complicated situations when the inhomogeneity exhibits an arbitrary shape and dimension (in particular, it may not be local) and the anomalous electric conductivity in the model varies arbitrarily, as, e.g., in regional models. In this case, as pointed out above, integral equation methods are of limited use. The only viable approach to such problems involves either a direct solution of differential equations with partial derivatives, or use of the so-called differential equation methods, the two main ones being the ﬁnite difference (FD) technique and the ﬁnite element (FE) technique. We will list the main features of their application to the problem of numerical modeling of 3-D magnetotelluric ﬁelds. 5.1.2.1. The FD technique The FD technique based on the FD approximation of derivatives, is used for the numerical solution of a differential equation with respect to the electric or magnetic ﬁeld in a certain spatial region that contains an inhomogeneity (see, eg., Jones and Pascoe, 1972; Lines and Jones, 1973; Jones, 1974; Jones and Lokken, 1975; Hibbs and Jones, 1978; Zhdanov and Spichak, 1980; Lam et al., 1982; Zhdanov et al., 1982; Yudin, 1980, 1983; Spichak, 1983a, b; Zhdanov and Spichak, 1989, 1992; Mackie and Madden, 1993; Mackie et al., 1993, 1994; Druskin and Knizhnerman, 1994; Smith, 1996a, b; LaBrecque, 1999; Newman, 1999; Spichak, 1999a, b, 2000; Weidelt, 1999; Weaver et al., 1999; Xiong et al., 1999b; Newman and Alumbaugh, 2000; Sasaki, 2001; Wang and Fang, 2001; Fomenko and Mogi, 2002; Newman et al., 2002; Weiss and Newman, 2002): DE r ðr EÞ þ k2 E ¼ 0 ðk2 ¼ iom0 s; Re k40Þ

ð5:3Þ

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or

r

1 r H þ iom0 H ¼ 0 s

ð5:4Þ

which follow directly from (5.1). The advantages of this technique compared to integral equation methods are its greater versatility, the simplicity of its numerical implementation, i.e., band structure of the matrix of resulting system of linear algebraic equations. The latter circumstance is vital, because it signiﬁcantly reduces the time and amount of operational computer memory required for solving the problem and, consequently, enables modeling the targets of large size or with complicated conductivity distribution in it. The main factors inﬂuencing the effectiveness of this method are as follows: choice of the appropriate governing equation; selection of the discretization scheme; approximation of the second derivatives in (5.3) or (5.4); specifying boundary conditions that are valid at non-indeﬁnite distance from the anomaly; conservation of currents and magnetic ﬁeld ﬂux; accuracy when solving the problems at frequencies approaching the static limit (similar problem is seen in the FE technique); and accuracy when solving the problems with sharp conductivity contrasts. Since the ﬁrst 3-D modeling results were obtained with this method by F. Jones (see references. above) a number of improvements were made that signiﬁcantly increased the effectiveness of FD technique. Zhdanov and Spichak (1980) suggested a balance technique for approximation of the FD equation in electric ﬁeld, which enabled the accuracy of the results to increase due to reduction from the unstable calculation of the second-order derivatives to the ﬁrst-order ones. Spichak (1983a) proposed to use a current divergence-free condition in the balance technique, which increases the accuracy of the results due to disappearance of the second term in Equation (5.3) in regions with zero conductivity gradient (see the details in Section 5.2 below) and also at the static limit. Zhdanov et al. (1982) and Spichak (1985, 1999a, 2006) introduced the asymptotic boundary conditions that greatly diminished the dimensions of the modeling domain and signiﬁcantly increased the accuracy of the forward modeling. Smith (1996a) proposed to use a staggered-grid method (pioneered by Yee, 1966) that guarantees automatic conservation of currents and magnetic ﬂux on the grid (though, at the cost of some inconvenience when computing some ﬁeld components on the Earth’s surface or the ﬁeld transformations involving the components speciﬁed on different grids). Davydycheva and Druskin (1999) and Weidelt (1999) extended this approach to the anisotropic media. Druskin and Knizhnerman (1994) and Druskin et al. (1999) developed a spectral Lanczos decomposition method (SLDM) with Krylov sub-spaces generated from the inverse of the Maxwell operator. SLDM enables acceleration of the forward modeling owing to possibility of getting the solution for the whole frequency range practically at the cost of solution for a single frequency.

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Smith (1996b) developed a static correction procedure that explicitly enforces the electric current and magnetic ﬁeld divergence-free conditions, which, in turn, increases the accuracy of the EM ﬁeld calculation when the frequency tends to zero. In order to accelerate the solution of the forward problem at low induction numbers (LINs), Newman et al. (2002) and Newman and Alumbaugh (2002) proposed a LIN pre-conditioner. It is based on splitting the electric ﬁeld into curl- and divergencefree projections that remove the null space of the discrete curl–curl operator in the solution process. A number of efﬁcient solvers and pre-conditioners are used at present to achieve both a good accuracy and high convergence rate of the iteration process, especially when high conductivity contrasts increase the condition number of the matrix of the system of linear equations (SLE) (see Fomenko (1999) and Varentsov (1999) for a comparative analysis of different pre-conditioners and solvers). Finally, Zhdanov et al. (1982) and Spichak (1999a) constructed internal criterions for controlling the accuracy of the forward modeling results (see Section 5.2.4).

5.1.2.2. The FE technique The ﬁnite element technique is especially useful for modeling the regions with complicated geometry or on relief Earth surface. Depending on its speciﬁc applications, this method is interpreted as either the method of weighted residuals or a variational procedure. Equivalency of the equations obtained with these two techniques provides the grounds for their joint consideration in the context of differential equation methods applied (see, e.g., Reddy et al., 1977; Pridmore, 1978; Pridmore et al., 1981; Boyse et al., 1992; Livelybrooks, 1993; Mogi, 1996; Haber, 1999; Sugeng et al., 1999; Zunoubi et al., 1999; Zyserman and Santos, 1999). In the work of Reddy et al. (1977) the method of weighted residuals (of the Galerkin type) was applied such that the entire region under study was divided into hexahedral units, piecewise continuous functions were adopted as basis functions, and the unknown ﬁeld components were approximated by third-degree polynomials. Modeling results of a homogeneous prismatic body embedded in the lower half-space showed that even with a comparatively weak contrast in electric conductivity (1:10) the results were accurate to within 10%. At the same time, the choice of more complex basis functions (and weights) that better describe the unknown function led to a dramatic deterioration of the properties of the matrix obtained by solving the SLE (Petrick, 1978). In the works of Pridmore (1978), Pridmore et al. (1981), and Livelybrooks (1993), the variational approach was used to obtain FE equations. Solution of equations (5.1) in the ﬁnite region V is equivalent to ﬁnding the stationary point of the energy functional: ZZZ ðr E r E k2 E EÞ dV ð5:5Þ F ðEÞ ¼ V

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(We are determining a stationary point, not the minimum of the functional F, because the differential operator on the right-hand side of (5.5) is not positively deﬁned and hence does not satisfy the conditions of the Minimum theorem.) As in the case of FD technique, one should be cautious in solving the variationally formulated problem (5.5) at low frequencies (say, o p 0.1 Hz), when the contribution of the second term on the right-hand side becomes comparable with the round-off errors. In addition, attempts to make use of the most prominent advantage of the ﬁnite element technique, namely, the possibility of dividing the region into elements of any conﬁguration, may involve oscillations or even divergence of the solution process (Pridmore et al., 1981). Haber (1999) proposed a potential-current formulation of the initial problem based on the Helmholtz decomposition. The matrix of the resulting SLE has a property of the diagonal dominance, which, in turn, allows its efﬁcient solution even for very high conductivity contrasts. An alternative way to overcome the above problem was proposed by Sugeng et al. (1999), who have reformulated the FE method using a single vector shape function at each edge of the grid instead of three scalar functions deﬁned in corner nodes. Because of this, one has to only solve for tangential components of the electric ﬁeld along the edges of cells. Thus, avoiding a solution for normal components of the ﬁeld allows modeling for high contrasts. Finally, Zyserman and Santos (1999) proposed a ‘‘mixed hybrid domain decomposed iterative nonconforming’’ method, which is based on the iterative decomposition of the model domain and solving the local linear systems of equations. This approach allows signiﬁcant reduction of memory and time required for SLE solution; however, the convergence of the appropriate procedure to the correct solution still has to be proved. It is worth mentioning that this method is similar to the Schwarz alternative iteration FE method developed in Yudin (1983) for modeling magnetotelluric ﬁelds. Despite the fact that, theoretically, ﬁnite elements are well suited to modeling the geological sections with complex distributions of electric conductivity, the ﬂexibility of the method is attained through considerable computing efforts associated with the use of complicated FEs. To characterize differential equation methods in general, it should be emphasized that they are generally more versatile compared to integral equation methods. However, to increase their computing efﬁciency, a number of problems discussed above has to be solved. 5.1.3. Mixed approaches One way to reﬁne the techniques of numerical modeling of 3-D electromagnetic ﬁelds is to apply hybrid (‘‘mixed’’ or ‘‘non-classical’’) approaches that blend the advantages of the differential and integral equation methods. This implies the differential equation method to be used inside the modeling domain, which allows considering a model with an arbitrary distribution of the electric conductivity, and integral relations between ﬁeld components to be employed at the boundary, which

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ensures the possibility of limiting the modeling domain to within an area that only slightly exceeds the inhomogeneity in dimensions. The ﬁrst step in this direction was by Weidelt (1975), who used the Green functions apparatus to remove the regions of the normal section above and below the inhomogeneities considered: ZZ e a E ðr0 Þ ¼ f½r G^ ðr0 jrÞ½n Ea ðrÞg dS ð5:6Þ S

where S is the surface of the discontinuity separating the layers, Ea the anomalous (secondary or scattered) ﬁeld (Ea E Eb ), and the points r0 are selected in the region to be removed. The same method was used by Yudin (1981b) Takacs and Turai (1986). In Yudin (1981b), the vertical dimensions of regions above and below the layer with inhomogeneities are consecutively reduced during the iterative procedure. This is performed by reﬁning the boundary conditions at the upper and lower boundaries of the model domain after analytical continuation of the ﬁeld spectra. The efﬁciency of hybrid schemes is further improved by reducing the modeling domain not only vertically but horizontally as well. Thus, in Petrick (1978), Pridmore and Lee (1980), Lee et al. (1981), Best et al. (1985), and Gupta et al. (1987), an area of limited dimensions is covered over a small distance by a ﬁnite element grid. Applying the ﬁnite element technique to the inner part of the domain and the integral equation technique to the outer part gives rise to two systems of equations: (1) FE equations in the inner nodes with the respective part of the matrix showing a characteristic band structure and (2) equations that relate the unknowns on the boundary and inner nodes (with the respective part of the matrix being ﬁlled) in accordance with the formula ZZZ e a sa G^ EðrÞ dV ð5:7Þ E ðr0 Þ ¼ where r0 lies on the boundary of the domain V. Hybrid schemes are commonly grouped into two types: direct and iterative. In direct hybrid techniques the matrix of the system is inverted directly, whereas in iterative techniques the values of the unknowns at the domain boundaries are at ﬁrst assumed to be known, and then the values in the inner nodes are computed, which are recalculated by (5.7) into boundary values. As it appears from the estimation performed in Lee et al. (1981), iterative hybrid schemes are generally more sparing in terms of the operational memory resources and CPU time used. However, even small spatial grids (10 10 10 ¼ 1000 nodes) require 2 106 operational memory words. So, an essential increase in the number of grid nodes can only be achieved by employing slow external memory, which, in turn, increases CPU time considerably. Therefore, despite certain advantages of hybrid systems, their practical application entails a number of difﬁculties. In particular, the requirements on computer

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resources remain rather high, convergence of the iterative procedure is not guaranteed (if the most resource-saving iterative scheme is used), and no internal criteria for the accuracy of the results obtained are used. (The last observation, which also applies to the rest of the numerical approaches just mentioned, is perhaps most topical for hybrid schemes). As follows from the above discussion of the various approaches, numerical modeling of 3-D electromagnetic ﬁelds in complex media is a rather painstaking exercise that involves the problems of constructing an effective algorithm and its numerical implementation. The latter circumstance is further aggravated by the fact that, in full-scale numerical modeling, requirements to computer resources, especially the volume of operational memory, exhaust the capacities of computers used in actual practice. Hence, where possible, it is rewarding, when posing the problem, to bear in mind its peculiarity to obtain the most effective solution of the narrow spectrum of issues being considered. Apparently, such an increase in efﬁciency can only be achieved at the expense of versatility. In some cases, however, this is justiﬁed. In particular, models in which a 3-D inhomogeneity is approximated by long, horizontally inhomogeneous thin sheets are of great practical importance. Such an approximation is valid if the thickness of the surface layer is small compared to that of the skin layer of the surface-layer material and to the depth of penetration of the ﬁeld into the underlying medium. In such models, the electric conductivity of the layer depends only on the two horizontal coordinates, whereas the ﬁelds themselves are three-dimensional. Numerical modeling based on the use of such models has been termed thin-sheet (or quasi-3-D) modeling. Most studies based on this approach (see, e.g., Dawson and Weaver, 1979; Weaver, 1979; Singer and Fainberg, 1985) make use of the Price–Scheinmann or Dmitriev boundary conditions (Berdichevsky and Zhdanov, 1984). Reﬁning this approach has afforded the modeling of inhomogeneous layers of ﬁnite thickness (Zhdanov and Tikhomirova, 1982a,b). It thus appears that further advances in the application of this approach may only be achieved by the use of ever more precise boundary conditions, which would considerably enlarge the spectrum of models to be studied. Finally, we will mention the original approach to forward numerical modeling, which is based on the method of trajectory integration in a state space (Dreizin et al., 1981). Despite the obvious advantage ensuing from the fact that CPU time does not increase with the dimensionality of the problem, the applicability of this approach is, unfortunately, restricted to rather simple models with a priori known Green function or, else, to models with very weak gradients in the electric conductivity. 5.1.4. Analog (physical) modeling approaches Analog, or physical, modeling implies a physical nature for the model of the primary ﬁeld source, the medium, and the inhomogeneity. Two basic approaches are distinguished here: the continuous media method and the electric circuit method (Tetelbaum and Tetelbaum, 1979). In the former approach, the model is deﬁned by the ﬁeld of an electric current in a continuous medium (Dosso, 1966), and the latter

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involves the use of electric circuits with concentrated parameters (Brewitt-Taylor and Johns, 1980). The continuous media method is based on the use of the electrodynamic similitude criterion (Stratton, 1941): mosL2 ¼ inv

ð5:8Þ

where L is the characteristic linear dimension of the model. Among the advantages of this approach, one should recall its simplicity and low cost, the uniqueness of the equipment, and the possibility of modeling the media with sharp electric conductivity contrasts (Kuznetsov, 1964; Dobrovolskaya et al., 1970; Berdichevsky et al., 1987). Some of these advantages, however, prove to be drawbacks if the issue is addressed in a broader context, namely, from the standpoint of electromagnetic ﬁelds modeling in real situations. The extremely small number of physical modeling installations all over the world virtually obviates their application as a tool for analyzing the observed ﬁelds, let alone the repeated use of the modeling results, which is only possible through coupling physical installations to computing facilities. In addition, one should note a number of technological difﬁculties, such as the choice of materials with a sufﬁciently wide range of electric conductivity variations but not subject to frequency dispersion, the painstaking technological implementation of models for multilayer media, the poorly developed modeling technique for use with hard materials and low-melting metals, and the difﬁculties arising in obtaining experimental curves over a broad frequency range. The other important approach in physical modeling is the electric circuit method, which in actual practice is realized by two modiﬁcations. The ﬁrst is based on discretization of the modeling domain followed by the representation of the elementary volumes obtained by means of elements of an electric circuit (replacement schemes) and measurement of voltages and currents in the circuit, which model the electric and magnetic ﬁelds, respectively. The other is based on obtaining replacement circuits directly from FD equations that describe the ﬁeld being modeled (Brewitt-Taylor and Johns, 1980). As pointed out in Tetelbaum and Tetelbaum (1979, p. 211), ‘‘the prime advantage of electric circuits is that they enable one to model three-dimensional ﬁelds, which are described by equations with a right-hand side.’’ Another important advantage of this approach is that it can be used in hybrid analog/numerical installations. Note that with this approach the use of the diacoptics concept of Kron (1972) may prove noteworthy. Despite the outward merits of the above approach, however, it is not free of the characteristic ﬂaws of both the physical modeling techniques (measurement errors and technical difﬁculties in constructing the models) and mathematical modeling approaches (e.g., errors ensuing from the discretization of the modeling domain and ﬁeld equations). To summarize the advantages and drawbacks of physical modeling just mentioned, it is worth noting that its possibilities are limited to comparatively simple conductivity models of the medium, which, needless to say, reduces the ﬁeld of its applicability.

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Mathematical modeling of such ﬁelds allows a more comprehensive understanding of their morphology than is yielded by physical modeling methods. In addition, the study of how sensitive the various components of the ﬁeld are to the variations in the model parameters, which necessitates multivariant calculations, can be performed in actual practice only on the basis of numerical modeling. Therefore, by virtue of the above reasons, mathematical modeling seems to be the most suitable tool for analyzing 3-D electromagnetic ﬁelds in complex geophysical situations. We have discussed the main techniques and approaches currently in use for modeling 3-D electromagnetic ﬁelds. Collating their advantages and drawbacks shows the differential equation methods to be the most versatile and best suited for modeling the broadest spectrum of practical situations. However, their application involves a number of theoretical, methodological, and computing problems that affect the modeling efﬁciency. In the following sections, these crucial issues will be addressed in the context of those approaches to the numerical modeling of 3-D electromagnetic ﬁelds that make use of the balance technique. We will also provide examples of test calculations of electromagnetic ﬁelds for some typical 3-D models – among others, for those included in the international project on the Comparison Of Modeling Methods for ElectroMagnetic Induction problems (COMMEMI) (Zhdanov et al., 1997).

5.2. BALANCE METHOD OF EM FIELDS COMPUTATION IN MODELS WITH ARBITRARY CONDUCTIVITY DISTRIBUTION Analysis and interpretation of the results of the array electromagnetic sounding of the Earth is often done by means of simpliﬁed (one- or two-dimensional) models of medium. It is thus interesting to study peculiarities of electromagnetic ﬁeld behavior in the Earth with three-dimensional conductivity distribution. Several approaches to the solution of this problem have been developed in the recent 10–15 years (see Section 5.1 and, for instance, a review paper by Zhdanov et al. (1997)), but the substantial computational difﬁculties arising in certain steps of realization of the proposed algorithms impede their practical application in electromagnetic ﬁelds calculation in three-dimensional inhomogeneous media. In publications of Zhdanov and Spichak, (1980, 1989), Spichak, (1983a, 1985, 1999a), basic principles of constructing an algorithm were formulated for this problem bearing on a moderate-speed computer with a limited core memory (for instance, mini-computers). In what follows, we will describe this algorithm and demonstrate its application to solution of the forward problem in 3-D axial symmetric situations. 5.2.1. Statement of the problem Let some domain O in the Earth’s crust or upper mantle be isotropic, nonmagnetic (m m0 is the permeability of free space), and has a three-dimensional

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distribution of electric conductivity sðx; y; zÞ that can be represented in the form 8 P 2 O1 > < sðzÞ; sðPÞ ¼

> :

sðx; zÞ; sðx; y; zÞ;

P 2 O2 P 2 O3

where O ¼ O1 [ O2 [ O3 , P ¼ Pðx; y; zÞ, with O3 c0. The electromagnetic ﬁeld in the domain O is induced by a plane wave vertically incident on the Earth’s surface. The time dependence of the ﬁeld is given by factor expðiotÞ. Given the conductivity distribution sðx; y; zÞ, it is necessary to determine electric and magnetic ﬁelds everywhere in the domain S ¼ O [ O0 , where O0 is the lower atmosphere adjacent to O. 5.2.2. Calculation of the electric field 5.2.2.1. Equations and boundary conditions Over periods of interest to geophysics, the ﬁeld in the domain S is quasi-stationary and satisﬁes the Maxwell equations: r H ¼ sE

ð5:9Þ

r E ¼ iom0 H

ð5:10Þ

Equations (5.9) and (5.10) yield the electric ﬁeld equation DE rðr EÞ þ k2 E ¼ 0

ð5:11Þ

where k ¼ ðiom0 Þ1=2 ; Re k40. Taking the divergence from the two sides of Equation (5.9) we derive sðr EÞ þ ðE; rsÞ ¼ 0

ð5:12Þ

With due account taken of (5.12), equation (5.11) takes the form DE þ rðE; r ln sÞ þ k2 E ¼ 0

ð5:13Þ

To determine the electromagnetic ﬁeld in the domain S we have to solve a boundary-value problem for the ﬁeld E satisfying the Equation (5.13) within S and then calculate H. Let us now consider the choice of boundary conditions at the boundaries of the domain S. The values of electric ﬁeld or of its normal derivative are not known beforehand. One of the possible approaches makes use of integral boundary conditions (Spichak, 1999a), but it is very difﬁcult to use them in practice because of their entailing considerable computing difﬁculties. In this connection it would be interesting to consider another type of boundary condition that are based on the implicit account of the character of the anomalous electric ﬁeld decaying far from the medium heterogeneities. In particular, the algorithm in question makes use of

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the asymptotic boundary conditions derived in Spichak (1985): @ 1 ikr þ r ðE Eb Þ ¼ 0 @r

ð5:14Þ

where Eb is the background electric ﬁeld corresponding to the case when s sðzÞ for all P 2 S; r is the distance between the points lying on the boundary of the domain S and the origin of the coordinates. Spichak (2006) showed that application of these boundary conditions at the ﬁnite distance from the anomalous body (in particular, when r o 0.5d, where d is the skin depth in the background structure) enables getting more correct results than using the hypothesis of secondary ﬁeld vanishing at this distance. Another effect of their application consists in essential diminishing of the modeling domain without loss of accuracy. 5.2.2.2. Discretization scheme To obtain a system of linear algebraic equations and its subsequent computeraided solution, a transition from continuous to discrete values is required. Various mechanisms for such a transition, considered in Section 5.1, are mainly based on (1) the application of Taylor series, (2) the variational formulation, and (3) integration of the primary equation. Each of these approaches has its advantages and respective ﬁelds of applicability. Thus, the ﬁrst technique applicable to differential equations in general case is used most often in deﬁning the order of approximation in difference schemes. The variational formulation is inviting in that, in its context, natural boundary conditions are a direct consequence of the relevant functional being stationary (Pridmore et al., 1981). Finally, the third approach, often referred to as the ‘‘balance technique,’’ is applicable in general case and yields particularly simple discrete schemes with internal boundaries and non-uniform grids (Spichak, 1983a). We will use this last technique to obtain discrete analogs of Equation (5.13). Provided that the electric conductivity in the region varies stepwise, derivatives of the function being sought for may undergo breaks. It is thus worthwhile to compute the values of the function itself at the nodes of a certain grid while specifying values of the electric conductivity function at the nodes of an intermediate grid (Brewitt-Taylor and Weaver, 1976) without worrying to satisfy internal boundary conditions. Similar idea is realized in the staggered grids approach mentioned above. To derive discrete equations for space grid nodes, we will proceed from the continuous vector function E to the discrete vector Ul;m;n , deﬁned at the nodes of the main grid. Integrating Equation (5.13) with respect to the volume of an elementary cell in the vicinity of the node (l,m,n) (Fig. 5.1), we obtain an equation for current balance: ZZ ZZ ZZZ rUl;m;n ds þ ðUl;m;n ; r ln sl;m;n Þ ds þ k2l;m;n Ul;m;n dV ¼ 0 ð5:15Þ Sl;m;n

Sl;m;n

V l;m;n

where l ¼ 1,2, y, L; m ¼ 1,2, y, M; n ¼ 1,2, y, N

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Fig. 5.1. Elementary cell of a spatial grid.

Substituting the derivatives of U and s into Equation (5.15) by FDs and approximating the integrals by the trapezium rule we arrive at a linear algebraic equation relating the values of the vector function U only in seven adjacent nodes: ^ ð0Þ

1

^ ð1Þ

^ ð2Þ

^ ð3Þ

Ul;m;n ¼ Dl;m;n ðDl;m;n Ul;m;n1 þ Dl;m;n Ul;m1;n þ Dl;m;n Ul1;m;n ^ ð4Þ

^ ð5Þ

^ ð6Þ

þDl;m;n Ulþ1;m;n þ Dl;m;n Ul;mþ1;n þ Dl;m;n Ul;m;nþ1 Þ ðl ¼ 1; 2; . . . L; m ¼ 1; 2; . . . M; n ¼ 1; 2; . . . NÞ ðiÞ D^ l;m;n

ð5:16Þ

(i ¼ 1,2, y, 6) are the matrix coefﬁcients having a size (3 3) and where determined by the grid geometry, distribution of the conductivity s, and EM ﬁeld frequency. Note that numerical approximation of Equation (5.13) has a number of advantages compared to the approximation of the corresponding second-order differential equation (5.11): the balance of currents is held within each cell of the spatial grid; the use of Equation (5.13) makes it possible to avoid the approximation of mixed second derivatives in (5.11) and explicitly enforces current divergencefree conditions at each grid node (compare with staggered grid approximation; Smith, 1996a); the approximation of the third term in Equation (5.13) is accurate enough in the vicinity of electric conductivity contrasts as well; and in the regions where s ¼ 0 or rs ¼ 0 the second term of the equation automatically vanishes and in approximation of (5.15) the total error decreases. The matrix of a corresponding SLE has a block-banded shape and is very sparse (Fig. 5.2).

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Fig. 5.2. Structure of matrix D: L,M,N – number of nodes in x, y, and z axes, respectively; ðiÞ Q ¼ L M N, D^ (3 3) -sub-matrix.

Such systems are solved more efﬁciently by iterative methods, which enable the most economical use of the CPU core memory. In particular, SLE (5.16) is solved using conjugate gradient technique BiCGstab (Steijpen et al., 1994) after its preconditioning by means of the diagonal (Jacobi) scaling. 5.2.3. Calculation of the magnetic field ~ can be determined directly from equation (5.4) (see, e.g., The magnetic ﬁeld H Weaver et al. (1999)) or, as noted above, it can be simply recalculated from the determined electric ﬁeld by its differentiation in terms of the ﬁnite-difference (or FE) approximation of Equation (5.10). In doing so, we have to overcome the difﬁculty arising in regions with large conductivity gradients, which may cause blunders in the calculation of the relevant electric ﬁeld derivatives and, eventually, false values of the magnetic ﬁeld. Particularly, in the calculation of the horizontal components Hx Hy and at the Earth’s surface according to formulae (5.10), the derivatives dE x =dz and dE y =dz elude a stable determination (Yudin, 1982). A possible way to tackle this problem lies in constructing a spline over a set of values of the grid function and then using the analytically obtained derivatives to compute the magnetic ﬁeld components from Equation (5.10). This technique gives satisfactory results in 2-D case when the solution involves only one component of the electric ﬁeld, but it proves to be unstable in three-dimensional case (Pridmore et al., 1981). Another approach lies in computing the anomalous magnetic ﬁeld by means of numerical integration of excess currents circulating in the anomalous region V a : Z m ð5:17Þ Ha ¼ G^ DsE dv, Va

where Ds ¼ s sb

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Despite the fact that this procedure is stable, to determine the magnetic ﬁeld the relevant Greens tensor must be known. If the electric ﬁeld was calculated using one of the integral equation techniques (see Section 5.1.1), then computing the magnetic ﬁeld would not require any additional calculations of the Greens tensor. However, if the computation is performed by means of a differential equation technique (see Section 5.1.2), determining the magnetic ﬁeld will additionally require computing the Green tensor, which involves considerable computation expenses. From this standpoint, the approach proposed in Zhdanov and Spichak (1992) appears to be more fascinating. In this approach, magnetic ﬁeld components are determined in two steps. First, in the centers of the grid cells the vertical component of the magnetic ﬁeld Hz is found, and then, using Hilbert transforms (see, for instance, Zhdanov, 1988), tangential components Hx and Hy are computed at the grid nodes: H x ðx0 ; y0 ; 0Þ ¼ H bx þ ð2pÞ1

ZZ

H z ðx; y; 0Þðx x0 Þ dx dy r3

S

H y ðx0 ; y0 ; 0Þ ¼ H by þ ð2pÞ1

ZZ

H z ðx; y; 0Þðy y0 Þ dx dy r3

ð5:18Þ

S

1=2 where r ¼ ðx x0 Þ2 þ y y0 2 ; the value of Hz is pre-calculated according to

formula (5.10); H bx and H by are the background magnetic ﬁeld components at the Earth’s surface. (Note that integral relations (5.18) are used in Weaver et al. (1999) as boundary conditions on the Earth’s surface.) The above algorithm for the forward problem solution was realized as a software package FDM3D (Spichak, 1983b) and used for modeling EM ﬁelds in 3-D media (Zhdanov and Spichak, 1992; Spichak, 1999a; see also Chapter 12).

5.2.4. Controlling the accuracy of the results 5.2.4.1. Criteria for accuracy In most cases, the accuracy of modeling EM ﬁelds can only be evaluated indirectly, because the existing control techniques provide, as a rule, necessary but not sufﬁcient conditions for the accuracy of the results. ‘‘External’’ and ‘‘internal’’ techniques are employed for accuracy control. External techniques include (a) comparison with the results of other authors and (b) comparison with the results obtained by other methods, including analytical ones. Thus far, these two criteria have been used chieﬂy for checking the correctness of solutions. The results of numerous comparisons, however, clearly show the above accuracy criteria to be insufﬁcient. Control techniques that enable the accuracy estimation by an internal means of one or another numerical approach should be applied.

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As pointed out in Poggio and Miller (1973), ‘‘the idea of numerical evaluation of the accuracy of a technique at times appears to be internally inconsistent, because it seems that numerical results cannot be used to conﬁrm their own correctness. In reality, there are several approaches that enable a manifest incorrectness of numerical results to be detected.’’ Such criteria include testing the following: (a) the reciprocity theorem; (b) the energy conservation law; (c) the accuracy to which the equations and boundary conditions are satisﬁed; and (d) convergence of the solution with decreasing size of grid cells, number of unknowns, etc. Finally, one may consider as a mixed type (in employing differential equation techniques) the criterion proposed in Zhdanov et al. (1982) based on an estimation of the accuracy to which the integral identity ZZZ e b G^ ðr0 jrÞsa EðrÞ dv Eðr0 Þ ¼ E ðr0 Þ þ iom0 Va

is satisﬁed by substituting in it the solution obtained by the FD or FE techniques. Unfortunately, numerical implementation of this criterion requires a time-consuming computation of Green tensor for a layered medium, which reduces its practical value. A more efﬁcient internal criterion can be based on satisfying the integral relations between components of the anomalous EM ﬁeld on the Earth’s surface (Spichak, 1999a): E ax ¼ G x E z iom0 GH ay ; E ay ¼ Gy E z þ iom0 GH ax

ð5:19Þ

where G is the free-space Green function and Gx and Gy are its derivatives over x and y. The developed program package FDM3D (Spichak, 1983b) based on the algorithm described above was tested on models that allowed an analytical solution and on models computed by other techniques in the framework of the COMMEMI project (Zhdanov et al., 1997). 5.2.4.2. Comparison with high-frequency asymptotic solution Indirect assessment of the computation accuracy of the FDM3D package was carried out in Spichak (1983a) using a high-frequency asymptotic solution for a solid sphere with radius a ¼ 2 km in a homogeneous space, excited by a plane wave (Berdichevsky and Zhdanov, 1984). The conductivity of the target is sT ¼ 10–4 S/ m, and that of the host space is sH ¼ 10–6 S/m, the period T ¼ 0.01 s. With o ! 1 and sT sH , for a point located above the solid sphere at a distance r from its center, we have E ax ¼ E bx ða=rÞ3 iom0 H by a3 =ð2r2 Þ where E bx and H by are the background electric and magnetic ﬁelds.

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Fig. 5.3. Comparison of numerical modeling results for a cube (1) with an analytical solution for an equivalent solid sphere (2).

The study involved numerical modeling for a cube of equivalent volume and 3.2 km on a side. Fig. 5.3 presents the results for a proﬁle through the center of the cube (model of ‘‘a solid sphere’’). One can see that with increasing distance from the center r the compatibility of the results increases. At the same time, the divergence of the curves (p 10%) at short distances from the center (r/a ¼ 2) is evidently due to the dissimilar geometries of the models. 5.2.4.3. Comparison with results obtained by other techniques Fig. 5.4 depicts the results of computing the horizontal component of the magnetic ﬁeld for the model 3D-1 (T ¼ 0.1 s; Hb ¼ ðH x ; 0; 0Þ) from the project COMMEMI (Zhdanov et al., 1997), which were performed in Zhdanov and Spichak (1992) using different techniques. The magnetic ﬁeld was determined from the electric ﬁeld by (1) ﬁnite-difference approximation of the Maxwell equation, (2) spline-interpolation of the electric ﬁeld values, (3) Hilbert transforms (5. 18), and (4) integration of the excess currents according to (5. 17). A comparison of these curves with the ‘‘statistical average’’ values, derived for this model from the results obtained by different authors and using a variety of techniques, shows that computing the tangential components of the magnetic ﬁeld by means of Hilbert transforms yields the best results. In the next section we will dwell on the particular case of three-dimensional models of the medium, where allowing for symmetry results in a sharp reduction of computations.

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Fig. 5.4. Graphs of the normalized horizontal component of anomalous magnetic ﬁeld computed from the electric ﬁeld by (1) ﬁnite-difference approximation of the Maxwell equation, (2) spline-interpolation of the electric ﬁeld values, (3) Hilbert transforms (5.18), and (4) integration of the excess currents according to (5.17).

5.3. METHOD OF THE EM FIELD COMPUTATION IN AXIALLY SYMMETRIC MEDIA Analysis and interpretation of electromagnetic ﬁeld anomalies on the Earth, investigation of the resolution of soundings as well as a number of other methodological problems encountered today in electromagnetics require calculation of many different models. These problems can be solved in principle by using the existing methods of numerical modeling of EM ﬁelds in media that include arbitrary three-dimensional inhomogeneities (see Section 5.1). However, in many cases, the objective can be achieved with reduced computer resources if we conﬁne ourselves to models of speciﬁc type of symmetry. In particular, axially symmetric threedimensional models reduce the vector problem to a series of independent tasks in a plane for two scalar functions (Zakharov, 1978). In this case, the reduction to a discrete system may be accomplished either by the integral equation method (Barashkov and Dmitriev, 1982) or by one of the differential equation methods (Zhdanov et al., 1984, 1990). In this section an algorithm for the forward modeling of quasi-stationary electromagnetic ﬁelds in axially symmetric three-dimensional media will be described following the latter papers.

5.3.1. Problem statement Consider an electromagnetic ﬁeld excited by a plane vertically incident wave in a layered medium that includes a three-dimensional axially symmetric inhomogeneity

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Fig. 5.5. Model of the geoelectric section (in cylindrical coordinates); O is the domain of modeling.

(Fig. 5.5). The medium is assumed to be isotropic and non-magnetic. The magnetic permeability in the whole space is equal to free-space permeability (m ¼ m0 ). Displacement currents, as earlier, are neglected, i.e., the ﬁeld is assumed to be quasistationary. The time dependence of the ﬁelds is deﬁned by the factor exp(iot). We will introduce a cylindrical coordinate system (r; j; z) whose vertical axis coincides with the axis of symmetry of the inhomogeneity and is positive vertically downwards. 5.3.2. Basic equations Following Zakharov (1978), represent the components of the vectors E,H as Fourier series: E r;j;z ¼

þ1 X

E ðnÞ r;j;z expðinjÞ; H r;j;z ¼

n¼1

þ1 X

H ðnÞ r;j;z expðinjÞ

ð5:20Þ

n¼1

Substituting these expansions into Maxwell equations (5.9), (5.10) we derive the following equations for harmonics: ðnÞ

in ðnÞ @H j ¼ sE ðnÞ H r @z r z

ð5:21Þ

@H ðnÞ @H ðnÞ r z ¼ sE ðnÞ j @z @r

ð5:22Þ

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V.V. Spichak and M.S. Zhdanov/3-D EM Forward Modeling Using Balance Technique ðnÞ

1 ðnÞ @H j in H ðnÞ ¼ sE ðnÞ H þ z @r r j r r

ð5:23Þ

ðnÞ

in ðnÞ @E j ¼ iom0 H ðnÞ E r @z r z

ð5:24Þ

@E ðnÞ @E ðnÞ r z ¼ iom0 H ðnÞ j @z @r

ð5:25Þ

ðnÞ

1 ðnÞ @E j in ¼ iom0 H ðnÞ E ðnÞ E þ z @r r j r r

ð5:26Þ

ðnÞ Let u ¼ E ðnÞ j and v ¼ H j . Using Equations (5.21), (5.23), (5.24), and (5.26) exðnÞ ðnÞ ðnÞ press the components E r ; E ðnÞ z and H r ; H z in terms of u and v:

^ ^ E ðnÞ r ¼ inDr u iom0 rDz v; H ðnÞ ¼ srD^ z u þ inD^ r v; r

^ ^ E ðnÞ z ¼ inDz u iom0 rDr v H ðnÞ ¼ srD^ r u þ inD^ z v,

ð5:27Þ

z

where r @ 1 r @ þ and D^ z ¼ D^ r ¼ a @r r a @z D^ r D^ z

!

^ ¼ are the components of the vector differential operator D and 2 2 a ¼ iom0 sr n . Substituting these expressions into (5.22) and (5.25) gives equations in u and v: ^ þ su inrot2 ðDvÞ ^ ¼ 0; div2 ðsrDuÞ

^ ¼0 ^ þ iom0 v inrot2 ðDuÞ div2 ðiom0r DvÞ ð5:28Þ

where div2 ¼ er

@ @ þ ez ; @r @z

rot2 ¼ er

@ @ ez @z @r

with er and ez being the unit vectors of the cylindrical coordinate system. It is evident that when the ﬁeld is excited by a plane wave it is sufﬁcient to solve the problem for harmonics n ¼ 1 (Barashkov and Dmitriev, 1982). If the normal ﬁeld is polarized linearly (with the magnetic ﬁeld in the j ¼ p=2 azimuth), the harmonics n ¼ 1 and n ¼ 1 are related by ð1Þ E ð1Þ j ðr; zÞ ¼ E j ðr; zÞ;

ð1Þ H ð1Þ j ðr; zÞ ¼ H j ðr; zÞ

ð5:29Þ

Hence, to determine the electromagnetic ﬁeld components within an axially symmetric three-dimensional model, it is sufﬁcient to deﬁne functions u and v corresponding to n ¼ 1 and then determine the required ﬁeld by (5.20) with reference to relations (5.27) and (5.29).

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127

5.3.3. Boundary conditions Without loss of generality, the modeling domain is assumed to be a rectangle O on the plane (r,z), whose left-hand side lies on the z-axis, the upper side is in the atmosphere, and the lower side is located in the underlying basement (Fig. 5.5). On the boundaries of the domain O the following boundary conditions are speciﬁed: (a) On the upper boundary in the atmosphere (and on the lower boundary of the domain if the underlying basement is highly resistive) the ﬁrst-order asymptotic boundary conditions for an anomalous ﬁeld are valid: @ @ @ @ b ðu u Þ ¼ 0; 1 þ r þ z ðv vb Þ ¼ 0, 1þr þz @r @z @r @z where u and v are the azimuthal components of the total ﬁeld harmonics, while ub and vb are those of the background ﬁeld. These conditions are readily established from the asymptotic boundary conditions (5.14). (b) At the interface of a highly conducting underlying basement, which can be roughly considered as a perfect conductor, the horizontal components of the electric ﬁeld are zero. This leads to the boundary conditions @v ¼0 @z (c) On the axis of symmetry, the exact relations u ¼ 0;

@u ¼ 0; @r

@v ¼0 @r

are satisﬁed. (d) On the right-hand boundary of the modeling domain, the total ﬁelds are locally approximated by a plane vertically incident wave. In this case, the boundary conditions are @u ¼ 0; @r

@v ¼0 @r

5.3.4. Discrete equations and their numerical solution To derive discrete equations a direct FE method (Norrie and de Vries, 1978) is used. In this case it enables to set up a conservative scheme around a nine-point pattern. 5.3.4.1. Discrete equations Introduce a grid S on a plane ðr; zÞ : ðri ; rj Þ 2 S; 1oioI; 1ojoJ. Unit cells are rectangles S kl ðk ¼ 1; 2; . . . ; I; l ¼ 1; 2; . . . ; JÞ with their vertices being in the middle of the cells of the grid S adjacent to grid points ðri ; rj Þ (Fig. 5.6). Integrating Equations (5.28) over an area of the cells S kl and using the Ostrogradsky–Gauss and Stokes 2-D formulae, we obtain the balance equations

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Fig. 5.6. Unit cell S k;l of the rectangular mesh.

Z

ZZ

^

srðD uÞB dl þ @Skl

Z

Z

Skl

^ iom0 rðDvÞB dl þ

@S kl

^ ðDvÞl dl ¼ 0

su ds in @S kl

ZZ

Z iom0 v ds in

S kll

^

ðD uÞl dl ¼ 0

ð5:30Þ

@Skl

where @Skl is the boundary of the cell S kl , 1 and l are the unit vectors directed along an outward pointing normal and a tangent to the boundary @Skl , respectively; the contour @Skl is transversed counterclockwise. We seek u and v as an expansion in terms of ﬁnite basis functions: uðr; zÞ ¼

I X J X

uij jij ðr; zÞ;

i¼1 j¼1

vðr; zÞ ¼

I X J X

vij jij ðr; zÞ

ð5:31Þ

i¼1 j¼1

where jij ðr; zÞ ¼ 0; if ðr; zÞe½ri1 ; riþ1 ½zj1 ; zjþ1 : Substituting expansions (5.31) into Equations (5.30) we write kþ1 X lþ1 X

ðAijkl uij þ Bijkl vij Þ ¼ 0

i¼k1 j¼l1 kþ1 X lþ1 X i¼k1 j¼l1

ðBijkl uij þ C ijkl vij Þ ¼ 0

ðk ¼ 1; 2; . . . ; I; l ¼ 1; 2; . . . ; JÞ,

ð5:32Þ

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129

where Aijkl ¼

Z

ZZ

^

sðr; zÞrðD jij Þ1 dl þ @Skl

sðr; zÞjij dS Skl

Bijkl

Z

^

ðD jij Þl dl

¼ in @Skl

2 6 C ijkl ¼ iom0 4

Z

ZZ

^

rðD jij Þ1 dl þ @Skl

3 7 jij dS5

S kl

Deﬁnite integrals entering the formulas for the coefﬁcients Aijkl ; Bijkl ; and C ijkl are evaluated by means of the ‘‘rectangle rule’’. 5.3.4.2. Basis functions As it is known, allowance for the ﬁeld behavior contributes to the accuracy of equation approximation. Assuming that in the each grid point neighborhood total ﬁelds vary linearly in the horizontal and exponentially in the vertical, it is possible to introduce the following basis functions: jij ðr; zÞ ¼ xij ðrÞzij ðzÞ,

ð5:33Þ

where

zij ðzÞ ¼

8 0; > > > shðkij ðzzj1 ÞÞ > > < shðk ðzj zj1 ÞÞ ; ij

> > > > > :

shðkþ ij ðzzjþ1 ÞÞ shðkþ ij ðzj zjþ1 ÞÞ

;

z zj1 zj1 z zj , zj z zjþ1

0; zjþ1 z qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ ¼ iom0 sij ; and kij ¼ iom0 sij , and sþ where ij and sij are the average conductivities in the upper and lower halves of a cell, respectively. Note that the derived basis functions show fairly good approximation properties. In particular, the background ﬁelds calculated for a 1-D conducting medium from a system of equations (5.32) with due account of the relation (5.33) coincides with that calculated analytically. It is worth mentioning, that in non-conductive medium (kij ! 0), the basis function (5.33) is reduced to bilinear one, which, in turn, corresponds to the geometry-only dependence of the ﬁeld. kþ ij

5.3.4.3. Numerical solution of discrete equations The system of linear algebraic equations resulting from discretization is solved by employing the Crout algorithm of expanding a matrix into the product of the

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V.V. Spichak and M.S. Zhdanov/3-D EM Forward Modeling Using Balance Technique

upper and lower triangular matrices (Tewarson, 1973). Below is a brief outline of the algorithm. ^ Represent the matrix AðN NÞ of the system in the form d A^ ¼ v

oT G

where d is a scalar, v a column vector, oT a row vector, and G^ an (N1)th-order square matrix. It is easy to see that for da0, the following representation d ^ A¼ v

0T 1 I^N1 0

d 0T 0 0 G

oT =d I^N1

0 0 holds, where the (N1)th-order square matrix G^ is deﬁned as G^ ¼ G^ vx=d, I^N1 is an (N1)th order-identity matrix, and 0 a zero column vector. At the next step, 0 the matrix G^ is expanded in the same way. As a result, at N steps, the initial matrix is expanded into the product of the upper and lower triangular matrices. Upon expansion, the lower and upper triangular systems of equations are solved. The system of linear algebraic equations resulting from discretization has a banded structure (the bandwidth M ¼ 4 þ 2 minðI; JÞ, where I and J are the number of grid points in the vertical and in the horizontal, respectively). The application of the Crout algorithm to this matrix is distinguished by the following feature. At each step of the algorithm all operators are executed over the matrix elements lying inside a square LðM MÞ, which slides diagonally downwards (Fig. 5.7). In doing so, we obtain the relevant column of the lower triangular matrix and a row of the upper triangular matrix. This permits matrix expansion by parts, utilizing a hard disk directaccess ﬁle and a small portion of the core memory.

^ Fig. 5.7. Structure of matrix AðN NÞ: N is matrix size, M is a bandwidth.

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Fig. 5.8. Model of a cylindrical insert with resistivity ri1 (after Berdichevsky and Dmitriev, 1976).

5.3.5. Code testing The above algorithm has been used to develop a code for forward modeling of magnetotelluric ﬁelds in the Earth containing an axially symmetric three-dimensional inhomogeneity. To test the software, the code FDMS-3D was used to calculate the model response in the DC asymptotic approximation (Berdichevsky and Dmitriev, 1976). The model (Fig. 5.8) consists of a thin layer with thickness h1 and a constant resistivity re1 (integral electric conductivity of the layer Se1 ¼ h1 =re1 ), an intermediate non-conducting layer with thickness h2 ðr2 ¼ 1Þ, and an ideally conducting underlying basement ðr3 ¼ 0Þ. The upper layer contains a cylindrical insert of radius a and a constant resistivity ri1 that matches the upper layer in thickness (integral electric conductivity S i1 ¼ h1 =ri1 ). For DC asymptotic approximation an analytical solution is available for such a model (Berdichevsky and Dmitriev, 1976), from which the following relations follow: E a ¼ F a E ba ;

a ¼ r; j

ð5:34Þ

where E r;j and E br;j are components of the total and background ﬁelds, respectively, and 8 i e > 1 < 1 SS1i S 0 r a e ; þS 1 1 Fr ¼ i e > : 1 þ S1i S1e a22 ; r a; r S þS 1

Fj ¼

8 > <

1

Si S e

1 S1i þS1e ;

> :1

1 Si1 S e1 Si1 þS e1

0 r a

1

2

ar2 ;

r a;

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The value of an anomalous magnetic ﬁeld for the DC asymptotic approximation is determined by the formula (Berdichevsky and Dmitriev, 1976) 1 Ha ez ¼ ðS 1 E S e1 Eb Þ, ð5:35Þ 2 where S 1 ¼ S i1 if 0 r a and S1 ¼ Se1 if r a; ez is the unit vector of the Cartesian coordinate system. Calculations were performed for a model with h1 ¼ 0.5 km, h2 ¼ 10 km, a ¼ 5 km, Se1 ¼ 500 S, Si1 ¼ 2500 S, s2 ¼ 105 S/m, and s3 ¼ 105 S/m for periods T ¼ 21, 84, and 360 s. Fig. 5.9 depicts normalized values of the azimuthal component of the electric ﬁeld for a number of periods, computed numerically using FDMS-3D and obtained by the asymptotic formula (5.34) (T - N). The diagram shows that at T ¼ 84 s, the curve E j departs from the asymptotic curve by no more than 2–3%, merely smoothing out the break at the boundary of the inclusion. The curve that corresponds to the period T ¼ 21 s differs signiﬁcantly from the asymptotic curve above the anomaly owing to the inﬂuence of the induction effect. However, beginning at r ¼ 5.5 km (r/a ¼ 1.1), the curves converge, and the discrepancy becomes no greater than 2–4%. Fig. 5.10 shows values of the normalized anomalous magnetic ﬁeld jH aj j=H aj;anal ð0Þj determined by means of the code FDMS-3D for periods 21, 85, and 360 s and using the asymptotic formula (5.35) (T-N).

Fig. 5.9. Normalized values of E j at j ¼ p=2 on the Earth’s surface for model of a cylindrical insert shown in Fig. 5.8; the solid curves represent the results calculated by the FDMS-3D program, the dashed curve designates the results obtained by the asymptotic formula.

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Fig. 5.10. Normalized values of H aj at j ¼ 0 on the Earth’s surface for the model of a cylindrical insert shown in Fig. 5.8; the solid curves represent the results calculated by the FDMS-3D code, the dashed curve designates the results obtained by the asymptotic formula.

Owing to the inductive inﬂuence, the curve H j at T ¼ 21 s differs considerably from the asymptotic one (by as much as 30% in the center of the anomaly). As the period increases, the inductive interaction decreases. Thus, the curve that corresponds to T ¼ 84 s departs from the asymptotic one by as much as 12%, and the curve that corresponds to T ¼ 360 s virtually replicates the asymptotic one, with a difference of 0.2% in the center of the anomaly, of 3% at r ¼ 7 km (r/a ¼ 1.4), and of 8% at the boundary of the anomaly (at r ¼ 5 km). The results of test calculations and practical experience with the FDMS-3D code demonstrate that the direct FE method with special basis functions is an effective means of numerical modeling of quasi-stationary electromagnetic ﬁelds in three-dimensional media exhibiting an axial symmetry. The FDMS-3D code does not require appreciable computer resources and applies equally to purely methodological calculations and to the solution of a fairly wide range of practical problems. Thus, we considered the 3-D forward modeling algorithms based on the balance technique. The corresponding computer codes enable calculations of EM ﬁelds in the models with a relief topography, mixed type of the conductivity structure (1-D/ 2-D/3-D), different 1-D layering at inﬁnities as well as at arbitrary level in the Earth and atmosphere. In Chapter 12 some methodological results obtained using this software will be considered.

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REFERENCES Avdeev, D.B., Kuvshinov, A.V., Pankratov, O.V. and Newman, G.A., 1997. Highperformance three-dimensional electromagnetic modelling using modiﬁed Neumann series. Wide-band numerical solution and examples. J. Geomagn. Geoelectr., 49: 1519–1539. Barashkov, I.S. and Dmitriev, V.I., 1982. Method of two-dimensional integral equations for calculating magnetotelluric ﬁelds in a layered medium containing an axially symmetric inhomogeneity Vych. metody i programmirovanie, 36 (in Russian). Moscow State University, Moscow, pp. 14–27. Berdichevsky, M.N. and Dmitriev, V.I., 1976. Basic principles of interpretation of magnetotelluric sounding curves in geoelectric and geothermal studies: Budapest, Acad.Kiado, 165–221. Berdichevsky, M.N., Moroz, I.P., Kobzova, V.M. and Bilinsky, A.I., 1987. Physical Modeling in Geoelectrics (in Russian): Naukova Dumka, Kiev, 84pp. Berdichevsky, M.N. and Zhdanov, M.S., 1984. Advanced Theory of Deep Geomagnetic Sounding. Elsevier, Amsterdam-London-NewYork-Tokyo, 408pp. Best, M.E., Duncan, P., Jacobs, E.J. and Scheen, W.L., 1985. Numerical modelling of the electromagnetic response of three-dimensional conductors in layered earth. Geophysics, 50: 665–676. Boyse, W., Lynch, D., Paulsen, K. and Minerbot, G., 1992. Nodal based ﬁnite element modeling Maxwell’s equations: IEEE Trans. Antennas Propagat., 40: 642–651. Brewitt-Taylor, C.R. and Johns, P.B., 1980. On the construction and numerical solution of transmission-line and lumped network models of Maxwell’s equations. Int. J. Num. Meth. Eng., 15, 2: 13–30. Brewitt-Taylor, C.R. and Weaver, J.T., 1976. On the ﬁnite-difference solution of twodimensional induction problems. Geophys. J. R. Astr. Soc., 47, 2: 375–396. Cerv, V., 1990. Modelling and analysis of electromagnetic ﬁelds in 3D inhomogeneous media. Surv. Geophys., 11: 205–230. Das, U.C. and Verma, S.K., 1981. Numerical considerations on computing the EM response of three-dimensional inhomogeneities in a layered earth. Geophys. J.R. Astr. Soc., 66, 3: 733–740. Davydycheva, S. and Druskin, V., 1999. Staggered grid for Maxwell’s Equations in 3-D anisotropic media: Three/Dimensions/Electromagnetics, SEG monograph., GD7, Tulsa, USA, 138–145. Dawson, T.W. and Weaver, J.T., 1979. Three-dimensional induction in a nonuniform thin sheet at the surface of a uniformly conducting earth. Geophys. J.R. Astr. Soc., 59, 3: 445–462. Debabov, A., 1980. On the computer modeling EM ﬁelds in inhomogeneous media (in Russian). Dokl. Akad. Nauk USSR, 250, 20: 326–331. Dmitriev, V. I., 1969, Elektromagnitnye polya v neodnorodnyh sredah (in Russian). MGU, M., 132pp. Dmitriev, V.I., Barashkov, I.S. and Mershchikova, N.A., 1985. Mathematical modeling MT ﬁelds in inhomogeneous media (in Russian). MGU, Moscow, 88pp.

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Chapter 6 3-D EM Forward Modeling Using Integral Equations D.B. Avdeev1,2 1 2

IZMIRAN, Russian Academy of Sciences, Troitsk, Moscow region, Russia Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland

6.1. INTRODUCTION Three-dimensional (3-D) electromagnetic (EM) modeling is based on various methods for numerical solution of Maxwell equations, ~ oÞE þ jext ; r H ¼ sðr;

r E ¼ iomH

ð6:1Þ

written here for monochromatic electric E(r, o) and magnetic H(r, o) ﬁelds. Here ~ o) ¼ sioe is the generalized complex conductivity of the model, m its mags(r, netic permeability, jext (r, o) the extraneous current density that excites the model, o the angular frequency, r ¼ ð@x ; @y ; @z Þ the gradient and the sign stands for the vector product, r ¼ ðx; y; zÞ. I intentionally do not mention whether conductivity s is a real or complex-valued function, or if it is an isotropic or an anisotropic. However, I will clarify this matter below in the text where that needs further consideration. It is of importance that the generalized conductivity s~ is 3-D, i.e. it is a function of the Cartesian coordinates x, y, z. This clearly means that the pulse methods of geoelectrics remain out of the scope of this paper. One usually uses the ﬁnite-difference (FD), ﬁnite-element (FE), or integral equation (IE) methods to solve Equations (6.1) numerically. A review of the methods is given in Avdeev (2005a,b; see also references herein). This paper is entirely devoted to the volume integral equation method, but there also exists the surface integral equation method (see Chew, 1999; among others). As a rule the surface IE Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40006-8

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method assumes that the conductivity does not change within each anomalous volume and therefore it imposes very strong limitations on the complexity of the models under consideration. Here I review, following Avdeev et al. (2002a), the volume IE approach and then present model examples for induction logging and airborne EM.

6.2. VOLUME INTEGRAL EQUATION METHOD The theory of the volume integral equation method, as it is applied to 3-D geoelectromagmetic problems, has been developed in the pioneering works of Harrington (1968), Dmitriev (1969), Raiche (1974), Hohmann (1975), Weidelt (1975), and Tabarovsky, 1975). The ﬁrst 3-D numerical solutions based on this method appeared in the mid-1980s (Wannamaker et al., 1984; Newman et al., 1986; Cerv, 1990; Wannamaker, 1991; Dmitriev and Nesmeyanova, 1992; Xiong, 1992). These ﬁrst solutions were found to be effective in simulating EM responses of one or a few compact bodies. Let me describe the essence of the integral equation method following the work by (Avdeev, 2002a), where this method is employed in the case of a 3-D anisotropic ~ oÞ ¼ s io ¼ diagðs~ xx ; s~ yy ; s~ zz Þ. Besides model with generalized conductivity sðr; this model, let me also introduce another so-called ‘‘reference model’’ with conductivity s~ 0 ðr; oÞ. It is essential that it is assumed that for the reference model we are able to effectively calculate the 3 3 dyadic for the electric-to-electric 0 ee 1 Gxx G ee G ee xy xz ee B ee ee ee C G^ 0 ðr; r0 ; oÞ ¼ @ G yx G yy G yz A ee ee Gzx G zy G ee zz and electric-to-magnetic 0

Gme xx me B me 0 ^ G G 0 ðr; r ; oÞ ¼ @ yx Gme zx

G me xy G me yy G me zy

1 G me xz C G me yz A G me zz

Green’s tensors. The reason why we introduce the reference model will become clear from what follows. At this stage the reference model is usually chosen to have pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ y2 ; oÞ or radial-symmetric ~ ~ ðz; oÞ, either axisymmetric s ð x layered s 0 0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 s~ 0 ð x þ y þ z ; oÞ conductivity. Explicit expressions for the aforementioned Green’s tensors are presented in Avdeev et al. (1997) for a layered uniaxial anisotropic model. In the case of axisymmetric models these are given in Zhang and Zhang (1998) and in Kuvshinov et al. (2002) for radially symmetric models, respectively. It should be noted here that if by pure chance we were able to effectively calculate the Green’s tensors for a 2-D or even for a 3-D model, then this model could be chosen as the reference one. Also the reference model can be chosen in many ways, for example, it may or may not coincide with the background model. Let us assume henceforth that the reference model is already chosen. The electric

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Eo ðr; oÞ and magnetic Ho ðr; oÞ ﬁelds in the reference model s~ 0 ðr; oÞ then satisfy the following Maxwell’s equations: r Ho ¼ s~ ðr; oÞEo þ jext ;

r Eo ¼ iomHo

ð6:2Þ

ee Since (as is assumed above) the calculation of the Green’s tensors G^ 0 ðr; r0 ; oÞ and me o 0 G^ 0 ðr; r ; oÞ is straightforward, it is easy to calculate the ﬁelds E and Ho. Indeed, by deﬁnition of the Green’s tensors, the ﬁelds can be presented as the following integrals Z Z he ee o 0 ext 0 0 o ^ H ðr; oÞ ¼ G 0 ðr; r ; oÞj ðr ; oÞ dv ; E ðr; oÞ ¼ G^ 0 ðr; r0 ; oÞjext ðr0 ; oÞ dv0 V ext

V ext

ð6:3Þ where V

ext

is the volume supporting the current jext ðr; oÞ; dv0 ¼ dx0 dy0 dz0

6.2.1. Traditional IE method Substracting Equations (6.2) from Equations (6.1), one obtains the following Maxwell equations for scattered ﬁelds: r Hs ¼ s~ 0 ðr; oÞEs þ jq ;

r Es ¼ iomHs

ð6:4Þ

where the scattered ﬁelds are Es ¼ E E0 ; Hs ¼ H H0 and where jq ðr; oÞ ¼ ðs~ s~ 0 ÞðEs þ Eo Þ.

ð6:5Þ

Comparing Equations (6.2) and (6.4),Rin a similar way to the derivation of Equation ee (6.3) one can derive that Es ðr; oÞ ¼ V s G^ 0 ðr; r0 ; oÞjq ðr0 ; oÞdv0 , and substituting in this integral the jq of Equation (6.5), one gets the traditional integral scattering equation (Dmitriev, 1969; Weidelt, 1975), Z ee ~ 0 ; oÞ s~ 0 ðr0 ; oÞÞEs ðr0 ; oÞ dv0 ð6:6Þ G^ 0 ðr; r0 ; oÞðsðr Es ðr; oÞ ¼ E0 ðr; oÞ þ Vs

R ee where the free term E0 ðr; oÞ ¼ V s G^ 0 ðr; r0 ; oÞðsðr ~ 0 ; oÞ s~ 0 ðr0 ; oÞÞE0 ðr0 ; oÞ dv0 and s the integration in (6.6) is done over volume V , where ðs~ s~ 0 Þ differs from zero. A discretization of the scattering Equation (6.6) yields the following system of linear equations: ^ ¼ E0 ðI^ GÞE

ð6:7Þ

~ s~ 0 and the electric ﬁelds Es ; E0 are constant provided that both the conductivities s; within each cell. Here, the n n matrix G^ is the discrete representation of the integral operator given on the right-hand side of Equation (6.6), n-vectors E and E 0 are discrete representations of the ﬁelds Es and E0 , respectively, I is the identity n n matrix, n ¼ 3Nx Ny Nz and Nx, Ny, Nz are the number of the cells along the axes of the orthogonal coordinate system. Thus the initial problem is reduced to the solution of a linear system of Equation (6.7).

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6.2.1.1. Comparison with other methods The FD and FE methods also produce systems of linear equations. From a computational point of view the key difference between these methods is due to the difference in properties of the system matrices. Let us discuss this further. The basic attraction of the IE method is that it allows the modeling volume to be conﬁned to the scattering volume Vs (where ðs~ s~ 0 Þ differs from zero). This leads to a dense, ^ As for FD and FE methods their matrices although sparser but compact matrix G. are larger. However this merit of the IE method is counterbalanced by the fact that the matrix G is non-Hermitian, dense and ill conditioned. Besides, as was correctly mentioned by Wannamaker (1991), accurate calculation of this matrix is a complex problem in itself. 6.2.1.2. Straightforward solution Moreover, the numerically effective solution of system (6.7) with the use of the ^ is a very time-consuming procedure and it is direct inversion of the matrix ðI^ GÞ possible only for a relatively small value of n (see Xiong, 1992), since such a direct inversion of this matrix depends on the cubic size of the problem EO(n3). I believe that this fact was the main reason for the common opinion that the IE method is effective only for relatively small, compact and simple anomalies. It was believed (some researchers still erroneously believe) that the IE method is non-effective for modeling more complex 3-D models. Recent modiﬁcations of the traditional IE method that will be described below are extremely effective and demonstrate a linear dependence EO(NxNy) on the horizontal size of the problem, and quadratic dependence EO(Nz2) on vertical size. 6.2.1.3. Neumann series An alternative way to solve Equations (6.7) is to summarize the inﬁnite Neumann series ^ 0 þ G^ 2 E 0 þ þ G^ m E 0 þ ^ 1 E 0 ¼ E 0 þ GE E ¼ ðI^ GÞ

ð6:8Þ

However, this approach is also not very efﬁcient, since numerical experiments show that the series diverges for the models with lateral contrasts of conductivity of 100 or more. Series (6.8) converges only for models with a weak contrast of conductivity. It is appropriate to mention here the efforts in the mid-1990s to solve Equation (6.7) approximately using a Born-type approximation (the ﬁrst two terms in series (6.8)) or on various modiﬁcations to the approximation (Habashy et al., 1994; Torres-Verdin and Habashy, 1994; Zhdanov and Fang, 1996). Interestingly, in spite of signiﬁcant problems concerning the accuracy of such solutions, they still appeal to some researchers because of their higher speed, as computational speed is vital especially for solution of inverse problems (see, for example, Tseng et al., 2003; Avdeev and Avdeeva, 2006).

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6.2.2. Modified iterative dissipative method The modiﬁed iterative dissipative method (MIDM) encouraged great interest and advances in the IE approach. As it is an extension of an iterative dissipative method (Singer and Fainberg, 1985), the method was originally presented by Singer (1993, 1995) for quasi-static EM ﬁelds and for isotropic media. Subsequently, it was extended to media with displacement currents (Pankratov et al., 1995; Singer and Fainberg, 1995) and anisotropy (Pankratov et al., 1997; Singer and Fainberg, 1997). Let me brieﬂy describe the gist of MIDM, following works by Avdeev (2002) and Avdeev et al. (2002a). Using a classical method for shifting the spectrum of an integral operator, one must add to both sides of Equation (6.6) the following term: ~ oÞ s~ 0 ðr; oÞÞEs ðr; oÞ 1=2 l2 ðr; oÞðsðr;

ð6:9Þ

s

and change variable (E ! w) using ~ oÞ þ s~ 0 ðr; oÞ Es ðr; oÞ þ ðsðr; ~ oÞ s~ 0 ðr; oÞÞEo ðr; oÞ wðr; oÞ ¼ 1=2 l1 ðr; oÞ sðr; ð6:10Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where l0 ðr; oÞ ¼ diagð Res~ 0t ; Res~ 0t ; Res~ 0z Þ. After some algebra one can obtain the so-called MIDM scattering equation Z ^ r0 ; oÞRðr0 ; oÞwðr0 ; oÞ dv0 wðr; oÞ ¼ wo ðr; oÞ þ ð6:11Þ Kðr; Vs

R ^ r0 ; oÞRðr0 ; oÞEo ðr0 ; oÞ dv0 and the inwith the free term given as wo ðr; oÞ ¼ V s Kðr; tegral operator (that is contracting one for any choice of the reference conductivity s~ 0 ) as Z ^ r0 ; oÞRðr0 ; oÞwðr0 ; oÞ dv0 wðr; oÞ Kðr; ð6:12Þ Vs

Here ^ r0 ; oÞ ¼ dðr r0 Þ1^ þ 2lðr; oÞG^ ee ðr; r0 ; oÞlðr0 ; oÞ Kðr; 0

ð6:13Þ

1 ~ oÞ s~ 0 ðr; oÞÞ sðr; Rðr; oÞ ¼ ðsðr; ~ oÞ þ s~ 0 ðr; oÞ

ð6:14Þ

the conjugate and real part of s~ 0 , respectively, d the Dirac’s delta s~ 0 and Res~ 0 are q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 R ^ function, w ¼ V s wðr; oÞ dv, and 1 is the identity 3 3 matrix. Inequality (6.12) guarantees that simple iteration Z ^ r0 ; oÞRðr0 ; oÞwðmÞ ðr0 ; oÞ dv0 ð6:15Þ Kðr; wðmþ1Þ ðr; oÞ ¼ wo ðr; oÞ þ Vs

converges to the solution of Equation (6.11) for any frequency and for any contrast ~ oÞ. An initial guess wð1Þ can be chosen arbitrarily. Numerical of conductivity sðr; experiments have proven that iteration (6.15) outperforms the traditional IE solution ^ of Equation (6.7). based on the straightforward inversion of the matrix ðI^ GÞ

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The ﬁrst results for 3-D modeling using Equation (6.15) were presented in (Avdeev et al., 1997). In this work the authors demonstrated the effectiveness of their numerical solution over a wide range of frequencies. They compared their results with those obtained by other IE and FD solutions (Newman et al., 1986; Wannamaker, 1991; Mackie et al., 1993; Alumbaugh et al., 1996) for various 3-D models and types of the ﬁeld excitation. Mackie and Watts (1999) presented comparisons between the MIDM solution (Avdeev et al., 1997) and FD solution (Mackie et al., 1994) for a case of a 3-D model with high conductivity contrasts. They concluded that the MIDM solution allows the EM ﬁelds to be simulated even for ore body models with very high contrasts of conductivity. In Avdeev et al. (1998) the numerical MIDM solution has been modiﬁed for the needs of airborne electromagnetics, while in Avdeev et al. (1999) the authors presented their numerical MIDM solution for induction logging problems in deviated wells. For realistic 3-D models with contrast as large as 10 000:1 or more (ore bodies, wells, etc.), the number of MIDM iterations of Equation (6.15) can typically reach a few hundred (see Fig. 6.1, ‘Simple iteration’ curves). Taking into account that a single iteration of Equation (6.15) for an adequately discretized model can require several minutes, total run time for the complete solution of Equation (6.11) can reach 10 h. In Avdeev et al. (2000, 2002a,b) the authors presented a novel, more

Fig. 6.1. Comparison of the IE solution based on the Krylov subspace iteration with that based on the simple iteration. (a) Relative residual. (b,c) Apparent conductivity responses versus iteration counts. (After Avdeev et al., 2002a.)

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149

effective approach based on a Krylov subspace iteration for the solution of scattering Equation (6.11). Let me now brieﬂy describe their approach. 6.2.2.1. Krylov subspace iteration By imposing a numerical grid on the model, one can easily derive discrete representations of Equations (6.11) and (6.15) in the form of the following system of linear equations: ^ ð1^ MðoÞÞ XðoÞ ¼ X0 ðoÞ

ð6:16Þ

^ XðmÞ ðoÞ Xðmþ1Þ ðoÞ ¼ X0 ðoÞ þ MðoÞ

ð6:17Þ ^ Here the n n matrix MðoÞ is a discrete representation of the integral operator that is presented on the right-hand side of Equation (6.11), and the n-vectors X(o) and X0(o) represent vector ﬁelds wðr; oÞ and wo ðr; oÞ, respectively. It is obvious that the iteration given in Equation (6.17) is nothing but a simple iteration for the solution of the following linear system: ^ AðoÞXðoÞ ¼ X0 ðoÞ

ð6:18Þ ^ ^ ^ ^ where A ¼ 1 M. It is noteworthy that the matrix AðoÞ is well preconditioned, although it is non-Hermitian and dense. In many cases, its condition number, ^ kA^ 1 k, is strongly bounded from above (Avdeev et al., 2000). Thus, it kðAÞ ¼ kAk follows that pﬃﬃﬃﬃﬃﬃ ^ Cl kðAÞ ð6:19Þ ~ oÞ of the model. For where Cl is the maximum lateral contrast of conductivity sðr; example, kðAÞ 102 for a model with a contrast as large as C l ¼ 104 . The wellpreconditioned matrix A^ of Equation (6.18) makes the IE method advantageous and better than other methods. Indeed, it is known (see, for example, Tamarchenko et al., 1999) that the condition numbers for the matrices of FD system can reach values of 108–1012, or even more. This requires that the FD matrices must be preconditioned. In Avdeev et al. (2000, 2002a,b) the authors, for the ﬁrst time, applied a generalized conjugate gradient method (van der Vorst, 1992; Zhang, 1997) in order to effectively solve system (6.18). They demonstrated that such an approach allows the solution of (6.18) to be accelerated dramatically (see Fig. 6.1) in comparison with the simple iteration given in (6.17). For monotonic convergence to the solution they applied a speciﬁc QMR-smoothing procedure, which was proposed in Zhou and Walker (1994). The iteration process is terminated when ^ ðmÞ ð6:20Þ rðmÞ ¼ X0 AX kX0 k1 0:003 Given the solution of the system (6.18) (wðr0 ; oÞ XðoÞ) and using expression (6.5), it is possible to calculate the current jq ðr; oÞ ¼ 2lðr; oÞRðr; oÞ wðr; oÞ þ lðr; oÞE0 ðr; oÞ ð6:21Þ

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Table 6.1. Computational statistics for a 3-D induction logging model.

Method Grid Nx Ny Nz ¼ M Frequency (kHz) Preconditioner Iterates-m Run timea(s) IE FD

31 31 32 ¼ 30,752 563,328 435,334 435,334

10, 1600, 5000 10 160 5000

MIDM LIN Jacobi Jacobi

7 17 6000 1200

2950 2121 5686 1101

Source: After Avdeev (2005a). a Times are presented for Pentium/350 MHz PC (IE code) and for IBM RS-6000 590 workstation (FD code).

Fig. 6.2. Comparisons of the responses obtained from the IE and FD solutions of a 451 deviated-borehole model. (a) The model, (b) 10-kHz, (c) 160-kHz and (d) 5-MHz responses. Responses for a vertical borehole model are also included for comparison. (After Avdeev et al., 2002a.)

within the volume Vs and ﬁnally to calculate scattered ﬁelds in any point ra , as Z

s

H ðra ; oÞ ¼ Z

V

s

V

s

¼

me G^ o ðra ; r0 ; oÞjq ðr0 ; oÞ dv0 ; ee G^ 0 ðra ; r0 ; oÞjq ðr0 ; oÞ dv0

Es ðra ; oÞ ð6:22Þ

Additionally if the total ﬁeld is needed, one can sum the ﬁelds deﬁned by (6.22) and (6.3). Alternative numerical solutions based on the modiﬁed iterative dissipative method were presented by Zhdanov and Fang (1997) and Singer et al. (2003).

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6.3. MODEL EXAMPLES 6.3.1. Induction logging problem Fig. 6.2 presents the induction logs for a 3-D model of a 451-deviated borehole. Curves are also shown for the case of a vertical borehole. The effect of the deviation

Fig. 6.3. (a) An airborne EM system over a vertical contact and an earth’s uplift; 900-Hz AEM responses, (b,c) vertical magnetic dipole excitation and (d,e) horizontal magnetic dipole excitation. (After Avdeev et al., 1998.)

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is clearly seen. Very good agreement is observed between the different solutions (typical discrepancies are less than a few percent). The computational statistics for this simulation are listed in Table 6.1. 6.3.2. Airborne EM example Fig. 6.3 shows a model of a vertical fault contact with surface topography. To the right of the fault, the earth has been thrust 10 m upwards. The IE responses obtained by Avdeev et al. (1998; solid lines) coincide with the FD responses of Alumbaugh et al. (1996; dashed lines) quite well. In order to demonstrate the topography effect, the responses without the thrust (dotted lines) are also shown.

6.4. CONCLUSION In this Chapter I brieﬂy described the volume integral equation method as it is today applied for solving 3-D geoelectromagnetic problems. The most advanced up-to-date modiﬁcation of this method uses a Krylov subspace iteration to solve the MIDM scattering equation. This approach has been implemented to develop a new generation of 3-D forward modeling codes for various EM applications, including (1) induction logging in deviated wells, (2) (under)grounded and airborne controlled-source EM, (3) magnetotellurics (MT), and (4) global induction studies. This new generation of computational codes has proven to be an effective way to simulate geoEM ﬁelds in complex 3-D environment. The codes developed: (1) work on PC and workstation platforms under Windows, Unix or Linux operating systems; (2) have been thoroughly checked against semi-analytical solutions (Chew et al., 1984; Liu, 1993) and veriﬁed by other 3-D IE (Wannamaker, 1991) and FD (Mackie et al., 1993; Alumbaugh et al., 1996; Newman and Alumbaugh, 2002; Fomenko and Mogi, 2002) solutions; (3) give accurate results for lateral contrast of electrical resistivity up to 100 000; (4) simulate the responses from DC up to 50 MHz frequency; (5) account for the induced polarization (IP) and displacement currents; (6) admit an anisotropy of electrical resistivity; and , ﬁnally, (7) allow running of large-scale models with up to 8 million cells. Acknowledgements Some results presented in this review were obtained in close collaboration of the author with Alexei Kuvshinov and Oleg Pankratov. The author thanks Greg Newman for fruitful collaboration, which improved the IE solution, and Brian O’Reilly for help with English.

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Chapter 7 Inverse Problems in Modern Magnetotellurics V.I. Dmitriev and M.N. Berdichevsky Moscow State University, Russia

The inverse problem in modern magnetotellurics using the plane-wave approximation of the source ﬁeld consists in the determination of the geoelectric structure of the Earth from a known dependence of magnetotelluric (MT) and magnetovariational (MV)response functions on the observation coordinates x,y,z ¼ 0 and the frequency o of the electromagnetic ﬁeld. The basic MT response functions are the impedance tensor Z xx Z xy Z^ ¼ ð7:1Þ Z yx Z yy deﬁned from relations between the horizontal components of the electric and magnetic ﬁelds (Berdichevsky and Zhdanov, 1984) E x ðx; yÞ ¼ Z xx H x ðx; yÞ þ Z xy H y ðx; yÞ E y ðx; yÞ ¼ Z yx H x ðx; yÞ þ Zyy H y ðx; yÞ and the apparent resistivities 2 rxy ¼ Z xy =om0 ;

2 ryx ¼ Z yx =om0

ð7:2Þ

ð7:3Þ

calculated from the components Zxy,Zyx of the secondary diagonal of the ^ impedance tensor Z. The basic MV response function are the tipper vector (the Wiese–Parkinson vector) W ¼ W zx 1x þ W zy 1y Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40007-X

ð7:4Þ

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deﬁned from relations between the vertical component of the magnetic ﬁeld and its horizontal components (Parkinson, 1983) H z ðx; yÞ ¼ W zx H x ðx; yÞ þ W zy H y ðx; yÞ

ð7:5Þ

and the magnetic tensor ^ ¼ M

M xx M yx

M xy M yy

ð7:6Þ

deﬁned from relations between the horizontal components of magnetic ﬁelds at two observation sites (Berdichevsky and Zhdanov, 1984), a ‘‘ﬁeld’’ site ðx0 ; y0 Þ and a ‘‘reference’’ site ðx00 ; y00 Þ: H x ðx0 ; y0 Þ ¼ M xx H x ðx00 ; y00 Þ þ M xy H y ðx00 ; y00 Þ H y ðx0 ; y0 Þ ¼ M yx H x ðx00 ; y00 Þ þ M yy H y ðx00 ; y00 Þ

ð7:7Þ

MT and MV inversions usually reduce to solution of the operator equations for the impedance tensor and tipper ~^ ^ y; z ¼ 0; o; sðx; y; zÞg ¼ Z Zfx;

ð7:8aÞ

~ Wfx; y; z ¼ 0; o; sðx; y; zÞg ¼ W

ð7:8bÞ

where Z^ and W are operators of the forward problem that calculate the impedance tensor and tipper from a given electric conductivity s(x,y,z), both the operators ~ ~ are the impedance tensor and tipper depend parametrically on x,y,o; Z^ and W determined on the sets of surface points (x,y) and frequencies o with errors dZ and dW. The electric conductivity s(x,y,z) is found from the conditions ~^ ^ y; z ¼ 0; o; sðx; y; zÞg ð7:9aÞ Z Zfx; dZ W ~ Wfx; y; z ¼ 0; o; sðx; y; zÞg dW

ð7:9bÞ

MT problem (7.8a)–(7.9a) and MV problem (7.8b)–(7.9b) should be mutually consistent. They are solved in the class of piecewise-homogeneous or piecewisecontinuous plane models excited by a plane wave vertically incident on the Earth’s surface, z ¼ 0. ~ y; zÞ such that misﬁts Both the inversions result in approximate distributions sðx; of the impedance tensor and tipper do not exceed errors, dZ and dW, in the initial ~ y; zÞ generate a set Sd of equivalent solutions of the data. The distributions sðx; inverse problem (7.8)–(7.9). MV problem (7.8b)–(7.9b) can be extended by solution of the operator equations for the magnetic tensor ~^ ^ Mfx; y; z ¼ 0; o; sðx; y; zÞg ¼ M

ð7:8cÞ

V.I. Dmitriev and M.N. Berdichevsky/Inverse Problems in Modern Magnetotellurics

~^ ^ y; z ¼ 0; o; sðx; y; zÞg dM M Mfx;

159 ð7:9cÞ

^ is the operator of the forward problem that calculates the magnetic tensor where M ~^ from a given electric conductivity s(x,y,z), it depends parametrically on x,y,o; M is the magnetic tensor determined on the sets of surface points (x,y) and frequencies o with errors dM. Errors in the initial data dZ, dW, dM include the measurement and model errors. The measurement errors are commonly random. They are caused by instrumental ~^ ~ ~^ noises, external interferences, and inaccuracies in the calculation of Z, W, M. Improvement in instrumentation and ﬁeld data processing methods decreases these errors. Presently, due to the progress in MT technologies, measurement errors are, as a rule, fairly small (the problems may be encountered near the sources of intense industrial disturbances). A main difﬁculty is related to model errors that as a result of inevitable deviation of numerical simulations from real geoelectric structures and real MT ﬁelds. As an illustrative example, we can cite the errors arising in 2-D inversion of data obtained above 3-D structures or the errors typical of polar zones, where the magnetic ﬁeld of ionospheric currents has a vertical component violating the plane-wave approximation. Model errors are systematic. They are usually larger than measurement errors. To estimate the model errors, we need a tentative mathematical modeling. Strategy and informativeness of the inverse problems depend on the dimensionality of models in use. The simplest inverse problem is 1-D inversion carried out in the class of 1-D models. It applies the mathematics of zero horizontal derivatives. Such a mathematics provides the local determination of the electrical conductivity along vertical proﬁles passing through observation points. The 1-D inversion evidently ignores distortions produced by horizontal geoelectric inhomogeneities. It is justiﬁed if horizontal variations in the conductivity are fairly small. Otherwise, it can miss real structures and give birth to false structures (artifacts). The transition to 2-D and 3-D inversions carried out in the classes of 2- and 3dimensional models enables a more or less adequate regard for the effects of horizontal geoelectric inhomogeneities, but calls for horizontal derivatives. This mathematics substantially complicates the inverse problem.

7.1. THREE FEATURES OF MULTI-DIMENSIONAL INVERSE PROBLEM Consider three distinguishing features of the multi-dimensional inverse problem. 7.1.1. Normal background When solving the multi-dimensional inverse problem, we face a contradiction between a ﬁnite area of MT and MV observations and a mathematical statement calling for conditions at inﬁnity. In forward problem, this contradiction can be

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easily removed through the embedding of the observation area into a reasonably constructed inﬁnite, horizontally homogeneous layered background (a normal background). In the inverse problem, the normal background of the medium under consideration is unknown and it should be chosen as a mathematical abstraction consistent with observation data and a priori geological and geophysical information. We believe that in the general 3-D case such a normal background can be introduced by the extrapolation of scalar invariants of the measured impedance tensors, for example, the invariant Zbrd (the ‘‘Berdichevsky impedance’’) or Zeff (the effective impedance). Let values of the impedance tensor Z^ be determined in an ðlÞ observation area S0 bounded by a contour C0 and let Z^ ; l ¼ 1; 2 ::: L be speciﬁed at L points of C0 (Fig. 7.1). The average value of the invariant Zbrd on the contour C0, i.e., on the boundary of the observation area, is found as 1 Z brd ¼ ant log L

L X l¼1

L Z xy Z yx 1X log 2 L l¼1 ðlÞ

log ZðlÞ brd ¼ ant log

ðlÞ

ð7:10Þ

Using a spline approximation, the values Zbrd are extrapolated in such a way that the condition Z brd ¼ Z brd is valid on a new boundary contour C1 and the derivative of Zbrd along the normal to C1 vanishes. Given these conditions, we assume that the impedance Z brd is close to the normal impedance Zn of a horizontally layered medium in the inﬁnite area Sn external to C1 and determine its normal conductivity

Fig. 7.1. Introduction of a normal background in the 3-D inversion.

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sn ðzÞ by 1-D inversion of the impedances Z brd . At the last stage, we perform 1-D inversion of the impedances Zbrd extrapolated in the transition zone St and ﬁnd slightly varying transition conductivity st ðzÞ between the observation area S0 and the area Sn. So, we get a model, in which a normal background and a transition zone embrace the observation area: 8 > < sðx; y; zÞ; P 2 S 0 sðPÞ ¼ st ðzÞ; P 2 St ð7:11Þ > : sn ðzÞ; P 2 Sn Likewise, the normal background is introduced using the effective impedances. To test this algorithm, we should make sure that an expansion of the transition zone St has no signiﬁcant effect on the results of MT and MV inversions in the central part of the observation area S0. A similar algorithm based on the averaging and extrapolation of Zbrd or Zeff can be applied in a 2-D approximation of elongated structures. Let observations be carried out along a transverse proﬁle P0 from y ¼ c0 to y ¼ c0 (Fig. 7.2). The average of the invariant Zbrd at the edges of the proﬁle is determined as 1 Z brd ¼ antlog fZbrd ðy ¼ c0 Þ þ Z brd ðy ¼ c0 Þg 2 Z jj ðc0 Þ þ Z ? ðc0 Þ þ Z jj ðc0 Þ þ Z ? ðc0 Þ ¼ antlog 4

ð7:12Þ

Using a spline approximation, the values Zbrd are extrapolated beyond the observation proﬁle P0 in such a way that the conditions Zbrd ¼ Z brd and @Z brd =@y ¼ 0 be valid at the points y ¼ c1 and y ¼ c1. The extrapolation frames the proﬁle P0 (c0 y c0 ) by the transition zones P0t ðc1 oyo c0 Þ; P00t ðc0 oyoc1 Þ and the inﬁnite normalized proﬁles Pn ðy c1 Þ; Pn ðy c1 Þ with the normal impedance Z n Z brd . After 1-D inversion of the impedances Zbrd in the transition zones and the normal impedance Z n Z brd on the normalized proﬁles, we get a model with a symmetric normal background:

Fig. 7.2. Introduction of a normal background in the 2-D inversion.

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8 sn ðzÞ; > > > > 0 > > < st ðzÞ; sðy; zÞ ¼ sðy; zÞ; > > > > s00t ðzÞ; > > : sn ðzÞ;

y c1 c1 oyo c0 c0 y co

ð7:13Þ

c0 oyoc1 y c1

Alternatively, we can determine a symmetric normal background by separate extrapolation of the longitudinal or transverse impedances, Z jj or Z ? . Introduction of a 2-D symmetric homogeneous background is reasonable, if the impedance values measured at the edges of the observation proﬁle P0 do not greatly differ from each other (for instance, on a platform with smooth topography). In the regions with strongly pronounced asymmetry (for instance, on the ocean coast or at foothills), we have to give preference to an inhomogeneous background characterized by the different normal impedances Z_ n ; Z€ n and the different normal conductivities s_ n ðzÞ; s€ n ðzÞ, which provide the best compliance with the real geoelectric structures bordering the observation proﬁle. But it should be mentioned that any 2-D asymmetric model can, by means of mirror-imaging mirror mapping, be reduced to a symmetric model with a homogeneous background. 7.1.2. On detailness of multi-dimensional inversion Compared to a 1-D model, a much greater number of parameters are required for constructing adequate 2-D and 3-D models. It is evident that multidimensional inversions are less stable. Proceeding to the 2-D or 3-D inversion, we aggravate the contradiction between the detailness of solution and its stability, that controls the resolution of the inversion (Berdichevsky and Dmitriev, 2002). To relax this contradiction, we have to ﬁt the solution detailness to the inversion resolution and smooth or schematize models of the geoelectric medium (to diminish a number of parameters). This complies with the nature of the geoelectromagnetism. Indeed, the electromagnetic ﬁeld observed on the Earth’s surface provides information on smoothed buried structures and their integral characteristics. 7.1.3. On redundancy of observation data Solving a 1-D inverse problem, we determine a real scalar function of conductivity s(z) from the scalar complex-valued Tikhonov–Cagniard impedance Z, i.e., from two scalar functions jZ j and argZ, which have different resolving power and can complement each other. Increasing the interpretation model dimensionality, we extent the amount of observation data. Solving a 2-D inverse problem, we separate the galvanic and induction effects, associated with the TM- and TE- modes, and reduce experimental data to twoprinciple complex-valued components of the impedance tensor and one-principle complex-valued component of the tipper oriented along and across the model strike. These components differ in both the stability with respect to subsurface

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distortions and the sensitivity to various target structures. They nicely complement each other, and their successive partial focused inversions can give the most complete information on the geoelectric structures sought for. The distinguishing feature of the 3-D inverse problem is the redundancy of observation data. In an isometric 3-D model, the galvanic and induction effects are inseparable, while a well-deﬁned direction of the model strike does not exist. Thus, solving a 3-D inverse problem, we should have to determine a real scalar function of conductivity s(x,y,z) from all six complex-valued components of the impedance tensor and tipper, Z xx; Zxy ; Z yx ; Z yy and W zx ; W zy , i.e., from 12 scalar functions jZxx j; Z xy ; Z yx ; Z yy ; arg Z xx ; arg Z xy ; arg Z yx ; argZyy and ReW zx ; ReW zy ; ImW zx ; ImW zy (to say nothing about four complex-valued components of the magnetic tensor). The paradox is dramatized by the fact that we poorly know the informativeness of these functions and cannot reasonably choose the efﬁcient strategy of their interpretation (not to mention the laboriousness and instability of such an interpretation). The best approach appears to be the scalarization of a 3-D inverse problem, i.e., the determination of conductivity s(x,y,z) from scalar invariants of the impedance tensor, (e.g., using the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ invariant Zeff ¼ Z xx Z yyqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z xy Zyx orﬃ Z brd ¼ ðZ xy Zyx Þ=2), and the tipper (e.g., using the invariant W ¼ W 2zx þ W 2zy ). This approach may include two levels: (1) MV inversion (i.e., inversion of the scalar invariants of the tipper and/or the magnetic tensor), that at low frequencies are free from subsurface distortions and can give information about deep structures, and (2) MT inversion (i.e., inversion of the scalars invariants of the impedance tensor), that contains errors due to subsurface distortions but can give information about structures that manifest themselves in galvanic anomalies. Note that the scalarization of the 3-D inverse problem (notwithstanding a substantial simpliﬁcation of the interpretation procedure) requires signiﬁcant computational resources, because two forward problems for two different polarizations of the primary ﬁeld should be solved at each iteration step in order to determine the impedance tensor and tipper. The required computational resources can be substantially reduced by using the telluric ^ derived from synchronous MT observations. On and magnetic tensors, T^ and M, this way, only one forward problem for a given polarization of the primary ﬁeld Ep, Hp is solved at each iteration step and the conductivity s(x,y,z) is found directly ^ p ; H ¼ MH ^ p computed at the from the magnetic (or electromagnetic) ﬁeld E ¼ TE Earth’s surface. The same approach can be realized using the electromagnetic ﬁeld synthesized from a known distribution of the impedance tensor or tipper.

7.2. THREE QUESTIONS OF HADAMARD 1. 2. 3.

Solving an inverse problem, one should answer three questions of Hadamard: Does the solution of this problem exist? Is it unique? Is it stable with respect to small errors in initial data?

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These questions determine the correctness of the inverse problem. If its solution exists and if it is unique and stable, the problem is well-posed (posed correctly). But if one of these conditions is violated, the problem is regarded as ill-posed (posed incorrectly), and it calls for special consideration. We will show that inverse problems of magnetotellurics are ill-posed. 7.2.1. On the existence of a solution to the inverse problem ~^ At the ﬁrst glance, this question looks simple, because the impedance tensor T, ~ and magnetic tensor M ^ measured on the Earth’s surface should the tipper W, correspond to the real distribution of conductivity in the inhomogeneous Earth. However, the experimental values of the impedance tensor and the tipper are inaccurate, and they may conﬂict with mathematical models. ~ ~ contain measurement and model errors dZ and dW. It is evident Let Z^ and W that the real distribution of conductivity in the Earth and the real MT and MV response functions do not belong to the chosen model class on which the inverse problem is deﬁned. Such an inverse problem does not have a rigorous solution. To remove this contradiction, the notion of quasi-solution is introduced: a conductivity distribution s(x,y,z) is said to be a quasi-solution to the inverse problem (7.8) if conditions (7.9) are satisﬁed, i.e., if the misﬁts of the impedance tensor and the tipper do not exceed errors in the initial data, dZ and dW, dM . The inverse problem (7.8)–(7.9) has a set, of quasi-solutions. From this set, we have to select a quasisolution that provides (at a given level of abstraction) the best approximation to the y; zÞ is called the exact real geoelectric structure. This conductivity distribution sðx; model solution. When solving the inverse problem, we endeavor to ﬁnd the exact model solution. Using the notion of the exact model solution, we can formalize the deﬁnition of be the impedance tensor and the measurement and model errors. Let Z^ and W tipper obtained from a model that belongs to the chosen model class and has the y; zÞ. Then, measurement errors are determined as conductivity sðx; ~^ ^ ~ ð7:14Þ dms dms Z ¼ Z Z ; W ¼ W W and model errors are determined as ^ ^ dmd Z ¼ Z Zfx; y; z ¼ 0; o; sðx; y; zÞg Wfx; y; z ¼ 0; o; sðx; y; zÞg dmd ¼ W W

ð7:15Þ

md ms md Setting dZ ¼ dms Z þ dZ and dW ¼ dW þ dW and applying the triangle rule, we reduce (7.14), (7.15) to the initial condition (7.9a,b).

7.2.2. On the uniqueness of the solution to the inverse problem We proceed from the following heuristic statement. The inverse problem has a unique solution if it is deﬁned on a given model class and the impedance tensor and the tipper belonging to this class are exactly determined over the entire Earth’s

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surface over the entire frequency range. This statement was proven in four partial cases. 1-D MT inversion Tikhonov (1965) proved the uniqueness theorem for 1-D MT inversion in the class of piecewise-analytical functions d(z). In our book, we present a simpliﬁed proof of the Tikhonov theorem for the case of a homogeneously layered model. Let d(z) be a piecewise-constant function of the depth z sðzÞ ¼ sm at zm1 ozozv ; m 2 ½1; M; z0 ¼ 0; zM ¼ 1; hm ¼ zm zm1 where sm and hm are the conductivity and thickness of the mth layer and zm the depth of its lower boundary. At the depth z ¼ zM1 , the model rests on an inﬁnite homogeneous basement of conductivity sM ¼ const. The admittance Y ðz; oÞ in this homogeneously layered model satisﬁes the Riccati equation dY ðz; oÞ þ iom0 Y 2 ðz; oÞ ¼ sðzÞ; dz with the boundary conditions ½Y ðz; oÞS ¼ 0;

z 2 ½0; zN1 ; o 2 ½0:1

ð7:16Þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sM Y ðzM1 ; oÞ ¼ ð1 þ iÞ 2om0

Using (7.16), we can easily derive a recurrent formula expressing Y m1 ¼ Y ðzm1 ; oÞ through Y m ¼ Y ðzm ; oÞ Y m1 ¼ bm

ðbm þ Y m Þ ðbm Y m Þe2ikm hm ðbm þ Y m Þ þ ðbm Y m Þe2ikm hm

ð7:17Þ

where km is the wavenumber of the mth layer rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ om0 sm km ¼ ð1 þ iÞ 2 and bm ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ km sm ¼ ð1 þ iÞ om0 2om0

Inverse of (7.17) yields a formula determiningYm through Y m1 (converting the admittance from the upper boundary of the mth layer in to its lower boundary) Y m ¼ bm

ðbm þ Y m1 Þ ðbm Y m1 Þe2ikm hm ðbm þ Y m1 Þ þ ðbm Y m1 Þe2ikm hm

ð7:18Þ

Let the admittance Y 0 ¼ Y ð0; oÞ be known at the Earth’s surface, while the conductivity s(z) be known within the interval 0ozozm . Then, the successive application of (7.18) provides the admittance Y m ¼ Y ðzm ; oÞ at a depth zm. Now, we prove the theorem of uniqueness, which is formulated as follows. If Y ð1Þ ðz; oÞ and Y ð2Þ ðz; oÞ are the solutions of problem (7.16) for sð1Þ ðzÞ and sð2Þ ðzÞ,

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ð2Þ ð1Þ ð2Þ then Y ð1Þ 0 ðoÞ Y 0 ðoÞ implies that s ðzÞ s ðzÞ. This theorem is proven ad absurdum. Assume that ð2Þ Y ð1Þ 0 ðoÞ Y 0 ðoÞ

ð7:19aÞ

sð1Þ ðzÞ sð2Þ ðzÞ at 0ozozm1

ð7:19bÞ

sð1Þ ðzÞasð2Þ ðzÞ at z4zm1

ð7:19cÞ Y ð1Þ 0

Y ð2Þ 0

and to the Then, applying (7.18) to (7.19a) and (7.19b) and extending ð2Þ depth zm1 , we obtain Y ð1Þ ðoÞ Y ðoÞ. Let us determine the high-frequency m1 m1 asymptotics of Y m1 ðoÞ. According to (7.17), rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sm Y m1 ðoÞ bm ¼ ð1 þ iÞ ð7:20Þ 2om0 o!1 ð2Þ ð1Þ ð2Þ Thus, the identity Y ð1Þ m1 ðoÞ Y m1 ðoÞ leads to sm ¼ sm , which contradicts the assumption (7.19c). Successively increasing m, we reach the model basement and obtain sð1Þ ðzÞ sð2Þ ðzÞ; z 0. The theorem of uniqueness is proven.

2-D MT inversion The next step was made by Weidelt (1978), who proved the uniqueness theorem for a 2-D model excited by an E-polarized ﬁeld. In this model, the electrical conductivity s(y,z) is supposed to be an analytical function. It was shown that simultaneous observations of horizontal components of the electric and magnetic ﬁelds, carried out over the entire frequency range 0ooo1 along an y-proﬁle of a ﬁnite length, provide the unique determination of s(y,z). The Weidelt theorem was generalized by Gusarov (1981), who considered a 2-D E-polarized model with the piecewise-analytical conductivity s(y,z). The Gusarov theorem states that the piecewise-analytical function s(y,z) is uniquely determined by the longitudinal impedance Z jj ¼ Z xy speciﬁed over the entire frequency range 0ooo1 on an inﬁnite y-proﬁle 1oyo1. All these proofs have their basis in the skin effect. Due to the skin effect, there always exists a high frequency such that the ﬁeld or impedance can be approximated by a high-frequency asymptotics depending on a local conductivity. Comparison of high-frequency asymptotics for various geoelectric structures suggests that different distributions of conductivity s correspond to different ﬁelds and different impedances. Unfortunately, the realization of this simple idea encounters signiﬁcant mathematical difﬁculties due to complexity of the determination of the ﬁeld highfrequency asymptotics in heterogeneous media. Resorting to intuition, the above proofs of uniqueness can be extended to the general 3-D case of MT inversions. It seems evident that the o- dependence of the impedance tensor ensures determination of the vertical variations in the conductivity, whereas its x, y-dependence characterizes the horizontal variations in the conductivity. Intuition suggests that measurements of the MT impedance made in a wide frequency range along sufﬁciently long proﬁles or over a sufﬁciently

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large area can provide information adequate for the reconstruction of the geoelectric structure of the region studied. 2-D MV inversion The uniqueness of the MV inversion for a long time was open to question. It seemed that the tipper characterizes horizontal heterogeneities of the medium, but cannot provide information about its normal layered structure because W zx ¼ W zy ¼ 0 in a horizontally homogeneous model. However, if the medium is horizontally inhomogeneous, the MV sounding can be considered as a common frequency sounding using the magnetic ﬁeld of a local embedded source. The latter is formed by any inhomogeneity Dsðx; y; zÞ in which an excess electric current is induced that spreads into the host medium. It is evident that this current and its magnetic ﬁeld depend not only on the structure of the inhomogeneity, Dsðx; y; zÞ, but also on the normal structure, sn ðzÞ. Thus, the solution sðx; y; zÞ ¼ sn ðzÞ þ Dsðx; y; zÞ of the MV inverse problem exists and we should elucidate whether it is unique. The theorem of uniqueness for the MV inversion was proven by Dmitriev (Berdichevsky et al., 1997). Let us consider a model shown in Fig. 7.3. In this model, a homogeneously layered Earth with the normal conductivity sðzÞ; 0 z D sn ðzÞ ¼ sD ; Dz contains a 2-D inhomogeneous domain S of conductivity sðy; zÞ ¼ sn ðzÞ þ Dsðy; zÞ, where Dsðy; zÞ is the excess conductivity. The inhomogeneity is striking along the xaxis, and the maximum diameter of its cross-section is d. The functions sn ðzÞand Dsðy; zÞ are piecewise-analytical. An inﬁnite homogeneous basement of conductivity sD ¼ const occurs at depth D. The model is excited by the plane E-polarized

Fig. 7.3. A layered model with a 2-D inhomogeneous bounded domain S.

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electromagnetic wave incident vertically on the Earth’s surface z ¼ 0. The Dmitriev theorem states that the piecewise-analytical distribution of conductivity sn ðzÞ PeS sðPÞ ¼ sn ðzÞ þ Dsðy; zÞ P 2 S is uniquely determined by exact values of the tipper W zy ðyÞ ¼

H z ðy; z ¼ 0Þ ; H y ðy; z ¼ 0Þ

1oyo1;

0 oo1

given on the Earth’s surface z ¼ 0 at all points of the y-axis from N to N over the entire range of frequencies o from 0 to N. The uniqueness theorem is proven in two stages. First, we derive the asymptotics of the tipper W zy ðyÞ at a large distance from the inhomogeneity S and show that it determines the normal conductivity sn(z). Then, with the known conductivity sn(z), we prove that the tipper uniquely determines the longitudinal impedance of the inhomogeneous medium. The anomalous magnetic ﬁeld Ha on the Earth’s surface can be represented as a ﬁeld produced in a horizontally homogeneous-layered medium by excess currents of density jx induced in the domain S. Normalizing Ha, we write Z _a H ay ðy; z ¼ 0Þ H 0y ðyÞ ¼ ¼ j x ðPo Þhy ðy; Po Þ dS H ny ðz ¼ 0Þ S Z _a H az ðy; z ¼ 0Þ ¼ j x ðPo Þhz ðy; Po Þ dS ð7:21Þ H 0z ðyÞ ¼ HN y ðz ¼ 0Þ S

where hy ðy; Po Þ; hz ðy; Po Þ are magnetic ﬁelds produced at a surface of a horizontally homogeneous medium by an inﬁnitely long linear current of the unit density ﬂowing at the point Po ðyo ; zo Þ 2 S in the x-direction. The functions hy ðy; Po Þ and hz ðy; Po Þ assume the form (Dmitriev, 1969; Berdichevsky and Zhdanov, 1984) i lim hy ðy; Po Þ ¼ omo z!0

Z1

cos lðy yo Þelz Uðl; z ¼ 0; zo Þl dl

0

i lim hz ðy; Po Þ ¼ omo z!0

Z1

sin lðy yo Þelz Uðl; z ¼ 0; zo Þl dl

ð7:22Þ

0

lz

where the factor e relates to the upper half-space z 0 and the function Uðl; z; zo Þ is the solution of the boundary problem d2 Uðl; z; zo Þ Z2 ðl; zÞUðl; z; zo Þ ¼ dðz zo Þ; dz2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Zðl; zÞ ¼

2

l iomo sN ðzÞ;

z; zo 2 ½0; D Re Z40

ð7:23Þ

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with conditions dUðl; z; zo Þ þ lUðl; z; zo Þ ¼ 0; dz dUðl; z; zo Þ ZD ðlÞUðl; z; zo Þ ¼ 0; dz qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ZD ðlÞ ¼

l2 iomo sD ;

z¼0 z¼h Re ZD 40

Let us turn to (7.22) and ﬁnd the asymptotics of the functions hy ðy; Po Þ and hz ðy; Po Þ at jy yo j ! 1. Given large jy yo j, harmonics of low spatial frequencies l make the major contribution to hy ðy; Po Þ, hz ðy; Po Þ. Expanding Uðl; z ¼ 0; zo Þ in powers of small l, we get dUðl; z ¼ 0; zo Þ þ Uðl; z ¼ 0; zo Þ ¼ Uðl ¼ 0; z ¼ 0; zo Þ þ l dl l¼0 whence, upon the substitution into (7.22) and integration, we obtain ! i Uðl ¼ 0; z ¼ 0; zo Þ 1 þO hy ðy; Po Þ ¼ omo ðy yo Þ2 ðy yo Þ4 ! 2i 1 dUðl; z ¼ 0; zo Þ 1 þO hz ðy; Po Þ ¼ omo ðy yo Þ3 dl ðy yo Þ5 l¼0 _a

ð7:24Þ

_a

In order to write the relations between H 0y and H 0z in the form containing the MT impedance, we introduce the functions dUðl; z; zo Þ V y ðzÞ ¼ Uðl ¼ 0; z; zo Þ; V z ðzÞ ¼ ð7:25Þ dl l¼0 The function V y ðzÞ is the solution of problem (23) at l ¼ 0. The problem for the function V z ðzÞ is obtained by differentiating (23) with respect to l and setting l ¼ 0. Then d2 V z ðzÞ þ iomo sðzÞV z ðzÞ ¼ 0 z 2 ½0; D dz2 dV z ðzÞ ¼ V y ð0Þ dz z¼þ0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dV z ðzÞ iomo sD V z ðDÞ ¼ 0 dz z¼D In this notation, V y ð0Þ i 1 þO hy ðy; Po Þ ¼ 2 om0 ðy yo Þ ðy yo Þ4 and

!

ð7:26Þ

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2i V z ð0Þ 1 hz ðy; Po Þ ¼ þO om0 ðy yo Þ3 ðy yo Þ5

! ð7:27Þ

Now return to (7.21) and determine the anomalous magnetic ﬁeld at large distances from the inhomogeneity. Let y yo 44d. Then, Z Z _a V y ð0Þ V y ð0Þ i j x ðPo Þ i i V y ð0Þ dS ¼ j x ðPo Þ dS Jx H 0y ðyÞ ¼ 2 2 omo omo ðy yS Þ omo ðy yS Þ2 ðy yo Þ S

S

and _a

H 0z ðyÞ ¼

2i V z ð0Þ omo

Z S

j x ðPo Þ 2i V z ð0Þ dS ¼ omo ðy yS Þ3 ðy yo Þ3

Z j x ðPo Þ dS S

2i V z ð0Þ ¼ Jx omo ðy yS Þ3

ð7:28Þ

where Z j x ðPo Þ dS

Jx ¼ S

is the total excess current in the inhomogeneity and yS is the coordinate of the central point of its cross section S. Thus, with regard for (7.26), we have _a

H 0z ðyÞ _a

H 0y ðyÞ

¼

2 V z ð0Þ 2 V z ð0Þ ¼ ðy yS Þ V y ð0Þ ðy yS Þ dV z ðzÞ=dzz¼0 _a

ð7:29Þ

_a

It is easy to show that the ratio H 0z =H 0y can be expressed through the normal impedance of the Earth. Let us introduce the function ZðzÞ ¼ iomo

V z ðzÞ dV z ðzÞ=dz

ð7:30Þ

It is seen from (7.26) that Z(z) satisﬁes the Riccati equation dZðzÞ sn ðzÞ Z 2 ðzÞ ¼ iomo dz

ð7:31Þ

with the boundary condition sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ iomo ZðDÞ ¼ sD We obtained the known problem for the impedance of a 1-D medium with the conductivity sn ðzÞ; 0 z D and sD ¼ const; z4D. The function Z(z) in the model under consideration evidently represents the normal impedance Zn(z). Setting Z(z) ¼ Zn(z) and taking into account (7.29)–(7.31), we ﬁnd the far-zone asymptotics

V.I. Dmitriev and M.N. Berdichevsky/Inverse Problems in Modern Magnetotellurics

iomo ðy yS Þ H 0z ðyÞ Z n ð0Þ ¼ _a 2 H 0y ðyÞ _a

jyyS j d

iomo ðy yS Þ H a0z ðyÞ ¼ 2 H a0y ðyÞ

171

ð7:32Þ j yyS j d

that coincides with the known expression for a remote inﬁnitely long linear current (Vanyan, 1965). normal impedance Zn(0) is connected with the ratio of the a _ a The _ components H 0z and H 0y of the anomalous magnetic ﬁeld, which can be determined from_values of the tipper Wzy known at all points of the y-axis from N to N. To a ﬁnd H 0y , we solve the known integral equation (Dmitriev and Mershchikova, 2002) _a

1 W zy ðyÞH 0y ðyÞ þ p

Z1 1

_a

H 0y ðy0 Þ y y0

dy0 ¼ W zy ðyÞ

ð7:33Þ

Then we compute _a

_a

H 0z ¼ W zy ð1 þ H 0y Þ

ð7:34Þ _a

_a

Knowing Wzy, we synthesize the normalized anomalous magnetic ﬁeld H 0y , H 0z and the normal impedance Zn from the far-zone asymptotics. With known _ a calculate _a H 0y , H 0z and Zn, we integrate the second Maxwell equation (the Faraday law) and continue the longitudinal impedance Z jj to the entire y-axis 9 8 Zy _ a = < E ðyÞ 1 x ð7:35Þ Z k ðyÞ ¼ Z iom H ðyÞ dy ¼ N a o 0z _ ; : H y ðyÞ 1 þ H 0y

1

jj

Thus, we ﬁnd Z from Wzy. A one-to-one correspondence exists between Zjj and Wzy. Therefore, we can apply the Gusarov theorem (Gusarov, 1981), stating that inversion of Z jj has a unique solution, and extend this result to inversion of Wzy. The uniqueness theorem for the longitudinal impedance Z jj reduces to that for the tipper Wzy. Moreover, these two theorems can be supplemented by the uniqueness theorem for the magnetic tensor. Return to a 2-D model shown in Fig. 7.3. Let the longitudinal impedance Z jj ðyÞ ¼ Z jj ðy; z ¼ 0Þ ¼ E x ðy; z ¼ 0Þ=H y ðy; z ¼ 0Þ be known at all points of the yaxis from N to N over the entire range of frequencies o from 0 to N. The electric ﬁeld E x ðy; zÞ is a solution of the problem @2 E x ðy; zÞ @2 E x ðy; zÞ @2 E x ðy; zÞ þ þ k20 ¼ 0; 2 2 @y @z @y2

1oyo1;

0 z4 1 ð7:36Þ

with boundary condition on the Earth’s surface E x ðy; z ¼ 0Þ ¼ Z jj ðyÞH y ðy; z ¼ 0Þ ¼ and absorption condition in the air

Z jj ðyÞ @E x ðy; zÞ iom0 @z z¼0

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pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E x ðy; zÞ E 0 eik0 z 0 as y2 þ z2 1

where k0 is the air wavenumber, Im k0 40, and E 0 the amplitude of the incident wave. It is well known that a problem of this kind has an unique solution continuously depending on the coefﬁcient Z jj ðyÞ in the boundary condition. Consequently, to the different impedances Z jjð1Þ ðyÞ andZjjð2Þ ðyÞ, the different electric ð2Þ ﬁelds E ð1Þ x ðy; zÞ and E x ðy; zÞ correspond. Does it mean that to different impedances, different magnetic ﬁelds on the Earth’s surface correspond. Let us give the proof by contradiction. The boundary problem for the electric ﬁeld can be rewritten as @2 E x ðy; zÞ @2 E x ðy; zÞ þ þ k20 E x ðy; zÞ ¼ 0; 1oyo1; 0 z4 1 @y2 @z2 @E x ðy; zÞ ¼ iom0 H y ðy; z ¼ 0Þ @z z¼0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E x ðy; zÞ E 0 eik0 z ! 0 as y2 þ z2 ! 1

ð7:37Þ

Solution to this problem exists and is unique. Hence, to identical magnetic ﬁelds ð2Þ H ð1Þ identical electrical ﬁelds correspond y ðy; z ¼ 0Þ H y ðy; z ¼ 0Þ, ð1Þ ð2Þ E x ðy; zÞ E x ðy; zÞ. Assume that to the different impedances Z jjð1Þ ðy; oÞ and Zjjð2Þ ðy; oÞ, the identical ð2Þ magnetic ﬁelds H ð1Þ y ðy; z ¼ 0Þ H y ðy; z ¼ 0Þ correspond. But from (7.37) it follows ð2Þ that in this case the identical electric ﬁelds E ð1Þ x ðy; zÞ E x ðy; zÞ also correspond to jj jj the different impedances Z ð1Þ ðy; oÞ and Z ð2Þ ðy; oÞ, which contradicts the statement derived from (7.36). So, we say that to different impedances Z jjð1Þ ðy; oÞ and Zjjð2Þ ðy; oÞ ð2Þ different magnetic ﬁelds H ð1Þ y ðy; z ¼ 0Þ and H y ðy; z ¼ 0Þ correspond. And taking into account the Gusarov uniqueness theorem for the longitudinal impedance Z jj ðy; oÞ, we state that to different conductivity distributions sð1Þ ðy; zÞ and sð2Þ ðy; zÞ, ð2Þ different magnetic ﬁelds H ð1Þ y ðy; z ¼ 0Þ and H y ðy; z ¼ 0Þ correspond on the Earth’s surface. This proves the uniqueness theorem for the diagonal component M yy of the ^ magnetic tensor M. Both methods, MT and MV soundings, have a common mathematical basis. The 2-D conductivity distribution is uniquely determined from exact values of TE impedances as well as from exact values of tippers or TE components of the magnetic tensor given on the inﬁnitely long transverse proﬁle in the entire frequency range. 7.2.3. On the instability of the inverse problem Inverse problems of magnetotellurics are unstable. The set Sd , characterized by small misﬁts of the impedance tensor and tipper, can contain equivalent solutions that strongly differ from one another and from the exact model solution. We illustrate this property of the inverse problem by the example of the 1-D inversion. The analysis is based on the theorem of stability of the S-distribution proven by Dmitriev in (Berdichevsky and Dmitriev, 1991, 2002).

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Recall that the S-distribution stands for a function Zz SðzÞ ¼

sðzÞ dz

ð7:38Þ

0

determining the conductance of the Earth on the interval ½0; z. The conductivity s is connected with the conductance S through the differential relation sðzÞ ¼ dSðzÞ=dz. The theorem of stability of the S-distribution consists of two statements. 1. The admittance Y ðoÞ ¼ Y ðz ¼ 0; oÞ measured at the Earth’s surface depends continuously on SðzÞ. Thus, the condition ð1Þ S ðoÞ Sð2Þ ðoÞ ð7:39Þ C implies that ð1Þ Y ðoÞ Y ð2Þ ðoÞ dðÞ L2

ð7:40Þ

where d ! 0 at ! 0. 2. The conductance SðzÞ is stably determined from the admittance Y ðoÞ ¼ Y ðz ¼ 0; oÞ measured at the Earth’s surface. Thus, ð1Þ S ðoÞ S ð2Þ ðoÞ ! 0 ð7:41Þ C if ð1Þ Y ðoÞ Y ð2Þ ðoÞ ! 0 L2

ð7:42Þ

Take the set of conductivity distributions obtained from the inversion of 1-D admittance sd 2 Sd ¼ fsðzÞ : Y~ ðoÞ Y ½o; sðzÞL dY g ð7:43Þ 2 where Y~ ðoÞ is the measured admittance, Y ½o; sðzÞ the operator calculating the admittance from a given distribution sðzÞ, and dY the error in the admittance. The ð2Þ theorem of stability of the S-distribution implies that, for any sð1Þ d ðzÞ and sd ðzÞ of the set Sd , the following condition is valid: z Z Zz sð1Þ ðzÞ dz sð2Þ ðzÞ dz ðdY Þ ð7:44Þ d d 0

0

sð1Þ d ðzÞ

C

sð2Þ d ðzÞ

where ! 0 as dY ! 0. If and meet condition (7.44), they are equivalent, i.e., they are characterized by closely related S-distributions and cannot be resolved by MT observations performed with an error dY . Such s-distributions are called S-equivalent distributions. We say that Sd is the set of S-equivalent distributions of the conductivity. In the framework of 1-D magnetotellurics, we can formulate the following generalized principle of S-equivalence: the conductance S characterizes the whole set Sd of equivalent solutions of the inverse problem. To specify the entire set Sd it is sufﬁcient to know its S-distribution.

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Differentiating the conductance SðzÞ, one intends to ﬁnd the conductivity sðzÞ. However, the immediate numerical differentiation of SðzÞ is an unstable operation generating a scatter in the distribution sðzÞ. The determination of sðzÞ from Y ðoÞ is evidently an ill-posed problem. It is easy to show, that essentially different distributions sð1Þ ðzÞ and sð2Þ ðzÞ exist corresponding to close distributions Sð1Þ ðzÞ and S ð2Þ ðzÞ, and thereby to close distributions Y ð1Þ ðoÞ and Y ð2Þ ðoÞ. As an example, consider a model with an inﬁnite homogeneous basement at a depth h. Let 0 for ze½z0 ; z0 þ Dh ð1Þ ð2Þ pﬃﬃﬃﬃﬃﬃ ð7:45Þ s ðzÞ s ðzÞ ¼ c= Dh for z 2 ½z0 ; z0 þ Dh where z0 þ Dhoh; while c and Dh are arbitrary positive constants. Then 8 for 0 z z0 > Zz <0 p ﬃﬃﬃﬃﬃﬃ S ð1Þ ðzÞ S ð2Þ ðzÞ ¼ ½sð1Þ ðzÞ sð2Þ ðzÞ dz ¼ cðz z0 Þ= Dh for z0 z z0 þ Dh > : pﬃﬃﬃﬃﬃﬃ c Dh for z0 þ Dh z h 0 ð7:46Þ The norms of deviations (45) and (46) are determined as 91=2 8 h =
N S ¼ Sð1Þ ðzÞ S ð2Þ ðzÞL3 ¼

8 h
91=2 =

½Sð1Þ ðzÞ S ð2Þ ðzÞ2 dz ; : 0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ c Dhðh z0 2Dh=3Þ

ð7:47Þ

Choosing large c and small Dh, the deviation N s can always be done arbitrarily large, and the deviation N S arbitrarily small. Consequently, arbitrarily differing conductivities can correspond to close conductances and close admittances. Let the medium contain a thin layer, whose conductance S0 is much smaller than the conductance S of the overlying layers. The conductivity of the layer can vary within wide limits constrained by the condition S 0 S, but these variations scarcely affect the admittance measured on the Earth’s surface. The 1-D inverse problem is unstable. Evidently, we have every reason to extend this conclusion to the 2-D and 3-D inverse problems. Compare, for example, a 2-D or 3-D model with a slowly varying boundary between two deep layers and a model with this boundary rapidly ﬂuctuating around its slow variation. Their MT and MV response functions observed on the Earth’s surface will virtually coincide, although these models are largely different. Inverse problems of magnetotellurics are unstable. An arbitrarily small error in initial MT and MV data can lead to an arbitrarily large error in the inversion of these data, i.e., in the conductivity distribution. Using the terminology of

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Hadamard, we state that the inverse problems of MTs are ill-posed. An immediate solution of an ill-posed (unstable) problem is generally meaningless, because it can yield results far from reality.

7.3. MT AND MV INVERSIONS IN THE LIGHT OF TIKHONOV’S THEORY OF ILL-POSED PROBLEMS The cornerstone of the MT and MV data interpretation is the theory of ill-posed problems. Its basic principles were formulated by A.N. Tikhonov (1963). Presently, methods of this theory have been developed rather comprehensively and are widely adopted in practice (Tikhonov and Arsenin, 1977; Lavrentyev et al., 1980; Glasko, 1984; Tikhonov and Goncharsky, 1987; Zhdanov, 2002). The Russian mathematical school headed by Tikhonov gave rise to a new doctrine of physical experiment encompassing various ﬁelds of science and technology. Following (Berdichevsky and Dmitriev, 1991, 2002; Zhdanov, 2002), we consider inverse problems of in light of Tikhonov’s theory of regularization, which provides a basis for developing the strategy of MT and MV inversions. 7.3.1. Conditionally well-posed formulation of inverse problem The interpretation of an unstable MT or MV inverse problem makes sense if a priori geological–geophysical information about the region under consideration is used and certain assumptions on its geoelectric structure are imposed. This is a way for transforming an unstable problem into a stable one. In the absence of a priori information restricting the scope of search, we can obtain only one of equivalent models or, at best, a model with a signiﬁcantly smoothed distribution of conductivity leveling out contrasts of sought-for structures. Thus, the transformation of an unstable problem into a stable one is attained by restricting the scope of search. Before searching, we have to decide where are we searching and what are we searching for (Gol’tsman, 1971). Considering a set Sd of equivalent solutions, we choose a compact subset SC d containing the exact model solution and comprising of solutions that are sufﬁciently close to the exact model solution (recall that a functional set is compact if any sequence of functions in this set contains a subsequence converging to a function also belonging to this set – the necessary condition of compactness of a set is its boundedness). The theory of regularization is based on the Tikhonov theorem on the stability of an inverse problem deﬁned on compact subset (Tikhonov and Arsenin, 1977; Berdichevsky and Dmitriev, 2002). This theorem is formulated as follows: if the error d in the initial information tends to zero, the solution of the inverse problem on a compact subset SC d converges to the exact model solution. An ill-posed inverse problem that has a unique solution and is stable on the compact subset SC d is called conditionally well-posed problem (or well-posed after Tikhonov), and the subset SC d is called a correctness set. Thus, the inverse problem ill-posed after Hadamard becomes wellposed after Tikhonov.

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The compact subset SC d (the correctness set) is chosen by the generalization of a priori geological–geophysical information (experimental evidences, reasonable hypotheses, general ideas as to the origin and conﬁguration of geoelectric structures). In essence, this means that a new geoelectric model is constructed on the basis of previous geological and geophysical models. The solution of an MT or MV inverse problem is efﬁcient if magnetotellurics provides a new information as compared to what was known before MT and MV observations. Naturally, the solution of an inverse problem should be preceded by analysis of a priori information (in conjunction with the visualization of MT and MV characteristics, which help to identify and localize geoelectric structures). In constructing the correctness set (the compact subset SC d ), i.e., in imposing restraints on the geoelectric structure of the medium, one should keep in mind that the condition d ! 0 is unrealizable in practice, because any initial information obtained by processing the ﬁeld measurements is never free from errors. Therefore, we should speak of a practical stability of a conditionally well-posed problem. The problem is regarded as practically stable if the width of the correctness set is such that, with real errors d, it consists of solutions that are sufﬁciently close to the exact model solution. The correctness set, within the solution to the inverse problem is sought, forms an interpretation model. The latter should incorporate modern ideas (hypotheses) on the stratiﬁcation of the medium and on the (homogeneous, inhomogeneous) disturbing this stratiﬁcation. The interpretation models of magnetotellurics are divided into two classes: (1) layered models, (2) locally inhomogeneous models. Layered model consist of a ﬁnite number of inﬁnite or wedging-out layers. This model class includes horizontally layered models, composed of homogeneous layers with horizontal boundaries, and quasi-horizontally layered models, in which the electrical conductivities of layers and their boundaries slowly vary in horizontal directions (easy dip, slight folding). A very important feature of the quasihorizontally layered models is the presence (or absence) of high-resistivity layers playing the role of galvanic screens. This property, characterized by the galvanic parameters of the model, determines the extent of near-surface galvanic anomalies and the sensitivity to deep conductive structures. Locally inhomogeneous models consist of a ﬁnite number of layers with breaks and sharp variations of their conductivity and boundaries. The models can include discontinuities, displacements, subductions, intersections, bounded bodies and channels of more or less complicated geometry. The construction of an interpretation model is based on a prior geological and geophysical information as well as on constraints obtained directly from the qualitative analysis of the observation data and hypothesis testing that narrow the choice of the correctness set and allow one to suggest an interpretation model described by a small number of parameters. The interpretation model should meet the following two requirements: (1) it should be informative (reﬂecting main properties of the geoelectric medium and containing target layers and structures), and (2) it should be simple (being

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determined by a small number of free parameters that ensure the practical stability of the inverse problem). It is evident that these conditions are opposite: the more informative the model, the more complicated it is. Thus, we have to choose an optimal model, that is, it should be reasonably informative yet sufﬁciently simple. This is a crucial point of the interpretation, predetermining not only the strategy for inversion, but also, to some degree, its result. At this stage of interpretation, the factors such as intuition of a researcher, his professional skill and academic experience, his understanding of the actual geological situation and goals of the MT survey, his adherence to traditions and willingness to deviate from these traditions all play a role. Constructing an interpretation model, a researcher is limited by a priori information, results of qualitative analysis of ﬁeld measurements, and hypothesis testing. Just in this sense we say that the interpretation of MT and MV data is effective under the condition of sufﬁciently complete a priori information constraining the search. The statement ‘‘the better we know the medium under consideration, the better we can determine its geoelectric structure’’ seems paradoxical. But it actually means that, solving the inverse problem, we improve and widen our knowledge of the structure of the medium, and therefore, the better was this structure known, the more meaningful and detailed are the new results. The amount of a priori information required for constructing an optimal interpretation model depends on the complexity of the medium studied and on the goals of the interpretation. Whereas rather detailed a priori information about the tectonics and geodynamics are required in rift or subduction zones, only very general ideas on the stratiﬁcation of the medium are sufﬁcient for stable platforms with gentle folding. Moreover, we can reject a priori information at the preliminary stage of interpretation and perform the smoothed Occam inversion.1 This simple transformation provides a gross geoelectric regionalization helpful for the identiﬁcation of zones of interest for further interpretation. Tikhonov’s theory of ill-posed problems offers two basic approaches to the interpretation of MT and MV data: (1) optimization method, and (2) regularization method (Berdichevsky and Dmitriev, 1991, 2002). We brieﬂy describe below these approaches.

7.3.2. Optimization method This approach is effective in studying simple media, described by a small number of parameters. Return to inverse problem (7.8)–(7.9) and assume that available prior information constrains a sufﬁciently narrow compact set M of admissible ~ solutions including the exact model solution. Let values of the impedance tensor Z^ ~ be known from observations. Then we can determine the and the tipper W approximate solutions s~ Z ðx; y; zÞ and s~ W ðx; y; zÞ of problem (7.8)–(7.9) by minimizing the misﬁt functionals

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~ ^ y; z ¼ 0; o; s~ Z ðx; y; zÞg I Z fs~ Z g ¼ Z^ Zfx; ^ ^~ ¼ inf Z Zfx; y; z ¼ 0; o; sðx; y; zÞg s2M ~ Wfx; y; z ¼ 0; o; s~ W ðx; y; zÞg I W fs~ W g ¼ W ~ Wfx; y; z ¼ 0; o; sðx; y; zÞg ¼ inf W s2M

ð7:48Þ

The misﬁt minimization procedure is usually iterative. A starting model is constructed through the parametrization of the interpretation model. Solving the forward problem for the starting model, we calculate the misﬁts between model and experimental values of the impedance tensor and the tipper. A new model, decreasing the misﬁts, is then chosen. The iterations are performed until the misﬁts ~ ~ If the misﬁts cannot be approach the level of errors in the initial values of Z^ and W. decreased to the level of errors in the initial data, this implies that the compact set M is overly narrow. In this case, we test successively widening compactum (e.g., we increase the density of subdivision of the model). A compactum on which the equation misﬁt is equal to the error in initial data is regarded as an optimal compact set. However, an overly wide compactum makes the problem unstable and can yield a solution that differs strongly from the exact model solution. This limits the practicality of the optimization method. It is obvious that separate inversions of the impedance and the tipper make sense if solutions s~ Z ðx; y; zÞ and s~ W ðx; y; zÞ are close to each other. Otherwise, MT and MV inversions call for correlation. We can, for instance, carry out the MT and MV inversions in parallel, minimizing the functional of total misﬁt 2 ~^ ^ y; z ¼ 0; o; sðx; y; zÞg Ifsðx; y; zÞg ¼ gZ Z Zfx; ~ Wfx; y; z ¼ 0; o; sðx; y; zÞg2 þ gW W ð7:49Þ and controlling the contributions of MT and MV inversions by means of weights, gZ and gW . Alternatively, we can accomplish successive partial inversions, minimizing the functionals of MT and MV misﬁts 2 ~ ^ y; z ¼ 0; o; sðx; y; zÞg I Z fsðx; y; zÞg ¼ Z^ Zfx; ð7:50Þ ~ Wfx; y; z ¼ 0; o; sðx; y; zÞg2 I W fsðx; y; zÞg ¼ W Adopting this strategy, we start with MV inversions, which is free from distorting effects of local subsurface inhomogeneities, and then proceed to MT inversion with a starting model, constructed from the results of MV inversion. 7.3.3. Regularization method Regularization of solutions substantially widens the possibilities of interpretation. Given a sufﬁcient amount of a priori information, this approach provides maximum geoelectric information consistent with the accuracy of ﬁeld observations

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and modeling. The main peculiarity of the regularization method is that the criterion for choosing an approximate solution is introduced directly into the algorithm of inversion. When solving the inverse problem, the compactum SC narrows around the exact model solution. The regularization method admits the introduction of any type of a priori information with a control of its inﬂuence on the solution of an inverse problem. What is more, the regularization method enables one to focus the inversion on target layers and structures. This approach is based on the regularization principle: the criterion for the selection of solution should be such that the inferred approximate solution tends to the exact model solution of the inverse problem, when the errors in the initial information tend to zero. The regularization principle for impedance and tipper inversions takes the form lim s~ Z ðx; y; zÞ ¼ s Z ðx; y; zÞ dZ !0 W

lim s~ ðx; y; zÞ ¼ s W ðx; y; zÞ

ð7:51Þ

dW !0

where s~ Z ; s~ W and s Z ; s W are approximate and exact model solutions of MT and MV problems, and dZ ; dW are errors in the initial data. The regularization principle is implemented with the help of a regularizing operator. The regularizing operator R of an inverse problem is referred to as a set of analytical and numerical operations that allows one to obtain an approximate solution satisfying the regularization principle. In inverse problems of geophysics, it is advantageous to use a regularizing operator R depending on a numerical parameter a>0, which is called the regularization parameter. As the error d in the initial data tends to zero, the regularization parameter a should also tend to zero lim a ! 0

ðimpedance inversionÞ

lim a ! 0

ðtipper inversionÞ

dZ !0

dW !0

ð7:52Þ

and the regularizing operator, when applied to the approximate response function, should yield the exact model solution ~ lim Ra Z^ ¼ s Z ðx:y; zÞ

ðimpedance inversionÞ

dZ !0

~ ¼ s W ðx; y; zÞ lim Ra W

ð7:53Þ ðtipper inversionÞ

dW !0

The MT and MV inversions reduce to constructing the regularizing operator Ra and determining the regularization parameter a consistent with the accuracy of observations. The approximation solution obtained in this way is stable with respect to errors in the initial data. It is called a regularized solution. Variational methods of constructing the regularizing operator are most widespread in geophysics. A stabilizing functional (a stabilizer), providing a criterion for the selection of admissible solutions, plays a key role in this approach. The stabilizer is usually written in the form

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Ofsðx; y; zÞg C

ð7:54Þ

where C is a positive constant. The functional Ofsðx; y; zÞg determines a compact set of functions sðx; y; zÞ 2 SC . The smaller the value of C, the narrower the set SC . Introducing (7.54), inverse problem (7.9) is formulated as a variational problem for a conditional extremum inf Ofsðx; y; zÞg ~^ ^ y; z ¼ 0; o; sðx; y; zÞg Z Zfx; dZ

ð7:55aÞ

W ~ Wfx; y; z ¼ 0; o; sðx; y; zÞg dW

ð7:55bÞ

So, we ﬁnd a minimum compactum SC consisting of functions sðx; y; zÞ that satisfy conditions (7.55a) and (7.55b). The set S of approximate solutions of such an inverse problem is the intersection of the compactum SC with the sets SdZ and SdW of equivalent solutions of MT and MV inverse problems S ¼ SC \ SdZ \ SdW

ð7:56Þ

It is convenient to replace the conditional-extremum problem by the unconditional-extremum problem inf Fa fsðx; y; zÞg

ð7:57Þ

where Fa is Tikhonov’s regularizing functional Fa fsðx; y; zÞg ¼ Ifsðx; y; zÞg þ aOfsðx; y; zÞg

ð7:58Þ

consisting of the misﬁt functional IðsÞ and the stabilizing functional OðsÞ. The solution of this inverse problem reduces to the minimization of Fa ðsÞ, that is, to the minimization of IðsÞ and OðsÞ. Whereas the initial problem (7.8)–(7.9) is unstable, the solution obtained by minimizing the functional Fa is stable with respect to small ~ ~ The point is that the functional OðsÞ narrows the class of variations in Z^ and W. possible solutions and stabilizes the problem. It is given the name the stabilizer. The structure of the misﬁt functional IðsÞ depends on the inversion strategy. It can be a functional (7.49) of total misﬁt, when carrying out the MT and MV inversions in parallel, or functionals (7.50) of partial misﬁts, when accomplishing successive inversions. The structure of the stabilizing functional OðsÞ depends on a priori restraints imposed on the inverse problem. It can be, for example, the requirement of Occam’s smoothness of sðx; y; zÞ satisﬁed in minimizing the functional Z ( 2 2 2 ) @s @s @s dx dy dz ð7:59Þ þ þ OðsÞ ¼ @x @y @z V

or the requirement of closeness of sðx; y; zÞ to a hypothetical model s0 ðx; y; zÞ satisﬁed in minimizing the functional

V.I. Dmitriev and M.N. Berdichevsky/Inverse Problems in Modern Magnetotellurics

Z OðsÞ ¼

fsðx; y; zÞ s0 ðx; y; zÞg2 dx dy dz

181 ð7:60Þ

V

The weight of the stabilizing functional, i.e., the degree of its effect on the solution of an inverse problem, is controlled by the regularization parameter a (Fig. 7.4). At large a, the minimization of Fa ðsÞ leads to the dominating minimization of OðsÞ, i.e., oversmoothes the solution or retains it near the a prior hypothetical model, ignoring results of observations. At small a, the minimization of Fa ðsÞ leads to the dominating minimization of IðsÞ, i.e., the stabilizing effect of OðsÞ is suppressed and an unstable incorrect solution is obtained. An optimum value of a providing a sufﬁciently small misﬁt and ensuring sufﬁciently strong stabilization of the solution is to be found. The regularization parameter a should be consistent with the error d in the initial information. The optimum value of a can be chosen by testing a monotonically decreasing sequence a1 4a2 4 4an . For each a, variational problem (7.57) is solved and the iterative sequence of solutions characterized by their misﬁt is determined. The parameter a ¼ aopt ; at which the misﬁt attains the error d in the initial information, is regarded as an optimum parameter. The optimum parameter of regularization provides a conductivity distribution ﬁtting best the exact model solution. This simple technique is applicable if the error d is well known. However, we commonly have a more or less gross estimate

Fig. 7.4. Dependence of the solution of an inverse problem on the regularization parameter, a; s is the exact model solution.

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dmin d dmax

ð7:61Þ

In this case, solutions consistent with various values of d from interval (7.61) are tested. Close solutions selected from the resulting set are averaged, providing an approximation to the exact model solution. If we know next to nothing of measurement and model errors, the parameter aopt cannot be chosen reasoning from the knowledge of solution misﬁts. In this case, a quasi-optimal value of the regularization parameter is determined. For example, aopt can be determined as a value a at which the solution of the problem signiﬁcantly deviates from requirements of the stabilizer (smoothness or closeness to the hypothetical model) but yet remains sufﬁciently stable. This heuristic method for the determination of aopt was proposed by Hansen (1998). It is based on the socalled L-representation. A monotonically decreasing sequence of regularization parameters a1 4a2 4 4an is tested and the misﬁt I a and the stabilizer Oa are determined for various a and a ﬁxed minimum of Tikhonov’s functional Fa . Fig. 5 presents, on a log–log scale, the Oa versus I a plot. This curve has typically the Lshaped form, with a fairly distinct bend separating a nearly horizontal branch with large I a and small Oa from a nearly vertical branch with small I a and large Oa . The exact model solution is best approximated by assuming that the central point of the bend, characterized by the largest curvature, deﬁnes the quasi-optimal parameter of regularization aopt. Acknowledgments We are grateful to P. Weidelt and U. Schmucker for discussions stimulating this work. This work was supported by the Russian Foundation for Basic Research,

Fig. 7.5. The L-representation.

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grants 02-05-64079, 05-01-00244 and 05-05-65082.

NOTES 1. William Occam (1285–1349) formulated a maxim known as Occam’s razor (assumptions introduced to explain a thing must not be multiplied beyond necessity).

REFERENCES Berdichevsky, M.N. and Dmitriev, V.I., 1991. Magnetotelluric Sounding of Horizontally Homogeneous Media (in Russian). Nedra, Moscow, 250pp. Berdichevsky, M.N. and Dmitriev, V.I., 2002. Magnetotellurics in the context of the theory of ill-Posed Problems: Investigations in Geophysics, 11, Society of Exploration Geophysicists, Tulsa, 215pp. Berdichevsky, M.N., Dmitriev, V.I., Novikov, D.B. and Pastutsan, V.V., 1997. Analysis and Interpretation of Magnetotelluric Data (in Russian). DialogMGY, Moscow, 161pp. Berdichevsky, M.N. and Zhdanov, M.S., 1984. Advanced Theory of Deep Geomagnetic Sounding. Elsevier, Amsterdam, 408pp. Dmitriev, V.I., 1969. Electromagnetic ﬁelds in inhomogeneous media. Proceedings of the Computational Center of Moscow University, Moscow, 166pp. Glasko, V.B., 1984. Inverse Problem in Mathematical Physics (in Russian). Nauka, Moscow, 111pp. Gol’tsman, F.M., 1971. Statistical Models of Interpretation (in Russian). Nauka, Moscow, 327pp. Gusarov, A.L., 1981. On uniqueness of solution of inverse magnetotelluric problem for two-dimensional medium (in Russian): Mathematical Models in Geophysics. Moscow University, pp. 31–61. Hansen, C., 1998. Rank-deﬁcit and discrete ill-posed problems, Numerical aspects of linear inversion. Department of Mathematical Modeling, Technical University of Denmark, Lyngby, 247pp. Lavrentyev, M.M., Romanov, V.G. and Shishatsky, S.P., 1980. Inverse Problem in Mathematical Physics and Analysis (in Russian). Nauka, Moscow, 286pp. Parkinson, W.D., 1983. Introduction to Geomagnetism. Scottish Academic Press, Edinburgh, 522pp. Tikhonov, A.N., 1963. On solution of ill-posed problem and the regularization method. Docl. Acad. Nauk. SSSR, 154, 3: 501–504. Tikhonov, A.N., 1965. Mathematical basis for electromagnetic sounding. Comput. Math. Math. Phys., 3: 207–211. Tikhonov, A.N. and Arsenin, V.Ja., 1977. Solutions of Ill-Posed Problems. W.H. Winston and Sons, , 258pp. Tikhonov, A.N. and Goncharsky, A.V. (Eds), (1987). Ill-Posed Problems in Natural Sciences. Mir Publishing House, Moscow, 343pp.

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Vanyan, L.L., 1965. Principles of Electromagnetic Soundings. Nedra, Moscow, 218pp. Weidelt, P., 1978. Entwicklung und Erprobung eines Verfahren zur Inversion zweidimensionalen Leitfaehigkeits in E-poarisation, Dissertation, Goettingen Universitaet, 161pp. Zhdanov, M.S., 2002. Geophysical Inverse Theory and Regularization Problems. Elsevier, Amsterdam, 609pp.

Chapter 8 Joint Robust Inversion of Magnetotelluric and Magnetovariational Data Iv.M. Varentsov Geoelectromagnetic Research Centre IPE RAS, Troitsk, Moscow Region, Russia

In this chapter, a general scheme for nonlinear inversion of natural electromagnetic (magnetotelluric, MT, and magnetovariational, MV) data in the important class of piecewise-continuous models of a medium is presented. Several alternatives of model parameterization are considered, the problem of Newtonian minimization is formulated in a joint space of inverted data and estimated model parameters, a robust functional metric is introduced for suppressing outliers, and other adaptive techniques stabilizing the inversion are discussed. Primarily, this approach has been implemented in the framework of a simple two-dimensional (2-D) piecewise-constant model parameterization (Golubev and Varentsov, 1993; Varentsov et al., 1996). At this stage, it yielded the ﬁrst experience in the robust electromagnetic (EM) inverse problem solution and demonstrated a reasonable trade-off between stability and resolution factors. The related 2-D inversion software has been extensively veriﬁed (Varentsov, 1998; Novozhinski and Pushkarev, 2001) and used in the practice of MT/MV methods (Varentsov et al., 1996; Vanyan et al., 2002; Berdichevsky et al., 2003, 2006). Later, this approach has been realized in the one-dimensional (1-D) case and extended to the class of two- and three-dimensional (3-D) models with a piecewise-continuous distribution of electrical conductivity (Varentsov, 2002, 2005). Adaptive parameterization schemes enabled an adequate description of both continuous smooth and discontinuous or sharply varying conductivity distributions. Robust misﬁt estimators resolvedconﬂicts in the inversion of multi-component datasets. Effective

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40008-1

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forward ﬁnite-difference algorithms extended data modeling capabilities (Zhdanov et al., 1997; Varentsov, 1999; Varentsov et al., 2002). The chapter is started with a formulation of principal methodical developments and then the effectiveness of speciﬁc inversion algorithms is illustrated within a number of synthetic datasets calculated for known geoelectric models. The class of 2-D problems is examined in detail, and ﬁrst 3-D solutions are outlined. A number of pragmatic issues, such as inversion of quasi-2-D data within 2-D inversion procedures, and diagnostics/correction of systematic data distortions (static shifts) in the course of inversion are specially considered.

8.1. ADAPTIVE PARAMETERIZATION OF A GEOELECTRIC MODEL 8.1.1. A background structure and windows to scan anomalies A geoelectric model with an isotropic distribution of the electrical conductivity is approximated by a background (normal) structure consisting of blocks of a ﬁxed geometry and ﬁnite-size scan windows with an arbitrary distribution of anomalous electrical conductivity. The normal geoelectric structure involves generalized a priori ideas of the model in its peripheral parts with a lack of data, and is usually expressed in the form of left-hand, right-hand, and bottom normal horizontally layered sections with ﬁxed thicknesses of layers and with resistivities constant within a layer. In the central part of the model, the normal structure can include ﬁnite blocks with ﬁxed geometry and piecewise-constant resistivity. Resistivities of layers, half-layers, and blocks of the normal structure are either ﬁxed on the basis of a priori knowledge or may be partly adjusted during the inversion process. The anomalous geoelectric structure within each window can be described by a set of discrete inversion cell parameters or by a sum of continuous parametric ﬁnite functions (Varentsov, 2002). The set of inversion cells deﬁnes the discrete subdivision of the scan window. Each cell can have a complex individual conﬁguration consisting of primary elements of the forward modeling mesh used within the inversion technique, while its resistivity as being an estimation parameter, is constant. The parametric ﬁnite functions express anomalous conductivity within the whole window and are zero outside it. The composite vector p 2 RNp of estimated model parameters consists of unknown resistivities of selected normal structure elements and all inversion cells together with unknown parameters of ﬁnite functions. Resistivity in real geoelectric media satisﬁes reasonably well the lognormal distribution, thus the use of log-resistivities is preferable in constructing the parameter vector, providing quasi-normal distribution of its components variations. 8.1.2. A priori model structure and constrains A priori information in the space of model parameters includes an initial model p0, its uncertainty intervals r p0 , correlation properties R^ p , and vectors of ‘‘common sense’’ limitations p, p+. A priori covariance matrix of parameters assumes the

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187

T

^ p ¼ rp R^ p rp , where T stands for transposition. The correlation matrix R^ p is form W 0 0 a unit matrix in conventional window with independent resistivity parameters (IR window) or in a window with ﬁnite functions (FF window). However, some other parameterization schemes may provide non-zero off-diagonal elements of correlation matrix R^ p (see Section 8.1.3). Thus, in general, this matrix includes both diagonal and ﬁlled blocks. In terms of the statistical inversion approach (Tarantola and Valette, 1982), a ^ p speciﬁes an optimal initial linear scaling of the vector priori covariance matrix W of parameters. Within the deterministic approach, this matrix still plays an important role in a priori weighting in the space of parameters. In the approach presented, the optimization of model parameters is held unconstrained; however, limitation vectors p and p+ are taken into account in the determination of the scalar step of Newtonian minimization to reject obviously implausible models (see Section 8.3.5). 8.1.3. Window with correlated resistivities of inversion cells This type of a scan window, that is, a window with correlated resistivity parameters (CR window) is introduced with the aim to constrain resistivity variations in close cells within the inversion process by specifying a ﬁlled a priori correlation matrix R^ p . For two CR cells in a certain window (with indices i, j and respective radius vectors of their geometric centres ri, rj), a correlation matrix element is determined in inverse proportion to the distance between the cell centres: R^ p ij ¼ f R ðdij =d R Þ, where dij ¼ ri rj and dR incorporates additional a priori constrains on the scale (‘‘diameter’’) of the structure examined in a given window, and the monotonically decreasing function fR can be chosen, for example, in the form expðxÞ (Tarantola and Nercessian, 1984; Varentsov, 2002). The introduced dR parameter can be constructed from two independent scale estimates, horizontal qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ and vertical: d R ¼ d 2X þ d 2Z . Some composite (average) dR value is to be used to set up correlation relations for two CR cells representing different windows. The use of the CR parameterization scheme provides a consistent variation of parameters in closely spaced cells, whereas their variation for distant cells remains virtually independent. However, this scheme does not impose any explicit limitations on the smoothness of spatial resistivity variation within the scan window after a number of inversion iterations, thereby providing prerequisites for resolution of both discontinuous and smooth structures. 8.1.4. Window with finite functions A one-dimensional ﬁnite function is deﬁned (Strakhov, 1978) as a continuous non-negative single-extremum function of a ﬁnite support [a,b]: F ðx; a; b; A; a; b; p; qÞ ¼ F ðA; a; b; p; qÞ ¼ A Cft½tðx; a; bÞ; p; q; a; bg Cðt; a; bÞ ¼ ð1 tÞa ð1 þ tÞb

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1 ð1 þ k ÞðexpðiptÞ kÞ t½t; p; q ¼ arg p ð1 þ kÞð1 k expðiptÞÞ 1 t 1; k ¼ p þ iq; k ¼ p iq; jkjo1 t ¼ tðx; a; bÞ ¼ ½2x ða þ bÞ=ðb aÞ With a ﬁxed support [a, b], such a function depends on the following ﬁve parameters: the amplitude A, form factors p and q (accounting for the excess and asymmetry), and edge factors a and b (accounting for the behaviour at the ends of the support). Fig. 8.1 exempliﬁes such a function of unit amplitude and [0, 1] support depending on the form factors p and q. This ﬁgure clearly demonstrates that in its limits, the ﬁnite function approaches a rectangle (at the ends of the support) or a delta-function. Based on these limit properties, ﬁnite functions can be used not only for the approximation of continuous conductivity variations, but also for localizing discontinuous structures of various scales. A multi-dimensional ﬁnite function (2- or 3-D) in a given window is constructed as a Cartesian product of corresponding 1-D functions; as a result, 2- and 3-D ﬁnite functions depend on 5 and 13 parameters, respectively. It is most appropriate to use ﬁnite functions to parameterize distributions of anomalous (excessive) conductivity and to further determine spatial resistivity distribution from them. In this case, it is natural to redeﬁne the amplitude parameter of the ﬁnite function as its integral over the entire support, yielding the integral conductivity of the anomaly studied (Zhdanov and Golubev, 1983; Varentsov, 2002). The CR and FF approaches to the parameterization of the geoelectric structure of scan windows obviously contain adaptive resources, providing the concentration of parameters in areas of interest and the capability to resolve both continuous and discontinuous structures. The instrument of ﬁnite functions looks capturing due to its compactness, but it introduces an additional non-linearity into the formulation of the inversion problem.

8.2. INVERTED AND MODELING DATA The data to be inverted are integrated into a real vector d0 2 RNd . This data vector may combine values of different EM ﬁeld transfer operators (Dmitriev and Berdichevsky, 2006; Varentsov, 2006): namely, the impedance Z^ or apparent re^ etc. The magsistivity r^ a , geomagnetic tipper Wz , horizontal magnetic tensor M, netotelluric phase tensor (Caldwell et al., 2004; see Section 11.2.1, Chapter 11, for more detailes) 1 P^ ¼ ReZ^ ImZ^ being free from period-independent galvanic distortions looks a reasonable substitute for the impedance phase data. These transfer operators are converted, through an appropriate rotation, to the inversion coordinate system. Their complex data may be represented in various ways (as Re, Im, Mod, log(Mod) or Arg) on

Iv.M. Varentsov/Joint Robust Inversion of Magnetotelluric and Magnetovariational Data

Fig. 8.1. Variety of forms of a 1-D ﬁnite function Fðx; 0; 1; p; q; . . .Þ depending on the excess value p (left panel) and the asymmetry q (right panel), the horizontal x-axis indicates function support.

189

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discrete sets of observation sites and periods. The period grid is normally selected as log-equidistant. An inevitable attribute of the data is the uncertainties in their determination in ^ d : fW d gii ¼ fsd g2 , and error the form of the a priori diagonal covariance matrix W 0 i ﬂoors (lower error limits) are conventionally applied to error estimates for speciﬁc components. The information about the data errors usually ensues from the stage of transfer function estimation (Egbert, 2002; Varentsov et al., 2003) and may not account for a number of factors, such as the adequacy of the inversion model to the studied geophysical medium, or the data-error contribution arising from the modeling and inversion calculations. Error ﬂoors approximate the level of unknown data-error factors. The importance of getting reliable data-error estimates and setting reasonable error ﬂoors increases in the inversion of multi-component datasets. The vector of modelled data, having a similar structure, is to be determined by a ^ generally nonlinear modeling operator d ¼ MðpÞ for all model parameters lying within the limits p, p+. The derivatives of the modeling operator f@d i =@pj g, known as model sensitivities are also required. Techniques of calculating the modeling operator and model sensitivities are essentially dependent on the inversion dimension and are discussed in Varentsov (1999, 2002), Varentsov et al. (2002), and Zhdanov et al. (1997). The ﬁnite difference approach is used in cases of both 2- and 3-D modeling, while 1-D model calculations associated with the effects of normal ﬁelds and boundary conditions are based on recurrent analytical representations (Pek, 1985). The success of joint inversion of various EM data components depends on how adequately the vectors d0, rd0 were speciﬁed. In a favourable case, the use of new data components that are more or less independent of the existing dataset increases the signiﬁcance of the inverted ensemble. The robust minimization procedure (see Sections 8.3.2 and 8.3.4) protects the solution from the inﬂuence of strong but local data misﬁt elements induced by contradictions in ﬁtting different data components. Most unfavourable is a systematic data bias (e.g. static distortions in apparent resistivities, see Section 8.3.6). The signiﬁcance of various data components can be adjusted (diminished or emphasized) in the joint inversion solution by increasing or decreasing the respective rd0 estimates by a common factor (Varentsov et al., 1996). In the case of 2-D inversion of quasi-2-D (3-D-distorted) data, it can also be advisable to extend rd0 estimates adaptively at speciﬁc sites and periods in a direct proportion to 3-D-factors (skews) of the correspondent transfer operator, thus penalizing the most distorted data elements.

8.3. INVERSION AS A MINIMIZATION PROBLEM 8.3.1. Minimizing functional A statistical trade-off between the requirements of obtaining an inversion solution close to a priori model, on the one hand, and a model response best ﬁtting the inverted data, on the other hand, can be achieved by minimization of an overall misﬁt in the joint space of inverted data and ﬁtted model parameters (Tarantola and

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Valette, 1982). This approach naturally takes a form of a minimization problem of Tikhonov’s regularizing functional (Tikhonov and Arsenin, 1977; Zhdanov and Varentsov, 1983): T½p ¼ T l ½p ¼ jjddjj2 þ ljjdpjj2 ¼ ddT dd þ l dpT dp ¼ I½p þ Ol ½p,

ð8:1Þ

where l is a scalar regularization parameter, L2 norms are initially considered, ^ 1=2 ½MðpÞ ^ ^ 1=2 ðp p0 Þ are weighted misﬁts of data and padd ¼ W d0 , dp ¼ W d p ^ d is diagonal, while the parameter weighting rameters, the data weighting matrix W ^ matrix W p is non-diagonal only in the presence of CR windows; I½p is a data misﬁt functional, while Ol is a stabilizer. The deterministic approach in the minimization of functional (8.1) looks more pragmatic, because it removes the hardly feasible requirement of checking a priori assumptions on the type of statistical distributions of variations in parameters and data spaces. At the same time, this approach can follow several evident statistical ‘‘recipes’’ and apply, for example, the non-linear conversion of the data and parameter vector elements into the form with quasi-normal variation at a comparable scale. Such transformations consisting, in particular, of the transition to the inversion of logarithmic apparent resistivity data for logarithmic model resistivities are favourable for improving the scaling of the minimization problem and increasing the stability of its solution. pﬃﬃﬃﬃﬃﬃﬃ ﬃ The weighted data misﬁt norm jjddjj ¼ I½p gives in Eq. (8.1) a relative mean square misﬁt estimate (RMS), dependent both on the misﬁt and error structure of the data. The RMS estimate is an effective tool to monitor the data ﬁt within a certain inversion problem solution, especially in the case of the joint inversion of the multi-component data ensemble. However, there is still a need to consider partial absolute misﬁt norms for different data components in the comparison of a number of inversion solutions, because the RMS measure is dependent on the ‘‘re-weighting’’ changes in the data errors and error ﬂoors discussed in the previous section. 8.3.2. Robust misfit metric The conventional RMS functionals in Eq. (8.1) written in L2 norm would be too sensitive to abnormally large dd, dp components caused by gross errors (outliers) in the inverted data and a priori information about model parameters. A drastic tool for suppressing such errors is a robust modiﬁcation of misﬁt norms intended to signiﬁcantly reduce the contribution of large outliers (Huber, 1981). The M-transformation of Huber consists in the transition from the L2 metric to a combined robust metric but switched for L1 for relatively large still equal to L2 for relatively small misﬁts, P ones. The functional F ðxÞ ¼ jjxjj2 ¼ i x2i is transformed then to the robust form 2 X jxi jpt xi ^ F R ðxÞ ¼ jjFðxÞ ð8:2aÞ x2 jj ¼ Rðxi Þ; Rðxi Þ ¼ 2 2 tjx j t jxi j4t i i ^ with the following diagonal matrix FðxÞ ¼ ji ¼ jii : 1; wi 1; ji ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

wi ¼ jxi j=t ð8:2bÞ 2wi 1 wi ; wi 41;

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Here the scaling parameter t expresses the scatter in fxi g and is robustly calculated using the median estimator of the mean mx and the scatter sx : t ¼ y sx ;

sx ¼ med f jx mx jg;

mx ¼ med fxg;

y ¼ const: 1

ð8:2cÞ

This robust approach has been primarily applied in MT studies for the stable minimization of quadratic functionals in the estimation of EM ﬁeld transfer operators (see reviews in Egbert (2002) and Varentsov et al. (2003)), and later extended to a case of the non-linear functional (8.1) in Golubev and Varentsov (1993) and Varentsov (2002). The suggested extension reduces to non-linear robust transform (8.2b) of the weighted misﬁt vectors dd, dp within the initial L2 formulation (8.1): ^ d dd; dpR ¼ F ^ p dp ddR ¼ F

ð8:2dÞ

The introduced robust metric (8.2) is most signiﬁcant in the data space, but is also relevant in the space of parameters, because it excludes gross errors in entering of a priori model information. This metric induces, according to Eq. (8.2d), a robust RMS estimate as a preferable data ﬁt measure. 8.3.3. Cycles of Tikhonov’s minimization The constructed nonlinear Tihkonov functional (8.1) is minimized in a two-cycle iterative procedure (Zhdanov and Varentsov, 1983; Varentsov, 2002; Zhdanov, 2002). The inner iterative cycle minimizes the functional at a ﬁxed value of the regularization parameter l, while the external cycle changes this parameter. Generally, l is successively decreased in a geometric progression, ensuring the greater stability at the beginning of inversion and providing higher model resolution at its ﬁnal stage. The internal cycle terminates on the solution p or the functional Tl stabilization, or on the excessive increase of the stabilizer Ol : I½p=Ol ½po. The outer cycle stops when stabilization criteria (substantially more rigorous compared with the inner cycle) are met or the minimizing functional reaches a certain level. A unity value of the regularization parameter implies statistical equivalence of two misﬁt norms in functional (8.1); however, the starting value of the regularization parameter is chosen one to two orders of magnitude higher, thereby ensuring the initial dominance of the stabilizer Ol . In its classical form, Tikhonov’s regularization scheme with a single scalar regularization parameter is quite a rough stabilization tool seriously limiting the resolution of the inversion techniques. However, within the developed approach, this scheme plays a signiﬁcant role only at the initial stage of the minimization process, whereas its signiﬁcance reduces with the subsequent decrease in the regularization parameter and enhances the role of ﬁner stabilization algorithms described below, which explicitly incorporate the structure of minimization techniques. 8.3.4. Newtonian minimization techniques The unconstrained nonlinear minimization problem of Tikhonov’s functional (8.1) for a ﬁxed regularization parameter is solved with the help of Newtonian

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schemes (Gill et al., 1982) in terms of its local quadratic approximation. Newtonian iteration is written in the general form as pkþ1 ¼ pk þ tk sk ; k ¼ 0; 1; . . .

ð8:3Þ

where the Newtonian search direction sk results from the solution of the Newtonian linear system ð8:4Þ H^ k sk ¼ gk with a positively deﬁnite Hesse matrix H^ k . This matrix in the Gauss–Newton scheme for the L2 functional norm has the following approximation by the weighted d M transform G^ of the model sensitivity operator G^ ¼ fG M ¼ @d k;i =@p g: k

k

d d p p GN H^ k H^ k ¼ ðG^ k ÞT G^ k þ lðG^ k ÞT G^ k ;

k;ij

d ^ 1=2 G^ M ; G^ k ¼ W k d

k;j

p ^ 1=2 G^ k ¼ W p

ð8:5Þ

and the right-hand-side vector takes the form: d p ¼ ðG^ k ÞT ddk þ l ðG^ k ÞT dpk gk ¼ gGN k

ð8:6Þ

System (8.4) has a universal form for both the conventional overdetermined problem (N d 4N p ) and the underdetermined case (N d oN p ). The calculation of the model sensitivity operator is a heavy computational problem even in the case of the FF parameterization with a relatively small number of parameters. However, the approach involving the explicit construction of Newtonian system (8.4) and its explicit stabilized solution ensures, given a moderate number of parameters (a few hundred in 3-D or few thousand in 2-D problems), a more reliable convergence, reducing the number of iterations and the overall volume of computational work. For a sufﬁciently large number of parameters, the system solution with an application of implicit iterative schemes (Mackie and Madden, 1993) becomes unavoidable. The advantage of quasi-Newtonian schemes (Gill et al., 1982) is the use of a relatively to a sequence of matrices n QN o simple iterative updating procedure applied ^ ^ H approximating the Hessian sequence H k without calculating of the senk

M

sitivities G^ k (now required only for the construction of the initial sequence element). In some cases, such an approach saves considerable computer resources, but this advantage can be invalidated by gross errors at initial iterations of the approximation. Section 8.4.3 presents an example of an effective hybrid approach, in which within the Gauss–Newtonian scheme the matrix H^ k is estimated according to (8.5) once in few iterations, and between these re-calculations, is used either without any changes or with quasi-Newtonian updating. When introduced in (8.1), the robust metric based on (8.2) changes formulas of the Newtonian iterative process. These changes become evident when the Newtonian procedure, i.e. the construction of the approximating quadratic form and the determination of its minimum, is applied to the new functional T R ½pk allowing for the non-linearity of transform (8.2a). Within the framework of the Gauss–Newton scheme, formulas (8.4)–(8.6) do not experience any structural changes. The misﬁts

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are transformed according to (8.2d), and the weighted sensitivities G^ k are treated in a similar way: R

d ^ d G^ d ; G^ k ¼ C k

R

p ^ p G^ p G^ k ¼ C k ^ with elements of the diagonal matrix CðxÞ ¼ ci ¼ cii deﬁned as 1; wi 1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ci ¼ 1 2wi 1; wi 41

ð8:7aÞ

ð8:7bÞ

The weights ji ; ci 2 ½0; 1 are nonlinear functions of the scaled absolute value wi (8.2b) of a speciﬁc misﬁt element and tend toward zero with its increase. The scaling parameter t in (8.2c) becomes dependent on the number of the Newtonian iteration: t ¼ tk . The robust metric in functional (8.1) also depends within certain limits on these changes, and the monotonic decrease in the scaling parameter tkþ1 tk must be imposed in order to prevent the false iteration divergence. The constant y in (8.2c) is chosen within a range 1–3 under the condition that the number of weights ji ; ci that are below unity should not exceed 20–30% of the dimensions of the related spaces but should signiﬁcantly differ from 0. The robust transform effect virtually vanishes at y45. The robust modiﬁcation of Newtonian schemes improves the convergence of iterations, both protecting the iterative process against outliers of external origin and suppressing internal noises arising in various blocks of the inversion algorithm. This accounts for a higher convergence rate of the robust modiﬁcation of the Gauss–Newton method as compared with its standard scheme, as is noted even in the inversion of synthetic datasets containing neither outliers nor appreciable uncertainties. 8.3.5. Solution of linear Newtonian system and choice of scalar Newtonian step System (8.4) is linearly scaled at each iteration to decrease its condition number and is further solved using stabilized procedures, such as the SVD factorization or modiﬁed Cholesky factorization (Gill et al., 1982; Lawson and Hanson, 1984), ensuring the positive deﬁniteness of the matrix H^ k in terms of computer arithmetic and thereby the actual convergence of Newtonian iteration (8.3). Threshold parameters of these factorization schemes are speciﬁed in accordance with the data misﬁt functional I½pk , ensuring high stability at the initial iterations and increasing the model resolution in the course of convergence. This scheme stabilizing the solution of the Newtonian system adaptively acts on the most unstable parameters and can neutralize the effect of signiﬁcant linear dependence between parameters. It becomes a primary stabilization tool with a sufﬁcient decrease of Tikhonov’s regularization parameter in (8.1). The Newtonian scalar step tk in Eq. (8.3) is initially set equal to unity, corresponding to the minimum of the related Newtonian quadratic form. Then the resulting parameter vector pkþ1 is checked for its conformity to the ‘‘common sense’’ constrains p, p+. If the vector does not comply with these restrictions, the step is successively decreased in geometric progression. This approach combines

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advantages of the rapid convergence of an unconstrained minimization method and the helpful possibility of using a priori information in the form of linear constrains. A forward problem is then solved for pkþ1 , and the new value of the minimizing functional T kþ1 is determined. If convergence is not attained, a smaller step is found from the 1-D minimization of the functional TðtÞ ¼ Tðpk þ tsk Þ held by the parabolic approximation method (Zhdanov and Varentsov, 1983; Varentsov, 2002). The approximating parabola is effectively constructed using the value of TðtÞ obtained for the current scalar step and the quantities Tð0Þ ¼ Tðpk Þ, T 0 ð0Þ ¼ T 0 ðpk Þ known prior to the application of this procedure. The parabola minimum (meeting the evident condition T 0 ð0Þ40) is found analytically, thus deﬁning the new value of the scalar step. This parabolic approximation may be repeated several times until the convergence is attained, but no more than two solutions of the forward problem are usually required at each iteration of the Newtonian process considered. 8.3.6. Multi-level adaptive stabilization The efﬁciency of the optimization procedure constructed above is determined by a reasonable combination of several techniques of the inversion solution stabilization at various stages of the procedure. These techniques protecting the solution against external and internal uncertainty sources of various origins are as follows. Rational parameterization of the medium (identiﬁcation of normal and anomalous structures of the model and the use of ﬁnite functions or correlated parameters) Linear (in accordance with a priori data and parameters weighting matrices) and non-linear (selection of logarithmic representation of parameters and apparent resistivity data) scaling Tikhonov’s regularization (with an adaptive decrease of the regularization parameter as a function of the convergence rate of the inversion iterations) Robust metric of functionals (neutralizing rare but strong outliers with the adaptation to median statistics of misﬁts) Stabilized factorization of Newtonian system (operating with an adaptive dependence on the level of the data misﬁt functional) Linear ‘‘common-sense’’ constraints (rejecting implausible models during the selection of the Newtonian scalar step) The adaptive implementation of most of these techniques ensures the convergence reliability and a high resolution of the approach proposed. An important factor stabilizing the inversion process is the suppression of systematic distortions in the inverted data. These are primarily static distortions of impedance amplitudes (apparent resistivities) caused by sub-surface galvanic effects of heterogeneities (beyond the scale of the inverse problem solved) and giving a strong bias in these data at a certain observation site over the entire range of inversion periods. Primary protection from static distortions consists in the data reweighting decreasing the signiﬁcance (up to the complete exclusion from the inversion) of apparent resistivities supported by the relevant focus on the inversion of impedance phases and magnetovariational data (Varentsov et al., 1996; Varentsov, 2002, 2005).

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Another simple and natural tool for correcting static distortions is an explicit ﬁt of resistivity in the subsurface cell directly under the given site (Vanyan et al., 2002). However, this technique implies that static distortions in various components of the apparent resistivity (e.g., in various 2-D polarizations) are self-consistent via the operator of the forward problem, which is sometimes at variance with their real nature. There are known approaches, such as Ogava (1999), providing for explicit incorporation of the static distortion coefﬁcients into the optimized vector of model parameters. However, in this case the static distortion coefﬁcients and parameters describing the near-surface geoelectric structure can be strongly equivalent, which can worsen the convergence of the inversion iterations. Here, an independent robust statistical estimation of static shift coefﬁcients within the inversion process analysing the sequence of log-ratios of the observed and modelled apparent resistivities at a certain site for the entire period grid fT l g is suggested:

Zl ¼ log rOBS ðT l Þ rMOD ðT l Þ ; l ¼ 1; . . . ; N T Median statistics of this sequence, namely, the mean M Z and the scatter sZ determined in accordance with Eq. (8.2c) indicate a period-independent static shift with non-zero value of the following transform: expðM Z Þ; jM Z j4csZ with a constant c 2 ½1; 3, mZ ¼ 0; jM Z jocsZ while a zero mZ for a large MZ usually indicates a shift only within a part of the inversion period range. The above analysis is performed independently for each apparent resistivity component at successive iterations, but its results become meaningful only when a proper initial ﬁt is achieved for the entire ensemble of inverted and modelled data. At this stage, it is worthwhile to make a decision on the correction of static distortions at certain sites and to divide the observed apparent resistivity data by estimated mZ factors. It is advisable to use this approach for the downweighted apparent resistivity data. 8.3.7. Post-inversion analysis An essential attribute of the inversion problem solution and a very important subject of a posteriori analysis is the estimation of uncertainties in the parameters of the ﬁnal model. Likewise, the inverted data obtained from the estimation of the EM ﬁeld-transfer operators are characterized by the vector pair d0, rd0 the inversion problem solution must also be represented by the vector pair p, rp. The rp estimate is conventionally constructed within the linear approach (Tarantola and Valette, 1982; Hjelt, 1992) specifying an approximate interrelation between the uncertainties M in data and parameters on the basis of the model sensitivity operator G^ in the vicinity of the inferred solution p. This approach involving the direct calculation of sensitivities and explicit factorization of the Newtonian system does not require any considerable computer resources in the proposed optimization procedure.

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Additional tasks at the post-inversion stage may be the estimation of the relative signiﬁcance of model parameters and the informativeness of elements and groups of data obtained by adequate averaging and scaling of elements of the model sensitivity operator (Tarantola and Valette, 1982; Hjelt, 1992; Zhdanov, 2002) as well as the check for the accuracy of the discrete approximation and linear assumptions in the forward and inverse problem solution algorithms. This constitutes, in essence, the formal solution of the inverse problem, giving further way to the informal analysis of the inferred results and the adjustment of the parameterization schemes, a priori assumptions, stabilization parameters, the structure of data vectors d0, rd0 , and other key features of the inversion problem formulation. The success of this informal stage is essentially dependent on the intuition and experience of the interpreter, as well as on the comprehensiveness and clearness in the representation of the results derived from the formal solution. The development of inversion methods increases the completeness of the formal inversion results, but the role of the human factor at the post-inversion stage remains signiﬁcant. The formal inversion cycle is repeated several times giving a set of acceptable inversion solutions accounting for the listed adjustments. It is hard to select an optimal model from this set, and to this extent it is suggested to consider the robust average of this model set giving a ‘‘mainstream’’ view on the common model features through a median mean mx and the resolution estimate through a median scatter sx as deﬁned in Eq. (8.2c). In this way, the improved a priori assumptions in the pair of vectors p0, rd0 may be deﬁned for the next series of inversion solutions.

8.4. STUDY OF INVERSION ALGORITHMS USING SYNTHETIC DATASETS Synthetic datasets calculated for known geoelectric structures serve as a valuable tool to verify various inversion techniques and to compare their efﬁciency, because their use ensures the possibility to estimate the model ﬁt not only in the data space but directly in the space of model parameters. The data simulation approach provides a high ﬂexibility in the analysis of complicated and even ‘‘exotic’’ data ensembles and permits an easy and precise data contamination with noise factors of different origin. A valuable experience in the use of synthetic datasets for the study of 2-D inversion techniques is gained in the frames of the international project COPROD2S (Varentsov, 1998, 2002, 2005; Ogawa, 1999). A dataset from this project will be used below to demonstrate the efﬁciency of model parameterization schemes introduced in Section 8.1. 8.4.1. Comparison of three model parameterization schemes in 2-D inversion Consider quite a simple block model of a conductive ﬂexure-type structure (left bottom panel in Fig. 8.2) indexed as 2S2 in the COPROD-2S project. The model

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Fig. 8.2. 2-D inversion of the true synthetic dataset with strong outliers in the model COPROD-2S2: left panels show ﬁnal models (resistivities, in Om; the horizontal scale is normal and the vertical scale is logarithmic, both in km; the observation sites are indicated by crosses) for FF, IR, and CR parameterization schemes (from top to bottom), right panels show correspondent misﬁt relative to the true geoelectric structure (also in Om); resistivities and their misﬁts for normal layers are given by numbers.

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resistivity drops to 1 Om in the centre of the ﬂexure, and the low-resistive half-layers stretch horizontally at different depths from the centre into the left and right normal sections. A four-component synthetic dataset is considered in this analysis, containing two pairs of apparent resistivities and impedance phases for two EM ﬁeld modes, namely, electric (EP) and magnetic (HP), and including 17 observation sites and 10 observation periods within a range 1–1000 s. Data elements at three random sites for each data component and period are strongly (by orders of magnitude) but randomly distorted; all other data elements are taken as modelled, and the data errors are all set above the modeling noise at 1% level (not accounting for outliers). The joint bi-modal 2-D inversion starts from the following initial model. The normal section has true layer depth, but layer resistivities are almost all adjusted. The anomalous model structure is approximated using the following three tools: a single IR window (with independent resistivities of cells), a single CR window (with correlated resistivities of cells), and three FF windows successively covering a heterogeneity from top to bottom with a single horizontal 1-D ﬁnite function in each window. The resistivities of the normal section layers, except those of the highly resistive background and the conducting basement, are also optimized. Initial values of optimizing window resistivities and those of the normal section layers are all set at 1000 Om, and the amplitude of the ﬁnite functions describing the excessive electric conductivity is selected negligibly small. In the ﬁrst two cases over 100 parameters are optimized, while in the latter case their number reduces to only 18. Inversion results are shown in the left panels in Fig. 8.2 for the three listed parameterization alternatives and the right panels of the ﬁgure present deviations of these solutions from the true geoelectric structure. The ﬁrst two inversion solutions obtained with IR (central panels) and CR (lower panels) windows yield very low robust RMS estimates of the data misﬁt, amounting to about 0.1 and 0.01, respectively; while the conventional RMS stabilizes at the high level above 4, well correspondent to the strength of the data outliers. Absolute deviations of the model resistivities are well below 1 Om in the conducting part of a window. In both cases

Fig. 8.3. Localization of geoelectric anomalies for the dataset COPROD-2S1 using the system of 26 vertical 1-D ﬁnite functions (resistivity section in Om; the axes are the same as in Fig. 8.2).

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Fig. 8.4. A series of three successive inversion models for the dataset COPROD-2S1 using the system of 11 2-D ﬁnite functions with a ‘‘condensing’’ support (resistivity section in Om; the axes are the same as in Fig. 8.2).

the distorting effect of data outliers is practically negligible, this proves the effectiveness of the robust metric applied. The IR-parameterization inversion requires more than 90 iterations, but even this number of iterations could not reduce appreciable errors in the resolution of the lower and left parts of the heterogeneity.

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Fig. 8.5. The ﬁnal inversion model (resistivity section in Om; the axes are the same as in Fig. 8.2) for the COPROD-2S1 dataset obtained using the ﬁnite function parameterization (top panel) and the true geoelectric structure (bottom panel).

Fig. 8.6. The central section (y ¼ 0) of the geoelectric structure studied with a 3-D inversion technique: model resistivities are given in Om, the horizontal x-axis has a normal scale, and the vertical scale is logarithmic, both in km; the observation sites are marked by crosses.

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Fig. 8.7. Vertical sections (from top to bottom: y ¼ 0.8; 0.4; 0; 0.4; 0.8 km) of the inverted 3-D model after the second iteration (resistivities in Om, the axes are the same as in Fig. 8.6, the half-space resistivity is indicated by a number).

The number of iterations for the CR parameterization is reduced by a factor of 1.5, but the model resolution accuracy markedly increases. The effect of the additional stabilization on the quality achieved in the CR solution is evident; moreover, it is accompanied by a rise in the effectiveness of computations. The solution with ﬁnite functions faces natural difﬁculties in describing discontinuous step-like structures by continuous functions. However, the model shown in

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Fig. 8.8. Comparison of the modelled (for the ﬁnal 3-D inversion solution, left) and the observed (right) data pseudo sections along the central proﬁle y ¼ 0, from top to bottom: xy apparent resistivity (Rxy) and impedance phase (PhZxy), yx apparent resistivity (Ryx) and impedance phase (PhZyx), tipper amplitude (|Wzx|); the horizontal axis represents x-coordinates (km), the vertical logarithmic axis indicates time periods (s).

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the upper panel of the ﬁgure is fairly close to the resulting CR model, the errors in model parameters are, on average, not larger than in the IR solution, the number of iterations does not exceed 30 (i.e. three times smaller than in the IR solution), and the required computer resources are more than an order of magnitude smaller (taking into account a ﬁvefold decrease in the number of ﬁtted parameters). Thus, the FF parameterization, being inferior in the solution quality to the CR parameterization, shows remarkable effectiveness at the localization stage of geoelectric anomalies.

8.4.2. 2-D inversion with numerous finite functions Here an example of inversion of another synthetic dataset, indexed as 2S1 in the COPROD-2S project (Fig. 8.10) is described. The inverted data ensemble consists of bi-modal apparent resistivities and impedance phases together with real and imaginary parts of the geomagnetic tipper at 61 observation sites for 11 periods within an interval from 2 to 10,000 s. The geoelectric structure in this case is more complicated, containing a large number of low-resistive local anomalies (lower panel in Fig. 8.5), but the dataset is denser and is deﬁned with the modeling accuracy (below 1%) without any outliers. The model includes in the resistive background seven successively deepening bodies 8 km in width ranging in thickness from 5 to 25 km and all having a low resistivity of 3 Om; in the left part of the section, these bodies are underlain by a sub-horizontal, 50-km wide and 17-km thick structure having a resistivity of 10 Om. Some of these structures are hardly visible in the dataset. Actually, the example under consideration serves to examine the ultimate possibilities of resolving multiple anomalies of electrical conductivity from detailed MT/MV data speciﬁed with a percent-level accuracy. This inversion is conducted in three successive stages. At the ﬁrst 1-D inversion stage, a set of the apparent resistivity and impedance phase curves in two polarizations is simultaneously inverted at three edge sites of the proﬁle. This analysis yields an initial estimate of the normal section structure with a very good resolution of the depth to the conducting basement. Thus this depth is further ﬁxed at its true level, while normal section resistivities are set with 100% error and further optimized together with the anomalous model structure. At the second stage, a 2-D inversion problem is solved with an aim to primarily localize the strongest geoelectric anomalies. The section is scanned by numerous (up to 26) vertical 1-D ﬁnite functions deﬁned within horizontally narrow ( 4 km) regions covering in two stages the depth range fromo1 to 60 km.This localization results in the identiﬁcation of seven low-resistive inclusions successively deepening from left to right and gives a weak indication of one more anomaly lying in the left part of the section under the well delineated bodies (Fig. 8.3). Fig. 8.9. Comparison of the modelled (for the ﬁnal 3-D inversion solution, left) and the observed (right) data pseudo sections along the edge proﬁle y ¼ 0.8 km, from top to bottom: xy apparent resistivity (Rxy) and impedance phase (PhZxy), xx and yy apparent resistivity (Rxx, Ryy), tipper amplitudes (|Wzx|,|Wzy|); the axes are the same as in Fig. 8.8.

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Fig. 8.10 (Continued).

At the ﬁnal stage of the inversion, the inferred anomalies are covered by supports of 2-D ﬁnite functions (upper panel in Fig. 8.4). Eleven ﬁnite functions and 57 optimization parameters are used in a series of successive inversion solutions. For each solution, the ﬁnite function supports are decreased when their edge columns or rows of cells reach resistivity values close to the host medium resistivity. After that the inversion repeats with smaller FF supports. This iterative approach is justiﬁed because continuous ﬁnite functions can effectively approximate rectangular discontinuous structures (already recognized at the localization stage) only at the boundaries of their supports (Fig. 8.1). The sequence of three inversion solutions Fig. 8.10. The synthetic dataset COPROD-2S1 in pseudo sections, from top to bottom: H-polarization apparent resistivity (R_HP) and impedance phase (PhZ_HP), E-polarization apparent resistivity (R_EP) and impedance phase (PhZ_EP), the geomagnetic tipper (ReWzx, ImWzx), the vertical magnetic ﬁeld (ReHz, ImHz) and the horizontal magnetic ﬁeld (ModHx,PhHx); the axes are the same as in Fig. 8.8.

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Fig. 8.11. Inversion models for the dataset COPROD-2S1 obtained within the CR parameterization, from top to bottom: for the apparent resistivity and the impedance phase, for the tipper data (Re,Im); for the combination of all four mentioned data components (resistivities in Om, the horizontal scale is normal and the vertical scale is logarithmic, both in km; the normal section resistivities are indicated by numbers).

with condensing FF supports is traceable in Fig. 8.4 from top to bottom. The comparison of the ﬁnal solution (upper panel in Fig. 8.5) with the true structure (bottom panel) shows fully adequate resolution of the ﬁve upper bodies and approximately outlines the two deepest anomalies (the right-most ones). The sub-

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Fig. 8.12. Inversion models for the dataset COPROD-2S1 obtained within the CR parameterization, from top to bottom: for the vertical magnetic ﬁeld (Re,Im); for the horizontal magnetic ﬁeld (Mod, Phase); see legend to Fig. 8.11.

horizontal deep structure below the left anomalies is also resolved on a qualitative level. 8.4.3. 3-D inversion example In the frames of a 3-D inversion problem (Varentsov, 2002), the advantage of using the ﬁnite function parameterization becomes the most obvious. The inversion of a regular array of synthetic MT/MV data over a simple geoelectric structure approximated by a single 3-D ﬁnite function is considered here. This model contains a uniform 2 2 2 km cube inclusion with low (1 Om) resistivity buried at a depth of 0.4 km in a homogeneous 100 Om half-space. The central cross section of the model is shown in Fig. 8.6. The inverted data are speciﬁed on a grid of 7 7 ¼ 49 observation sites covering the inclusion in a plan view. Logarithms of apparent resistivities for all the four components of the impedance tensor, the phases of two off-diagonal impedances and the amplitudes of both tipper components are inverted (8 data components in total) at periods of 0.1, 1, and 10 s. These data are complicated (just like the dataset in Section 8.4.1) by randomly distributed

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Fig. 8.13. The inversion model for the dataset COPROD-2S1 obtained within the CR parameterization for the six-component combination of the E-polarization apparent resistivity, impedance phase, tipper and horizontal magnetic ﬁeld (top panel) compared with the true geoelectric structure (bottom panel); see legend to Fig. 8.11.

sparse high-amplitude outliers, obviously visible in the pseudo sections of the inverted data (Figs. 8.8, 8.9, right panels). The data errors are all set above the modeling accuracy at 1% level. One parameter of the host medium (log-resistivity of the half-space) and one 3-D ﬁnite function deﬁned within the inclusion (13 parameters more) are used for the parameterization of the unknown geoelectric medium in the model. The initial half-space resistivity is twice bigger (200 Om), and the initial integral conductance of the ﬁnite function is 20 times smaller than the correspondent true values. The convergence of the inversion iterations is very fast. Even after the second iteration, the half-space resistivity is estimated with 20% accuracy; the same is the accuracy level in the resistivity resolution for the top cells of the heterogeneity (Fig. 8.7). At the next iteration, the half-space resistivity gets its true value with 1% accuracy. The robust procedure successfully suppresses the effect of the outliers after a few initial iterations. The ﬁnite function ﬁrst assumes symmetry and concentrates the excessive conductivity in the upper part of the inclusion and then tends towards a rectangular shape.

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Finally, after 15 iterations the resolved model practically coincides with the structure given in Fig. 8.6. The convergence in this 3-D inversion solution is visibly faster than in a correspondent 2-D inversion solution (with ﬁve data components given at one proﬁle) taking more than 20 iterations for the same level in the model resolution. The robust data RMS decreases well below 0.1, while the conventional RMS stabilizes near 3 due to the inﬂuence of strong data outliers. The data ﬁt for the selected data components is shown in Fig. 8.8 for the central proﬁle (y ¼ 0) and in Fig. 8.9 for a proﬁle above the edge of the heterogeneity (y ¼ 0.8 km). All the features of the observed data not distorted by the outliers are precisely ﬁtted. The inversion quality is also proved by a prominent horizontal symmetry of the modelled data. The whole inversion solution takes a few minutes running in a graphical interaction. Almost the same accuracy is obtained in the procedure with the hybrid estimation of the system matrix in Eq. (8.4): the most time-consuming calculation of M the sensitivity operator G^ k is made once every four iterations and then this operator remains unchanged during the three subsequent iterations. In this case, the decrease of the robust data RMS is not as regular and smooth but is still as monotonous and fast as in the preceding case, while the overall inversion time reduces nearly three times.

8.4.4. Resolution of a system of local conductors using the CR-parameterization Let us now return to the COPROD-2S1 dataset to investigate the effectiveness of the CR parameterization. The initial normal section model is the same as in Section 8.4.2. The anomalous structure is scanned in one triangular window with 516 inversion cells (Fig. 8.11), which includes all the previously localized anomalies (Fig. 8.4). The dataset (Fig. 8.10) is now extended with the inclusion of the horizontal Hx and vertical Hz magnetic ﬁelds both scaled by the normal horizontal magnetic ﬁeld. All the data components are free from any superimposed noise factors. The success of the CR parameterization scheme in the comparative study (Section 8.4.1) gives grounds to attain in this case an acceptable solution in a single run of the inversion procedure, in contrast to the multi-step ﬁnite function inversion (Section 8.4.2). However, the increase of the parameters number and the strong interference of the H-polarization data responses from the top anomalies sharpen the solution instability. Serious complications are met in the inversion of the data ensembles containing H-polarization components causing problems in the resolution of the both the structures closest to the observation surface and those that are deepest. Such complications are common for a number of modern inversion techniques and also strongly inﬂuence the joint inversion of bi-modal data ensembles. Finer grids and greater stabilization efforts are required to overcome them. In this respect the advantage of a compact ﬁnite function parameterization, giving in this model an acceptable solution (Fig. 8.4) for the bi-modal dataset (including both E- and H-polarization apparent resistivities and phases), should be strongly emphasized.

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At the same time, the inversion of different combinations of the E-polarization data within the CR parameterization scheme runs quite effectively for this dataset and model. In this case, an important task is to compare the model resolution achieved for different E-polarization data components inverted separately, as well as to outline the problems and advantages in the joint inversion of the multicomponent data ensembles. This analysis is particularly promising in the view of recent proofs of the information equivalence of various ideal (given continuously over the whole inﬁnity proﬁle and period range) E-polarization responses, namely, the impedance, tipper, and normalized horizontal and vertical magnetic components (Vanyan et al., 1998; Berdichevsky et al., 2003; Dmitriev and Berdichevsky, 2006). Fig. 8.11 shows three inversion models obtained for the E-polarization set of apparent resistivity and impedance phase, for the geomagnetic tipper and for the joint four-component data ensemble. The model derived from the tipper data seems to be as good or even preferable to the model based on the impedance data. Thus, the statement of the ideal information equivalence of the impedance and tipper data applies to the practical dataset with the density and accuracy typical for highquality proﬁle observation systems. Naturally, the joint inversion of the impedance and tipper combination improves the both partial solutions causing no contradictions in the inversion convergence. Fig. 8.12 displays the comparison of the inversion solutions for the single vertical and horizontal magnetic ﬁelds. Both the solutions give a model resolution comparable with that for the tipper data. The inversion of the horizontal magnetic data looks preferable in the resolution of the deepest anomalies in the right part of the model section. Finally, Fig. 8.13 presents the resulting model of the joint inversion of the most complicated six-component dataset including the impedance, the tipper and the horizontal magnetic data in comparison with the true geoelectric structure. This solution gives the best resolution and does not suffer from the contradictions in the data ﬁt in the course of inversion iterations. 8.4.5. Reduction of strong data noise and static shifts Finally, let us return to the model 2S2 (Fig. 8.4) to learn the effectiveness of the considered 2-D inversion procedure in overcoming noise factors of different nature, including normal random noise, strong but sparse random outliers, and periodindependent static shifts. A new six-component COPROD-2S2S dataset includes all the listed noise factors. It consists of bi-modal impedance and tipper data ( Fig. 8.14) at the same sites and periods as in the previous COPROD-2S2 dataset (Section 8.4.1). The previous section clearly demonstrates the effectiveness of the robust metric in the elimination of the outliers. Fig. 8.15 compares three inversion Fig. 8.14. The synthetic dataset COPROD-2S2S in pseudo sections, from top to bottom: H-polarization apparent resistivity (R_HP) and impedance phase (PhZ_HP), E-polarization apparent resistivity (R_EP) and impedance phase (PhZ_EP), the geomagnetic tipper (ReWzx, ImWzx); the axes are the same as in Fig. 8.8.

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Fig. 8.15. Inversion models for the dataset COPROD-2S2S obtained within the CR parameterization, from top to bottom, left column: for the six-component dataset with the downweighted apparent resistivities; for the four-component dataset containing only the impedance phase and the tipper data; right column: for the six-component dataset with the corrected apparent resistivities, the true geoelectric structure; see legend to Fig. 8.11.

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Table 8.1. Estimation of the static shift coefﬁcients of the apparent resistivity for the dataset COPROD2S2S.

E-Polarization Data site index Static shift coefﬁcients True Estimated

3

4

5

15

28

29

30

0.20 0.20

0.040 0.040

0.20 0.20

0.25 0.24

3.33 3.38

11.1 11.1

3.33 3.14

H-Polarization Data site index Static shift coefﬁcients True Estimated

4

5

6

16

29

30

31

0.10 0.11

0.010 0.010

0.10 0.10

0.625 0.75

10.0 8.7

100 68

10.0 8.7

solutions outlining alternative approaches to the reduction of the most severe static shift distortion. These solutions start from the same initial model as deﬁned in Section 8.4.1. The ﬁrst approach is based on the joint inversion of the whole dataset with sufﬁciently downweighted apparent resistivities; in the second solution the apparent resistivities are completely excluded from the data ensemble and only impedance phases and tipper data are inverted. Both the solutions look quite effective in spite of a substantial (by several tens of percent) noise-to-signal ratio. The third inversion solution implements the static shift correction procedure (Section 8.3.6) for the apparent resistivity basing on the results of the ﬁrst inversion solution, and further inverts the whole six-component dataset without downweighting of the corrected apparent resistivity. The static shift correction procedure works perfectly after the ﬁrst 10 iterations. The estimated static shift coefﬁcients are listed in Table 8.1 in comparison with the true values realized in the dataset. It should be emphasized that all the approaches strongly reduce static shift distortions, completely exclude data outliers and are stable to the inﬂuence of the normal noise. The structure of the best conducting ﬂexure blocks is resolved to an accuracy of a few percent, though the overall model resolution is somewhat lower than in the case with less distorted data (Section 8.4.1).

8.5. CONCLUSIONS A general approach for the joint magnetotelluric and magnetovariational data inversion has been formulated in terms of models of piecewise-continuous geoelectric media, and several basic aspects of its implementation have been analyzed, primarily, the adaptive parameterization of the models and the stable Newtonian minimization of the Tikhonov’s misﬁt functional in a robust metric. Various examples of synthetic dataset inversion are presented. These examples demonstrate reliable performance of the derived algorithms in both the simplest but instructive

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3-D application and relatively complex 2-D problems, e.g. those including a large number of anomalies and a strong data noise factors. Various methodological alternatives for localizing and detailed resolution of the heterogeneities of the geoelectric medium are outlined within this approach. The example in Section 8.4.4 demonstrates the encouraging resolution of the joint MT/MV method in the separation of numerous closely spaced conductors in the presence of multi-component data characterized by percent-level accuracy and obtained from a detailed proﬁle system of observations within a wide period range. In the same example, singlecomponent MV inversions look quite compatible with the impedance inversions. A number of problems require further studies; primarily, the search for better criteria to terminate the inversion iterations, the enhancement of a posteriori analysis, subtle improvements in the implementation of ﬁnite function and CR-parameterizations, and problems of the numerical efﬁciency in calculating model sensitivities. Acknowledgments I am grateful to M. Berdichevsky, V. Cerv, N. Golubev, J. Pek, M. Zhdanov, E. Sokolova, and N. Baglaenko for the long-term joint experience in the inversion studies. I appreciate the participation of Y. Ogawa, P. Pushkarev, and M. Smirnov in the inversion of the datasets of the COPROD-2S project. This work was supported by the grants RFBR 95–05-15066, 98-05-65411, 03-05-64646, DFG-RFBR 03-05-04002, and INTAS 97-1162.

REFERENCES Berdichevsky, M.N., Dmitriev, V.I., Golubtsova, N.S., Mershchikova, N.A. and Pushkarev, P.Yu., 2003. Magnetovariational Sounding: New Possibilities. Izvestiya, Phys. Solid Earth, 39, 9: 701–727. Berdichevsky, M.N., Dmitriev, V.I., Golubtsova, N.S., Mershchikova, N.A. and Pushkarev, P.Yu., 2006. Magnetovariational method in deep geoelectrics. Electromagnetic Sounding of the Earth’s Interior, Chapter 2. Caldwell, G.T., Bibby, H.M. and Brown, C., 2004. The magnetotelluric phase tensor. Geophys. J. Int., 158: 457–469. Dmitriev, V.I. and Berdichevsky, M.N., 2006. Inverse Problems in Modern Magnetotellurics. Electromagnetic Sounding of the Earth’s Interior, Chapter 7. Egbert, G.D., 2002. Processing and interpretation of EM induction array data. Surv. Geophys., 23: 207–249. Gill, P., Murray, W. and Wright, M., 1982. Practical Optimization. Academic Press, London. Golubev, N.G. and Varentsov, Iv.M., 1993. Algorithms of robust inversion of MT sounding data for 2-D geoelectric structures. EGS General Assembly (Abstracts), Ann. Geophys., 11, Suppl. Part I: 48.

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Hjelt, S.E., 1992. Pragmatic inversion of geophysical data. Lecture Notes in Earth Sciences, 39, Springer, Berlin, 262pp. Huber, P., 1981. Robust Statistics. Wiley, New York. Lawson, C.L. and Hanson, R.J., 1984. Solving Least Squares Problems. PrenticeHall, Englewood Cliffs. Mackie, R.L. and Madden, T.R., 1993. 3-D magnetotelluric inversion using conjugate gradients. Geophys. J. Int., 115: 215–229. Novozhinski, K. and Pushkarev, P.Yu., 2001. The efﬁciency analysis of programs for 2-D inversion of magnetotelluric data. Izvestiya, Phys. Solid Earth, 37: 503–516. Ogava, Y., 1999. Constrained inversion of COPROD-2S2 dataset using model roughness and static shift norm. Earth Planets Space, 51: 1145–1151. Pek, J., 1985. Linearization methods of interpreting MT and MV data. Travaux Geophys., 33: 199–326. Strakhov, V.N., 1978. On the parameterization problem in gravity inversion (in Russian). Izvestiya, Fizika Zemli, 6: 39–49. Tarantola, A. and Nercessian, A., 1984. 3-D inversion with blocks. Geophys. J. R. Astron. Soc., 50: 1618–1627. Tarantola, A. and Valette, B., 1982. Generalized nonlinear inverse problems solved using the least squares criterion. Rev. Space Phys., 20: 219–232. Tikhonov, A.N. and Arsenin, V.Ya., 1977. Methods of Solution of Ill-Posed Problems. Wiley, New York. Vanyan, L.L., Berdichevsky, M.N., Pushkarev, P.Yu. and Romanyuk, T.V., 2002. A geoelectric model of the Cascadia subduction zone. Izvestiya, Phys. Solid Earth, 38, 10: 816–845. Vanyan, L.L., Varentsov, Iv.M., Golubev, N.G. and Sokolova, E.Yu., 1998. Derivation of simultaneous geomagnetic ﬁeld components from tipper arrays. Izvestiya, Phys. Solid Earth, 34, 9: 779–786. Varentsov, Iv.M., 1998 The COPROD-2S project web site. (http://user.transit.ru/

igemi/c_2s_p0.htm). Varentsov, Iv.M., 1999. The selection of effective FD solvers in 3-D EM modelling schemes. Proceedings of 3DEM-2 International Symposium, University of Utah, Salt Lake City, 201–204. Varentsov, Iv.M., 2002. A general approach to the magnetotelluric data inversion in a piecewise-continuous medium. Izvestya, Phys. Solid Earth, 38, 11: 913–934. Varentsov, Iv.M., 2005. Robust methods of joint inversion of MT/GDS data in the piecewise-continuous media (in Russian). Electromagnitnye issledovania zemnykh nedr. Nauchny Mir, Moscow, 54–75. Varentsov, Iv.M., 2006. Arrays of simultaneous EM soundings: design, data processing and analysis. Electromagnetic Sounding of the Earth’s Interior, Chapter 10. Varentsov, Iv.M., Engels, M., Korja, T., Smirnov, M.Yu. and BEAR WG, 2002. A generalized geoelectric model of Fennoscandia: a challenging database for long-period 3-D modelling studies within the Baltic Electromagnetic Array Research (BEAR) project, Izvestya. Phys. Solid Earth, 38, 10: 855–896.

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Varentsov, Iv.M., Golubev, N.G., Gordienko, V.V. and Sokolova, E.Yu., 1996. Study of the deep geoelectric structure along the Lincoln Line (EMSLAB experiment): Izvestiya. Phys. Solid Earth, 32, 4: 124–144. Varentsov, Iv.M., Sokolova, E.Yu., Martanus, E.R., Nalivayko, K.V. and BEAR WG, 2003. System of EM ﬁeld transfer operators for the BEAR array of simultaneous soundings: Methods and Results, Izvestya, Phys. Solid Earth, 39, 2: 118–148. Zhdanov, M.S., 2002. Geophysical inverse theory and regularization problems. Elsevier, Amsterdam, 609pp. Zhdanov, M.S. and Golubev, N.G., 1983. Use of the ﬁnite function methods for the solution of the 2-D inverse problem. J. Geomagn. Geoelectr., 35: 707–722. Zhdanov, M.S. and Varentsov, Iv.M., 1983. Interpretation of local 2-D electromagnetic anomalies by formalized trial procedure. Geophys. J. Roy. Astr. Soc., 75: 623–638. Zhdanov, M.S., Varentsov, Iv.M., Weaver, J.T., Golubev, N.G. and Krylov, V.A., 1997. Methods for modelling EM ﬁelds (results from COMMEMI project). J. Appl. Geophys., 37, 3–4: 133–271.

Chapter 9 Neural Network Reconstruction of Macro-Parameters of 3-D Geoelectric Structures V.V. Spichak Geoelectromagnetic Research Center IPE RAS, Troitsk, Moscow Region, Russia

Three-dimensional inversion of EM data in terms of a ‘‘cell by cell’’ conductivity distribution is a challenging problem both from theoretical and computational points of view. Many geophysicists hope that a breakthrough in this direction will enable them to solve practical problems, which require three-dimensional interpretation of incomplete and noisy data. However, in spite of achievements in 3-D inversion of EM data (Smith and Booker, 1991; Mackie and Madden, 1993; Ellis, 1999; Portniaguine and Zhdanov, 1999a,b,c; Spichak et al., 1999a; Xie and Li, 1999; Zhdanov and Fang, 1999; Newman and Alumbaugh, 2000; Sasaki, 2001; Newman et al., 2002; Zhdanov, 2002) it becomes evident that interpretation of real data requires a variety of tools to be used depending on the volume and quality of both the data and prior information available (Spichak, 1999; Spichak et al., 1999a). Unfortunately, application of methods mentioned above requires the geophysicist to know in advance one-dimensional (1-D) layering and to supply an initial guess (expressed in deterministic or statistical terms) on the 3-D conductivity distribution in the region of research. Prior information often comes from other geophysical methods and we have to be able to incorporate it in a ﬂexible way into the inversion procedure. On the other hand, sometimes geophysicists have only an idea about the type of unknown conductivity distribution (e.g. horst, graben, fault, etc.). In such a case, none of ‘‘regular’’ inversion techniques can transform EM data into a resistivity image. They are also inefﬁcient for a multiple inversion of data in the frames of the same class of models (e.g. in the monitoring mode) since do not ‘‘remember’’ an inversion way already found. Finally, inversion of very noisy data

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40009-3

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(say, if the level of noise is 30, 50 or even 100%, which often is the case in practice) by these methods may give results, which will be very far from the reality. Hence, it is important to develop basically new approaches to the interpretation that would overcome or at least reduce the difﬁculties mentioned above. An alternative approach especially useful in this situation could be developed in the framework of the artiﬁcial intelligence paradigm. The methods of data interpretation, based on the analogy with the function of the human brain’s neural network have proven to be successful in the solution of the image-reconstruction problems in many ﬁelds of science. A pattern of recognition method, namely, the artiﬁcial neural network (ANN) technique, has become especially popular during past few years. The following properties of ANNs are helpful in their successful applications: – ANNs are very effective for the solution of nonlinear problems, – ANNs can conclude from incomplete and noisy data, – ANNs admit the interpolation and extrapolation of the available database, – ANNs provide a means for the synthesis of separate series of observations to obtain an integral response, which allows a joint interpretation of diverse data obtained by different geophysical methods, – ANNs enable simultaneous data processing, thereby essentially reducing the computation time, particularly when special chips are employed, – The time necessary for ANN recognition depends on the dimension of space of the unknown parameters rather than the physical dimension of the medium, which makes ANNs particularly promising for the interpretation of the class of 3-D geoelectric structures. A review of the ANN paradigms and the detailed analysis of their application to various geophysical problems is given in Raiche (1991). The ANN methods were used in geoelectrics for 1-D inversion (Sen et al., 1993; Hidalgo et al., 1994; Poulton and Birken, 1998). Parameters of 2-D structures were estimated from synthetic and real time-domain electromagnetic data in Poulton et al. (1992a,b). Finally, in Spichak and Popova (1998, 2000) a ﬁrst attempt to apply the ANN approach to the inversion of electromagnetic data in the class of 3-D geoelectric structures is made based on ideas formulated in Spichak (1990).

9.1. BACKPROPAGATION TECHNIQUE To solve the inverse problem Spichak and Popova (1998, 2000) used one of the so-called ‘methods of learning with a teacher’ namely the error BackPropagation (BP) technique (Rumelhart et al., 1988; Schmidhuber, 1989; Silva and Almeida, 1990). Such an approach implies two stages of the inversion procedure: the training of the network and testing, or recognition (the inversion itself). At the learning stage, the ‘teacher’ speciﬁes the correspondence between chosen input and output data, which is similar to the mechanism of the training of a man. The analogy with the human brain also consists in the similarity of some functional elements of the biological neural system to the nonlinear system ‘data – parameters of the target’ modeled by ANN (its elements are also called ‘neurons’). In both cases, the system

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Fig. 9.1. Three-layered artiﬁcial neural network (perceptron).

could be considered as a n-layered network in which every neuron of one layer is somehow connected with the neurons of other layers. A signal comes to the input layer of neurons from outside the system, whereas its magnitude at the neurons of other layers depends on the signal magnitudes and connection weights of all associated neurons of the previous layer. Moreover, similar to the biological systems, the net response of an artiﬁcial neuron is described by a nonlinear function. In spite of the application of BP technique has become rather routine procedure (see, in particular, the references above), it is worth to specify the main elements of the scheme. A three-layered ANN (Fig. 9.1) consists of the layer of input neurons (data), the layer of hidden neurons (their number, generally speaking, is arbitrary and could be adjusted in order to reﬂect the complexity of the system, see Section 9.3), and the layer of output neurons (unknown parameters of the geoelectric structure). The propagation of the input signal via a network occurs in the following way. The input signal xi comes to each ith neuron of the input layer. It is equal to the correspondent element of the input vector, composed of the values of the measured electromagnetic ﬁeld (or their transforms) at a number of periods. Every kth neuron of a hidden layer receives summary input signal ykinp from all neurons of the input layer X yinp wik xi ð9:1Þ k ¼ i

where wik are the connection coefﬁcients (weights) between the input and hidden layers and the summation is carried out over all input neurons. The signals ykinp are transformed by each kth neuron of the hidden layer into the output signals ykout by the

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neuron ‘‘activation functions’’ Gkh h inp yout k ¼ G k ðyk Þ.

ð9:2Þ

Then the signals propagate from a hidden layer to the output one and for each jth neuron of the output layer we obtain ! X u out uj ¼ G j wkj yk ð9:3Þ k

where uj are the output signals at the output layer, wkj the connection weights between the hidden and output layers, and Gju activation functions for neurons at output layer. (The activation functions are considered usually as the same for each neurons of the same layer.) Meanwhile, in Section 9.3.1 it will be shown that individual choice of activation function type for different output neurons may improve the recognition of some model parameters. At the training stage the actual output signals uj are compared with known ‘correct answers’ ujt, which correspond to given input signals, and a standard error X ðup;j utp;j Þ2 ð9:4Þ Erp ¼ j

is calculated for each pth learning sample; here the summation is carried out over all neurons of the output layer. The term ‘learning sample’ means a pair ‘calculated synthetic EM data or their transforms at a number of periods – corresponding set of model parameters’. Such input–output pairs are deﬁned by the ‘teacher’ and compose the ANN training sequence. The total error to be minimized is !1=2 1X Erp ð9:5Þ Er ¼ P p where the summation is performed over all P learning samples. The connection weights wik and wkj are the parameters that determine the signal propagation through the network and, therefore, the ﬁnal error. BP is actually a gradient descent technique minimizing the error Er by means of adjusting the connection weights DwðnÞ ij ¼ a

@ Er @ wij

ð9:6Þ

where Dwij(n) is the increment of the weight matrix at the nth step of the iteration process and a a nonnegative convergence parameter called learning rate. In order to accelerate the process an inertial term proportional to the weight change at the previous step (n1) is often added to the right-hand side of (9.6) DwðnÞ ij ¼ a

@ Er þ bDwijðn1Þ @ wij

ð9:7Þ

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223

where b (0pbp1) is the inertial coefﬁcient called ‘learning momentum’. The momentum can speed up training in very ﬂat regions of the error surface and supresses the weight oscillations in steep ‘valleys’ or ‘ravins’ (Schiffman et al., 1992). Learning starts from small random values of the weights. The input signal comes via network to the output. The output signal of the output layer is compared then with the desired value and the misﬁt is calculated. If it exceeds predetermined small number, the signal propagates back through the network to the input, and so on. This procedure is fulﬁlled for the whole learning pool and ends upon reaching a user-speciﬁed threshold value Eps (EroEps) named further ‘a teaching precision’. The testing process uses the ANN interpolation and extrapolation properties. Unlike the training procedure requiring many steps of back and forth throughnetwork movements of the signal, the recognition one requires only one passage of the recognizable signal from input to output layer and uses the connection weights speciﬁed at the learning stage. The ﬁnal set of output values may be treated as a result of the testing data inversion in a given model class.

9.2. CREATION OF TEACHING AND TESTING DATA POOLS So that the ANN learns the correspondence between data and desired geoelectric parameters, it is ﬁrst necessary to formulate the hypothesis on the class of inversion models (for instance, dike, geothermal reservoir, magma chamber, oil or gas deposit, etc.). Note that we mean only the assumption on the class of models for which the solution is sought, rather than considerably more stringent constraints on the parameters of 1-D layering and/or target geometry used in the applications of other inversion methods. This may be difﬁcult in the general case, if we have no initial guess about the type of the geoelectric model to be searched for, but quite possible in some practically important cases. The recognition of the crustal dikes from the surface measurements of the electromagnetic ﬁeld is an example for such a formulation of the problem. It is easily parameterized, and the inversion is reduced to the determination of a few macro-parameters of the target itself as well as of the host medium. To apply the ANN method, it is necessary to create ﬁrst a fairly representative database (consisting from either synthetic or real data) for it’s training. Let us consider, for example, a model of 3-D dipping dike in the basement of a two-layered earth with the dike in contact with the overburden (Fig. 9.2), used in Spichak and Popova (1998,2000) as a ‘class-generating’ one. It is characterized by the following parameters of the dike and the host medium: thickness of the upper layer (H1); the conductivity contrast between two layers (C1/C2); the conductivity contrast between the dike and the hosting layer (C/C2); and the depth of the upper edge of the dike (D), it’s width (W), length (L) and the dip angle in the plane xOz (A). It was supposed for simplicity that the upper boundary of the dike always lies at the interface between the ﬁrst and the second layers, so that D ¼ H1, and that the conductivity of the second layer is ﬁxed: C2 ¼ 0.01 S/m. Thus, six parameters of 3-D geoelectric structure (D (H1), W, L, A, C1/C2 and C/C2) were to be reconstructed.

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Fig. 9.2. Cross-section of a three-dimensional model, containing the dike buried in a second layer of a two-layered earth.

The following sampling of unknown parameters was actually used in forward modeling: D (H1) ¼ 50, 200 m; C1 ¼ 0.00333; 0.01, and 0.03 S/m; C2 ¼ 0.01 S/m ( ¼ ﬁxe); C ¼ 0.0002; 0.001; 0.003333; 0.01; 0.02; 0.034; 0.06; 0.1; 0.17; 0.3 and 0.5 S/m; W ¼ 16.65; 25; 50; 66.6; 100 and 200 m; L ¼ 16.65; 25; 50; 66.6; 83.25; 100; 125; 200; 250; 330; 500 and 1000 m; A ¼ 01 (1801), 451, 661, 901, 1141 and 1351. Note that due to computation time restrictions not all possible combinations of parameters’ values mentioned above were used for creation of the synthetic database. In particular, the total amount of calculation was narrowed by conditions like D/W ¼ 1, 2, 3; L/W ¼ 1, 5; ‘‘basic’’ values of the conductivity contrast C/C2 ¼ 2, 10, 50; and so on. In order to create a synthetic database the software package FDM3D (Spichak, 1983) was used, which has proven to be efﬁcient in the solution of forward and inverse 3-D magnetotelluric problems (Spichak, 1999). All calculations were carried out for two primary ﬁeld polarizations within the period range typical for audiomagnetotellurics: T ¼ 0.000333; 0.001; 0.00333; 0.01; 0.0333 and 0.1 s. Note that due to a well-known electrodynamic similitude relation (Stratton, 1941) o s L2 ¼ inv,

ð9:8Þ

where o is the frequency, s the electrical conductivity and L the geometrical scale, the same synthetic data base could be used for ANN inversion also in other period range and geometrical parameters’ scale satisfying (9.8).

9.3. EFFECT OF THE EM DATA TRANSFORMATIONS ON THE QUALITY OF THE PARAMETERS’ RECOGNITION Since ANN architecture is of great importance for the recognition of the model parameters, a comprehensive study was carried out in Spichak and Popova (2000) aimed at ﬁnding the appropriate values of the following parameters of the ANN: types of activation function for hidden and output layers as well as for neurons at the output layer, number of neurons in a hidden layer, the effect of a second hidden layer. Finally, a teaching precision was estimated that enabled to get reasonable inversion results.

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225

In order to save the computation time (without loss of generality) the data base used in these experiments were narrowed in comparison with the total one: only 90 synthetic data sets randomly selected from the total data base were used for teaching, while 10 randomly selected for testing. The ANN architecture in this experiment was as follows: the input layer consisted from 80 neurons, hidden layer consisted from 20 neurons, while the output one–from neurons corresponding to 6 model parameters that are to be recognized. The threshold level (Eps) for rms errors in teaching was equal to 0.0075. The learning rate was equal to 0.01 and momentum–to 0.9. In the process of teaching the rms errors were used to estimate the misﬁts between the calculated and ‘‘true’’ responses, so the total error for the test set for all parameters was determined as follows: " #1=2 X 1 Err ¼ err2n;j ð9:9Þ N test N par n;j where err(n, j) ¼ [target (n, j)neural (n, j)]/[max (j)min (j)], (j ¼ 1,y, Npar; n ¼ 1,yNtest); j ¼ number of the neurons in the output layer corresponding to the jth model parameter; n ¼ number of the tested sample; Npar ¼ number of the output neurons ( ¼ 6); Ntest ¼ number of the testing data sets; min (j), max(j) ¼ minimum and maximum values of jth parameter in the teaching pool; neural (n, j) ¼ the recognition result for jth parameter in nth testing sample; target (n, j) ¼ target value of jth parameter in nth testing sample. In order to estimate the quality of the ANN inversion of the synthetic MT data (when the true result is known in advance) the relative error averaged over all testing samples was calculated for each jth unknown parameter 1 X targetj;n neuralj;n 100% ð9:10Þ Errj ¼ targetj;n N test n

9.3.1. Types of the activation function at hidden and output layers Since the type of activation function used is crucially important for a proper simulation of the behavior of a real system, some experiments are to be conducted before using ANN for interpretation of the real data. In spite of any monotonically increasing and continuously differentiable function may be used as an activation function for BP type networks, the most commonly used ones are sigmoidal functions: hyperbolic tangent 1 1 ez ez 1 ¼ ð1 þ tanhðzÞÞ ¼ 1 þ GðzÞ ¼ ez þ ez 2 2 1 þ e2z

226

V.V. Spichak/Neural Network Reconstruction of Macro-Parameters

and ‘‘logistic’’ function G(z) ¼ (1/1+ez). Their derivatives G0 (z) ¼ G(z)(1G(z)) have a Gaussian shape that helps stabilize the network and compensate for over correction of the weights (Caudill, 1988). ANN interpolates the parameters of the model using these activation functions quite satisfactorily, but it completely failes to extrapolate their values because the neural output for logistic and hyperbolic tangent functions lies in the interval [0,1]. In this situation no values apart from it are achievable, so no real extrapolation could be carried out. Probably, the simplest way to overcome this difﬁculty consists of using the linear activation function at the output layer. Comparative testing of neural networks with linear and hyperbolic tangent activation functions revealed that network with nonlinear outputs reasonably extrapolated low values of the conductivities, but failed in extrapolation of high conductivities and, vice versa; network with linear outputs reasonably extrapolated high values of the target conductivity but failed in extrapolation of its low values. Basing on this preliminary experience, the effects of the following two types of the activation function for the neurons at the output layer were compared Glin ðxÞ ¼ 0:5ð1 þ xÞ ðlinear functionÞ ð1 þ tanhðxÞ; xo0 mix G ðxÞ ¼ 0:5 ðmixed functionÞ ð1 þ xÞ; x40

ð9:11Þ ð9:12Þ

The ANN used for experiments had the same type of activation function at each neuron of the hidden or output layer, so, in order to estimate effects of different activation functions it was necessary to teach independently six ANNs each having only one output neuron, corresponding to the appropriate model parameter. All neurons at the hidden layers had the same mixed activation function, while each (single!) output neuron of appropriate ANN had activation function depending on the nature of corresponding model parameter: linear (9.11) or mixed (9.12) for dimensional parameters (D, W, L) of the model and only mixed for the conductivity contrasts (C/C2, C1/C2) and dip angle (A). Table 9.1 demonstrates the recognition results for two types of the activation function used and two ways of teaching/testing mentioned above: (1) six ANNs (three of them having ‘‘linear’’ and others – ‘‘mixed’’ activation function); (2) six ANNs (all ‘‘mixed’’) and (3) one ANN (‘‘mixed’’). (Here and after average relative errors and their bars (in %) are placed in the round brackets through the semicolon for all model parameters). Table 9.1. The results (in terms of averaged relative errors and bars (both in %)) of the model parameters recognition for two types of activation function and two ways of testing.

N

D

C1/C2

W

L

A

C/C2

1 2 3

(5.5; 16.3) (6.7; 14.6) (4.9; 16.1)

(11.8; 16.8) (11.8; 16.8) (9.0; 15.4)

(5.5; 16.3) (6.7; 14.6) (4.9; 16.1)

(4.8; 14.5) (5.4; 13.0) (4.3; 14.4)

(7.3; 1.1) (7.3; 12.1) (7.1; 7.9)

(23.6; 24.7) (23.6; 24.7) (33.2; 35.3)

V.V. Spichak/Neural Network Reconstruction of Macro-Parameters

227

It could be concluded from the comparison of ﬁrst two rows of Table 9.1 that under the condition that the activation function at output layer has a linear part the recognition errors for all parameters are reasonable and practically do not depend on the type of the activation function used. The errors even insigniﬁcantly decrease (though, less than at 2.8% except C/C2) if all model parameters are reconstructed by the same ANN teached to be able to recognize all model parameters (third row). It is important to note, however, that in spite of the total time of teaching in the latter case is much less than in the former one, the best recognition accuracy for the dike conductivity contrast C/C2 is achieved if this parameter is recognized independently (in a ‘‘partial solution’’ mode) by ANN having only one output neuron and was taught in an appropriate way. In the latter case the relative error may decrease at 10% (from 33.2% to 23.6%). 9.3.2. Number of the neurons in a hidden layer Unfortunately, there is no general theory on dependence of the recognition errors on the number of neurons at a hidden layer. Meanwhile, the approximation properties of ANN are improved when the number of hidden neurons increases. In particular, Yoshifusa (1991) has proved that nonlinear perceptron with one hidden layer can approximate any continuous function with given precision if the number of hidden neurons tends to inﬁnity. The recognition ability of ANN increases when the number of hidden neurons increases if it deals with familiar data (in particular, those used for training). On the other hand, it may even decrease if ANN deals with unknown testing data because in the general case it’s recognition ability (called ‘‘generalization property’’) depends in a complex way on it’s architecture (number of hidden layers, number of neurons, type of activation function, etc.), volume and structure of the training data pool, etc. Therefore, the optimal number of hidden layers and hidden neurons are usually found by trial-and-error technique (Baum and Haussler, 1989; Kung and Hwang, 1988; Soulie et al., 1987). Spichak and Popova (2000) studied the effect of the number of neurons in a hidden layer on the accuracy of the model parameters’ recognition by means of testing the same data sets as were used in the previous section. The ANN architecture was 80-Nh-6, where Nh is a number of neurons in a hidden layer. The values of Nh were assigned to be as follows: 10, 20, 30, 40, 50. The teaching precision was equal to 0.0075. Fig. 9.3 shows the dependence of the accuracy of the model parameters’ recognition (in terms of relative errors averaged over all testing data sets and appropriate bars (both in %)) on the number of neurons in a hidden layer. Here and after (if otherwise is not mentioned): (a) depth of the dike and thickness of the upper layer (D (H1)), (b) conductivity contrast of the upper layer (C1/C2), (c) width (W) of the dike, (d) conductivity contrast of the dike (C/C2), (e) length (L), (f) dip angle (A). It is seen from the Fig. 9.3 that the relative errors for four parameters (D, W, L and C1/C2) are generally less than 3–4%, while the maximum relative errors for

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Fig. 9.3. The dependence of the recognition errors on the number of neurons in a hidden layer (Nh).

C/C2 and A is around 14%. The total average error for all six parameters ranges from 3.9% to 5.5%. In spite of the recognition errors are not very sensitive to the number of neurons in a hidden layer, the minimal total error is achieved at Nh ¼ 40. It is worth to note that for all numbers of Nh the standard deviations of the relative errors for W, L and C1/C2 were small enough (o5%), while they reached 15% for C/C2 and A. This result indicates that the reconstruction of the dike’s depth, conductivity contrast and dip angle is less stable procedure than the recognition of other unknown parameters.

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229

9.3.3. Effect of an extra hidden layer In order to estimate whether two hidden layers are better than one, a second hidden layer was incorporated into the ANN. Basing on the results of the previous experiments the number of the neurons in a ﬁrst hidden layer was ﬁxed at Nh ¼ 40, while the number of neurons in a second one varied. So, a new ANN architecture was 80 – 40 – Nh (2) – 6, where Nh (2) was successively equal to 10, 20, 30 and 60. Fig. 9.4 shows the results of recognition of model parameters in comparison with the case when ANN consists of only one hidden layer with optimal number of neurons (Nh (1) ¼ 40, Nh (2) ¼ 0). The total average error is minimal (4.3%) when Nh (2) ¼ 0 and is maximal if Nh (2) ¼ 60 (mean value equals to 7.7%). The behavior of all relative errors’ graphs do not change signiﬁcantly in comparison with the case of only one hidden layer (Fig. 9.4). Thus, it can be concluded that the architecture of the ANN is quite adequate to the complexity of the problem considered, so, addition of the second hidden layer effects like simple increasing of the total number of neurons in intermediate layers, the total number Nh ¼ 40 in all hidden layers being the optimal value. 9.3.4. Threshold level To select the optimal value of the teaching precision (Eps) the parameters of the model were reconstructed using ANNs having the same architecture, but was taught using different values of stopping criterion Eps (0.005; 0.0075; 0.01; 0.02; 0.05). Fig. 9.5 shows the dependence of the relative recognition errors and their bars (in %) on the threshold value Eps (in horizontal axis a Log10 Eps is put in inverse order). It is seen that the average errors and bars for all model parameters (besides the dip angle of the dike) sharply decreases, when Eps tends to zero and stabilize when it becomes less than 0.01.

9.4. EFFECT OF THE INPUT DATA TYPE The results of ANN parameters recognition depend on the data type used (both for teaching and inversion itself), the volume and structure of the teaching data pool, etc. In order to estimate the effect of data used on the results of inversion the following ﬁle types of input data were studied: (1) Normalized components of electric and magnetic ﬁelds jE y;x j jE ny;x j jH x;y j jH nx;y j ; jE ny;x j jH nx;y j (2) (3) (4) (5)

Components of not normalized electrical ﬁeld parallel to the polarization of the primary ﬁeld (Re E x;y , Im E x;y ); Only apparent resistivities raxy , rayx ; Apparent resistivities raxy , rayx and phases jxy ; jyx of impedance; and Only phases jxy ; jyx of impedance.

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Fig. 9.4. The dependence of the recognition errors on the number of neurons in a second hidden layer (Nh(2)).

Table 9.2 shows the recognition results for ﬁve groups enumerated above. It is seen that the recognition of the dimensional parameters (D, W and L) is carried out with errors less than 0.5% independently on the type of input data used. The dip angle (A) is determined fairly well (the maximum error 4.14% corresponds to using only apparent resistivities and the minimum – to the case, when only impedance phases are used). The errors of the conductivity contrast (C1/C2) estimation are bigger and range from 3.28% to 8.43%. They are maximal (in contrast to the

V.V. Spichak/Neural Network Reconstruction of Macro-Parameters

231

Fig. 9.5. The dependence of the recognition errors on the rms error achieved in the teaching process (Eps).

previous case), when only impedance phases are used and minimal when only apparent resistivities are used. Finally, the dike conductivity contrast (C/C2) recognition errors are the biggest and vary from 24,45% (apparent resistivities and phases) upto 67.71% (only electric ﬁelds). In average, the groups 4 (apparent resistivities and phases) and 5 (only phases) give the best recognition results for all searched dike parameters.

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Table 9.2. Relative errors (in %) of parameters’ recognition for ﬁve types of input data.

N

D

C1/C2

W

L

A

C/C2

1 2 3 4 5

0.10 0.13 0.04 0.27 0.02

3.84 7.20 3.28 4.14 8.43

0.10 0.15 0.05 0.40 0.02

0.13 0.17 0.04 0.43 0.02

0.41 1.77 4.14 1.63 0.24

40.77 67.71 39.05 24.45 26.81

Thus, we conclude that, ﬁrst, the conductivity contrasts are determined worse than other model parameters; second, the recognition errors of the dimensional parameters (D, L and W) practically do not depend on the data type, while the conductivity contrasts (especially, of the dike) essentially depend; and, third, the phases of impedance could be used to determine not only the geometrical parameters, but also the conductivity distribution. The latter conclusion enables to reduce (at least, twice in comparison with routinely used two components of apparent resistivity and two components of the impedance phase) the volume of data required for magnetotelluric data inversion in 3-D media.

9.5. EFFECT OF THE VOLUME AND STRUCTURE OF THE TRAINING DATA POOL 9.5.1. Effect of size The dependence of the accuracy of the model parameters recognition on the number of data sets in the teaching database was studied in Spichak and Popova (2000). Totally, 120 data sets were used in these experiments: 12 of them were used for testing, while the rest 108 samples – for teaching. Synthetic electric and magnetic ﬁelds were, in accordance with the results of the previous section, ﬁrst, transformed to the apparent resistivities and phases. In order to reduce a huge volume of data being processed during teaching, the data were compressed then by 2-D Fast Fourier Transformation (FFT) (with the use of ﬁrst 5 pairs of sine and cosine coefﬁcients). Thus, the whole number of input neurons in the network was N ¼ 10NtrNT, where Ntr is the number of ﬁeld components or transformations used for the inversion, and NT is the number of periods. 2-D FFT of the data was carried out on the grid 32 32 for each period. The testing data sets were not changed during this experiment, while the data sets used for teaching were selected from the rest of the data sets in a random way so as their number was successively equal to 54 (50% of the total data base), 65 (60%), 76 (70%), 86 (80%), 92 (85%), 97 (90%), and 103 (95%). The results of the parameters’ recognition are shown in Fig. 9.6. It is well seen that the average errors for all parameters decrease with increasing the number of data sets used for teaching and stabilize (bars for almost all parameter errors decrease) when the volume of the training database becomes about 90–100. Thus, increase of the teaching data base volume up to 90–100 data sets may improve both the accuracy and robustness of the recognition of all the

V.V. Spichak/Neural Network Reconstruction of Macro-Parameters

233

Fig. 9.6. The dependence of the recognition errors on the number of synthetic data sets (N) used for teaching ANN.

model parameters. The latter circumstance is very important from the point of view of interpretation of a single data set, which is usually the case in practice. 9.5.2. Effect of structure It could have been expected that the larger the volume of the training data base better the results of unknown parameters recognition; however, the effect of the

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Table 9.3. The model parameter’s recognition errors and bars for 5 randomly selected teaching data pools.

N

D

1 2 3 4 5

(2.2; (4.6; (4.5; (4.9; (4.9;

1.3) 4.6) 6.8) 5.6) 5.6)

C1/C2

W

(2.6; (3.4; (5.4; (2.8; (2.6;

(1.0; (1.7; (2.5; (1.4; (2.0;

2.7) 2.7) 6.2) 2.3) 3.1)

L 0.7) 2.1) 2.3) 1.6) 3.9)

(1.3; (1.3; (2.3; (1.1; (1.4;

A 0.8) 1.5) 1.9) 1.2) 2.0)

(3.3; (4.9; (5.5; (5.7; (3.7;

C/C2 2.5) 3.3) 4.6) 8.3) 2.3)

(7.1; 7.3) (8.8; 9.8) (10.3; 11.6) (11.3; 10.7) (12.0; 8.7)

data base structure is much less clear. This item was studied in Spichak and Popova (2000) in two different ways: ﬁrst, the effect of the random selection of the training data sets from the data base was estimated; second, the inﬂuence of the ‘‘gaps’’ in the database used for teaching was studied. In both cases the testing data set was not changed and consisted of the same 12 data sets as in the previous section. 9.5.2.1. Random selection of synthetic data samples In order to estimate the effect of a random selection of the training data sets the teaching was carried out using 97 data sets randomly selected from the database. The architecture of the ANN was the same as above. The results of the model parameters’ recognition for ﬁve different random formations of the training data sets are represented in Table 9.3. Table 9.3 indicates that for each model parameter both the errors and their bars are quite reasonable (the errors of the dike width (W) and length (L) recognition being the smallest, while the error of the dike conductivity contrast (C/C2) recognition being the biggest) and do not vary depending on the random selection of the data sets in the training data base. Thus, a random selection of the data sets used for training practically does not effect the results of the recognition. This implicitly indicates that the data base consisted from even 97 data sets is of good enough quality for teaching the ANN to recognize the model parameters. 9.5.2.2. Gaps in the training data base It is important to estimate the effect of gaps in the data base on the recognition errors. In order to study this item an artiﬁcial damage (decrease of the number of teaching values) of the training data base was made separately for each model parameter. In order to estimate the effects of the gaps in the teaching data pool on the recognition abilities of the ANN, the following experiments for four characteristic model parameters (A, D, C1/C2, C/C2) were carried out. First, the removal of an appropriate group of the data sets (consisted from all data sets with one of the parameter values) from the teaching data was made. Then, two testing data pools were created: one (unchanged in all experiments) consisted from 12 reference data sets randomly selected from the total database and second one consisted from those data sets that were deleted from the teaching data, the latter being corresponded to

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Table 9.4. Effect of the gaps in the training data sets on the results of testing 12 reference data sets (in the case of linear output activation function).

Gaps in

D

C1/C2

W

L

A

C/C2

A C1/C2 D C /C2 No gaps

(70; 125) (23; 61) (136; 161) (5.1; 6.1) (3.2; 8.2)

(10; 17) (36; 52) (33; 65) (3.6; 3.4) (4.3; 5.1)

(44; 106) (2.4; 3.2) (88; 131) (4.9; 10) (1.9; 2.1)

(30; 71) (1.6; 1.4) (58; 86) (4.6; 9.5) (1.9; 2.3)

(17; 25) (10; 19) (16; 23) (7.8; 9.1) (4.1; 3.2)

(56; 87) (179; 388) (33; 39) (23; 23) (23; 17)

Table 9.5. Effect of the gaps in the training data sets on the results of testing 12 reference data sets (in the case of mixed output activation function).

Gaps in

D

C1/C2

W

L

A

C/C2

A C1/C2 D C/C2 No gaps

(67; 149) (19; 26) (149; 197) (8; 16) (8.7; 11.4)

(9; 13) (67; 98) (16; 24) (4; 5) (5.9; 6.0)

(48; 127) (8; 12) (84; 149) (6; 15) (2.2; 3.2)

(34; 87) (8; 13) (58; 100) (6; 16) (2.1; 2.9)

(14; 21) (12; 20) (18; 29) (8; 16) (4.7; 5.3)

(29; (37; (32; (17; (23;

32) 35) 32) 14) 31)

the following values of the model parameters: A ¼ 66, C1/C2 ¼ 0.333, D ¼ 50 m, C/C2 ¼ 3.4 and 6. The architecture of ANN was the same as in the previous sections except the type of the activation function at the output layer. Tables 9.4 and 9.5 represent the results of the model parameters’ recognition for two types of the output activation function – linear and mixed, correspondingly – when the testing data pool consists from the 12 reference data sets mentioned above. The last rows in both tables contain for comparison the recognition errors for the case when no artiﬁcial gaps are created in the teaching data pool. Similarly, the Tables 9.6 and 9.7 represent the results of the model parameters’ recognition for two types of the output activation function – linear and mixed, correspondingly – when the testing data pool consists of data sets removed from the teaching data pool. Analysis of Tables 9.4–9.7 results in the following conclusions: 1. In the case of using ‘‘damaged’’ training data pool recognition errors depend on the type of the output activation function (on the contrary to the case of ‘‘homogeneous’’ training data (Table 9.1). The mixed function is more preferable to determine the conductivities and dip angle, while the linear function gives better results for dimensional parameters. It enables to diminish the recognition errors of C/C2 even if the number of grading of C1/C2 and A in the training data base is insufﬁcient. 2. Lack of training data sets in C1/C2 turns to be more crucial for the recognition of C/C2 than the lack of values of C/C2 themselves. 3. Lack of training data sets in dip angle A effects the recognition error of D even more than the recognition error of A itself.

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Table 9.6. Effect of the gaps in the training data sets on the results of testing deleted data sets (in the case of linear output activation function).

Gaps in

D

C1/C2

W

L

A

A C1/C2 D C/C2

(127; 153) (22; 40) (261; 145) (35; 74)

(6; 10) (104; 16) (69; 80) (12; 34)

(117; 152) (6; 8) (219; 102) (32; 37)

(80; 103) (7; 7) (152; 63) (30; 34)

(41; (32; (45; (28;

C/C2 32) 31) 43) 46)

(58; 60) (297; 492) (80; 66) (59; 53)

Table 9.7. Effect of the gaps in the training data sets on the results of testing deleted data sets (in the case of mixed output activation function).

Gaps in

D

C1

W

L

A

A C1/C2 D C/C2

(178; 246) (56; 31) (320; 202) (31; 31)

(9; 13) (200; 0.3) (28; 19) (6.0; 7.8)

(142; 181) (19; 20) (221; 129) (36; 28)

(100; 122) (15; 18) (155; 82) (34; 29)

(34; (21; (46; (32;

C 24) 22) 35) 24)

(29; (89; (66; (27;

25) 72) 19) 25)

The biggest recognition errors of all model parameters occur if the number of data sets are insufﬁcient in D. On the other hand, the smallest errors occur when the number of data sets corresponding to C/C2 is close to other parameters’ grading. Thus, it could be concluded that in order to get good results of ANN recognition the numbers of the intervals (grading) in each parameter in the training data pool should be close to each other. Since it is often difﬁcult to follow this recommendation in practice, it is, generally, better to use different types of the activation function for two groups of parameters: mixed for essentially ‘‘non-linear’’ parameters (C/C2, C1/C2 and A) and linear – for dimensional ones (D, W and L). 4.

9.5.2.3. ‘‘No target’’ case It is important to estimate also the ability of the ANN to recognize the situations when no anomalous body is embedded into the layered earth. To this end the dependence of the ‘‘no dike’’ recognition on the number of corresponding synthetic data sets in the training data pool was studied. Twelve data sets corresponding to the ‘‘no dike’’ case were randomly selected from the total data base and then successively shared by teaching and testing data pools as follows: 4/8, 6/6, 8/4 and 10/2. Thus, the teaching data pool included consequently 4, 6, 8 and 10 data sets corresponding to the absence of the dike, while the testing data consisted only from the ‘‘no dike’’ data sets (8, 6, 4 and 2, accordingly). The ANN architecture was the same as in the previous sections. The model parameters recognition errors and their bars depending on the number N of the ‘‘no dike’’ data sets in the training data pool are represented in Fig. 9.7((a) – D (H1), (b) – C1/C2, (c) – C/C2, (d) – total average relative error).

V.V. Spichak/Neural Network Reconstruction of Macro-Parameters

237

Fig. 9.7. The dependence of the recognition errors on the number of synthetic data sets, corresponding to ‘‘no fault’’ models (N), in the teaching data pool: (a) – D (H1), (b) – C1/C2, (c) – C/C2, (d) – total average relative error.

Fig. 9.7 shows that the average errors and bars for the model parameters sharply decrease if the number of the ‘‘no dike’’ data sets in the teaching data pool increases. The recognition errors of the upper-layer parameters (C1/C2 and H1) manifestate similar behavior: they become quite reasonable if the number of ‘‘no dike’’ data sets included to the teaching data pool is at least 8 (6,7% of the total number of the teaching data sets used). On the other hand, even if the number of ‘‘no dike’’ data sets in the training data pool further increases the relative error of the conductivity contrast C/C2 recognition remains at the level of 50% (Fig. 9.7(c)), which is evidently not sufﬁcient for classiﬁcation of such inversion results as ‘‘absence of the dike’’. This situation could be probably improved due to further increasing of the ratio of appropriate data sets in total data pool used for teaching ANN. Fortunately, in all experiments (see Tables 9.8 and 9.9 for test examples) the resulting values of the dike width (W) and length (L) were close to zero, so, even if other parameters (in particular, the dike conductivity contrast C/C2) do not indicate the ‘‘absence of the dike’’, these two conditions could be used as reliable indicators of such a situation.

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Table 9.8. The recognition results for the ‘‘no dike’’ case when the host medium is homogeneous (C1/ C2 ¼ 1).

Target Result Err(%)

D (m)

C1/C2

W (m)

L (m)

A (deg)

C/C2

0.0 0.0 0.0

1.00 1.98 1.7

0.0 0.0 0.0

0.0 0.0 0.0

0.00 3.21 0.0

1.00 0.59 40.2

Table 9.9. The recognition results for the ‘‘no dike’’ case when the host consists of two layers (C1/C2 ¼ 3).

Target Result Err(%)

D (m)

C1/C2

W (m)

L (m)

A (deg)

C/C2

50 40 4.2

3.00 3.00 0.1

0.0 0.0 0.0

0.0 0.0 0.0

0.00 0.53 0.0

1.00 1.53 53.9

9.6. EXTRAPOLATION ABILITY OF ANN As it was mentioned above, the most important abilities of ANN consist of interpolation and extrapolation of the parameters’ values. Numerous testing carried out in the previous sections demonstrated good interpolation ability of ANN. Special tests were made in order to estimate it’s extrapolation properties with respect to the dike’s depth (D), dip angle (A), length (L) and the conductivity contrast (C/C2). Appropriate model parameters values to be recognized due to extrapolation were preliminarily removed, if existed, from the teaching data pool. If necessary, all data sets corresponding to smaller or larger values of the parameters considered were also removed. Based on the results obtained in Section 9.3, the recognition (inversion) was carried out for xy and yx components of the apparent resistivities and the impedance phases. Tables 9.10C9.12 indicate the extrapolation abilities of the ANN with respect to the dike depth (D) (Table 9.10), dip angle (A) (Table 9.11) and conductivity contrast (C/C2) (Table 9.12). Tables 9.10C9.11 demonstrate good enough results of extrapolation in D and A. The recognition errors for these parameters equal to 22.8% and 19.7%, correspondingly, each of them being underestimated. However, it is worth mentioning that extrapolation in D results in a rather big error in C/C2 (51%), that could be explained by relatively rare grading in D and, vice versa, dense grading in C/C2 in the teaching data pool (see sub-section above). Table 9.12 shows the results of extrapolation in conductivity contrast of the dike (C/C2). The recognition of the conductivity contrast as well as of other parameters in this test is quite reasonable. Similar test of the extrapolation abilities of ANN in C/C2 (C/C2 ¼ 50, while values of other parameters vary) shows that the results are

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239

Table 9.10. The recognition errors for extrapolation in D.

Target Result Err(%)

D (m)

C1/C2

W (m)

L (m)

A (deg)

C/C2

300 230 22.8

1.00 1.00 0.0

200 220 10.9

200 170 12.4

66.00 67.34 2.0

10 4.9 51

Table 9.11. The recognition errors for extrapolation in A.

Target Result Err(%)

D (m)

C1/C2

W (m)

L (m)

A (deg)

C/C2

200 190 2.3

0.333 0.334 0.35

200 190 2.3

150 140 2.1

135.0 10.4 19.7

50 46.7 6.6

Table 9.12. The recognition errors for extrapolation in C/C2.

Target Result Err(%)

D (m)

C1/C2

W (m)

L (m)

A (deg)

C/C2

50 50.1 0.2

3.0 2.82 6.0

50 50.1 0.2

50 50.1 0.2

66.0 59.9 9.2

50.0 37.1 25.9

very little affected by the location of the point selected for extrapolation regarding the parameter space used for teaching. Thus, we can conclude that an ANN created possesses good interpolation and extrapolation properties.

9.7. NOISE TREATMENT The situation dramatically changes, if some artiﬁcial noise is added to the synthetic testing data. Fig. 9.8 shows the extrapolation recognition results when synthetic data are mixed with 30%, 50% and 100% Gaussian noise. In spite of the recognition errors for all parameters averaged over all testing samples generally increase when the noise level in the data increases, in some cases their behavior is quite unusual. In particular, the recognition errors for dip angle and the dike conductivity contrast often diminish, when the noise level increases, which results in ‘‘plateaus’’ in averaged errors graphs well seen in Fig. 9.8. Nonmonotonic increase of the recognition errors when the level of noise increases up to 100%, indicates, that the routine methods of the data-noise reduction may not necessarily lead to an improvement of the ANN inversion results, so development of a special noise treatment methodology is required. Spichak and Popova (1998) found that if the testing and teaching data had the same noise level the recognition errors were minimal. However, since it is often

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Fig. 9.8. The effect of the noise level (Noise) in the testing data on the recognition errors for the case of extrapolation in C/C2 (see also Table 9.12).

difﬁcult to estimate properly the noise level in the data and mix training data with equivalent synthetic noise, application of the ANN inversion methodology to the real data may lead to erroneous results. In order to overcome this difﬁculty a special study of the noise effect on the results of the model parameters’ recognition was carried out (Spichak, 1999).

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241

Fig. 9.9. Relative recognition errors for all model parameters depending on the level of Gaussian noise added to the testing data. Numbers (0%, 10%, 20% and 0%+10%+20%) mean the ‘‘noise spectrum’’ in the teaching data.

Fig. 9.9 demonstrates graphs of the recognition errors for all unknown parameters of the dike model (see Fig. 9.2) depending on the level of Gaussian noise added to the testing data. It is seen that minima for all curves are attributed to the noise levels in testing data equal to those in the teaching data. However, since the level of noise in the real data is not known in advance, it is important to know how to mix the teaching data with artiﬁcial noise. Many experiments were carried out in (Spichak et al., 1999b) aimed at ﬁnding the best way of the teaching data pre-processing. It was revealed, in particular, that teaching the

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ANN by synthetic data mixed with a noise of different levels ranging from 0% (no noise) to some level, which exceeds the maximal one in the data, enables to diminish greatly the recognition errors and results in robust parameters’ recognition even if the noise levels in testing and teaching data are different. In other words, prior knowledge about the noise level in the data becomes not necessary. This is demonstrated by the graphs in Fig. 9.9 marked as (0%+10%+20%). In this case the recognition errors for all model parameters do not exceed 5–10% even if the level of noise in the testing data reaches 50%. Thus, ANN paradigm enables inversion of insufficient and very noisy electromagnetic data, if the data correspond to the model class familiar to the ANN. On the contrary to other inversion techniques, no initial guess regarding the parameters of 1-D layering or the geometry of the model is required. It is worth to note that in spite of the teaching process based on forward modeling is a common feature of practically all known inversion methods, ANN approach to inversion differs from others by the fact that here the teaching and recognition/inversion procedures are separated in time, that, in turn, enables practically instantaneous inversion of real data even using micro-computers. Moreover, the ability of the ANN taught once to recognize parameters in the same model class enables to diminish the effective cost of one inversion as much as the number of data inversion using the same ANN is carried out. Unfortunately, the recognition abilities of ANN are restricted by it’s ‘‘education’’ level: the more models (classes) of different geological situations are familiar to it the better will be the inversion results. Meanwhile, the same database could be used for teaching in the case when another frequency range and geometrical parameters’ scale satisfy electrodynamic similitude relation (9.8).

9.8. CASE HISTORY: ANN RECONSTRUCTION OF THE MINOU FAULT PARAMETERS This study was aimed at application of the ANN Expert System MT-NET, developed and trained earlier (Spichak and Popova, 2000), to 3-D interpretation of scalar controlled source audio-magnetotelluric (CSAMT) data from a northern part of the Minou fault zone, Kyushu Island, Japan (Spichak et al., 2002). 9.8.1. Geological and geophysical setting The Minou fracture zone is an active fault system and runs from the East to the West, North Kyushu, Southwest Japan (Chida, 1981). The Minou fault is one of the components of the Imari-Matsuyama Line (IML) situated at the boundary between the northern and central parts of Kyushu Island and is considered as a source of the great earthquake that occurred in 679 A.D. (Matsumura, 1990). The fault runs on the northern foot of the Minou mountain range which is composed mainly from metamorphic rocks and located at the boundary between the mountain area and an alluvium plain extending along Chikugo river (Fig. 9.10).

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243

Fig. 9.10. Geological map of the Minou fault zone. The rectangle restricts the CSAMT survey area. IML—Imari-Matsuyama Line.

A prominent low gravity anomaly is recognized at the northern side of the fault beneath the plain (Mogi et al., 1997). The anomaly extends to the East along the IML (Komazawa and Kamata, 1985). The depth to the metamorphic formation is decreased abruptly on the northern side of the fault and the maximum depth of it was estimated to be approximately 1.5 km (Mogi et al., 1997). This structure could be considered as a graben buried in a thick Quaternary sediment. The gravity and resistivity data revealed that the steep southern slope of the graben dipping northward is located 0.5–1.0 km to the north from the active fault line marked by the topographic discontinuity in the western part of the Minou fault zone (Mogi et al., 1997). At the eastern part of the area an outcrop of the metamorphic rocks is observed, though the granite-metamorphic rock boundary is not clear. No marked displacement of the basement was detected by the gravity and resistivity data measured above the presently active fault. It seemed to have been active at the southern wall of the graben before alluvium deposition. The active part ‘‘moved’’ then to the south and a new activity has been started at the topographic discontinuity. This fault displacement suggests that extensional movement has been prevailing in this area during the Quaternary period. Thus, the geology of the Minou fault zone is rather complicated and has to be studied by different methods. In particular, in order to determine the geoelectric structure beneath the northern part of the Minou area a scalar CSAMT survey was carried out. Another aim of the measurements was to examine the applicability of the CSAMT survey to the resolution of an inclined fault zone. 9.8.2. CSAMT data acquisition and processing CSAMT data collection was arranged in four major measurement lines, which cover the surveyed area around the fault zone (Fig. 9.11). The transmitting bipole

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Fig. 9.11. Site location map (the rectangle deﬁnes the northern part of the Minou fault area). Txtransmitter, L1-L4 – survey lines.

Tx was oriented in nearly North-South direction and located about 6 km away from the nearest proﬁle L1. In the northern part of the region (outlined by the rectangle in Fig. 9.11) only scalar measurements were conducted, so the electric ﬁeld Ex was measured in the North-South direction (parallel to the bipole) and the magnetic ﬁeld Hy was recorded in the East-West direction. The measurement points were aligned at about 500 m intervals in each line. Along the proﬁle L4 the intervals were reduced to 250 m and twice as many measurement points as along other proﬁles were placed. The length of the Tx bipole was about 1.9 km.The transmitter used (model CH97 T, Chiba Electric Research, Japan) generally creates 9 kW output power. The contact with the ground was established by 100 copper electrodes at each end of the bipole. The impedance of the source was about 20 O between the two ends, so that a current of 15 amperes was achievable at lower frequencies. The frequency of acquisition ranged from 2 Hz to 16384 Hz in a binary progression. Calculation of the apparent resistivities and phases in each site was carried out using weighting according to their coherencies followed by selective stacking by means of the ‘‘jacknife’’ procedure. Fig. 9.12 shows, for example, the apparent resistivities and phases for the sites located along proﬁles L1. The quality of the collected data is fairly good; however, in the northern part of the surveyed area there were some registered data contaminated by cultural noise. The behavior of the apparent resistivity and phase curves indicates that the geoelectric structure of the northern part of the Minou fault zone could be considered as quasi horizontally layered with the resistivity of the ﬁrst layer close to

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245

Fig. 9.12. Graphs of the apparent resistivities (in O m) and phases (in degrees) with standard errors versus frequency (in Hz) for the sites located along the proﬁle L1.

100 O m. Another characteristic feature of the geoelectric structure revealed from apparent resistivities and phases is that the resistivity in the area between the proﬁles L1 and L3 slightly decreases with depth, so, a presence of relatively low resistive zone at some depth could be expected. 9.8.3. 3-D imaging Minou fault zone using 1-D and 2-D inversion The interpretation of the data was carried out for frequencies higher than 8 Hz, so that the distance between the transmitter and the sounding locations was at least four skin-depths, which is far enough for a plane wave approximation of the source ﬁeld and application of MT ﬁeld equations. In order to get an idea about 3-D resistivity distribution in the northern part of the region basing on the scalar CSAMT data measured along proﬁles L1–L4 (see Fig. 9.11) two independent ways of data inversion were used: (1) Bostick transforms beneath each site followed by 3-D synthesis of the results, and (2) 2-D inversion along proﬁles.

9.8.3.1. Synthesis of Bostick transforms It is often necessary to invert measured data when there is practically no prior information on the resistivity distribution in the area. In this case the inversion methods based on usage of the prior data are not applicable. In this situation, it is

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Fig. 9.13. Volume apparent resistivity image based on 3-D synthesis of Bostick transforms. (Axis O points in the North–South direction.)

useful for a preliminary estimation of the resistivity distribution to construct a three-dimensional image of the medium based on MT ﬁelds or their transforms. In order to get a 3-D image of the resistivity beneath the northern part of the Minou fault zone Bostick transforms of the apparent resistivity component raxy calculated beneath each site along proﬁles L1–L4 (outlined by the rectangle in Fig. 9.11) were compiled using 14 frequencies ranging from 8 Hz up to 16384 Hz in a binary progression. Fig. 9.13 shows a volume apparent resistivity image constructed based on synthesis of 1-D Bostick transforms. It is seen that although the general 3-D resistivity distribution in the northern part of the Minou area is rather complicated, it has two features: ﬁrst, the angle of inclination of the resistivity ‘‘layering’’ to the horizontal plane is approximately 201 and, second, the resistivity decreases downward so that a dipping low-resistivity zone (resistivity less than 30 O m) beneath 500–800 m can be detected. Fig. 9.14 demonstrates explicitly the isosurface of the apparent resistivity equal to 35 O m. It is clear that the boundary between the relatively conductive and resistive zones lies at the depth about 500 m in the northern and at the depth about 800 m – in the southern part of the region. Fig. 9.15 demonstrates vertical slices of the apparent resistivity in the xOz plane beneath each proﬁle. The low-resistivity zone (depth 300 m, width 200 m, length o1000 m, angle of inclination 301, resistivity o30 O m) is seen here beneath the northern edge of the proﬁle L2 (Fig. 9.15b). 9.8.3.2. 2-D inversion results In order to construct a series of initial 2-D models a multi-layered 1-D smoothness-constrained Occam’s inversion of the apparent resistivity and phase data was

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247

Fig. 9.14. Apparent resistivity isosurface (r ¼ 35 O m). (Axis O points in the North–South direction.)

ﬁrst carried out for each site. The earth was considered to be composed of 50 artiﬁcial layers and their thicknesses were assumed to be known and ﬁxed. The unknown resistivities of the layers were recovered using Tikhonov’s regularization. The results of 1-D inversions carried out beneath each site were then linearly interpolated in order to construct initial models for 2-D TM mode inversion along the proﬁles L1–L4 perpendicular to the fault strike direction. The ﬁnite element forward modeling and Powell’s hybrid optimization technique (Powell, 1970) were used to reﬁne the initial models in the framework of 2-D inversion procedure. The low-resistive zone detected earlier by synthesis of Bostick transforms is here more ﬂat (Fig. 9.16): the depth at the northern part varies from 550 m at proﬁles L2 and L3 (Fig. 9.16b, c) to 850 m at proﬁle L1 (Fig. 9.16a) the ‘‘dip angle’’ being around 101. The low-resistive zone, looking like a seam, (the resistivity contrast between the surrounding medium and the seam is approximately 2–5) has a width around 200 m, extends from the North to the South at the depth approximately 600 m, lies mainly in the area beneath the proﬁles L1–L3 and is discontinuous beneath the proﬁle L4. 9.8.4. ANN reconstruction of the Minou geoelectric structure The previous section showed that 3-D imaging based on synthesis of 1-D Bostick transforms and 2-D inversion results reveal a relatively conductive seam dipping in the southern portion of the surveyed area. However, the resistivity image corresponding to the scalar CSAMT data can differ from the ‘‘true’’ 3-D resistivity distribution theoretically corresponding to more complete tensor CSAMT data that could be collected in this part of the Minou area. Moreover, a 3-D resistivity distribution resulting from synthesis of 1-D apparent resistivity transforms and 2-D inversion implicitly implies a local 1-D of the medium beneath each site and 2-D beneath each proﬁle, respectively. Meanwhile, presentation of imaging (inversion) results using continuous function distributions independently of the quantity and quality of the data used gives a false impression that the number of degrees of

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Fig. 9.15. Apparent resistivity cross-sections beneath proﬁles L1 (a), L2 (b), L3 (c) and L4 (d) based on syntheses of Bostick transforms.

freedom available is sufﬁcient to determine tens or even hundreds of unknown model parameters. That is why, alternative ways of model parameter reconstruction given insufﬁcient and noisy data as well as adequate inversion results presentation (in particular, in terms of discrete parameters’ values) are to be considered. It is difﬁcult to expect success in these directions within the framework of the well-known paradigm of geophysical data interpretation, which identiﬁes it with the solution of a nonunique inverse problem. Instead, one of the artiﬁcial intelligence

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249

Fig. 9.16. Resistivity cross-sections beneath proﬁles L1 (a), L2 (b), L3 (c) and L4 (d) based on 2-D TM Occam’s inversion.

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V.V. Spichak/Neural Network Reconstruction of Macro-Parameters

techniques is used, which treats the data inversion as ﬁnding unknown model parameters based on a similarity principle. Application of this approach to the data inversion requires advance teaching of ANN using real or synthetic data and this process may take some hours of CPU (provided that appropriate databases are already created). However, if the model class speciﬁed by the interpreter is already ‘‘familiar’’ to ANN (that is, ANN is already trained), reconstruction of the model parameters could be carried out practically instantaneously. 9.8.4.1. ANN recognition in terms of macro-parameters According to the imaging and 2-D inversion results demonstrated in the previous sections, the geoelectric structure of the northern part of the Minou fault zone was considered to be close to the model class of the type ‘‘dyke buried in a twolayered earth’’. So, MT-NET taught in advance (Spichak, Popova, 2000) was applied reconstruction of the geoelectric parameters enumerated above basing on scalar CSAMT data collected in the northern part of the Minou area. Apparent resistivities rxya and phases jxy measured along proﬁles L1–L4 for ﬁve frequencies (8, 32, 128, 256, 1024 Hz) almost identical to those in the training data were used for ANN reconstruction. In order to eliminate the dependence of the ANN inversion results on the noise in the data a special pre-processing of the teaching samples was used (Spichak et al., 1999b). In particular, we were interested in estimating the depth (D) to the upper edge of the target; the ratio (C1/C2) of the electric conductivities of the ﬁrst and second layers; ratio (C/C2) of the electric conductivities of the target and hosting layer; width (W), length (L) and dip angle (A) of the target in the plane xOz (Fig. 9.17). Table 9.13 shows the recognition results for two data sets used separately: apparent resistivities rxya and phases jxy (data Set 1) and only apparent resistivities rxya (data Set 2). It is seen from Table 9.13 that inversion of only apparent resistivities (data Set 2) leads to underestimation (compared to the values obtained using data Set 1) of all except two model parameters (C1/C2 and A). In other words, adding phases to the apparent resistivities changes the estimates of many model parameters, the biggest improvements being attributed to the ﬁrst layer conductivity (from 6.59 to 0.36) and, to a lesser extent, to the dip angle (from 541 to 451). It is worth mentioning in this connection that ANN reconstruction of the model parameters using only phases could give precise results (at least, for synthetic data) as in the case of inversion of both apparent resistivities and phases (Spichak, 1999). Thus, the Table 9.13. Results of ANN interpretation of the northern part of the Minou fault data (1-rx,y, jxywere used both; 2jxy only).

Data sets used

D (m)

C1/C2

W (m)

L (m)

C/C2

A (degrees)

1 2

308 263

0.36 6.59

386 330

925 800

5.92 3.40

45 54

V.V. Spichak/Neural Network Reconstruction of Macro-Parameters

251

following model parameters were determined by ANN: RHO1 ¼ 100 O m, RHO2 ¼ 36 O m, RHOt ¼ 6 O m, D (H1) ¼ 310 m, W ¼ 390 m, L ¼ 925 m, A ¼ 451, where RHO1, RHO2 and RHOt are the resistivities of the ﬁrst and second layers, and the target, respectively. 9.8.4.2. Testing ANN inversion result In order to test the results of ANN recognition we have carried out the forward modeling by means of the same forward solver that was used for generation of the synthetic database. Electromagnetic ﬁelds for the model obtained using ANN were calculated for two polarizations of the primary ﬁeld at six periods mentioned above. The resulting apparent resistivities rxya and phases jxy as well as observed data were interpolated then to the same rectangular grid with equal number of nodes in each direction ( ¼ 16) and the rms misﬁt was determined by the following formula: 8 2 2 391=2 ra r~ a 2 j j = < ~ xy 1 5 xy xy xy 4 D¼ ð9:13Þ þ a dj~ xy ; :2N xy N T dr~ xy Here NT is the number of periods ( ¼ 6), Nxy the number of interpolation points ~ xy are standard deviations in apparent resistivity and (16 16 ¼ 256), dr~ axy and dj phase, respectively, and the Euclidean norms are calculated as follows: "N #1=2 xy X NT X f ðPi ; T j Þ2 f ¼ i¼1 j¼1

where Pi are the interpolation points. The total rms misﬁt determined according to formula (9.13) was equal to 0.91. (Note that the expected value of this misﬁt is equal to 1.0, which corresponds to the case, when the misﬁts in apparent resistivities and phases are equal to corresponding standard errors in the data.) This means that the best-ﬁtting model reconstructed by ANN belongs to the guessed model class formed by ‘‘dykes buried in the two-layered earth’’, on the one hand, and to the equivalence class formed by all models giving rms misﬁt (9.13) less than 1.0, on the other. Note that if we used for interpretation more ‘‘educated’’ ANN (or many ANNs taught in advance toward one type of models each), the misﬁt (0.91) could be, probably, further diminished. However, striving for this rms misﬁt may become less than the noise level in the data and may lead to unjustiﬁed overestimation of the data and, on the contrary, underestimation of the initial geophysicist’s guess expressed in terms of the expected geoelectric model type. It is worth mentioning in this connection that the knowledge about the noise level in the data enables to use it as a stopping criterion in the framework of deterministic inversion methods that imply the rms misﬁt minimization procedure or to incorporate it explicitly into the inversion process as in the context of the Bayesian probabilistic inversion (Spichak et al., 1999a). The ANN parameters’ recognition, in distinction to the deterministic methods mentioned above, does not

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use the rms misﬁt minimization procedure, at least, in the framework of a single backpropagation scheme (see Section 9.1), so, the only way to compare the appropriate misﬁt with the standard error in the data consists in forward modeling for the model parameters estimated by means of ANN. 9.8.5. Discussion and conclusions Thus, we have interpreted the data measured in the northern part of the Minou area using (1) 3-D imaging based on synthesis of 1-D Bostick transforms, (2) 2-D Occam’s inversion carried out along four proﬁles, and (3) ANN recognition in terms of the macro-parameters. In doing so, we made the following assumptions on the character of the resistivity distribution in the domain of search: (1) local onedimensionality, (2) two-dimensionality (no resistivity changes in the East-West direction), and (3) belonging to the model class of the type ‘‘an inclined dyke buried in a two-layered earth’’, respectively. Comparison of the resistivity models reconstructed by means of ﬁrst two techniques (see Figs. 9.15 and 9.16, respectively) reveals some common features of the geoelectric structure of the northern part of the Minou fault zone. First, the resistivity decreases in the northern direction and from the surface to depth. Second, the layering of the geoelectrical structure is not horizontal with the dip angle being varied from 101 to 251. Finally, a relatively conductive zone (with resistivity less than 30 O m) is detected in the domain beneath the proﬁles L1, L2 and L3 (Figs. 9.15b and 9.16a–c). The ANN inversion of the same data set results, as was mentioned above, in a 3-D resistivity model of an inclined dike buried in a two-layered earth (see Row 1 of Table 9.13): RHO1 ¼ 100 O m, RHO2 ¼ 36 O m, RHOt ¼ 6 O m, D (H1) ¼ 310 m, W ¼ 390 m, L ¼ 925 m, A ¼ 451. 3-D model parameters reconstructed by ANN differ from those estimated by means of 1-D and 2-D interpretation tools that do not take into account 3-D effects. Three-dimensionality of the resistivity distribution in the Minou area is well seen from the comparison of 2-D resistivity cross-sections beneath the proﬁles L1–L4 (Fig. 9.16). In particular, the ‘‘length’’ of the low-resistive zone in the EastWest direction determined by ANN (925 m) matches very well with it’s value that could be estimated visually from the Fig. 19.16(a–c), taking into account that the distance between each neighbouring proﬁles is 450–500 m. Thus, ANN interpretation of scalar CSAMT data measured in a northern part of the Minou area was carried out and the resulting 3-D resistivity distribution was formulated in terms of six macro-parameters of an inclined dike model. The model represents the northern dipping slope of a graben inferred by gravity anomaly in this part of the Minou fault zone (Mogi et al., 1997). Comparison of the resulting resistivity images obtained by different interpretation techniques (Bostick transforms, Occam’s 2-D inversion in TM-mode, ANN recognition) indicates that ANN reconstruction of 3-D model macro-parameters gives appropriate results even if incomplete (for instance, scalar instead of tensor CSAMT) and noisy data are inverted.

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253

The best-ﬁtting model reconstructed by ANN belongs to the guessed model class formed by ‘‘dikes buried in the two-layered earth’’, on the one hand, and to the equivalence class formed by all models giving rms misﬁt less than the noise level in the data, on the other. The ANN model parameters’ reconstruction could be very effective when the geoelectric model searched for is among the model classes used for MT-NET education. In this case fast 3-D interpretation of even incomplete and noisy data could be carried out in the ﬁeld. Another promising ﬁeld of MT-NET application consists in interpretation of the monitoring data, since in this case the ANN learns itself by real data and, therefore, no prior time-consuming forward modeling is required. The studies provided demonstrate that CSAMT data (even incomplete and noisy) can resolve the parameters of the inclined fault zone. The ability of the ANN to teach itself by real geophysical (not only electromagnetic) data measured at the same place during sufﬁciently long period gives an impetus to use this approach for interpretation of the monitoring data in terms of the target macro-parameters. Acknowledgements The author gratefully acknowledges Drs T. Mogi, I. Popova, K. Fukuoka, T. Kobayashi and H. Shima for fruitful collaboration in the framework of the joint project supported by OYO Corporation (grant EM-95).

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Smith, J.T. and Booker, J.R., 1991. Rapid inversion of two- and three- dimensional magnetotelluric data. J. Geophys. Res., 96, B3: 3905–3922. Soulie, F., Gallinari, P., Le, C., and Thiria, S., 1987. Evaluation of neural network architectures on test learning tasks. IEEE Proceedings 1st International Conference on Neural Networks, pp. 653–660. Spichak, V.V., 1983. The FDM3D software package for the numerical modeling of 3-D electromagnetic ﬁelds (in Russian). Algorithms and Programs for the Calculation and Inversion of the Electromagnetic Fields in the Earth, Moscow, IZMIRAN, pp. 58–68. Spichak, V.V., 1990. A general approach to the EM data interpretation using an expert system. Proceedings. X Workshop on EM Induction in the Earth, Ensenada, Mexico. Spichak, V.V., 1999. Magnetotelluric Fields in 3-D Geoelectrical Models. Scientiﬁc World, Moscow, 204pp. Spichak, V.V., Menville, M. and Roussignol, M., 1999a. Three-dimensional inversion of the magnetotelluric ﬁelds using Bayesian statistics. Three Dimensional Electromagnetics, SEG Monograph, GD7, Tulsa, USA, pp. 406–417. Spichak, V.V., Fukuoka, K., Kobayashi, T., Mogi, T., Popova, I. and Shima, H., 1999b. Neural – network based interpretation of insufﬁcient and noisy MT data in terms of the target macro – parameters. Proceedings 2nd International Symposium of Three Dimensional Electromagnetics, Salt-Lake City, USA, pp. 297–300. Spichak, V.V., Fukuoka, K., Kobayashi, T., Mogi, T., Popova, I. and Shima, H., 2002. Artiﬁcial neural network reconstruction of geoelectrical parameters of the Minou fault zone by scalar CSAMT data. J. Appl. Geophys., 49: 75–90. Spichak, V.V. and Popova, I.V., 1998. Application of the neural network approach to the reconstruction of a three-dimensional geoelectrical structure. Izv. Phys. Solid Earth, 34: 33–39. Spichak, V.V. and Popova, I.V., 2000. Artiﬁcial neural network inversion of MT data in terms of 3-D earth macro – parameters. Geophys. J. Int., 42: 15–26. Stratton, J.A., 1941. Electromagnetic Theory. McGraw-Hill, New York, 615pp. Yoshifusa, I., 1991. Approximation of functions on a compact set by ﬁnite sums of a sigmoid function without scaling. Neural Networks, 4: 817–826. Xie, G. and Li, J., 1999. A New Algorithm for 3-D Nonlinear Electromagnetic Inversion. Three Dimensional Electromagnetics, SEG monograph GD7, Tulsa, USA, pp.193–207. Zhdanov, M.S., 2002. Geophysical Inverse Theory and Regularization Problems. Elsevier Publ., Amsterdam-Boston-London, 609pp. Zhdanov, M.S. and Fang, S., 1999. Three-dimensional quasi-linear electromagnetic modeling and inversion. Three-Dimensional Electromagnetics, SEG Monograph. GD7, Tulsa, USA, pp. 233–255.

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Part III: Data Processing, Analysis, Modeling and Interpretation

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Chapter 10 Arrays of Simultaneous Electromagnetic Soundings: Design, Data Processing and Analysis Iv. M. Varentsov Geoelectromagnetic Research Center IPE RAS, Troitsk, Moscow Region, Russia

The topics of this chapter are simultaneous arrays of electromagnetic (EM) soundings of the Earth using natural ﬁelds, systems of their relevant transfer operators and speciﬁc methods for the processing, analysis and interpretation of their data applied in studies of geoelectric structure of the Earth interior. A review of the progress in this ﬁeld of research can be found in (Berdichevsky and Zhdanov, 1984; Egbert, 2002; Varentsov, 2003). The author’s intention is to present a series of important methodical and applied results obtained in the very recent years within frameworks of big simultaneous array EM sounding projects such as EMSLAB, BEAR, EMTESZ-Pomerania, etc.

10.1. SIMULTANEOUS SYSTEMS FOR NATURAL EM FIELDS OBSERVATION Simultaneous observation systems imply simultaneous recording of EM (or only geomagnetic) ﬁeld at a set of points {ri}. The analysis of the collected data can be performed in both the time domain and the domains of spatial and frequency spectra. In a frequency domain, the initial data ensemble for the interpretation is the system of transfer operators of EM ﬁelds, which relate the spectra of its various components at the same one or at different sites and are independent, under certain conditions (Berdichevsky and Zhdanov, 1984) on the ﬁeld excitation model. These _ operators comprise the local ones (single-site), namely, the impedance Z and the tipper Wz: Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40010-X

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^ i ÞHh ðri Þ; H z ðri Þ ¼ Wz Hh ðri Þ, Eh ðri Þ ¼ Zðr _

_

and the inter-site ones, the magnetic and electric (telluric) tensors M , T and the simultaneous tipper Sz: ^ rb ÞHh ðrb Þ; Eh ðrÞ ¼ Tðr; ^ rb ÞEh ðrb Þ; H z ðrÞ ¼ Sz ðr; rb ÞHh ðrb Þ Hh ðrÞ ¼ Mðr; that relate the simultaneous EM ﬁeld data components at the observational grid with horizontal ﬁelds Hh ðrb Þ; Eh ðrb Þ at the base site. It is used to conﬁne the analysis to the calculation of two-site operators with respect to a comparatively small number of reference sites (Varentsov and Sokolova, 2003; Varentsov et al., 2003). A complete system of transfer operators permits a control of their accuracy by ‘‘tran^ i ; rb Þ Mðr ^ b ; ra Þ or Sz ðri ; r0 Þ ¼ Wz ðri Þ Mðr ^ i ; r0 Þ ^ i ; ra Þ ¼ Mðr sitive’’ relations like Mðr and a more reliable reconstruction of simultaneous spectra of EM ﬁelds for a given model of excitation in the interpretation tasks. When a single reference site located ^ i ; r0 Þ; Sz ðri ; r0 Þ is considered. It can be at r0 is used, a system of operators Mðr determined even without a complete synchronization of observations from a set of pairs of simultaneous measurements with respect to a common base site. In this way, however, it is difﬁcult to control the noise at certain sites and, especially, to ensure an optimal base site selection. A trade-off here is the hybrid approach with simultaneous measurements for separate subsets (clusters) of the array including common reference sites. Spatial structure of such clusters should facilitate an efﬁcient use of multi-site schemes for estimating the transfer operators, and availability of common reference sites should provide the synchronization of estimates for the whole array of measurements. As examples of completely simultaneous systems of soundings, the EMSLAB and BEAR experiments can be cited. In the former one (Wannamaker et al., 1989; Varentsov et al., 1996, Vanyan et al., 2002), proﬁle EM soundings on the US Paciﬁc Coast were carried out simultaneously with the adjacent sea bottom array. In the latter case (Varentsov et al., 2002; Varentsov et al., 2003), a subcontinental network of magnetotelluric (MT) and magnetovariational (MV) soundings had been deployed within boundaries of the Baltic Shield and the Svalbard archipelago. Hybrid (multi-cluster) measurement systems were implemented within the framework of EM project in Andes (Soyer and Brasse, 2001) and in the EMTESZPomerania array sounding experiment (Varentsov et al., 2005; Brasse et al., 2006) in the northwest Poland and northeast Germany (Fig. 10.1) across the TransEuropean Suture Zone (TESZ). In Pomerania, long-period measurements were carried out with several systems of ﬁve-component MT stations (German, Polish and Swedish; up to 17 instruments spaced by 10–350 km) that were running simultaneously for 1–6 weeks at a sampling rate of 1–2 s during six ﬁeld campaigns in 2002–2005, with a synchronization to the three repeated ﬁeld base sites (P08 ¼ S27, G7 and P55) and to the nearest geomagnetic observatories (HLP, BEL and NGK). Efﬁciency of hybrid systems is deﬁned by the accuracy of estimating the spatial transfer operators and their time stability, in particular, of the quality of inter-site ^ used in reducing the estimates for different ‘‘clusters’’ to a selected operators M common base (see Section 10.3). This approach permits one to avoid the over

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Fig. 10.1. The EMTESZ-Pomerania array of EM soundings (Varentsov et al., 2005): Gxx (triangles), German; Pyy (circles), Polish; Szz (squares), Swedish long-period MT sites; crosses, Czech, German and Swedish audio MT sites; NGK and HLP (boxes), geomagnetic observatories; dashed lines, DC railways.

concentration of instruments, to plan ﬂexibly the array development, to apply effective modern data processing procedures for noise suppression based on simultaneous observations and, ﬁnally, to estimate reliably a joint set of arraywide inter-station and local transfer functions (TF). It is, at last, possible to reconstruct two-site operators and simultaneous spectra of magnetic ﬁelds from arrays of single-site estimates of the tipper assuming the potentiality of the magnetic ﬁeld in the nonconducting atmosphere. In 2-D case, the ^ i ; r0 Þ and Sz ðri ; r0 Þ from a proﬁle array Wz ðri Þ is provided by a reconstruction of Mðr direct solution of integral equation with a Hilbert kernel (Vanyan et al., 1998). In a general case, iterations are built on the basis of 3-D Hilbert transforms that relate horizontal components of the ﬁeld with the vertical one (Banks et al., 1993; Egbert, 2002). One of the latest examples of such an approach as applied to the study of Carpathian region is presented by Cerv et al. (2002).

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10.2. MULTI-SITE SCHEMES FOR ESTIMATION OF TRANSFER OPERATORS With the deployment in 70-ties of digital measurement systems capable of highaccuracy synchronization, an understanding had matured of the importance of simultaneous measurements for the purposes of noise suppression in traditional MT and MV soundings. The remote reference data-processing techniques, RR, of twosite soundings (Goumbau et al., 1978 etc.) had become a powerful means in the reduction of local noise and systematic distortions in local transfer operators Z^ and Wz, and the robust methods of their estimation had further extended up-to-date capabilities in the suppression of intense noise effects. As a result, the estimation accuracy for ‘‘noisy’’ records has increased substantially in simultaneous observation systems. Contemporary approaches to the estimation of transfer operators are based on the principles of generalized harmonic analysis and robust methods of linear estimation in a frequency domain (see the reviews in Egbert, 2002; Varentsov et al., 2003 etc.). For the preprocessed records of the EM ﬁeld, a sequential Fourieranalysis of the sections of MT process is carried out as well as stacking of linear equations (relating the spectral components of EM ﬁelds) on a series of sections for each interval of estimation periods. The stacked vastly redundant systems of linear equations are solved entirely or in parts with a sufﬁcient stabilization. Consider the resources of data synchronism in estimating the impedance within the scope of pragmatic scheme (Varentsov et al., 2003, 2005) based on obtaining partial TF estimates for EM ﬁeld spectra at large enough (from 1 to 256 Ksamples) overlapping record sections (time windows), their sorting by coherence and other criterions and ﬁnal multi-level robust averaging. SS Partial single-site (SS) estimate of the impedance has a form Z^ i ¼ ðS^ HH Þ1 S^ HE where i is the number of the record section, and matrices S^ HH and S^ HE are determined by mutual spectra and auto-spectra of EM ﬁeld (Semenov, 1998) in the vicinity of the estimation period. In the case of uncorrelated noise and signal in an additive model, this estimate is free of the upward bias of impedance amplitudes caused by auto-spectra of the noise. It, however, depends on the noise in magnetic auto-spectra that shifts the amplitudes downwards. To remove the ‘‘magnetic’’ bias Goumbau et al. (1978) proposed to observe simultaneously the horizontal magnetic RR ﬁeld R at a remote site. Within this approach, a two-site RR-estimate arises: Z^ l ¼ 1 ^ ðS^ RH Þ S^ RE that is efﬁcient for a high-quality linear relation R ¼ MH and independent noises in R and H. Here, the matrices S^ RH and S^ RE are composed of only ^ does not enter into the the mutual spectra of the ﬁeld, and the value of operator M estimate explicitly but only the quality of this linear relationship is important. The rejection (sorting) of partial SS- and RR-estimates is performed in cases of low multiple coherence (low quality of linear relations) and high input coherence (strong linear polarization of magnetic channels). The rejection criteria are formulated for both the separate periods (local rejection) and the entire period range on average (global rejection) (Varentsov et al., 2003). Rejection of the sections showing

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intense low coherent noise facilitates the tuning of the robust averaging methods to the extraction of highly coherent MT signal. Averaging of the selected estimates is primarily carried out over the sections of a record (ﬁrst level, multi-period and multisection). Then a multi-window averaging follows for a series of time windows of various widths (second level), providing a speciﬁc protection against nonstationary noise events with relatively narrow spectra. For RR-estimates also the third level of robust multi-RR averaging is suggested, to be applied over a series of different remote sites (Varentsov et al., 2003). Multi-RR data-processing technique has got its approval in determination of impedances and tippers in a large-scale BEAR experiment. At each BEAR site, as many as 3–4 to 9 remote sites are used located at different azimuths and spaced by distances from 100–150 to 700–900 km. Such a scheme of data processing permitted not only the suppression of instrumental and industrial noise at short periods, but it also ensured an increase in the stability of the results in the medium-and longperiod ranges thus protecting, in particular, from the distorting effects of the subpolar inhomogeneous source ﬁeld (Varentsov and Sokolova, 2003; Varentsov et al., 2003). Other examples of multi-site robust schemes are discussed by Oettinger et al. (2001) and Egbert (2002). In all cases, the sufﬁcient quality of remote sites is required, in particular, the stability of horizontal linear relations of the magnetic ﬁeld and the low spatial correlation of the noise should be provided. In practice, however, propagation of EM noise often shows regional-scale features with the level of spatial coherence of the noise being not worse, and sometimes even better, than for MT signal. In that case the choice of remote sites is difﬁcult, and criteria for the coherence-based rejection becomes less efﬁcient. Such a situation is met within the EMTESZ-Pomerania experiment, where a strong and highly coherent industrial EM noise is mainly produced by the DC railways (Palshin, Smirnov, and EMTESZ-Pomerania WG, 2005). In these conditions the available RR-estimators sometimes became ineffective, and the conventional coherency sorting started to be dangerous. There is a strong need to equip data-processing algorithms with speciﬁc tools directly ﬁghting against highly coherent noise effects. New possibilities for the development of multi-point schemes are opened on the ^ they allow for the contrast way of controlling the spatial structure of operator M; between the smoothness of variations of horizontal magnetic ﬁelds from MT source and their sharpness for close noise sources, electric dipoles having the impedance other than that of a plane wave. Within the scope of RR-technology, it is proposed (Ritter et al., 1998) to reject from robust averaging partial estimates of Z^ and Wz ^ from the unit under substantial deviations of the related partial estimates of M ^ tensor I. This approach has been extended in RRMC data-processing technique (RR with MC, the ‘‘magnetic control’’): conditions on the rejection of partial estimates by input and multiple coherences are complemented by a series of new criteria taking into account the spatial structure of horizontal magnetic ﬁeld (Varentsov, 2005; ^ 0 ðr; rRR Þjj of partial ^ rRR Þ M Varentsov et al., 2005). In particular, variability jjMðr; estimates of the magnetic tensor (that links the sounding site r with the remote site ^ 0 is restrained; rRR ) relative to its a priori (and, implicitly, unit) expectation M

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^ rRR Þ, and the explicitly ensured is the high mutual coherence of estimates Mðr; condition of reciprocity of partial estimates (similar to the impedance-admittance criterion often used in single-site MT data processing) is introduced, i.e. the misﬁt ^ rRR ÞMðr ^ RR ; rÞ Ijj ^ is restrained. These criteria are used both locally (for each jjMðr; period) and globally (on average over the entire range of periods). Then a multilevel robust averaging of the selected estimates described above is carried out. ‘‘Magnetic’’ criteria of rejection permit also obtaining of efﬁcient ﬁnal estimates of ^ itself by the two-point MC-scheme, while RR-technologies here require at tensor M ^ least three simultaneous sites (main, reference and remote). Analysis of operator M in the process of Z^ and Wz estimation is of essential importance also for the removal of errors of time series synchronization at different sites because such errors introduce easily distinguishable and correctable hyperbolic (in period) phase distortions of its principal and determinant components. Finally, it should be stressed that the tipper estimates appears to be the most sensitive to the coherent noise inﬂuence. Their amplitudes in this case rise up to and even higher than the unit level. That is why in the tipper RRMC estimation I also suggest to limit the norm of its partial estimates as a forth MC criterion, just as it was done in Garcia et al. (1997) for the control of subpolar source distortion effects. Fig. 10.2 presents the results of SS, RR and RRMC estimation of the impedance at the site S07 at the P2 proﬁle, located 8 km apart from the DC railway (Fig. 10.1). The most disturbed ‘‘northern’’ impedance Zxy exhibits high distortions of SS estimates (separate black lines with crosses): the phase at periods of 10–1000 s is strictly shifted to the zero, and the apparent resistivity has a false maximum. The conventional RR estimates for different remotes are shown in the left panel. They are practically similar to SS responses for three closest sites S06, S08 and S05 (at distances of 9, 10 and 18 km) and are still disturbed for remote sites S13–S16 (at distances of 60–90 km). Only for remotes P07 or P08 (at distances of 160 and 195 km) noises seem to be suppressed. The cluster of RR responses at sites S13–S16 still visibly differs from the cluster for the most distant sites P07, P08. The RRMC algorithm effectively eliminates the distortions for most of considered remotes (starting from S05 remote in 18 km from the railway), rejecting up to 90–95% of partial estimates (Fig. 10.2, right panel). The subsequent multiRRMC robust averaging for the several remotes at 18–200 km distance permits to obtain reliable ﬁnal results. Even the average of nearby remotes at 18–90 km gives an acceptable apparent resistivity estimate. But note the obvious difference between the long-period RR and RRMC phases even for the most distant remotes at P07 and P08. We really need such far remotes and the MC sorting to get reliable phase estimates.

10.3. TEMPORAL STABILITY OF TRANSFER OPERATORS Temporal stability of transfer operators is deﬁned by quite a number of properties of the ﬁeld source, EM noise, measurement equipment and procedures of their estimation from the observed time series. The inﬂuence of three latter factors

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Fig. 10.2. Simultaneous multi-site estimation of the impedance component Zxy (log-amplitude at the top panels, in mV/(nT km); phase at the bottom ones, in degrees; logarithmic horizontal axes indicate periods, in seconds) at the site S07 (Fig. 10.1.) in the EMTESZ-Pomerania experiment: left panels demonstrate conventional RR estimates for a series of remote sites (from the nearest, S06, S08 to the most distant P07, P08); right panels present corresponding results of the new RRMC scheme; the simplest and most distorted single-site (SS) responses are given in all panels.

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can be studied and minimized in the process of rational setup of the experiment and data-processing technologies, while the effects of excitation are liable only to indirect estimation. Quite a satisfactory convergence of a series of estimates of Z^ and ^ (with a spread at a few percent level) from separate records 1–2 week long is M obtained in a majority of sites in the subpolar BEAR experiment (Varentsov et al., 2003; Varentsov and Sokolova, 2003) over a period range from a very few tens of second up to 3–6–12 h notwithstanding the close location of inhomogeneous auroral sources. Of course, this result does not exclude distorting effects of excitation at time scales exceeding the 2-month duration of the experiment (Varentsov and Sokolova, 2003). The spread of estimates in mid-latitude conditions, according to the results of the EMTESZ-Pomerania experiment, is even less (in spite of the high level of EM noise in that region), and their stability is also traceable up to periods of 6–12 h. Virtual coincidence of estimates of these operators over a series of diurnal records obtained with the Phoenix equipment, is noted in quite a number of simultaneous soundings in the Central Russia. Naturally, the use of multi-site estimation schemes increases the time stability of the results owing to better noise suppression. The situation is different for the tipper – in an overwhelming majority of cases time stability of its estimates is limited to periods up to 3 h. When synchronizing the arrays of transfer operators in the large-scale experiments like the EMTESZ-Pomerania with a few ﬁeld campaigns, it is important to ^ in the course of repeated measmonitor the temporal variations of operator M urements at permanent ﬁeld sites and in the nearest observatories available. Results of such an analysis in Pomerania based on two-month-long records are shown in Fig. 10.3: displayed on the left panel is the convergence of period-dependent am^ (BEL, HLP) linking two close observatories, and plitude estimates of operator M ^ on the right panel is M (G7, HLP) between the ﬁeld base and the nearest observatory. Temporal deviations of maximum and minimum invariant amplitudes (see details of these invariants in Section 10.4) lie within a few percent limit, correspond to the order of conﬁdence intervals of estimation and are by orders of magnitude ^ in the TESZ zone (Fig. 10.4). smaller than the anomalies of maximum amplitude M ^ This allows a reliable recalculation of operators M in the process of the reduction of estimates obtained for separate simultaneous clusters to the common base.

10.4. METHODS FOR THE ANALYSIS AND INTERPRETATION OF SIMULTANEOUS EM DATA Development of interpretational approaches in the method of simultaneous sounding started from the analysis of spatial-frequency distributions of EM ﬁeld spectra. Mathematical apparatus had been deﬁned by the potentiality of geomagnetic ﬁeld in nonconducting atmosphere, properties of EM ﬁeld Green operators in layered media and approximate boundary conditions at thin conducting layers (Berdichevsky and Zhdanov, 1984). Interpretation was based on the separation of surface and deep EM anomalies within a given horizontally layered geoelectric

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^ Fig. 10.3. Convergence of the repeated estimates of the maximal and minimal amplitude components of the horizontal magnetic operators MðBEL; HLPÞ (left ^ panel) and MðG7; HLPÞ (right panel) which connect observatory (BEL, HLP) and ﬁeld (G7) magnetovariational bases in the EMTESZ-Pomerania project; the logarithmic horizontal axis indicates periods (s), the observation intervals (months.year) are given in the legend.

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Fig. 10.4. The pseudo section of the maximal amplitude invariant of the horizontal magnetic tensor for a base site P8 (Fig. 10.1.) along the proﬁle P2 in the EMTESZ-Pomerania project; the horizontal axis represents proﬁle coordinates of observation sites (in kilometers, with an origin at the Polish–German border); the vertical logarithmic axis gives periods (s); selected site names are indicated above the section.

section with a subsequent separate study of these anomalies. For surface anomalies, quasi-analytic methods of direct forward inversion were applied (Schmucker, 1970; Avdeev et al., 1990), and for deep anomalies – the methods of analytic continuation and ﬁtting (Zhdanov and Varentsov, 1983; Berdichevsky and Zhdanov, 1984; Zhdanov et al., 1986; Baglaenko et al., 1996). With this approach many important results justifying the use of simultaneous sounding schemes were obtained. However, with the increase in the data accuracy, the oversimpliﬁcation of the horizontally layered normal model of the medium becomes more and more evident. Currently, a transition takes place to separation and study of conductivity anomalies against horizontally inhomogeneous normal section (Varentsov et al., 1996; Varentsov, 2002, 2005) using the developed schemes of 2-D and 3-D inversion. However, the experience in solving inverse problems in such models with the use of simultaneous EM data is rather limited – the most of modern means for the inversion are intended for the analysis of impedances and, rarely, tippers. True, in 2-D approximation it is possible to solve the inverse problem using simultaneous components of geomagnetic data without leaving the scope of a standard means for impedance inversion. In this way, the halvanically distorted Z amplitudes are replaced by the amplitudes of the ‘‘induction’’ impedance Zind (Vanyan et al., 1997) synthesized from geomagnetic data (electric ﬁeld is determined by integration of the vertical magnetic ﬁeld and is divided by the horizontal magnetic ﬁeld). Comparison of the phases Zind and Z provides a control of the accuracy. Induction impedances are constructed in the EMSLAB experiment from both the

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^ and Sz (Vanyan et al., 1997) and the explicitly estimated transfer operators M simultaneous geomagnetic ﬁelds reconstructed from the array Wz (Vanyan et al., 1998). Another way of constructing the induction impedance is the method of spatial derivatives recently reviewed by Egbert (2002). Though, this technique was developed since a long while ago, now it has got new information resources in large-scale experiments like EMSLAB and BEAR. ^ and Sz (and deﬁned by them simulExamples of explicit ﬁtting of operators M taneous components of the geomagnetic ﬁeld) in horizontally inhomogeneous models are still singular (Varentsov, 2002, 2005; Cerv et al., 2002; Pajunpaa et al., ^ Wz and 2002). However, the notions justifying the joint inversion of operators Z, ^ are weighty: the tensor M ^ is reliably estimated at long periods, is stable against M, the surface distortions and introduces a minimum nonlinearity into the inverse ^ to different elements of geoproblem solution. Sensitivity of the data Wz and M electric structure can differ signiﬁcantly, particularly at sites with minimal Wz amplitudes. Encouraging model resolution in the inversion of 2-D synthetic data sets ^ or Sz components is demonstrated (Varentsov, 2006) as well as containing single M even better perspectives to jointly invert these data together with the impedance (especially, phase) responses. ^ necessarily involves determination of its rotation Interpretation of the tensor M invariants. The rotation procedure for this operator to a common coordinate frame generally depends on two clock-wise rotation angles a,b, relative to the observation frames at a ﬁeld and a base sites, correspondingly _ _ _ _ _ cosðjÞ sinðjÞ RðjÞ ¼ ð10:1Þ M ða; bÞ ¼ RðaÞ M ð0; 0Þ RðbÞ; sinðjÞ cosðjÞ and by its means, the horizontal magnetic ﬁelds and their linear relations may be further studied in a common coordinate system. The simplest system of the horizontal magnetic tensor rotation invariants (extremal azimuths, amplitude-phase values and 3-D parameter, namely, horizontal magnetic skew, skewM) is determined by the minimization in the course of rotation (10.1) of the amplitude of tensor’s nondiagonal elements (in analogy with the classic Swift’s formalism for the impedance tensor) M xy ða; aÞ2 þ M yx ða; aÞ2 n 2 2 2 ¼ 2 M 2 ð0; 0Þ þ M 1 ð0; 0Þ cos2 2a þ M 3 ð0; 0Þ sin2 2a min Re M 1 ð0; 0ÞM 3 ð0; 0Þ sin 4a ¼ a where M 1 ¼ ðM xy þ M yx Þ=2; M 3 ¼ ðM xx M yy Þ=2;

M 2 ¼ ðM xy M yx Þ=2 M 4 ¼ ðM xx þ M yy Þ=2

and the asterisk stands for complex conjugation.

ð10:2Þ

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^ – tensor principle directions deﬁned This minimization yields the azimuths of M by an angle gM : 1 2ReðM 1 M 3 Þ gM ¼ arctg 4 j M 3 j 2 jM 1 j 2

ð10:3Þ

which is obtained in a sequence with a p=4 period. It is necessary to compare the amplitudes M xx ðgM þ kp=4; gM þ kp=4Þ; k ¼ 0; 1; 2; 3 to select proper maximal and minimal azimuths from this sequence. The maximal and minimal amplitudes and phases are naturally obtained with a rotation (10.1) to correspondent maximal and minimal azimuths. Finally, the skew- parameter takes the form skewM ¼

jM 2 j . jM 4 j

ð10:4Þ

^a¼M ^ I^ display Similar invariants for the anomalous (perturbation) tensor M anomalous properties of the horizontal magnetic tensor and specify ‘‘perturbation’’ vectors or ellipses (Schmucker, 1970; Pajunpaa et al., 2002; Varentsov et al., 2005) complementing the induction vectors in the MV imaging of geoelectric structures. An obvious interpretation of perturbation ellipses is possible when the geoelectric structure well-around the base site can be considered as 1-D. In this case (Varentsov et al., 2005), ellipses are almost zero within areas of quasi-1-D structure close to that at the base site, take a circular form above quasi-1-D structures different from the structure at the base site, stretch as a bar around 2-D structures and keep a general ellipse form in the presence of 3-D effects. In the EMTESZ-Pomerania experiment, it is possible to achieve a good agreement between invariant azimuths of tipper and, especially, horizontal magnetic tensor with halvanically undistorted principal directions of impedance phase tensor (Varentsov et al., 2005; Brasse et al., 2006). This result indicates a presence of a 2-D ^ tensor in given structure of the deep conductors. Interpretation resources of the M experiment are illustrated in Fig. 10.4 by a pseudo section of maximal amplitude along the P-2 proﬁle Gdansk-Frankfurt/Oder (Fig. 10.1). The maximal direction at the proﬁle slightly varies around the azimuth 301NE being quite close to the proﬁle azimuth of 42.51NE. The pseudosection looks simple, smooth but informative: two prominent (up to 100%) positive deep anomalies appear at the edges of the TESZ with a maximum at 1000–2000 s periods and indicate two strong quasi-2-D current systems in the crust; the amplitude decrease is observed at short periods pointing at the resistive sedimentary center of the TESZ; a high homogeneity of the data is traced at the Precambrian Craton (on the right) and some more inhomogeneous character is seen at less consolidated Paleozoic Platform (on the left).

10.5. CONCLUSIONS Simultaneous EM soundings open new perspectives for noise suppression in multi-site estimation of the impedance and tipper, and provide additional

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interpretation possibilities for analyzing the spatial transfer operators. On the way of implementation of these possibilities, efﬁcient algorithms for simultaneous processing of MT/MV data are present, problems of constructing the arrays of simultaneous transfer operators are analyzed, schemes for their invariant analysis are suggested and possibilities of the inversion of simultaneous data sets are demonstrated. Acknowledgments The author thanks his colleagues in the EMTESZ-Pomerania Working Group for their efforts in the implementation of the project and for the possibility to illustrate the contents of this Chapter with the ﬁrst project results. The author is grateful to the members of EMSLAB and BEAR Working Groups for the common experience in joint analysis of simultaneous data. The research was supported by grants INTAS 97-1162, RFBR-DFG 03-05-04002 and NATO EST.CLG 980126.

REFERENCES Avdeev, D.B., Dubrovsky, V.G., Singer, B.Sh., Pankratov, O.V. and Fainberg, E.B., 1990. Film interpretation of deep EM soundings in Central Turkmenia. Izv., Phys. Solid Earth, 26: 819–825. Baglaenko, N.V., Varentsov, Iv.M., Gordienko, V.V., Zhdanov, M.S., Kulik, S.N. and Logvinov, I.M., 1996. Geoelectric model of the Kirovograd anomaly from geomagnetic data. Izv., Phys. Solid Earth, 32, 4: 341–351. Banks, R.J., Irving, A.A. and Livelybrooks, D.W., 1993. The simulation of magnetic variation anomalies using single station data. Phys. Earth Planet. Int., 81: 85–98. Berdichevsky, M.N. and Zhdanov, M.S., 1984. Advanced Theory of Deep Geomagnetic Sounding. Elsevier, Amsterdam, 408pp. Brasse, H., Cerv, V., Ernst, T., Hoffmann, N., Jankowski, J., Jozwiak, W., Korja, T., Kreutzmann, A., Neska, A., Palshin, N., Pedersen, L.B., Schwarz, G., Smirnov, M., Sokolova, E. and Varentsov, Iv.M., 2006. Probing electrical conductivity of the Trans-European Suture Zone. Eos Trans. AGU, 87, 29: 281–287. Cerv, V., Kovacikova, S. and Pek, J., 2002. Modeling of conductivity structures generating anomalous induction at the eastern margin of the Bohemian Massif and the W Carpathians. Acta Geophys. Pol., 50, 4: 527–546. Egbert, G.D., 2002. Processing and interpretation of EM induction array data. Surv. Geophys., 23: 207–249. Garcia, X., Chave, A.D. and Jones, A.G., 1997. Robust processing of MT data from the auroral zone. J. Geomagn. Geoelectr., 48: 1451–1468. Goumbau, W.M., Gamble, T.D. and Clarke, J., 1978. MT data analysis: removal of bias. Geophysics, 43: 1157–1162. Oettinger, G., Haak, V. and Larsen, J.C., 2001. Noise reduction in MT time-series with a new signal-noise separation method and its application to a ﬁeld

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experiment in the Saxonian Granulite Massif. Geophys. J. Int., 146: 659–669. Pajunpaa, K., Lahti, I. and Olafsdottir, B., 2002. Crustal conductivity anomalies in central Sweden and SW Finland. Geophys. J. Int., 150: 695–705. Palshin, N.A., Smirnov, M.Yu., and EMTESZ-Pomerania WG, 2005. Characterization of the cultural EM noise: study of geological structures containing well-conductive complexes in Poland. Publ. Inst. Geophys. Pol. Acad. Sci., Warszawa, C-95, 386: 87–96. Ritter, O., Junge, A. and Dawes, G.J.K., 1998. New equipment and processing for MT remote reference observations. Geophys. J. Int., 132: 535–548. Schmucker, U., 1970. Anomalies of Geomagnetic Variations in the SW United States. University of California Press, Berkeley. Semenov, V.Yu., 1998. Regional conductivity structures of the Earth’s mantle. Publ. Inst. Geophys. Pol. Acad. Sci., C-65, 302: 122. Soyer, W. and Brasse, H., 2001. A magneto-variation study in the central Andes of N Chile and SW Bolivia. Geophys. Res. Let., 28, 15: 3023–3026. Vanyan, L.L., Berdichevsky, M.N., Pushkarev, P.Yu. and Romanyuk, T.V., 2002. A geoelectric model of the Cascadia subduction zone. Izv., Phys. Solid Earth, 38, 10: 816–845. Vanyan, L.L., Varentsov, Iv.M., Golubev, N.G. and Sokolova, E.Yu., 1997. Construction of MT induction curves from a proﬁle of geomagnetic data to resolve electrical conductivity of the continental asthenosphere in the EMSLAB experiment. Izv., Phys. Solid Earth, 33, 10: 807–819. Vanyan, L.L., Varentsov, Iv.M., Golubev, N.G. and Sokolova, E.Yu., 1998. Derivation of simultaneous geomagnetic ﬁeld components from tipper arrays. Izv., Phys. Solid Earth, 34, 9: 779–786. Varentsov, Iv.M., 2002. A general approach to the magnetotelluric data inversion in a piecewise-continuous medium. Izv., Phys. Solid Earth, 38, 11: 913–934. Varentsov, Iv.M., 2006. Joint robust inversion of magnetotelluric and magnetovariational data. Electromagnetic Soundings of the Earth’s Interior (Chapter 8). Varentsov, Iv.M., Engels, M., Korja, T., Smirnov, M.Yu. and BEAR WG, 2002. A generalized geoelectric model of Fennoscandia: a challenging database for long-period 3-D modeling studies within the Baltic Electromagnetic Array Research (BEAR) project. Izv., Phys. Solid Earth, 38, 10: 855–896. Varentsov, Iv.M., Golubev, N.G., Gordienko, V.V. and Sokolova, E.Yu., 1996. Study of the deep geoelectric structure along the Lincoln Line (EMSLAB experiment). Izv., Phys. Solid Earth, 32, 4: 124–144. Varentsov, Iv.M., Sokolova, E.Yu. and BEAR WG, 2003. Diagnostics and suppression of auroral distortions in the transfer operators of the EM ﬁeld in the BEAR experiment. Izv., Phys. Solid Earth, 39, 4: 283–307. Varentsov, Iv.M., Sokolova, E.Yu. and EMTESZ-Pomerania WG, 2005. The magnetic control approach for the reliable estimation of transfer functions in the EMTESZ-Pomerania project. Study of geological structures containing well-conductive complexes. Poland, Publ. Inst. Geophys. Pol. Acad. Sci., Warszawa, C-95, 386: 67–80.

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Varentsov, Iv.M., Sokolova, E.Yu., Martanus, E.R. and EMTESZ-Pomerania WG, 2005. Array view on EM transfer functions in the EMTESZ-Pomerania project. Study of geological structures containing well-conductive complexes. Poland, Publ. Inst. Geophys. Pol. Acad. Sci., Warszawa, C-95, 386: 107–121. Varentsov, Iv.M., Sokolova, E.Yu., Martanus, E.R., Nalivayko, K.V., and BEAR, WG, 2003. System of EM ﬁeld transfer operators for the BEAR array of simultaneous soundings. Methods and results: Izv., Phys. Solid Earth, 39, 2: 118–148. Wannamaker, P.E., Booker, J.R., Filloux, J.H., Jones, A.G., Jiracek, G.R., Chave, A.D., Tarits, P., Waff, H.S., Egbert, G.D., Young, C.T., Stodt, J.A., Martinez, M.G., Law, L.K., Yukutake, T., Segawa, J.S., White, A. and Green, A.W., 1989. Magnetotelluric observations across the Juan de Fuca subduction system in the EMSLAB project. J. Geophys. Res., 94, B10: 14111–14126. Zhdanov, M.S., Golubev, N.G. and Varentsov, Iv.M., 1986. 2-D model ﬁtting of a geomagnetic anomaly in the Soviet Carpathians. Ann. Geophys., 4: 336–341. Zhdanov, M.S. and Varentsov, Iv.M., 1983. Interpretation of local 2-D electromagnetic anomalies by formalized trial procedure. Geophys. J. Roy. Astr. Soc., 75: 623–638.

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Chapter 11 Magnetotelluric Field Transformations and their Application in Interpretation V.V. Spichak Geoelectromagnetic Research Center IPE RAS, Troitsk, Moscow Region, Russia

Analysis of the behavior of magnetotelluric (MT) ﬁelds in 3-D conductivity models is performed mainly by studying the frequency dependence of the apparent resistivity and impedance phase at selected points (see, e.g., Vanyan et al., 1984; Berdichevsky et al., 1987). The non-uniqueness and mathematical incorrectness of the solution of the inverse geoelectric problem, which are attributed to the insufﬁcient quality and quantity of underlying electromagnetic data, seem to be the main reason of the emergence of a large group of methods, techniques, approaches and, ﬁnally, of heuristic procedures used to interpret measured electromagnetic data. One of the main means employed to this end is the transformation of data into forms more convenient to analyze. It is evident from the above that it is hardly possible to provide a complete coverage of all the practical methods of electromagnetic ﬁeld transformation and, in addition, to cite numerous examples of data interpretation relying thereon. Many of them can be found in the publications by Rokityansky (1982), Kaufman and Keller (1981, 1983), Berdichevsky and Zhdanov (1984), Vozoff (1972, 1985, 1991), Zhdanov and Keller (1994). The overview presented in this chapter is conﬁned to the consideration of those approaches to the analysis of magnetotelluric data that are based on linear relations between components of the ﬁeld measured at a single or several sites on the Earth’s surface. In doing so, only a few examples of data interpretation will be given, restricted primarily to the works that were published in less-accessible journals (see also a review paper by Spichak, 1990).

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40011-1

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11.1. LINEAR RELATIONS BETWEEN MT FIELD COMPONENTS According to the theory of linear relations between components of the MT ﬁeld, developed by Berdichevsky and Zhdanov (1984), for a fairly large class of geomagnetic variations (pulsations, bay-like disturbances, solar quiet diurnal changes (Sq) and worldwide magnetic storms (Dst)) we can write ^ EðrÞ ¼ ZHðrÞ

ð11:1Þ

HðrÞ ¼ Y^ EðrÞ

ð11:2Þ

^ EðrÞ ¼ TEðr 0Þ

ð11:3Þ

^ HðrÞ ¼ MHðr 0Þ

ð11:4Þ

where Z^ ¼

Z xx Z xy Z yx Z yy

;

Y^ ¼

Y xx Y xy Y yx Y yy

;

T^ ¼

txx txy tyx tyy

;

^ ¼ M

mxx mxy

myx myy

are the magnetotelluric operators (impedance, admittance, telluric and magnetic, respectively) which do a linear transformation of the ﬁeld, while r and r0 are the radius-vectors of observation points at the surface. Berdichevsky and Zhdanov (1984) also consider the so-called induction operators J^ effecting transformations of the magnetic ﬁeld as a whole and its parts into each other uv

Hu ðrÞ ¼ J^ Hv ðrÞ

ð11:5Þ

where 2

uv J^

uv 3 J uv xx J xy 6 uv 7 ¼ 4 J uv yx J yy 5 uv J uv zx J zy

u; v ¼ t; n; a; e; i (t corresponds to the total ﬁeld, n the normal (background) part, a the anomalous one, e the external ﬁeld, i the internal ﬁeld). Within these designations, we arrive at the induction vector (Parkinson, 1959; Wiese, 1962; Schmucker, 1970): an Wan ¼ J an zx dx þ J zy dy

ð11:6Þ

(where dx and dy are the unit vectors in the x and y directions, correspondingly) and perturbation vectors (Schmucker, 1970) an p ¼ J an xx dx þ J yz dy ;

an q ¼ J an xy dx þ J yy dy

ð11:7Þ

Relations (11.1) through (11.7) are also known as transfer functions. The conditions under which the above linear transforms hold are outlined in Dmitriev and Berdichevsky (1979), Gough and Ingham (1983), and Berdichevsky

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277

and Zhdanov (1984). Variations of these relations in time are treated in review papers by Beamish (1982) and Kharin (1982). Menvielle and Szarka (1986) expose various causes of errors in their calculation, while Egbert and Booker (1986), Larsen et al. (1996) and Egbert (1997) discuss the robust methods used to estimate transfer functions reducing the systematic and random noise. The application of linear MT transforms to electromagnetic data interpretation was developed in two main directions: (1) Determination of apparent parameters of the geoelectric section at each point, which is based on frequency (time) characteristics of transforms, and (2) Studies of structural features (size, boundaries and even conductivity of an anomalous zone), which are based on theoretical examination of the properties of particular transforms, or most frequently on heuristic formulas.

11.2. POINT TRANSFORMS OF MT DATA Transforms of electromagnetic ﬁeld components taken at a single point on the surface make it possible to establish the parameters of the conductivity distribution corresponding to a certain speciﬁc vicinity of this point. In the following, these ﬁeld transforms will be called (following Spichak, 1990) ‘‘point transforms.’’ It should be noted that the method suggested by Goubau et al. (1978) for carrying out measurements at an additional reference point enhances the quality of estimation of the transforms, but this question goes beyond the scope of our consideration. Obviously, the above deﬁnition of the considered class of transforms is rather conventional since it includes the notion of the ‘‘point vicinity,’’ which has no clear physical meaning. Indeed, the electromagnetic ﬁeld measured at a certain point may carry information about more or less distant regions of the Earth’s interior. The problem of deriving this information from the ﬁeld characteristics or from its point transforms (at least, deﬁning the degree of their ‘‘locality’’) calls for speciﬁc investigations (see, for example, the paper by Bahr, 1991). The following subsections will describe the point transformations most commonly used in practice (impedance, apparent resistivity, induction vectors) and give some examples of their application to data interpretation. 11.2.1. Impedance transforms In regions where the distribution of conductivity in the Earth is approximated fairly well by the function s (z), a horizontal electric ﬁeld at any point on the surface can be found, as is known, from the magnetic ﬁeld by simply multiplying the latter by a frequency dependent coefﬁcient Z: Hx Ex 0 Z ¼ ð11:8Þ Ey Hy Z 0 Here Z ¼ Zxy ¼ Zyx is the Tikhonov–Cagniard impedance (Tikhonov, 1950; Cagniard, 1953). Tikhonov (1965) proved the uniqueness of determination of

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parameters of a one-dimensional horizontally layered section from the dependence Z(o), but it is evident that when the conductivity at a depth z is determined from a ﬁnite set of values Z(oi), the result is ambiguous. Weidelt (1985) constructed sets of extremal models (as a sequence of thin layers of different conductivities), setting upper and lower limits on s(z) which depend on the quantity and quality of available impedance data. Sometimes, it is reasonable to employ other forms of the impedance transformation instead of (11.8). For example, Fiskina et al. (1985) interpreted most advantageously deep electromagnetic data using the magnetic ﬁeld transform @H z Z ¼ iomo H z =divs H ¼ iomo H z = @z (where divsH is the surface divergence), suggested earlier by Berdichevsky et al. (1969). Berdichevsky et al. (1983) have established by model calculations that the transform allowing for the exponential nature of magnetic ﬁeld variations in a layer of sea water (Berdichevsky and Vanyan, 1969) yields better results, compared to those obtained conventionally. If the geoelectric section at a speciﬁed point differs from a one-dimensional one, transform (11.8) can give, nevertheless, an approximate notion of the function s(z), which is basic to the so-called ‘‘formal interpretation’’ (see, for instance, the monographs by Vanyan, 1965 and Berdichevsky, 1968). When the conductivity distribution in the vicinity of an observation point is twodimensional, the impedance tensor is reduced, by turning the coordinate system about the vertical axis, to the form (Word et al., 1970): 0 Zxy ^ Z¼ ð11:9Þ Z yx 0 with Zxy+Zyx6¼0 (unlike (11.8)). The four degrees of freedom of transform (11.9) (two complex numbers Zxy and Zyx) are associated with four physically meaningful values: two principal values of impedance and, correspondingly, two preferred directions of a structure, one of which coincides with the local axis of two-dimensionality. This does not rule out, however, the possibility of interpreting such situations from other parameters. For instance, Kovtun and Vardanyants (1985) suggested that two-dimensional anomalies should be interpreted using the frequency characteristic of a relative value of the impedance modulus: E;H E;H aE;H ¼ jZ E;H 1 j jZ 2 j=jZ 2 j

where ZE,H and ZE,H are the impedances for E- and H-polarizations of the ﬁeld at 1 2 two points above an anomaly (1-closer to the center). For the model concerned (Fig. 11.1a), the position of a maximum (Tmax) of the frequency characteristic aE proved to be independent to 1-D section parameters or location of points 1 and 2 (Fig. 11.1b) but instead is mainly governed by the integral conductivity of the insert cross-section: G aE 1; 2 106 T max ðS=mÞ for the range of L, ds and dc values given

V.V. Spichak/Magnetotelluric Field Transformations

279

Fig. 11.1. Frequency characteristics aE and aH for the insert model: (a) model (d1 ¼ 1 km; ds ¼ dc ¼ 10 km; L ¼ 75 km, r1 ¼ 7 O m, rs ¼ 50 O m, rs ¼ 100 O m); (b) relationship of aE (solid line) and aH(broken line) for the chosen point of normalization z2: (1) at inﬁnity, (2) outside the insert (2.5 km from the edge), (3) above the insert (5 km from the edge); (c) Ts dependence of aE (solid line) and aH (broken line): (1) 5 108 O m2; (3) 5 10 O m2 (after Kovtun and Vardanyants, 1985).

by the inequality 0,3rL/(ds+dc)r3. The value aH reﬂects the degree of conductive coupling of the sedimentary section with a conducting body and permits to deﬁne the setup of the conducting zone more closely. Fig. 11.1c plots the behavior of aH versus T with different values of parameter Ts ¼ dsrs for the case where point 1 is just above the center of the body, while point 2 is far off the structure. Variations in the transverse resistivity of the insert Ts make aH change considerably but do not practically affect aE. It is remarkable that aH is related to the parameters of the section and insert within the DC-approximation, when the insert can be replaced by equivalent parallel - connected resistors (Fig. 11.1a): aH ¼ R1 =ðR1 þ R þ 2R? s Þ, where R1 ¼ r1 L=d 1 , R ¼ rs L=d c , R? s ¼ 2rs d s =L. If the medium is locally three-dimensional, the impedance does not reduce to the form (11.9). In this situation, the interpretation may involve forms of the impedance transform, which are invariant under the coordinate system rotating about the vertical axis: Z inv1 ¼ 0; 5ðZxx þ Z yy Þ

ð11:10Þ

Z inv2 ¼ 0; 5ðZxy Z yx Þ

ð11:11Þ

Z inv3 ¼ ðZ xx Z yy Z xy Z yx Þ1=2

ð11:12Þ

Their application in practice provides ‘‘averaged’’ parameters of a local 1-D section. Ingham (1988) shows, using a 3-D model as example, that the interpretation of highly conductive anomaly (in particular, determination of the depth to its upper edge) by means of transform (11.11) reduces the extent of the error compared to conventional 1-D interpretation. Hermance (1982), Sule and Hutton (1986) and Ranganayaki (1984) employed transform (11.12). Particularly, in the latter work,

280

V.V. Spichak/Magnetotelluric Field Transformations

Fig. 11.2. A comparison of geologic structure with /jDet/ pseudo-section along the same line (after Ranganayaki, 1984).

the jjDet j j arg Zinv3 j pseudo-section produced along the proﬁle concerned coincides virtually with the geologic section (Fig. 11.2). Lilley (1993) and Szarka et al. (2000) have introduced another two rotational invariants of impedance Zinv4 ¼ 0; 5½ðZ xx þ Zyy Þ2 þ ðZxy Z yx Þ2 1=2

ð11:13Þ

Z inv5 ¼ ðZ 2xx þ Z 2xy þ Z2yx þ Z 2yy Þ

ð11:14Þ

V.V. Spichak/Magnetotelluric Field Transformations

281

In the latter paper, ﬁve different functions [|f(Z)|, Re(f(Z)), Im(f(Z)), f(Re Z) and f(Im Z)] of three rotational invariants f determined by (11.11), (11.12) and (11.14) were studied. Several researchers developed another approach to estimation of horizontal characteristics of a medium at an observation point, implying physically meaningful characteristics of a local geoelectric section are selected from impedance tensor components in an amount equal to the number of its degrees of freedom (for instance, in a locally 3-D medium this number is eight). Thus, Eggers (1982) used, with this view in mind, bi-orthogonal analysis in a plane and found eigenvalues and eigenvectors of the matrix Z^ on the assumption that EH ¼ 0. Since the actual electromagnetic ﬁelds in 3-D media do not always comply with this assumption, this formulation of the problem will not ensure at all times extreme values of impedance. Using the factorization method, Spitz (1985) constructed two internal systems of coordinates for Z^ which are deﬁned by two angles of rotation. However, as was noted by the author himself, it is not yet clear from the mathematical standpoint which of the two established coordinate systems should be preferred in data interpretation. Counil et al. (1986) introduced, along with the starting basis in which all measured data are speciﬁed, two more real bases – ‘‘electric’’ and ‘‘magnetic’’ – in which the electric and magnetic ﬁelds are linearly polarized, respectively. The linearly polarized electrical ﬁeld is associated with electric-type impedance, and the direction in which it attains its maximum value is called by the authors, the direction of maximum current. Similarly, the linearly polarized magnetic ﬁeld is in correspondence with the magnetic-type impedance, and the direction in which its maximum is achieved has been given the name of the direction of maximum induction. These directions are shown in Fig. 11.3 within a thin-sheet model, as an example. The thin layer consists of two half-layers of differing conductivities ðs1 4s2 Þ. At two points taken near the contrast interface, the direction of maximum induction and maximum current are deﬁned by a speciﬁed distribution of conductivity. La Torraca et al. (1986) examined the eigenvalues of the impedance tensor by means of the singular value decomposition. The authors arrived at eight parameters deﬁning it uniquely (two complex eigenvalues and four angles). The transforms of Z^ as well as of Y^ in three-dimensional space were analyzed by Yee and Paulson (1987a) and Tzanis (1988a). The former authors called it canonical decomposition, while the latter referred to it as a generalized rotation method. Despite certain differences in detail, the two methods are very close in the bulk of mathematics used to solve a particular problem. It is based on a group of rotations deﬁned on a set of unitary matrices (2 2) in three-dimensional space. As a result, the initial coordinate system transforms into two internal systems by directions (angles of rotation) yE, jE; yH and jH (they are not necessarily mutually orthogonal and horizontal). In these systems, components of the ﬁelds E and H are interrelated and form two modes of the electromagnetic ﬁeld inside the Earth. In the two abovementioned studies, the authors compare in detail their own methods of analysis of the tensor to those developed by their predecessors (Eggers, 1982; Spitz, 1985;

282

V.V. Spichak/Magnetotelluric Field Transformations

M Fig. 11.3. Induction arrow I, maximum induction vM 0 and maximum current X0 directions at stations close to the boundary between two half sheets of conductivities s1 and s2(s1>s2) (after Counil et al., 1986).

Counil et al., 1986; La Torraca et al., 1986) and show that these ﬁndings are attainable, under certain speciﬁc assumptions, within the general canonical approach. Yee and Paulson (1987b) employ canonical decomposition to analyze the telluric operator (11.3), while Tzanis (1988b) applies the generalized rotation method to the interpretation of MT data obtained in Northern England and Southern Scotland (Banks and Beamish, 1984). A glance at Fig. 11.4 shows at the site PW estimated to display a locally three-dimensional distribution of conductivity, the structure parameters established by Tzanis (1988a) (Fig. 11.4b) differ substantially from those determined using the conventional rotation technique (Word et al., 1970) (Fig. 11.4a). However, the most general and elegant approach to the impedance tensor decomposition was proposed by Caldwell et al. (2004). It is based on the use of the socalled ‘‘phase tensor’’ P^ determined as a relation between the imaginary and real parts of the impedance tensor Z^ ( ¼ X^ þ iY^ ) 1 P^ ¼ X^ Y^

ð11:15Þ

As a second-rank 2-D tensor, it can be expressed by three independent coordinate invariants Pinv1, Pinv2 and Pinv3 determined similarly to (11.10)–(11.12). Caldwell et al. represent the phase tensor using the following three functions of the coordinate invariants (Bibby, 1986) Pmin ¼ ðP2inv1 þ P2inv2 Þ1=2 ðP2inv1 þ P2inv2 P2inv3 Þ1=2 Pmax ¼ ðP2inv1 þ P2inv2 Þ1=2 ðP2inv1 þ P2inv2 P2inv3 Þ1=2

V.V. Spichak/Magnetotelluric Field Transformations

283

Fig. 11.4. (a) the conventional analysis results for site PW and (b) the UD results for site PW (after Tzanis, 1988b).

_

skew angle b ¼ 1=2tan1 Pinv2 =Pinv1 (where Pmin and Pmax are principal values of P) and also the angle depending on the coordinate system a ¼ 1=2 tan1

P12 þ P21 P11 P22

where Pij (i,j ¼ 1,2) are the phase tensor elements in the Cartesian coordinate system (x1, x2) (Fig. 11.5).

284

V.V. Spichak/Magnetotelluric Field Transformations

Fig. 11.5. Graphical representation of the phase tensor. The lengths of the ellipse axes, which represent the principal axes of the tensor, are proportional to the principal (or singular) values of the tensor. If the phase tensor is non-symmetric, a third coordinate invariant represented by the angle b is needed to characterize the tensor. The direction of the major axis of the ellipse, given by the angle ab, deﬁnes the relationship of the tensor to the observer’s reference frame or coordinate system (x1, x2) (after Caldwell et al., 2004).

The MT phase tensor is very useful for analyzing the effects caused by local inhomogeneities and regional structures. In particular, where galvanic effects produced by heterogeneities in subsurface conductivity distort the regional MT response, the phase tensor preserves the regional phase information. Calculation of the phase tensor requires no assumption about the dimensionality of the underlying conductivity distribution and is applicable where both the heterogeneity and the regional structure are 3-D. For 1-D regional conductivity structures, the phase tensor is characterized by a single coordinate invariant phase equal to the 1-D impedance tensor phase. If the regional conductivity structure is 2-D, the phase tensor is symmetric with one of its principal axes aligned parallel to the strike axis of the regional structure. In the 2-D case, the principal values of the phase tensor are the transverse electric and magnetic polarization phases. The orientation of the phase tensor principal axes can be determined directly from the impedance tensor components in both 2-D and 3-D situations. In the 3-D case, the phase tensor is non-symmetric and has a third coordinate invariant that is a distortion-free measure of the asymmetry of the regional MT response. 3-D model studies show that the orientations of the phase tensor principal axes reﬂect lateral variations in the underlying regional conductivity structure. Let us mention ﬁnally the method of the MT data analysis by means of the socalled ‘‘Mohr circles’’ (Lilley, 1976). It is a convenient tool used to depict MT impedance information considering the in-phase and quadrature parts of MT tensors separately. In this way, such concepts like ‘‘three-dimensionality,’’ ‘‘skew’’ or ‘‘anisotropy’’ are given quantitative expression on a diagram (Lilley, 1993).

V.V. Spichak/Magnetotelluric Field Transformations

285

In Section 11.3, we will refer again to impedance transformation in the context of its application (along with other ﬁeld transformations) to solve some methodological problems of data interpretation. 11.2.2. Apparent resistivity type transforms As a rule, in the interpretation of realistic data one employs second- and even third-level transforms derived from the basic ones (say, (11.1)–(11.4)) through certain algebraic, differential or integral transformations. These transforms are intended for determining as accurately as possible the parameters of a local 1-D section at an observation point. It is also desirable that they enable higher depth resolution and be clear and transparent. The ﬁrst transform of this kind seems to have been suggested by Cagniard (1953) ra;jzj ¼ 1=ðm0 oÞjZj2

ð11:16Þ

Applying (11.16) to real data at a single point, we evaluate the apparent resistivity ra at a depth corresponding to the skin depth 2ra 1=2 ha ¼ m0 o Transform (11.16) is now basic to the interpretation carried out by the MT sounding method. Subsequently, modiﬁcations of transform (11.16) as well as of its transformation were suggested (Niblett and Sayn-Wittgenstein, 1960; Molochnov, 1968; Schmucker, 1970; Weidelt, 1972; Yakovlev et al., 1975; Molochnov and Sekrieru, 1976; Bostick, 1977; Vanyan et al., 1980; Le Vyet Zy Khyong and Berdichevsky, 1984; Murakami, 1985; Szarka et al., 2000) to ensure enhanced resolution of the geoelectric section parameters in a particular period range. A comparative analysis of these transforms can be found in the works by Weidelt et al. (1980), Jones (1983), Spies and Eggers (1986), Schmucker (1987), Szarka et al. (2000). Spies and Eggers consider, in particular, the behavior of the apparent resistivity curves calculated according to formula (11.16) as well as by means of the following impedance transforms: ra;ReZ ¼

ra;jZj ¼

2 ðReZÞ2 ; m0 o 2 jZj2 ; m0 o

ra;ImZ ¼

ra;ImðZ 2 Þ ¼

2 ðImZÞ2 m0 o

1 ImðZ 2 Þ m0 o

They show, using two two-layer models with resistivity contrasts 100 and 0.01 as an example (Fig. 11.6), that the curve calculated from the real part of the impedance ‘‘behaves’’ better than others do (fewer oscillations in the transition zone, maximum speed of convergence to the resistivity of the underlying rock), whereas the curve calculated from the imaginary part of the impedance ‘‘behaves’’ worse all the others. Nevertheless, the authors stress that one should not overrate the obtained

286

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Fig. 11.6. Apparent resistivity graphs for two-layered model (after Spies and Eggers, 1986): a r2 =r1 ¼ 100; b r2 =r1 ¼ 0:01; 1 ra;ReZ ¼ m 2o ðReZÞ2 ; 2 ra;ImZ ¼ m 2o ðImZÞ2 ; 3 ra;jZj ¼ m 2o jZj2 ; 4 0 0 0 ra;ImðZ2 Þ ¼ m 1o ImðZ2 Þ: 0

results and draw conclusions about the resolution of particular techniques of Earth sounding just on the basis of apparent resistivity curves. A simple visual analysis of apparent resistivity maps and of sounding curves averaged over and around a conductive (and a resistive) heterogeneity embedded in a homogeneous half space has shown that the imaging properties appear to depend much more on the apparent resistivity deﬁnition than on the rotational invariant itself (this corresponds to ﬁndings of Szarka et al. (2000)). Except for the very short period range corresponding to the oscillating section of the sounding curves, a robust and regular behavior of the imaging parameters was observed. The authors conclude that in this so-called ‘‘normal’’ period range the 3-D imaging properties seem to be the best if the apparent resistivity is derived when using the function ^ i.e. when computing the rotational invariant with the real parts of the four Re Z, ^ a rapid impedance tensor elements. For apparent resistivities derived from Re Z, convergence over lateral resistivity contrasts and oscillations with small amplitude over homogeneous areas are actually observed in the apparent resistivity maps. In the period domain, they are characterized by a maximum rate of convergence to the underlying resistivity at longer periods and by a reasonably small standard deviation, even in presence of a subsurface disturbing body. At the same time, it is noteworthy that a joint analysis of the ra-curves and the behavior of the impedance phase carrying additional information about the local geoelectric structure (see, for instance, Fischer, 1985), improves the efﬁciency of local 1-D interpretation of 2-D and 3-D structures (Vaghin and Kovtun, 1981; Ranganayaki, 1984; Fischer and Schnegg, 1986; Schnegg et al., 1986; Schnegg et al., 1987; Weidelt and Kaikkonen, 1994). The general principles underlying this approach as well as numerous examples of interpretation are outlined in the work by Berdichevsky and Dmitriev (1976). As noted earlier, the most comprehensive information about the horizontal distribution of conductivity in the vicinity of an observation point is derived by examining all the elements of the impedance tensor (admittance or other linear transforms of the electromagnetic ﬁeld components). If each component of the

V.V. Spichak/Magnetotelluric Field Transformations

287

tensor Z^ is subject to transformation (11.16) we get an apparent resistivity tensor ! raxx raxy R^ a ¼ ð11:17Þ rayx rayy The possibility of interpreting MT data by means of transform (11.17) was discussed by Vozoff (1972), Kaufman and Keller (1983) and many other researchers (see the references cited therein). Note that by analogy with the ‘‘apparent resistivity’’-type transforms, one can use ‘‘apparent conductivity’’ sa. A procedure for interpreting apparent resistivity curves plotted for a section having 2-D and 3-D conductivity anomalies is designed according to the purpose of the investigations. If it is necessary to establish an averaged 1-D section (say, in the global MT sounding; Vanyan and Shilovsky, 1983; Singer and Fainberg, 1985), the differences between the actual ra-curves and those corresponding to a local 1-D section are interpreted as their distortions. The latter are analyzed by numerical and physical modeling of elecrtomagnetic ﬁelds in 2-D and 3-D media in the papers of Ting and Hohmann (1981), Wannamaker et al. (1984a), Park (1985), Berdichevsky et al. (1984a, 1984b, 1984c, 1987). An alternative way to interpret actual data may involve a transform like (11.14) to the principal values of impedance obtained via its decomposition (Yee and Paulson, 1987a; Groom and Bailey, 1989, 1991). In locally 3-D media, this approach may help avoiding errors possible in the case of 1-D and even 3-D interpretation of apparent resistivity curves. Note, ﬁnally, a useful approximate relationship between the apparent resistivity and anomalous MT ﬁelds introduced by Portniaguine and Zhdanov (1999) 1=2 ln ra þ ija E a =E b H a =H b

ð11:18Þ

where ln ra is so-called ‘‘log-anomalous apparent resistivity’’ (ln r ln rb ), ja the anomalous phase, Ea and Ha the anomalous MT ﬁelds, Eb, Hb and rb the background ﬁelds and apparent resistivity, corresspondingly. According to the authors’ estimates, relation (11.18) holds with an accuracy 0.01 if the maximum value of the log-anomalous apparent resistivity ra is less than 0.2. 11.2.3. Induction and perturbation vectors Horizontal conductivity gradients in the vicinity of an observation point can be evaluated by transforming the horizontal components of the magnetic ﬁeld (11.5). Parkinson (1959), Wiese (1962), Schmucker (1970) introduced convenient graphic representations of operators, determined by relations (11.6) and (11.7), in the form of ‘‘vectors’’ or ‘‘arrows.’’ The review papers by Gregory and Lanzerotti (1980), Meyer (1982) and Gough and Ingham (1983) offer a detailed treatment of various representations of inductive operators, their relationship and application to MT ﬁeld interpretation. There are many publications devoted to this subject. For instance, Labson and Becker (1987) consider the behavior of induction arrows in 2-D models of contacts (over the VLF range), while Fischer and Weaver (1986) use them to compare the

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thickness of the continental and oceanic lithosphere. Ingham et al. (1987) studied the geoelectric pattern under the Cordilleras, Pajunpaa (1986) and Korja et al. (1986) in the Baltic shield, while Menvielle and Tarits (1986) investigated the Rhine–Graben conductivity anomaly. Schmucker (1970) and Bailey et al. (1974) suggested a ‘‘hypothetical event analysis’’ for data interpretation. The method comes about as follows. Employing transform (11.6) one can predict the vertical component of the magnetic ﬁeld specifying a background magnetic ﬁeld of ﬁxed polarization and intensity. Beamish and Banks (1983) employed this approach to produce contour maps for three components of an anomalous magnetic ﬁeld when they studied the conductivity anomaly in the north of Great Britain. Chamalaun et al. (1987) established a 2-D strike of a geoelectric structure in the northwest of India. The interpretation of MT data can be made more efﬁcient by studying the spatial-frequency characteristics of transformations (in particular, induction operators) on typical 2-D and especially 3-D models. Lam et al. (1982) and Wannamaker et al. (1984b) examined the behavior of induction arrows in the case where the horizontal-layered section contains a 3-D conductivity anomaly. Nienaber et al. (1983) and Chen and Fung (1988) were interested in the behavior of the real and imaginary induction arrows above the edge of a conducting plate. Chen and Fung (1985) studied imaginary arrows versus inducing ﬁeld frequency by 2-D modeling. They discovered, in particular, a characteristic period Tc in which the phase difference between the components Hz and Hx is zero. The same conclusion has been reached by Beamish (1985) who analyzed the frequency dependence of an anomalous vertical ﬁeld in the British Isles. The period dependence of induction vector azimuths was also considered by Beamish and Banks (1983), Beamish (1987) and Chen and Fung (1988). Jones (1986a) studied the frequency dependence of magnetic transfer functions uv J uv zx and J zy (11.5) using 2-D modeling. The author showed that at sufﬁciently high frequencies, induction arrows composed of these components with u ¼ t; v ¼ t, n, may behave anomalously with their heads showing the direction away from highly conducting zones. It is better, therefore, for the interpretation to involve aa the transfer functions J aa zx and J zy calculated only from anomalous components of the magnetic ﬁeld, which conﬁrms the conclusion of Summers (1981). The parameter R ¼ H az =H ax was helpful in the interpretation of two-dimensional structures carried out by Ingham et al. (1983), Chen and Fung (1986), Jones (1986b) and Beamish (1987). In the latter work, model and practical examples demonstrate that the R-ratio may provide the location of horizontal boundaries of an anomaly (Fig. 11.7) as well as its upper edge from the intersection of rays traced from each observation point downward at an angle deﬁned by the equality ctg y ¼ R (Fig. 11.8). As noted earlier, induction operators are also useful in the interpretation of real magnetotelluric data (say, by means of hypothetical event analysis). In doing so, the accuracy of the estimation of a 3-D conductivity function is generally limited by the fact that here we deal with point transforms of the ﬁeld. The transition to integral transforms naturally allowing for the coupling between electromagnetic ﬁeld

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289

Fig. 11.7. Anomalous ﬁeld ratio R for the symmetric 2-D model. One-sided results plotted against distance (y) from the center (y ¼ 0) of the rectangular conducting prism. Results at three periods are shown: (1) 1000 s, (2) 100 s and (3) 10 s. (a) Real part of R and (b) Imaginary part of R (after Beamish, 1987).

Fig. 11.8. Line radials constructed using Re R determined from the anomalous ﬁeld for the same model for two periods (a) 1000 s and (b) 10 s. The inner rectangle shows one-half of the buried conducting region (after Beamish, 1987).

components measured at different points of the surface, however, calls for synchronous observations, which are difﬁcult to do in practice. To surmount this difﬁculty, Beamish and Banks (1983) suggested that a common reference point should be used for data recording. The limitation of this solution to the problem lies in the fact that the presence of anomalous horizontal ﬁelds at reference points shifts the results. This approach was subsequently reﬁned by Banks (1986), who reduced vertical magnetic ﬁeld components to a single instant of time by means of couplings ^ Another solution to the problem between the components of the tensors J^ and M. has originated with Parkinson (1990). It involves step-by-step deﬁnition of the vertical magnetic ﬁeld at the surface within an ever-increasing accuracy via Gilbert transforms (with known magnetic transfer functions). Fig. 11.8 presents magnetic

290

V.V. Spichak/Magnetotelluric Field Transformations

Fig. 11.9. Anomalous ﬁelds for the 2-D model shown at the bottom. The solid line has been computed using the Brewitt–Taylor and Weaver 2-D code, the dots have been calculated from the transfer functions. Reading from top to bottom: in-phase horizontal, quadrature horizontal, in-phase vertical, quadrature vertical (after Parkinson, 1990).

ﬁeld plots for a 2-D model (E-polarization), which have been obtained either by direct calculations or by the iterative procedure (six iterations) suggested by the author. Fig. 11.9 displays a very good agreement of the plots for the vertical ﬁeld component and a small discrepancy between the horizontal component plots (which may be attributed to the calculation error). It is worth noting that Dmitriev and Mershchikova (2002) have suggested a way for the magnetic ﬁeld reconstruction from the induction vectors known at a number of sites at the earth surface (see Section 11.4 for more details) that enables interpretation of non-synchronous MT data using their integral transformations.

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291

11.3. EXAMPLES OF THE USE OF MT FIELD POINT TRANSFORMS FOR THE INTERPRETATION 11.3.1. Dimensionality indicators While interpreting electromagnetic data measured at a single point or over an area, simultaneously or separately, an attempt is often made to estimate the dominant size of a studied conductivity anomaly as well as to divide an observed ﬁeld into parts consistent with various mechanisms of their formation. Some of these questions can be answered by resorting to the point transforms of the ﬁeld discussed in a previous section. Beamish (1986) used the MT sounding data obtained in Southern Scotland and in the north of Great Britain to study whether it is possible to divide the response observed at a single point into 1-, 2- and 3-D parts by means of the dimensionality indicators put forward by Kao and Orr (1982) or by the traditional ones (skew, ellipticity and eccentricity). An analysis has revealed that skew as well as the dimensionality indicators of Kao and Orr yield fairly reliable estimates. Ranganayaki (1984) investigated, with this aim in mind, along with the skew some more parameters, and established that their contour maps at the Earth’s surface provide an estimate of the predominant size of a geoelectric structure at measurement points. Meanwhile, Hermance (1982) had earlier shown by numerical calculations on a thin-sheet model that electromagnetic ﬁeld anomalies and the parameter skew are always correlated quite clearly. ^ Yee and Having applied canonical decomposition to the telluric operator T, Paulson (1987b) showed that in 2-D geoelectric structures the information contained within this operator is ﬁtted by ﬁve (rather than eight in a 3-D case) parameters. This fact is basic to the procedure suggested by the authors for separating ^ 2- and 3-D contributions to T. Iliceto et al. (1986) proposed that an indicator of two- and three-dimensionality should be provided by the parameter R¼

jtxy þ tyx j jtxx þ tyy j

ð11:19Þ

^ For the sake of illustration, where tab ða; b; ¼ x; yÞ are the elements of the tensor T. the authors performed numerical calculations for several typical 2-D models. Thus, Fig. 11.10 depicts the pseudo-sections of R produced for a graben model. It is evident from the ﬁgure that the pseudo-section of this parameter yields a fairly good approximation of the conductivity pattern. Zhdanov and Spichak (1992) used three-dimensionality indicators proposed by Swift (1967), Kao and Orr (1982) and Bahr (1988) in order to verify the justiﬁcation of a 2-D interpretation of MT data along the Lincoln Line (in the frameworks of the EMSLAB experiment).

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Fig. 11.10. R(o) pseudo-sections for the Graben model (1:100 resistivity ratio) with reference base at left inﬁnity (a) and located at ‘‘b’’ (b), (c) and (d) show pseudo-sections of the ‘‘step’’ model with resistivity ratios 10:1000 and 1000:10, respectively (after Iliceto et al., 1986).

11.3.2. Local and regional anomalies Another important problem successfully solved in terms of point ﬁeld transforms is the determination of the regional strike of a structure and identiﬁcation of a local disturbance against its background. To this end, Banks and Beamish (1984) took the frequency dependence of the azimuths of real inductive vectors at various points of the surface. In this way, ranges of periods (and, accordingly, of space coordinates) were established over which currents induced in the Earth are determined by the local and regional patterns of conductivity. Menvielle and Tarits (1986) examining the Rhine–Graben conductivity model had virtually to decide upon one of the two explanations of the magnetic ﬁeld

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anomaly – by local induction in a 2-D structure or by static deviation of telluric currents by poorly conducting crystalline masses (regional structure). To ﬁnd the answer to these questions, the authors resorted to the notion of adjustment distance of the inductive mechanism: for l2 =S420 (where l2 ¼ 2=ðmosÞ and S is the crosssectional area of the anomaly) they decide on the second mechanism, while for l2 =So20 they tend to the ﬁrst one. Their theoretical considerations have been conﬁrmed experimentally: the curves for the moduli and the phases of induction vector at two different points at T>1000 s coincide up to a constant factor. Zhang et al. (1987), Bahr (1988) and Groom and Bailey (1991) studied the properties of the impedance Z in a long-wave approximation, using for this purpose a model consisting of a near-surface local inhomogeneity and a regional structure. Zhang et al. (1987) claim that the regional strike is characterized by the direction ^ at which the elements of the columns of Z are proportional and their ratios b ¼ Z xx =Z yx and g ¼ Zyy =Z xy are real and independent of the period. The local strike is noted for the direction at which the impedance diagonal elements are proportional and the parameter a ¼ Zxx =Zyy is real, negative and independent of the period T. To separate the effects of local disturbance and regional induction, Bahr (1988) has elaborated a method of telluric vectors. It relies on the information about the impedance phases elements. Fig. 11.11 plots phases of all the impedance tensor elements versus the coordinate system chosen. At a2 ¼ 471, the phases Fxy and Fxx corresponding to the unit vector ey, are close, whereas the other two are not. When a1 ¼ 591, the phases corresponding to ex are identical. This circumstance underlies a method of determining the regional strike. Within this method, a system of coordinates is chosen to correspond to a1, and instead of four impedance phases one employs two phases of telluric currents, "

ðImZ xx Þ2 þ ðImZyx Þ2 tgjx ¼ ðReZ xx Þ2 þ ðReZyx Þ2

#1=2

"

;

ðImZ xy Þ2 þ ðImZ yy Þ2 tgjy ¼ ðReZ xy Þ2 þ ðReZ yy Þ2

#1=2 ð11:20Þ

which are subsequently examined. Under the same model, when the frequency is low enough that the inductive response can be neglected, Groom and Bailey (1989) decompose the data to obtain seven parameters per frequency: regional strike, two parameters describing the effects of the local electric ﬁeld distortion (twist and shear) and two complex regional impedances. 11.3.3. Constructing resistivity images in the absence of prior information In actual practice, one often faces with the necessity of inverting the measured data when there is practically no prior information about the resistivity distribution. In this case, ‘‘regular’’ methods of inversion based on the use of prior data do not work. The key to the solution of this problem was found as early as in (Tikhonov, 1950, 1965), which substantiated the possibility of constructing a 1-D geoelectric

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Fig. 11.11. Phases of the elements of the impedance tensor (bottom) and phases of the ‘‘telluric vectors’’ of site WAL, T ¼ 1 min, at a stepwise coordinate transformation (after Bahr, 1988).

section under each observation point based on MT data measured on the surface for a series of frequencies. Despite the development of 2- and 3-D inversion methods, to this day, the approach based on the synthesis of 1-D resistivity proﬁles has remained an effective tool for the target imaging in the absence of prior information. Necessity of such an approach increases in the 3-D case, when the measured data is deﬁcient, prior information is too scanty, and computer codes implementing the inversion of MT data in the 3-D Earth are very slow. In this situation, a unique practical recourse, which is especially helpful for a prompt tentative estimation of the resistivity distribution, lies in constructing a 3-D

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295

image of the medium based on the MT ﬁelds or their transformations: ^ i ðr; oj Þ ð j ¼ 1; 2; . . . ; N o Þ F~ i ðr; ðzapp Þj Þ ¼ TF

ð11:21Þ

where Fi are the components of the MT ﬁeld measured on the surface for No frequencies, T^ the transforming operator, F~ the MT ﬁeld image, r the radius-vector of the observation point, oj the frequency and (zapp)j the apparent depth corresponding to this frequency. ^ 2 , where Z^ is the impedance, then F~ takes ^ ¼ 1 jZj Note that if, for example, TF m0 o on the meaning of the apparent resistivity. At the same time, it is possible to imagine other transforms of the observed ﬁeld which make no clear physical sense, but which enable one to obtain a focused image of the medium (it is interesting to note in this connection the probability tomography method introduced in Mauriello and Patella (1999a, 1999b), which results in the so-called ‘‘conductivity anomaly occurrence probability function’’). It is obvious that the quality of such an image, based on the synthesis of 1-D apparent resistively proﬁles, depends essentially on the ﬁeld components used or their transformation. In particular, when two dissimilar data sets have identical information contents, it is possible to obtain two completely different images of the medium. Therefore, it is important to ﬁnd the components of the MT ﬁeld and their transforms that ensure the best focusing of the geoelectric structure. For this purpose, let us consider the prism model (1 1 1 km; 0.2 S/m) embedded at a depth 1 km in a half-space of 0.01 S/m. The electric and magnetic ﬁelds as well as the impedances, the apparent resistivity and other functions of the MT ﬁeld, were calculated for this model for a series of periods in the range 0.1–10 s. Under each point of the surface, the apparent resistivity proﬁles were constructed, and then, using a 3-D spline interpolation, a 3-D conductivity image was synthesized (Spichak, 1999). Fig. 11.12a shows the lower half-space image obtained from the apparent resistivity as the model ﬁeld transformation. The resistivity was calculated from an invariant of the impedance known as ‘‘determinant’’ (see Formula 11.11). Fig. 11.12b demonstrates that the boundaries of the conductive insert are well determined, with apparent resistivity values in the vicinity of the insert ranging from 2 to 10 O m (the electric conductivity varies accordingly from 0.5 to 0.1 S/m), and in the host medium, from 90 to 97 O m (the electric conductivity varies accordingly from 0.011 to 0.01 S/m), which departs from the true value by no more than 10%. Comparison of these results with those given in Spichak et al. (1999) shows that, although the precision of the electric conductivity reconstruction can be much worse than for a full inversion, the geometrical features of the anomaly are determined quite satisfactorily. Now, let us consider an example of an application of the imaging method just described that is closer to real practice. For this purpose, take a probable crustal conductivity model in which the highly conductive target (s ¼ 1 S/m) is located at 1 km depth in a low-resistivity subsurface layer (s1 ¼ 0.2 S/m, h1 ¼ 2 km) underlain by a high-resistivity basement (s2 ¼ 0.002 S/m) (Szarka and Menvielle, 1999) (Fig. 11.13).

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Fig. 11.12. The distribution of apparent resistivity obtained from model data that were calculated for the prism model (1 1 1 km, 0.2 S/m) embedded in the homogeneous half-space of 0.01 S/m: (a) 3-D resistivity image and (b) pseudo-section under the central proﬁle (after Spichak, 1999).

Fig. 11.13. A model of a highly conductive target located in a low-resistive near-surface layer: (a) vertical cross-section (y ¼ 1.5 km) and (b) horizontal cross-section (z ¼ 1.05 km) (after Szarka and Menvielle, 1999).

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297

Fig. 11.14. Horizontal slices of a 3-D resistivity image of the lower half-space: (a) z ¼ 0 and (b) z ¼ 1.05 km (after Spichak, 1999).

The MT ﬁeld for this model was calculated for periods T ¼ 0.01; 0.03; 0.1; 0.3; 1.0; 3.0; 10.0; 30.0; and 100.0 s for two polarizations of the primary ﬁeld. Then, for each period, the apparent resistivity corresponding to the impedance determinant was calculated, and under each node of the surface grid, the apparent resistivity proﬁles were constructed. Their synthesis was carried out by means of a 1-D spline interpolation, yielding a 1-D image of the lower half-space, whose 2-D horizontal projections are shown in Fig. 11.14. The analysis of the image, thus, constructed shows that the geometrical parameters of the 3-D insert, just as in the previous example, are determined well enough. The contours of the target are well deﬁned on the surface (Fig. 11.14a) and less so at a depth z ¼ 1.05 km, where the conductive target (Fig. 11.14b) is actually located. At the same time, while the apparent resistivity in the projection of the anomaly to the surface is 48 O m, which is only 4% less than the background value, at the depth z ¼ 1.05 km, it equals 27–28 O m; although this is two times less than the background value, it is still much greater than the true value, 1 O m. For comparison, an electric conductivity distribution at the same depth, obtained for this model by means of the Bayesian inversion (Spichak, 1999), is shown in Fig. 11.15. It clearly demonstrates that in this case the minimal resistivity value in the target zone equals 1.36 O m, which departs from the true value by only 36%. On the other hand, the contour of the anomaly is less distinct than in the case of imaging using the apparent resistivity (Fig. 11.14a, b). Note that in the latter case, in both the vertical section y ¼ 1.5 km and in the vertical section y ¼ 0 the depth of the image and its geometry are strictly consistent with the model (Fig. 11.16). Therefore, the above example demonstrates how a 3-D image can be constructed from MT data when we have no prior information on the geoelectric section. In this case, the depth and geometrical parameters of the target are determined quite conﬁdently.

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Fig. 11.15. A horizontal slice (z ¼ 1.05 km) of the 3-D conductivity distribution obtained by Bayesian inversion (after Spichak, 1999).

We have considered only some examples of using point transformations of the MT ﬁeld for interpretation of actual data. While point transformations are indispensable in the analysis of non-synchronous ﬁeld records, in the presence of MT data recorded synchronously at several points (or even arrays of points) we can employ interpretation methods based on accurate integral ﬁeld transforms.

11.4. INTEGRAL TRANSFORMS The solution of many MT problems involves integral transforms of the ﬁeld (forward problems, data processing, etc.). In this section, however, we will conﬁne ourselves only to the electromagnetic ﬁeld transforms directly related to data interpretation. 11.4.1. Division of the MT field into parts It is often useful to divide MT data recorded synchronously at a number of sites into external and internal, background and anomalous, surface and deep ones in accordance with their origin. Another fruitful idea based on the integral transforms of the ﬁeld, is concerned with the analytic continuation of it down or up from the surface where the observations are made.

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299

Fig. 11.16. Vertical cross section of the 3-D resistivity image of the lower half-space: (a) y ¼ 1.5 km and (b) y ¼ 0 (after Spichak, 1999).

Berdichevsky and Zhdanov (1984) expose comprehensively the transformation methods in which the division can be regarded as linear ﬁlters performing the following integral transforms: (a) division of the magnetic ﬁeld into external and internal parts;

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division of the magnetic ﬁeld into background and anomalous parts; division of the magnetic ﬁeld into surface and deep parts; and separation of the major part of a deep anomaly. In a 2-D case, for instance, this set of operators includes two types of the matrix ^ operators B^ and C. (1) The operator B^ affects the magnetic ﬁeld: (b) (c) (d)

1 ^ H ðx ; z Þ ¼ BHðx; 0Þ ¼ 2p T

0

0

Z1

^ x ; z0 Þhðkx Þeikx x dkx bðk

1

Z1 ¼

b G^ ðx0 x; z0 ÞHðx; 0Þ dx

ð11:22Þ

1

where H and h are the magnetic ﬁeld and its spectrum at the surface, respectively: H¼

Z1 hx H x ikx x ;h ¼ ¼ e dx Hz hz Hz

Hx

1

(2)

b G^ and b^ stand for the kernel of integral transforms and its characteristic, respectively: " # Z1 b b bxx bxz ikx x G G b 1 xx xz G^ ¼ e ¼ dkx 2p bzx bzz G bzx G bzz " # 1 H Tx T is the transformed ﬁeld. H ¼ H Tz The operator C^ affects the sum vector of the magnetic ﬁeld and extraneous current and the transformation has the form

1 ^ 0Þ ¼ H ðx ; z Þ ¼ CFðx; 2p T

0

0

Z1 1

^ x ; z0 Þf ðkx Þeikx x dkx ¼ Cðk

Z1

c G^ ðx0 x; z0 ÞFðx; 0Þdx

1

ð11:23Þ where F and f are the sum vector and its spectrum, respectively 2 3 2 3 2 3 hx Hx Z1 H x 6 7 6 7 6 s 7 ikx x F ¼ 4 I sy 5; f ¼ 4 I sy 5 ¼ 4 I y 5e dx 1 Hz hz Hz c ^ and G ; c^ are the kernel of integral transform and its spectral characteristic, respectively " c c c # Z Gxx G xy G xz c 1 1 cxx cxy cxz ikx x ^ ¼ G ¼ e dkx Gczx G czy G czz 2p 1 czx czy czz

V.V. Spichak/Magnetotelluric Field Transformations

301

Thus, the division and continuation operators act asb linear ﬁlters with specc tral characteristics b, c and spatial characteristics G^ ; G^ . For example, basing on relation (11.22) we obtain e;i

He;i ¼ B^ H a;b ; etc:, Ha;b ¼ B^ H where He, Hi are external and internal ﬁelds, correspondingly; Ha, Hb – anomalous and background ﬁelds. Using linear transforms of spatial spectra of the ﬁeld, Shabelyansky (1985) constructed operators performing these transformations in a 3-D medium. Zhdanov et al. (1987) employed such operators to divide the magnetic ﬁeld observed on

Fig. 11.17. Geomagnetic ﬁelds of VCM anomaly (E-polarization) for T ¼ 1800 s: (a) observed magnetic ﬁelds Hx, and Hz (b) results of Hx separation: a-anomalous part, s-surface, d-deep, and (c) results of Hz separation (after Zhdanov et al., 1987).

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the Voronezh crystalline massif into a background and anomalous, surface and deep one. In a similar manner, Zhdanova (1986) divided ﬁelds measured at the bottom of the sea (within 2-D ocean models excited by an H-polarized ﬁeld). Zhdanov and Shabelyansky (1988) solved the problem of dividing the electromagnetic ﬁeld taken at the sea bottom into a normal and anomalous parts in the presence of an extraneous ﬁeld source for a plane model of the Earth. Berdichevsky and Yakovlev (1984) derived a pair of integral transforms relating the electric and magnetic components of an anomalous ﬁeld observed at the surface. The authors discussed the applicability of these transforms to the solution of several interpretation problems. Another type of the ﬁeld integral transforms relies on the body of Stratton– Chu-type integrals (Zhdanov, 1988). For instance, within this approach formulas for the division of the ﬁeld, recorded over arbitrary surface into anomalous and background parts, look as follows (Zhdanov and Spichak, 1983): Ea ðr0 Þ ¼ 1=2Eðr0 Þ þ Ke0 ðr0 Þ þ DKe ðr0 Þ Eb ðr0 Þ ¼ 1=2Eðr0 Þ þ Ke0 ðr0 Þ þ DKe ðr0 Þ m Ha ðr0 Þ ¼ 1=2Hðr0 Þ þ Km 0 ðr0 Þ þ DK ðr0 Þ b m H ðr0 Þ ¼ 1=2Hðr0 Þ þ K0 ðr0 Þ þ DKm ðr0 Þ

ð11:24Þ

where Ke;m 0 ðr0 Þ ¼

ZZ n

h e o i e iom0 G^ 0 r0 =r0 ½n H þ rG^ 0 r0 =r0 ½n E ds0

S

Km 0 ðr0 Þ ¼

ZZ n

h m o i m iom0 G^ 0 r0 =r0 ½n H þ rG^ 0 r0 =r0 ½n E ds0

S e Eb, Hb and Ea, Ha are the background and anomalous ﬁelds, accordingly;G^ 0 and m G^ 0 are the tensor Green functions of an inhomogeneous medium of the electric and e;m e;m e;m e;m magnetic types, respectively; DG^ G^ G^ 0 ; G^ ðr0 =r0 Þ-Green function of the homogeneous medium with conductivity s ¼ sðr0 Þ. Thus, in synchronous electromagnetic observations we have at our disposal a fairly large arsenal of techniques permitting effective data analysis.

11.4.2. Transformation of the field components into each other In many applications, it is useful to be able to convert MT ﬁeld components measured synchronously at a number of sites at the plane Earth’s surface into another ones. An appropriate basement for such transformations is given by solution of the Poisson’s equation in the atmosphere regarding anomalous MT ﬁeld

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303

components (Dmitriev and Mershchikova, 2002). The same result was obtained by Zhdanov (1988) using the Stratton–Chu apparatus _h

E ax ðx0 ; y0 ; zÞ ¼ iomG H ay

ð11:25aÞ

_h

E ay ðx0 ; y0 ; zÞ ¼ iomG H ax

ð11:25bÞ

_h

H ax ðx0 ; y0 ; zÞ ¼ G x H az

ð11:25cÞ

_h

H ay ðx0 ; y0 ; zÞ ¼ G y H az _h

_h

H az ðx0 ; y0 ; zÞ ¼ Gx H ax G y H ay

_h

_h

ð11:25dÞ ð11:25eÞ

_h

where integral operators Gx , G y and G x are determined as follows: _h

Z1 Z

_h

G x f ðx0 ; y0 Þ ¼ G x f ¼ 2

@G h f dx dy @y

1 _h

Z1 Z

_h

Gy f ðx0 ; y0 Þ ¼ G y f ¼ 2

@Gh f dx dy @y

1 _h

_h

Z1 Z

G f ðx0 ; y0 Þ ¼ G f ¼ 2

G h f dx dy

1

Gh ¼

1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 4p ðx x0 Þ þ ðy y0 Þ2 þ z2

These formulas are used, in particular, for calculation of the magnetic ﬁeld from the electric ﬁeld determined by forward modeling (Zhdanov and Spichak, 1992) and also for internal testing the results of the MT ﬁeld computation (Spichak, 1999) (see in this connection Sections 5.2.3 and 5.2.4 in the Chapter 5, accordingly). 11.4.3. Synthesis of synchronous MT field from impedances and induction vectors In order to solve the inverse problem of magnetotellurics it is useful to deal with MT ﬁeld components synchronously determined at the Earth’s surface. This enables, ﬁrst, to use more sophisticated and fast algorithms of the data interpretation and, second, to reduce the geological noise by means of the subsequent upward analytical continuation of the anomalous ﬁeld (Spichak, 2001). Dmitriev and Mershchikova (2002) solve the problem of reconstructing a magnetic ﬁeld from its impedance and tipper.

304

V.V. Spichak/Magnetotelluric Field Transformations

11.4.3.1. Magnetic ﬁeld synthesis from known impedance Let us rewrite relation (11.2) in the form H x Y xx E x Y xy E y ¼ 0 H y Y yx E x Y yy E y ¼ 0

ð11:26Þ

where Y xx , Y xy , Y yx and Y yy are the admittance elements determined from the impedance Z

Z

Y xx ¼ Z2yy ;

Y xy ¼ Z2xy

inv3

Y yx ¼

inv3

Z Z2yx inv3

;

Y yy ¼

Zxx Z2inv3

and Z inv3 – impedance invariant determined according to (11.12). Substituting (11.25a) and (11.25b) into (11.26), we obtain a couple of integral equations on the unknown components of the anomalous magnetic ﬁeld h H ax iomðG^ ðH ax H ay ÞÞ ¼ H bx þ Y xx E bx þ Y xy E by h H ax iomðG^ ðH ax H ay ÞÞ ¼ H by þ Y yx E bx þ Y yy E by

ð11:27Þ

It is worth mentioning that the knowledge of the background ﬁeld is not necessary. Indeed, if the background magnetic ﬁeld is polarized, for example, in OY axis (i.e. H bx ¼ 0, H by ¼ 1, E bx ¼ Z b H by , E by ¼ 0), we can introduce normalized anomalous magnetic ﬁelds hx ðx; yÞ ¼

H ax ; H by

hy ðx; yÞ ¼

H ay H by

Now we can rewrite the system of equations (11.27) as follows: h

hx iom ðG^ ðhx hy ÞÞ ¼ Y xx Z b h hy iom ðG^ ðhx hy ÞÞ ¼ Y yx Z b 1

ð11:28Þ

After determining the horizontal components of the anomalous magnetic ﬁeld from integral Equation (11.28), it is possible to ﬁnd the vertical component from (11.25e). 11.4.3.2. Magnetic ﬁeld synthesis from known tipper Similarly, if we know the tipper components Wzx and Wzy, it is possible to synthesize the magnetic ﬁeld using the relations (11.25c, d) _h

_h

H az W zx G x H az W zy G y H az ¼ W zx H bx þ W zy H by

ð11:29Þ

As in the previous case, it is not necessary to know the background ﬁeld. After calculation of the vertical component of the anomalous magnetic ﬁeld, it is possible to determine the horizontal components using (11.25c) and (11.25d) and, if

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305

necessary, to compute subsequently the electrical ﬁeld (using (11.25a, b)) and, ﬁnally, the impedance. Thus, an analysis of various ﬁeld transformations and corresponding methods of interpretation has indicated certain trends in this ﬁeld of investigation. Firstly, there is a trend to hybridization and combination of transformation methods used for interpretation; secondly, data interpretation tends more and more clearly to rely on mathematical methods and procedures adequate to a problem (particularly, this is true of point transforms); thirdly, on transition from 1- and 2- to 3-D interpretations we obviously need a better insight into the behavior of electromagnetic ﬁelds and their transformations in typical 3-D media.

Acknowledgements The material of this chapter is based on the review paper of the author and is included with kind permission of Springer Science and Business Media.

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Zhdanov, M.S., 1988. Integral Transforms in Geophysics. Springer, New York, 367pp. Zhdanov, M.S. and Keller, G., 1994. The Geoelectrical Methods in Geophysical Exploration. Elsevier, Amsterdam, 873pp. Zhdanov, M.S., Maksimov, V.M., Gruzdev, V.N. and Shabelyansky, S.V., 1987. Results of separation of the ﬁeld of bay disturbances of the Voronezh Cristalline Massifs: Methods of solution of direct and inverse geoelectrical problems (in Russian). IZMIRAN, Moscow, 100–107. Zhdanov, M.S. and Shabelyansky, S.V., 1988. Separation of the Electromagnetic Field at the Bottom of Seas and Oceans into Normal and Anomalous Ones with due Refence to a Hydrodynamic Source (in Russian): Preprint 25 (779). IZMIRAN, Moscow, 15pp. Zhdanov, M.S. and Spichak, V.V., 1983. Stratton—Chu-Type Integrals for Inhomogeneous Media and Some of Their Applications to Geoelectrical Problems (in Russian): Nathematical Modeling of Electromagnetic Fields. IZMIRAN, Moscow, 4–25. Zhdanov, M.S. and Spichak, V.V., 1992. Mathematical Modeling of Electromagnetic Fields in 3D Inhomogeneous Media (in Russian). Nauka Publication, 188pp. Zhdanova, O.N., 1986. Ways of the spatial analysis of EM ﬁelds at the sea bottom (in Russian). Fundamentalnie problemi morskikh elektromagnitnikh issledovanii, IZMIRAN, Moscow, pp. 108–122.

Chapter 12 Modeling of Magnetotelluric Fields in 3-D Media V.V. Spichak Geoelectromagnetic Research Center IPE RAS, Troitsk, Moscow Region, Russia

In this chapter, on the basis of the approaches to solving the forward and inverse problems considered in the previous chapters, certain methodological issues of practical interest are investigated. In Section 12.1, a case study of a model for the Achak gas-condensate ﬁeld is used to assess the feasibility of detecting hydrocarbon deposits from magnetotelluric data, and a methodology for interpreting MT data in the case of low-contrast target is proposed. Section 12.2 presents a comparative analysis of magnetotelluric (MT) ﬁelds in alternative models for deep geoelectric structure of the Transcaucasian Region, constructed on the basis of prior information and data obtained by means of geothermics, gravimetry and MT methods. Although electromagnetic data for the area in question are of an extremely nonsystematic nature, their analysis has revealed certain regularities in the behavior of the electric and magnetic ﬁelds, as well as Wiese vectors, which made it possible to put forward some hypotheses concerning the geoelectric structure of the region. To prove or disprove them, four alternative models of electric conductivity distribution in this region were constructed, which differ in the type of near-surface conductivity structure and in the presence or absence of a conducting channel linking the Black and Caspian seas. A comparative analysis of the numerical modeling results has enabled an assessment of (1) the resolving power of the MT method with respect to the type of near-surface structure of the electric conductivity and (2) the impact of a highly conducting layer (asthenosphere) on the MT ﬁelds observed on the surface. In Sections 12.3 and 12.4, three dimensional (3-D) electric conductivity models of volcanoes are used for study of feasibility of their inner-structure visualization, Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40012-3

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and of monitoring the electric conductivity of the melt in a magma chamber on the basis of MT data measured on a relief surface. The studies indicate that the best resolution in complex geological media is shown by impedance phases as well as by the real and imaginary parts of the ‘‘electric ﬁeld’’-type transform. Based on the results obtained, methods for the volcano inner-structure imaging and for monitoring the electric conductivity within a magma chamber are proposed. Section 12.5 presents a study of the audio MT method resolving power (within corresponding frequency range) for detecting groundwater salinization by seawater and mapping the salinized areas. The problem is solved by means of a Bayesian inversion of synthetic MT data on the Earth’s surface at a number of frequencies. Expert estimates of the salinization probability are taken into account as a priori data. As a result, a posteriori distribution of electric conductivity in the survey area is obtained, which shows the boundaries of the salinized area – in both vertical and horizontal projections – to be established in a reliable manner.

12.1. A FEASIBILITY STUDY OF MT METHOD APPLICATION IN HYDROCARBON EXPLORATION Among a wide range of experimental studies aimed at investigation of natural and artiﬁcially induced seismic, deformational and ﬂuid-dynamical processes in a pay strata and hosting rock and in the upper part of the section, a considerable part belongs to a complex of geophysical studies. Despite the fact that, for the purpose of potential hydrocarbon reservoirs localization, those most commonly used are the seismic methods, they are poorly suited for ﬂuid content determination. In addition, employment of seismic methods is ineffective within regions of complex geological structure and relief surface. In this connection, since recently, the electromagnetic methods of hydrocarbon reservoirs exploration and mapping is becoming addressed more and more often. Among these, of growing popularity is the MT method (David et al., 2002; Watts et al., 2002) owing to its high efﬁciency and capability of determination of the hydrocarbon deposit parameters in complexly structured media. Meanwhile, most of the geophysical studies carried out up to date were conﬁned to the construction of 2-D geoelectric cross section that does not provide an adequate idea of a hydrocarbon deposit extension and, correspondingly, does not allow a correct estimation of the deposit potential. As it is known, methodology of MT sounding data interpretation is most fully developed for 1-D and 2-D situations, while the real medium is evidently three-dimensional. In this view, evaluation of the resolving power of MT method with respect to the three-dimensional model of the hydrocarbon deposit that is most adequate to the reality is important. Besides, the existing methodology of the ﬁeld studies in hydrocarbon deposit exploration is not essentially unlike the conventional methods for other structures exploration, but it is characterized by more strict requirements to the accuracy and detailness of the data, which, eventually, results in the rise of cost of the ﬁeldwork.

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Therefore, another important task is the development of new methodological approaches to magnetotelluric data interpretation that would make it possible to increase the efﬁciency of oil and gas deposit exploration by means of magnetotelluric sounding. 12.1.1. Statement of the problem Let us estimate the resolving power of the MT method as to oil and gas deposits by studying the behavior of MT ﬁelds in a three-dimensional model of electric conductivity of Achak gas condensate deposit located in the North-East of Turkmenistan (Spichak and Larionova, 1990; Zhdanov and Spichak, 1992). Soundings with controlled sources are in wide use in oil and gas prospecting. At the same time, methods based on natural ﬁelds (in particular, MT) are much less applied, though they are more economic. MT method is chosen for this region because the measurements of MT ﬁelds could be the most effective on the territory with surface dry sands bedding, hard grounding conditions and several high-resistive screens. In this connection, it was important to assess the resolving power of MT method in three-dimensional modeling of the structure under study. Besides, the existing technique of ﬁeldworks in beds prospecting does not differ in essence from usual techniques of other structures prospecting but requires higher accuracy and more detailed data. All this ﬁnally rise the cost of investigations. That is why another important task is to design new methodological ways of MT data interpretation, which could increase the efﬁciency of oil and gas deposits prospecting by means of MT sounding. 12.1.2. Numerical modeling To solve these problems, numerical calculations of simpliﬁed three-dimensional electrical conductivity models of Achak gas-condensate deposit were carried out in Spichak (1999a, 2006a), and a series of transformations of a synthetic MT ﬁeld were analyzed. Below, the main results obtained in these publications will be discussed. The electric conductivity model adequately reproducing the basic features of a geological structure in this region, consists of a 3-D relatively low-conductive embedment with the size of 20 10 0.15 km located at a depth of 1.4 km in the upper layer of a three-layer section (Fig. 12.1).The electric conductivity of the inclusion is 0.5 S/m. Electric conductivities of layers, from the upper down, are 0.78, 0.04 and 0.002 S/m, respectively. The calculations were made for six periods: T ¼ 5, 10, 15, 20, 25 and 30 s. As the model has two vertical planes of symmetry, only one fourth of its part was calculated, which allowed an essential run time reduction. According to the calculation results, amplitudes of ﬁeld components practically do not contain any information about the anomaly. On the other hand, contour plots of phases of various ﬁeld components show that the phases do carry sufﬁcient information (see in this connection Section 9.4, Chapter 9).

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Fig. 12.1. Geoelectric model of the Achak gas-condensate deposit: (a) XOZ plane section and (b) XOY plane section.

Fig. 12.2. Phase contour map of the normalized component of an electric ﬁeld jE~ y ð10Þ (T ¼ 5 s, E b ¼ ð0; E by ; 0Þ).

The analysis of the phase contour maps of the normalized component of an electric ﬁeld jE~ y (jE~ y ¼ jE y jE by ; where jE by is the phase of the background electric ﬁeld) shows that at all frequencies the phase jE~ y increases above the inclusion, compared with the background value jE by . Besides, jE~ y contours at all periods are following the outline of the target of interest (Fig. 12.2 and 12.3), and on its boundaries that are perpendicular to the electric ﬁeld direction in the incident wave, the phase gradient is higher than along the boundaries coinciding with this direction. This, apparently, is due to the inﬂuence of charges accumulated at the edges of the insert (compare with the result of reconstruction of the resistive target, obtained in Spichak et al. (1999)). The anomalous ﬁeld is also seen on maps of phase contours of normalized horizontal magnetic ﬁeld component jH~ x ( jH~ x ¼ jH x jH bx , where jH bx is the phase of the background magnetic ﬁeld). Spatial distribution of this phase is more complex than the electric ﬁeld component phase considered above, although the

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Fig. 12.3. Phase contour map of the normalized component of an electric ﬁeld jE~ y ð10Þ (T ¼ 10 s, E b ¼ ð0; E by ; 0Þ).

Fig. 12.4. Phase contour map of the normalized component of an magnetic ﬁeld jH~ x ð10Þ (T ¼ 10 s, Hb ¼ ðHbx ; 0; 0Þ).

absolute value of jH~ x is lower than that of jE~ y (Fig. 12.4). It is important to note that at all frequencies the changes in the sign of the magnetic ﬁeld phase (jH~ x ) are attributed to the embedment edges perpendicular to the magnetic ﬁeld direction in the incident plane wave. At the same time, no period-dependent variations are observed in the contour shapes. Thus, the phase behavior of this ﬁeld component also gives an idea about the location of the embedment horizontal boundaries perpendicular to the primary magnetic ﬁeld. Therefore, more complete information about horizontal boundaries of an anomalous region can only be obtained from a combined interpretation of phases of mutually perpendicular components E~ y and H~ x , corresponding to the directions of electric and magnetic ﬁeld in the primary ﬁeld.

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The analysis of the H z phase shows that above the inclusion along its edges perpendicular to the magnetic ﬁeld polarization, the maximum values of jH z are observed, being most intensive at a frequency corresponding to the embedment depth. At the same time, the relevant contours are so complex-shaped that their further analysis is too complicated. Let us now consider the behavior of some transforms of the synthetic electromagnetic ﬁeld, which are often used for real data interpretation (see Chapter 11 and references therein). Analysis of the contour maps of apparent resistivity shows that absolute variations in this parameter in the considered model are too small (2–3%) to be distinguished in the ﬁeld measurements since the noise level is usually higher. Therefore, it is worthwhile to depart from the ordinary scheme of the MT data interpretation based on apparent resistivity curves, and to consider other possibilities as, for example, those associated with the creation of induction vector maps and pseudosections (apparent structures). In Fig. 12.5 and 12.6, maps of real and imaginary induction vectors for periods T ¼ 10 and 20 s are shown. Despite the vector lengths being small (which is due to weak conductivity contrast between the anomalous embedment and the enclosing rock), their directions keep no doubts in the horizontal location of the center of

Fig. 12.5. Map of the real induction vectors (T ¼ 10 s).

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Fig. 12.6. Map of imaginary induction vectors (T ¼ 20 s).

inclusion. The lengths of real and imaginary vectors vary with period, with their maxima being observed at T ¼ 10 s, where the skin depth is approximately equal to the depth of the upper edge of the target. The position of the embedment center and its contours in a vertical plane could be roughly estimated from pseudosection maps constructed using the imaging technique discussed above in Chapter 11. Note that, using the ﬁeld phases and some of their functions, it is possible to create images with visible geometrical contours of anomalous embedment. In particular, in the considered model the pseudosections of electric and magnetic ﬁeld phases (Fig. 12.7 and 12.8) and impedance phases (Fig. 12.9) allow better resolution of the structure than those of apparent resistivity. As seen from these maps, the contours of an anomalous zone could be detected from both ﬁeld and impedance phases. Thus, the analysis of MT ﬁeld behavior in three-dimensional model of Achak gas-condensate deposit allows drawing conclusions as follows: 1. An ordinary scheme of MT data interpretation on the basis of apparent resistivity curves is inefﬁcient here. 2. Horizontal position of the center of anomalous zone can be determined for all periods using maps of real and imaginary induction vectors. 3. Horizontal boundaries of anomaly are rather well contoured by maximum gradients in phases of electrical and magnetic ﬁelds perpendicular to the corresponding borders.

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Fig. 12.7. Horizontal electric ﬁeld component phase pseudosection (jE~ y ð10Þ, Eb ¼ ð0; E by ; 0Þ).

Fig. 12.8. Horizontal magnetic ﬁeld component phase pseudosection (jH~ x ð10Þ,Hb ¼ ðH bx ; 0; 0Þ).

The depth of the embedment can be estimated (a) from the skin depth for the period where the phase gradient reaches its maximum on the horizontal borders of anomaly and (b) from maps of pseudosections of electric and magnetic ﬁeld and impedance phases. It is worth mentioning in summary, that, although the obtained results apply to the speciﬁed model of an electric conductivity, the conclusions drawn can be useful in practice at a step of planning the MT sounding and at interpretation of the measured data as well. 4.

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Fig. 12.9. Impedance phase jxy pseudosection.

12.2. TESTING HYPOTHESES OF THE GEOELECTRIC STRUCTURE OF THE TRANSCAUCASIAN REGION FROM MT DATA The geoelectric structure of the Transcaucasian region is complex and insufﬁciently studied. Although the telluric and MT ﬁelds were measured at various points (Bukhnikashvili et al., 1969; Gugunava, 1988), the construction of meaningful models of the crustal conductivity is impeded by the insufﬁcient amount of measured electromagnetic data and sparse measurement points. On the other hand, numerical modeling of the ﬁeld in terms of hypothetical models of electric conductivity in the study region might answer some questions of the ‘‘What would happen if ...?’’ type. The results of such studies make it possible, on the one hand, to reject hypotheses that obviously contradict the available data and, on the other, to gain constraints on the inﬂuence of various structural elements on electromagnetic ﬁelds observed at the Earth’s surface. In this connection, answers to the following two questions are of interest: What is the resolution of magnetotelluric data with respect to the type of Transcaucasian shallow conductivity structure? What can be the inﬂuence of high-conductivity channel presumably pairing the Black and Caspian seas on the electromagnetic ﬁelds observed at the surface? 12.2.1. Geological and geophysical characteristics of the region The internal structure of the Transcaucasian region was studied by geothermal, gravimetric and electromagnetic methods. Although electromagnetic data available for this region are very fragmented, their analysis revealed some regular features in the behavior of electric and magnetic ﬁelds and Wiese vectors (Gugunava, 1988):

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In both long- and short-period intervals, the electric ﬁeld is polarized mainly quasi-linearly. 2. Contour lines of the electric ﬁeld are elongated in the E-W direction. 3. The intensity of the electric ﬁeld reaches a maximum near the central part of the Georgian block (the Dzirul Massif) and decreases toward the Black Sea. 4. Plots of components of the magnetic ﬁeld measured on three N-S proﬁles (Likhauri-Khaishi-Tyrnyauz, Bogdanovka-Zegduleti-Urozhainoe and Tabakhmela-Dusheti-Sunzha) and normalized to its value at the Dusheti Observatory as a reference point (Fig. 12.10) indicate that the amplitude of the vertical component reaches its maximum values in the axial zone of the Caucasus Range, whereas the amplitudes of the horizontal components vary insigniﬁcantly within the study region, reaching a maximum near the western slope of the Georgian block. 5. The Wiese vectors in the entire region are generally directed southward, slightly deviating toward the west. 6. Near the eastern coast of the Black Sea, the Wiese vectors are nearly parallel to the coast; i.e., the coast effect is virtually absent. The available experimental data have led to well-founded suggestions concerning both the regional distributions of main physical parameters of the crust in the study region and their controlling factors. According to Gugunava (1988), the Caucasus crust contains the following electrically conducting structures: the sedimentary complex 0–15–20 km thick; relics of magma chambers in the form of an oblong ellipsoid within the Greater Caucasus at depths of about 20 km as well as isolated lenses within the Lesser Caucasus at depths of 10–20 km; the electrically conducting crustal asthenosphere underlying the entire Transcaucasian region (0–25 km thick), with maximum thicknesses beneath the Greater and the Lesser Caucasus. Combined analysis of electromagnetic data of deep MT sounding and results of 3-D temperature calculations showed that an anomalous conducting zone or a crustal conducting horizon, is present within the lowermost units of the basaltic layer. Apparently, it is conﬁned to crustal regions consisting of partially molten water-saturated granites and basalts heated to 600 1C. This hypothesis was used to construct a map of the longitudinal conductance of the crustal inverted layer of the Caucasus (Gugunava, 1988) (Fig. 12.11). Although the above evidence indicates a complex geoelectric structure of the region, it is possible to clearly identify the main structural elements controlling the spatial distribution of natural electromagnetic ﬁelds in the Transcaucasian region: the sedimentary cover, the Black and Caspian seas and the Dzirul Massif (Georgian block). Moreover, as noted above, it is possible to assume the presence of a crustal conducting channel at a depth of 40–50 km connecting the Black and Caspian seas. To conﬁrm or reject the hypotheses of the shallow and deep distributions of the crustal electric conductivity in the Transcaucasian region, alternative models of electrical conductivity in this region were constructed in Spichak et al. (1988) and numerical modeling of electromagnetic ﬁelds was performed in Zhdanov and Spichak (1992) and Spichak (1999, 2006b). 1.

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Fig. 12.10. Plots of normalized components of the magnetic ﬁeld at periods of (1) 7–15, (2) 61–100, and (3) 100–130 min measured on N-S proﬁles: (a) Likhauri–Khaishi–Tyinyauz, (b) Bogdanovka–Zegduleti–Urozhainoe, and (c) Tabakhmela–Dusheti–Sunzha (Gugunava, 1988).

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Fig. 12.11. Map showing the longitudinal conductance (in Siemens) of the crustal inverted layer in the Caucasus (Gugunava, 1988).

Fig. 12.12. Simpliﬁed geometric scheme of the Transcaucasian region; 1-10 , 2-20 , and 3-30 are magnetotelluric sounding proﬁles (Spichak, 2006b).

12.2.2. Alternative conductivity models Here, we address the most simpliﬁed 3-D models of the electric conductivity in the region, involving only its main structural elements. Fig. 12.12 presents a schematic geometric model of the study region, in which the Black and Caspian seas are represented by rectangles, the sedimentary cover is bounded to the north and south by broken lines, and the horizontal contours of the Dzirul Massif are shown by a dashed line (the rectangle). As seen from this diagram, the modeled region is symmetrical relative to the vertical plane that passes through the axial OO0 line. Fig. 12.13 shows schematically vertical cross sections through the OO0 line of four models of electric conductivity. The models in Fig. 12.13a and Fig. 12.13c are intended to examine the effect of two types of shallow conductivity structure (the Dzirul Massif and the Georgian block, respectively); the models shown in

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Fig. 12.13. Distributions of conductivity in the vertical cross-section YOZ through the central proﬁle OO0 (Fig. 12.12) in the following four models of the Transcaucasian crustal conductivity: (a) Dzirul Massif, (b) Dzirul Massif+channel, (c) Georgian block, and (d) Georgian block+channel.

Fig. 12.13b and Fig. 12.13d are useful for estimating the contribution of the highconductivity asthenospheric layer (channel?) connecting the Black and Caspian seas to the electromagnetic ﬁeld at the Earth’s surface. Note that the choice of the simpliﬁed models constructed here is related to two circumstances. On the one hand, in view of the insufﬁcient amount of measured electromagnetic data in the Transcaucasian region, the construction of a 3-D regional conductivity model that would be as close as possible to reality is very difﬁcult, if not impossible. On the other hand to answer the above questions, it is enough to analyze hypothetical (but three-dimensional!) models that are not overloaded with inessential details.

12.2.3. Numerical modeling of magnetotelluric fields Numerical modeling, which is a versatile research tool, is particularly effective for testing hypotheses because it is considerably less time-consuming and expensive than ﬁeld experiments. If modeling results are at variance with the results of in situ observations, this indicates a low reliability of the hypothesis. On the other hand, their good agreement increases the plausibility of the hypothesis. In other words,

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mathematical modeling is not a substitute for a real experiment but contributes to its rational substantiation. Calculations of electromagnetic ﬁelds in the models of electric conductivity described above were performed with the FDM3D program (Spichak, 1983) on a spatial grid, with sides along the Ox, Oy and Oz axes equal to 1100, 900 and 100 km, respectively. Calculations were performed at three periods (T ¼ 10, 100 and 1000 s) under the assumption that the electric ﬁeld is polarized linearly in the direction of the Oy axis, coinciding with the axial line OO0 (Fig. 12.12). Some results of these calculations are presented below. 1. Notwithstanding the quasi-stationarity of the geoelectric structure, the electric ﬁeld at the surface in all of the four models is, on the whole, three-dimensional due to lateral geoelectric heterogeneities (the Dzirul Massif and the relief of the Georgian block). This conclusion agrees with the results of thin-layer modeling of the low-frequency telluric ﬁeld in the Transcaucasian region obtained by Vanyan et al. (1989). 2. The vector of the horizontal component of the synthesized magnetic ﬁeld in all four models is directed predominantly toward the south, insigniﬁcantly deviating from this direction only near the coast. This result leads to the important conclusion that the behavior of magnetic ﬁelds in the study region is controlled by systems of currents located mainly in the vertical plane. This is consistent with the experimentally conﬁrmed E-W direction of the currents in the region. 3. Model calculations indicate that the behavior of the horizontal components of the electromagnetic ﬁelds and the vertical component of the magnetic ﬁeld at the surface can be used to determine the type of the near-surface conductivity structure in the region: lower values of the horizontal components of the electromagnetic ﬁeld and higher values of the vertical components of the magnetic ﬁeld in the central part of the region reﬂect the presence of the Dzirul Massif in the sedimentary sequence; vice versa, higher values of the electromagnetic horizontal components and lower values of the vertical component of the magnetic ﬁeld indicate that the near-surface structure of the electric conductivity is controlled by the Georgian block. This conclusion can be illustrated by the behavior of the horizontal components of the magnetic ﬁeld in the coastal zone of the Black Sea. Fig. 12.14 plots horizontal components of the magnetic ﬁeld normalized to its ‘‘normal’’ value on the proﬁle 1-10 (Fig. 12.12), nearest to the aforementioned real proﬁle Likhauri–Khaishi–Tyrnyauz. As seen from these plots, the extreme values of the H~ x and H~ y components are conﬁned to the northern and southern boundaries of the Black Sea coast. The observed anomalies of both components are largest in models (c) and (d) and smallest in models (a) and (b) (Fig. 12.13). This result can be interpreted as follows. The anomalies in the horizontal components of the magnetic ﬁeld near the eastern coast of the Black Sea are determined by the intensity of currents in the vertical plane in the near-surface layer. As easily seen from comparison of the vertical cross sections of models (a), (b), (c) and (d) (Fig. 12.13), only one vertical system of currents can form in the sedimentary sequence in the latter two models, whereas the current ﬂowing in the former two models beneath the Dzirulskii Massif splits, forming a second current loop around the massif.

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Fig. 12.14. Normalized horizontal components of the synthetic magnetic ﬁeld (T ¼ 1000 s) at the Earth’s surface on the N-S proﬁle 1-10 (Fig. 12.12): (a–d) models shown in Fig. 12.13.

This reduces the intensity of the electric ﬁeld at the surface, which in turn leads to a decrease in the intensity of the horizontal components of the magnetic ﬁeld. The difference between the shallow conductivity structures is reﬂected most clearly in the behavior of the vertical component of the magnetic ﬁeld. Numerical modeling results show that the presence of the high-resistivity Dzirul Massif (with its top lying tentatively at a depth of 0.5 km) in the sedimentary sequence of the Transcaucasian region is determined from the characteristic features of the current ﬂow effect: the local maximums of the vertical component of the magnetic ﬁeld. In the case when the near-surface conductivity structure is due to the Georgian block, the ﬂow effect virtually disappears. This difference in the behavior of Hz is well illustrated in Fig. 12.15, mapping contours of the normalized difference a c H H ð12:1Þ D ¼ a z c z 100% ð H z þ H z Þ=2

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Fig. 12.15. Contour map of the normalized difference (in percent) between the moduli of the component Hz in models (a) and (c) (Fig. 12.13) at the period T ¼ 100 s.

4.

where H az and H cz are the vertical components of the magnetic ﬁeld in models (a) (Dzirul Massif) and (c) (Georgian block), respectively; the broken lines delineate the same areas as in Fig. 12.12. Analysis of the behavior of the synthesized electric ﬁeld at the surface of the region shows that the inﬂuence of the high-conductivity channel depends signiﬁcantly on the shallow structure of electric conductivity. For example, if the basic element is the Georgian block (Fig. 12.13c, d), part of the current is diverted into the channel, thereby decreasing the intensity of the horizontal electric ﬁeld at the surface by about 15% (Fig. 12.16). If the conductivity structure is controlled by the Dzirul Massif (Fig. 12.13a, b), lying within the sedimentary sequence, the behavior of the horizontal electric ﬁeld at the surface is mainly inﬂuenced by the vertical current loop forming around the massif. The conducting channel, whose upper boundary occurs at

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Fig. 12.16. Contour map of the normalized difference (in percent) between the moduli of the component Ey in models (d) and (c) (Fig. 12.13) at the period T ¼ 100 s.

a depth of 48 km below the surface, is virtually unresolved in the ﬁeld contour maps. Fig. 12.17 shows the contour map of the normalized difference between the amplitudes of the horizontal component of the electric ﬁeld in models (b) (Dzirul Massif+channel) and (a) (Dzirul Massif). As seen from this map, the ﬁeld remains virtually constant in the western part of the region and varies insigniﬁcantly in its eastern part (between the Dzirul Massif and the Caspian Sea). Thus, the comparative analysis of the numerical modeling results and their comparison with experimentally established facts lead to the conclusion that the most probable model of electric conductivity in the study region is model (b), including the Dzirul Massif in the sedimentary sequence and a conducting channel connecting the Black and Caspian seas (see Fig. 12.18).

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Fig. 12.17. Contour map of the normalized difference (in percent) between the moduli of the component Ey in models (b) and (a) (Fig. 12.13) at the period T ¼ 1000 s.

12.2.4. Conclusions 1.

This study has led to the following conclusions: The behavior of the electric ﬁeld in all four models generally indicates a 3-D structure of the ﬁeld, which is due to the presence of geoelectric heterogeneities in the upper crust.

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The behavior of magnetic ﬁelds in the study region is controlled by current systems located mainly in the vertical N-S plane. 3. The behavior of the horizontal components of the electric and magnetic ﬁelds and the vertical component of the magnetic ﬁeld at the surface can be used to determine the type of the near-surface conductivity structure in the region. 4. The inﬂuence of the high-conductivity channel on the MT ﬁeld depends signiﬁcantly on the near-surface structure of electric conductivity. 5. The most probable model of electric conductivity in the region under study is a model in which the sedimentary sequence contains the Dzirul Massif and a conducting channel connecting the Black and Caspian seas. Of course, the latter conclusion can be modiﬁed as new experimental data become available. It can be hoped that results of 3-D MT surveying of this region in conjunction with other geophysical data will prove effective for constructing its 3-D geoelectric model and reﬁning the macro-parameters of its shallow and deep structure. 2.

12.3. MT IMAGING INTERNAL STRUCTURE OF VOLCANOES MT ﬁelds are widely used to study the geodynamic processes in geothermal and volcanic zones (Fitteman et al., 1988; Park and Torres-Verdin, 1988; Mogi and Nakama, 1990; Spichak, 1995; Mauriello et al., 1997; Di Maio et al., 1998; Manzella et al., 1999, 2000; Matsushima et al., 2001, Spichak et al., 2004, 2006) owing to their deep penetration into the earth and ability to resolve the parameters of complex geological media. However, in most of the cases, 1-D or 2-D interpretation tools have been used. Meanwhile, long-term forecast of the Earth’s activity should be evidently based on the knowledge of the deep three-dimensional volcanic structure as well as on our ability to interpret the measured data properly. At a ﬁrst glance, the construction of a 3-D geoelectric model as well as the monitoring of crucial parameters would require synchronous MT measurements carried out at sites regularly distributed over the Earth’s surface. However, forward modeling indicates that if one monitors only the conductivity variation within a locally homogeneous area in the Earth (for instance, geothermal reservoir or magma chamber) it may be sufﬁcient to interpret properly data measured even at one site (Spichak, 1999a). Since very few 3-D MT surveys are available currently in the world, only computer simulation of MT-ﬁeld behavior may provide a proper basis for effective 3-D MT imaging and monitoring of active zones in the lithosphere. For example, Newman et al. (1985) modeled a homogeneous prism in a layered earth with a 3-D integral equation method to study the detectability of a magma chamber, whereas Moroz et al. (1988) built a more elaborate analogue model to study the distortion of MT ﬁelds due to a volcanic cone. 12.3.1. Simplified model of the volcano Spichak (1999b, 2001) studied the MT response in volcanic environments using a 3-D simpliﬁed geoelectric model of a Hawaiian volcano Kilauea. This model was constructed in 1989 (with Prof. George Keller) to determine whether the MT

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Fig. 12.18. Vertical cross section of 3-D geoelectrical model of the Kilauea volcano (Spichak, 2001).

method could detect the internal processes that had previously been found with TDEM measurements (Jackson and Keller, 1972). The model represents a marine shield volcano, characterized by a low and ﬂat summit formed by homogeneous basaltic rocks (Fig. 12.18). Its ﬂanks stretch down into the ocean; the conductivity of the ocean water was taken to be 3.6 S/m. The volcano’s summit, 0.5 km thick, is formed by basaltic lavas with conductivity s ¼ 0.001 S/m. There is a small layer, 0.8 km thick, with conductivity s ¼ 0.01 S/m at the boundary between the air and ocean. Below are porous volcanic lavas, which are characterized by a high content of brine/seawater (this zone is 1.7 km thick and has a conductivity s ¼ 0.17 S/m). At 3 km depth from the volcano summit, there are dense lava formations 5.5 km thick with conductivity s ¼ 0.01 S/m, underlain by crystalline crust with conductivity s ¼ 0.001 S/m. The conductivity distribution in the model was considered to be symmetrical with two vertical planes of symmetry, so only one-quarter of a 3-D grid was used for calculations. 12.3.2. Synthetic MT pseudosections MT ﬁelds for this model were synthesized by the program package FDM3D (Spichak, 1983) for two polarizations of the primary ﬁeld at four periods

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Fig. 12.19. jxy( 101) pseudosection for volcano model shown in Fig. 12.18.

T ¼ 0.1, 1, 10 and 100 s. Then, a number of the MT-ﬁeld transformations were calculated and analyzed at different levels in the atmosphere to ﬁnd those which are most sensitive to the parameters of the model (Spichak, 2001). Since the geoelectric structure is strongly screened by seawater, direct estimation of the conductivity distribution from apparent resistivity pseudosections is very difﬁcult. In contrast, 3-D isosurfaces or 2-D contour maps of the transforms based on the impedance phases and on the in-phase and quadrature parts of the horizontal electric ﬁelds turned to be the best for imaging the interior of the volcano (Figs. 12.19–12.22). In particular, Figs. 12.19 and 12.20 show the vertical cross-sections of the volcano overlapped by the contour maps of isolines of the transformed impedance phases (jxy and jdet , correspondingly), constructed for the plane at a height 0.5 km above the summit of the volcano. Although the values assigned to these isolines could hardly be interpreted in terms of the rock physical properties, their spatial gradients clearly indicate the location of the magma chamber and of the conductive formation above it. It is worth mentioning that the gradient of jdet delineates not only the magma chamber but also the ﬂanks of the volcano (Fig. 12.20). This result agrees with earlier ﬁndings by Park and Torres-Verdin (1988) on the interpretation of the impedance phases.

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Fig. 12.20. jdet( 101) pseudosection for volcano model shown in Fig. 12.18.

Fig. 12.21 shows ReZyx isolines. They indicate a position of the lower boundary of the magma chamber and even the interface between the dense volcanic rocks and crystalline crust. Transforms of the in-phase and quadrature parts of the horizontal electric ﬁeld component parallel to the incident electric ﬁeld (Figs. 12.22 and 12.23) turn out to be even more sensitive to gradients of conductivity. Fig. 12.22 indicates the vertical cross section of the model overlapped by the map of the isolines of the Re Ey transformation. The isolines condensation correlates with gradients of the conductivity with the local extrema marking the upper and lower edges of the magma chamber. Isolines of the transformations of Im Ey are shown in Fig. 12.22. There is a strong maximum located at the lower boundary of the magma chamber delineated by the isolines. A 3-D pseudostructure of the medium can be fairly seen from Im Ey volume image (Fig. 12.24). Thus, construction of the 3-D pseudostructures based on joint interpretation of transforms of the in-phase and quadrature parts of the horizontal electric ﬁeld and the impedance phases appears to be a useful tool for delineating the geometric parameters of the complex volcanic environments. This modeling supports the analytical ﬁndings of Szarka and Fischer (1989), explaining the behavior of the MT-ﬁeld transformations at the Earth’s surface in terms of the distribution of subsurface currents.

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Fig. 12.21. ReZyx( 103) pseudosection for volcano model shown in Fig. 12.18.

12.3.3. Methodology of interpretation of the MT data measured over the relief surface The calculation of the transformations used in the previous section is a very important practice. They could be determined at the relief Earth’s surface from the electric and magnetic ﬁeld measured there. However, multiple effects of the geological noise caused by near-surface inhomogeneities may greatly distort corresponding MT-ﬁeld transformations and, consequently, the interpretation results obtained using some imaging techniques. One way to overcome this difﬁculty consists of foregoing upward analytical continuation of the data to the artiﬁcial reference plane located in the non-conductive atmosphere higher than the top topographic point (Spichak, 2001) by means of the integral transformation of the Stratton–Chu type (Berdichevsky and Zhdanov, 1984): ZZ a fðn Ea ÞrG þ ½n Ea rG þ iom0 ½n Ha Ggds0 E ðrÞ ¼ S

ZZ

a

H ðrÞ ¼

fðn Ha ÞrG þ ½n Ha rGgds0

ð12:2Þ

S a

a

where E and H are the anomalous electric and magnetic ﬁelds, correspondingly, determined at the reference plane, G ¼ 1=ð4pjr rjÞ – is the Green’s function of free

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Fig. 12.22. ReEy( 101) pseudosection for volcano model shown in Fig. 12.18 (primary electric ﬁeld is directed in OY axis).

space, S the surface of measurements, r the radius-vector of the points belonging to this surface and n is a unit vector normal to the surface S pointing outwards. The analytical continuation of anomalous MT ﬁeld from the Earth’s surface to the atmosphere could prove to be unique and, therefore, continued data could be used for the interpretation. The corresponding numerical calculations are straightforward and stable. Another important item concerns the appropriate height of the reference plane. After many simulations it was found (Spichak, 1999b) that its optimal value is 250–500 m above the volcano summit. At lower heights, the pseudosections become distorted mostly by the nearest parts of the topography and by the noise (natural and artiﬁcial), whereas at greater heights, details in geoelectrical structure may be lost. The choice of the best height for a given structure requires special investigations. The synthetic MT ﬁelds and their transforms, calculated for a 3-D model of a volcano of Hawaiian type have indicated that impedance phases as well as in-phase and quadrature parts of the electric ﬁeld components are the most sensitive to the structure. Hence, the following procedure of volcano inner structure visualization can be outlined (Spichak, 2001): 1. Upward analytical continuation of MT data to an artiﬁcial reference plane located 250–500 m higher than the highest topographic point. from a reference frame (X, Y, 2. Construction of continued ﬁeld transformations pﬃﬃﬃﬃ Z) to a new one (X, Y, zapp (or log T or T ), where zapp is an apparent depth).

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Fig. 12.23. Im Ey( 101) pseudosection for volcano model shown in Fig. 12.18.

Fig. 12.24. Im Ey( 101) isosurfaces for the volcano model shown in Fig. 12.18.

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Spline-interpolation of constructed 1-D vertical proﬁles on a regular threedimensional grid. 4. Computer tomography of geoelectric structure by construction of transformation isosurfaces in the domain of search, limited horizontally by the size of the data array and vertically by the largest skin-depth. Thus, the model studies show, that for a visualization of the volcanogenic zones it is necessary to use not only magnetic, but also electric fields. This makes the MT method indispensable for studying their geoelectric structures. 3.

12.4. SIMULATION OF MT MONITORING OF THE MAGMA CHAMBER CONDUCTIVITY 12.4.1. Geoelectric model of a central type volcano Spichak (2001) studied the effect of melt condition in a magma chamber on MT data collected at the Earth’s surface. The probable volcanic ediﬁce was represented by a cone with a basis diameter of 30 km and heights 1,3 and 5.0 km (Fig. 12.25). It was also supposed that the cone is made of volcanic rocks with 1000 O m resistivity. Below there is 2.5 km thick layer of volcano-sedimentary rocks with resistivity r ¼ 20 O m. Crystalline rocks of the crust bottom with 1000 O m resistivity and 45–50 km thickness underlie it. At a depth of 2.5 km from the Earth’s surface (disregarding of volcanic construction) there is cubic magma chamber (5 5 5 km), ﬁlled with a basalt melt of r ¼ 2 O m and connected with a crater by a channel with the diameter of 0.6 km. 12.4.2. Detection of the magma chamber by MT data At ﬁrst, the estimation of MT data resolving power on detection of the magma chamber was conducted for the case in which the model has the magma channel (i.e., contrary to the case considered in the previous section). For calculation of synthetic magnetotelluric ﬁelds, the software package FDM3D (Spichak, 1983) was used. The calculations were made for periods ranging from 1 to 1000 s. As previously, the pseudosections of various ﬁeld components and their transformations were constructed. Fig. 12.26 shows the pseudosection of the impedance phase constructed by the procedure described in the previous section. It can be observed that in spite of the cone inﬂuence, all basic elements of geoelectric structure (magma chamber, channel, cone and 1-D layering) are adequately contoured by isolines of this transformation. Effective monitoring in cases when the observations are made on a relief surface and data contain noise (natural or artiﬁcial) requires, at least, foregoing modeling. After the preliminary three-dimensional model is determined, it is necessary to recognize those components of MT ﬁeld (or its functions), whose measuring (or evaluation) allows a reliable monitoring of the selected parameters.

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Fig. 12.25. Geoelectric model of a central type volcano (vertical cross-section under the central proﬁle). The ﬁgures show an electric conductivity values in S/m).

Fig. 12.26. jxy( 101) pseudosection in XOZ plane.

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12.4.3. Estimation of MT data resolving power with respect to the conductivity variations in the magma chamber To understand how the melt state in a magma chamber can be monitored using MT data measured on a relief surface, a simulation was carried out as follows (Spichak, 2001). For the model shown in Fig. 12.25, the MT ﬁeld was ﬁrst calculated for the conductivity of the magma chamber from 0.1 to 0.5 S/m. Then they were continued upwards to an artiﬁcial reference plane allocated 0.5 km above the highest point of the topography. On this plane, the maps of isolines of differences of the ﬁeld component transformations, and also of apparent resistivity corresponding to two different values of electric conductivity of the magma were constructed. Let us consider how the increase in the electric conductivity in the magma chamber from 0.1 up to 0.5 S/m inﬂuences the components of electric and magnetic ﬁelds. In Fig. 12.27a,b, the difference maps of normalized amplitudes of the magnetic ﬁeld component H x at a period T ¼ 1 s are represented in the case of cone absence (a) and cone presence (h ¼ 1.3 km) (b). In both cases, relative changes of the ﬁeld amplitudes (up to 24% – for a cone absence and up to 5% – for a cone with h ¼ 1.3 km) have local character and are manifested in the neighborhood of the magma vent. On both maps there exist the sign of change of the component amplitude coinciding with the sign of change of the electric conductivity in the chamber as zones, where they are opposite (in Fig. 12.27a, b – domains with negative value of relative change). This is caused by distinct character of the free-space attenuation of the anomalous ﬁeld for different values of electric conductivity in the chamber. As the period increases (in Fig. 12.28a, b the difference maps for T ¼ 10 s are represented), the sizes of area in which it is possible to detect the change in the state of melt as well as the value of this change (up to 60% in a neighborhood of the vent) increases. At the same time, dependence of a sign of the ﬁeld component amplitude alteration on the measurement site (evidenly negative for monitoring data inter-

Fig. 12.27. Isolines of normalized amplitude differences (in %) of the component H x at a period T ¼ 1 s:(a) in the case of the cone absence and (b) at its presence.

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Fig. 12.28. Maps of difference isolines (in %) of normalized amplitudes of magnetic ﬁeld component H x at the period T ¼ 10 s: (a) in the case of the cone absence and (b) at its presence.

pretation) remains. Difference contour maps constructed for the amplitude of an electric ﬁeld component Ey behave similarly (Fig. 12.29a, b). The results for the most expressive MT-ﬁeld components in TE-mode were demonstrated above. With change of the primary-ﬁeld polarization, the conﬁguration of differences contour maps (in particular, location of domains in which the sign of differences is the same) also changes. Unfortunately, interpretation in terms of the apparent resistivity components does not improve this situation, since appropriate difference maps have similar peculiarities (by the same reason). From these results, it is evident that for reliable interpretation of monitoring of the electric conductivity variation in the magma chamber it is desirable to use some scalar function of the MT ﬁeld, which takes into account horizontal ﬁeld components measured at different polarizations of the external ﬁeld. One (and, perhaps, unique) such function is ‘‘an apparent resistivity’’ (or ‘‘an apparent conductivity’’), based on the impedance determinant. Indeed, it naturally takes into account all horizontal components of the MT ﬁeld measured at different polarizations of the primary ﬁeld, and its change has mainly the same sign as the variation of the resistivity (conductivity) in a magma chamber (as shown below). In Fig. 12.30, the differences contour maps of normalized values rdet at the period T ¼ 10 s are represented: (a) for the cone absence and (b) for the cone presence. As it is visible from the ﬁgure, irrespective of availability of the cone, the difference of values of the selected function in both cases has the same sign as the corresponding difference of resistivities in a magma chamber. Moreover, its values depend only on the horizontal distance to the center of the ‘‘crater’’ (at least, in a zone, which diameter is three times larger than the horizontal sizes of the magma chamber) that can essentially simplify interpretation of monitoring results. To eliminate inﬂuence of the magma channel on the behavior of the isolines in a difference map, the same model without the channel was considered. In Fig. 12.31,

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Fig. 12.29. Maps of difference isolines (in %) of normalized amplitudes of electric ﬁeld component E y at the period T ¼ 10 s: (a) in the case of the cone absence and (b) at its presence.

Fig. 12.30. Map of difference isolines (in %) of normalized values rdet on the period T ¼ 10 s: (a) in the case of the cone absence and (b) with its availability.

the difference maps of rdet for the model without the cone at the period 10 s corresponding to an increase in conductivity in the magma chamber from 0.1 to 0.2 S/ m (a), and to 0.5 S/m (b) are represented. The analysis of maps represented in Fig. 12.31 indicates the maximum effect is observed in the area restricted by the horizontal sizes of the magma chamber (20–25% – in the case (a) and 42–52% – in the case (b)); the minimum effect (lower than 5%) is observed outside an area, three and ﬁve (cases (a) and (b), correspondingly) times exceeding the horizontal sizes of the the magma chamber.

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Fig. 12.31. Map of differences isolines (in %) of normalized values rdet at a period T ¼ 10 s for model represented in a Fig. 12.25 without the magma channel at diminution of resistivity in the chamber: (a) in 2 times and (b) in 5 times.

Comparing Fig. 12.31b with Fig. 12.30a shows that the magma channel strengthens the effect of variation of magma-melt conductivity approximately by three times. Moreover, it increases the diameter of a zone of reliable monitoring, and, secondly, reduces the period threshold, sufﬁcient for detection of even small variations of electric conductivity in the magma chamber.

12.4.4. ‘‘Guidelines’’ for MT monitoring electric conductivity in a magma chamber From the preceding it follows that for reliable monitoring of a melt state by MT method it is worth to follow the next ‘‘guidelines’’ (Spichak, 2001): (1) to construct a geoelectric model of the region; (2) to create difference maps – (templates) similar to those given in Fig. 12.31a, b; (3) to locate the sensors on the surface at a distance from the estimated center of the magma chamber no more than three times exceeding its horizontal diameter; (4) to make measurements, whenever possible, of long-period ﬁeld variations and at different primary-ﬁeld polarizations; (5) to continue analytically the measured MT ﬁelds upwards to a reference plane (by the procedure described above in Section 12.3.3); (6) to create difference maps by means of function rdet from the continued MT ﬁeld at the reference plane; and (7) to ﬁnd relevant variations of a melt resistivity using templates. The use of the upward analytical continuation allows not only to ﬁlter the data, as in the case of imaging, but also to reduce the data to a ‘‘common denominator’’, which is important for removing the relief effect on the transform templates and, consequently, for proper interpretation of the monitoring data (at least, in the framework of the procedure above).

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The modeling results indicate that the apparent resistivity function derived from the impedance determinant is most suitable for adequate interpretation of measurements carried out for the purpose of monitoring. Finally, the results shown in this section point to the opportunity of remote MT monitoring of volcanoes and other targets of difﬁcult access. Thus, synthetic MT ﬁelds and their transforms, calculated for 3-D geoelectric models of volcanoes, indicate that the impedance phases as well as in-phase and quadrature parts of the electric field components provide the best imaging of the volcanic interior. On the other hand, the apparent resistivity function derived from the impedance tensor determinant seems to be the most suitable parameter for adequate interpretation of measurements carried out with the purpose of monitoring. It is worth mentioning that the use of the intepretation methodology discussed in this section should make it possible to monitor the conductivity variations in some locally homogeneous zone of the Earth’s crust (in particular, in the magma chamber) based on appropriate measurements carried out even in one site properly located with respect to the position of the target. It is obvious that the actual situations are much more complicated than the models considered. However, the methodologies suggested following the results of computer simulation may help, ﬁrst, to avoid unnecessary interpretation errors due to noise (both geological and instrumental) in the data, irregularity and lack of observation points in regions of difﬁcult access; and, second, to make MT survey planning more scientiﬁcally substantiated.

12.5. SIMULATION OF MT MONITORING THE GROUND WATER SALINITY The problem of freshwater salinity in lakes and other reservoirs becomes more and more pressing, and therefore development of effective methods for remote monitoring their salinity becomes more and more urgent. Owing to the fact that the electric conductivity of fresh water varies with its salt content, one of the possible ways to solve this problem is to arrange a remote MT monitoring of an electric conductivity. Although the details may vary depending on the case speciﬁed, the key features of the approach can be demonstrated on a typical coastal aquifer salinity problems shown in Fig. 12.32 (Custodio, 1985). 12.5.1. Statement of the problem It is necessary to detect the event of seawater intrusion into the freshwater aquifer and, if possible, to map the boundary of the salted water using audio-MT sounding. The geoelectric model relevant to this problem is given in Fig. 12.33 (Spichak, 1999a). The values of s ¼ 0.05 and 0.4 S/m correspond to an electric conductivity of the ground saturated with fresh and salted water, accordingly. The electric conductivity of the seawater is assumed to be equal to 4 S/m.

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Fig. 12.32. Different situations in continental and island aquifers. The fresh, salt and mixed water bodies are indicated (Custodio, 1985).

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Fig. 12.33. Geoelectric model relevant to problem of groundwater salinity with a marine water: (a) vertical section in a symmetry plane and (b) horizontal cross-section in a low layer of the domain of search, border of which is designated by a dashed line.

12.5.1.1. The data The model MT data were calculated using the software package FDM3D (Spichak, 1983) at frequencies 50, 100 and 200 Hz for two polarizations of a primary ﬁeld. It was assumed that in practice only ground measurements of the data are available, therefore in the inverse problem solution, only those values of synthetic electric ﬁelds measured on the ground (within region limited by a surface projection of the horizontal boundaries of anomalous domain) were taken into account. 12.5.1.2. Prior information In the solution of the problem stated it was assumed that 1-D layering in the region of interest is known from the data of other geophysical methods; the region of a possible ground water salinity is conﬁned to an area marked by a dashed line in Fig. 12.33; and

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Fig. 12.34. Vertical (a) and horizontal (b,c,d) sections of a posterior distribution of an electric conductivity in the domain of search.

within the anomalous domain, electric conductivity of the water can take values sFW ¼ 0.05 S/m (conductivity of the ground saturated with freshwater) and sSW ¼ 0.4 S/m (conductivity of the ground saturated with salt water), which corresponds to the uncertainty of an expert estimation of the fresh water salinity.

12.5.2. Modeling of the salt water intrusion zone mapping by audio-MT data The inverse problem was solved using the INVERS-3D program package where the Bayesian approach proposed in Spichak et al. (1999) has been realized. Since the constructed electric conductivity model has a vertical plane of symmetry (Fig. 12.33), the inversion was made only for a half of a model. The anomalous

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domain was assumed to consist of 32 subdomains with constant conductivity within each. The inversions results were obtained in 37 iterations of an exterior cycle. Fig. 12.34 represents vertical (a) and horizontal (b,c,d) slices of a posterior distribution of an electric conductivity at depths 2, 4 and 6 m, respectively. In the lowest layer of ‘‘salted water’’ (from 6 to 8 m, see Fig. 12.33) the posterior electric conductivity did not exceeded an average prior estimation (0.225 S/m). On the other hand, three high layers were detected and contoured quite well. In Fig. 12.34 the boundaries of the salted zone both in vertical and in horizontal projections are distinctly seen. Thus, the obtained result enables to make the following conclusions: 1. The audio-MT sounding method can be successfully applied in monitoring the changes of physical properties in the near-surface layer (ﬁrst 10 m) that lead to variations in its electric conductivity. 2. The method of MT data inversion based on a Bayesian statistics is an effective tool for the solution of problems that require formalized expert estimations. In this Chapter we have considered methodological questions, the answers to which are wider than being case-descriptive only: they are important not only for understanding the MT ﬁeld behavior as this applies to a speciﬁc given problem, but also for the effective interpretation of MT data in similar situations. Acknowledgments The author is grateful to G.E. Gugunava, who proposed testing various hypotheses on the geoelectric crustal structure of the Transcaucasian region. This study was partly supported by INTAS, grant 03-51-3327.

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Gugunava, G., 1988. Interrelation of some geophysical ﬁelds and deep structure of the Caucasus (in Russian). Doct. Dissertation, Tbilisi, 316pp. Jackson, D.B. and Keller, G.V., 1972. An electromagnetic sounding survey of the summit of Kilauea Volcano, Hawaii. J. Geophys. Res., 77: 4957–4965. Manzella, A., Mackie, R. and Fiordelisi, A., 1999. MT survey in the Amiata volcanic area: a combined methodology for deﬁning shallow and deep structures. Phys. Chem. Earth (A), 24, 9: 837–840. Manzella, A., Volpi, G. and Zaya, A., 2000. New magnetotelluric soundings in the Mt. Somma-Vesuvius volcanic complex: preliminary results. Ann. Geoﬁs., 43, 2: 259–270. Matsushima, N., Oshima, H., Ogawa, Y., Takakura, S., Satoh, H., Utsugi, M. and Nishida, Y., 2001. Magma prospecting in Usu volcano, Hokkaido, Japan, using magnetotelluric soundings. J. Volcanol. Geotherm. Res., 109: 263–277. Mauriello, P., Patella, D., Petrillo, Z. and Siniscalchi, A., 1997. Mount Etna structural exploration by magnetotellurics. Acta Volcanol, 9, 1/2: 141–146. Mogi, T. and Nakama, K., 1990. Three-dimensional geoelectrical structure of geothermal system in Kuju volcano and its interpretation. Geotherm. Res. Council Trans., 14, II: 1513–1515. Moroz, Y.F., Kobzova, V.I., Moroz, I.P. and Senchina, A.F., 1988. Analogue modeling of MT-ﬁelds of volcano (in Russian). Vulkanol. Seismol., 3: 98–104. Newman, G.A., Wannamaker, P.E. and Hohmann, G.W., 1985. On the detectability of crustal magma chambers using the magnetotelluric method. Geophysics, 50: 1136–1143. Park, S.K. and Torres-Verdin, C., 1988. A systematic approach to the interpretation of magnetotelluric data in volcanic environments with applications to the quest for magma in Long Valley, California. J. Geophys. Res., 93: 13265–13283. Spichak, V.V., 1983. Program package FDM3D for numerical modeling of 3Delectromagnetic ﬁelds (in Russian). Algorithms and Programs for Solving Direct and Inverse Problems of EM-Induction in the Earth, IZMIRAN, Moscow, pp. 58–68. Spichak, V.V., 1995. Three-dimensional electromagnetic imaging of volcanoes. Per. Mineral., 64: 273–274. Spichak, V.V., 1999a. Magnetotelluric Fields in 3-D Geoelectrical Models (in Russian). Scientiﬁc World, Moscow, 204pp. Spichak, V.V., 1999b. Imaging of volcanic interior with MT data. Three Dimensional Electromagnetics, SEG monograph, GD7. Tulsa, USA, pp. 418–425. Spichak, V.V., 2001. Three-dimensional interpretation of MT data in volcanic environments (computer simulation). Annali di Geoﬁsica, 44, 2: 273–286. Spichak, V.V., 2006a. Estimation of MT data resolution in respect to the hydrocarbon exploration (in Russian). Geoﬁsika, 1: 39–42. Spichak, V.V., 2006b. Comparative analysis of the hypotheses of the geoelectric structure of the Transcaucasian region from Magnetotelluric data. Izvestiya, Phys. of the Solid Earth, 42, 1: 60–68.

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Spichak, V., Borisova, V., Fainberg, E., Khalezov, A. and Goidina, A., 2006. 3-D EM tomography of the Elbrus volcanic region by magnetotelluric and satellite data (in Russian). Vulkanol. Seismol. (in press). Spichak, V., Gugunava, G. and Akopjants, S., 1988. Three-dimensional modeling of the geoelectrical structure beneath the Caucasus region. Proc. IX Workshop on electromagnetic induction in the Earth and Moon, Dagomys, USSR. Spichak, V. and Larionova, T., 1990. Modeling of 3-D conducting target surrounded by more conductive medium. Proc. X Workshop on EM induction in the Earth, Ensenada. Spichak, V.V., Menville, M. and Roussignol, M., 1999. Three-dimensional inversion of the magnetotelluric ﬁelds using Bayesian statistics,Three Dimensional Electromagnetics, SEG monograph, GD7. Tulsa, USA, pp. 406–417. Spichak, V., Yamaya, Y. and Mogi, T., 2004. ANN modeling of 3-D conductivity structure of the Komagatake volcano (Hokkaido, Japan) by MT data. Proc. IV Int. Symp. MEEMSV – 2004, LaLonde Les Maures, France, 121–122. Szarka, L. and Fischer, G., 1989. Electromagnetic parameters at the surface of conductive halfspace in terms of the subsurface current distribution. Geophys. Trans., 35: 157–172. Vanyan, L., Demidova, T., Egorova, I., Konnov, Yu. and Yanikyan, V., 1989. The inﬂuence of the sea and the sedimentary cover on the low frequency telluric ﬁeld in the Caucasus (in Russian). Geoﬁzicheskii zhurnal, 1: 70–72. Watts, M.D., Savvaidis, A., Karageorgi, E. and Mackie, R., 2002. Magnetotellurics applied to Subthrust Petroleum Exploration in Northern Greece. Proc. EAGE 64th Conference & Exhibition, Florence, Italy. Zhdanov, M.S. and Spichak, V.V., 1992. Mathematical Modeling of Electromagnetic Fields in 3-D Inhomogeneous Media (in Russian). Nauka, Moscow, 188pp.

Chapter 13 Regional Magnetotelluric Explorations in Russia V.P. Bubnov1, A.G. Yakovlev1, E.D. Aleksanova1, D.V. Yakovlev1, M.N. Berdichevsky2 and P.Yu. Pushkarev2 1

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North-West Ltd., Moscow, Russia Geological Faculty of Moscow University, Russia

13.1. INTRODUCTION Electromagnetic (EM) geophysical methods (telluric current method, magnetotelluric sounding, frequency sounding, and transient sounding) have been used in the erstwhile USSR to study a deep structure of sedimentary basins and of the consolidated crust since the 1950s. Tectonic schemes of the principal sedimentary basins of the USSR were constructed and several large hydrocarbon deposits, for example, the Urengoy gas ﬁeld, were discovered using the telluric currents method and magnetotelluric soundings, in combination with other geophysical methods. In the 1970s and 1980s, extensive magnetotelluric data characterizing the electrical conductivity of the Earth’s crust were collected, and maps of crustal anomalies of electron-conducting and ﬂuid origin were constructed. A review of major results obtained up to the 1990s is presented in Berdichevsky (1994). The review shows that a strong scientiﬁc community of researchers, applying EM methods to study the Earth, appeared in the country. In the 1990s, due to economic problems, EM investigations were reduced. However, an abrupt expansion began in 2000 (Berdichevsky et al., 2002), caused by the depletion of established resources and increase of prices for hydrocarbons and other mineral resources.

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40013-5

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Nowadays, EM methods, ensuring an exploration depth of more than 100 m, are widely adopted in Russia in three main ﬁelds: regional exploration; oil and gas prospecting; and solid mineral prospecting. Regional surveys are conducted at the request of the Russian Ministry of Natural Resources, while hydrocarbon and other mineral prospecting is being mainly funded by private companies holding licenses for particular regions. During recent years, the third area of application associated with studies of the upper few 100 m by means of the high-frequency (audio) magnetotelluric method has been developing rapidly. Audio-magnetotellurics proved to be one of the most efﬁcient geophysical methods for the exploration of ore minerals and kimberlite pipes (Alekseev et al., 2004). Regional geophysical land surveys in Russia are performed along separate proﬁles ranging from a few hundreds to several thousand kilometers in length and running through deep boreholes. The locations of the most extensive proﬁles, called geo-traverses, are shown in Fig. 13.1. Investigations along regional proﬁles provide information about the deep structure of vast regions and help solve applied tasks such as the prognosis of oil-and-gas content in sedimentary basins and the location of promising solid mineral zones. In active tectonic regions, data required to study geodynamic conditions and to predict seismic activity are collected. The combined application of geophysical methods is characteristic for regional surveys. The combination includes CDP (common-depth-point) seismic, EM, gravity and magnetic prospecting, and other methods. Seismic prospecting plays the

Fig. 13.1. Location map of geotraverses (1) and regions considered in the paper (2). 1 – Soligalich aulacogen, 2 – Tokmov arch and Melekes depression, 3 – Kotelnich arch and Kazansko-Kazhimsky aulacogen, 4 – Voronezh anteclise, 5 – Pre-Caspian syneclise, 6 – Karpinsky swell, 7 – Western Caucasus forelands, 8 – Central Caucasus, 9 – ‘‘Uralseis’’ proﬁle, 10 – 1-SB proﬁle, 11 – 3-SB proﬁle, 12 – 2-DV proﬁle.

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leading role – in most cases it determines the location of geological boundaries rather precisely. Other methods, in particular EM, supplement this data with information about the physical properties of rocks characterizing their lithology, ﬂuid content, rheological state, etc. The total length of regional proﬁles studied by EM methods each year is about 3000–4000 km, while the spacing between the sites is 1–3 km.In the European part of Russia, surveys are performed mainly by the State enterprise Spetsgeoﬁzika; in the Caucasus, by the State enterprises ‘‘Kavkazgeolsyemka’’ and ‘‘GEON Centre’’; in Siberia, by the State enterprises Irkutskgeoﬁzika and Eniseygeoﬁzika. Among private companies, the most active are North-West Ltd. and CEMI Ltd. In this paper, we present some results obtained within a few last years by North-West Ltd. in cooperation with the organizations mentioned above and the Geological faculty of Moscow University.

13.2. OBSERVATION TECHNOLOGY The basic regional EM method is the magnetotelluric (MT) method. MT provides the largest exploration depth and is inexpensive and mobile, as it does not require an artiﬁcial ﬁeld source. Different kinds of equipment are used for measurements. In the USSR, CES-2 receivers and their later modiﬁcations were applied. In the 1990s domestic CES-M, SGS, EIN, AKF, and other kinds of equipment were widely used in Russia. Since 2000, regional MT surveys have usually been conducted by means of receivers produced by the Canadian company Phoenix Geophysics Ltd. This equipment is characterized by high sensitivity and broad dynamic range, unattended operation, synchronization using the GPS satellite system, reliability, and simplicity. The MT method is applied in three ways: high-frequency (frequencies from 20,000 to 1 Hz, 1-km spacing between sites); standard (periods down to 5,000 s, 3-km spacing); and low-frequency (periods down to 50,000 s, 10-km spacing). At observation sites, either all ﬁve components of the natural electromagnetic ﬁeld (Ex, Ey, Hx, Hy and Hz) or only the two electric-ﬁeld components (Ex and Ey) are measured. In the latter case, magnetic ﬁeld records obtained at adjacent sites are used. As a rule, a receiver at some reference site operates synchronously with the receivers at a proﬁle. A difﬁcult problem of MT soundings is connected with industrial electromagnetic inductive and galvanic noises. The inductive noise is caused by electric power lines. The galvanic noise caused by current leakages from electriﬁed railroads is usually more intense. If resistive layers are present producing gradual attenuation of the electric ﬁeld when moving away from the railroad, this noise source inﬂuences the measurements performed several tens of kilometers away. Fig. 13.2 shows that near an electriﬁed railroad, the galvanic noise caused by the electric circuit between the locomotive and the nearest power substation dominates the weaker MT signal at high frequencies. Note that this ‘‘noise’’ can be used to acquire information

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Fig. 13.2. Observed and modeled apparent resistivity curves near the Moscow–Kazan electriﬁed railroad. 1 – Observed curves, 2 – result of forward modeling using plane wave source, 3 – the same using horizontal electric dipole as a source, 4 – zones where apparent resistivity is inﬂuenced by the electriﬁed railroad ﬁeld.

about resistive layers (Aleksanova et al., 2003). With the increase in the distance to the railroad, galvanic noise diminishes, and MT curves return to normal. If industrial noise is very strong, controlled-source measurements are performed. In most cases, time-domain soundings with coaxial transmitting and receiving loops are used. Frequency-domain soundings having very high tolerance to the industrial noise are still seldom applied. They require a large distance between transmitter and receiver, and if the medium changes signiﬁcantly in this interval then simpliﬁed (one-dimensional, 1-D) approaches to data interpretation become inapplicable.

13.3. MT-DATA PROCESSING, ANALYSIS AND INTERPRETATION As a rule, MT data processing is performed in remote reference mode, allowing the suppression of uncorrelated noise. In addition, robust statistical approaches are used to increase the reliability of results. Rejection of data values according to

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different criteria, such as dispersion relations between apparent resistivity and impedance phase, gives considerable improvement. Manual editing of the impedance and tipper response function plays an important role. This stage is necessary because automated processing algorithms often do not allow the suppression of industrial noise or at least require time-consuming adjustment of parameters. Manual editing is used to eliminate both outliers and stable branches of response functions caused by industrial ﬁeld sources. Another problem is associated with the distortion of MT curves by local subsurface inhomogeneities, producing uninterpretable geoelectric noise. This noise appears as a static shift of apparent resistivity curves along the vertical axis. If we have a dense observation network, this noise can be reasonably decreased by the spatial smoothing of apparent resistivity at some period and further shift of apparent resistivity curves to this smooth level. Another way to normalize MT curves is to adjust them to the levels of the time-domain sounding curves obtained when using a magnetic excitation and magnetic measurements of the EM ﬁeld. If geoelectric noise is suppressed insufﬁciently, the interpretation is performed with the priority of impedance phases and tipper, which become free from subsurface distortions with lowering frequency. MT data interpretation is performed in terms of Tikhonov’s theory of ill-posed problems. The most important stage of interpretation is the construction of an interpretational model combining all possible inverse problem solutions. The interpretational model is based on a priori information about the medium and on the MT data analysis. In the course of data analysis, pseudo cross sections of MT and magnetovariational parameters characterizing dimensionality of the medium are constructed. In addition, we determine the principal values and directions of the impedance tensor and analyze impedance polar diagrams and induction arrows showing the location and strike of resistivity structures. Impedance tensor decomposition methods describing the relation between regional and local structures are also applied. As a result of data analysis, the acceptable dimensionality of inversion methods is determined: usually one- or two-dimensional (2-D). In regional investigations, three-dimensional (3-D) inversion methods are not applied because observations are performed along separate proﬁles. However, to verify the reliability of 1- and 2D approaches, 3-D modeling is used to study 3-D effects and possible errors. Data interpretation is usually performed in two stages. At the ﬁrst stage, rough smoothed-structure inversion is applied. At the second stage, we deal with piecewise-uniform models to deﬁne the resistivity structure more precisely. All MT data components are used for interpretation, although their simultaneous inversion is not always effective because of their differing sensitivity to resistivity structures and differing robustness against 3-D distortions. We suppose that in regions with complicated geoelectric conditions, better results can often be obtained using a succession of partial inversions with tipper and impedance phases priority (see Chapter 2 for details), although this approach is still rarely used in industrial surveys. Interpretation is concluded by a geological and geophysical analysis of the resistivity models obtained. At this stage, EM results are considered together with

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other geophysical data. Specialists in the integrated application of geophysical methods as well as geologists are involved in this work.

13.4. CASE HISTORIES 13.4.1. East-European craton We start the review with some results obtained at the East-European craton where a large number of MT soundings were performed within the last few years. In this region, the following geoelectric complexes are present (from top to bottom): inhomogeneous Mesozoic – Cenozoic (rather conductive); Upper Devonian – Carboniferous including mainly carbonate rocks (resistive); mainly terrigenous, including Meso- and Neo-Proterozoic and Devonian rocks, saturated by mineralized water (conductive); metamorphic basement consisting of Archean and Paleo-Proterozoic rocks (resistive). New geoelectric information about the Moscow syneclise, the largest tectonic structure of the craton, was obtained along proﬁle IV of the RIFEY exploration program (region 1 in Fig. 13.1). The proﬁle consisting of 160 MT sites has length of 650 km.The resistivity cross section constructed using borehole and seismic information (Fig. 13.3) includes the basement depression – the Soligalich aulacogen and the superimposed uplift in sediments (Bubnov et al., 2003). Owing to the resistive layer that resists the ﬂow of transverse electric currents, this uplift strongly inﬂuences the transverse impedance data (TM-mode). At the same time, the longitudinal impedance (TE-mode) provides information about deeper layers and reveals conductive Meso- and Neo-Proterozoic and Devonian rocks. Their total rock thickness in the Soligalich aulacogen is about 2–3 km, and their low resistivity indicates good

Fig. 13.3. Resistivity cross section of the Moscow syneclise, proﬁle IV of the ‘‘Rifey’’ program. 1 – Boreholes, 2 – electrical logging results, 3 – seismic boundaries.

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reservoir properties. The resistive basement consists of large blocks of different resistivity. On the sides of the Moscow syneclise, the basement is presented by resistive, probably Archean rocks. In the central part of the syneclise, it is more conductive, due to the Paleo-Proterozoic rocks present. Fig. 13.4 presents the resistivity cross section of the Tokmov arch and the Melekes depression (region 2 in Fig. 13.1). Here, the resistive crystalline basement lies at a depth of approximately 2 km.The use of borehole and seismic data revealed quite a number of layers in the sedimentary cover. It is notable that horizontal variations in the resistivity were revealed. The valuable information that supplements seismic data is that the resistivity diminishes from west to east reﬂecting the increase in porosity and ﬂuid mineralization. The next example demonstrates the ability of the MT method to locate reefs in the junction zone of the Kotelnich arch and the Kazansko–Kazhimsky aulacogen (region 3 in Fig. 13.1). Here, the integrated interpretation of seismic and MT data was performed to supplement the cross section with geoelectric parameters based on seismic data. Within large lithological complexes potentially productive of oil and gas, several zones presumably containing reef traps were revealed using seismic data. To verify and reﬁne this result, variations of layer conductance determined using MT data were studied. Fig. 13.5 shows characteristic fragments of geological cross sections predicted from seismic data, and graphs of conductance of the appropriate lithological complexes. In the layers between P1 and C2vr seismic reﬂectors, as well as between C2vr and C1jp reﬂectors, the conductive anomalies correlate well with supposed reefs. These anomalies are explained by the high porosity and permeability of reefs with compared with host rocks. 2-D inversion of MT data obtained in the Voronezh anteclise (region 4 in Fig. 13.1), where sediment thickness is small, revealed striking conductive anomalies in the consolidated crust (Fig. 13.6). Here, the resistivity decreases to fractions of an ohm m (O m), allowing these anomalies to be explained by graphitization of PaleoProterozoic rocks. They are of practical interest as zones of probable ore mineralization. One of them is connected with the deep fault outlined according to geological data.

Fig. 13.4. Resistivity cross section of the Tokmov Arch and the Melekes depression. (1) – Seismic boundaries.

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Fig. 13.5. Fragments of geological cross section obtained using seismic data and graphs of total conductance of the named layers (junction zone of Kotelnich Arch and Kazansko–Kazhimsky aulacogen). 1 – Limestones, 2 – prospective reefs.

Fig. 13.6. Resistivity cross section along the proﬁle in the Voronezh anteclise.

Now, we move to the northern part of the Pre-Caspian syneclise (region 5 in Fig. 13.1). This area is promising for hydrocarbons, and salt-dome structures are common here. Fig. 13.7 displays the resistivity cross section obtained using 2-D inversion of MT data along one of the proﬁles oriented across the structures. In the

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conductive sedimentary cover, resistive salt domes approximately 6 km thick are easily seen. Some of them have a mushroom-like shape, producing salt overhang. Such zones in terrigenous rocks above the salt layer can be oil and gas traps. As the result of interpretation, areas of high and low conductance of the complex beneath the salt layer were also revealed. Accordingly, they correspond to zones of mainly terrigenous and carbonate composition. Delineation of carbonate bodies in this complex is an important task, because in similar areas to the East in Kazakhstan, such bodies contain large hydrocarbon deposits. To conclude the review of recent MT investigations of the East-European craton, we consider the result obtained at its southern ﬂank in the Karpinsky swell area (region 6 in Fig. 13.1). The observations were performed along a 190-km proﬁle comprising 71 MT sites (Berzin et al., 2005). On the basis of MT data and a priori geological and geophysical data analysis, a conclusion was drawn about strong horizontal inhomogeneity of the medium. A large isometric subsurface depression ﬁlled by sediments is superimposed on regional elongated (quasi-2D) structures. In this case, quasi-longitudinal (TE mode) impedance suffers from galvanic distortions that are much larger than the effect of deep structures. In contrast, quasi-transverse (TM mode) impedance has a low sensitivity to deep structures, although it contains information about shallow ones. In this situation, the deep conductive anomalies were studied using tipper data weakly distorted by the inﬂuence of isometric near-surface inhomogeneities and quite sensitive to deep conductive structures.

Fig. 13.7. Resistivity cross section along the proﬁle in the Pre-Caspian syneclise. 1 – Layer above the salt, 2 – salt domes, 3 – top of the layer below the salt, 4 – layer below the salt, 5 – basement top, 6 – basement.

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The cross section obtained by means of 2-D inversion of tipper and transverse impedance data is shown in Fig. 13.8. The cross section includes two conductive zones. The upper conductor constructed using transverse impedance occurs at approximately a 1-km depth. These are terrigenous Cretaceous and Cenozoic sediments, mainly clays. Beneath them there are more resistive, mainly carbonate rocks. The lower conductor occurs at a depth of about 15 km.It probably represents the southeastern extension of the Donbass conductivity anomaly (Rokityansky et al., 1989) covered by thick young sediments. The total conductance of this anomaly is several thousand Siemens, and it can be associated with the presence of both electron-conducting minerals and increased ﬂuid content. 13.4.2. Caucasus, the Urals, Siberia, and North East Russia In the Greater Caucasus mountains and in the Caucasus forelands, MT measurements were recently conducted along 10 proﬁles of a total length of 2000 km.Consider the proﬁle in Western Caucasus forelands. It stretches from the Black sea to the Scythian plate, crossing the Caucasus Mountains and the Kuban depression (region 7 in Fig. 13.1). Fig. 13.9 displays a geophysical cross section along the proﬁle based on 2-D MT data inversion results and seismic data. Its remarkable feature is that at the northern border of the Kuban depression, an unexpected deep trough ﬁlled with conductive (supposedly terrigenous Jurassic) rocks is revealed. Let us also consider the proﬁle in the central part of the Greater Caucasus, crossing the Elbrus mountain (region 8 in Fig. 13.1). The resistivity cross section (Fig. 13.10) clearly displays the transition from the folded belt of the Greater Caucasus to the Scythian plate and the associated gradual increase of sediment thickness (Arbuzkin et al., 2003). Within the limits of these tectonic structures, the Hercynian basement is heterogeneous, and the most complicated geoelectric situation is observed in the tectonic block of the Greater Caucasus. Known tectonic disruptions are seen as conductive zones, possibly because they are ﬂuid-saturated. A small conductive anomaly at a 2–8 km depth beneath the Elbrus volcano is interpreted as a magma chamber; at a depth of approximately 30 km another conductive anomaly is revealed, possibly connected with the magma center. In the Southern Urals, a regional MT survey was conducted along the 510-km Uralseis proﬁle (region 9 in Fig. 13.1). Measurements at 500 sites were performed (Kulikov et al., 2005). Three domains were marked out in the resistivity structure of the Southern Ural: Western Ural, being a part of the East European craton edge; Eastern Ural formed by Paleozoic volcanic and Plutonic basic and ultrabasic complexes; and Trans-Ural, which is part of the Kazakhstan Caledonian plate. According to MT data 2-D inversion results, the Earth’s crust is resistive beneath the East-European craton and the Kazakhstan plate, and conductive between them (Fig. 13.11). The southern Urals show a divergent structure. In its western part, nappes and thrusts moved westwards, and in the eastern part they moved eastwards. The most striking conductivity anomalies are associated with the Main Ural fault and the

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Fig. 13.8. Typical MT curves and resistivity cross-section of the Karpinsky swell. 1 – rXY, 2 – rYX, 3 – ReWZY, 4 – Im(WZY).

362 V.P. Bubnov et al./Regional Magnetotelluric Explorations in Russia Fig. 13.9. Typical MT curves and resistivity cross section of the Kuban depression and zones. 1 – Observed TE curves, 2 – observed TM curves, 3 – geological boundaries according to seismics, 4 – tectonic disruptions according to seismics.

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Fig. 13.10. Resistivity cross section along the proﬁle in Central Caucasus.

Fig. 13.11. Resistivity cross section of the Southern Ural (‘‘Uralseis’’ proﬁle) and the results of seismic data interpretation.

Zuratkulsky, Zapadno–Uraltaussky, and Kartalinsky faults. Here, the resistivity of rocks goes down to a few O m, probably characterizing their ﬂuid saturation. Chrome and gold deposits of the Magnitogorskaya metallogenic zone occur in areas where these deep faults rise to the surface. In the Magnitogorskaya and Trans-Ural zones, crustal conductive layers were also revealed. A conductor in the ﬁrst zone occurs at 15–25 km depth; it is about 30 km thick and its conductance is

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Fig. 13.12. Resistivity cross section along part of the 1-SB proﬁle to 60 km depth. 1 – Low sensitivity zone.

above 1000 Siemens (S). A crustal conductor of the Trans-Ural zone dips eastward from the Kartalinsky fault, its conductance exceeding 150 S. A signiﬁcant geophysical event of recent years was a study of the Earth’s crust in the Asian part of Russia along geotraverses 1-SB, 2-SB, 3-SB and 2-DV (Fig. 13.1). Fig. 13.12 shows the 800-km-long resistivity cross section along the 1-SB geotraverse (Aleksanova et al., 2005). The cross section starts at the West-Siberian plate, crosses the Yeniseisky range, and ends at the Siberian craton (proﬁle 10 in Fig. 13.1). Within the limits of the West-Siberian plate, the conductance of the sedimentary cover reaches 1000 S. Against this background, the details of the resistivity structure of the consolidated crust are uindistinguishable. In the Yeniseisky range area beneath the resistive Proterozoic metamorphic rocks, a conductive zone of unknown nature is present. MT data analysis demonstrated that it has a complicated 3-D structure, so that 1- or 2-D data interpretation is not acceptable here. At the Siberian craton within the Baikitskaya anteclise, a conductive layer is clearly seen. Its resistivity is approximately 100 O m and its thickness is about 10–15 km.Possibly the nature of this anomaly can be explained by ﬂuid presence in disintegrated rocks in the brittle–ductile transition zone. Currently, a special study of crustal conductivity structures in oil-and-gas provinces is being performed. In this connection, the mid-crustal conductive layer detected in the Baikitskaya anteclise (where the Yurubcheno–Tokhomskoe oil ﬁeld, the largest in Eastern Siberia, is situated), and also in the region of the gigantic Romashkinskoye oil ﬁeld at the East-European craton, can be of great practical interest. A cross section of the sedimentary cover of the Siberian craton along a 700-kmlong part of the 3-SB geotraverse (proﬁle 11 in Fig. 13.1) is displayed in Fig. 13.13. In the south of the proﬁle, within the Irkenyevsky aulacogen, conductive MesoProterozoic rocks are present at 7–11 km depth. According to borehole data from the adjoining Baikitskaya anteclise, these are mainly carbonates. Their low resistivity is probably caused by high porosity and mineralized water content. Within

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Fig. 13.13. Resistivity cross section along part of the 3-SB proﬁle to 15 km depth, constructed using MT and seismic data. 1 – Seismic boundaries in the sedimentary cover, 2 – top of the AR-PR1 complex, 3 – top of the PR1 complex, 4 – faults according to seismic data.

Fig. 13.14. Resistivity cross section along part of the 2-DV proﬁle (North-East Russia) and the results of seismic data interpretation. 1 – Moho boundary, 2 – reﬂecting boundaries, 3 – reﬂecting surfaces, 4 – fault zones.

the Kureiskaya syneclise, conductive layers that probably include reservoir rocks were also revealed. The 2-DV geotraverse crosses the Magadan and Chukotka regions (proﬁle 12 in Fig. 13.1). Three variations of the MT method (low frequency, standard, and high frequency) were applied, and spacing between sites was 1 km (Berzin et al., 2002). To date, more than 2000 soundings have been performed. Fig. 13.14 displays the resistivity cross section along the southern part of the geotraverse. In the Koni–Murgalskaya fold system and in the Okhotsko–Chukotsky volcanogenic belt, several deep conductive anomalies were outlined. There is a strong correlation of these anomalies with areas in which the intensity of seismic reﬂections from the Moho

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boundary is small. Possibly, they are connected with paleo-subduction zones and correspond to permeable rocks that provided the migration of mantle ﬂuids to the Earth’s surface. Major gold and silver deposits of the region are situated in the vicinity of these anomalies. Further north, in the Yano–Kolymskaya fold system, conductive anomalies mainly correspond to areas with thick sediments or graphitized rocks in the upper crust.

13.5. CONCLUSION MT investigations essentially expand the existing ideas about the structure and geodynamics of the Earth’s interior, based on the results of drilling and of seismic, gravity, and magnetic studies. MT investigations provide unique information about the structure and reservoir properties of sedimentary complexes, the state of active geodynamic regions, the graphitization and ﬂuid regime of the consolidated crust, and the permeable and ﬂuid-saturated crustal zones. The generalization of all electromagnetic data obtained on the territory of Russia is currently being performed. Maps of sediment conductance and other parameters of large sedimentary complexes and lithospheric conductive layers are being constructed (Sheinkman et al., 2003; Feldman et al., 2005). Acknowledgements The authors wish to acknowledge A.V. Lipilin, Head, Department of ROSNEDRA Federal Agency, for the support of regional electromagnetic explorations. We are also grateful to I.S. Feldman, A.V. Pospeev, A.K. Suleimanov, V.V. Belyavskiy, V.V. Lifshits, and other leading experts of industrial geophysical companies for fruitful collaboration, as well as to V.A. Kulikov, E.V. Andreeva, A.G. Morozova, D.A. Alekseev, and other specialists of North-West Ltd. for taking part in the studies considered. The scientiﬁc effort of authors from Moscow University was supported by RFBR (project 05-05-65082). P.Yu. Pushkarev also thanks INTAS for support (project 03-55-2126).

REFERENCES Aleksanova, E.D., Bubnov, V.P., Kaplan, S.A., Lifshits, V.V., Pospeev, A.V., and Yakovlev, A.G., 2005. Deep magnetotelluric studies along geotraverses in the East-Siberian craton (in Russian): Abstr. 7th V.V. Fedynsky Geophysical readings, Moscow. Aleksanova, E.D., Kulikov, V.A., Pushkarev, P.Yu. and Yakovlev, A.G., 2003. Application of electromagnetic ﬁelds created by electriﬁed railroads for electromagnetic soundings (in Russian). Izvestiya VUZov (Geology and Prospecting), 4: 60–64.

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Alekseev, D.A., Kulikov, V.A., Yakovlev, A.G., Grebnev, V.P., Koryavko, A.I. and Matrosov, V.A., 2004. Application of AMT method for mineral prospecting (in Russian). Prospect. Preserving Interiors., 5: 40–44. Arbuzkin, V.N., Kampaniets, M.A., Andreeva, E.V., Morozova, A.G., Yakovlev, A.G. and Yakovlev, D.V., 2003. Magnetotelluric soundings along a proﬁle in Elbrus region (in Russian): Abstr., 5th V.V. Fedynsky Geophysical Readings, Moscow. Berdichevsky, M.N., 1994. Role of geoelectric methods in hydrocarbon and deep structural investigations in Russia. Geophys. Trans., 39: 3–33. Berdichevsky, M.N., Fox, L., Yakovlev, A.G, Bubnov, V.P., Kulikov, V.A. and Pushkarev, P.Yu., 2002. Russian Oil and Gas Geoelectric Surveys: Abstr. 16th Workshop on Electromagnetic Induction in the Earth, Santa Fe. Berzin, R.G., Suleimanov, A.K., Berdichevsky, M.N., Yakovlev, D.V., Andreeva, E.V., Sborshchikov, I.M. and Yakovlev, A.G., 2002. Results of electromagnetic prospecting in the southern part of 2-DV proﬁle (in Russian). Proc. All-Russian Conference ‘‘Geodynamics, Magmatism and Minerageny of Northern Paciﬁc Continental Margins’’, 1, Magadan. Berzin, R.G., Suleimanov, A.K., Filin, S.I., Bubnov, V.P., Aleksanova, E.D., Yakovlev, A.G. and Pushkarev, P.Yu., 2005. Electromagnetic explorations using MT method along the ‘‘Morozovsk-Kamyshev’’ proﬁle’’ (in Russian). Proc. 5th and 6th V.V. Fedynsky Geophysical Readings, Moscow, pp. 185–189. Bubnov, V.P., Aleksanova, E.D., Morozova, A.G., Yakovlev, A.G. and Andreeva, E.V., 2003. Results of electromagnetic prospecting using MT method along proﬁle IV of the ‘‘Rifey’’ exploration program in the Moscow syneclise(in Russian): Abstr. 5th V.V. Fedynsky Geophysical Readings, Moscow, pp. 110–111. Feldman, I.S., Lipilin, A.V., Shpak, I.P. and Erinchek, Yu.M., 2005. Geological interpretation of electromagnetic prospecting results on the territory of the European part of Russia (in Russian): Abstr. 7th V.V. Fedynsky Geophysical Readings, Moscow, pp. 31–32. Kulikov, V.A., Yakovlev, A.G., Morozova, A.G., Svistova, E.L. and Kamkov, A.A., 2005. Deep resistivity cross-section along ‘‘Uralseis’’ proﬁle (in Russian): Proc. 5th and 6th V.V. Fedynsky Geophysical Readings, Moscow, pp. 180–184. Rokityansky, I.I., Ingerov, A.I. and Baisarovich, M.N., 1989. Donbass conductivity anomaly (in Russian). Geophys. J., 3: 30–40. Sheinkman, A.L., Narskiy, N.V. and Lipilin, A.V., 2003. Map of the total conductance of the sedimentary cover on the territory of the European part of Russia, scale 1 : 2 500 000. Abstr. Intern. Geophysical Conference ‘‘Geophysics of the XXI Century – the Leap into the Future’’, Moscow.

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Chapter 14 EM Studies at Seas and Oceans N.A. Palshin Shirshov Institute of Oceanology, RAS, Moscow, Russia

A comprehensive review of marine EM studies is not the main purpose of the chapter; the latter could be found in recent papers by Edwards (2005) and Baba (2005). The authors’ intention is to outline main trends in marine EM studies and to list key experiments and most important results obtained within the last 5–10 years. The technique of EM studies in seas and oceans differs in certain aspects from the technique used on the land. The measurements of EM ﬁeld are carried out on the sea ﬂoor, which accounts for the differences in the excitation method and EM ﬁelds measurement. Conductivity structure of the ocean crust differs from the continental one, which also accounts for certain peculiarities of marine EM studies. At present, two groups of methods are used: one utilizing a controlled source in frequency and/or time domains and the second employing natural EM ﬁelds induced by magnetosphere–ionosphere current systems. The ﬁrst group of methods is used to study the Earth crust at the depth of several kilometers, while the second method is applied in studies of the ocean crust and the upper mantle. Unfortunately, the scope of EM sounding in seas and oceans has reduced considerably in 1990s, which can be attributed to the overall reduction in funding of fundamental research aimed at studying the deep sea and ocean structure. Certain progress in deep seaﬂoor EM studies is seen in the Northwestern Paciﬁc and the Northeastern marginal seas due to signiﬁcant ﬁnancial support of academic marine researches by Japanese government. Nevertheless, on the whole the outlook of future development of EM methods will most probably be deﬁned by possibilities of their effective application in oil and Corresponding author: e-mail: [email protected], [email protected]

Methods in Geochemistry and Geophysics, Volume 40 V.V. Spichak, Editor r 2007 by Elsevier B.V. All rights reserved. ISSN: 0076-6895 DOI: 10.1016/S0076-6895(06)40014-7

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gas geophysics and also on ecological and engineering studies, especially on the shelf.

14.1. CONDUCTIVITY STRUCTURE OF SEA AND OCEAN FLOOR 14.1.1. Background conductivity structure of the ocean crust and upper mantle The main factor that is to deﬁne conductivity of the ocean crust as opposed to the continental one, is the sea water, that penetrates practically into the whole crust due its porosity and penetrability as well as a great number of subvertical faults. As a result, small-scale subsurface inhomogeneities, that cause hardly traceable distortions on land (static shift), are practically absent on the sea ﬂoor. Electric conductivity of sea ﬂoor sediments is only in 1.5–4 times lower than that of the sea water (e.g. Edwards, 2005), and the second lower layer, consisting of basalt pillow laves has a conductivity about 0.1 S m. The lower part of the crust formed by less porous massive basalts is characterized by lower conductivity ranging from 0.0001 to 0.03 S/m. The Moho boundary is the lowest limit of the sea water penetration, caused by the process of serpentinization in the lower part of the ocean crust. Thus, only ﬂuids of the mantle origin can exist in the upper mantle of the ocean (Anderson, 1989), excluding subduction zone and back-arc basins (Hyndman and Peacock, 2003). One more peculiarity of the ocean crust is an anisotropy of its electric conductivity. Close to the spreading axis the basalt laves erupt along relatively narrow elongated weakened zones. Thus inhomogeneous anisotropic structures are formed, where conductivity in the direction parallel to the rift zone is greater than in the perpendicular direction, while in the vertical direction it is greater than in the horizontal one. The degree of its anisotropy as well as the average conductivity of the ocean crust decrease with the age of the crust (e.g. Shaw, 1994). The ocean mantle, which its upper boundary at a depth of 5–7 km is characterized, according to the data available, by extremely low values of conductivity (e.g. Cox et al., 1986; Chave et al., 1990), up to about 105 S/m. Further, at greater depths, the increase in temperature and pressure also cause the conductivity gradually. In several regions of the Paciﬁc Ocean a conducting ‘‘asthenosphere’’ layer in the depth interval from 60 up to 120 km was detected, the existence of which can be accounted for by partial melting of rocks of the upper mantle. At the depth of 200–300 km the difference in conductivity between the ocean and continents is most likely to smooth out (Palshin, 1988). 14.1.2. Principle objectives of marine EM studies For many years, there was mostly an academic interest to studying the conductivity structure of the oceanic crust and upper mantle. These efforts were focused mainly at active plate boundaries: rifts and subduction zones. The targets of these studies are conductors associated with thermal and tectonic evolution of the ocean crust and upper mantle (e.g. Palshin, 1996; Heinson, 1999; Baba, 2005).

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Recently, the industry demands shifted hydrocarbon explorations from the continents offshore, making the continental shelves a focus for geophysical activity, and offshore hydrocarbon detection and assessment became the main objectives of marine geophysics. Thus, the main targets of hydrocarbon offshore EM explorations are relatively resistive petroleum, free gas and gas hydrates (Edwards, 2005). One of the main goals of EM marine studies is a better understanding of gas hydrate spatial distribution and its content in seaﬂoor sediments. Gas hydrates are a mixture of methane and water in solid state. They are widely spread in sediments of the coast zones practically of all seas and oceans. The zones and the depth interval of their propagation are controlled by the temperature and pressure. Gas hydrates are stable in oceanic sediments over the ﬁrst hundred meters below seabed in shelf zones where the sea depth does not exceed 400 m. Usually, the amount of gas hydrates increases with the increase of distance from the sea ﬂoor and gradually reaches a certain maximum, and after that, due to the phase transition gas hydrates becomes instable. This lower border can be deﬁned rather accurately by means of seismic methods, while the upper smooth border is practically not seen by reﬂection seismics, which makes impossible to estimate the amount of gas hydrate by seismic methods alone. That may be done only by resorting to conductivity data, since hydrates, being in solid state, act similar to ice, i.e. they replace the sea water in sedimentary pores and reduce considerably their conductivity. At present gas hydrates are considered as source of hydrocarbons alternative to the traditional deposits of oil and natural gas. The estimates of hydrocarbon deposits, obtained by the leading specialist in gas hydrates Keeth Kvenvolden (Kvenvolden, 1993), show the gas hydrates to contain more than 50% of all organic hydrocarbons on the Earth, which is twice as much as in coal, oil and natural gas taken together. One more important aspect in gas hydrates studies in sea ﬂoor sediments is a green house effect. At present the atmosphere contains a very small portion of methane (of its global content in geosphere), it is likely that its emanation from the sediments into the atmosphere might enhance the green house effect. Therefore, the studies of sea ﬂoor sediments in terms of gas hydrate distribution and deposits seen to be one of the main goals in the Earth sciences.

14.2. INSTRUMENTATION FOR MARINE EM STUDIES One of the main distinctions of marine EM studies is, of course, the necessity to use special equipment both to excite and observe EM ﬁelds on the sea ﬂoor, since the sea water is a well-conducting media. Some characteristics of modern acquisition systems to carry out marine EM studies are described below. 14.2.1. Seafloor controlled source frequency and transient EM sounding The instrumentation designed for marine controlled source electromagnetics (CSEM) is considered to have most striking differences from traditional on-land ones. In seaﬂoor CSEM sounding horizontal electric dipoles are used as a rule both

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as a transmitter and as a receiver. The main principle of sea ﬂoor sounding is that the EM wave, propagating in the seawater layer decays, while in the less conductive seaﬂoor the EM ﬁeld propagates in the direction close to the horizontal one, like in geometric sounding (horizontal skin-effect). The depth resolution of the method is deﬁned to a great degree by the separation between the transmitter and receiver of the EM ﬁeld (e.g. Vanyan and Palshin, 1993). To carry out seaﬂoor CSEM sounding, an instrumentation consisting of two horizontal electric dipoles, one of which is used as a transmitter, and the other (or others) as receiving antennas, is employed. To excite EM ﬁeld, relatively low frequencies not exceeding 10 Hz are used. In transient modiﬁcation of CSEM method, a square waveform signals are used. There are two modiﬁcations applied for different penetration depth. One modiﬁcation is used to study the uppermost sediments at a depth up to several hundred meters, here a single-towered system is used (see Fig. 14.1). When using such towed system, the separation range is approximately 50–500 m and the whole system is towed in direct contact with seaﬂoor sediments (Schwalenberg et al., 2005). In the case when studies are carried out at shallow waters, an alternative EM technique with both transmitting and receiving dipoles towed at the water surface could be applied (Fig. 14.2). This modiﬁcation is especially effective in low-salinity basins (e.g. Black, Caspian and Baltic Seas and Obskaya Guba) where water conductivity is not as high as in the ocean. Depth resolution of the method developed by Russian prospecting company ‘‘Sibisrkaya Geoﬁzicheskaya Nauchno-proizvodstvennaya Companiya’’ averaged hundreds of meters. The method utilizing both transient and induced polarized technique was successfully applied in various basins (Davidenko et al., 2005).

Fig. 14.1. Geometry of the inline dipole–dipole conﬁguration. A current signal is produced by an onboard transmitter and sent through the coaxial winch cable to the transmitter bipole on the sea ﬂoor. Two receiver dipoles at distances r1 and r2 record the signal after it passes through the seawater and the sediments. A heavy weight (pig) attached to the front of the system keeps the array on the seaﬂoor while moving along the proﬁle. Moving the ship and taking in the winch cable pulls the array forward and causes a vertical movement of the pig. Solid and dotted line present the winch cable in idle and moving state, respectively. The wheel represents the curve over which the marine cable appears to move while in motion (modiﬁed from Schwalenberg et al., 2005).

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Fig. 14.2. General layout of CSEM transient and induced polarization studies with surface towed multiseparation transmitter–receiver system aimed for hydrocarbon deposits detection in shallow water (modiﬁed from Legeido et al., 2005).

In deep sea modiﬁcation of the method (Shina et al., 1990), it is necessary to vary separation between the transmitter and receivers within wide limits (up to 10 km or even more). In such case ocean bottom electrometers (OBE) are deployed to observe the electric ﬁeld, while towed above the seaﬂoor horizontal transmitting dipole is used as a source (see Fig. 14.3). The employment of autonomous OBE makes it possible to apply both azimuth and radial sounding geometry that considerably increases the possibilities of the method, in particular it makes possible to detect the anisotropy of conductivity (e.g. Vanyan and Palshin, 1996). At present this CSEM modiﬁcation is also used to investigate carbohydrates on the sea ﬂoor at a depth up to several kilometers (Ellingsru et al., 2002). It should be mentioned that the implementation of CSEM advances serious requirements to the equipment of research vessels: the availability of a power unit to ensure the supply signal, a large powerful winch and a special shielded power cable, etc. 14.2.2. Measurements of variations of natural EM fields on the seafloor To measure long-period natural ﬂuctuation of EM ﬁelds on the sea and ocean ﬂoor is a rather challenging task in terms of technical procedure. In order to measure EM ﬁelds on the sea ﬂoor it is necessary to have instrumentation with accuracy of 0.05 nT for the magnetic ﬁeld and 0.01–0.02 mV/km for the electric ﬁeld. Besides, measurement of long-period electric ﬁelds is additionally complicated by the polarization of electrodes. In sea ﬂoor measurements, when electrodes are grounded through the sea water, self polarization even of the best samples of electrodes exceeds the useful signal at least by an order and is characterized by aperiodicity in its long-period variations.

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Fig. 14.3. General conﬁguration of CSEM frequency sounding using transmitting dipole towed above the seabed and autonomous OBE. Dark shaded layer in sediments represents target resistive layer, arrows show the EM wave propagation paths (modiﬁed from MacGregor, 2006).

To carry out long-term measurements of low-frequency electric ﬁeld on the sea and ocean ﬂoor, autonomous OBE are employed, where special mechanical devise is used – the salt bridge (‘‘chopper’’), that makes it possible to separate a desired signal against the background of slowly varying electrode potentials of much higher intensity (e.g. Filloux, 1987; Petitt et al., 1992; Constable et al., 1998). Measurements of magnetic ﬁeld on the sea and ocean ﬂoor are not that different on the whole from the land measurements. In seaﬂoor measurements, suspended optic-mechanical or ﬂuxgate sensors as well as induction coils are applied. In Fig. 14.4, a principal scheme of the autonomous ﬂoor MT instrument is shown. Beside autonomous ocean bottom instruments, the electric ﬁeld in seas and oceans is measured by submarine cables, which are rather effective tools for measuring electric ﬁeld. The advantage of such systems is evident: the longer the receiving line, the higher the level of a signal, while disturbances caused by electrodes and conditions of grounding do not increase (Lanzerotti et al., 1993; Palshin. 1996; Fujii and Nozaki, 1997). In addition to existing telecommunication cables, specialized relatively short submarine cable-based horizontal electric dipoles, the socalled Earth’s electric ﬁeld observation systems (EFOS), could be installed in the key locations at the seaﬂoor (see Fig. 14.5). The deployment of EFOS is carried out with the help of remotely operated vehicle (ROV). A prototype with a 10 km long cable (EFOS-10) was installed by JAMSTEC’s deep-tow and ROV technology in Daito ridge, West Philippine Basin (e.g. Utada et al., 2005).

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Fig. 14.4. Line drawing of seaﬂoor MT instrument. A 60 kg concrete anchor held beneath the center of the package sinks the device to the seaﬂoor. The anchor is released by the acoustic unit on receipt of a command code, and the device rises to the surface with the help of the glass ﬂotation spheres. The electric dipole arms are 5 m lengths of 5 cm diameter polypropylene pipes terminated with silver–silver chloride electrodes. Dipole cables run along the insides of the tubing (modiﬁed from Constable et al., 1998).

Fig. 14.5. Earth’s electric Field Observation System (EFOS) deployed at the Philippine Sea in 2004–2005. A recording device and a 10 km were installed in NE direction by towing from the ship using ROV (modiﬁed from Utada et al., 2005).

14.3. SOME RESULTS OF EM SOUNDING IN SEAS AND OCEANS The sea and ocean ﬂoor studies by EM methods are far from being extensive. The number of measuring points amounts to hundreds of such sites, whereas most of studies are concentrated mainly in the Paciﬁc Ocean and its marginal seas, while

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the Atlantic Ocean and especially the Indian and Arctic Oceans remain practically an outﬁeld. Most of the studies had a fundamental academic character, and only recently EM methods have been used for commercial purposes in prospecting natural resources (e.g. Fisher, 2005).

14.3.1. Studies of gas hydrates in seabed sediments of continental slopes Among the ﬁrst experiments using the transient EM sounding method were the studies at the western coast of Canada (Cascadia margin). They started in 1998 and were aimed at testing the method and obtaining estimations of gas hydrate content independent of seismic methods. The experiments were conducted in a well-studied region with exploring wells (Ocean Drilling Project) that revealed gas hydrates. As a result, the maps of conductivity for different horizons were obtained, which, by means of logging data and Archie law, were then recalculated into maps of percent concentration of gas hydrates in sediments (Edwards, 1997). In 2000, similar studies were carried out off the coast and they also made it possible to map the gas hydrate layer in the thick sediments (Ellinsgru et al., 2002). The most striking results were obtained in 2004 in Cascadia margin where CSEM measurements were carried out along proﬁles crossing the well-known blank seismic zones associated with vent zones (see Fig. 14.6). Pronounced conductivity anomalies were found and gas hydrate concentration was estimated to be of 50–60% (Schwalenberg et al., 2005). These results, demonstrating high efﬁciency of CSEM sounding method in gas hydrates studies, proved that the method could be successfully used in solving both research and exploration tasks, while CSEM prospecting technique in this case could be treated as a ‘‘direct’’ method, which makes it possible to estimate directly gas hydrates content within the deposit.

Fig. 14.6. ( a) Seismogram showing BSR occurrence along the proﬁle and seismic blank zones (Riedel et al., 2002). (b) Bulk resistivities derived from CSEM data show anomalous resistivities exceeding 5 O m over background resistivities between 1.1 and of 1.5 O m. The anomalous areas coincide spatially with the surface expression of the blank zones, indicated in the ﬁgure (modiﬁed from Schwalenberg et al., 2005).

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Fig. 14.7. Inverse model, parameterized in terms of layer interface depths, of TM-mode data between 0.001 and 0.1 Hz from 3-D numerical model of Gemini with a 10 O m basement at 8 km depth. White line shows salt outline as interpreted from 3-D prestack depth-migrated seismic data (modiﬁed from Hoversten et al., 2000).

14.3.2. Studies of buried salt dome-like structures There is one more example of employment of EM method for exploration tasks, i.e. seaﬂoor magnototelluric (MT) sounding, carried out in the Mexican Bay. The possibilities of base MT soundings were demonstrated at Gemini salt structure in the Mexican Bay, where soundings were carried out at 45 sites along two proﬁles crossing the structure. Impedance tensors were calculated easily within the range from 0.003 to 1 Hz and the quality of calculated tensors is comparable with the quality of on-land MT soundings. To increase reliability of two-dimensional inversion, the surface of the salt anticline known from seismic data was ﬁxed. Such joint approach yielded a conductivity structure (see Fig. 14.7), where the base of the salt structure is well deﬁned. At present this method is resorted for commercial aims. Traditionally, the attention of marine geologists and geophysicists is attracted to mid-ocean rift zones with active volcanic and tectonic processes, which differs from those on the continents. One of the main tasks is detecting and mapping of a magma chamber beneath the axial part of ocean rifts, and the EM methods played the leading role in solving this task. 14.3.3. The Reykjanes axial melt experiment: Structural synthesis from electromagnetics and seismics (RAMESSES project) In 1993 a complex geophysical experiment, that involved seaﬂoor CSEM and MT sounding and seismic studies, was carried out on the Reykjanes Ridge

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(the northern part of Middle Atlantic Ridge at 571N). MT soundings were performed at ﬁve stations along the proﬁle, crossing the axial part. The EM ﬁeld measurements were conducted during 20 days. Impedances were obtained in the range from 30 to 1000 s due to relatively small depths. Simultaneously, CSEM frequency sounding and seismic studies were carried out along several proﬁles crossing the axis of the rift and parallel to the axis (Heinson et al., 2000). As a result of such a unique experiment, a consistent petrophysical model of a slow spreading rift zone was constructed (see Fig. 14.8).

14.3.4. Seafloor MT soundings of the Eastern-Pacific rise at 91500 N The EM experiment was carried out in 2000 by American geophysicists with MT stations, designed for studying the sedimentary oil-bearing basins (see above). The studies were conducted along a proﬁle where prior seismic studies had been performed (Key and Constable, 2000). As a result of 2-D interpretation based on the TE method as being the most sensitive to the conductive objects, a model was constructed coinciding well with previously obtained distribution of seismic velocities (see Fig. 14.9).

Fig. 14.8. Combined interpretation based on data from seismic, controlled-source electromagnetic, and magnetotelluric experiments. Seaﬂoor bathymetry and seismic layer boundaries are quantitatively accurate based on swath bathymetry measurements and seismic modeling. Similarly, estimates of electrical resistivity, porosity, melt content, and temperature are quantitative estimates based on modeling and interpretation. Note the 10:1 break in scale at 10 km depth (modiﬁed from Heinson et al., 2000).

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Fig. 14.9. Conductivity structure of magma chamber at Eastern-Paciﬁc rise at 91500 N. MT sites are shown as black diamonds. Seismic velocity perturbation contours for DVp ¼ 1.9, 0.7, -0.5 and 0.3 km/s from Dunn et al. (2000) are shown for comparison (modiﬁed from Key and Constable, 2002).

14.3.5. Mantle electromagnetic and tomography experiment This is one of the most signiﬁcant deep see geophysical experiments when seaﬂoor MT soundings and seismic tomography were implemented together in the southern part of the Paciﬁc Ocean. In 1996–1997 by joint efforts of American, Canadian, French, Japanese and Australian geophysicists, the seaﬂoor MT soundings along two latitudinally extended proﬁles crossing Eastern-Paciﬁc rise along 171 south latitude were carried out. A total of 47 base stations were recording the EM ﬁeld variations during a half-a-year period. The experiment was preceded by a seismic tomography experiment along the same proﬁles. The main task of the MELT experiment was to study the upper mantle and to test two controversial hypotheses regarding the structure of the upper mantle of ocean rift zones (Evans et al., 1999; Baba et al., 2006). A two-dimension TM inversion and TE polarization were performed both separately and together. However, they failed to yield a consistent model – the discrepancy of bimodal inversion considerably exceeds the discrepancy of solutions obtained separately for TM and TE modes. According to the authors, this proves considerable anisotropy of conductivity of the upper mantle. The results are still being under interpretation, and the ﬁnal model of the upper mantle according to the MELT results has not been published yet.

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Summarizing the above results, we may say that by means of EM methods it was possible to show that the sizes of magma chamber in axis parts of ocean rift zones are small, not exceeding several kilometers in altitude; the magma chambers are located almost completely within the thin ocean crust. The partial melting in magma chamber ranges from 1 to 20%. The so-called ‘‘McKenzie model’’, according to which the rift zones look like narrow subvertical zones, got its convincing conﬁrmation by seaﬂoor EM experiments.

14.4. DEEP SEAFLOOR EM STUDIES IN THE NORTHWESTERN PACIFIC Electromagnetic observations on the ﬂoor of the ocean surrounding Japan were carried out under the Ocean Hemisphere Project (OHP) network, Stagnant Slab project, and related collaborative studies with other institutions (Fig. 14.10). These experiments contribute to the study of semi-global scale mantle dynamics in the region by conjunct analysis of the result and studying the respective tectonic settings (Baba, 2006). Super deep sounding using submarine cables. Super deep MT soundings allow obtaining impedance values up to periods of several days, and thus ﬁll the gap

Fig. 14.10. Seaﬂoor observation sites deployed through OHP network, Stagnant Slab and related projects (crosses), superimposed on bathymetry map around Japan. Triangles are geomagnetic stations on land and lines indicate submarine cables for electric ﬁeld measurements (modiﬁed from Baba, 2006).

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existing between the impedances retrieved from MT and GDS soundings, which is of great importance for studies of a deep structure of the upper mantle. The results obtained in various regions of the Earth, i.e. in the passive margins of the Atlantic Ocean, in the Baltic Sea located on the continental shelf, and in the Japan Sea and the Northwestern Paciﬁc, indicate that an obvious similarity exists in the deep geoelectric structure of the upper mantle of the Earth, starting from the depths exceeding 200–300 km (Palshin and Sigray, 2003; Santos et al., 2003; Utada et al., 2003, 2005; Nikiforov et al., 2004). These results convincingly speak in favor of the concept of the reference proﬁle, which is based on the assuption of spherical symmetry in the electric conductivity of the upper mantle of the Earth. Philippine Sea experiment. Six ocean bottom electromagnetometers (OBEMs) were deployed on the seaﬂoor of West Philippine Basin, Parece-Vela Basin, and Mariana Trough, in November 1999 and recovered in July 2000. One-dimensional (1-D) electrical conductivity structure of the upper mantle was estimated for each site. The experiment demonstrated its high potential in studying the geothermal structure and water and melt content in the upper mantle (Seama et al., 2006). Sea-floor electro-magnetic station (SFEMS). SFEMS was developed to provide continuous seaﬂoor observations of absolute geomagnetic total ﬁeld strength with the same accuracy as in the data from land-based observatories, geomagnetic ﬁeld vector and horizontal electric ﬁeld (Toh et al., 1998). SFEMS has been in operation at 41107’03’’N, 159155’43’’E in northeastern Paciﬁc Ocean since August 2001. The data collected so far demonstrate that the seaﬂoor observatory can contribute to the increase of the spatial resolution of the existing geomagnetic observatory network in the middle of the northwestern Paciﬁc, where long-term geomagnetic data are missing (Toh et al., 2004). Mariana experiment. Mantle dynamics associated with plate subduction, arc volcanism and back-arc spreading has been investigated through seaﬂoor MT experiments in central Mariana region. Pilot survey was conducted using 10 OBEMs in 2001–2002. The obtained data were inverted and the resultant electrical conductivity model suggested that the melt generation process at back-arc spreading axis is similar to that of normal oceanic spreading. Further experiment at 40 sites with 47 instruments started in December 2005 under international collaboration. Japan Sea experiment. This experiment was aimed to imaging the back-arc mantle beneath eastern Japan Sea. Six OBEMs were utilized for it in 2002–2003. Useful data were acquired from four OBEMs and analyzed together with the data on land. The obtained conductivity model exhibits high-conductivity zone, suggesting relation with the root of ‘‘hot ﬁngers’’, which is known as cluster-like distribution of volcanoes and low-velocity anomalies in northeastern Japan. Northwest Pacific experiment. Since 2003, one-year-long seaﬂoor MT survey using OBEMs has been conducted within several cruises in northwest Paciﬁc Ocean to investigate electrical conductivity of the upper mantle and transition zone. In this area, low-velocity anomaly was revealed by global seismic tomography, and very young (within 1 Ma) intra-plate volcanism was found. One of the goals of this experiment is to elucidate the relationship between these phenomena. The data were collected at seven sites, so far. The analysis is continuing now.

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Stagnant Slab experiment. As a part of Stagnant Slab project, a long-term semiglobal scale seaﬂoor MT experiment using OBEMs has been planned to image mantle transition zone beneath the Philippine Sea where the subducting Paciﬁc plate seems to be stagnated above the lower mantle. Eleven OBEMs were deployed in October 2005. The sites cover northern West Philippine Basin, Shikoku Basin and northern Parece-Vela Basin with 500 km spacing. The survey is just the ﬁrst phase of the experiment. The one-year-long measurement will be iterated three times to acquire long enough data for imaging the transition zone.

Acknowledgements Many colleagues promptly provided reprints and pre-prints. I would like to express my gratitude to the following colleagues (in alphabetic order): Kieshi Baba, R. Nigel Edwards, Lucy MacGregor, Alexander Petrov, Katrin Schwalenberg and Hisashi Utada.

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Subject Index

1-D inversion 220 2-D bimodal inversion 31 3-D 105–106, 109–110, 112, 114, 116, 121, 133, 220, 223–224, 232, 255, 275, 281, 284, 295, 297, 331–334 3-D axially symmetric 105 3-D imaging 286 3-D inversion problem 209 A priori information 186 accuracy control 121 activation 222, 224–227, 235–236 anisotropy 370 ANN interpolation 223 ANN inversion 224–225, 239–240 ANN recognition 220 ANN training 222 ANNs 220, 226, 229 anomalous geoelectric structure 186 anomalous properties of the horizontal magnetic tensor 270 antenna polarization effect (APE) 68 apparent resistivities 157, 275, 277, 285–287, 295, 297 aquifer salinity 344 aquifers 344–345 Archie law 376 artiﬁcial intelligence paradigm 220 artiﬁcial neural network (ANN) 220–221, 223–227, 229, 234–239, 242, 253 asymptotic boundary conditions 110, 118, 127 asymptotic formula 132–133 axially symmetric three-dimensional 124, 126, 131 BackPropagation 220 balance technique 105, 110, 116, 118, 133

basis functions 111, 128–129, 133 BEL 266 Biot critical frequency 82 Biot equations 80, 82, 87 Bostick transforms 246–248, 252 boundary conditions 110, 113–114, 117–118, 121–122, 127 C-responses 10 compact subset 175 conditionally well-posed problem 176 connection weights 221–223 coordinate invariant 282, 284 COPROD-2S1 dataset 211 COPROD-2S2 dataset 213 COPROD-2S2S dataset 213 correctness set 176 databases 220, 223–224, 234, 242, 250 differential equation methods 105, 107, 109, 111–112, 116, 124 dikes 223, 227–229, 231–232, 234, 236–239, 241 electrodynamic similitude 224, 242 electrodynamic similitude criterion 115 EMTESZ-Pomerania array sounding experiment 260 exact model solution 164 extrapolation 220, 223, 226, 238–239 fast EM ﬁelds 84–87, 90 fault 219, 237, 242–244, 246–247, 250, 252–254 FF window 187 ﬁnite 186 ﬁnite element technique 111, 113 ﬁnite-difference 120, 123–124 ﬂuids 370 forward modeling 224, 247, 251–253

386

Subject Index

frozen rocks 72 frozen-in 85 gas hydrates 371 gas-condensate 313, 316 Gauss–Newton scheme 193 Gaussian noise 239, 241 generalized harmonic analysis 262 geoelectric noise 31 geomagnetic storms 9 global conductance map 6 global EM induction 4 Green functions 113 Green’s tensors 107–108, 121–122, 144–145 hidden layers 221–222, 224–229 highly coherent industrial EM noise 263 Hilbert transforms 121, 123–124 HLP 266 horizontal magnetic tensor rotation invariants 269 hydrocarbon 313–314 hydrocarbon reservoirs 314 imaging 286, 294–295, 297, 319, 331, 333, 335, 343–344, 349 impedance 275–282, 284–287, 293, 295, 297, 303–305 impedance tensor 157 impedance tensor decomposition 282 induced polarization (IP) effect 68 induction vector 288, 290 integral equation methods 105–106, 108–110, 112, 124 integral transform 285, 288, 290, 298–300, 302 international project COPROD-2S 197 interpolation 220, 238–239 interpretation 219–220, 225, 233, 253, 255, 314–315, 317–320, 331, 333–336, 341, 343–344, 348–349 interpretation model 33 inversion solution stabilization 195 inversions 219–220, 223–224, 229, 232, 237–238, 242, 245, 247, 249–255, 350 Krylov subsurface iteration 148–149

L-representation 182 Lame equation 80 layered model 176 locally inhomogeneous models 176 macro-parameters 223, 252–253 magma chamber 314, 331, 333–334, 338, 340–344 magnetic tensor 158 magnetotelluric (MT) method 27, 185 magnetotelluric data 313, 349 magnetotelluric data interpretation 315 magnetovariational (MV) method 27, 185 marine EM studies 369 measurement errors 159 methane 371 method of partial inversions 33 misﬁt functionals 178 model errors 159 model sensitivity operator 193 modeling 242, 321, 325–327, 331, 334, 338, 344, 347, 349–350 models 315 modiﬁed Cholesky factorization 194 modiﬁed iterative dissipative method (MIDM) 147 monitoring 219, 253, 314, 331, 338, 340–341, 343–344, 348 MT problem 158 MT–MV geoelectric complex 27 multi-dimensional ﬁnite function 188 Multi-RR data-processing technique 263 multicriterion 31 MV problem 158 MV–MT complex 29 neurons 220–222, 224–229 Neumann series 146 Newtonian iteration 193 Newtonian linear system 193 Newtonian scalar step 194 Newtonian search direction 193 normal background 160 normal geoelectric structure 186 numerical modeling 313, 329, 349

Subject Index

Occam’s 249, 252 Occam’s inversion 177, 246, 252 ocean bottom electrometers 373 ocean circulation 4 ocean crust 369 ocean effect 13 one-dimensional ﬁnite function 187 optimization method 177 parallel (joint) inversion 31 partial misﬁts 180 penetrability 370 phase tensor 282–284 piecewise-continuous models 185 point transform 277, 288, 291, 298, 305 porosity 370 quasi-Newtonian schemes 193 receiver 372 recognition 220, 222–242 regularization method 177 regularization parameter 179 regularization principle 179 regularizing functional 180 regularizing operator 179 relative mean square misﬁt 191 relief surface 314, 335, 338, 340 remote reference data-processing techniques 262 repeated measurements 266 resolving power 313–315, 338 rift zone 370 robust methods of linear estimation 262 robust metric 192 robust modiﬁcation of Newtonian schemes 194 rotational invariant 280–281, 286 RRMC data-processing technique 263 Sq variations 7 salinity 344, 346 salt bridge 374 satellite measurements 3 scattering equation 145 SE logging 86, 88, 90, 96 SE transfer 93 SE transfer functions 86–87, 90, 96

387

sea ﬂoor sediments 370 sea water 370 seismoelectric transfer functions 85 serpentinization 370 similarity principle 250 simultaneous array EM sounding 259 simultaneous observation systems 259 skin-effect 372 stabilizer 180 stabilizing functional 179 static shift coefﬁcients 196, 215 sub-surface galvanic effects 195 subduction zone 370 submarine cables 374 subvertical faults 370 successive (partial) inversions 31 superparamagnetic (SPM) effect 68 surface integral equations 106 SVD factorization 194 synthetic data sets 197 teaching 223–227, 229, 232, 234–237, 239, 241–242 teaching data 238 teaching database 232 temporal stability of transfer operators 264 theorem on the stability 175 three-dimensional 106, 114–116, 120, 123–124, 133, 219, 224, 255, 314, 325–326, 331, 338, 349 three-dimensional axially symmetric 124 three-dimensional interpretation 219 three-dimensional model 315, 319, 338, 350 three-dimensional modeling 315 tides 4 Tikhonov’s regularizing functional 191 tipper 303–304 tipper vector (the Wiese–Parkinson vector) 157 total misﬁt 180 training data 227, 233–237, 240, 250 training database 232 transfer 277, 288 transfer operators of EM ﬁelds 259 transmitter 372

388 volcanoes 313, 331–339, 344, 348–350 volume integral equations 106 volume integral equation method 143 water springs 65 weighted data misﬁt 191

Subject Index

window with correlated resistivity parameters (CR window) 187 window with ﬁnite functions 187 window with independent resistivity parameters (IR window) 187