Earth Soundings Analysis - Processing vs Inversion

334 Pages·2004·3.832 MB·English

by  J. Claerbout

EARTH SOUNDINGS ANALYSIS:
Processing versus Inversion
Jon F. Clærbout
Cecil and Ida Green Professor of Geophysics
Stanford University
(cid:1)c March 23, 2004

Contents
0.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 ConvolutionandSpectra 1
1.1 SAMPLEDDATAANDZ-TRANSFORMS . . . . . . . . . . . . . . . . . . 1
1.2 FOURIERSUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 FOURIERANDZ-TRANSFORM . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 CORRELATIONANDSPECTRA . . . . . . . . . . . . . . . . . . . . . . . 14
2 DiscreteFouriertransform 21
2.1 FTASANINVERTIBLEMATRIX . . . . . . . . . . . . . . . . . . . . . . 21
2.2 INVERTIBLESLOWFTPROGRAM . . . . . . . . . . . . . . . . . . . . . 25
2.3 SYMMETRIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 SETTINGUPTHEFASTFOURIERTRANSFORM . . . . . . . . . . . . . 34
2.5 TWO-DIMENSIONALFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 HOWFASTFOURIERTRANSFORMWORKS . . . . . . . . . . . . . . . 43
2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Z-plane,causality,andfeedback 47
3.1 LEAKYINTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 SMOOTHINGWITHBOXANDTRIANGLE . . . . . . . . . . . . . . . . 51
3.3 CAUSALINTEGRATIONFILTER . . . . . . . . . . . . . . . . . . . . . . 54
3.4 DAMPEDOSCILLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 INSTABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 MINIMUM-PHASEFILTERS . . . . . . . . . . . . . . . . . . . . . . . . . 68
CONTENTS
3.7 INTRODUCTIONTOALL-PASSFILTERS . . . . . . . . . . . . . . . . . 69
3.8 PRECISIONEXHAUSTION . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.9 MYFAVORITEWAVELET . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.10 IMPEDANCEFILTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Univariateproblems 77
4.1 INSIDEANABSTRACTVECTOR . . . . . . . . . . . . . . . . . . . . . . 77
4.2 SEGREGATINGPANDSCROSSTALK . . . . . . . . . . . . . . . . . . . 77
4.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 HOWTODIVIDENOISYSIGNALS . . . . . . . . . . . . . . . . . . . . . 85
4.5 NONSTATIONARITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6 DIPPICKINGWITHOUTDIPSCANNING . . . . . . . . . . . . . . . . . 94
5 Adjointoperators 101
5.1 FAMILIAROPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 ADJOINTDEFINED:DOT-PRODUCTTEST . . . . . . . . . . . . . . . . 109
5.3 NORMALMOVEOUTANDOTHERMAPPINGS . . . . . . . . . . . . . . 111
5.4 DERIVATIVEANDINTEGRAL . . . . . . . . . . . . . . . . . . . . . . . 122
5.5 CAUSALINTEGRATIONRECURSION . . . . . . . . . . . . . . . . . . . 123
5.6 UNITARYOPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.7 VELOCITYSPECTRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.8 INTRODUCTIONTOTOMOGRAPHY . . . . . . . . . . . . . . . . . . . . 127
5.9 STOLTMIGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Modelfittingbyleastsquares 131
6.1 MULTIVARIATELEASTSQUARES . . . . . . . . . . . . . . . . . . . . . 131
6.2 ITERATIVEMETHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3 INVERSENMOSTACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4 MARINEDEGHOSTING . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.5 CGMETHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
CONTENTS
6.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7 Time-series analysis 155
7.1 SHAPINGFILTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2 SYNTHETICDATAFROMFILTEREDNOISE . . . . . . . . . . . . . . . 161
7.3 THEERROR FILTERFAMILY . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4 BLINDDECONVOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.5 WEIGHTEDERRORFILTERS . . . . . . . . . . . . . . . . . . . . . . . . 172
7.6 CALCULATINGERRORFILTERS . . . . . . . . . . . . . . . . . . . . . . 175
7.7 INTERPOLATIONERROR . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8 Missing-data restoration 181
8.1 INTRODUCTIONTOALIASING . . . . . . . . . . . . . . . . . . . . . . . 181
8.2 MISSINGDATAINONEDIMENSION . . . . . . . . . . . . . . . . . . . . 183
8.3 MISSINGDATAANDUNKNOWNFILTER . . . . . . . . . . . . . . . . . 187
8.4 2-DINTERPOLATIONBEYONDALIASING . . . . . . . . . . . . . . . . 193
8.5 AFULLYTWO-DIMENSIONALPEFILTER . . . . . . . . . . . . . . . . 204
8.6 TOMOGRAPHYANDOTHERAPPLICATIONS . . . . . . . . . . . . . . 208
8.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9 Hyperbolatricks 213
9.1 PIXEL-PRECISEVELOCITYSCANNING . . . . . . . . . . . . . . . . . . 213
9.2 GEOMETRY-BASEDDECON . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10 Spectrumandphase 227
10.1 HILBERTTRANSFORM . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.2 SPECTRALFACTORIZATION . . . . . . . . . . . . . . . . . . . . . . . . 234
10.3 ABUTTERWORTH-FILTERCOOKBOOK. . . . . . . . . . . . . . . . . . 241
10.4 PHASEDELAYANDGROUPDELAY . . . . . . . . . . . . . . . . . . . . 245
10.5 PHASEOFAMINIMUM-PHASEFILTER . . . . . . . . . . . . . . . . . . 248
CONTENTS
10.6 ROBINSON’SENERGY-DELAYTHEOREM . . . . . . . . . . . . . . . . 250
10.7 FILTERSINPARALLEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
11 Resolutionandrandomsignals 255
11.1 TIME-FREQUENCYRESOLUTION . . . . . . . . . . . . . . . . . . . . . 256
11.2 FTOFRANDOMNUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . 261
11.3 TIME-STATISTICALRESOLUTION . . . . . . . . . . . . . . . . . . . . . 263
11.4 SPECTRALFLUCTUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 268
11.5 CROSSCORRELATIONANDCOHERENCY . . . . . . . . . . . . . . . . 273
11.6 SMOOTHINGINTWODIMENSIONS . . . . . . . . . . . . . . . . . . . . 275
11.7 PROBABILITYANDCONVOLUTION. . . . . . . . . . . . . . . . . . . . 277
11.8 THECENTRAL-LIMITTHEOREM . . . . . . . . . . . . . . . . . . . . . 278
12 EntropyandJenseninequality 281
12.1 THEJENSENINEQUALITY . . . . . . . . . . . . . . . . . . . . . . . . . 281
12.2 RELATEDCONCEPTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
13 RATionalFORtran==Ratfor 287
14 SeplibandSEPsoftware 291
14.1 THEDATACUBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
14.2 THEHISTORYFILE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
14.3 MEMORYALLOCATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
14.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
14.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
15 Notation 297
15.1 OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
15.2 SCALARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
15.3 FILTERS,SIGNALS,ANDTHEIRTRANSFORMS . . . . . . . . . . . . . 298
15.4 MATRICESANDVECTORS . . . . . . . . . . . . . . . . . . . . . . . . . 299
15.5 CHANGESFROMFGDP . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
CONTENTS
16 Interactive,1-D,seismology programed1D 301
16.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
17 TheZplaneprogram 303
17.1 THESCREEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
17.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Index 307
CONTENTS
CONTENTS
FREEWARE, COPYRIGHT, LICENSE, AND CREDITS
This disk contains freeware from many authors. Freeware is software you can copy and give
away. But it is restricted in other ways. Please see author’s copyrights and “public licenses”
alongwiththeirprograms.
Asyousawonthecopyrightpageandwillfindintheelectronicfiles,myelectronicbookis
copyrighted. However,theprogramsIwrotethatdisplaythebookanditsfiguresareavailable
toyouundertheGNUpubliclicense(seebelow). Ihavesignedovercopyrightofthebooktext
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We (you and I) areindebtedto manypeople whohavegenerously contributed softwareto
the public good. I’ll mention here only those outside the Stanford University research group
whosecontributions arewidelyused andon whichwedeeplydepend:
TEX DonKnuth,StanfordUniversity
LATEX LeslieLamport,StanfordResearch Institute
ratfor77 OzanYigit,Arizona,andWes Bauske,IBM
ratfor90 Bob Clapp
dvips Tomas Rokicki,StanfordUniversity
I feel sure thelist of valuablecontributors is much longer. I am afraid I may have overlooked
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My electronic book is free software; you can redistribute it and/or modify it under the
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1BlackwellScientificPublications,3CambridgeCenter,Cambridge,MA02142
CONTENTS
PREFACE TO THE ELECTRONIC BOOK
Reproducibility
Each figure caption is followed by an [R] or an [NR] which denotes Reproducible or Not
Reproducible. To actually burn and rebuild the illustrations you will need to have “seplib”
installedatyour site.
SEPsoftware
Contained on the CD-ROM distribution are two interactive programs, ed1D and Zplane. I
originally wrote these programs in Sunview, an interactive software development platform
from Sun Microsystems. Fortunately, Steve Cole converted them to the X Window system,
using the X toolkit and Xlib graphics, so they are now available on machines from many
manufacturers. Unfortunately,in1998, wedonothavethemcompiledforour mainmachines
atSEP,linuxPC’sandSGI.
Acknowledgement
This textbook itself was updated in minor ways since the 1991 CD-ROM was produced. The
electronic document, however, is greatly enhanced through systems improvements made by
Martin Karrenbach, Steve Cole, and Dave Nichols. Most of the features described in this
prefacewereabsentorincompletein1991.