Magnetotellurics in the Context of the Theory of Ill-Posed Problems

229 Pages·2002·6.55 MB·English, Russian

by  Keller, George Vernon, Dmitriev, Vladimir Ivanovich, Berdichevskiĭ, M. N

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Series: Investigations in Geophysics, Volume 11
Michael R. Cooper, Series Editor
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Magnetotellurics in the
Context of the Theory
of Ill-Posed Problems
Mark N. Berdichevsky and Vladimir I. Dmitriev
Translated and Edited by
G. V. Keller
Society of Exploration Geophysicists
Tulsa, Oklahoma, U.S.A.
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Library of Congress Cataloging-in-Publication Data
Berdichevsky, M. N. (Mark Naumovich)
Magnetotellurics in the context of the theory of ill-posed problems / Mark N. Berdichevsky
and Vladimir I. Dmitriev ; translated and edited by G. V. Keller.
p. cm. – (Investigations in geophysics ; no. 11)
Includes bibliographical references and index.
ISBN 1-56080-106-9
1. Magnetotelluric prospecting. 2. Differential equations, Partial—Improperly posed
problems. I. Dmitriev, Vladimir Ivanovich, 1932– II. Keller, George Vernon, 1927–
III. Title. IV. Series.
TN269 .B42 2001
622'.153—dc21 2001020119
ISBN 978-0-931830-46-4 (Series)
ISBN 978-1-56080-106-1 (Volume)
© 2002 by Society of Exploration Geophysicists
All rights reserved. This book or parts hereof
may not be reproduced in any form without per-
mission in writing from the publisher.
Published 2002
Reprinted 2010
Printed in the United States of America
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Contents
Preface ix
BasicNotations xiii
I GeneralStatements 1
1 Ill-posedProblemsinGeophysics 3
1.1 DirectandInverseGeophysicalProblems 3
1.2 ElementsofFunctionalAnalysis 4
1.3 ConditionallyCorrectStatementofInverseProblemsinGeophysics 6
1.4 SolutionofanInverseProblembyOptimization 8
1.5 SolutionofanInverseProblemUsingtheMethodofRegularization 9
1.6 TheMethodofRegularizedOptimization 12
1.7 AdaptiveRegularizingOperators 12
1.8 ConstructionoftheInverseOperator 13
1.9 TheBackus-GilbertMethod 15
1.10ProbabilisticStatementoftheInverseProblem 18
2 TheMagnetotelluricField 20
2.1 BasicTypesofMagnetotelluricVariations 20
2.2 CanGalvanicCouplingBetweentheIonosphereandtheEarthBeIgnored? 23
2.3 TheMagnetotelluricFieldinaQuasi-stationaryApproximation 24
2.4 Quasi-stationaryMagnetotelluricFieldintheFrequencyDomain 26
3 StatementoftheMagnetotelluricProblem 27
3.1 TheTikhonov-CagniardTransformations 27
3.2 TwoApproachestoMagnetotelluricProblems 28
3.3 ExplorationandDeepMagnetotelluricSurveys 29
II One-dimensionalMagnetotellurics 31
4 ASphericalModeloftheEarth 33
4.1 SpectralDecompositionoftheFieldinaSphericalModel 33
4.2 SpectralImpedanceoftheEarthinaSphericalModel 34
4.3 SpectralImpedanceinaHomogeneousSphericalModel 35
4.4 SpectralImpedanceinaLayeredSphericalModel 35
4.5 TheTikhonov-CagniardOperatorsforaSphericalModel 36
4.6 TheTikhonov-CagniardTransformationsasSpatialFiltration 37
4.7 IsItPossibletoIgnoretheEarth’sSphericity? 37
5 APlaneModeloftheEarth 39
5.1 SpectralDecompositionoftheFieldinaPlaneModel 39
5.2 SpectralImpedanceoftheMagneticTypeinaPlaneModel 40
5.3 SpectralImpedanceoftheElectricTypeinaPlaneModel 41
5.4 GeneralCase:TheFieldContainsBothModes 42
5.5 TheTikhonov-CagniardImpedance 42
5.6 AsymptoticsofSpectralImpedances 43
5.7 TheWait-PriceCriterion 43
5.8 TheTikhonov-CagniardMagnetotelluricOperatorinaPlaneModel 44
5.9 TheTikhonov-CagniardMagnetovariationalOperatorinaPlaneModel 46
5.10EffectiveResponsesoftheTikhonov-CagniardFilters 46
5.11TheTikhonov-CagniardTransformationsforaUniformMagneticField 48
v
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vi Contents
5.12TheTikhonov-CagniardTransformationsforaFieldwithLinearSpatialVariations
ofHorizontalMagneticComponents 48
5.13TheTikhonov-CagniardTransformationsforaFieldwithQuadraticSpatialVariations
ofHorizontalMagneticComponents 49
5.14AccuracyoftheTikhonov-CagniardModel 49
5.15PracticalUseoftheTikhonov-CagniardModel 51
6 Tikhonov-CagniardImpedance 54
6.1 BasicModelofMagnetotelluricSounding 54
6.2 TheMagnetotelluricFieldintheTikhonov-CagniardModel 54
6.3 TheLawofTotalCurrent 55
6.4 PhysicalInterpretationoftheImpedance 55
6.5 DeterminationoftheImpedancefromtheBoundaryValueProblemforthe
ElectromagneticField 57
6.6 TheLayeredModel 57
6.7 TheGradientModel 58
6.8 DeterminationoftheImpedancefromtheRiccatiEquation 60
6.9 ReciprocalModels 61
6.10High-frequencyAsymptoticsoftheImpedance 63
6.11Low-frequencyAsymptoticsoftheImpedance 64
6.12ImpedanceonthePlaneofComplexFrequency 65
6.13FrequencyDependenceoftheImpedance 68
6.14RelationsBetweenRealandImaginaryPartsoftheImpedance 69
6.15RelationsBetweenModuleandArgumentoftheImpedance 71
6.16PrincipleofSimilitude 72
6.17Admittance 73
6.18MagnetotelluricTransformationintheTimeDomain 75
6.19DeterminationoftheImpedanceintheSea 77
7 Transformations 79
7.1 ConversionofImpedancetoaFormUsefulforInterpretation 79
7.2 SkinEffectinLayeredMedia 79
7.3 SummingorScanning? 81
7.4 ApparentResistivityoftheEarth 81
7.5 Apparent-resistivityCurves 84
7.6 AsymptotesofApparent-resistivityCurves 87
7.7 PropertiesofApparent-resistivityCurvesforaK-typeModel 88
7.8 PropertiesofApparent-resistivityCurvesforanH-typeModel 91
7.9 EvaluatingGeoelectricParametersfromApparent-resistivityCurves 94
7.10ApparentConductivityoftheEarth 95
7.11TheNiblett-WittgensteinTransformation 96
7.12TheMolochnov-VietTransformation 98
7.13ControlledTransformation 100
7.14TheDepthofMagnetotelluricInvestigation 102
7.15GlobalMagnetovariationalSounding 103
7.16MagnetotelluricProfiling 107
8 TheInverseMagnetotelluricProblem 108
8.1 One-dimensionalDefinitionoftheInverseMagnetotelluricProblem 108
8.2 ExistenceandUniquenessofaSolutiontotheMagnetotelluricProblem 109
8.3 EquivalentGeoelectricMedia 110
8.4 InstabilityoftheMagnetotelluricInverseProblem 112
8.5 ComparisonCriteria 112
8.6 SplineApproximationofMTData 114
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Contents vii
8.7 TheOptimizationMethod 116
8.8 TheMethodofRegularizedOptimization 119
8.9 TheS-method 120
III MultidimensionalMagnetotellurics 123
9 MagnetotelluricTransferFunctions 125
9.1 ImpedanceofaHorizontallyInhomogeneousMedium 125
9.2 PolarDiagramsoftheImpedanceTensor 129
9.3 PrincipalValuesandDirectionsoftheImpedanceTensor 130
9.4 SeparationofLocalandRegionalEffects 133
9.5 TheWiese-ParkinsonMatrix(theTipper) 136
10 AnalysisofMTData 139
10.1QuestionsandStagesinMagnetotelluricInterpretation 139
10.2AnalysisoftheImpedanceTensor 140
10.3AnalysisoftheWiese-ParkinsonMatrix 141
10.4AnalysisofApparentResistivity 142
11 MultidimensionalMagnetotelluricInversion 146
11.1MultidimensionalDefinitionoftheInverseMagnetotelluricProblem 146
11.2Quasi-one-dimensionalInversion 147
11.3Two-dimensionalInversion 148
12 MagnetotelluricsinGeologicalService 167
12.1ABitofHistory 167
12.2MagnetotelluricsinExplorationforOilandGas 168
12.3MagnetotelluricsinStudiesoftheEarth’sCrustandUpperMantle 173
12.4GeoelectricModelsoftheJuandeFucaSubductionSystem 174
12.5GeoelectricModeloftheKirghizTienShan 178
12.6GeoelectricModeloftheBaikalRift 187
12.7InterpretationofGlobalMagnetovariationalSounding 197
IV Summary 203
13 MagnetotelluricsintheTwenty-FirstCentury 205
13.1RestrictionsofMagnetotellurics 205
13.2TheFutureofMagnetotellurics 206
References 208
Index 215
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About the Authors
Mark N. Berdichevsky
After receiving a degree from Moscow University, he went on to earn a degree of Doctor of Philosophy
in 1954 and in 1967, Doctor of Science from the Institute of Physics of the Earth, Moscow. For 20 years
his research at the Institute of Geophysical Prospecting of the Soviet Ministry of Geology focused on
geoelectromagnetism and novel methods of electrical and electromagnetic prospecting. In 1968 he joined
the Geophysical Department of Moscow University. He and his team have made numerous contributions
to the theory and methodology of magnetotelluric and magnetovariational sounds, and he has made
significant contributions to the literature on the subject. His current interests include deep geoelectrical
investigations. He is a member of the Russian Academy of Natural Sciences and Polish Academy of
Sciences.
Vladimir I. Dmitriev
A graduate of Moscow University, he received the degree of Doctor of Philosophy in 1959, and Doctor
of Science in 1967 from the Institute of Physics of the Earth, Moscow. Since 1962, he has been with the
Department of Computing Mathematics and Cybernetics of Moscow University. His areas of research
include mathematical physics, electrodynamics, theory of inverse problems, theoretical geoelectrics
and geothermy, and theoretical engineering. He and his team have made numerous contributions to the
theory and methodology of magnetotelluric and magnetovariational soundings. He has made significant
contributions to the literature in monographs and papers in geophysical and mathematical journals. He is
a member of the Russian Academy of Natural Sciences.
viii
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Preface
TothememoryofAndreyN.TikhonovandLouisCagniard,pioneersofmagnetotellurics
In1950,A.N.Tikhonovpublishedapaper,“Onde- computationalelectrodynamics,andinterpretationphi-
terminationofelectriccharacteristicsofdeeplayersof losophyandtechnology.
the earth’s crust” in the proceedings of the Academy The mathematical basis for MTS was developed by
of Sciences of the USSR (Doklady, Akademia Nauk A.N.Tikhonov,V.I.Dmitriev,P.Weidelt,U.Schmucker,
SSSR). In this paper, Tikhonov examined the rela- C.Swift,G.Hohmann,K.Vozoff,R.Parker,J.Weaver,
tionsbetweenthehorizontalcomponentsE , H of the andD.Oldenburg.
x y
magnetotelluricfield(thenaturaltime-varyingelectro- The earliest magnetotelluric experiments were con-
magnetic field of cosmic origin), and introduced the ducted by M. N. Berdichevsky, N. V. Lipskaya, A. M.
impedance Z = E /H as a quantity characterizing Alekseev,T.Cantwell,F.Bostick,andD.Rankin.
x y
the electric conductivity of the earth’s interior. A one- There were several crucial moments in the develop-
dimensionalmodeldisregardingthelateraleffectswas mentofmagnetotellurics.
usedforimpedanceinterpretation.Inthisway,thefeasi- The question of physical validity of the Tikhonov-
bilityofsoundingtheearththroughthemagnetotelluric Cagniard one-dimensional model seemed to be the
observationsatasinglepointontheearth’ssurfacewas mostcontroversialone.Thediscussionwasopenedby
demonstrated,andnewinformationaboutconductivity J. Wait and A. Price, who referred to the strong hor-
inthemantlewasobtained. izontal nonuniformity of the external magnetic field
Thissimpleideagaveimpetustothedevelopmentofa and pointed to the necessity of serious limitations for
newgeophysicalmethodcalledmagnetotelluricsound- MT sounding. An even more critical viewpoint was
ing,orMTsounding,orsimply MTS.Thismethodisa that of D. N. Chetaev, who suggested abandoning the
variation of frequency sounding. With all its strengths Tikhonov-Cagniardmodelandreplacingitbyamodel
and weaknesses, it has found wide utility in commer- with horizontally propagating waves of both electric
cial electric exploration and deep geoelectric investi- andmagnetictypes.SupportfortheTikhonov-Cagniard
gations. A new branch of geophysics, given the name modelwasprovidedbyT.Madden,P.Nelson,C.Swift,
magnetotellurics,hascomeintobeing. V. I. Dmitriev, M. N. Berdichevsky, L. L. Vanyan, and
Magnetotellurics aroused considerable interest J.Hermance.Thecontroversieswereratheruseful,since
amonggeophysicistsallovertheworld,sinceitoffered it became evident that the Tikhonov-Cagniard model
greatnewpotentialinstudyingnotonlysediments,but approximates fairly well the magnetotelluric relation-
the earth’s crust and upper mantle as well. One of the shipsobservedinmiddle-andlow-latituderegionswith
most interesting and encouraging events at that time horizontallyhomogeneouslayering.
wasthepaper,“Fundamentaltheoryofthemagnetotel- Much time was expended in development of data-
luricmethodingeophysicalprospecting,”publishedby processing methods that provided stable and accurate
Louis Cagniard in 1953 in GEOPHYSICS. Cagniard determination of impedance. The earliest experiments
transformedthefrequencyresponsesoftheimpedance displayed large (occasionally very large!) scatter in
intocustomaryapparent-resistivitycurves,andshowed impedance values, amounting to tens or even several
that magnetotellurics could serve very nicely for fast hundred percent. M. N. Berdichevsky and T. Cantwell
geophysical surveys over vast terrains. He believed attributed this scatter to the effects of horizontal geo-
in the rapid success of magnetotellurics and claimed electric inhomogeneities, and proceeded from scalar
that MTS would help electric prospecting out of the measurementstotensorones.Thevalidityofthetensor
deadlockinwhichithadbeentrappedbythelimitations approachreceivedstrongsupportfromextensiveobser-
causedbyapplyingdirectcurrent. vationsovermanyyears.So,theproblemwasreducedto
Theideasof TikhonovandCagniardwerewellahead eliminationofindustrialandmodelnoises.Itssolution
ofthegeophysicsofthefirstpostwardecade.Nowonder layinthesphereofmathematicalfiltering,robuststatis-
thatittookalmost20yearstobringMTSintopractice. tics, and deterministic and statistical control. Today,
Geophysicists of many countries were involved in the thanks to the efforts of B. S. Svetov, V. Y. Semenov, I.
workthatwascarriedonintheory,fieldinstrumentation, M.Varentsov,T.Gamble,A.Chave,J.Larsen,G.Egbert,
ix
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x Preface
and T. Ernst, we have a number of methods that yield theory,weeliminateoratleastsmooththelateraleffects,
theimpedancetensorwithafairdegreeofaccuracy. therebyclearingthewayforaone-dimensionalinterpre-
The transition to tensor-based data processing re- tation of MT data. However, such a normalization of
sulted in a major increase in the amount of informa- MTdataisnotalwaysreliableandnearlyalwaysresults
tion. Now, at each observation site, we arrive at an inthelossofsomepartofinformation.Thechallenging
infinite multitude of apparent-resistivity curves, de- problem of present-day magnetotellurics is, therefore,
pending upon the orientation of the coordinate axes. transitiontotwo-andthree-dimensionalinterpretation.
All this information can either be averaged using ten- This trend of development is found in the studies by
sor rotational invariants, or systematized and concen- V. I. Dmitriev, A. S. Barashkov, I. S. Barashkov, M. S.
tratedontheoptimumdirectionsbythemethodsrelated Zhdanov,V.V.Spichak,I.M.Varentsov,N.G.Golubev,
totheproblemonmatrixeigenvaluesandeigenvectors E. B. Fainberg, B. S. Singer, K. Vozoff, P. Weidelt, S.
(M.N.Berdichevsky,F.Bostick,C.Swift,K.Vozoff,D. Park, J. Smith, J. Booker, D. Oldenburg, S. Constable,
Eggers, C. LaTorraca, T. Madden, E. Yee, K. Paulson, G. Egbert, and R. Mackie. In the past few years, con-
andM.Menvielle). siderable progress has been made in the mathematical
Another difficulty was caused by local near-surface technologyoftwo-dimensionalinterpretation.Whatwe
inhomogeneitiescreatinguninterpretablenoisethatin- debate now is the strategy of the 2-D interpretation.
terfereswiththestudyofregionalstructures.Advances Progress toward the three-dimensional problem we re-
in this field are associated with techniques suggested late to development of the 3-D methods using the hy-
by G. Hohmann, K. Bahr, R. Groom, and R. Bailey pothesistestmode(V.V.Spichak,M.N.Berdichevsky,
thatmakeitpossibletoseparatethelocalandregional andV.I.Dmitriev),andquasi-one-dimensionalmethods
effects. reducing the three-dimensional inversion to an itera-
Many troubles were caused by the one-dimensional tive sequence of one-dimensional inversions corrected
interpretation that ignored the distorting effects of the by the three-dimensional misfit (V. I. Dmitriev, A. S.
earth’shorizontalinhomogeneities.Moreoftenthannot, Barashkov,D.Oldenburg,J.Booker,andT.Smith).
its internal inconsistency and geological unauthentic- Andfinally,weoffersomewordsabouttheinforma-
ity were brought clearly into view. Looking through tiveness of magnetotellurics. The questions are “What
old journals, we find numerous examples of naive in- is the robustness of MT interpretation?” and “What is
terpretation that ignored these field distortions. Such theresolutionofMTinterpretation?”Butthesefunda-
naivete led to dramatic impairment of MT accuracy mentalquestionsgofarbeyondmagnetotellurics.They
and even to the emergence of false structures. If, for aresubjectsofthegeneraltheoryofill-posedproblems.
instance, the depth to the conductive mantle indicated Theinversemagnetotelluricproblemisunstableand,
from the one-dimensional interpretation of MT data hence,itisill-posed.Anarbitrarilysmallerrorininitial
variesfrom50to250kmalongaprofile10-kmlong,one MTdatacancauseanarbitrarilylargeerrorinthegeo-
cansupposethatthisindicationisaphantomproduced electricstructuresobtainedbytheinversion.Ofcourse,
by the lateral effect of a near-surface inhomogeneity. thereisthesameindeterminacyininterpretationforany
Bythemid-1970sithadbecomepainfullyobviousthat methodofelectromagneticsounding,or,ingeneral,for
the lateral effects (primarily of a galvanic nature) are anygeophysicalmethod.Asolutiontoaninverseprob-
“the main villains of the magnetotelluric piece,” and lem is geophysically meaningful if we use a priori in-
that magnetotelluric interpretations should be started formationaboutthemedium,andlimittherangeofthe
withdiagnosticsoftheseeffectsandcorrections(where search.F.M.Gol’tsmandeterminedexactlythecruxof
possible). Thus, it became necessary to have a theory the matter, “Before starting to search, we should first
dealing with typical distortions of MT data and sug- decidewhatwearegoingtosearchforandamongwhat
gestingmethodsfortheircorrection.Thedevelopment wearesearching.”Obviously,theeffectivenessofmag-
of the distortion theory was favored by advances in netotelluric interpretation depends on the amount of a
computational electrodynamics by V. I. Dmitriev, V. V. priorigeologicalandgeophysicalinformationthatmay
Spichak,E.B.Fainberg,B.S.Singer,D.B.Avdeev,I.M. beavailable.
Varentsov,O.V.Pankratov,A.V.Kuvshinov,V.Druskin, The cornerstone of magnetotelluric interpretation is
P.Weidelt,P.Wannamaker,J.Weaver,G.Hohmann,and the theory of regularization of ill-posed problems. Its
R.Mackie.Thetheoryofdistortionswasdevelopedby centralthemewasformulatedbyA.N.Tikhonovatthe
M. N. Berdichevsky, V. I. Dmitriev, A. A. Kaufman, beginningofthe1960s.Today,methodsusingthisthe-
A. Jones, J. Weaver, F. Bostick, P. Wannamaker, and oryhavebeendevelopedwithsufficientcompleteness.
S. Park. Using criteria and methods of the distortion Inourview,anewscienceofinterpretationofphysical
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