Time Series Analysis and Inverse Theory for Geophysicists

272 Pages·2004·3.759 MB·English

by  Gubbins D.

TIME SERIES ANALYSIS AND INVERSE THEORY
FOR GEOPHYSICISTS
Geophysicists make measurements at the Earth’s surface, and from aeroplanes and satel-
lites,inordertoinferstructureinsidetheEarth.Thedigitalrevolutionnowprovidesuswith
vast datasets that are interpreted by using sophisticated processing methods. This unique
textbook provides the foundation for understanding and applying those techniques com-
monlyusedingeophysicstoprocessandinterpretmoderndigitaldata.Thedigitalrevolu-
tionisonlyadecadeold,butithaschangedallaspectsofgeophysicalfieldmeasurement.
Sophisticated methods are needed to maximise the information contained within a mod-
erndataset,andthesemakeheavydemandsonthemathematicalexpertiseofthemodern
geophysicsstudent.
Thedataanalyst’stoolkitcontainsawiderangeoftechniquesthatmaybedividedinto
two main groups: processing, which concerns mainly time series analysis and is used to
separatethesignalofinterestfrombackgroundnoise;andinversion,whichinvolvesgen-
erating some map or physical model from the data. These two groups of techniques are
normally taught separately, but are here presented together as Parts I and II of the book.
PartIIIdescribessomerealapplicationstakenfromtheauthor’sexperienceingeophysical
research.Theyincludecasestudiesinseismology,geomagnetismandgravity.
This textbook gives students and practitioners the theoretical background and prac-
tical experience, through case studies, computer examples and exercises, to under-
stand and apply new processing methods to modern geophysical datasets. Suitable
for undergraduate- and graduate-level courses, these methods are equally applicable to
other disciplines. Files needed for the computer exercises are available on a website at
http://publishing.cambridge.org/resources/0521819652.Solutionstotheexercisesarealso
availabletotutorsthroughthissamewebsite.
DAVID GUBBINSwasawardedaPh.D.fromtheDepartmentofGeodesyandGeophysics,
UniversityofCambridgein1972.AfterseveralyearsspentasaresearcherintheUniversity
ofColorado,theMassachusettsInstituteofTechnologyandtheUniversityofCaliforniaat
Los Angeles, he returned to Cambridge University where he held the position of Assis-
tant Director of Research until 1989. In 1989 he joined the School of Earth Sciences in
theUniversityofLeedswhereheisnowProfessorofGeophysics.ProfessorGubbinsisa
memberoftheSocietyofExplorationSeismologistsandtheRoyalAstronomicalSociety,
andhasbeenelectedaFellowoftheAmericanGeophysicalUnion,oftheRoyalSocietyof
London,andoftheInstituteofPhysics.Hiscontributionstogeophysicsaremainlyingeo-
magnetismandseismology.Hisworkhasbeenrecognisedthroughnumerousprestigious
awardsandvisitinglectureshipsincludingtheMurchisonMedaloftheGeologicalSociety
ofLondonandtheGoldMedaloftheRoyalAstronomicalSociety.ProfessorGubbinshas
taught mathematics courses to science students for over 35 years and is also the author
of the acclaimed textbook Seismology and Plate Tectonics (Cambridge University Press,
1990).

TIME SERIES ANALYSIS AND
INVERSE THEORY FOR
GEOPHYSICISTS
DAVID GUBBINS
DepartmentofEarthSciences,
UniversityofLeeds
CAMBRIDGE UNIVERSITY PRESS
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São Paulo, Delhi, Dubai, Tokyo, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521525695
© David Gubbins 2004
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2004
Third printing 2008
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Gubbins, David.
Time series analysis and inverse theory for geophysicists / David Gubbins.
p. cm.
Includes bibliographical references and index.
ISBN 0 521 81965 2 – ISBN 0 521 52569 1 (paperback)
1. Earth science – Mathematics. 2. Time-series analysis. 3. Inversion (Geophysics) I. Title.
QC809.M37G83 2004
550´.1´51 – dc22 2003055730
ISBN 978-0-521-81965-7 Hardback
ISBN 978-0-521-52569-5 Paperback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party Internet Web sites referred to in
this publication, and does not guarantee that any content on such Web sites is,
or will remain, accurate or appropriate. Information regarding prices, travel
timetables, and other factual information given in this work are correct at
the time of first printing but Cambridge University Press does not guarantee
the accuracy of such information thereafter.
Dedication
Tomyteachers,RodneySpratley,JohnHills,BrianPippard,andtothememoryof
TeddyBullard
v

Contents
Preface pagexi
Acknowledgements xiii
Listofillustrations xiv
1 Introduction 1
1.1 Thedigitalrevolution 1
1.2 Digitalrecording 3
1.3 Processing 5
1.4 Inversion 7
1.5 Aboutthisbook 10
Exercises 12
PartI Processing 15
2 Mathematicalpreliminaries:thez-anddiscreteFouriertransforms 17
2.1 Thez-transform 17
2.2 ThediscreteFouriertransform 21
2.3 PropertiesofthediscreteFouriertransform 26
2.4 DFTofrandomsequences 34
Exercises 36
3 Practicalestimationofspectra 40
3.1 Aliasing 40
3.2 Spectralleakageandtapering 44
3.3 Examplesofspectra 49
Exercises 53
4 Processingoftimesequences 57
4.1 Filtering 57
4.2 Correlation 63
4.3 Deconvolution 65
Exercises 69
vii
viii Contents
5 Processingtwo-dimensionaldata 73
5.1 The2DFouriertransform 73
5.2 2Dfiltering 75
5.3 Travellingwaves 77
Exercises 80
PartII Inversion 83
6 Linearparameterestimation 85
6.1 Thelinearproblem 85
6.2 Least-squaressolutionofover-determinedproblems 89
6.3 Weightingthedata 92
6.4 Modelcovariancematrixandtheerrorellipsoid 100
6.5 Robustmethods 103
Exercises 107
7 Theunder-determinedproblem 110
7.1 Thenullspace 110
7.2 Theminimum-normsolution 112
7.3 Rankingandwinnowing 113
7.4 Dampingandthetrade-offcurve 115
7.5 Parametercovariancematrix 117
7.6 Theresolutionmatrix 121
Exercises 123
8 Nonlinearinverseproblems 125
8.1 Methodsavailablefornonlinearproblems 125
8.2 Earthquakelocation:anexampleofnonlinearparameter
estimation 127
8.3 Quasi-linearisationanditerationforthegeneralproblem 130
8.4 Damping,step-lengthdamping,andcovarianceand
resolutionmatrices 131
8.5 Theerrorsurface 132
Exercises 135
9 Continuousinversetheory 138
9.1 Alinearcontinuousinverseproblem 138
9.2 TheDirichletcondition 139
9.3 Spread,errorandthetrade-offcurve 142
9.4 Designingtheaveragingfunction 144
9.5 Minimum-normsolution 145
9.6 Discretisingthecontinuousinverseproblem 147
9.7 Parameterestimation:themethodsofBackusandParker 149
Exercises 154
Contents ix
PartIII Applications 157
10 Fourieranalysisasaninverseproblem 159
10.1 ThediscreteFouriertransformandfiltering 159
10.2 Wienerfilters 161
10.3 Multi-taperspectralanalysis 164
11 Seismictraveltimesandtomography 170
11.1 Beamforming 170
11.2 Tomography 177
11.3 Simultaneousinversionforstructureandearthquakelocation 183
12 Geomagnetism 188
12.1 Introduction 188
12.2 Theforwardproblem 189
12.3 Theinverseproblem:uniqueness 190
12.4 Damping 193
12.5 Thedata 196
12.6 Solutionsalongthetrade-offcurve 199
12.7 Covariance,resolutionandaveragingfunctions 203
12.8 Findingfluidmotioninthecore 207
Appendix1 Fourierseries 213
Appendix2 TheFourierintegraltransform 218
Appendix3 Shannon’ssamplingtheorem 224
Appendix4 Linearalgebra 226
Appendix5 Vectorspacesandthefunctionspace 234
Appendix6 Lagrangemultipliersandpenaltyparameters 241
Appendix7 Filesforthecomputerexercises 244
References 246
Index 250