Exploration Seismology

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rsBN 0-521-46282-7

CANNBRIDGE UNIYERSITY

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E,xploration Seismology E

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R.E.SHERIFF Pro.fessor,Geostiences DePartment, University oJ Houston, Houslon,Texas

L. P.GELDART Former Coortlinator, Canudian International Development Agency Progrum Jbr Brazil

ClvrnnrDGE UNIVERSITY

PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 IRP ' SA 4 0 W e s t 2 0 t h S t r e e t ,N e w Y o r k , N Y 1 0 0 1 1 - 4 2 1 I U l0 Stamford Road, Oakleigh, Melbourne 3166, Australia @ Cambridge University Press 1982, 1995 First published 1982 Second edition 1995 Printed in the United States of America Library of Congress Cataloging-in- Publicatittn Data Sheriff. Robert E. Exploration seismology / R. E. Sheriff' L P. Geldart. 2nd ed. p. cm. Includes bibliographical references. ISBN 0-521-46282-7.- ISBN 0-521-46826-a(pbk ) l. Seismicprospecting.

I. Geldart, L. P II. Title.

1994 TN269.S52415 622'.1592-dc20

94-4153 CIP

A catalogrecord for this book is availablefrom the British Library ISBN

Hardback 0-521-46282-7 0- 521-46826-4PaPerback

Contents

Preface Mathematical conventionsand symbols Generalrules and definitions Latin symbols Greek symbols I

2

Introduction Overview L I Outline of seismicmethods l.l.l Seismicreflectionmethod 1.1.2Seismicrefractionmethod 1.2 History of seismicexploration 1.2.I Historicalsources 1.2.2Preliminaryevents |.2.3 Early applicationsto petroleum exploration 1.2.4The GeophysicalResearch Corporation 1.2.5Other activitiesin the 1920s 1.2.6Early geophysicalcasehistory 1.2.7Developmentof the geophysicalcontractingindustry 1.2.8Evolution of reflection equipmentand methods 1.2.9Reproduciblerecording,the common-midpointmethod, and nonexplosivesources L 2 . l 0 R e c e nh t istory 1.3 Geophysicalactivity 1.3.1The future of exploration seismology 1.3.2History of seismicactivity 1 . 3 . 3D a t a f o r l 9 9 l 1.4The literatureof exploration seismology References Theory of seismicwaves Overview 2.1 Theory of elasticity 2.1.1Waveson a stretchedstrine 2 . 1 . 2S t r e s s 2 . 1 . 3S t r a i n 2.1.4Hooke'slaw 2.1.5Elasticconstants 2.1.6Strainenergy 2.2Wave equations 2.2.1Scalarwaveequation 2.2.2Yector waveequation 2.2.3Waveequationincluding source term

xi xiii xiii xiii xv I I 2 2 z J J J

J

8 9 l0 l3 l4

l8 2l z3

23 24 26 28 3l JJ JJ JJ JJ

35 36 J I

38 38 39 3 9 40 40

2.2.4 Kir chhoff's theorem 2.2.5Plane-wavesolutions 2.2.6 Spherical-wave solutions 2.3 Generalaspectsof waves 2.3.1 Harmonic waves 2.3.2Waveinterference 2.3.3Huygens'principle 2.4 Body waves 2.4.1 P-wavesand S-waves 2.4.2 Displacementand velocity potentials 2.4.3Waveequationin fluid media 2.4.4Boundary conditions 2.4.5Wavesfrom a sphericalsource 2.5 Surfacewaves 2 . 5 . 1G e n e r a l 2.5.2Rayleighwaves 2.5.3Stoneleywaves 2.5.4 Love waves 2.5.5Tube waves 2.6 Anisotropic media 2.6.I Typesof anisotropy 2.6.2Transverseisotropy 2.6.3Waveequationfor transversely isotropic media 2.7 Effectsof the medium on wave propagation 2.7.1 Energydensityand geometrical spreading 2.7.2Absorption 2.7.3Relativeimportanceof absorptionand spreading 2.7.4Dispersion;group velocity 2.7.5Reflectionand refraction; Snell'slaw 2.8 Diffraction 2.8.1Basicformulas 2.8.2 Diffraction effect of part of a plane reflector 2.8.3Time-domain solution for diffraction 2.8.4Diffraction effectof a halfplane 2.8.5Using Huygens'principle to construct diffracted wavefronts Problems References

3 Partitioning at an interface Overview 3.I Application of boundary conditions

4l 4l A 1 AL

43 43 ^ a

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44 44 44 46 47 41 47 49 49 49 50 52 53 55 55 56 56 57 57 59 60 60 62 63 63 64 65 66 68 68 1l It IJ

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CONTENTS 7.I Determining location 7 . 1 . 1L a n d s u r v e y i n g I .1.2 Marine positioning 7 .1 . 3R a d i o p o s i t i o n i n g 7.1.4Transitsatellitepositioning 7.1.5Global PositioningSystem (GPS) 7.1.6Acousticand inertial positioning 7.1.7 Locating the streamer 7.2 Impulsiveland energysources 7.2.1The desiredsource 7.2.2 Explosivesourcesin boreholes 7.2.3Largeimpulsivesurfacesources 7.2.4Small surfacesources 7.3 Nonimpulsiveenergysources 7.3.1Vibroseis /.-1..Z)OSle

7.3.3Choiceof land sources 7.4 Marine equipment 7.4.I General 7.4.2Bubble effect 7.4.3Air guns 7.4.4Implodersand other marine sources 7 . 4 . 5C h o i c eo f m a r i n es o u r c e s 7.5 Detectors 7.5.I Theory of geophones 7.5.2At-the-geophone digitization 7.5.3Hydrophones 7.5.4Streamers 7.5.5Matchinghydrophoneand geophonerecords 7.6 Recording 7.6.I Amplifier requirements 7 . 6 . 2R e c o r d i n g instruments 7.6.3Analog recording 7.6.4 Digital representation 7.6.5Digital instruments 7.6.6Display Problems References Reflectionfield methods Overview 8.I Basicconsiderations 8.1.1Data acquisition 8 .1 . 2C r e wo r g a n i z a t i o n 8.1.3Environmentaland safety considerations 8.1.4Conductof a field survey 8.2 Field operationsfor land surveys 8.2.1The program 8.2.2Permitting 8.2.3Laying out the line 8.2.4Field procedures 8.3 Field layouts 8.3.1Spreadtypes 8.3.2Singlefoldrecording 8.3.3Common-midpointmethod

vll

191 l9l 192 193 194 195 191 198 198 198 199 204

20s 206 206 210 210 2tl 211 213 214 214 217 2t8 218 223 223 225 22s 226 226 226 221 229 230 ZJJ

234 236 239 239 239 239 239 240 240 241 241 241 241 242 z+J z+J

244 244

8.3.4Practicalconstraintsand specialmethods 8.3.5Array concepts 8.3.6Uniform linear arrays 8.3.7Weighted(tapered)arrays 8.3.8Areal arrays 8.3.9Practicalconstraintson arrays 8.3.10Spatialsamplingrequirements 8.3.1I Extendedresolution 8.4 Selectionof field parameters 8.4.I Noise analysis 8.4.2Determiningfield parameters 8.4.3Field testing 8.5 Defining the near surface 8.5.1Uphole surveys 8.5.2Near-surfacerefraction 8.6 Marine methods 8.6.I Conventionalmarine operations 8.6.2Shallow-waterand obstructed operauons 8.6.3Profiling methods operations 8.7 Transition-zone 8.8 Data reduction 8.8.1Field processing 8.8.2Elevationand weatherins corrections 8.8.3Picking reflectionsand preparingcross-sections Problems References Data Processing Overview 9.1 Transforms 9.1.1Integraltransforms 9.1.2Fourieranalysisand synthesis 9.1.3Fouriertransforms 9.1.4MultidimensionalFourier transforms 9.1.5Radon (r-P) transforms 9.1.6 Implementationof transforms 9.2 Convolution 9 . 2 . 1T h e c o n v o l u t i o no P e r a t i o n 9.2.2 Sampling,interPolating,and aliasing 9.2.3Filtering by the earth 9.2.4W aterreverberationand deconvolution 9.2.5 Multidimensionalconvolution 9.3 Correlation 9.3.I Cross-correlation 9.3.2Autocorrelation 9.3.3Normalizedcorrelation 9.3.4VibroseisanalYsis 9.3.5Multichannelcoherence 9.3.6Sign-bitrecording 9.4 Phaseconsiderations 9.5 Deconvolutionand frequencY filterine

245 247 247 250 250 251 252 253 253 253 254 255 256 256 256 258 258

260 260 260 261 261 261 266 268 zt)

275 275 216 276 27'l 21'7

218 218 218 2't9 279 281 283 284 285 285 285 285 286 281 288 289 290 292

CONTENTS

viii 9 . 5 . 1G e n e r a l 9.5.2Deterministicinversefiltering 9.5.3Deghostingand recursive filtering 9.5.4 DeghostingbY combining geophoneand hYdroPhonerecords (Wiener)filtering 9.5.5 Least-squares 9.5.6Whitening 9.5.7Predictive(gaPPed)deconvolution 9.5.8Other types of deconvolution 9.5.9Waveletprocessing filtering 9.5.l0 FrequencY 9.5.1| Time-variantProcessing 9.5.I 2 Choosingdeconvolution parameters 9.5.I 3 Multichannel deconvolution 9.6 Automaticstaticsdetermination 9.6.1Interrelationof staticsand normal-moveoutcorrectlons model 9.6.2The surface-consistent the Maximizing 9.6.3 Powerof the stackedtrace 9.6.4Refractionstatics 9.7 Velocityanalysis(velocitY spectrum) 9.7.I Conventionalvelocityanalysis 9.7.2Velocitypanels 9.7.3PickingvelocitYanalYses 9.7.4Usesand limitationsof velocity analyses 9 .7.5 Horizon velocitYanalYsis 9.8 Preservationof amplitude information 9.9 Apparent-velocity(2-D) filtering 9.10Stacking 9 . 1 0 . 1G a t h e r s 9.10.2DMO (dip-moveout) correction 9 . 1 0 . 3M u t i n g 9. 10.4Common-midPointstacking 9. 10.5Weightedstacking 9.10.6Diversity stacking 9.10.7Simplanstacking 9.1I Other processingtechniques 9 . 1l . l r p t r a n s f o r mP r o c e s s i n g (slant stacking) 9.11.2IntelligentinterPolation 9.11.3AutomaticPicking analYsis 9.11.4Complex-trace to rePositiondata 9.12 Processes 9.12.1Introduction 9.I 2.2 Kirchhoff (diffraction-stack) migration 9.12.3Migration in the frequencydomain wavenumber methodof 9.12.4Finite-difference u are-equationmigration 9 . 1 2 . 5D e p t hm i g r a t i o n 9 .I 2 . 6 H y b r i d m i g r a t i o n

9.12.7Relativemerits of different migration methods 9.12.8Resolutionof migrated sections 9.12.9Other migrationconsiderations 9. l3 Data-processingProcedures sequence 9. I 3.I Typicalprocessing and 9.13.2lnteractiveProcessing workstations inversion 9.l4 Generalized Problems References

292 292 292 293 293 295 298 298 299 300 300 I0

302 303 303 303 303 305 305 306 306 309 309 3 lr 3ll 313 315 316 316 316 319 320 321 322 322 324 324 324 325 325 326 326 327 329 330 JJJ

334

Geologicinterpretationof reflectiondata Overview 10.1Basicgeologicconcepts 10.LI Generationand migrationof hydrocarbons 10.1.2Typesof traPs I 0.2 InterpretationProcedures 10.2.1 FundamentalgeoPhYsical assumPtions 10.2.2Collectionand examination of data 10.2.3Pickingreflections 10.2.4Mapping reflectinghorizons 10.2.5Deducinggeologichistory g e l ld a t a i n t o a n 1 0 . 2 . 6l n t e g r a t i n w interpretation 10.2.7Workstations from 10.2.8Drawingconclusions reflectiondata color 10.2.9Displaytechniques; 10.3Evidencesof geologicfeatures 10.3.1ConcePtsfrom structural geology I 0 . 1 . 2B a l a n c i n gs e c t i o n S 1 0 . 3 . 3F a u l t i n g 10.3.4Foldedand flow structures 1 0 . 3 . 5R e e f s 10.3.6Unconformities 10.3.7Channels 10.3.8StratigraphictraPs 10.3.9Integrationwith other geophysicaldata 10.4Modeling 10.4.1 lntroduction 10.4.2Physicalmodeling 10.4.3ComPutermodeling 10.4.4Syntheticseismograms I 0.4.5Ray-tracemodeling 10.5Lateral variationsin velocity 10.5.1Gradualchanges 10.5.2Suddenchanges I 0.6 Three-dimensional interpretationof 2-D data I 0.7 Stratigraphicinterpretation 1 0 . 7 . 1I n t r o d u c t i o n 10.7.2SequencestratigraPhY

134 334 335 335 335 340 340 343 346 349 349 350 350 351 J)J

353 353 356 357 359

359 361 JOJ

363 364 364 310 37r -r/o

182 385 386 388 389 390 390 390 391 392 392 392 392 395 398 398 398 401

CONTENTS

10.7.3Time significanceof reflections 10.7.4Depositionalmodels 10.7.5Systemtracts 10.7.6Seismic-facies analysis I 0.7.7Reflection-character analysis 10.8Hydrocarbon indicators 10.9Crustal studies Problems References 1l

Refraction methods Overview l l.l Field techniques I l.l.l In-linerefractionprofiling I 1.1.2Broadsiderefractionand fan-shooting l l . l . 3 G a r d n e r 'm s e t h o do f defining salt domes I 1.1.4Marine refraction I 1.2Refractiondata reductionand processing I L3 Basic-formulainterpretation methods I 1.3.1Using basicformulas I L3.2 Adachi'smethod I 1.3.3Generalizedreciprocal method (GRM) I 1.4Delay-timeinterpretation methods 1.4.1Delay time 1.4.2Barry'smethod 1.4.3Tarrant'smethod 1.4.4Wyrobek'smethod I 1.5Wavefrontinterpretation methods I 1.5.I Thornburgh'smethod I 1.5.2Hagedoorn'splus-minus method I 1.5.3Hales'graphicalmethod I 1.6Geologicinterpretationof refractiondata Problems References

t2 3-D Methods Overview 12.l 3-D acquisition 1 2 . 1I. A c q u i s i t i o nr e q u i r e m e n t s 1 2 . 1 . 2M a r i n e3 - D a c q u i s i t i o n 1 2 . 1 . 3L a n d 3 - D a c q u i s i t i o n 12.23-D processing 12.3Displayof 3-D data 12.4Interactive3-D interpretation 12.53-D interpretation Problems References

l 3 Specializedtechniques Overview

1X

403 404 405 409 4t2 415 418 4t9 420 425 42s 425 425 111

427 428 429 433 433 433 434 439 439 439 440 441 442 442

13.1Exploration with S-waves l3.l.l Why explorewith S-waves 13.1.2.S-waverecordingon land 13.1.3S-waverecordingat sea 13.1.4Processing and displaying S-wavedata 13.1.5Interpretationand useof S-wavedata 13.1.6 S-wavebirefringence I 3.2 Three-componentrecording 13.2.1Acquisition | 3.2.2Polarizationfi ltering 13.3Channelwaves(normal-mode propagation) 13.4Vertical seismicprofiling (VSP) 1 3 . 4 . 1G e n e r a l 13.4.2VSP typesand their uses 13.4.3Recordinga VSP 1 3 . 4 . 4V S Pp r o c e s s i n g 13.4.5VSP planning I 3.5 Seismictomography 13.5.1General I 3.5.2Tomographicconcepts 13.5.3Solutionfor a limited number of discretecells I 3.5.4Cross-holemeasurements 13.6Time-lapsemeasurements 13.7Boreholestudies I 3.7.I Salt-proximitysurveys 13.7.2Sonic waveformlogging I 3.7.3Boreholeteleviewer 13.8Passiveseismicmethods 13.9Joint inversion I 3.I 0 Geostatisticalmethods Problems References

t4 Specializedapplications 442 443 446 446 448 451 451 451 451 452 453 457 459 460 461 466 467 471 4'7|

Overview 14.I Engineeringapplications 14.l.l Objectivesof engineering work 14.1.2Refractionsurveyson land 14.1.3Reflectionsurveyson land 14.L4 Marine engineering surveys 14.2Coal geophysics 14.2.1Objectivesof coal geophysics 14.2.2Propertiesof coal 1 4 . 2 . 3L o n g w a l m l ining 14.2.4Surfaceseismicmethods 14.2.5In-seammethods 14.2.6Miscellaneousaspectsof coal geophysics I 4.3 Groundwater, environmental, archaeological,and geothermal applications I 4.4 H ydrocarbon-reservoir applications 14.4.1Introduction

411 471 471 474 474 415 476 476 476 483 483 487 487 487 488 489 492 492 492 493 496 491 499 500 500 500 500 s00 502 502 s02 503 505 505 505 505 505 506 506 508 508 508 508 508 509 5t2 512

5t2 512

CONTENTS l4.4.2The nature of hydrocarbon reservoirs I 4.4.3Reservoirdelineation I 4.4.4Reservoirdescription 14.4.5Reservoirsurveillance Problems References

l 5 Backgroundmathematics Overview l 5 . l S u m m a r i eos f b a s i cc o n c e p t s l5.l.l Determinants 15.1.2Vectoranalysrs 15.1.M 3 a t r i xa n a l y s i s 15.1.4Seriesexpansions 15.1.5Complexnumbers 15.1.6Method of leastsquares I 5.1.7Finite differences 15.1.8Numericalsolutionof differentialequations 15.l.9 Partialfractions 15.2Fourier seriesand Fourier transforms 15.2.1Fourierseries I 5.2.2Fourier integral I 5.2.3Fourier transforms I 5.2.4Multidimensional Fourier seriesand transforms I 5.2.5Specialfunctrons 15.2.6Theoremson Fourier transforms I 5.2.7Gibbs' phenomenon I 5.2.8Convolutiontheorem I 5.2.9Cross-correlationtheorem I 5.2.l0 Autocorrelation l 5.2.1I Multidimensional convolution | 5 . 2 . 1 2R a n d o mf u n c t i o n s I 5.2.I 3 Hilbert transforms 15.3Laplacetransform 1 5 . 3 . 1I n t r o d u c t i o n 15.3.2Theoremson Laplace transforms 15.4Linear systems 15.4.I Introduction

15.4.2Linear systemsin seriesand parallel 15.5Digital systemsand z-transforms 15.5.1Samplingtheorem I 5.5.2Convolution and correlation of sampledfunctions 15.5.3z-transforms I 5.5.4Calculationof z-transforms; Fast Fourier Transform I 5.5.5Applicationof z-transforms to digital systems I 5.5.6Phaseconsiderations I 5.5.7Inteeralrelationfor inverse z-transforms 15.6Cepstrumanalysis 15.7Filtering 1 5 . 7 . 1I n t r o d u c t i o n 15.7.2Filter synthesisand analysis 15.7.3Frequencyfiltering 15.7.4Butterworth filters 15.7.5Windows 15.7.6Optimum filters Problems References

513 514 5t4 515 515 515 517 517 5r7 517 518 519 522 522 523 527 529 530 531 531 532 532 533 533 538 539 540 541 542

547 547 547

548 548 549 550 554 554 555 555 555 556 55'7 558 559 563 566

Appendices A

List of abbreviationsused

569

B

Trademarksand proper namesused

569

C

Randomnumbers

570

D

Units

570

E

Decibelconversion

5t\

F

Typicalinstrumentspecifications and conventions

571

542
543 )4)

545

G

A seismicreport

571

545 s46 546

H

Symbolsusedin mapping

572

Index

515

Preface

Many improvementshave occurred in the seismic method since the publication of the first edition in 1982.Conceptsthat were of academicconcern then havesincebecomepracticaltools and someof the new concepts we have added may become tomorrow's tools. We want this book to be a referencework as well as a textbook and guide for practicing geophysicists. Thesethree objectivesare not alwayscompatibleancl readerswill often skip over portions that do not fit their current needs,hopefully later referring back to skipped portions. For those readers who do skip about, we have cross-referenced sections,equations. and figures.For thosewho are rustywith theii mathematics, we expressconceptsin words as well as by equatlons. Our specialinterestis in the interpretationof geophysicaldata, but an interpreterneedsto havea thorough understanding ofgeophysicalprinciplesin order to determinethe validity of his data and the possibility that featureshe seesare artifacts of acquisitionor processing.To this end, we have tried to emphasize seismicfundamentals. We give a systematicderivation of relationships from first principles,exceptlor a few caseswhere the derivationsare excessivelylengthy or involve higher mathematics,in which instanceswe refer the readerto other sources.As the prefaceto our first edition states, 'A reader willing to take the mathematicson faith should be able to jump over the equationsand still seethe implicationsof the mathematicalconclusrons, lvhich we haveendeavoredto explain in words rather than merely letting the equations speak for themselves.We have been encouraged by Sir Harold Jeffreys'prefaceto The Earth (1924): Ifthe geologist cannotfollowa partofthe book,I hopehe riill omit it andgo on.. .. Mathematically trainedreaders, *ith fewexceptions, will do thesame. Our treatment of seismic theory has been expanded.We haveexpandedpartitioning at interfaces, the heart of most seismicapplications,and made it a separatechapter.Anisotropy, AVO, Stoneleywaves, and tube waves,merelymentionedin our first edition, are now discussedin more detail. 3-D methods and -1-D interpretation now occupiesan entire chapter. Techniquessomewhatout of the mainstream.suchas

VSP, S-wavemethods, and channel wavesare now expanded into a full chapteron specializedtechniques, and we have added new topics such as tomography and geostatistics. One chapteris devotedto nonpetroleum applications,including not only coal and engineering seismicwork, but also the growing areas of groundwater,environmental,and reservoir seophvsics. Our final chapteron mathematicalbackg-roundis again intended to refresha reader'sforgotten mathematical concepts. Terms definedare indicatedby italics.precisionof terminology is often one of the clearestindicationsof a person'sdegreeof understandingand we havemade specialeffort to defineand usethe specializedvocabulary of seismologyprecisely. This book is structuredto be as consistentas possible. Mathematicalconventions,definitions,and the symbolsusedthroughoutthe book are listedimmedi_ ately following this preface.We havetried to conform to acceptedpracticesand nomenclature,but we have had to usesomesymbolsfor severalpurposes. Each chapterbeginswith an overviewto orient the reader as to the various topics to be discussedand their relationshipto each other. All chaptersexcept the first end with problems;theseendeavorto eluci_ dateaspectsnot treatedin the text and developproofs and additional relationships.Each problem has been designedto illustratea specificpoint and we have included hints where we anticipate studentsmay have difficulty in knowing how to proceed.Enlarging the book'sfiguresto usewith the problemsinvolvingthem should not prove difficult with the ready availability of enlargingcopiers. We acknowledgethe assistanceof many peonle in the preparationof this book and expressour thanks to them. In addition to thosecited in our first edition (Harry Mayne, Dan Skelton, Bill Laing, O. Leenhardt, Bruce Frizelle, Howard Taylor. Thomas Thompson,and Willis Reed),we particularlythank Barbara Barnes,Leslie Denham, Brian Evans, Tony Lauhoff, Dereck Palmeq and Margaret Sheriff for their help with this edition. We also expressappreciation to the University of Houston GeosciencesDepartment and the Curtin University of Technology (whereone of us (RES) worked on this book while the Haydn Williams Fellow). R. E. Sheriff L. P. Geldart

Mathematical conventions and symbols

General rules and definitions Generalfunctions C, g(l) c(/)* h(/) g(O, G{,a) C,{,")

d"o(')

Functionof the discretevariablet : nA, n : 0 , * 1 ,- r 2 , . . . . Functionofa continuousvariabler Convolutionof g(r)with /r(r) Functionsinvolvingthecepstrum transform:log G(ro): G(al e+ C(o Transformof g(r) to a function of frequencyz the argumentsy or to indicateFouriertransform,s Laplace transform,z z-transform Correlationof g(r) with ft(t)as functionof the time shift .r; a cross-correlation if g * h, 6,rG) : autocorrelation

Specialfunctions box,,It]

Boxcar of unit height and width a, centeredat t :0

comb[r] sgn[r] sinc[r] step[l]

Seriesof equallyspacedunit impulses Signof t : -l for I < 0, +l for l > 0 (1/r)sin I Unit stepfunction,step(l): 0 for r < 0, +lforr>0 Unit impulseat / : 0

6[4,6,

Mathematical conventions Approximately equal to Denotes corresponding functions in different domains, the arguments v, e, s, z indicating the type of transform; thus, g(t) e qv) and g(r) e G(r,r)indicate Fourier transforms, g(r) e+ G(r) a Laplace transform, g, e G{z) a z-transform; g(x, "y) <+ G(( 0) a Radon transform; lowercaseletters indicate the time domain, uppercasethe frequency domain. J Denotes a time sequenceconsisting of the la,b, c, Q elements a, b, c, d, with the superscribed arrow indicating the value associatedwith t : 0 (b in this instance);values not otherwise specifiedarc zero. A Vector quantity, magnitude is lAl A . B ,A x B Scalar and vector products ofA and B I Matrix with elementsa.. y'r Transposeof matrix d e

arg(o) tl

v v2

v0 v.A V X A det(a)

a{}

Argument of
D { } D-'{ } E { } exp(x) g(0+), g(0-) Valueof g whenapproaching0 from right, left q4, Ge r) Complexconjugateof G(z) P (e,) Probability of e, Re{g(r)}, Im{g(t)} Real,imaginarypartsof g(t) Absolutevalueor modulusof w, W lwl,lWl i, i, z Time derivativesof x, y, z ,a Productaoa(r2a3'' an llo, i=0

* L s, i:0 ls-

S u m g*o B ,* ' " + g , Sumofgooverappropriate valuesofk

Latin symbols a

a. b. c a,, b, a,, b,, c,, A, B A, B, C A(t) A(v), B(v)

.t,.8 c /-c.t'

C D

Element spacing, rate of increaseof velocity with depth, area, width of a boxcar Constants Angles of incidence Fourier-series coefficients Constants Amplitudes of waves or of displacements Amplitude of envelope,amplitude

attributedto reflectivity Amplitudespectrum potential Amplitudesof displacement functions,matrices Specificheat at constantvolume,pressure Incompressibility: 111r, volume fraction of clay Displacement potential function for diffraction, distance

xiv D., D* ei et

E E", EU E*, E, l(t), f ,

"fft),"fr "f,(t), "f : .f"(t) F

MATHEMATICAL CONVENTIONS

Depth of source,weathering Error in fth output Impulse responseof sequenceof reflectors Sum of errors squared, Young's modulus, energy.energydensity.energyratio Elevation of surface,datum Fraction of energy reflected,transmitted Filter in time domain, function of I High-pass filter Low-pass filter Responseof filter to unit step in time domain Array response,constant, magnitude of lorce, depth-conversionfactor

F(u),r(
glt), g, g,\t)

s(o G(u), G(or), qz) h

function Acceleration of gravity Seismictrace in time domain, input trace. function of I Quadrature trace Cepstrum of G(o) or G(;) Seismictrace in frequency domain Damping factor, distance to reflector or refractor, depth, thickness Output in time domain Desired output, complex trace Magnetic field strength, depth

h(t), h, h(t), tt, H H(v), H(a), H() Output in frequencydomain 7(v),.Vla), Desired output in frequency domain i Current. index io Angle of raypath at source point i, Inversefilter in time domain i,i,k Unit vectors in x-, .y-,z-directions I Intensity Transform ol inversefilter 4") (- l;'" J j Index k Constant. bulk modulus. index K Force on geophonecoil per unit current. effectiveelastic modulus ( Distance in Radon transform (,m,n Direction cosinesrelative to ,r-, l-, r-axes e , ,( " , ( o C r i t e r i ai n o p t i m u m f i l t e r i n g L Kinetic energy per unit volume, length. prediction lag, self inductance Lu Mr Delay at location k due to structure, normal-moveout error m Mass m,n Constants, exponential decay parameters, integers M Effective elastic modulus. width of frinse zone n Integer, number of layers

.vo

l.

AND

SYMBOLS

Impulse responseof near-surfacezone Noise p, p' Raypath parameter P Power p, Impulse responseof waveshapemodifying factors in the earth Pressure O Quality factor, probability r Distance. integer,radius, radial coordinate r,s Receiver.sourcecoordinates r,R Radius.resistance rr Additive noise R,R, Reflection coellicient ol lth interface, displacementpotential function for reflection R' Resistivity R*.S^ Delay due to geophone. source at location k R(u),R(
s

s,,

T

\1.{THEMATICALCONVENTIONS AND SYMBOLS

Il' , i . .\, \l u). X(
responseof water layer, downgoing waveform S-wave acoustic impedance Displacement, distance, offset Critical distance,crossoverdistance lor a refraction Imaginary part of Fourier transform : -(sine translorm) Depth, :-transform parameter P-wave acoustic impedance

Greek symbols

I, LN

\,, tr, p F '' F,' F* v Io

v,1.t) vN

,:, -r, B -. -i, 'r(f) .rtu). 1(o) I6 6 . 6. d(1)

fl

a,,.a,, e i i I I H

H, H, O r

K\

P-wavevelocity, angle, angle of approach Fourier-seriescoefficients S-wave velocity, angle Phaseor phase difference,specific heat ratio, potential function, sourcedensity Phaseshilt Instantaneousphase Phasespectrum Geophone transductionconstant, measureof simplicity Angle of S-wave,delay time, logarithmic decrement Delay time associatedwith shotpoint, geophone

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Introduction

Overview Exploration seismologydeals with the use of artificially generatedelasticwavesto locatemineral deposits (including hydrocarbons,ores, water, geothermal reservoirs,etc.), archaeologicalsites, and to obtain -eeologicalinformation for engineering.Exploration seismologyprovidesdata that, when usedin conlunction with other geophysical,borehole,and geological data and with conceptsof physicsand geology,can provideinformation about the structureand distribution of rock types.Usually,seismicexplorationis part of a commercialventureand, hence.economicsis an ever-presentconcern. Seismicmethods alone cannot determine many of the features that make for a profitable venture and, even when supplementedby other data, a unique interpretation is rarely evident. Seismicexploration usually stops long before unambiguousanswersare obtained and beforeall has been learned that might possibly be learned, becausein judgment further information is better obsomeone's tained in some other way, such as by drilling a well. Seismicmethodsare in continual economiccompetition with other methods. Almost all oil companiesrely on seismicinterpretation for selectingthe sitesfor exploratoryoil wells.Despite the indirectnessof the method- most seismic work results in the mapping of geologicalstructure rather than finding petroleumdirectly- the likelihood of a successfulventureis improvedmore than enough to pay for the seismicwork. The enormous detail produced by 3-D techniqueshas opened up a huge reservoir engineering potential. Likewise, seismic methods are important in groundwatersearchesand in civil engineering,especiallyto measurethe depth to bedrock in connection with the construction of large buildings, dams, highways, and harbor surveys, and to determine whether blasting will be required in road cuts, if potential hazards such as limestone caves or forgotten mine workings underlie building sites,if tunnels or mine drifts are likely to encounter water-filled zones, or if faults are present that might be hazards to a nuclear power plant. On the other hand, seismic techniques have found little application in direct exploration for minerals becsuse they do not produce good definition where interfaces between different rock types are highly irregular. However, they are useful in locating features such as buried channels in which heavy minerals may be accumulated.

Exploration seismologyis an offspring of earthquake seismology.When an earthquake occurs, the earth is fractured and the rocks on opposite sidesof the fracturemove relativeto one another.Sucha rupture generatesseismicwavesthat travel outward from the fracturesurface.Thesewavesare recordedat various sites using seismographs.Seismologistsuse the data to deduce information about the nature of the rocks through which the earthquakewavestraveled. Exploration seismicmethods involve basically the sametype of measurements as earthquakeseismology. However,the energysourcesare controlled and movable and the distancesbetweenthe sourceand the recordingpoints are relativelysmall.Much seismicwork consistsof continuouscoverage, wherethe responseof portions of earth is sampledalong lines of successive profile. Explosivesand other energysourcesare used to generatethe seismicwavesand arraysof seismometers or geophonesare usedto detectthe resultingmotion of the earth. The data are usuallyrecordedin digital form on magnetic tape so that computer processingcan be usedto enhancethe signalswith respectto the noise,extract the significantinformation, and display the data in such a form that a geological interpretationcan be carried out readily. The basic techniqueof seismicexplorationconsists of generatingseismicwavesand measuringthe time required for the wavesto travel from the sourceto a seriesof geophones,usuallydisposedalong a straight line directedtoward the source.From a knowledgeof traveltimesto the various geophonesand the velocity of the waves,one attempts to reconstructthe paths of the seismicwaves.Structuralinformation is derived principally from paths that fall into two main categories:head-waveor refractedpathsin which the principal portion of the path is along the interfacebetween two rock layers and hence is approximately horizontal, and reflectedpaths in which the wave travels downwardinitially and at somepoint is reflectedback to the surface,the overallpath being essentiallyvertical. For both types of path, the traveltimesdepend upon the physicalpropertiesofthe rocks and the attitudesof the beds.The objectiveof seismicexploration is to deduceinformation about the rocks, especially about the attitudesofthe beds,from the observedarrival times and (to a limited extent)from variationsin amplitude, frequency,and waveform. A brief outline of the seismicreflectionand refraction methods is given first (gl.l); this explanationig-

2 norescomplicationsand variations,which are the subjectsof futurechapters. Exploration seismologyis a fairly young activity, having begun only about 1923.The history of seismic exploration is summarized in $l.2. The seismic method is by far the most important geophysicaltechniquein termsof capitalexpenditure(gl.3) and number of geophysicistsinvolved. The predominanceof the seismicmethod over other geophysicalmethodsis due to various factors, the most important of which are the high accuracy,high resolution,and greatpenetration of which the method is capable.Seismicliterat u r e i s d i s c u s s eidn S l . 4

l.l Outline of seismic methods l.l.I Seismicreflectionmethod Seismictechniqueshavechangedconsiderablywithin recentyearsand many variationsexist.The technique describedin what follows providesa background to the understanding of subsequent discussions; the reasons for various stepsand various modificationsof techniques will be describedin subsequent chapters. Assume a land crew using an explosivecharge as the energy source.The first step after determining properlocationsis the drilling of a verticalhole in the earth at the sourcepoint, the hole diameterbeing perhapsl0 or l2 cm and the depthusuallybetween6 and 30 m. A chargeof I to 25 kg of explosiveis armed with an electricblastingt'up and then placednear the bottom of the hole. Two wires extendfrom the cap to the surface,where they are connectedto a blaster, which is usedto sendan electricalcurrent through the wires to the cap, which then explodes,initiating the explosionof the dynamite(the sftol). Two t'ables2 to 4 km long are laid out in a straight line extending each way from the hole about to be fired. The cablescontain many pairs of electricalconductors,eachpair terminatingin an electricalconnector at both endsofthe cable.In addition.eachoair of wiresis connectedto one of severaloutletsspacedat intervalsof 25 to 100m along the cable.Severalgeophones(seismometers) are connectedto each of these outlets so that each pair of wires in the cablecarries the output energyof a groupof geophonesback to the recording instruments.Becauseof the small spactng betweenthe geophonesin the group attachedto one pair of wires,the whole group is approximatelyequivalent to a singlefictitious geophonelocatedat the center of the group.Usually,48 or more geophonegroups are located at equal intervals along the cable.When the dynamite charge is exploded, each geophone group generatesa signal that dependsupon the motion of the ground in the vicinity of the group.The net result is the generationof signalsfurnishing information about the ground motion at a number of regularly spacedpoints (the groupcenters)along a straight line passingthrough the source. The electricalsignalsfrom the geophonegroupsgo

INTRODUCTION to an equal number of amplifiers.Theseamplifiersincreasethe overall signal strengthand partially eliminateQt'ilterout) parls of the input deemedto be undesirable.The outputs from the amplifiers along with accuratetiming signalsare recordedon magnetictape and on paper records.Thus, the recordeddata consist of severaltrace.s,eachtrace showinghow the motion of one geophone group varies with time after the sourceactivation. The data are usually processedto attenuatenoise vis-a-visreflectedenergybasedon characteristicsthat distinguish them from each other, and the data are displayedin a form suitablefor interpretation. Events,that is, arrivalsof energythat vary systematically from trace to trace and that are believedto represent reflectedenergy,are identified on the records. The arrival times (the interval betweenthe sourcernstant and the arrival of the energy at a geophone group, also known as the traveltinte)of theseevents are measuredfor variousgeophonegroups.The location and attitudeofthe interfacethat gaverise to each reflectionevent are then calculatedfrom the arrival times.Seismicvelocityentersinto the calculationol the locationand attitudeofthe interfaces. The results are combinedinto cross-sections and contour maps that representthe structure of the geological interfacesresponsiblefor the events.Patternsin the seismic data are sometimesinterpretedin terms of stratigraphic features or as indicators of hydrocarbons. However,the presenceor absenceof hydrocarbonsor other mineralsis usually inferred from the structural information. We have introduced a number of terms used in a specializedsensein seismicwork (indicatedbyitalic'.s), for example,sourcepoint,group, trace,events,and arrival time. Exploration seismologyabounds in such technicalterms.We shallhenceforthuseitalicsto indicatethat we are defininga term; we shall follow the definitions given in the Ent'yclopedit'Dictionary oJ'Explorution Geophysit's (Sheriff, l99l) for seismicterms and the Glo,ssary o/ Geology(Batesand Jackson,1987) for geologicterms. 1.1.2SeismicreJractionmethod The principal differencebetween reflection and refraction methods is that for refraction, the distance betweensourceand geophonesis large relativeto the depths of the interfacesbeing mapped, whereasit is small or comparableto the depthsfor reflection.Consequently,the travel paths in refraction work are predominantly horizontal, whereasfor reflection work, they are predominantly vertical. Head wavesor refractions(see$3.5)enter and leavea high-velocitybed at the critical angle and only a bed with velocity significantly higher than any bed above it can be mapped.Consequently,the applicationsof refraction methods are more restrictedthan those of reflection. (It should be noted that refraction is used in two different sensesin seismology,to refer to the bending

HISTORYOF SEISMICEXPLORATION t'rfraypathsdue to changesin velocity and in the present senseof involving head waves.The classicalmapping of high-velocitymassessuchas salt domesis also classedas a refractionmethod,although refraction at the critical angle is not necessarilyinvolved; see .il1.1.2.) Refraction exploration generally involves greater distancesthan reflectionwork, so strongersourcesare required. Because distributed in-line geophones nould attenuatethe head wavesthat havean appreciable horizontal componentof motion, geophonesare either bunchedtogetheror distributed perpendicular to the source-geophoneline. Otherwise,however,the sameequipmentcan often be used.

1.2 History of seismic exploration I .). I Historicalsources This accountis basedmainly on articlesby Barton 11929);Heiland (1929a, 1929b): Mintrop (1931); Shaw Bruckshaw,and Newing (193l); Rosaiieand ; e G o l y e r( 1 9 3 5 ) R ; o s a i r e( 1 9 3 5 ) L ; eet L e s t e r( 1 9 3 2 ) D ( 1 9 3 8 ) ;W e a t h e r b y( 1 9 4 0 ) ;V a j k ( 1 9 4 9 ) ;S c h r r e v e r ( 1 9 5 2 )B ; c G e ea n d P a l m e r( 1 9 6 7 ) E ; o r n ( 1 9 6 0 )M ; lki n s ( 1 9 7 0 ) ;L a i n g a n d S e a r c y( 1 9 7 5 ) ;O w e n ( 1 9 7 5 ) ; ; w e e (t 1 9 7 8 ) ; G r e e(n1 9 7 9 )B ; a t e sG P e t t y( 1 9 7 6 )S , ask e l l . a n d R i c e ( 1 9 8 2 ) ,a n d K a r c h e r( 1 9 8 7 ) ,s u p p l e mentedby conversations with individualswho were personallyinvolvedin earlygeophysical work. Recent historicalarticlesthat offer interestingglimpsesinto seismic history include Barrington ( 1982); Clark ( 1 9 8 2 , 1 9 8 3 , 1 9 8 4 a , 1 9 8 4 b ,1 9 8 5 , 1 9 9 0 a ,1 9 9 0 b ) ; ; r o u b a s t (a1 9 8 2 ,1 9 8 3 a1, 9 8 3 b1, 9 8 3 c , M a y n e( 1 9 8 2 )P 1 9 8 4 ,1 9 8 5 a 1 , 9 8 5 b ,1 9 8 6 a ,1 9 8 6 b ,l 9 9 l ) ; R o b e r t s o n ( I 986);Robinson( I 985);Sheriff( l 985, 1988);Wilcox ( 1 9 9 0 )K ; e p p n e r( 1 9 9 1 )a, n d P r o f f i t t( 1 9 9 1 ) . I .2.2 Preliminary events Geophysical explorationfor oil beganwith the torsion balance,which was developedby Baron Roland von Eotvosabout 1888(Vajk, l9a9).Althoughgravitysurveyswith the torsion balanceweremadein Europeon a limitedscale,beginningabout 1900,to map geologic structures,the first extensivesurveysfor petroleum objectiveswere in the United Statesand Mexico in the 1920s.In December1922,a surveyof the known Spindletopsalt dome in Texasgavea gravity anomaly, but subsequentsurveyswere disappointinguntil 1924 when the Nash Dome was discovered.This resulted in the first geophysicaloil discoveryin January 1926. Through 1929 sixteen salt domes found by torsion balancesurveyssubsequentlyresultedin hydrocarbon (Sweet,1978). discoveries The theory of seismicwavesmight be dated from Robert Hooke's law enunciatedin 1678.but most of the theory of elasticity was not developeduntil the 1800s.Baron Cauchy'smemoir on wavepropagation won the Grand Prix of the French Institute in 1818

3 and S. D. Poissonshowedtheoretically the separate existenceof P- and S-wavesaround 1828.C. G. Knott (1899)presenteda paper on the propagationof seismic waves and their reflection and refraction, and Emil Wiechert and Karl Zoeppritz (1907) published their work on seismicwaves.Lord Rayleigh (1885), A. E. H. Love in l9l1 (seeLove,1927),and R. Stoneley (1924)developedthe theoriesofthe surfacewaves that bear their names. Robert Mallet (1848, 1851) began experimental seismologyby measuringthe speedof seismicwaves using black powderas the energysourceand a disturbanceof the surfaceof a bowl of mercuryas the detector. Mallet obtainedvery low velocities;probably low sensitivityallowed him to seeonly the later cyclesof Rayleighwaves,then unknown. H. L. Abbot (1878) measuredP-wavevelocitiesusing essentiallythe same type of detectors but a very large explosion. John Milne (1885)and T. Gray useda falling weight as a source (as well as explosives)in a seriesof seismicwavestudiesusing two seismographs in line, probably the first seismicspread.Otto Hecker (1900)usednine mechanicalhorizontal seismographs in line to record both P- and S-waves. The possibilityof employingthe seismographto define subsurfaceconditions was first put forward by Milne in 1898(Shaw Bruckshaw, and Newing, l93l). As an earthquake wavetravelsfiom stratato strata,if we studyits reflection andchanging velocityin transit,wc may oftenbeledto thediscovery ofcertainrockystructures burieddeepbeneath our vieq aboutwhichwithoutthehelpof suchwavesit wouldbe hopeless everto attainanyknowledge.... Earthquakes are giganticexperiments whichtell theelastic moduliolrocksastheyexistin nature, andwhen properlyintcrpreted mayleadto thepropercomprehension of manyill-understood phenomena. L. P.Garret in 1905suggestedthe useof seismicrefraction to find salt domes but suitableinstruments had not yet beendeveloped(DeGolyer,1935). 1.2.3Early upplit'utionsto petroleumexploration After the sinking of the Titanic by an icebergin 1912, ReginaldA. Fessendenworked on inventionsfor iceberg detection. Among the methods was the use of acousticwavesin water,and an outcome of this was the first (U.S.)patent(fig. Ll) on the applicationof seismicwavesto exploration,applied for in 1914and issuedin 1917,entitled "Method and apparatusfor locating ore bodies."Fessenden's patent said: Theinventiondescribed hereinrelates to methods andapparatuswhereby,beinggivenor havingascertained two or moreof the followingquantities, i.e.,time,distance, intensity and medium,oneor moreof the remainingquantities maybe determined. He proposed using sourcesand detectorsin waterfilled holes,and locatingore bodiesby both the useof peflectionsfrom them and by variations they introduce in traveltimemeasurementsbetweenholes.His

INTRODUCTION

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was apparently only brought into application efforts about 1925. Mintrop in 1919 applied for a German patent on "Method for the determination of rock structures," which was issued in 1926. Mintrop's patent said: Where the problem is to obtain . . . the approximate composition of the strata,the divining rod has beenusedas is well known. However,. . . it has not yet been possibleto ascertain a connectionofunique meaningbetweenthe indication of the divining rod and the geologicparticularitiesof the subsoil.... Accordingto my invention... the connection of mechanicalwaveswith the characteristicpropertiesof the strata is much more immediate . . . mechanicalwavesare artificially generated... by detonating a certain amount of explosives, their elasticpropagationthrough the variousformations is recordedby a seismometerlocated at a suitable distance. . . from the recordsof the latter the velocitiesof the various wavesand the depth to which they penetrated can be determined.which allows conclusionsas to the succession,thickness,density as well as the direction of the strike and dip of rock formations.

tB

John William Evans and Willis B. Whitney in 1920 "Improvements applied for a British patent on in and relating to means for investigating the interior of the Earth's crust" which was issued in 1922. Their patent said:

//n y'rro
First page of Fessenden'spatent.

patent was subsequentlychallenged(unsuccessfully) by (amongothers)Mintrop (1931)becauseFessenden used "acoustic" waves rather than "seismic" waves and becausehis use of boreholesfor sourcesand detectorsdid not accord with subsequentpractice. Ludger Mintrop in Germanyin l9l4 deviseda seismograph with which he could make observationsof explosion-generated waveswith suflicientaccuracyto make explorationfeasible. The Germansand Allies both experimentedduring World War I with the useof threeor more mechanical seismographsto locate enemy artillery, but airwaves generallyprovedmore satisfactorythan seismicwaves for this purpose.Among those involved in theseexperiments were Mintrop and the Americans R. A. Fessenden,E. A. Eckhardt, W. P. Haseman, J. C. Karcheq and Burton McCollum. Thesesix were predominant in the developmentof commercialapplication of seismicwavesafter the war. McCollum attributed the idea of applying seismic methods to petroleum exploration to Haseman (unpublished "Recollections re McCollum" by R. L. Palmer).Mintrop's work was clearly independentand Fessenden

The presentinvention . . . is characterizedin that the sound waves. . . are receivedsimultaneouslyor approximatelyso at a plurality (at least two . . .) of receivingstations. . . for the following reasons:Even in the simplestcasewhen it is known that the stratum to be examinedis horizontal there are two unknown quantitiesnamely(l) the averagevelocity of the reflectedwave . . . and (2) the depth of the reflecting stratumand thereloretwo equations... and two observations are consequentlynecessary. Despite their rather complete grasp of reflection seismology, this patent does not figure prominently in subsequent developments, which concentrated on refraction. Udden (1920) wrote in the Bulletin of the American Association oJ Petroleum Geologists (AAPG) (and illustrated with fig. 1.2): it ought to be possible,with presentrefinementsin physical apparatusand their use, to construct an instrument that would record the reflectionsof earth wavesstarted at the surlace, as they encounter such a well-marked plane of differencein hardnessand elasticityas that separatingthe Bend and Ellenberger formations (in North-central Texas).. . . A seismicwavemight be startedby an explosion at the surlaceof the Earth, and a record of the emergedreflection of this wave . . . might be registeredon an instrument placed at some distance from the point of explosion. . . . It ought to be possibleto notice the point at which the first reflection from the Ellenberger appears.. . . With a map of the surface of the Ellenberger,it seemsto me that millions ol dollars worth of drilling could be eliminated. ln 1920, the Geological Engineering Company was founded by Haseman, Karcher, Eckhardt, and McCollum to apply seismic exploration to finding petroleum. Karcher had recorded a seismic reflection

HISTORY OF SEISMIC EXPLORATION

Fig. 1.2 Reflection expected from the contact between the Bend formation and the underlying Ellenburger limestone. (From Udden. 1920.)

from wavesgeneratedby artillery at the Indian Head test range in Maryland in l9l7 and in a quarry (fig. l.l6a) in Washington,D.C., in 19l9 (Karcher,1987). They convertedan oscillographinto a thrqe-tracerecorder and constructed electrodynamicgeophones from radiotelephonereceivers.In June 1921,Karcher, Haseman, I. Perrine, and W C. Kite at Belle Isle (Oklahoma City) obtained a clear reflectionfrom the contact betweenthe Sylvan shaleand the Viola limestone (fig. 1.3).About five months of reflection and refraction experimentationwere carried out. One experiment involved dropping dynamite from an airplanein an attemptto obtain more nearlyplanewaves (Karcher had tried using aerial fireworksas a source in l9l9). The company ran out of funds when a surplus of oil forced the price down to l5llbarrel. The principals returned to their former jobs, except for McCollum. He agreed to settle with the company's creditors in return for the company's patents and equipment. During 1920-1, Mintrop (Keppner, l99l) shot refraction lines across two known salt domes in northern Germany and discoveredanother,the Meissendorf dome, although it had no commercial significance.In 1921,he founded Seismosto do geophysicalexploration and subsequentlywrote a number of pamphletspromoting refractionexploration.In 1922, Seismostried seismicmethods in Swedenfor mining objectivesand in Holland for coal mapping. EverettLee DeGolyer wrote on October 3,1922, to J. B. Body in London (the following threeextractsare from DeGolyer's papers in the library of Southern Methodist University,Dallas): Youwill remember thatduringthepastsummerDr. Barton, of the AmeradaPetroleumCorporation'sstaff,spentsome time in Europereceivinginstructionin the useof Eiitvos torsionbalances, andwhiletheremadeseveral visitsto Germanyto investigate other physicalmethodsof approachto geologicproblems. Oneof themethodswhichinterestedhim verymuch,and rrhichhe seemedto think had considerable possibility,was

the Seismic method. . . . I should like to suggestthat it be called to Dr Erbh [Shell's]attention with the recommendation that he consider its availability for use in the Mecatepec-Papantla District [Mexico]. . . . Body wrote to DeGolyer on December 14, 1922: You will have seenmy cableNo. 88. . . . "Negotiating

with Seismosfrom Hannover for using Mexico their method measuringwith seismographtransmissionwavescausedby explosionsthereby determining depth positions subterranean Tamasopoalso outline salt domes.Method gavesatislactory resultscentral Europe and are assuredcan be used Mexican conditions.Our intention is sendout party. . . ." Seismos party I began work in the Golden Lane area of Mexico for Mexican Eagle (Shell) in 1923. The contract for this work provided: Seismosbind themselvesto organizean expeditionin order to carry out the . . . investigations.. . . This expeditionshall consist of 2 seismologistsand I mechanic,all of them experts with thoroughly up-to-datetechnical knowledgeand possessing suchzeal and senseof duty as is requisitefor the successof their work. [They were]to be equippedwith two complete seismicfield-stationswith the necessaryinstruments . . . to carry out in Mexico during 25 days effectual observations.. . ina geologicallyknownterritory. . . . Upon arrival the expedition shall confer with the local manager . . . who shall decidewhereand when their operationsshall be carried out and to the solutions of which geologicalproblemssameshall be applied,on the understandingthat as far as purely scientificquestionsare concerned. .. they shall usetheir own discretion.. . . Compensation was to be U.5.$600 for the two seismologists together and $150 for the mechanic during the time the expedition was in Mexico and $500 for the instruments. If Mexican Eagle so elected, the contract "by could be replaced a new contract concluded for an indefinite period and for observations in Mexican regions geologically unknown." In this case, monthly compensation was to be increased to $800 and $250 for the men and $1000 for the instruments. The geological problem involved finding high-velocity lime-

INTRODUCTION

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phone). (b) First depth section,at Vines Branch, Oklahoma, August 9, 1921. (c) First seismic structure map! near Ponca City, O k l a h o m a . S e p t e m b e rl 9 2 l .

stone reefsunder a shalecover,a situation for which refraction appearedto be ideal. Seismos party 2 began work in Oklahoma and Texas for Marland Oil Company (a predecessorof Conoco), also in 1923. Seismosparty 1 moved to Texasto work for Gulf Oil rn 1924and in Junediscoveredthe Orchard Dome southwestof Houston, which is usuallyconsideredto be the first seismic(refraction) hydrocarbon discovery,a claim disputed by McCollum (see what follows). Early refraction records are shown in figs. l.4a and 1.4b.

The Seismoscrewsuseda mechanicalseismograph (fig. 1.5) consistingof a masssuspendedby a horizontal leaf spring with a natural frequencyof about l0 Hz. The only amplification was mechanicaland optical, and recordingwas done by directing a beam of light onto a mirror connectedto the massby a hair so that the mirror rotated when the massmoved,and then onto a strip of photographicpaper moved by a hand crank turned by the observer.Shotpoint-toseismographdistancewas surveyedand a blastphone (fig. 1.6) was used to record the airwave to find the

end continue again at the left end. TB : shot instant obtained by radio and R : refraction arrival. (b) Refraction record, Texas,June 1932.

Fig. 1.4 Early refractionrecords.(Courtesyof Conoco.)(a) Recordobtainedwith mechanicalseismograph,1924 or 1925; therecordingwashelicalabouta drum so that tracesat theright

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pulling on the hair and rotating the mirror. (From Malamphy, 1929.)

INTRODUCTION

Fig.1.6 Blastphone usedto detect theairwave for determining shot-detector distance. The diaphragm d is a pie tin and the shot instant. (Subsequently, radio was used to determine the shot instant and the airwavearrival to find the shot-to-detectordistance.)The overall sensitivity and precision were low and profiles were only 3t,/z miles long, which gave limited penetration so that Seismoscrewsmisseda numberof domesat moderate depths.L. P. Garrett of Gulf Oil, for whom Seismos was working, developed the fan-shooting method ($l I . I .2) about I 925,which increasedthe effectiveness in locatingsalt domes.By 1929,the refractionmethod had found fifty salt domes that resultedin hydrocarbon discoveries(Sweet, 1978).During the same period, "geology and accident" discoveredone dome (Barton 1929:616). Followingthe failure of the GeologicalEngineering Companyin 1922,McCollum obtainedthe backingof Atlantic Refining and formed McCollum Geological Exploration to carry out refractionwork. New equipment wasbuilt, and in 1924both reflectionand refraction was usedin the Tampico areaof Mexico.The first well drilled on a seismiclocation, La Gatero No. 4, was dry, although the seismicprediction was correct. In May 1924, the Zacamixtle 199 well in the Golden Lane area succeededin finding oil, to dispute the claim of the Orchard Dome in Texas as the first seismic discovery. However, the Mexican well was noncommercialat the time becauseof its remote location (Owen, 1975).In 1928the Atlantic-McCollum joint venture was dissolved;McCollum and Atlantic divided the four sets of instruments and McCollum formed McCollum Exploration Company.

transducer a carbon granule microphone. (Photographed at the Museum of the Geophysical Society of Houston.)

The Marland Oil Company had supported two months of the l92l GeologicalEngineeringCompany reflection experimentation(which was unsuccessful) and had brought Seismosparty 2 to Texasin 1923. The Seismosparty failed to find any salt domes for Marland. In 1925, Marland hired Haseman, Eckhardt, Eugene McDermott and others to develop a more sensitiveelectrical seismograph.The Marland field party began exploration in 1926, replacing the Seismosparty. The equipmentworked well, but Marland never recorded a salt dome discovery. 1.2.4 The GeophysicalResearchCorporation DeGolyer was at first disappointedwith the refraction method, but successby Seismoscrews working for Gulf changedhis mind and he beganto searchfor personnel to develop seismicmethods. He learned of Karcher's 1921experimentsand in May 1925, Amerada, Rycade (an Amerada subsidiary)and Karcher fiormed the Geophysical Research Corporation (GRC). They acquiredFessenden's patent and his servicesas consultant. GRC built an electricalseismographthat was much more sensitivethan the Seismosmechanicalseismograph. The detector was a variable-reluctancetype and the amplifier was resistance-coupled using a vacuum tube. The oscillographused two galvanometers and the recording film was hand-cranked.Timing lines were obtained by shining a light through slits attachedto the prongs of a 50-Hz tuning fork. The

HISTORYOF SEISMICEXPLORATION time-break (shot instant) was transmitted by interrupting a CW (continuous-wave) transmitter. GRC fielded sevenfield parties in 1926and refraction explorationgreatlyexpanded.A GRC crew under E. E. Rosaire was forbidden to shoot profiles more than 3t/z miles in length, which had becomethe standard distancesince it had been successfulfor Mintrop, but the observers"got lost" and discoveredthe Port Barre salt dome (Sweet, 1978). Thereafter, the standard distancebecame6 miles. Refraction at the time was used as a reconnaissance method and was usually followed by detailing with a torsion balance(and later gravimeter)survey. GRC party 6, an experimentalcrew,tried reflection work in Kansasin 1926.They soon moved to Texas and obtained usable reflection records from the caprock of the Nash salt dome.Other GRC partiesalso experimentedat recording reflections.ln 1927,party 6 moved to the SeminoleBasin of Oklahoma, an area ideally suited to reflection work, where they soon found a structure that becamethe first discoveryby the reflectionmethod, the Maud Field (in 1928).This successwas quickly followed by others,and by 1930, the reflectionmethod beganto take over from the relraction method.Early reflectionrecordsare shownin fis.1.16.

Fig. 1.7 The Petty prototype geophone. The "steady mass" m rs on a long beam hinged (h) at the left and supported by a strong spring s. At the right end, the beam is attached to one plate of an air-gap condenserc. Movement of the case with respect to the steadymasschangesthe separationof the condenser

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9 1.2.5 Other activitiesin the 1920s Humble Oil Company, at the instigation of Wallace E. Pratt, establisheda geophysicaldepartment in 1924 under Dr. N. H. Ricker, and the following year fielded two refraction crews using mechanical seismographs designed by O. H. Truman (Carlton, 1946). These crewsbeganusing a telephoneline to carry the timebreak, but beforethe end of 1925,they usedradio for both communicationsand transmissionof the timebreak. Frank Rieberln 1924obtainedfunding (from General Petroleum, Standard Oil of California. Associated, and Shell) for a refraction survey in the San Joaquin valley of California. This survey was unsuccessfulin obtaining deep information. Rieber carried out other surveysin California in 1921,8,but his company failed in 1930.ln 1932,he beganwork on reflection instruments,and in subsequentyearsintroduced a number of instrumentalinnovations. ln 1925,the Petty GeophysicalEngineeringCompany was formed by Dabney E. Petty and Olive Scott Petty (and other family members). They felt they could easily improve on Mintrop's mechanicalseismograph, and in 1926fielded a crew equipped with condenser-type geophones(fig. 1.7)and vacuum-tube amplifiers.They used string galvanometers(fig. 1.8)

plates, the change in capacity being proportional to the displacement. The dimensions are 48 X 32 x 15 cm. (photographed at the Museum of the Geophysical Society of Houston.)

10

Fig. 1.8 String-galvanometerharp. Currents pass through fine wires (some are broken on this harp) held taut by small springs. causing the wires to be deflected in a magnetic field. Shadows

and a camerawith photographicpaper pulled along by a springmotor, shadowscastby the moving strings on the paper were recorded.The Pettysdid appreciable experimentationto find a quicker, easierway to locatesalt domes.They discoveredthat a salt forerunby its amner (fig. 1.9),which could be distinguished plitude, could be usedto tell if a salt dome had been encountered even without knowing the shot-todetectordistance.They also found that the Rayleighwave pattern changedwhen a salt dome intervened and usedthis fact when they could not get a readable P-wave.The increasedsensitivityof their equipment and their interpretationalingenuity allowed them to surveywith smallershotsthan others used. ln 1927,the first well velocity measurementswere made. A geophonelowered5000ft (1500m) into a Gulf well in Kansas recordedthe traveltime from a shot at the surface.Also in 1927.C. A. Heiland establishedthe first coursein explorationgeophysicsat the ColoradoSchoolof Mines. McCollum successfullymapped the Barbers Hill Dome by reflectionin 1928using 100detectorsat each station spacedto attenuatehorizontal waves,but the use of multiple detectorswas too cumbersomeand so was abandoned;it was revived about the mid-1930s with four to six detectorsper group. In the late 1920s,seismicexplorationbeganto move abroad: to Persia (Iran) and Venezuelain 192'7,to Australia in 1929.to the NetherlandsEast Indies in 1930. Donald C. Barton (1929), who subsequentlybecame first president of the Society of Exploration Geophysicists(SEG), describedthe early methods: method,a trooprather Forworkwith themirage[refraction] of onefiringunit,two, threeor four recommonlyconsists ceivingunits,a squadof hole diggers,a chiefof party,a "landman," a crewof surveyandin somecases a calculator, . . . Thefiringunit. . . is .ors,andin somea hole-fillingcrew. ... to firethecharge, apparatus with thenecessary equipped

INTRODUCTION

of the wireswerefocusedonto photographicpaper.A stringat the recordis shownin fig. I . I 6e.(Photographed galvanometer Societyof Houston.) Museumof the Geophysical

with meteorologicalapparatusand with a sendingand receiving wirelessset,which is usedto communicatewith the receiving units and to send out the instant of the explou n i t . . . i s e q u i p p e dw i t h a s e i s m o sion... . The receiving graph, . . . a wirelesssendingand receivingset,and meteorologicalapparatus.... To set up a station,a 3 inch hole is dug to a depth of 3 ft, the geophonecableis reeledout and the geophoneis dropped down the hole. . . . Each receiving unit ... signalsthat tentativelyit is ready to receive.... When all . . . have signaledtheir o.k. for firing, the firing mastersetshis wirelesssendingout a continuouswavenote, . . . waits a standardshort interval and then fires the charge. The wirelesskey is held down by a circuit which goesaround the dynamite.The explosioninstantaneouslybreaksthe circuit, causesthe releaseof the wirelesskey and the instantaneous [sic] cutting off of the wirelessnote. . . . The average chargesused ... range from 40 to 250 lbs. For the same shots,SeismosGesellschaftwould use two to three times as largeacharge.... In the reflectiontype ofshooting, the chargeis very much smaller.. . . The practice. . . [is to placethe] main charge17 to 25 ft down a 6 inch hole and the auxiliary chargeat the surface.The latter is usedto producean air wave.The holes are dug with hand augers... . The distancebetweenthe fir... rangesfrom 1.2 to 1.8 ing point and the seismograph times the depth of the lormation which it is desired to m a p . . .. A scout from a rival companynot uncommonly is set to watch the troop and to report their activity to his company and especiallyto report anything to indicate that possibly they may havepickedup a salt dome. He often getsto be on good terms with the troop, but at critical momentsthey go to all sortsof strategyto outwit him. 1.2.6 Early geophysical case history The exploration of Cameron Meadows field, a salt dome in Louisiana, documented by McGuckin (1945), provides interesting comparisons of early exploration methods. Attention was first called to this "sulphur gas" seeps ("boils") in the area because of "no recognizable physiographic marsh. The area has

HISTORY OF SEISMIC EXPLORATION

11

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I I

I I

II

lxJII Utl.ihIlit Iil ll u IL,illul

Fig. 1.9 Refraction recording, 1930.(a)Normal(nosalt)record;(b) recordshowingsaltforerunner; (c) recordingtruckin

use about this time. (Parts (a) and (b) are from petty, 1976; parl (c) is courtesy of Haliburton Geophysical Services.)

expression. . . [exceptthat] Old North Bayou. . . may possiblybe deflectedslightly," accordingto the Louisiana GeologicalBulletin No. 6 in 1935. A seriesof geophysicalsurveyswere carried out from 1926to 1943.They had different objectivesand covereddifferentareas.Hence,the maps shownin fig. l.l0 were to different scalesbut they have been enlargedor reducedto facilitatecomparisons. The SeismosCompany used its one-channelmechanical seismographto shoot refraction over this areain 1926as part ofa larger survey.Seismosgenerally built tiny isletsof sandbagsto support its detector abovethe water.The report said: "The surfaceindications around the shellhill of tent location 145.its sud-

den uplift out of a very deep marsh, and a gas seep south of it, lead to the conclusion that it migtrt Ui causedby the presenceofharder layers.However,the investigations. . . show clearly that no harder layers occur down to considerabledepths.. . . there may be a possibility to find a structurein sections2I and 2g of this region."The main part of the CameronMeadows field was subsequentlydiscoveredin section 21. Seismosinterpreted(correctly) SW-dip southwestof the recordingstation(fig. l.l0a).'source-detectordistances of less than 3.5 miles did not provide very deeppenetration. A reconnaissancetorsion balance survey in l92j (fig. 1.10b)indicateda large closedgravity minimum

t2

INTRODUCTION

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the same Fig. 1.10 A sequence of maps(to approximately domebasedon theearlygeoscale)of theCameronMeadows (FromMcGuckin,1945:1-16.)(a) Seismos physical surveys. reconmechanical refractionmap,1926.(b) Torsion-balance (c)Geophysical gravitygradients. naissance map,1927, showing Research Corp.(GRC)refractionfan map,1928;the filled-in refractionmap,1929. areasindicateleads.(d) GRCcorrelation (i) Map (e)Geophysical map,1933. Service, Inc.,dip reflection wellsthatencountered of thetop of saltlromdrilling,showing

salt.(g) Top-saltmap from McCollumrefractionshootinginto a well, 1942.(h) Top-saltprofile from McCollum refractionsurvey into a well and from well data;the deepestmap ofthe 1942 Pettycontinuousprofiling surveyis alsoindicated.(i) PettyGeophysicalEngineeringCo. continuousprofiling reflectionmap to definefaulting on the south flank, 1942.O RobertH. Ray,Inc., gravimetersurvey,1943,showingalso residualgravity (dashed contours).

centeredin section24 at the left edgeof the fig. l. l0 maps.An unsuccessfulwell was drilled on this minimum; the Cameron Meadows area lay 3 miles to the east on the easternflank of the minimum. A refraction fan shot in 1928from SP-9(fig. l.l0c) indicated a salt lead (in the direction of the blackedin area), but a secondfan from SP-78was interpreted as failing to confirm the lead, and it was concluded that no salt dome was present.However,additional fans shot to averagedistancesof 8 miles in 1929(one in the upper left corner offig. l.l0c) suggesteda salt dome in section21. A follow-up correlationreflection survey lasting I 7 days in July 1929(fig. I . l0d) mapped the closureon which the discoverywell, completedin 1931, was drilled. A dip-method survey (fig. 1.10e)

wasrun in 1933to serveas a guide for further drilling. A map of the top salt basedon 15 wells is shown in fig. l.l0f. The jump correlationsin the correlationand dip-methodsurveysprecludedthe mapping of faults. Becausethe refraction fans had suggestedan extension to the southeastwhere the seepswere located, a seriesof radial lines were shot in 1942into a geophone located near the salt in an abandoned well on the northeast flank of the dome (the method is describedin $11.1.3).The resultingfnap of the salt dome, shown in fig. l.l0g, can be comparedwith fig. offig. l.l0h showsthe general l.l0f. The cross-section agreementbetween seismicand well results. Continuous profiling was carried out in 1942 to map the southeasternnosing that had been indicated

HISTORY OF SEISMIC EXPLORATION

l3

'1,

by shootinginto the well. This surveydefinedfaulting and the deepestmap, a phantom severalthousandfeet above the salt, is shown in fig. 1.10i. No rerlections were recordedin the mainly shalesectionbelow this phantom, although the shale section also contains someprospectivesands. ln 1943,a gravimetersurvey (fig. 1.10j)conducted over the areaconfirmedthe earliertorsion balanceresults.It showedthe producing areaat the SE-edgeof a residualgravity anomaly. 1.2.7 Developmentof the geophysicalcontracting industry Burton McCollum in 1922 applied for a patent on "Method and apparatusfor determiningthe contour of subterraneanstrata," which was issued in 1928 along with two other patentson variationsof seismic methods. McCollum sold two of his patents to the TexasCompanyin 1928and sevenmore between1929

and 1935.These and other patents were transferred to the Texas Development Company, which tried to collect royalties from others but was mainly unsuccessful. In 1934, they sued Sun Oil Company for patent infringement.Almost the entire petroleumindustry joined with Sun in the defense;the matter was settledout of court in 1937.The settlementinvolved companies forming a SeismicImmunities Group and granting each other royalty-freelicensesof their patents and of patents for which they might file within a year of withdrawal from the group. Initially, 64 patents were involved, including 2 of Mintrop's, l0 of McCollum's,2 of Harvey C. Hayes',8 of Fessenden's, and 2 of Karcher's. Severalpayment schemescould be elected;one involved a lifetime payment of $10,000 per party, a party being defined as either (a) a single recording unit with no more Ihan 12 traces or (b) up to four recording units where shot-to-detector distance exceeded2 miles (to cover respectivelyreflection and refraction work). The Mayne CDP patent (see

14 $1.2.9)was one of the last important patentsinvolved before the group disbanded entirely. ln 1929, a new Amerada president decided that GRC would no longer do reflection work for other companies (Karcher, 1987). Although Petty and McCollum offered independent alternativesto accomplish geophysicalwork and some oil companies, such as Humble and Gulf, operatedtheir own crews, the oil companiesin generalencouragedthe formation of more new geophysicalenterprises.Thus, the early 1930ssaw the advent of many geophysicalcontractors,includingmost that dominatetoday.Most of thesewere formed by people who left GRC since it dominatedthe industry until then; some of theseare shownin fig. l.l L In addition,a few companies(such as Rogersand General) were formed by people who left other companiessuch as Petty, and still other companies(such as Heiland) were formed without clear connectionsto precedingindustry. Most of the major geophysicalcontractorsthrough the 1980scan be traceddirectly back to spinoffsfrom GRC. Haliburton GeophysicalServiceswas the successorto GeophysicalServiceInc. (GSI), which was lormed in 1930(aswell as successor to other companies,including Petty).Schlumbergeris successor to Seismograph ServiceCorp.(SSC)formedin 193l (and to Seismosand Prakla). Teledyneis successorto Independent Exploration formed in 1932. Western Geophysical Company, formed in 1933, is still independent;it has now taken over Haliburton Geophysical Services. Grant Geophysics is successor to United Geophysical,formed in 1935.Compagnie Generale de G6ophysique (CGG) is successorto SGRM, which began refraction work in France in | 930. The field work of Seismosdeclinedto zero in 193I . but Seismosrevived refraction work in Germany in 1934and beganreflectionwork about the sametime. Prakla was foundedin 1936and subsequentlymerged with Seismos to form Prakla Seismos.The rapid growth of explorationgeophysicswas almost entirely due to privateenterprisewith intenserivalry and competition and extremesecrecybetweenthe individual companiesinvolved.From the early 1930sto the early 1990s,no singlecompany dominatedgeophysicalexploration. However, by 1994 only two large contracting companies,Schlumbergerand Western,continued (seefig. 1.24).

INTRODUCTION

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'Amerada F-ig. Lll The Tree," a diagram drawn in 1950 to show geophysicalcontracting companies formed by people who left GRC. (W J. Zwart and K. M. Lawrence located this historic document.)

Fig. l.l2 Growth of multichannel recording. The starting point in l914 is Mintrop's first portable seismograph.The vertical scale gives the first known instance of use of numbers of channels.The use ofgroup recorders in the 1980seffectivelyremoved an upper limit on the number of channelsexcept for operational practicality. In one instance, 4000 channels reputedly were used.

1.2.8Evolutionof refiectionequipmentand methods The first GRC reflection work in 1926 employed the same two-galvanometerarrangementused in refraction work, but a third galvanometerwas addedsoon. A four-channelsystemwas built in 1928and before long six-channelinstrumentswerein use;the standard was 6 to 8 in 1937and by 1940most crewswere l0 to l2 channel.The number of channelshas continuedto increase(fig. l.l2). For many years after World War II, 24 channelswere standard,then in the late 1960s.

48 channelsbecamecommon, and today (1994)most crewsuse 120 to 240 channelsand someuse appreciably more. Digitization at the geophone(97.5.2)has now effectivelyremovedlimits,onthe numberof channels that can be recorded. The mechanical seismographwas soon replaced with electricalgeophonesand vacuum-tubeamplifiers. The early electrical geophoneswere mostly of

HISTORY OF SEISMICEXPLORATION three types: capacitance, variable-reluctance,and moving-coil electrodynamic;oil damping was generally used.Early electricalgeophones(fig. 1.13)had to havehigh sensitivitybecauseof the high noiselevelof availablevacuum tubes. For the variable reluctance and moving-coil types, this meant large magnetsbecauseof the low permeabilityof the magneticmaterials then available.As better magneticmaterials and lower-noisevacuum tubes becameavailable,the electromagneticgeophoneincreasedin sensitivityand decreasedin weight (from some 15 kg to a few hundred grams),electromagneticdamping replacedoil damping, and the electromagnetictype eventuallybecame dominant (for land work). As a result of these improvements,multiple geophonesper channelbecame practical; this usagewas introduced in 1933and was common practiceby 1937. The gain of early instrumentswas constantand repeatedshotswere usually required so that reflections at severalarrival times could be mapped.Sometimes the gain was manually changedduring the recording by the operatorturning a switch.About 1932,automatic gain control was developed,first by changing the grid biaswith time after the time-break,laterby a feedbackcircuit.Amplifiersincreased in gain,sophist i c a t i o n( i n i t i a ls u p p r e s s i o n a .u t o m a t i cg a i n .m i x i n g . etc.),and reliability.A lO-channel recorderfrom 1931. the first to change frequency responsewith time, rs shownin fig. 1.14.Sometiming wheelsare shownin fig. l.l5 and reflectionrecordsare shownin fig. 1.16. About 1950, recording instrumentsbecame sufficiently reliablethat the observercould "do a day's work ratherthan instrumentrepair and adjustment." The Southwestern IndustrialElectronics(SIE) Company'sP-l I recordingsystemwas a major advancein

F i g . l . l 3 E a r l y g e o p h o n e si n t h e M u s e u m o f t h e G e o p h y s i c a l S o c i e t y o f ' H o u s t o n . T h e g e o p h o n e sw e i g h ( l e f t t o r i g h t , b a c k

l5 reliability.A chronologyof someinstrumentdevelopmentsis givenin table 1.1. The need for weatheringcorrections ($8.8.2)was recognizedvery early and shallow refraction shots were often made for this purpose.The first mapping was done by correlating reflectionson widely separatedprofiles(fig. l.l7). Barton (1929)wrote: a depthdetermination is madeby eachshotandto mapthe dip,foldingor faultingofthe surface, . . . it is necessary only to scatter"shots"over the areato be mappedand draw structure-contours or profilesfrom the results.Practically the applicationof the methodis somewhat uncertain. ... Theimpossibility of recognizing thereflecting bedis a seriousdisadvantage.... The correlationmethod did not work well in the Gulf Coast becausethe area lacks distinctivereflections. ln 1929.T. I. Harkins noticedthatabnormal stepouts [dipmoveout see94.1.2] wererathercharacteristic of the (Darrowdome)areaand that theseabnormalstepoutsreversed. He correctlyattributedthisphenomenon to dippingbeds.[Rosaire andAdler 19341 Soon dip shootingwas carriedout along continuous lines of traverse. Although from the earliestdays crewscarried out surveysin water-coveredareas,the methodswere basically those for land crews,improvisedfor use in water.Petty(1976)describes a surveyin Chacahoula Swamp,Louisiana,in 1926(fiC. l.l8), and in 192'/, GRC fielded two crews for work in water-covered areas (Rosaireand Lester, 1932).Sidney Kaufman (personalcommunication),in lollowing up an onshorelead in 1938,took his Shellshallow-water crew seaward4 miles into 65 feet of water. Survevorson-

r o w )6 . 1 8, . 7 , 7 . 9 , 6 .(7f r,o nrto w )8 . 8a, n d0 . 8k g .A r r h el o w e r right is a modern phone(30 g) for comparison.

F ' i g .l . 1 4 P e t t y l 0 - c h a n n e lr e c o r d e rf r o m 1 9 3 1 .V a c u u m t u b e s "harp" H V amplified the current, which then passedthrough a (of the type shown in fig. 1.8): light fiom a source L passedby the harp and was focusedonto photographic paper in the takeup

m a g a z i n eM . A t i m i n g w h e e lT ( s e ef i g . l . l 5 ) d r i v e n b y a c l o c k interrupted the light to give the timing lines. (Photographed at the Museum of the Geophysical Society of Houston.)

Early timing wheels ("paddle wheels"). The wheels Fig. l.l5 were rotated by a clock motor so that they cut the light beam to produce timing line shadows. C)bserversoften had individual-

ized wheels that characterized their records. (Photographed the Museum of the Geophysical Society of Houston.)

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Fig. 1.16 Early reflection records. (a) First record made in a rock quarry near Washington, D.C., April 12, 1919,by Karcher. Time increasesfrom right to left; the upper trace is the timebreak; the lower traces are the geophone responseat different gains. Three reflections are marked. (From Schriever,1952.)(b) Record showing reflection from caprock of the Nash Dome,

Texas, 1926; the charge is I pound; the offset is 950 ft. (From Weatherby, 1940.) (c) 1929 three-channel Oklahoma record showing Viola reflecrion. (From Weatherby, 1945.)(d) 1929 rec_ ord from Cimarron anhydrite in Kansas. (e) 1934 Californra record with AGC. (From Weatherby, 1940.) (f) l93g srrinegalvanometer record lrom Mississippi.

INTRODUCTION

l8

Table 1.1 Chronologyof seismicinstumentation and methods l9l4 l9l7 1921 1923 1925

1926 1927 1929 l93l

1932 1933 1936 1939 1942 1944 194-l 1950 1951

Mintrop'smechanicalseismograph patent on seismicmethod Fessenden Seismicreflection work by Geological EngineeringCo. Refraction exploration by Seismosin Mexico and Texas Fan-shooting method Electrical refraction seismograph Radio used lor communications and/or timebreak Reflection correlation method First well velocity survey Reflectiondip shooting Reversedrefractionprofiling Use of uphole phone Truck-mounteddrill Automatic gain control Interchangeablefilters Use of multiple geophones pcr group Riebersonograph;first reproducible recording Use of closedloops to checkmisties Record sections Mixing Large-scalemarine surveying Use of largepatterns Marine shootingwith Shoran Common-midpointmethod+ Medium-rangeradionavigation

1952 1953 1954 1955 1956 196l-2 1963 1965 1967 1968 1969 19'71 1972 1974 19'15 19?6 1984

1985 1986 1988 I 989 1990

Analog magnetic recording* Vibroseisrecording* Weight-dropping Continuousvelocity logging Moveablemagneticheads Centraldata processing Analog deconvolutionand velocity filtering Digital data recording* Air-gun seismicsource Depth controllerson marine streamer Binary gain Velocity analysis Transitsatellitepositioning Instaneousfloating-pointamplifier statics Surface-consistent Bright spot as hydrocarbon indicator Digitization in the field Seismicstratigraphy Three-dimensionalsurveying Image-ray migration (depth migration) Amplitude variation with offset Determiningporosity from amplitude DMO (dip-moveout)processing Interpretationworkstations Towing multiple streamers S-waveexoloration Autopicking of 3-D volumes Dip and azimuth displays Acoustic positioningof streamers GPS satellitepositioning

*The acceptanceofthese methodsis shown in fig. 1.21. Dates are approximate;secrecyand competition often in-

volved developmentand use of the samefeature by several companieswithout public disclosure.

shore directed locations and this imposed the 4-mile limit. The survey was conducted from three 35-foot fishing boats.The instrumentswere eight-channelusing one land geophoneper channel bolted to an l8inch steelplate to keep it upright on the seafloor. Extensivemarine operationsdid not appear until 1944when Superior and Mobil beganrefraction fanshootingfor salt domesoffshoreLouisiana(Jack Lester, personal communication).A survey to map the offshore extension of Los Angeles basin fields was also carried out about this time (C. C. Bates,personal communication). A surveyor onshore gave lnstructions to keep lines straight while wire paying out through a countergavethe distance;buoyswere setto indicate locations. As work progressedfarther offshore,the chaining continuedon a compassbearing. Sightingon shot plumes(watersentup into the air by the shot explosion)both visually and with radar was also used. Surveying was the principal operational problem and often constitutedthe major cost. Shoran came into use about 1946,followed by Raydistabout 1951.The early refraction and reflection work used geophonesplanted on the bottom. About 1946' reflection work began using a l2-channelbottom drag

cable with gimble-mountedgeophones.The floating streamerwasfirst usedin 1949-50.Both the radionavigation methodsand the floating streamerwere based Proffitt (1991)sumon World War II developments. work. marizesthe history of marine I .2.9 Reprodut'ible recording, the common-midpoint sources method,and nonexplosive "Sonograph" Frank Rieber (1936) proposed the method of recordingseismicdata (fig. L 19) so that it "played back." His oscillographrecordedin could be variable density on film. On playback, variations in the intensity of a light beam that passedthrough the film weredetectedby a photocell.Rieberusedthe Sonograph to determinethe variation of reflectionamplitudewith apparentdip. DespiteRieber'spioneeringwork, reproduciblerecording did not becomepractical until the introduction of magnetic-tape recording. Commercial recording and playback equipment became available about 1952.The principal advantageof magnetic-tape recordingwas thought to be the ability to replaywith different filters.About 1955,moveableheadsallowed

HISTORY OF SEISMIC EXPLORATION

t9

static and normal-moveout corrections ($6.1) to be applied. The growth of analog magnetic-taperecordingis shownin fig. 1.21. A very important postwardevelopmentwas the use of record sectionsfor interpretation.Individual seismic recordshad been laid out adjacentto each other in the interpretationprocessfor a long time (fig. 1.20), but the large sizeofindividual recordsand variations in paperspeedand developingquality madeit difficult to obtain a synopticview.Normal moveout($4.1.1), irregularitiesin recordingor spreads,and the wiggletrace display mode added to the difficulties. Gulf Oil and Carter (now part of Exxon) and perhapsShell apparently led in developing variable-density or rariable-areadisplayswith uniform horizontal scale .rnd display amplitude.Carter bought Rieber'sequrpment for this use(amongothers)about 1946. Common-midpoint(CMP, originally calledCRP or -'ommon-reflectionpoint, later CDP or commonjepth point; see $8.3.3)recordingwas inventedby Harry Mayne (Petty Geophysical)in 1950as a way of .lttenuatingnoisethat could not be handledby the use ..i arrays.Magnetic-taperecordingmadeCMP practi-rl. and CMP recordingbeganabout 1956,but it did

not becomeusedextensivelyuntil the early 1960s(fig. l.2l) when its ability to attenuatemultiples ($6.3.2) and other kinds of noiseled to rapid adoption. Today, its useis nearly universal. Magnetic-taperecording also permitted the addition of tracesand thus the use of weaker sourcesbecause records from severalweak sourcescould be added togetherto get the effect of a strongersource. McCollum introduced the use of a dropped weight, the Thumpeq as a seismicsourceabout 1953.Weightdropping expandedseismicwork in areasof difficult shothole drilling, such as West Texas,and in desert areaswherewater for drilling is scarce. A variety of surfacesourcesfor use on land were also developedbesidesweight-dropping.The mosr ingeniousof these,the VibroseisrM method(see97.3.1;a list of trademarksand the companiesthat hold them is given in app. B) was developedby John M. Crawford, William Doty, and Milford Lee and first usedin 1953.Surfacesourcesare now usedfor about half the land work and Vibroseisis the predominant surface source.Severalalternativesto the use of dynamite as a source in the marine environmentwere developed about 1965.They were generallycheaperand more

::g LI7 Portion ofa dip map that resulted from dip shooting, .ruary 1935. Dips were expressed in feet per mile and the

arrow lengths indicated the spacing for 50-ft contours. (Courtesy of Conoco.) N

t

Ol

-J-

I

I oLtGocEXa oto

CONTINENTAL OIL COMPANY SEISMOSSTRUCTUREMAP ru ABBEVILL4 u l.16'

ull-l:ttJ_g

frm_eurall__ l-+tuPcdu-m Gntmttttil! svdt? 6ru Il5l

Fv

r.

t€

q

INTRODUCTION

20

.7r

"..$'$

Fig. 1.18 Early refraction work. (From Petty, 1976.) (a) D. E. Peity washes a reiraction record in Chacahoula Swamp; the

efficient and adaptable to CMP recording. Consequently, they rapidly replaceddynamite as a marine source(fig. l.2l). In addition,they wereenvironmentally acceptablebecausethey did not injure marine life. Although some sophisticatedplayback processing was done with magnetic-taperecordingand somedig-

t

*rf r{

,.t *

geophone is on the cypress stump in the background. (b) Petty (in boat) with his crew in Chacahoula Swamp.

ital processingwas done on analog data, the full potential of data processingwas not achieveduntil digital recording was introduced in the 1960s.Digital recordingnot only resultedin higher fidelity, but also in the large-scaleapplication of the digital computer in the processingand interpretation of seismicdata' "digital revolution" was probably the most farThe

HISTORY OF SEISMIC EXPLORATION

Rieber's sonograph, 1936. Field data were recorded Fig. l.l9 on fiI. in variable-density mode. In playback, the total light through a slit was summed to give a single output trace By ohanging the slit angle, data with various angles of approach (alsoialied apparent dip; see$4.1.2)could be emphasized,each

reachingdevelopmentin seismicexplorationsincethe pioneeringdays. For example,obtaining useful data in the North Seais almostimpossiblewithout deconvolution($9.5). I .2.I0 Recanthistor)) Sincethe developmentof the common-midpoint,Vimethods,a succession broseis,and digital-processing have expanded many improvements incremental of lbld the amount of geologicinformation extractable from seismic data. In consequence,the seismic method is being applied in ways not previouslycontemplated. The prevailingconcept prior to the 1970swas that noise was so much stronger than the seismicsignal that only structural information could be extracted from seismicdata on a practicablebasis;consequently, the emphasiswas almost exclusivelyon noiseattenuation in order to obtain more accuratetraveltimes.Recognition in the early 1970sthat a hydrocarbonaccumulation sometimeschangedthe reflectivity enough "direct" detection and mapping led to to allow its more accuraterecordingand interpretationof amplitude information. In the mid-1970s,recognitionof depositionalpatternsin seismicdata further changedattitudes and gave rise to the belief that much of what had previously been regarded as noise was actually geologic signal. The developmentof threedimensional(3-D) methodsin the late 1970sbeganto resolvethe interpretation ambiguitiesinherent in in-

2l

slit angle giving an additional trace. Thus, the sonogram record displayed amplitude in the angle-of-approach versus arrivaltime domain. (Two views from advertisements in Geophysit's, vols. I and 2.)

terpolating between seismiclines and also provided significantnoisereduction.3-D revealsso much structural and stratigraphicdetail that its use in reservoir engineeringis growing very rapidly (seefig. l.2l). It had long beenrecognizedthat the combined use of S- and P-waveswould permit extractingmore useful geologic information than from P-wavesalone. The developmentof S-wavesourcesresultedin appreciableexperimentationin the early 1980s,the primary objectivesbeing hydrocarbon and lithology identification. Measurement of amplitude variation with offset(AVO), which givesmuch the sameinformation as the combined use of S- and P-wavesbut at lower cost, resultedin somecurbing of S-wavestudies.Today,S-waves(and three-componentrecording)are being studied mainly to measure fracture orientation and intensity. The l9l7 Fessendenpatent contemplatedseismic measurementsin boreholes,and in the late 1920s,a boreholegeophonebegan to be used to definea salt dome flank (as an extensionof the refractionmethod) and to measureseismicvelocity.Gal'perin (1974) describedSovietdevelopmentof vertical seismicproflling (VSP)during the 1960-70period. The useof borehole measurementsfor purposes beyond simply velocity measurementexpandedin the early 1980s. Tomographicborehole-to-boreholeand borehole-tosurfacemethodshavebeenundergoingrapid development sincethe mid-1980sbut are still experimental. Three-dimensionalsurveyingrequiresmore channelsfor its efficientexecutionand involvesmore logis-

INTRODUCTION

22

* ,g

recFig. 1.20 Early record section made by splicing individual and orEs together.The records are made with a l0-trace camera

tical problemsin the field. Thesehaveled to increased ur. of digitization at the geophone and telemetry methodsfor recordinginformation' Marine seismicdata acquisitionhas been rendered and more emcient by towing more than one source that more than one streameroffset by paravanesso pass single a on severallines of data can be acquired sysof a ship. The use of improved radiopositioning ' precithe in refinement in continual temshai resulted seisrion oifo.utlng acquisitionpoints at sea'Marine satTransit of use mic exploration ttui made extensive Global the uses extensively ettltepositioningand now poritlonlng Sys6m (GPS),both developed.bythe U'S' Navy. The-Gi'S is also being usedfor land surveying' Ships involved in 3-D work now have very elaborate potitio"lng systems for determining, in real time *itttln u fe-wmeters,the locations of all of the survey (sourceand eachhydrophonegroup in each ;l;;;.,t coordistreamerfor everyenergyrelease)in the local to order in needed is nut. tytt.-. Thi; information ifthe advise to also and properly data bin and correct

lower marked hoare exceptionally uniform for the period The of Conoco') iizon is clearly iut by a large fault (Courtesy

surveyis obtaining the requisiteuniformity of coveragewhile still on the ProsPect. in The power of computersincreasedenormously and decreased costs time, theiame the late 1980s;at practical ilr. tp..a increasedso much that it became to piovide interpreters with their own. computers interpretation' Work1*oikrtotionr) to help with problems data-handling the of stations solve many an interof much so consumed have that historically p..t.f. time, allowing more cross-checkingand disptaying of data in waysthat make featuresof interest -oi. uitibt., and generallyallowing for a more completeand accurateinterpretation' proThe interpretercan also carry out simple data even become computers cessingat the workstation.As more iowerful and rapid, it is expectedthat.interpretcharac.r, *lit be ableto tailor processingmore to the This i..itti"t and problemsln their particular data' and processors -uJ to-.*ttut "lot. the gap between processing digital with iniJtpt.,..t that developed about 1960(fie.l'22\. enorShips acquiring marine 3-D data carry an

I

GEOPHYSICAL ACTIVITY

-\

1OO"/o

i,

lj

t

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,

Pi

Fi .' 5 , , ,

5O"/"

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, tl

b,t

I

ri.'.,,,*, t

9 r

'sl

Oo/o

.r-"?" .t?'

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1960

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1990

Fig. l.2l Percentageofseismic activity involving various techniques. (Data from SEG annual Geophysit'ul At'tivity Reports, pre-1981 data are for U.S. activity, post-1980 for worldwide ac-

tivity,3-D datafromDutt, 1992,adjusted according to judgmentexpressed in Goodfellow, 1991.)

mous amount of computer power.Marine acquisition is highly automatedand shipboardcomputersare capable of elaborateprocessing.Workstationspermit a preliminary interpretationto be made quickly. The major usersof seismicdata, especially3-D but also VSP and other seismic methods, have been changingfrom the exploration to the production departmentsof oil companies.The precisionand detail of 3-D seismic data make it an increasinglycosteffectivetool in field developmentand production. It is becomingincreasinglyrecognizedthat most reservoirs are quite heterogeneous and seismicdata provide about the only way of determininghow they change horizontally.This is especiallyimportant in the marine environmentwheredrilling and investmentcosts are so large,but it is also true on land. The use of seismic methods in coal studies has grown rapidly sincelongwall mining becamethe major extraction method for deep coal deposits(S14.2). Someof the most rapidly growingareasof seismicapplicationsare in engineering($14.l), groundwater,and wastedisposal(S14.3).

creasesin the productivity (and cost) of a field creq especiallyof a marine crew. Much uncertainty existstoday as to the future of geophysicalactivity. The majority view is that new technologywill continueto createopportunitiesin the geophysical industryand the levelofpetroleumactivity will not lall much farther; however,a return to the high activity of the 1980sis thought to be unlikely. Petroleum-related opportunities for geophysicists should continueat about the current leveland opportunities in nonpetroleum areas should continue to grow. Although seismiccrew activity has been long regardedas a leadingindicatorof the healthof the petroleurn (and energy)sectoqit may be losing validity as an economicpredictor becauseof changesin technology, attitudes,and world politics. The petroleum industry is beginningto shift drilling funds into 3-D seismology.Environmentalconcernsare changingthe demand part of the hydrocarbon supply/demand equation.For years,the United Statesdominatedthe petroleumindustry,but in the 1970s,much of the economic control shifted to the OPEC countries.In the 1980s, rising North Sea production changed the supply/demandbalance in Europe, and recently the revelation of enormous reservesand decayedinfrastructurein the former Soviet Union has introduced new uncertainty.Future changesmay be dominated by unpredictablecrises,just as in the past appliedgeophysicswas very much affectedby eventssuch as the cold war and conflictsin the Middle East. Petroleumand geophysicalservicecompaniesare

13 Geophysical activlty I .3. I Thefuture of exploration seismology The petroleum industry has long dominated applied geophysics.A graph of the number of seismiccrews searchingfor oil and gas in the United States(fig. 1.23)presentsa discouragingpicture. However,much of this declinein the number of crewsis offset bv in-

INTRODUCTION

z+

t9703

slnplr pott-3leck (c.10Ea) PROCESSING Concurrantprocaaalngtnd Int.g.at.d rt th! d.sklop

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COLLABORATIVE

ACQUSI

Multldlrclpllnary l$ms F{an, ecqul?a, procasr, F€n, dtta con@rttntly and Inlfptal lo ldmtlly

optlmC Flremottrt

Fig. 1.22 Concept showing the specializationthat followed the advent of digital processingand the projected unifying resulting

from the increased Power of workstations. (From Landmark Graphics Corp., 1992.)

reorganizing(seefig. 1.24) and repositioning to accommodateexpectedchanges.Major petroleumcompaniesare shifting their interestsoutside the United Statesand Canada, where they perceivegreater opportunities.This has left behind a wealth of opportunities,but small companieshavebeenslow to take advantageof thesebecauseof restrictedfunding. An often-promulgatedconcept is that the Earth's petroleum resourceshave topped out and an exponential decline is inevitable.Past predictions of the timing of this declinehave invariablybeen wrong becausegenerallythey havenot allowedfor technologic improvements.Lindseth(1990)found that the amount of oil discoveredper well has been nearly constant (fig. 1.25)rather than decliningsharplywith time; the periodic arrival of new technologypresumablyarrests the declines.Thus, new technology(which almost everyone predicts) should continue to sustain the geophysicalindustry.

-ot includeequipmentcostsor costsof studentlabor), and totals lump both small-scaleand large-scaleoperthe reports are regardedas relaations.Nevertheless, tively accurateand permit judgments with regard to trends.The latest annual report (as of early 1994)is for the year l99l (Riley, 1993).In addition,monthly reports of seismiccrew activity are publishedin The LeadingEdge.The following is basedon thesereports and Dutt (1992). The number of seismiccrewsis shown in fig. 1.23b along with the number of wildcat wells drilled. The number of seismiccrewswas long regardedas a leading indicator of the petroleum-industryeconomy,but major changesin the early 1990shave changedthis situation (see also fig. 1.27). The sharp upturns in prices and activity in the 1970sand early 1980sresulting from the Middle East oil crisisdid not produce comparableincreasesin hydrocarbondiscoveries(see fig. 1.25). The mean wellheadcostsof oil and natural gas adjusted for inflation are shown in fig. 1'23a.Economics along with technologyhave been the governingfactors in seismicactivity. A surplus of oil about 1937 produced a decline in activity that lasted until the United Statesbecameinvolved in World War II. A doubling of petroleumpricesbetween1945and 1948 resulted in an increasein seismicactivity' However, major finds of oil in the Middle East after World War II resultedin another world surplus of oil and prices dropped.From 1948to 1973,the price of petroleum

1.3.2History o/ seismicactivitY Geophysical activity is tabulated each year by the Societyof Exploration Geophysicists(SEG). Admittedly,not all activity is reported (the most important omission has been activity in the [former] Soviet Union and China), someis reportedon differentbases (some reports include only acquisition and some include processingand/or interpretation), accounting practicesdiffer (reported university work often does

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INTRODUCTION

26

Drcrra Ad$ (honclnnlc,) -i LlttonRao'jrc..o'o[wr|tmAd8#wcrEmoco9hFictl(DEsser30%rutbn?0%) OcoSonc-,t --5Hdliburtoo Oeophylicrl l-U.tUUnnmOcoptydol

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industryln contracting in thegeophysical Fig.1.24 Changes and wastakenoverby Schlumberget SSC(Raytheon) lale1992, OnlymajorconwastakenoverbyWestern. Haliburton in 1993.

remained almost constant and activity generallydeclinedfor most of this period, the declinebeingslowed and occasionallytemporarily reversedby new developments. Natural gas reservespeaked about 1970 und, fo. a while thereafter,explorationhad gas rather than oil as an objective.ln 1913-4,oil pricesincreased sharply as a result of the limiting of suppliesby the OPEC (Organizationof PetroleumExporting Countries) cartel. The price increasesas well as policy changescausedby uncertaintiesabout dependenceon foreign suppliesstimulatedseismicexploration' Conrecernsabout secureoil suppliesdiminishedas large "safe" other serveswere found in the North Sea and areas and geophysicalactivity went into the decline (especially in the United States)that continuestoday' ' Activity outsidethe United Statesincreasedrather steadily until 1958 and then leveledoff. During the period 1958-74the geographyofactivity changedseveral times in responseto political and economicfactors and discoveriesin new areas.Activity in Latin America declined sharply after 1959becauseof discouraging results and political changes in several countiies. The discovery of significant hydrocarbon reservesin North Africa, the beginningof North Sea exploration,nationalizationthreats in lndonesia,the opining of tropical African watersto exploration,and repeatedpolitical disruptionsin North Africa and the Uiaate East wereprobably the most significantof the events.Internationalactivity since l98l (fig. l'26) did not declinenearly as steeplyas in the United States, and since 1987,activity outside North America has beenincreasingsteadilY' The history of seismicactivity is also illustratedin

tracting companies are shown, In a sense,this updates the Ame r a d a T r e e s h o w n i n f i g . 1 . 1l . ( F r o m D u t t , 1 9 9 2 . )

frg. 1.27. The number of fleld crews (especiallyon tana) nas beendecliningsincethe early 1980sbut the volume of new data acquiredsince 1988has been increasingbecauseofthe shift from 2-D to 3-D data acquisition. 1 . 3 . 3D a t a f o r l 9 9 l Expenditures for geophysicaldata acquisition and processingin l99l were U.S'$ 2250 million (Riley' 1993).While the activity data are not strictly comparable with previousperiods,they indicatethat expenditures were up 3"/ufrom 1990 and 54o/ofrom 1987, when expendituresplunged after the Middle East oil of the peak expencrisis;however,they are only 53o/u dituresof U.S.$4168million in 1982.Theseexpenditures are shownby objectivesin table | .2 and by areas in table1.3and fig.1.26.Activity in the United States continued to declineby 25% of the 1990figure, now standingar 160Aof worldwideactivity,comparedwith in 1987' doublethis percentage Seismic work constituted almost 97"k of the reported geophysicalexpenditureand99o/oofthis work irad petroleum objectives.Although table l'2 shows only 13.6%of expenditureshad petroleum development (as opposed to petroleum exploration) objectives, iiley-(1993) gives severalreasonswhy he believes this figure is considerably unders{rted' Goodfellow (1991: 62) in the report for 1990 stated "we believethat a considerablylarger proportion that . . . was actually spent on development ' ' ' [probably] 24 percent)'cornparedwith a reported 6'20h'Several recint spectacular successesin defining reservoirs

GEOPHYSICALACTIVITY

27

Table 1.2 Worldwidel99I expenditures by objectives(U.5.$ x 103) Object

Land

Transition

Marine

Petroleum Exploration Development Minerals Environmental Engineering Geothermal Groundwater Oceanography Research Totai-Percent

1 189500. 252600. 1 61 0 0 . 3 100. 34 400. 400 2 200. 10. 9 000. 1 507300. 62.0

6 100 27 900

8 1 79 0 0 . 49 300.

Airborne

6 600. 1l 900. 140. 15.

Borehole

I 860 900. 690. 180.

900. 140. 900. 500. 869500. 35.7

300. 34 300. t.4

80. I 8 700. 0.8

320. 4 000. 0.2

Percent

96.7 8 3 I. 13.6 1.2 0.1 1.4 0.1 0.1 < 0.1 0.4

Nole. Becauseof differencesin the manner of reporting, there are minor inconsistencies in some of the numbersin this and the followins tables.

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ber of crews(fig. 1.27a).Increased explorationactivity in the late 1970sbrought about by threatenedsupplies from the Mideast produced a substantialincreasein unit costs(fig. 1.28),but unit costshavebeendeclining since 1980,the decreasebeing especiallyimpressive when adjustedfor inflation. Nonpetroleum seismicstatisticsare given in table 1.5.Thesefiguresare probablymuch more incomplete than the petroleum statistics becausemany small companies are involved. Environmental and engineering work generallyinvolve detailed surveyingof small areas,whereasoceanographicwork is apt to involve cursory surveysof large tracts of ocean, and crewsmay rangefrom 2 to 3 peopleto as many as 50 or so. Nevertheless, the statisticsdo give someidea of the scopeofgeophysicalwork and work in theseareas probablywill increasesignificantlyin the yearsahead. Regarding the seismicsourcesused for acquiring

Prudhoo Bay

Cumulativeexploralory wells (x 10,000) Fig. 1.25 Oil-finding eflrciencyindicated by new oil lound per well drilled. (After Lindseth, 1990.)

have been attributed to this type of work (Sheriff, 1992).Oil companiesare now resurveyingolder finds with newerseismictechnologyin both the marine and land environments,resulting in increasingtheir production and proven reservessubstantially(Abriel et a l . ,l 9 9 l ) . Table 1.4 lists petroleumseismicactivity statistics. Although many different kinds of work are averaged together in these statistics,the figures are probably representative ofthe costsand produtivity ofpresentday full-size geophysicalcrews. Marine work producesmuch more data than land work and the ratio of the volume of new marine data to new land data (f9. | .27b)is quite differentfrom the ratio of the num-

Table 1.3 WorldwideI99l expendituresbv areas (u.s.$ x lqt)

Area United States Canada Mexico South America Europe Africa Middle East Far East Australia-New Zealand International Total

Costs

3'71200. 133800. 44 600. 28'7600. 566700. 236800. 3 1 23 0 0 . 202000. 86 300. l8 900. 2250nU

Percent oftotal

Percent change 1990-l

t6.4 5.9 2.0 12.7 25.r 10.5 13.8 8.9

-36 -5 +700 +47 +58 +9 +ll9 -24

3.8 0.8

-27 -t2.

il

INTRODUCTION

28 E a s t e r nH e m i s P h e r e

W e s t e r nH e m i s P h e r g

2500

2000 c

:

r 1500

? f

1000

'77

'79

'81

' 83

'85

'87

'89

'91

Fig. 1.26 Expenditures on geophysicalpetroleum exploration and development by areas. (After Riley, 1993 ) (a) Western Hemisphere; the areas from the bottom upward represent the

Table 1.4 Petroleuml99l seismicactivity statistics Transitionzone

Land

Marine

Acquisition costs

(u.s.$x l0) Line miles Line kilometers Crew-months Averagemiles/ month Averagekm/ month Averagecost/ month

(u.s.$) Averagecost/ mile (U.S.$) Averagecost/ km (U.S.$) Cost of 3-D/ mile' (U.S.$) Cost of 3-D/ km' (U.S.$)

1 M2 t00. 242000 389000 2'780

34000. 24't00 39700 61

867200. I 367000 2 199000 94'7

87

405

| 440

140

651

2 320

519000 557000

9 1 60 0 0

4 600

I 400

630

2 860

870

390

36000

27 300

13900

l0 500

the land miles,explosivesconstituted34.6o/o,vibrators 59.2%,landair guns0.6%,and weightdrop 0.1%'Air guns were the only marine sourcereported for petro-

United States,Canada, South America, and Mexico (b) Eastern Hemisphere; the areas from the bottom upward represent Eu,op., Afti"u, the Middle East, Australia, and the Far East'

leum work. The percentageof land work that was 3-D cannot be calculatedbecausesome numbersare reported in line miles and some in squaremiles surveyed.Although some marine work was reported in squaremiles surveyed,l2.5ul' of the line miles reported were 3-D and so the total percentageexceeds this value. 1.4 The literature of exploration seismolos/ A. S. Eve and D. A. Keys (1928)wrote in their preface "we to Applied Geophysics, know of no book in English which deals with the theoretical and practical sidesof all of the many schemesof exploration now available."This was only 4 yearsafter the first discovery of hydrocarbonsbasedon seismicrefraction. Eve "in 1928 there were thirty or and Keys noted that 'troops' at work . . . eachconsisting groups or , more of three to five trained men, with an equal number of helpers."Extreme secrecywas common at this time and their book givesonly a brief sketchof methods. "blacklbox" elements As late as the early 1950s,some remained,that is, detailsas to how they worked were not disclosed. Literature on earthquakeseismologyprecededthat dealing with prospecting applications. H. Jeffreys' classicThe Earth appearedin 1924(3rd ed. in 1952). L. D. Leet's Practical Seismologyand Seismic Pros-

LITERATURE OF EXPLORATION SEISMOLOGY

r2,000 I 1,000 r0,000 9,000 1.000 !

7.000

>

5,OOO

3 s.ooo a.000 !,000 2.000 r,000

'77

' 79

! r

' lJ

' !5

laor

(a)

PercentMarineLine-Miles 5a

^

6J

55

6a

7t

t.aoo

I r.zoo 3 r.ooo E

loo

.5 -

600 a00

79

' t|

'!J

t5

'a7

t9

(b)

Frg. 1.27 Worldwide land and marine acquisition. The stippled area indicates land, the diagonal slashed marine. (From Riley, 1993.)(a) Number of field crew-months; (b) volume of data acquired.

pecting(1938)combined earthquakeand exploration >eismology. Although geophysicalliteratureis publishedin several languages,the seismologistwho reads English is especiallyfortunatein that almost all important refer3ncesare in this language.Most of the important papersand books that haveappearedin other languages :ave either English equivalentsor English transla:rons.Furthermore, most of the important technical pubrapersare containedin twojournals, Geophysics, .rshedby the Society of Exploration Geophysicists SEG), and GeophysicalProspecting,published by the

29

European Association of Exploration Geophysicists (EAEG). The Society of Economic Geophysicists was founded in Houston in 1930;the name was changed that sameyear to the Societyof PetroleumGeophysicists and in 1931to the Societyof Exploration Geophysicists.It continuesto be the largestprofessional geophysicalsocietytoday.The societybeganpublication of Geophysics in 1936.Prior to this, paperswere published in issuesof the AAPG Bulletin and Physics; many of the most important papersprior to 1936were republished in Early GeophysicalPapersin 1947. The European Association of Exploration Geophysicists was founded in l95l and beganpublishing Geophysical Prospectingin 1953. The unrefereedmagazinespublishedby thesetwo societies,The Leading Edge and the f irsl Break, provide surveyarticles,interpretationcasehistories,and information about newer topics. The Canadian and Australian Societiesof Exploration Geophysicsalso publish journals that are more along the lines of The Leading Edge and Ihe First Break than of Geophysics and GeophysicalProspecting.Otherjournals that often contain important articles are published in Europe, India, and elsewhere. The Bulletinof the AmericanAssociation of Petroleum Geologistsoften contains rmportant papers on interpretive applications of geophysics.The geophysicalliteratureof basicseismology also often contains papers of interestto exploration seismologists. The most important non-Englishjournalsare Russianand Chinese. A CumulativeIndex of Geophysicsis published every few years (most recently as a supplement to the March 1990issueof Geophysics); it lists the papersin the publicationsof most of the foregoingsocietiesexcept for those of the American Associationof Petroleum Geologists.The cumulativeindex is also availableon a computerdisk, which also liststhe expanded abstractsofpapers given at the annual meetings.This computer disk can be searchedfor key words. The most important papers from Geophyslcsare reprinted in the 25th and 50th anniversaryvolumes(Classicpapers of the past 25 years,1985)and important exploration seismicpapers from various journals are reprinted in three volumes of the Treatiseof Petroleum GeologyReprint Serles(Beaumont and Foster,1989). A seriesof 14 (as of 1994) reprint volumes dealing with various subjectsand a number of other geophysics books are published by the Society of Exploration Geophysicists. A multitude of books on various aspectsof seismic explorationare availabletoday.In the first edition, we were able to list most of the important books on aspects of seismicexploration, but today there are so many that it is not feasibleto do this. Many are referencedin subsequentchapters.Seismictechnologytoday embracesso much signal processingand computer technologyas well as geologythat a readinglist would include many works that are not specifically geophysical.Particular mention should be made of

= 6

E

4ooo

A o

!l1E66 1966

1075 costs adjusted for inflation. (Data from SEG Geophysital Activity Reports.)

F i g . 1 . 2 8 S e i s m i cc o s t s p e r m i l e ( 1 9 8 0 a n d p r e - 1 9 8 0 ,W e s t e r n Hemisphere; post-1980, worldwide). The dotted curves show

Table 1.5 Nonpetroleuml99l seismicactivity Surveytype

Typework

Expenditures(U.S.$x 103)

Minerals

P-wave reflection S-wavereflection Refraction P-wave reflection S-wavereflection P-wave reflection S-wavereflection Refraction Reflection Refraction Passive Reflection P-wave reflection S-wavereflection Refraction

5 290. I 050. t70. 820. 60.

Environmental Engineering

Groundwater Geothermal Oceanography Research

J l .

n a

12. 410. I 10. 0.3 920. 4 910. I 50. 15.

Cost/mile

Costikm

3 150. 6 900. 2 900. 7 300.

1 960. 4 300. I 800. 4 600.

I 500. 1*0. 6 100. 4 300 I 600.

940. 780. 3 800. 2 600. I 000.

90. 2 900. 50000. l5 000.

60. I 800. 31000. 9 300.

REFERENCES the safety and environmental guides published by the International Association of Geophysical Contractors (IAGC). References Abbot,H. L. 1878.On the velocityof transmission of earth waves. Amer.J. Sci.Arts,Ser.3, 15:178-84. Abriel,W.L., P S.Neale,J.S.Tissue, andR. M. Wright.1991. Modern technology in an old area. Bay Marchand revisited. The Leading Edge, 10(6):21,35. Barrington, lo

T. 1982. Cecil Green. The Leading Edge, l(l):

/4.

Barton,D. C. 1929.The seismicmethodof mappinggeologic structure.In Geophysical Prospecting, pp. 572,624.New York: AmericanInstituteof Mining and MetallurgicalEngineers. Bates,C. C., T. F. Gaskell,and R. B. Rice. 1982.Geophysics in the Afairs of Man Oxford: PergamonPress. Bates,R. L., and J. A. Jackson.1987,Glossaryof Geology,3d ed. FallsChurch,VA: AmericanGeologicalInstitute. Beaumont,E. A., and N. H. Foster.1989.Geophysics I: Seismic .Vethods;Geophysics II: Toolsfor SeismicInterpretation;Geopht'.sics III: GeologicInterpretationof SeismicData,Treatiseof PetroleumGeology,ReprintSeriesNos. 12, 13,and,14.Tulsa: .{mericanAssociationof PetroleumGeologists. E[rrn, W T. 1960.A review of geophysicalinstrumentation. Geophysics, 25277-91. - arlton, D. P. 1946.The History of the Geophysics Department. Houston:HumbleOil and RefiningCo. -lark, R. D. 1982.Gerald Westby.The LeadingEdge,l(l): .rrk. R. D. 1983.SidneyKaufman. The LeadingEdge,2(7\: ::7. :rk. R. D. 1984a.T. I. Harkins. The LeadingEdge,3(4\: r t8. ..r-k.R. D. 1984b.C. Hewiu Dix. The Leading Edge,3(8\: .

1 1

.:k. R. D. 1985.EndersRobinson.The LeadingEdge,4(2): - 10. ,:i. R. D. 1990a.Theodor Krey. The LeadingEdge,9(4); : lt. .:i. R. D. 1990b.Kenneth E. Burg. The Leading Edge, * l t ) ) :l 3 1 6 . .::. papersof the past25 years.1985.Geophysics, 50: 179'l:lrer, E. 1935. Notes on the early history of applied geo!:;s in the petroleum industry. Trans. Soc.Pet. Geophys.,6: r Reprinted in Early Geophysical Papers of the Society of ,ttion Geophysiclsts,pp. 245-54. Tulsa: Society of Explo: Geophysicists,1947.) J B. 1992.Seismicoveryiew.New Orleans: Howard, Weil, -rsse.Friedrichs Inc. . T A, 1970. A Brief History of Gulf's Geophysical Pros'.- Pittsburgh: Gulf Researchand Development Co. r. S . and D. A. Keys. 1928. Applied Geophysics. Cam.. Cambridge University Press. :::r. E. I. 1974. Vertical Seismic Profiling Tulsa: Society : .ration Geophysicists. . - ,.ri. K. 1991.Geophysical activity in 1990. The Leading Itlll):45-72. , H. 1979. John Clarence Karcher. 1894 1978. father ', -::ection seismograph.Geophysics,44: 1018-21.

3l Hecker,O. 1990.Ergebnisse de Messungvon Bodenbewegungen bei einer Sprengung. Gerland'sBeitrrige zur Geophysik,4: 98-104. Heiland,C. A. 1929a.Moderninstruments and methodsof seismic prospecting.ln Geophysical pp. 625-53.New Prospecting, York: American Institute of Mining and Metallurgical Engineers. Heiland, C. A. 1929b.Geophysicalmethods of prospectingPrinciplesand recentsuccesses. Quart. Col. Sch.Mines,A(\,t, Jeffreys, H, 1952.TheEarth,3ded.Cambridge:CambridgeUniversityPress. Karcher, J. C. 1987.The reflection seismograph:Its invention and usein the discoveryof oil and gasfields.TheLeadingEdge, 6(11):l0-20. Keppner,G. 1991.Ludger Mintrop. The LeadingEdge, l0(9): 2 t-8. Knott, C. G. 1899.Reflexionand refraction of elastic waves, with seismologicalapplications.Phil. Mag., 48:64-97. - The First Fifty Laing, W E., and F. Searcy.1975.Geophysics lears.Houston:Conoco. Landmark Graphics Corp. 1992.The ComingReunionof Seismic Interpretationand ProcessingHouston: Landmark Graphics Corp. Leet,L. D. 1938.PracticalSeismologyand SeismicProspecting. New York: Appleton-Century. Lindseth,R. O. 1990.The new wavein explorationgeophysics. TheLeadingEdge,9(12)t9-15. Love, A. E.H. 1927. Some Problemsof GeodynamicsCambridge:CambridgeUniversityPress. Malamphy,M. C. 1929.Factorsin designof portablefield seismographs.Oil llteekly,March 22, 1929. Mallet, R. 1848.On the dynamicsof earthquakes; beingan attempt to reducetheir observedphenomenato the known laws of wave motion in solids and fluids. Irans Roy.Irish Acad., 21:50-106. Mallet,R. 1851.Secondreporton the factsofearthquakephenomena.BAAS,2l: 272-320. Mayne,W. H. 1982.The evolutionof geophysical technology. TheLeadingEdge,l(l):75-80. McGee,J. E., and R. L. Palmer.1967.Earlyrefractionpractices. ln SeismicRefractionProspecting, A. W. Musgrave,Ed. Tulsa: Societyof ExplorationGeophysicists. McGuckin,G. M. 1945.History of the geophysical exploration of the CameronMeadowsDome,CameronParish.Louisiana. Geophysics, l0: l-16. Milne, J. 1895.Seismicexperiments. Trans.Sei,s.Soc.Jpn., B: l-82. Mintrop, L. 1931.On theHistoryof theSeismicMethodfor theInvestigationof Underground Formationsand Mineral Diposits.' Hanover,Germany:Seismos. Owen,E. W. 1975.Trekof the Oil Finders:A History o! Explorationfor Petroleum,AAPG Memoir 6. Tulsa:American Association of PetroleumGeologists. Petty,O. S. 1976.SeismicReflections. Houston: Geosource. Proffitt,J. M. 1991.A historyof innovationin marineseismic data acquisition.TheLeadingEdge,l0(3):24 30. Proubasta,D. 1982.O. S. Petty.TheLeadingEdge,l(7): 16-24. Proubasta,D. 1983a.John Hollister.The LeadingEdge,2(7): 14 t9. Proubasta,D. 1983b.Henry Salvatori.TheLeadingEdge,2(8\: t4-22.

)z

Proubasta,D. 1983c.John Crawford. The LeadingEdge'2(12): t6-26. Proubasta,D. 1984.Remembranceof geophysicalthings past' TheLeadingEdge,3(10'):32-8' Proubasta,D. 1985a.SvenTreitel. TheLeadingEdge'4(2\:24 8' Proubasta,D. 1985b.Harry Mayne. The Leading Edge' 4(7): t8 24. Proubasta,D. 1986a.Erik Jonsson.The Leading Edge' 5(6)" t4-23. Proubasta,D. 1986b.Enders Robinson and the shot heard round the geophysicalworld. TheLeadingEdge,3(9):14 17' Proubasta,D. 1991.Maurice Ewing. The LeadingEdge,l0(3)" 15 20. Rayleigh,Lord. I 885.On wavespropagatedalongthe planesurfaceof an elasticsolid.Proc.LondonMath Soc.,17:4-11' Rieber,F 1936.A new reflectionsystemwith controlleddirecl: 97-106. tional sensitivity.Geophysics, activityin l99l Riley,D. C. 1993.Specialreport:Geophysical The LeadingEdge,l2z 1094-1I 17. Robertson,H. 1986.EveretteLee DeGolyer.TheLeadingEdge, 5(ll):l4 21. Robinson,E. A. 1985.A historicalaccountof computerresearch in seismicdata processing,1949-1954.The Leading Edge,4(2\:40-5. Rosaire,E. E. 1935.On the strategyand tacticsof exploration for petroleum.J. Soc.Pet. Geophys.,6t1l-26 (Reprintedin Eariy GeophysicalPapersof the Societyof Exploration-Geophysicrsri pp. 255 70. Tulsa: Societyof ExplorationGeophysicists,1947.) Rosaire.E. E.. and J. L. Adler. 1934.Applicationsand limitationsof dip shooting.Bull. AAPG, 18: 19 32. Rosaire,E. 8., and O. C. Lester,Ir. 1932.Seismologicaldiscoverv and partial detail of Vermillion Bay salt dome.Bull' AAPG'

INTRODUCTION

Papersof the Society 16: 5l-9. (Reprintedin Early Geophysical pp. 381-9. Tulsa: Societyof Exof Exploration Geophysicists, 1947.) plorationGeophysicists, Schriever,W. 1952.Reflectionseismographprospecting How l7't 936-42. it started.Geophysics, Shaw H.. J. M. Bruckshaw and S. T. Newing' 1931.Applied London: His Majesty'sStationeryOffice. Geophysics. technologythrough Sheriff,R. E. 1985.History of geophysical in Geophysics.Geophysics,50':2299-2408' advertisements and interpretationoi seismtcreSheriff,R. E. 1988.Processing flection data: An historical pr6cis. The Leading Edge' 7(l): 40 2. Dictionaryof ExplorationGeo' Sheriff,R. E. 1991.Encyclopedic physics,3d ed.Tulsa:Societyof ExplorationGeophysicists. Tulsa: Societyof Sheriff,R. E., ed. 1992.ReservoirGeophysics. Exploration Geophysicists. Stoneley,R. 1924.Elasticwavesat the surfaceof separationof two solids.Proc. Roy.Soc.( London),A-106:416-28. ProspectrzgSudbury Sweet,G. E. 1978.History of Geophysical England:Spearman. of a newmethodof makingunUdden,J. A. 1920.Suggestions Bull. AAPG,4: 83 5. (Reprintedin dergroundobservations. 16:7l 5-16.) Geophysics, 14:6-9' Vajk, R. 1949.Baron Roland Eotvos.Geophysics, Weatherby,B. B. 1940.History and developmentof seismic 5: 215-30. prospecting.Geophysics, in Oklahoma' B. B. 1945.Early seismicdiscoveries Weatherby, s, l0t 345-67. Geophysic Wiechert,8., and K. Zoeppritz.1907.Uber Erdbebenwellen' der Wissenschaften Nachrichrenvonder Ki)niglichenGesellschaft pp. 415-549.Berlin. zur Gdttingen, of the seconddecadeof seisWilcox,S. W. 1990.Reminiscence TheLeadingEdge,9(8ll42-5. mic prospecting.

2

Theory of seismicwaYes

Overview

fronts and raypathsare introduced,as is the more general Huygens'principle approach. The two forms of the wave equation that had been derivedearlierare relatedto two typesof disturbances that can travelthrough the body ofsolids ($2.4).These involve changesin volume (P-waves)and rotations (S-waves).Discussion of potential functions, from which particle displacementsand velocitiescan be derived, follows.At interfaces,both stressesand particle displacementsmust be continuous; these boundary conditions are discussedin $2.4.4. Surface waves are examined next. Rayleigh waves are important becauseof the ground-roll noise that they produce on seismicrecords.Love, Stoneley,and tube wavesare encounteredoccasionally. Most seismictheory assumesthat media are isotropic, that is, their propertiesare the sameregardless of the direction of measurement.Anisotropy ($2.6)of severaltypes has beenobserved;howevet anisotropic effectsare usually small. The most important exceptions requiring study are those of transverseisotropy becauseof layeringand fracturing. Section2.7 examineswhat happensto seismicbody wavesas they travel in the earth. Intensity decreases becauseofgeometricalspreading(divergence)and absorption (and partitioning at interfaces;seechap. 3). Divergenceis the most important factor affectingthe change of intensity for the first few kilometers,but eventuallyabsorptionbecomesdominant. Absorption increasesapproximatelylinearly with frequencyand hencechangesthe waveshapewith distance.Various expressionsfor absorption are interrelated.Dispersion and the conceptsofgroup and phasevelocityare discussed,although dispersion is not an importqgt factor in seismicexploration. Reflection and refraction are discussedin $2.7.5. Diffraction ($2.8),the scatteringof wavesat discontinuities, involves somewhat complex mathematics. However, the construction of diffraction wavefronts using Huygens'principle is fairly straightforwardand nonmathematical.

The seismicmethod utilizesthe propagationof waves through the earth. To introduce the basicconceptsof wave motion, we flrst discusswaveson a stretched string($2.1.1)and introducedefinitionsofphase,frequency,wavelength,and other terms dealingwith periodicity.Becausewavepropagationdependsupon the elasticpropertiesof the rocks, we next discusssome of the basicconceptsof elasticity.(For more thorough treatments,seeSaada,1974,or Landau and Lifshitz, 1986.) The sizeand shapeof a solid body can be changed by applying forcesto the externalsurfaceofthe body. Theseexternalforcesare opposedby internal forces, which resistthe changesin sizeand shape.As a result, the body tendsto return to its original condition when the externalforcesare removed.Similarly,a fluid resists changes in size (volume) but not changes in shape.This property of resistingchangesin srzeor shapeand of returning to the undeformedcondition when the externalforcesare removedis calledelasticity. A perfectlyelasticbody is one that recoverscompletelyafter beingdeformed.Many substances including rocks can be consideredperfectly elasticwithout appreciable error provided the deformations are small.as they are in seismicsurveys. The theory of elasticity relatesthe forces that are applied to the external surface of a body to the resulting changesin size and shape.The relations between the applied forces and the deformations are most convenientlyexpressedin terms of the concepts of stressand strain. Strain, a changein shapeor dimensions,is generallyproportional to the stress(force per unit area) that producesit, as statedin Hooke's law. The constantof proportionality is called an elastic constant, or modulus, and moduli for different types of stressand strain are interrelated. Section 2.2 concerns seismic-wavemotion. Newton's secondlaw of motion, that an unbalancedforce on a massproducesan acceleration,is used to derive two forms of the waveequation.The waveequationis expressedin vector as well as the more conventional scalarnotation. Methods of including a sourceof disturbance and Kirchhoff's theorem are also given in this section. Plane- and spherical-wave solutions to the wave equation are given next. Waves are disturbances that travel through the medium. The concepts of wave-

2.1 Theory of elasticity 2.l.I Waveson a stretchedstring As an introduction to seismicwavesin three dimensions, we consider the one-dimensionalwave in a stretched string becausemany basic conceptsof wave JJ

THEORY OF SEISMIC WAVES

)+

motion can be more simply illustrated in this way. Parts of the following discussionwill be treated later in a broadercontext. We assumean ideal case where the mass of the string per unit length, p, is negligiblysmall in comparison with the tension,t, in the string, that the string when at rest is along the x-axis,and that the displacements,rf, which are parallel to the y-axis, are small in comparisonwith the length of the string so that angles o., and o., are also small (fig. 2.la). Becausethese anglesare not equal,the tensionproducesa net force in the y-direction (the net force in the x-direction is negligible)on an elementof the string, Ax, equal to t(sin c, - sin c,) - r(tan ct2- tan c,) - t(d{/dxl-, d$/dxl,,) - t A(drl/dx). Newton's secondlaw of motion statesthat this force equals the product of the massp Ax and the acceleration02'1il6t2. Dividing both sidesby roAx and taking the limit as Ax -+ 0 givesthe one-dimensionalwaveequation:

or P.ol /-cta3t

, J-__\./i

: (plt)0' ,$l0tz: (llW)02\tl0t,, (2.1) 02$10x2 where V = (rlp.)",(comparewith eq. (2.45)).Equation (2.1)showsthat Zhas dimensionsof distance/time,or velocity.The waveequation relatesvariation in space (the left side)with variation in time (the right side). The general solution of eq. (2.l) (also called d'Alembert'ssolution;see$2.2.5)is ,1,(x,t) = t, k - Vt) + 'ltr(x + Vt), (22) whererf , and rf, are arbitrary functions,rf , is a disturbancemoving in the positivex-directionwith increase of time, rf, a disturbancemoving in the negativexdirection, and V is the velocity of propagationalong the string (seethe following). Fourier analysis($9.1.2) showsthat any waveform (within reason)can be representedby a superposition of harmonic (sinusoidal)waves,so we do not losegenerality by confining our attention to harmonic waves. Thus, we considera harmonic solution of eq. (2.I ) in the form r.p= I cos [(2,n/\)(x * Vt)].

vlt.

F,aqucnct

I

-

P c ri o d

(b)

--T----:

l

l

tFWovalane

wov.numbrr =

th....+l

a;r..L;gi;(c)

(2.3)

The waveform is harmonic with r! varying between * A and - A ; A is the amplitude.If we look at the wave passinga fixed point in space(fig. 2.lb), period 7is the time betweensuccessive repetitionsof the waveform; frequencyv : llT is the number of wavesper unit time. If we look at the waveformat somemoment of time (fig. 2.lc) the distancebetweensuccessive repetitions of the waveform is the wavelength)\ and l/L is the wavenumber or number of wavesper unit distance. Multiplying IIT and llltby 2rr, we get the angularfrequencya : 2nlT : 2rv and the angular wavenumber r : 2rl}'. Becausev is the number of wavespassinga fixed point per unit time and eachwavehas length tr, velocity Z must be given by the equation V:

r+- PrriOd --l !

Q.4)

The argument of the cosine in eq. (2.3), namely,

(d) Fig. 2.1 Waves on a stretched string. (a) Portion of the string showing the relation betweendisplacementand tension; (b) representation of the wave in time; (c) representation of the wave in space;(d) the effect of change in mass/unit length.

(2rllr) (x - Vt1 : r(x - Vt): (rcx - ot), is calledthe phase.ln eq. (2.3), the phaseis zero at the origin; at times we add a fixed phase angle ^yoso that the phase becomesKr - r,-lt+ "yo.

THEORY OF ELASTICITY

I

35

Returning to the stretchedstring, if the mass/unit length changesabruptly from p, to p2 at somepoint, say, x : 0 (fig. 2.ld), certain boundary conditions ($2.4.4)must hold, namely,both the displacementand the y-componentof the tension in the string must be continuous,that is, neither changesin value as we go through the junction. These conditions can be expressedby the equations *,.r, : t,,rn,, r(dr!/dx),"r,: r(drlr/dx).,rn,.

(2.s)

We take the incident waveas l, cos (r,x - ol) coming from the left and the wavepassingon to the right (the transmittedwave) as l, cos (rrx - tot); however,we cannot satisfyeqs.(2.5)with thesetwo wavesonly and we must postulatea reflectedwave going to the left, l, cos (x,x + cor).Substitutinginto eqs.(2.5),we find that the boundaryconditionswill be satisfiedprovided A,+ A,: A,,

*,A,) ",)',: iit,.

I

(2.6a)

]

Equations(2.6a) can be solved for A,and A,: R : A , lA , : ( r z - x , ) / ( x *, r , ) , I

T= rq.,lA,:Z^,4*rl*,.y,

i

(2.6b)

where R and T are called the reflectioncofficient (or refectivily) and the transmissioncofficient, respectively(seealso93.2). If the string is fixed at x : 0, the effectis the same as if p, : -; then T : 0, so no wave is transmitted, and R : +1, which meansthat the reflectedwave is the sameas the incident one exceptthat the direction of travel is reversed.The two wavesinterfere($2.3.2) at the fixed end to produce perfect cancellation, hence,zero movement(node).If both ends are fixed, perfectcancellationmust occur at both ends,so these are nodes. When a string fixed at both ends is vibrating at its lowestfrequency,calledthe fundamental(u"), the displacement has its maximum amplitude at the midpoint (antinode). The wave pattern is fixed, so the wave is said to be stationary,or standing.If the string length is L, L : ),/2 and v,,: VII\ : Vl2L. The strins

Fig. 2.2 Componentsof stresson facesperpendicularto the x-axis.

can vibrate in a number of patterns called modes or eigenstates,the frequencies being harmonics (multi_ ples)of the fundamental,that is, vi = nvo,n : 1,2,3, . . . . In eachcase,the endsofthe string are nodesand L: nttl2: (2n)|t14. If the left end of the string is fixed and the right end free, we set K2 = 0 and get R = -1. The end of the string is an antinode, L = |t14, v^ : Vl4L, and the harmonicsare u : (2n + l)vo and,L : (2n + l)\14. The two casesof a string fixed at one end only and fixed at both endsare analogousto organ pipesilosed at one end only and closedat both ends(Logan, l9g7; s e ea l s oS 1 3 . 3 ) . 2.1.2 Stress S/ressis definedas force per unit area. Thus, when a force is applied to a body, the stressis the ratio of the force to the area on which the force is applied. If the force variesfrom point to point, the stressalso varies, and its value at any point is found by taking an infinitesimally small elementof area centeredat the point and dividing the total force acting on this area by the magnitudeof the area.If the force is perpendicularto the area, the stressis said to be a normal .r/re$ (or pressure).In this book, positive valuescorrespondto tensile stresses(the opposite convention of signs is sometimesused).When the force is tangential to the element of area, the stressis a shearings/ress.When the force is neither parallel nor perpendicularto the elementof area, it can be resolvedinto components parallel and perpendicularto the element;hence,any stresscan be resolved into component normal and shearingstresses. If we consider a small elementof volume inside a stressedbody, the stressesacting upon each ofthe six facesof the elementcan be resolvedinto components, as shownin fig. 2.2 for the two facesperpendicularto the x-axis.Subscriptsdenote the x-,1,-, and z-axes, respectively, and o,.*denotesa stressparallel to the y_ axis actingupon a surfaceperpendicularto the x-axrs. When the two subscriptsare the same (as with o.,), the stressis a normal stress;when the subscriptsare different (as with o,,), the stressis a shearingsiress. When the medium is in static equilibrium, the stressesmust be balanced.This meansthat the three stresses,o,,, or,, and o,.., acting on face OABC must be equal and opposite to the correspondingstresses shown on opposite faceDEFG with similar relations for the remaining four faces.In addition, a pair of shearingstresses, suchas oy,,constitutea coupletending to rotate the elementabout the z-axis,the magnitude ofthe couple being force X lever arm = (o u dy dz) dx. If we consider the stresseson the other four faces, we find that this couple is opposed solely by the couple due to the pair of stresseso"" with magnitude (o,, dx dz) d): Becausethe elementis in equilibrium,

THEORY OF SEISMIC WAVES

JO

the total moment must be zero; hence o", : or". In general,we must have or:

oii'

Au oy

(2.7\

r

2.1.3Strain When an elasticbody is subjectedto stresses, changes in shapeand dimensionsoccur.Thesechanges,which are called strains,can be resolvedinto certain fundamental types. Consider rectanglePQRS in the xy-plane (seefig. 2.3). When the stressesare applied,let P move to P', PP' having componentsu and v. If the other vertices Q, R, and S have the samedisplacementas 4 the rectangleis merelydisplacedas a whole by the amounts u and v,'in this case,there is no changein sizeor shape, and no strain exists.However,if u and y are different for the different vertices,the rectanglewill undergo changesin sizeand shape,and strainswill exist. Let us assumethat u : u(x,y) and y : r(r, y). Then the coordinatesof the verticesof PQRS and P'Q' R'S' are as follows: P(x, y): P'(x * u, y + v); Q@ + dx, y):

Q'(** d x* u + 4 a *,y * v *#*) ' S(x,y + dy): s'(" + u + !! dy,y +dy + r * j,ar)' R(x+dx,y+dy):

R ' ( " +d x * u + 4 d x . \ i , o r ,

y+dy*,*3.1a"+jjcr). In general,the changesin u and y are much smaller than the quantitiesdx and dy,' accordingly,we shall assumethat the terms (duldr), (6ulAfl, and so on are small enough that powers and products can be neglected.With this assumption,we seethe following: l . PQ increases in length by the amount (6ul0x) dx and PS by the amount (lvl0y) dy; hence 6ul0x and 6vl0y are the fractional increasesin length in the direction of the axes. 2 . The inflnitesimal angles6, and 6, are equal to \vl6x and 6ul0y,respectively. J . The right angleat P decreases by the amount El+E2:3vl3x+AulAy. 4. The rectangleas a whole has been rotated counterclockwisethrough the angle (6, 6r)12: (6vl3x - \ula)12. Strain is definedas the relativechange(that is, the fractional change)in a dimensionor shapeof a body. The quantities 6ul3x and 0vl6y are the relative increasesin lengthin the directionsofthe x- and y-axes, and are referred to as normal strains. The quantity 6vl6x i \ulEy is the amount by which a right anglein

sf -

au,.:Tar-"'|

,uj,---------l

iI

I

dy

t l rl

i

I

La=- Itv-

P ' I

---_______rt+-

l+-dx

. l-or

_J:'

F-l

I Fig. 2.3

J --\-au

ia'

u->\

I -N

Analysis of two-dimensional strain

the xy-plane is reducedwhen the stresses are applied, hence,is a measureof the changein shapeof the medium; it is known as a shearingstrain and will be denotedby the symbol e.,. The quantity (6vl6x dul6y)12,which representsa rotation of the body about the;-axis, does not involvechangein size or shapeand henceis not a strain; we shall denote it by the symbol 0-. Extending this analysis to three dimensions,we write (r.r,4 w) as the componentsof displacementof a point P(x, y, z). The elementarystrainsare thus

du Normal strains a," : -' dx 0v 8.. : ^ ' dy 6w.

(2.8)

4,,:

dz

: - y Shearingstrains 8,.,: Er, ' d !" x d !., E y " - - E : -'0y 4*4' 6z'

a,r:ar,:!-! dz

(2.e\

dx

In addition to thesestrains,the body is subjectedto simple rotation about the three axesgiven by ^

H

0":

e :

dwldv :

-

dvtdz

2

6ul6z - 6wl6x

(2.10)

z lvldx - 6ul0y

Equations(2.10)can be written in vectorial form (see $ l 5 . 1 . 2 ( aa)n d 1 5 . 1 . 2 ( c ) ) :

@ : 0 , i + e j + 0 , k : V x 6,

2

( 2.n)

THEORY OF ELASTICITY

J I

changein volume per unit volume A is . 0 u A : e . . * e ,' -, . * e - - : - - ' * ! * dx 0y

dz

(2.r2)

2.1.4Hooke'slaw

c 6

Stnin

-+

Ruptur

Tim.

_.+

In order to calculatethe strainswhen the stressesare known, we must know the relationshipbetweenstress and strain. When the strainsare small, this relation is given by Hookes /anl which statesthat a given strain is directly proportional to the stressproducingit. The strainsinvolvedin seismicwavesare usuallylessthan l0 8 exceptvery near the source,so that Hooke'slaw holds. When several stressesexist, each produces strains independentlyof the others; hence,the total strain is the sum of the strainsproducedby the individual stresses. This meansthat eachstrain is a linear function of all of the stressesand vice versa.This linearity has important implicationsthat will be utilized later: It allows us to representcurved wavefrontsas a superpositionof plane waves,for example,in p-r transforms($9.1.5 and 9.1l.l ), to expressa reflected wavetrainas a superpositionof individual reflectrons (theconvolutionalmodel),and to justify manyaspects of seismicdata processing. In general,Hooke's law leads to complicatedrelations. Stress and strain can both be regarded as second-order(3 x 3) matricesso that the Hooke'slaw proportionality relatingthem is a fourth-ordertensor. Stressand straincan alsobe lookedon as (l X 6) matrices(as in eq. (2.l5)) and the Hooke'slaw proportionality as a 6 x 6 matrix whoseelementsare elastic constants(Landau and Lifshitz, 1986:32 5l). Symmetry considerationsimmediatelyreducethe number of independentconstantsto 21. However,when the medium is isotropit',that is, when propertiesdo not dependupon direction, it can be expressedin the following relativelysimpleform (Love, 1944:102): o,, : \A * 2p,e,, (i : x, y, z), (2.13)

(D)

Fig. 2.4 Stress strain.time strain; (b) strain versus time.

U:v.r.

relationships. (a) Stress versus

o,,: 2p"e,,

(2.14)

Theseequationsare often expressedas a matrix equation, o : Ce: o.r*

where( : ui + vj + wk is the vector displacementof point P(x, y), and i, j, k are unit vectorsin the x-, y-, :- directions,respectively. The changesin dimensionsgiven by the normal strains result in volume changes when a body is stressed.The change in volume per unit volume is called the dilatationand representedby A. If we start with a rectangularparallelepipedwith edgesdx, dy, and d: in the unstrainedmedium. in the strainedmedium the dimensionsare dx(l + s,,), dy(l * e,.,),and d:(l + e--),respectively;hencethe increasein volume is approximately(e.. * e,,, * e,_)d,t dy dz. Because the original volume was (dx dy dz), we see that the

(i,j : x, y, z; i + j).

o"" C,,

o., oy o,,

I \ + 2 p \ I 0 0 0 \ tr+2pI 000 I \ \ + 2 p 0 0 0 0 0 p 0 0 0 0 0 p 0 0 0 0 0

l%,

1",, lu..

0 0 0 18r' p Ile,,

1"",

(2.rs) The equationis sometimeswritten e : So, whereS : C r. Components of C (or S) are sometimescalled stffiess (or compliance)components. The quantities\ and p are known as Lam6'sconstants. If we write e,, : o,,lp",it is evident that e,, is

T H E O R YO F S E I S M I CW A V E S

38 smaller the larger p is. Hence, p is a measureof the resistanceto shearingstrain and is often referred to as the modulus o/' rigidity, incompressibility,or shear modulus. Although Hooke'slaw has wide application,it does When the stressis increased not hold for largestresses. beyond an elastic limit (fi5. 2.4a), Hooke's law no longerholds and strainsincreasemore rapidly.Strains resulting from stressesthat exceedthis limit do not entirely disappear when the stressesare removed. With further stress, a plastic yield point may be reachedat which plastic flow begins and the plastic yieldingmay resultin decreasingthe strain.Somematerials do not pass through a plastic flow phase but rupture first. Rocks usually rupture at strains - 1 0 - 3 - 1 04 . Some materialsalso havea time-dependentbehavior to stress(fig. 2.ab). When subjectedto a steady stress,such materialscreep until eventuallythey rupture.Creepstraindoesnot disappearifthe stressis removed. 2.L5 Elasticconstants Although Lame's constantsare convenientwhen we are usingeqs.(2.13)and (2.14)other elasticconstants are also used.The most common are ktung'smodulus (E), Poisson's ratio (o), and the bulk ntodulus(ft) (the ratio symbol o is more or lessstandardfor Poisson's the subscriptsshould preventany confusion with the stressou). To define the first two, we considera medium in which all stressesare zeroexcepto,-. Assumdimensions ing o,. is positive(that is, a tensilestress), parallelto o., will increaseand dimensionsnormal to this meansthat e.. is positive(elono,, will decrease; gation in the x-direction)wherease,.,and €--are negative. Also, we can show(seeproblem2.la) that e,, : e--.We now defineE and o by the relations E:

o,,/e,,,

o : - e , , , / D . .: - e - - l e , , ,

(2.16) (2.11)

with the minus sign insertedto make o positive. To define the bulk modulus k, we consider a medium acted upon only by a pressure0; this is equivalent to the stresses o . r : o . , . . :o , . : 0 . o . , . , : o , , ,: o , r : - 0 , Pressure0 causesa decreasein the volume AT and a dilatation L : LYIT:, k is definedas the ratio of the pressureto the dilatation that it causes,that is,

k:

-911,,

( 2 .l 8 )

with the minus sign insertedto make k positive.Somel/k, is usedas an elasticcontimesthe compressibility, stant rather than the bulk modulus. By substitutingthe precedingvaluesin Hooke'slaw, we can obtain the following relations betweenE, o, and k and Lam6's constants,\ and p (seeproblems 2.lb and2.1c):

-

F

p(3\ + 2p) :

(2.re)

A+Lr (t=

A

2(}' + P)'

k : l (13 \ + 2 u , t .

(2.20) (2.21)

In nonviscousfluids, the shear modulus pr : 0, and hencek : \. Becausewe have not previouslygiven a specificname to L, we may call it thefuid incompressibility. By eliminating different pairs of constants among the three equations,many different relations can be derivedexpressingone of the five constantsin termsof two others(seeproblem2.2). The elasticconstantsare definedin sucha way that they are positive numbers.As a consequenceof this, o must have values between0 and 0.5 (this follows from eq. (2.20),becauseboth \ and p are positiveand hence\/(\ + p) is lessthan unity). Valuesrangefrom 0.05 for very hard, rigid rocks to about 0.45 for soft, poorly consolidatedmaterials.Liquids have no resistanceto shearand hencefor them p : 0 and o : 0.5. For most rocks, 4 k, and p lie in the range from 20 to 120GPa (2 x 10",to 12 x l0'0 N/m' ), trgenerally being the largest and p the smallest of the three. Tablesof elasticconstantsof rocks havebeengivenby Birch (1966).(Seealsoproblem2.4.) an isotropic Most of the precedingtheory assumes medium. In fact, rocks are usually in layers with differentelasticproperties,thesepropertiesoften varying with direction. Nevertheless, in discussingwave propagation,we generallyignore suchdifferencesand treat sedimentaryrocks as isotropic media; when one does so, the results are useful and to do otherwise leads to extremelycomplex and cumbersomemathematical equations,exceptfor the caseof transversely isotropicmedia,that is, media in which the properties are the samein one plane but different along the normal to the plane. Some rocks, especiallyshales,are transverselyisotropic,and more importantly, a series of parallel beds,eachof which is isotropic,but where the properties vary from bed to bed, behavesas though it is transverselylisotropic (Postma, 1955; Uhrig and van Melle, 1955).Anisotropyis discussed in $2.6. 2.1.6 Strain energy When an elastic medium undergoes deformation, work is done and an equivalentamount of potential energy is stored in the medium; this energy is intimately relatedto elasticwavepropagation. If the stresso,- resultsin a displacemente,., we assume that the stressis increaseduniformly from zero to o....,and hencethe averagestressis o,,/2. Thus, -E : work done per unit volume : energyper unit volume : o,,e,12.

WAVE EQUATIONS

39

Summing the effectsof all the independentstresses and using eqs. (2.13) and (2.14) gives (Love, 1944: 100) , :

o,u,,

:))

: j(o..",,

62u

p;, : unbalancedforce in the x-direction on ot- a unit volume

* orr.8r,,* o,,8,, * o..".."

* o,,,8.,, *

Newton's secondlaw of motion statesthat the unbalancedforce equalsthe masstimes the acceleration; thus, we obtain the equation of motion along the xaxls:

o--*er.)

: l' t[ +- ( \ A + 2 P " e ,*, t*e) ), ", 1 , - |I .

u*i'1

: jf 4' + p(r.l. + ef, + e?_) + lp(ei,+ ef- + e1.).

(2.22)

Notethat eq.(2.22)gives 0 E l 6 e , , :} ' A + 2 p e , , : o . . , : o\,., 6El0e,,: p,e\,. hence, 3E/6x,, : o ,

_ d o . , T, d o . , . dr 6y

(i,j : x, y, z).

(2.23)

lto,, ^A:u _ , do,, -t'df dr 0y

o , , +, 0 o , .d. r , a,

o , *

6o O ' ' d r

Because thesestresses areopposite to thoseactinson therearface,thenet (unbalanced) stresses are r,x

'-%,0r, dx

{o--r' 0x'

dx

ao'.. dx

6x

0y

do,_

0z

, r') 6:l

: (\ + r,)la+ u.v.r,

r, )St

where V2a is the Laplacian of u : A2ulAx2* 62ul0y,-t (seeeq. (15.14)).By analogy,we can write the 32ul0z2 equationsfor y and w.' o-v

p '

At2

_.,.

Similarexpressions hold for the other faces;hence,we hnd for the total force in the direction of the x-axis the expression do,. , do., T

: ^:1t p"v2u * *u1 C:.:;.

oo.,d"

Thesestresses act on a face having anarea(dy d;) and affectthe volume (dr dy d:); hence,we get for the net lorcesper unit volume in the directionsof the rc_.y_. and :-axes the respectivevalues 9%o 0x'

dz

: ^:i. .[,i].(,11 .':;) . (,';;: .::;)l

Up to this point, we havebeendiscussinsa medium in staticequilibrium.We shallno*..-ou.lthis restric_ tron and considerwhat happenswhen the stresses are not in equilibrium.In fig.2.2,we now assumethat the stresses on the rear faceof the elementof volumeare as shown in the diagram but that the stresseson the liont face are, respectively,

'.o', or.

, dtr,-_

: ^:: * ,*qu";. -'r] . *or";.

2.2.1 St'uhr v:uveequation

O o" d x , ox

(2.24)

dz

wherep is the density(assumedto be constant).Simi_ lar equationscan be written for the motion alons the y- and z-axes. Equation (2.24) relatesthe displacementsto the stresses. We can obtain an equationinvolvingonly dis_ placements by using Hooke's law to replace the stresses with strainsand then expressing the strainsin terms of the displacements, using eqs. (2.g), (2.9), ( 2 . 1 2 )(. 2 . 1 3 )a, n d ( 2 . 1 4 )T. h u s .

2.2Wave equations

o,. *

do,,

62w

A

A

: ( t r + *)o-+rV)r'.

:

;}A

p - " : ( r +p ) ' - * p V r r y .

(2.261 (2.271

dt"

To obtain the waveequation,we differentiatethese three equationswith respectto x, y, and z, respectively,and add the resultstogether.This gives i ) '( a u d r du lA2L a,A a,A\ + - \l : ( \ + P ) l + | 'P ^ , 1 " -+ 3: / dy ilr\dx \dr2 6y, 0:2|

+ -*v,F! + 9l + ar\ \ax

thatis,

,Yr,:: (r + 2p)V'A

oy

ozl'

THEORY OF SEISMIC WAVES

o'o, *19;u4: l

|

where

(2.28)

cr' : (\ + 2St")lp. ) By subtractingthe derivativeof eq. (2.26)with respectto z from the derivativeof eq.(2.27)with respect to y, we get

in two ways in general: (a) include in the wave equation terms that representthe forces generatingthe wavesor (b) surroundthe point of observationP by a closed surface 9 and regard the effect at P as being given by a volume integral throughout the interior of I to take into accountsourcesinsideg plus a surface integral over I to give the effectof sourcesoutsideI (see$2.2.4).To apply the first method, we note that eqs. (2.25),(2.26), and (2.27) are equivalentto Newton's secondlaw, and thesethree equationsare combined in eq. (2.31).Therefore,a sourcecan be taken into account by adding to the right-hand side of eq. (2.31)the term pF, where F is the externalnonelastic force per unit mass(often calledbodylbrce) that gives riseto the wavemotion. Thus,eq. (2.31)becomes

,#(:;-9:.,'(X-y), that is,

L a'e.: V2o. 9' ur' where

rr-(

p ; ; : (\ + p) VA + p V'( + pF.

(2.2e)

9'= p,lp.

Taking the divergenceand curl of eq.(2.34)and using e q .( 1 5 . 1 4a) n d p r o b l e m1 5 . 7g i v e s

^:orvrA+v.F, By subtractingappropriatederivatives,we obtain srmilar resultsfor 0,,and 0-. Equations(2.28) and (2.29) are differentexamplesof the waveequation,which we can write in the generalform I drU v2 af

= vr,lr.

(2'30)

where V is a constant. 2.2.2 Vectorwaveequation The waveequationcan also be obtained using vector methods. Equations (2.25), (2.26), and (2.21) are equivalentto the vectorvtaveequation: ^)y

(2.31)

dt"

If we take the divergenceof eq. (2.31) and use eqs. (2.12)and (15.14)we get eq. (2.28).Takingthe curl of eq. (2.31)and usingeq. (2.11)and problem 15.7gives the vector waveequationfor S-waves(see$2.4.1),

(2.32) which is equivalentto the three scalarequations, I'

A " 2"A,

p' at'

: v20,

1 i: x , y . _ - ) .

(2.33)

2.2.3 Waveequationincludingsourceterm The foregoing discussionof the wave equation has made no mention of the sourcesof the waves,and in fact, the equations discussedare only valid in a source-freeresion. Sourcescan be taken into account

(2.3s)

ar@ : B r v r o + v x F l 2 . at2

(2.36)

Theseequationsare difficult to solveas they stand. The solution is greatly simplifiedby using the Helmholtz separationmethod, which involves expressing both ( and F in terms of new scalarand vector functions.Thus.we write

(:v0+Vx1,

0-( p= : (I + p) VA + p"V'(.

(2.34)

oI-

F:VY+Vxq

V'x:0, V'O:0.

(2.31) (2.38)

Then, usingproblem 15.7,we obtain

.\:V'(=V'0. ) 20:rVx(=-V,x, l V.F=V:Y. I

VxF:-V2O.

(2.3e)

)

Substitutingin eqs.(2.35)and (2.36),we get

V , ( c , ' V , d + Y - #: r) ,

v ' ( e ' v ' x + o - #:)t Wheneverd, X, I or O contain powersof x, y, and z higher than the first, theseequationscan only be satisfied for all valuesof x, 1 and z if the expressions inside the parenthesesare identically zero at all points. Becausea linear function ofx, y, and z correspondsto a uniform translation and/or rotation of the medium, we can ignore this possibility and write (Savarensky, 1975:199)

a'0 : af

ctrVrd * Y,

(2.40)

WAVE EQUATIONS

a1 :

41

B' V'1+ o.

dt"

(2.41)

tainedidenticalresultsfor S-waves.Thus,the previous equationsrefer to either P- or S-waves.

2.2.4 Kirchhoff's theorem

2.2.5 Plane-wave solutions

Method (b) referredto in 92.2.3is in fact an extension of method (a). It uses the superposition concept (which followsfrom the linearity expressed in Hooke's law). We regard the wave motion at a point P as the superpositionof the wavesfrom all sourcesR within somevolumeV surroundingP plus the wavesradiated by points Q on the surfaceI surroundingthe volume (which takes into account any disturbances from sourcesoutside the volume). We adjust the times for thesesourcesso that their effectsall arrive at P at the sameinstant ln. We take Y(x, y, z, lo) in eq. (2.40) as the sourcedensity (body force/unit volume) inside g and specify6@, y, z, /o) lor eachpoint on the surface lf , t o, and to being the retardedtimes (tt)- rll), where Z is the velocity,and r is the distancebetweenP and the sourcesR or Q, that is, rlV is the time for the wave to travel from R or Q to P Thus, we specifythe wave n.rotionat different points at different times such that the wavesfrom all points arrive at P at the same instirnt /0. The result, known as Kirchhofl''stheorem(or fbrmula)(Ewing,Jardetzky, and Press,1957:l6), is

Let us consider first the case where r! is a function only ofx and /, so that eq. (2.30)reducesto

lndr,(.r', y,:.t,,): () r" IJI

I d,u_ a,{, ln Af

0x2

(2.4s)

Any function of (x - Vt),

* :jk

- v0,

(2.46)

is a solution of eq. (2.45)(seeproblem 2.5a)provided that r! and its first two derivativesare finite and continuous. This solution (known as d'Alembert's solution) furnishesan infinite number of particular solutions (for example, ek(\ vt\,sin (x - Vt), (r - V03, wherewe must excludepoints at which thesefunctrons and their first three derivativesceaseto exist or are discontinuous).The answerto a specificproblemconsistsof selectingthe appropriatecombinationof solutions that also satisfiesthe boundary conditions for the problem. A body waveis definedas a "disturbance"that travels through the medium and carries energy (Logan, 1987:230).In our notation,the disturbanceg is a volume changewhen rf : A and a rotation when I : 0,.Obviously,the disturbancein eq. (2.46)is traveling along the x-axis. We shall now show that it travels with a speedequalto the quantity Z In fig. 2.5athe certainpart ofthe wavehas reached point P,,at time to. If the coordinateof P,, is r,,, then the value of g at P,, is t,, : .f(xu- Vt(,).lf this same portion of the wavereachesP, at time /0 + Al, then we havefor the value of rf at P,

- to,a!"'t . JI{(;Xi;)t:il

. (i)[:l]].'

t) 4)\

rhere I is the outward-drawnunit normal, and the .quarebracketsdenotefunctionsevaluatedat point Q .,: tinle to: tu - rl V, [$l is often referredto as a re'.tnladpotential.lf we assumethat eachsourceemits ':.hericalwaves($2.2.6)of the form (l/r)e j,il(r/r,)(see -;s (2.55)and (2.56)),eq. (2.42)becomes(Savaren. , . r .1 9 7 52: 3 4 )

r;dr,(.r, !, :, t,,) :

(;)JII O*

.

,(243) lJ {,[:l] rorj,,!]av

(l/r)er.(lo-l/'r

:

But, becausethis is the sameportion of the wavethat was at Puat time lo, we must haverlr,,: r.!,,that is, xn - Vto: xn * Ax - V(to+ Lt). Thus, the quantity Izis equal to AxlAt and is therefore the speedwith which the disturbancetravels.The reciprocalof velocity,l/Z is calledslowness. A function of (rt+ Vt), for example,* : g(x + Vt), is alsoa solutionof eq. (2.45).It denotesa wavetraveling in the negativex-direction. The generalsolution o f e q .( 2 . 4 5 ) , - Vt) + g(x + Vt), S : l("x

':re in the integrand, ,)] :

V, :,/[ru * A,x V(to+ Lt)|.

€ejot0,

(:

(llr)e

:-,tv,

(2.44) : .'.'is the angularfrequency(seeg2.l.l). J:crrusewe startedfrom eq. (2.40),eqs.(2.42)Io - -.-' rrrevalid for P-waves(see$2.4.1).However,we : iust as well havestartedfrom eq. (2.41)and ob-

(2.41\

representstwo wavestravelingalong the x-axis in oppositedirectionswith velocityZ The quantity r + Vt (or a constanttimes theseexpressions;see$2.1.1)is the phase.The surfaceson which the wave motion is the same,that is, the surlaces on which the phase has the same value, are known as wavefronts.In the casewe are considering, '! is independentof y and;, and so the disturbanceis

THEORY OF SEISMIC WAVES

A1

the same everlrvhere on a plane perpendicular to the x-axis; the wavefront is therefore plane and the wave Note that the wave is travelingin the is a plane u.'cve. direction normal to the wavefront:this holds for all wavesin isotropicmedia.A line denotingthe direction of travel of the waveenergyis called a raypath. Plane waves are easier to visualize and to treat mathematicallythan more complicatedwaves.Moreover, curved wavefronts can be approximated as closelyas desiredby a superpositionof plane waves. It is convenientat times to havean expressionfor a plane wave traveling along a straight line inclined at an angle to each of the axes.Assumethat the wave is traveling along the x'-axis, which has direction cosines({, m, n) relativeto the -x-,y-, and :-axes (fig. 2.6).Then,at a point P on the,t'-axisat a distance-r' from the origin, we have

Fig. 2.5

lllustrating the velocity ofa wave

x':(xlmy*nz, wherethe coordinatesof P are (-t, l :). Then, q :.f((v t my * nz - v0 +g((xlmylnz

+ vt).

(2.48) (, : coso, m: cos02

2.2.6 Spherical-u,avesolut ions

a:coS0r

In additionto planewaves,we shallhaveoccasionto use another important type of wave, the spherical x,aye,where the wavefrontsare a seriesof concentric sphericalsurfaces.We expresseq. (2.30)in spherical coordinates(r e,6;, where0 is the colatitude,and $ the longitude(seeproblem2.6b). 1d2S: ,. dt2

(""'il) J, .,i, 'i[iI (":Y). a.fl. +

sinrOddrl

Fig. 1.6

W a v ed i r e c t i o n n o t a l o n g a n a x t s

Fig. 2.7

Relation between spherical and plane waves.

e.4s)

We consideronly the specialcasewhen the wavemotion is independentof 0 and $, henceis a function only of r and l. Then we get the simplifiedequation

'"':Y:,:i," (":Y)

(2.50)

A solutionof the foregoingequationis \ 1: ( t l r ) J Q - V t )

( 2 . 5)1

(seeeq. (2.46\).Obviously, ,11:(llr)g(r+Vt) is alsoa solutionand the generalsolutionofeq' (2.50) (seeproblem2.5c)is '$ : (llrV(r - Vi) + (1lr)g(r+ Vt),

(2.52)

in which the first term representsa wave expanding outward from a central point and the secondterm a wavecollapsingtoward the central point. When r and I are fixed, (r - Vtl is constant and hencerl is constant.Thus, at the instant l, the wave

has the samevalue at all points on the sphericalsurface of radius r The sphericalsurfacesare therefore wavefrontsand the radii are rays.Obviously,the rays

GENERAL ASPECTSOF WAVES

l a

+J

are normal to the wavefrontsas in the caseof plane rvaves.(This is not always the case in anisotropic media.) As the wave progressoutward from the center,the radius increasesby the amount Z during each unit of time. Eventually,the radius becomesvery large and the portion ofthe wavefrontnear any particular point rvifl be approximatelyplane.If we considerfig. 2.7, we seethat the error that we introduce when we replace the sphericalwavefront PQR by the plane wavefront P'QR' is due to the divergencebetweenthe true direction of propagationgiven by the direction of the radius and the assumeddirection normal to the plane. By taking OQ very large or PR very small (or both), we can make the error as small as desired.Because planewavesare easyto visualizeand also the simplest to handle mathematically,we generallyassumethat conditions are such that the plane-waveassumption is valid.

23 General aspects ofwaves 2.3.I Harmonicy'uves In $2.2.5and 2.2.6,we discussed the geometricalaspectsofwaves, that is, how they dependon the space We now considerthe time dependence coordinates. of wavemotion. The simplesttime variation that a wavecan havels lrurmonit(sinusoidal), equivalentto simpleharmonic motion.ln general,wavesaremorecomplexthan this, but the methodsof Fourieranalysis({i 15.2)allow us to representalmostany complexwaveas a superposition of harmonic waves.Harmonic waves,becauseof their simplicity,can be regardedas the time equivalent of plane wavesin space. Adding nl2 to the phasein eq. (2.3)changescosrne to sine. so harmonic wavescan be written in either form. Some of the commonest forms are the following: t : I cos [(2rrl\Xx - Vt)] : A cos <(.r - Vt) : I cos(rr - t'rt) : I cos2tr(xl),,* vt) : A cos2r(xllt - tlT"1 : I cosa(xlV - t), { : I cos x(d,x+ my } nz - Vt) : I sin [rc({x + my I nz - Vt) + nl2],

(2.s3)

I es+t

q : @lr) cosrc(r- V0 + (B/r)cosr(r + Vt). (2.55) Equation(2.53)represents a plane wave travelingin the t,r-direction, eq. (2.54) a plane wave moving along a straightline with directioncosines((, m, n), and eq. (2.55) a sphericalwave expandingfrom and collapsingtowardthe origin. Equation(15.45)enablesus to combinethe cosine and sine expressionsfor a harmonic wave;thus, if we wnte r! :

lgroltrr+,n|+nzrr

4 -

Aejdtttv o,

(2.56)

we can get either the cosineor sineform by taking the real or imaginary part of r.f. The quantities((, m, r) in eq. (2.54) representthe direction cosinesof the ray. In problem 15.9a, we show that (2 + m2 * n2: l. Although ordinarily each of the cosineshas a maximum value of unity, satisfying the waveequationrequiresonly that the sum of the squaresbe unity. If we admit pure imaginarynum"direction cosines"can be greater bers, some of the than ,unity.Let us take in fig. 2.6, 0, : j0. 0, : :n. 0, : i n - j0. 0 beingreal and positive:then d:cosj0:cosh0, m:0, n : cos (:rr - je) : sinj0 : j sinh 0, () i m1 + n2 : cosh20 - sinh: 0 : l, *

:

Ag

(o:/t') sinh0ej(ol(.!/t]coshe rl.

(2.57)

This representsa plane wave travelingparallel to the x-axis with velocity Vlcosh 0 < V and amplitude Ae @rnsinhH. If we had taken0, : _j0, this would give a wavetravelingin the x-direction with amplitude decreasingupward in the negativez-direction.Because the amplitude decreasesexponentiallywith z, these wavesare calledevanescentwaves.We shall refer asain t o t h e s ew a v e si n N 2 . 7 . 5 . In explorationseismology,the rangeof frequencies recorded with appreciableenergy is generally from about 2 to 120 Hz, and the dominant frequencieslie in a narrower range from 15 to 50 Hz for reflection work and from 5 to 20 Hz for refraction work. Becausevelocitiesgenerallyrange from 1.6 to 6.5 km/s, dominant wavelengthsrangefrom about 30 to 400 m for reflection work and from 80 to 1300 m for refraction. 2.3.2 Wuvainterf'crent'e If two wavesare superimposed, they interferewith eachother; the interferenceisconstrut'liue if they tend to add and destructiveif they tend to cancel.When the two wavesare harmonic and have the samefrequen(her.rce ciesand wavelengths the samevelocities), their amplitudessometimesadd togetherand sometimes cancel(at leastpartially); thus, they form a new wave of the samefrequencyand wavelengthwith different amplitudeand phase-shifled. When severalharmonrc waveswith different amplitudes,frequencies,and/or wavelengthsare addedtogether,tht resultsare usually very complex; constructiveinterferenceoccurs when the phasesare nearlythe same(e.g.,$2.7.4and 13.3). otherwise destructiveinterferenceresults in at least someattenuation.lf the wavesare not harmonic, they can be resolvedby Fourier analysis(99.1and 15.2) into harmonic componentsthat can then be addedto determinethe nature of the interference. If we add two harmonic wavesof equal amplitudes (A) and velocities but slightly different frequencies (seeproblem2.1),Ihesum is B cos (rox - ont),where B : 2A cos (Arcx - Ato/), rcoand
THEORY OF SEISMIC WAVES

44 the values for the two waves.We regard -Bas the vartable amplitude of the resultant wave; at a fixed point, -r2A at the rate of L'al2n times per -Bvaries between second,that is, slowly in comparison with the wave frequencyaro.This phenomenonis calledbeating. 2.3.3 Huygens'principle The solutionsof the waveequationgivenby eqs.(2.47) and (2.52)are restrictedto plane and sphericalwaves. On the other hand. the Kirchhoff formula is valid for any type of body wave.As expressedin eq. (2'42) (assumingno sourcesinside9), it statesthat the effectat a point P is the sum of effectsthat took place earlier at all points on a surfaceI enclosing4 allowancebeing made for the time for theseeffectsto travel from 9 to P. Thus, each point on I behavesas though it were a new wave source. To obtain Huygens'principle,we take I coincident with that portion of the wavefrontthat we wish to take into accountin finding the effectat ^8 and then complete the closed surfaceby passingit through space where the effect has not yet arrived so that $ rs zero over this part. Huygens' principle is important in understanding wave travel and is often useful in drawing successive positionsof wavefronts.Huygens'principlestatesthat every point on a wavefrontcan be regardedas a new sourceof waves.The physicalrationalebehind this is that each particle located on a wavefront has moved from its equilibrium position in approximately the same manner. that the elastic forces on neighboring particlesare therebychanged,and that the resultant of the changesin force becauseof the motion of all the points on the wavefront thus begins to produce the motion that forms the next wavefront.In this way, Huygens' principle helps explain how information about seismicdisturbancesis communicatedin the earth. Specifically,given the location of a wavefront at a certain instant, future positionsof the wavefront can be found by consideringeach point on the first wavefrontas a new wavesource.In fig. 2.8, lB is the wavefrontat the time lu and we wish to find the wavefront at a later time /,, + Ar. During the interval At, the wave will advancea distance V L't, V being the velocity (which may vary from point to point)' We select points on the wavefront, Pp P2,P,' and so on, from which we draw arcs of tadri V Al. Provided we selectenoughpoints,the envelopeof the arcs (l'B') will defineas accuratelyas we wish the position of the

wavefront at time t0 + Al. Except on the envelope,the elemental waves interfere destructively with each other so that their effectscancel.When lB is plane, and V constant,we needdraw only two arcs and the straight-linetangent to the two arcs definesthe new wavefront. Note that Huygens' principle gives only phaseinformation; it doesnot give amplitudes' 2.4 Body waves 2.4.1 P-wavesand S-waves Up to this point, our discussionof wave motion has beenbasedupon eq. (2.30).The quantity r! has not been defined;we have merely inferred that it is some disturbancethat is propagatedfrom one point to another with speed Z However,in a homogeneousisotropic medium,eqs.(2.28)and(2.29)must be satisfied. We can identity the functionsA and 0,with rf and conclude that two types of wavescan be propagatedin a homogeneousisotropicmedium, one corresponding to changesin the dilatation A and the other to one or more componentsof the rotationgivenin eq' (2.1| ). The first type is variously known as a dilatational, longitudinul, irrotatknul, t'ompressional,or P-wave, the latter namebeinggivenbecausethis type is usually the first (primary)eventon an earthquakerecording. The secondtype is referredto as the 'vheur,transverse, it is usuallythe second rotutional,or S-nave(because major eventobservedon earthquakerecords).The Pwavehas the velocityo in eq. (2.28)and the S-wave the velocityB in eq. (2.29),that is,

":(^*tu\'':(\'''

(2.58)

B = \(p*/ ) " ,

lr sqr

\

p

/

where M is the P-l'rive ntotlulus.Becausethe elastic constantsare alwayspositive,ct is alwaysgreaterthan B. Using eq. (2.20),we seethat

(2.60

: : ( ^ f r * ) ' " : ( 0 r ' - - ' " ) ")

(seefig. 2.9). As o decreasesfrom 0.5 to-0, P/ct increasesfrom 0 to itr maximum value, l/r/2; thus, the velocity of the S-waverangesfrom 0 to 70% of the velocity of the P-wave. For fluids, p is zero and henceB is also zero; therefore S-wavesdo not propagatein fluids. Using eq. (2.21),we seethat for a fluid, tr' : k, hence, o : lklp)'''

Fig. 2.8

Using Huygens'principle to locate new wavefronts.

\Pt

(2.6r)

The seismicvelocity in actual rocks depends on many factors, including porosity, lithology' cementation, depth, age, pressureregime, interstitial fluids, etc.. which are discussedin chap' 5. The velocity of water-saturatedsedimentary rocks is generally in the 1.5to 6.5 km/s range,increasingwith loss of porosity,

BODY WAVES

4\

a.lp 0.5 5.0

0.4

3.0 2.0

n a

1.5 1.0

v

o.2

0.4

0.6

9ta Fig.2.9

B/cras a function of Poisson'sratio o and k/p.

mum compression (maximum A) at the wavefront D,' particle velocity is zero at each of thesepoints. We can visualizethe plane-wavesituation by imagining that the radiusin fig. 2.10has becomevery large so that the wavefrontsare practically plane surfaces. The displacementswill be everywhereperpendicular to theseplanesso that there will no longer be convergenceor divergenceof the particlesof the medium as they move back and forth parallel to the direction of propagationof the wave.Sucha displacementis longitudinal, which explains why P-wavesare sometimes called longitudinal waves.P-wavesare the dominant wavesinvolvedin seismicexploration.A planeP-wave is illustratedin fig. 2.11a. To determine the motion of a medium during the passageof an S-wave,we return to eq.(2.29)and consider the casewherea rotation 0,, which is a function of x and t only, is being propagatedalong the x-axrs. We have

L a'q,: a'9,,

g' AP

i)x2

Because 0v 3u 0v z H' _0: x O y d x from eq. (2.l0), we seethat the wave motion consists

Fig. 2.10

Displacements for a spherical P-wave.

cementation,depth, and age. (Velocity versusdepth relationsfor three situationsare shown in fig. 11.28.) The velocity of P-wavesin water is approximately1.5 km/s. P-wave velocity is lowered, often markedly, *hen a gasreplaceswater as the interstitialfluid. This is especiallyimportant in the near-surface,generally above the water table, where a low-velocity layer tLVL, also called the weatheredlayer) typically has a relocity in the 0.4 to 0.8 km/s range,occasionallyas lorv as 150m/s, sometimesas high as 1.2 km/s. Let us investigatethe nature of the motion of the medium correspondingto the two types of wave motron. Consider a sphericalP-wave of the type given bv eq. (2.51).Figure 2.10showswavefrontsdrawn ar quarter-wavelength intervals, r being chosen so that x I7 is a multiple of d2. The arrows representthe di:ection of motion of the medium at the wavefront. The medium is undergoingmaximum compressionat B fthat is, the dilatation A is a minimum) and mini-

W W W W W (a)

(b)

Fig. 2.11 Motion during passageof plane body waves.(After Earth,2d ed., by F. Pressand B. Siever,p.424. Copyright 1974 by W H. Freeman and Company; reprinted with permission.) (a) P-wave; (b) S-wave.

THEORY OF SEISMIC WAVES

46 solely of a displacementv of the medium in the ydirection, v being a function of both x and t, Because u is independentof y and z, the motion is the same everywherein a plane perpendicular to the x-axis; thus, the case we are discussingis that of a plane S-wavetravelingalong the x-axis (fig. 2.I 1b). We can visualizethe foregoing relations using fig. 2.12.When the wavearrivesat -Eit causesthe medium in the vicinity of P to rotateabout the axis Z'2" (par' allel to the;-axis) through an anglee. Becausewe are dealingwith infinitesimalstrains,s must be infinitesimal and we can ignore the curvatureof the displacementsand considerthat points suchas P' and P" are displacedparallel to the y-axis to the points Q' and Q". Thus, as the wavetravelsalong the x-axis,the medium is displaced transverselyto the direction of propagation,hencethe name transversewave.Moreover,becausethe rotation variesfrom point to point at any given instant, the medium is subjectedto varying shearing stressesas the wave moves along; this accounts for the name shearwave. Becausewe might havechosento illustrate0,,in fig. 2.12insteadof 0-, it is clear that shearwaveshavetwo degreesof freedom,unlike P-waves,which have only one - along the radial direction. In practice,S-wave motion is usually resolvedinto componentsparallel and perpendicularto the surfaceof the ground, these When are known respectivelyas Sf1- and SV-w'aves. the wave is traveling neither horizontally nor vertically, the motion is resolvedinto a horizontal [Sl{ component and a component in the vertical plane through the direction of propagation. Henceforth' S-wavewill mean SZ-waveunlessotherwisenoted. Becausethe two degreesof freedomof S-wavesare independent,we can havean S-wavethat involvesmotion in only oneplane,for example,Sll or SZmotion; such a wave is said to be plane-polarized.We can also havea wavein which the SH and S Z motion havethe same frequencyand a fixed phasedifference;such a wave is elliptically polarized. Polarization of S-waves is a factor in their explorationuse (see$13.1).Note that we cannot havea sphericallysymmetricalS-wave (analogous to the P-wave illustrated in fig. 2.10). S-waveamplitude must vary with direction. In the caseof a medium that is not homogeneous and isotropic, it may not be possibleto resolvewave motion into separateP- and S-waves.However,inhomogeneitiesand anisotropy in the earth are small enough that the assumption of separate P- and S-wavesis valid for practical purposes.

Fis..2.12 Rotation of medium during passageof an S-wave

and 1(x, y, z, t), which are solutionsof the P- and Swaveequations,respectively, and which are so chosen that u, v, w (or i, v, *) can be found by differentiation. A simpleexampleof suchfunctionsis the following:

V0:( :(ui+ui+ ruk),

X:0, so that

(2.62)

a0

,' r : q 0

0y'

6x'

This procedureis valid only if it correspondswith A being a solution of the P-waveequation.Because( is a solutionand A : V' ( : V'0, A is alsoa solution (becausederivativesof a solution are also solutions). SettingX : 0 is equivalentto sayingthat S-wavesdo not existand this choiceof potential functionsis suitable for discussingwavemotion in fluids. For wavemotion in three-dimensionalsolids,d and 1 can be definedso that

r : v ( o +*) - o'*u

(2.63)

This ensuresthat A and 0 are solutionsofthe P- and S-waveequations,respectively(seeproblem 2.9a). For two-dimensionalwavemotion in the xz-plane, g and 1 can be definedby

(:V0+Vx1,

x:-xi,

2.4.2Displacementand velocitypotentials Solutionsof the waveequationssuch as those in eqs. (2.48) and (2.52) furnish expressionsfor A and e. Howevet often we needto know the displacementsa, v. "lr'.or the velocities i : duldt, i, w, and referenceto eqs. (2.8) to (2.12) shows that these are not easily found given only values of A and 0,. This difficulty is often resolvedby using potential functions6@, y, z, t)

d0 0z

u:

ad

^ t dx

ax ^'t dz

w

=99_ax dz

dx

,,*, ]

It is easy to show that eqs. ( 2 . 1 2 )a f i ( 2 . 1 1 )c a n b e expressedas

A:V.(:V'd, I

2@: V x (: V'Xi, )

(2.6s)

BODY WAVES

A 1

so that A and @ are again solutions of the p- and S-waveequations(seeproblem 2.9b). Becausethe wave equationsare still valid if both sidesare differentiatedwith respectto time /, it follows that velocity potentialswill be obtained in each of the preceding.casesif u, v, w, and ( are replaced with n, n, w, and i.

2.4.3 Waveequation influid media In fluids, only P-wavesare propagatedand we are generally interestedin pressurevariationsrather than displacementsor velocities,as in solid media. Equation (2.62) can be expressedin terms of pressureg. We redefined in the form

VO:ri+ij +wk, fi:#,

etc.(2.66)

In eq. (2.24),we set o.., : or,, : o," : 0,

o,._,: 0r, : o_:

-g;

hence,using eq. (2.24),we get d:u Ag p. . : - ; = acceleration along the x-axis, (2.67) dx dl'

and similarly for the y- and z-axes.Adding the three componentsof accelerationgives .1A -V9' P V"^Y : dI

Ignoring the additiveconstantdue to hydrostaticpressure(becausewe are interestedonly in pressurevariatrons).

^ a ?J': -p

d :JopQ. d

(2.68)

if we consideronly harmonic wavesof the form -

6

Ae:,",--'t

:

lgt
(seeeq. (2.56)).Thus, both g and I sarisfythe p-wave equationas in eq. (2.28),the velocity reducingto a : (klp)t/2in fluids(seeeq. (2.61)). In the caseof a gas, k dependsupon the way the uasis compressed,isothermallyor adiabatically(that 15.with no transferof heat during the wave passage). For sound wavesin air, the compressionis essentially .rdiabaticso that the pressureand volume obey the iaw, .')I'" : constant,

^l = ('rl(', - 1.4 for air, (2.69)

'* herec.,and c' are the specificheatsat constantpres.ure and volume, respectively(Hsieh, 1975: 54-5; Lapedes,1978).Equation(2.18)can be written LeP

, A

:

-

Vdg

L1ft1r d1/" ,.hereA9 is thepressure changecreatedby thewave.

By using the adiabatic law, logarithmic differentiation of eq. (2.69) gives k = "y0, and hence o:

lyTlp)v2

(2.70)

2.4.4 Boundary conditions When a wavearrives at a surfaceseparatingtwo media having different elasticproperties,it gives rise to reflectedand refractedwavesas describedin 93.1.1.The relationshipsbetweenthe various wavescan be found from the relationsbetweenthe stressesand displacementson the two sidesof the interface.At the boundary between two media, the stressesand displacements must be continuous. Two neighboringpoints R and S, which lie on oppositesidesof the boundaryas shownin fig.2.13will in generalhavedifferent valuesof normal stress.This differenceresults in a net force that acceleratesthe layer between them. However, if we choose points closer and closer together,the stressvaluesmust approach each other and in the limit when the two points coincideon the boundary,the two stresses must be equal. If this were not so, the infinitesimallythin layer at the boundary would be actedupon by a finite force and hencehave an accelerationthat would approach infinity as the two points approacheachother. Becausethe same reasoningapplies to a tangential stress,we seethat the normal and tangentialcomponents of stressmust be continuous (cannot change abruptly) at the boundary. The normal and tangentialcomponentsof displacement must also be continuous.If the normal displacement were not continuous,one medium would either separatefrom the other,leavinga vacuumin between, or elsewould penetrateinto the other so that the two media would occupy the samespace.If the tangential displacementwere not continuous, the two media would move differently on opposite sides of the boundary and one wtruld slide over the other. Such relative motion is assurnedto be impossibleand so displacementmust be continuous. When one or both of the solid media are replaced by a fluid or a vacuum, the boundary conditions are reducedin number(seeproblem2.10). \ 2.4.5 Wavesfroma sphericalsource The potentialfunction6: Qr)flt - rlI)is a solution to the waveequationwhen thereis sphericalsymmetry (seeeq. (2.51));hence,the radial displacementrz(ar) is

u(r, t):'# : -(lr)t( - ;)

.0;,v(-;)l

(2.11)

(using eq. (2.62) with the x-axis in the radial direction). For harmonic waves,the two terms have equal importance at a distancer : |tl2rr, but the first term decaysrapidly in importanceat greaterdistances.The

THEORY OF SEISMIC WAVES

48

^l o" ^,1o,,

"'f o,,

.tl Fig.2.14 Radial displacementinvolvesshapedistortion. When the radial displacement decreaseswith distance from source S, sectot ABDC becomes thinner and approaches a rectangular plate.

{o"

of normalstress. Fig.2.13Continuity

second term is the far-field ffict, wheteas the nearfietd effectdependson both terms. This distinction is important when calculating a far-field waveshape from near-field recordings.Note that radial motion involves shape distortion (fig. 2.14) and therefore shearstrain. Equation (2.11) can be usedto derivethe wavemotion createdby symmetricaldisplacementof the medium outward from a point source.When the wave is createdby very high pressures,as in an explosion of dynamite, the wave equation is not valid near the source becausethe medium does not obey Hooke's law there; this difficulty is usually resolvedby surrounding the sourceby a sphericalsurfaceof radius r,,such that the waveequationis valid for r > ro' then specifyingthe displacementor pressureon this surface due to the source. Let us consider the case where the displacement u(r, t) is to be found, given the displacement uo(t) of the surfacer: rn.We let 6 : t (r ro)/I/and write

Note that the lower limit of the integral means that t : 0 is the instant at which the wavefirst reachesthe surfacero, uo(t)being zero beforethis. To carry the calculationfurther,we must know uo(/). Let us approximatean explosionby the expression uo(t) : ke-'l

: 0

o'dt J'Q):-rnve'"','!i,',uker'. rtrc- e ut). \vk 1s Vlru - a - r)l V We replaceI in this expressionby ( : t - (r and eq. (2.73)becomes rok dd : l' ,-'o'" - ee 's ulr. tl : Ar r(Vlrn - a) Lro

-Y-e-'u'u+!r-"r] r r l

(
-

then

t):#: -l)trr,.I,Wl Q73) u(r, -[J,nt, * ,)rfff uo(tt:

(2.74)

Using thesevaluesand multiplying both sidesof eq' by the integrating factor evtt'o,we get Ql\

d dl

: [.''"/(r)l ,"' *'frr] , , r \ - , J el,' olof r0dl : f(t):

I L -ro Vuo(!)en,t,o,

- r o V e - r ' r ' o u o ( / ) e " ' ' od t ' ( 2 ' 7 5 ) Jo

o'] e .,6)

Then

S0 0: Qtr)l(0, ( > 0 , r = , r l e . l Z ) : 0 I

At r : rn, (: t and u(r,t) : uoQ),whereuo(t)depends on the specificsource:

I ?3:"

( ' , - , n , , -, o . , . ) . _..r,k , r(Vlro - a\ \r,, ,

(2.77) r ))

ro. (2.78)

the latter equationgiving the far-field solution. The fact that eqs. (2.77) ar,d (2.78) are valid only for ( > 0 (seeeq. (2.72))merely meansthat u(r, t) ts zero until t : (r - r^)ll\ that is, until the disturbance reachesthe point. At this instant, ( : 0 and u(n t) : k(rolr); hence the initial displacementis the same as that of the surface r0 except that it is reduced by the factor rJr, that is, u(n t) falls off inversely as the distance(see$2.7.1and eq. (2.109).Moreover,u:0 at / : - and also when (seeeq. (2.77)) V(llro - llr)e ltr'o+ (Vlr - a)e-'t : Q, that is. when t:

t -- t n " + V

V(r-r^) I ln Vlrn-a ror(a-Vlr)

SURFACE WAVES

49

Provided Vlro) a > Vlr, this equationhas a real positive root and u(r, l) will vanish, that is, the displacement must changesign. BecauseVlroislarge in practice and Vlr rapidly becomessmall, in general the unidirectionalpulsein eq. (2.76)givesrise to an oscillatory wave. By using different expressionsfor ao(l)in eq. (2.75) or by specifyingS|,,(t),the pressureat the cavity, we can investigatethe wave motion for various spherically symmetricalsources(seeBlake, 1952;Savarensky, 1975: 243-55). By finding the limit as a in eq. (2.76)goesto zero(seeproblem2.12),wegetthe result lor a unit step,step(r);then the resultsfor other inputs can be found using convolution techniques (see $r 5 . 4 .)l. 2.5 Surface waves 2.5.I General The waveequationsfor P- and S-wavesin terms of the potentialfunctionsof eq. (2.64)are V:S :

(l/ar) Ar{,ldtr,

YtX, : (118' ) l' yrl6t', YtXr:

(llBr) \ryrl6tr,

(p-wave)

componentsexist (S^F1-motion is parallel to the xyplane and so is not involved in the boundary conditions) and adjust their amplitudes to satisfy the boundary conditions. Appropriate potentialsare +

:

Ae

vn't,

Xt,

:

Be

neejKtx

vn),

(2.82) wherez and n must be real positiveconstantsso that the wave decreasesin amplitude away from the surface; V^ is, of course, the velocity of the Rayleigh wave.Substitutingg and 1,, in eqs.(2.79)and (2.80) glves m2 : (.1- V' ^la' ),

n' : (l - Vr-lgr).

(2.83)

Becausem and n are real, li < g < o, so that the velocity of the Rayleighwave is lessthan that of the S-wave. We next apply the boundary conditions.Using the resultsof problem2.1l, we get for : : 0

(2.19)

=trv:d* 2pflt - l'*-'):, I \r,:' dx d:l

a

{Strz-wave.;(2.80)

';_1,, : r : u(,o^:!o=ilr) J

f fZ.S+l

o.,,

(SI/-wave) (2.81)

wherethe S-wavepotential hasbeenreplacedwith the l'unctions1,. and XH correspondingto SZ- and Sl1components.If we considerplane wavestravelingin the directionof the -t-axisin an infinitehomogeneous medium,solutionsof theseequationsare of the form elx(\ ''t, V : o or B. However,othersolutionsarepossiblewhenthe infinitemediumis dividedinto different media.When the.r1-planeseparates two media,solutions of the form e'(:eiK(\ '?)existundercertainconditions. Thesesolutionscorrespondto plane waves travelingparallelto the,rr-axiswith velocity V and amplitude decreasingexponentiallywith distancefrom the ,ry-plane(in a semiinfinitemedium; see$2.5.2to 2.5.4). Such wavesare called sur/at'ewavesbecause theyare "tied" to the surfaceand diminishas they get farther from the surface.

mEe)K(\

Substitutingeq. (2.82)into the foregoingand setting : : 0 gives l(|t + 2p")m,- \ll

+ 2jnp"B: 0

and -2jmA + (n'+ l)B:

0.

We can useeqs.(2.58),(2.59),and (2.83)to write the first result in the form (2P'-

Vil'l + 21n$18: 0.

Eliminating the ratio BIA from the two equations gives l (2 - V]l$)(n2 + 11: 4*n; hence,

2.5.2 Ra"vleigh v,uves The most important surfacewavein explorationseismology is the Ra1'lr,ighv'ave,which is propagated along a free surfaceof a solid. Although a "free" surlbce means contact with a vacuum, the elastic constants and density of air are so low in comparison with valuesfor rocks that the surfaceof the earth is approximatelya free surface.Groundrol/ is the term commonly usedfor Rayleighwaves. We take the free surfaceas the x.y-planewith the :-axis positive downward. The boundary conditions (ti2.4.4)require that ct,": 0 : o,, at z : 0 (seeproblem 2. l0), that is, two conditions must be satisfied, and so we require two parameters that can be adjusted.Therefore,we assumethat both P- and SV-

vi-

89' Vi+ Q4 - l6B'la')$av] + l 6 ( B r l a ,- l ) p u : O .

(2.85)

Becausethe left side of eq. (2.85)is negativefor V^: 0 and positive for V*: +8, a real root must exist between thesetwo values,this root giving the Rayleigh wave velocity Z^. However,we cannot find this root without knowing p/ct. For many rocks, o - t/q, that is, (B/ct), - t/z from eq. (2.60).If we use this value, the three roots of eq. (2.85) are Vt*: 49r,2(l r l/.,F)82. Because V^lg must be lessthan unity, the only permissiblesolution IS

Vi:2(r - l/{3)8.,

or

V*: 0.9t99.

THEORY OF SEISMIC WAVES

50 W e n o w f i n d t h a t \ l a : 0 . 5 3 1l,n : 0 . 8 4 8n, : and BIA: +1.468j;hence, 0 yr:

:

0.393,

6r.

vRt)i, le-0.848K2 ejKtx 0 3 e 3 x ue j K ( r r R ' ) . l.468jAs

Using eq. (2.64),we get for the displacementsat the surface rRl). w : 0.620r
-

lre,,,,^'ejK(f

t's1), t

Xt

:

Ble,r(--ejK(\

Medium (2):

rs1)'

:

Are^.^'ejK(r

l/st),

y, :

fl'gnr*zgi"G vst'),

wherem,, n, are real,positiveconstants,and Z, is the velocity.Substitutingd,, X, in eqs.(2.79)and (2.80), we find that ml : | - 1V,la,)2.

ni : | - (V,lp)'

(i : 1,2), (2.88)

Becausem, and ti, are real. Z. must be less than the smallerof B, and B.. The boundary conditions required that w, : wt, Lt,: tt1,o--,lr: o,.l:, o,--l': o.-1.at: : 0' The results of problem 2.11 show that theseconditionslead to equationsinvolving first and secondderivativesofthe potentials,; being setequal to zero after the differentiation; consequently,all terms will have the factors ejK(\ t'sr)and either r or r:, and thesefactorswill cancel out, so we can ignore them; at the sametime, the exponential term in z will become unity. Moreoveq differentiationwith respectto r and z is equivalentto multiplyingthe potentialsby jrc a\d +miK or +niK, respectively;by eliminatingrc,differentiationbecomes equivalentto multiplying by j and lm,, tn,. The four boundaryconditionsnow give fr,A, - jB,: -m.A. - iB., jA,4-n,B,: jA. - n.8., \,(-l

+ m | ) A , + 2 p , ( m l A ,- i r , B , ) : \, (- 1 + m!)A' + 2P.(m1A'+ jn.B,),

p.,f2jm,A,+(nl+l)B,l : p,.l-2jm,A, + (nl + l)8,1. If we transfer all terms to the left-hand side, we equations($15.1.1); havea set of four homogeneous thesehave a nontrivial solution only if the determinant of the coefficientsvanishes.Settingthe determinant equal to zero and using eqs.(2.58),(2.59),and (2.88),we get the following equation for Vr: V ! l ( p , - p , ) ' - ( p , m ,* p . m , )( p , n ,* P , n , ) . + 4 4 0 t , - p J [ p , ( l - m , n , )- p , ( l - m r n . ) ] + 4 ( p , - F r ) t ( 1 * m , n , )( l * m r n , ): 0 .

(2.8e)

Equation (2.89)was first given by Stoneley(1924). Scholte(1947)studied the propertiesof the equation and found that a solution alwaysexistswhen one of the media is a fluid but, when both media are solids, a solution exists only when B, - B, and the ratios prlprand p,/p, fall within the narrow limits shown in fig. 2.17. Thesewavesare a type of generalizedRayleigh wavesand are usually called Stoneleywaves. Stoneleywavesare often presentin borehole seisfrequenciesare far mic surveys($2.5.5).Stoneley-wave below those used in sonic logging ($5.4.3),but they fall within the frequencyrange of VSPs ($13.4)and Stoneleywavesconstitutean important sourceof coherentnoisein VSP surveys.

Directionof wavepropagation

Surface

f

A Y

l

- L

\<

l0'

20'

( .

)

7) I +

I

U

30'

40' 50t

60'

70'

: ,.lergh waves.(a) Cross-section showing motion on ,:J diminishing with depth for a semiinfinite solid. , :--.rrrtlon of a particle on the surfaceof a semiinfinite .-rl motion of a particle on the surface (from How, Ralleigh-wave motion from a small explosion as -:rred geophones;motion changesfrom retrograde , : . r b o u t4 0 f t ( a f t e r D o b r i n , l 9 5 l ) .

I

fl I

I /

0

80'

e

g0'

e

100'

t (d)

THEORY OF SEISMIC WAVES

52

n2xz--) 0 We must have q, real and positive so that e as z -) +-; therefora, Vt 1 Br. However,qr is unrestricted becausez is finite in the upper layer. Applying the boundary conditions,we have at z : - h , o r , : 0 : p r s , , l ' s o 6 v , l 0 z: 0 f r o m e q ' ( 2 ' 9 ) ' that is,

0.5 0.4 o 0.3

Ae ,r*n- Berr*r'- O.

0.t 0

Fig. 2.16 ratio, o.

0.2

0.8

0.6 0.4 VEIICITY RANO

: cr.,lz,vr: v2,so At z : 0, o,,"11 -lL,\,C, tLfl,(A B):

1,0

Rayleigh-wave velocity, Z^, as a t'unction of Poisson's

and A + B :C . s 2tvh,b : prtl,/lr,tl,, theseequations

By setting6: become

a A- B : 0 , A-B+bC:0, A+B-C:0.

I I )

(2.er)

For thesehomogeneousequationsto havea nontrivial solution.the determinantof the coefficientsmust vani s h ( s e e$ 1 5 . 1 . 1t)h, a t i s ,

l,

l'

-l

-t

0l

,l: u ,

-'l lr +l

o3

o.r

,lrr, Fig. 2.17 Conditions for the existenceof a Stoneleywave; solutions exist within the shaded area. (Alier Scholte' 1947.)

2.5.4Lovewuves Love waves(Love, l9l l) are surfacewavesconsisting of SH-motion parallelto an interface'They exist only when a semiinfinitemedium is overlain by an upper layer of finite thicknessterminating at a free surface' We take the lower interface as the -.ry-planeand the -h; the SHfree surface as the parallel plane z : of component only The y-direction. is in the motion displacementis v, so ( : vi in eq.(2'11),hence,v satrsfies the wave equation and we can dispensewith the potential1" of eq. (2.81)and use v instead. The boundary conditions (see problem 2'10) require that o., : 0 &t z : -h and that o,.- and v be continuousat z : 0; thus, we need three parameters to adjustand so we write the following: M e d i u m( l ) : -h = z < 0. vil), v, : (Aeu*' ",u39 rr*)el<" Medium (2): V, :

Q!-nz*'giK\r

- vLt),

0 S

Z <

+-.

Substituting vr and v2in the wave equdtions gives

r?:r-(vJg)"\ \tr:l-(vLlg,)' . )

(2.e0)

-b : (l - a)l(l + a) :(l :

(grr*r, -

- e ' r r . r ) / ( l* e : ' ' " ' ;

g-rr"i')/(grr.r' + e n'd')

: t a n hr | K f t : - ( p . r l r l p , 1 , ) . But tanh -t is positive for all real valuesof x, also 1, is real and positive; therefore,rlr must be imaginary' = that is, rl, : j(, where ( is real. Becausetanh jx j tan x, we now get pzrlz: p,( tan (rcft.

(2.92)

From eq. (2.90),we have

r e | f i : l - r l i: l + ( ' , sothat Il > B,.Thus, 9,> 4> 9,, and the S-wavevelocity must be higher in the deeper layer than in the surface layer, ( then being in between the two velocities. Becauser : 2nl\ : .14, as the frequency increasesfrom zero, tan r{h increasesand approaches infinity; thus for eq. (2.92) to hold, as the frequency increases,( must approach zero and If must approach its ( B,. Conversely,as K approacheszero, approaches maximum value and trfapproachesBr. Hence,at high frequencies, the Love-wave velocity approaches the velocity ofS-wavesin the surfacelayer,and as the frequency approacheszero, the Love-wave velocity approachesthe S-wavevelocity in the lower layer (Dobrin.1951).

SURFACE WAVES

)J

The expressionfor v, can be written v, : :

(Ae:*L' + Be iKLz)etk(x-vL' jKLz)ejK(xvLn A(ejKLz + ae

on using eq. (2.91).Therefore, v, : :

/(gi*tz

+

e-2jK{re-jK(z)ejK(x-

AlejKAd+h) 1

vLt)

g-jx((z+l)]gjx(x- th- vLt)

: 2Alcos r((z _r h)lst*e tn_vo

(2.93) Az

on taking the real part of the amplitude. We seethat vr vanisheson planeswhere r((z + h) : (r + ll2)r

I

(r integral) (2.94)

(recall that ft is positiveand z is negativein the upper fayer);theseplanesare callednodalplanes(see$2.I . I ). Nodal planesare characteristicof normal-modepropagation ($13.3)and indeed Love waves can be explainedin terms of normal-modepropagation(Grant : l-5). a n d W e s t .1 9 6 5 8 Fig. 2.18

2.5.5 Tubewaves Waves traveling in a fluid-filled borehole or on the walls of a borehole in the direction of the axis (tube waves)are of considerableinterestin velocity surveys in wells ($5.4.2),in vertical seismicprofiling ($13.4), and in sonic logging ($5.4.3). Because they have mainly only I degreeof freedom(along the axis),their amplitude decreasesslowly with distance.Sometimes severalmodesoftube wavesare presentand often the mechanismsof their generationand the natureof their motion are not clear. Tube waveshave the potential of furnishing infiormationabout the elasticproperties and permeabilityof the surroundingformations. Most tube-waveenergy travels axially, but radial motion is also involved in some modes. A pressure geophoneor one hanging freely in the borehole will sensethe maximum tube-waveeffectsin the borehole fluid, whereasa geophoneclamped to the borehole wall will sensemuch smallermotion. The classicaltube wave is merely a P-wavepropagating in the fluid, the borehole wall expandingand contractingas the pressurewavepasses.We assumea homogeneousfluid in a cylindrical boreholepenetratisotropicmedium (fig. 2.18).Using a homogeneous ing I for the pressureand lt for the displacement, Newton'ssecondlaw,net force : massX acceleration, applied to a volume element of the fluid, V : rr2 Az,is

uo;|. (1?^') trr:- -(prr'^') ag 0z

62w AP

F r o m e q .( 2 . 1 8 ) ,

8l : -kL:

-kAYl\ .

Changes involved in passageofa tube wave.

The changein volume A1/ is due to expansionboth along the axis and radially,that is, LY : rr2'! u * (2nru,)a,2, 0z wherea, is the changein the radius of the hole. Thus, we get

: -k (uo'; *'?) QP

(2.e6)

Lamb (1960:$157)derivedthe followingrelationbetweenu, and I for an annulusofinner and outer radii r and R, where c, o, and p are respectivelyYoung's modulus, Poisson'sratio, and the shear modulus for the annulusmaterial:

u, _0 (l + q)(Rl+ l) r

E

]o,'

R2-l

If we let R -r -, we obtain for a cylindrical hole in an infinite medium u,lr :0(l

+ o)lE:012p"

(usingproblem2.2).Substitutionin eq. (2.96)gives

ei lt ; *l \*_:/ - -0^w' dz

and substitution of this result in eq. (2.95)givesthe waveequation:

"n; : e)*,r 'r::(f * 1)' rzrr

White (1965: 153-6; 1983: 139-91) discussestube

(2.9s) ' *atuesin sreaterdetail. (1981)

discusstwo other tubeCheng and Toksoz wavemodes.One is a Stoneleywave($2.5.3)propagating along the boreholewall and dying awayexponentially in the formation surroundingthe borehole;this

THEORY OF SEISMIC WAVES

\4

is the dominant tube-wavemode in VSP work. The other tube-wave mode is pseudo-Rayleigh waves, guidedwaves($13.3)confinedlargelyto the fluid, also dying away exponentially in the surrounding formation. Both wavesare dispersive(52.7.4). Cheng and Toksozcalculateddispersioncurvesfor both modes(fig. 2.19a).The Stoneleywaveis slightly dispersivewith both group and phasevelocitiesclose to 0.9ct.,where o, is the P-wavevelocity in the borewavescanhole fluid (seeeq. (2.61)).Pseudo-Rayleigh not existbelow a minimum frequency(wherethetr velocity equals the S-wavevelocity of the surrounding rock, B,) and their group velocity passesthrough a minimum, which resultsin an Airy phase(see$13.3 and fig. 13.19).Severalmodes may exist (see eq. (13.1)).Pseudo-Rayleigh wavesare not a factor in ordinary seismicwork (fig. 2.19a showsa l0-kHz lowfrequencycutoff), but they are involvedin sonic logging. At higher frequencies,the velocitiesof both Stoneley and pseudo-Rayleighwaves approach the Swave velocity in the medium surrounding the borehole. "synthetic microCheng and Toksoz calculated one of which seismograms"for variouscircumstances, is shown in fig. 2.19b;fig. 2.19cshowsan observed waveform. Hardage(1985)discusses the role of tube wavestn VSP surveys.Figure 2.20a shows progradeelliptical motion in an axial plane.The radial motion is zero at the center of the hole and maximum at the borehole wall, where it is continuous(fig. 2.20b),but it decays rapidly in the surroundingformation. The axial component of motion is relativelyconstantin the fluid but is discontinuousat the boreholewall whereits amplitude decreases by a factor as largeas severalhundred. This explainswhy geophonesshould be clamped to the boreholewall. Tube wavesare reflectedat impedancechanges,just as other acousticwavesare ($3.2).When the borehole area changesfrom a, to a., the refleccross-sectional tion (R) and transmission(7n)coefficientsare (Hard a g e ,1 9 8 5 : 8 6 - 7 ) R-azQ2t

at, al

T:

2:,

(2.98)

a2+ el

(comparewith eqs.(3.14)and (3.15)).At the top of the borehole fluid and the bottom of the hole, R : - I and *1, respectively. Tube wavesare also reflected at a geophonesondeand wherecasingchanges.Figure 2.21 showsseveralreflectedtube waves. Tube wavescan be generatedby almost anything that disturbs the borehole fluid. The most common sourceis a Rayleighwave passingover the top of the borehole;thus, tute wavesare uncommon in maririe VSP surveysand, in land surveys,lowering the bprehole fluid level often lessenstube-wavegeneration. Tube wavesinitially have the same spectrum as the generatingsourceand their spectrumchangesslowly becausethereis little absorptionin the boreholefluid.

t

9 UJ

-- Phose -

Group

ieuoo- z> \ \ Royleigh

o Ld

N J

(r z

Sloneley'

ro

20

(kHz) FREQUENCY

ANISOTROPIC MEDIA

))

I OOIVNGONG TUBEI\IAVE

| ,.r,01,...0 BoR€HoLE

I

lA/\rAf

variation of seismic velocity with the direction in which it is measured or with wave polarization ($13.1.6).The generalelasticitymatrix relating stress o, to strain eo,(the generalizedform of the 6 X 6 matrix in eq. (2.I 5)) can contain at most 2l independent constants becauseof symmetry considerations,but Winterstein(1990: 1084-5)saysthat only l8 of these can be truly independent.The number of independent constants depends on the symmetry of the system (Love, 1944:99). A number of different types of symmetry (symmetry systems)can exist. Classically,eight systemsare defined (Love, loc. cit.; Landau and Lifshitz, 1986; Saada, 1974),but some writers define subsystemsas well; for example,Winterstein(1990:1083 5) lists ll systemsplus subsystemsin discussingcracks.Anisotropy typesare associatedwith the symmetrysystems. At seismicwavelengths,however,the only anisotropy types reported are transverseisotropy (hexagonal symmetry), orthorhombic anisotropy, and monoclinic anisotropy. Transverseisotropy involves elastic properties that are the samein any direction perpendicularto an axis but are different parallel to this axis. Two important types of transverseisotropy are observed;that with a nearly vertical symmetry axis (thin-layer anisotropy) and that with a nearly horizontal axis (azimuthalanisotropy)(Bush and Crampin, 1987).Transverseisotropy is the most important type of anisotropy encountered;it is discussedfurther in 92.6.2. Orthorhombicanisotropyis equivalent to a superposition of thin-layer anisotropy and azimuthal anisotropy. It arises becausea vertical fracture system has

ELASTICROCK MATERIAL PARTICLEMOTION

il1ffifi-l lvvr|/\r

l---

I I H A R D I IF O R M A T I O N

| 630 |620 r6to

6 4 2

IISOFTII FORMATTON

r650 I640

r q

sEcoN0s

to

DOUELE CASING

500 2 r/R (c)

rooo o u F

: l0 Wave motion for a tube wave. (From Hardage, 1985: -! ) (a) Prograde elliptical motion of fluid particles during .,ge of a tube wave (ellipticity is greater than shown here). r',ul and radial displacementsfor hard formation, v : 82 .,:d lc) lor soft formation.v : 74H2.

3 rsoo I I

o

2ooo

2 500

.:quently, tube wavesoften have appreciableenr the signalrangeevenafter considerabletravel.

A

CEMENI

\nisotropic media l. pes of anisotropy ' . ,,nl ls a generalterm denoting variation of a - .. propertydependingon the directionin which - :,rsured.Seismicanisotropy is evidencedby a

Fig.2.21 VSP record showing severaltube waves. Tube wave (1) is generatedat the baseofthe surfacecasing; (2) is generated at the surface by a Rayleigh wave; (3) is a reverberationofwave (2) betweenthe well sondeand the surface;and (4) is a reflection f r o m t h e b o t t o m o f t h e b o r e h o l e .( F r o m H a r d a g e . 1 9 8 5 : 8 8 . )

THEORY OF SEISMIC WAVES

56 been superimposedon a horizontally layeredsystem' VSP data from the Paris Basin have beeninterpreted using an orthorhombic model (Bush and Crampin. 1987;MacBeth, 1990).Layering anisotropyis usually much stronger than fracture anisotropy so that the overalleffectmay be difficult to distinguishfrom thinlayer anisotropy.Monoclinic anisotopy can be produced by superimposingtilted fractureson a layered medium (Schoenbergand Muir, 1989)' Examplesof monoclinic anisotropyhavebeenobservedin the field (Crampin, McGonigle, and Bamford, 1980; Wintersteinand Meadows,1990). The stress-strainrelationshipsrequire 5 independent elasticmoduli for transverseisotropy, 9 for orthorhombic anisotropy,and l3 for monoclinic anisotropy, comparedwith only 2 for the isotropiccase. 2.6.2 Transverseisotropy Taking the z-axisas the axis of symmetry,Love (1944'. 160-l) showedthat for transverseisotropy, Hooke's law reducesto the following: o., : (tr -F 2p )e,., * \,,".,r,* \re-,, (2.9e) o,,,.: \,,e.. + (\ + 2*)er,* I'.e--, o-- : \re., * \r€,,, + (\r + 2pt)",,,

o...: l-L€,,, I or-: p*8,.,, r o-, :

p*e-,,

(2.100)

(Crampin, l98l). For horizontal thin-layer refringence anisotropy, the two wavesare the 4SP-waves(that is, quasi-S-waveshaving displacementparallel to the symmetryaxis) and SR-waves(displacementin radial directions).For azimuthalasymmetry,they are sometimes called qSV- and Sl1-waves. In anisotropicmedia, pure S- and P-wavesmay exist only in certain directions.In transverselyisotropic media, SZ- and P-modesof propagationare coupled (see$2.6.3).Wavefrontsare not in generalorthogonal to the directionsof wave propagatton.Phasevelocity is velocity perpendicular to a surface of constant phase(a wavefront), and group velocity,the velocity with which the energytravels($2.7.4),is in a different direction (see fig. 2.23). The surfaces for SZwavefrontsmay havecusps. Anisotropy is often described by the fractional differencebetweenthe maximum and minimum velocities for a given wave surface,i.e., (V^ - V^^)|Z-^*, sometimesby the ratio of maximum and minimum velocities, V^ulV^-. Uhrig and van Melle (1955)give a table showing anisotropyvaluesof 1.2to 1.4for rocks at the surface at depthsof 2.1 to 2.4 km and 1.1to 1.2for sediments in west and central Texas.Stoep(1966)found average valuesbetween1.00and 1.03for TexasGulf Coast sediments.S6gonzacand Laherr6re (1959) obtained to 1.08to 1.12for valuesfrom 1.00for sandstones limestonesand l.l5 to 1.20for anhydritesfrom the northern Sahara.

)

wherethe five independentconstantsare \ and p., tr'r and p' and p*. Layering and parallel fracturing tend to produce transverseisotropy. A sequenceof isotropic layers (suchas sedimentarybedding)producesthin-layeranisotropy for wavelengthsappreciablylarger than the layer thicknesses(), > 8d where d is layer thickness; seeEbrom et al., 1990).The symmetryaxis is perpendicular to the beddingwith the velocitiesof P- and Swaves that involve motion parallel to the bedding larger than those involving motion perpendicularto the bedding. The velocity parallel to the bedding is greaterbecausethe higher-velocitymemberscarry the first energy,whereasfor wave motion perpendicular to the bedding, each member contributesin proportion to the time takento traverseit. Nonhorizontal fracturing and microcracksproduce azimuthal anisotropy with a symmetry axis perpendicular to the fracturing (fractures often are somewhat parallel and vertical).The velocity of wavesthat involvemotion parallel to the fracturing (S') is larger than that of waveswith motion perpendicularto the fracturing (S,). If the motion is neither parallel nor perpendiculaito the fracturing, an S-wavesplits into i*o *uu.t with orthogonal polarizations(fig. 2'22): one (S ) traveling at the S' velocity,the other (S') at the S, velocity; this is calledshear-wavesplitting ot bi-

isotropicmedia 2.6.3 Wuveequationfor transversely When media are not isotropic, the mathematicsbecome more complex the more anisotropic the medium. However,the case of a transverselyisotropic medium can be treated without great difficulty. We consider wavesin the rz-plane, where the symmetry axis is along the z-axis.Derivativeswith respectto y are zero, but S-wavesmay involve motion in the ydirection.We substituteeqs.(2.99)and (2.100)into eq. (2.24), and using eqs. (2.8) and (2.'7),we get the waveequationsfor transverselyisotropicmedia: 62u '3P

d o .' . + d o , " 0: 0x

a"'l : u [ , ^' + 2 '' t' a du x + ^ .' d : l + arL

. #)] i,[-.(::

.. 02w = (\ + 2pf p*91 + (r, * p ' ) ^dz; ,dx #,+ I A,, 6zv d:v P ^ , : l r t . , t F * . , . ' cJI'

( 2 .l 0 l ) (2.102)

d:-

dX'

31u rJ:N .d2w Par. : (\- + **) u" A, * F"a", * 32rt'

(\-+2P)

dz-

.

(2.l 03)

t

EFFECTS OF THE MEDIUM ON WAVE PROPAGATION

57

thesebecome (V2 - afi(' - o*znz)(A - (oi - o*' + g*')(t : 0, lV' B*2n2lnB (V2-a2rn2-o*zgz)nA + l V ' - ( o l - o * ' * 9 * ) n ' - 9 * z ( z l ( B: 0 . Eliminating A and B givesthe following quadratic equationin Z2: [Vt

- (oi - ct*2 + g'k2)(2- g*'rtln

(v, _ ul(, _

o*z4z)(

: _ V : _ ( o i _ o * , * 9*)n'- B*z(zl( Fig. 2.22 S-wave propagation in a lractured medium with cracks oriented N45"W For an S-wavetraveling parallel to fracturing, the velocity (Sr) is slower for a component involving motion perpendicular to the fracturing than for one involving motion parallel to the fracturing (velocity S,).

Note the dependence in eqs. (2.101)and (2.103)on derivativesof both u and w; P- and SZ-wavesare said to be coupled.The Sf/-wave governed by eq. (2.102) is, however,independentof the other two. We simplify the problem by assuminga plane wave travelingin the xz-planein the direction ofincreasing ,r and decreasing:,the anglebetweenthe raypathand the x-axis being 0. We now usethe potential functions of eq.(2.64)in the form d:ler-t.

1:fsr-i

where ( x - n z ) l V- t ,

g:

(:cos0.

n:sin0.

Then

,:4*lx dx

0z

: (;)u^

- nB)e:-t,

, : u Q - 4 = -(:,\r^ * (81s,-,' 6z

dx

Whenwe substitutetheseinto eqs.(2.101)and (2.103), the following factors appear in every term and hence can be ignored:jall\ (jor),, and ej-(.Equations(2.101) and (2.103)become lpV'( lpV'n lpV'n + [pV(

(L + 2p,)f3- (tr,+ 2p"*)h' lA - (\, + 2p - \, - p*)('n - p"*n3lB: 0, (\, * 2pr)r3 - (\,- + 2p,*)(' n]A + (p* - 2p",)(n2- p,*(3lB : 0.

c,*t : (\, + 2p,*)lp,

ctl : (}", + 2p,,)lp, F*' : l-r*/p,

a2,.n2_ d i 2 ( z ) n

or [V'-

( o i - c t * 2+ 9 * 2 ) ( ' - B * ' n z y

X (V2 -

o'rn' -

o*z(z)42

- p'l2c21 + lV' - (oi - o*' + g,*2)n2 x(V2 - ai(, - o*zrz1( : 0. (2.104) The solution hasbeengivenby Stoneley(1949),Grant and West (1965:42),and White (1965: 46).The roots are always real and positive and approach o and B of eqs. (2.58) and (2.59) as the anisotropy approaches zero. When the wave is traveling vertically, ( : 0, n : l, and V : ctr or B* for verticallytravelingP- or SI1waves.When ( : l, n : 0, V : o, or B*, corresponding to horizontally traveling P- or S/I-waves.However,when the wave is traveling at an angle to the vertical, the roots are complicatedfunctionsof the elastic constantsand the motion is not separatedinto distinct P- and S-waves. 2.7 Effects of the medium on wave propagation 2.7.1 Energy density and geometricalspreading Probably the single most important feature of any wave is the energyassociatedwith the motion of the medium as the wavepassesthrough it. Usually,we are not concernedwith the total energy of a wave but rather with the energy in the vicinity of the point where we observe it; the energydensity is the energy per unit volume. Consider a sphericalharmonic P-wave for which the radial displacementfor a fixed value of r is given Dy u:

Writing ai : (\, + 2p.,)lp,

(V2 -

A c o s( o t * ^ y ) ,

where 1 is a phase angle. The displacement r'tranges flrom -A to *A. Becausedisplacementvaries with time, each element of the medium has a velocity, ,i :

THEORY OF SEISMIC WAVES

58

Wavefront at time I + al

Wavelront at lime t Fast direction

Slor/ direction

I Wlocity function

(b)

(a)

(c) Fig.2.23 Wavefrontsin anisotropicmedia (a) Applicationot' Uuygens'principleto an anisotropicmediumillustratesdirectionind magnitud.diff...n..t betweenphaseand groupvelocities.(b) FerLat'sprincipleappliedto a reflectionfor a coincident sourceand receivershowsthat a reflectionmay not occur at a right angleto the reflector.(c) Sl/-wavefronts(surfacesof

6ul0t, andan associatedkinetic energy.The kinetic energy 6Eo contained within each element of volume 6.1/is EQ : l{PEl/)r;" The kinetic energyper unit volume is 6.E,,

#:

, .r * or t u ': l o o ." A' sin' ( 1;'

This expressionvaries from zero to a maximum of ipaltA'. The wave also involves potential energy resulting from the elasticstrainscreatedduring the passageof the wave. As the medium oscillatesback and forth'

isotroptcmeconstantphasefor a point source)in transversely are not ellipdia are eliiptical;however,P- and Sl/-wavefronts axis of vertical with V,. Vn) instances. special in excepi tical ,y-rn.try. Z. is group velocity as a function of the anglewith the symmetrYaxts

the energyis convertedback and forth from.kinetrcto potentia'i forms, the total energy remaining fixed' iWh.n u particle is at zero displacement,the potential energyii zero and the kinetic energyis a maximum' and;hen the particle is at its extremedisplacement' the enersv is all potential. Becausethe total energy - e'iuals ti'e maximum value of the kinetic energy,the , .n.tgy density E for a harmonic wave is E:

r o t r 2 A 2- 2 n z p v 24 2 .

).'

(2.l0s)

Thus, we seethat the energy density is proportional to the first power of the densityof the medium and to it. ,..orrd^power of the frequencyand amplitude of the wave.(SeeBraddick, 1965,for a different derivation of eq. (2.105).)

EFFECTS OF THE MEDIUM ON WAVE PROPAGATION

59

We are also interestedin the rate of flow of energy and we define the intensity as the quantity of energy that flows through a unit areanormal to the direction of wave propagationin unit time. Take a cylinder of infinitesimal cross-section,69, whose axis is parallel to the direction of propagationand whose length is equal to the distancetraveledin the time, Et. The total energyinsidethe cylinder at any instant t is EV 6t 6g: at time I + Er all of this energyhas left the cylinder through one of the ends.Dividing by the area of the end of the cylinder,Eg, and by the time interval, 61, we get 1, the amount of energypassingthrough unit area in unit time: r : EV

(2.106)

Fig.2.24

Dependence of intensity upon distance.

For a harmonic wave,this becomes t :

)pVarA,

: )qzp/y2[2.

(2.101)

In fig. 2.24, we show a sphericalwavefrontdiverging from a center O. By drawing sufficientradii, we can definetwo portions of wavefronts,g, and gr, of radii r,''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''and rr, such that the energy that fows outward through the spherical cap 9, in I second must be equal to that passingoutward through the spherical cap 9, in 1 second(becausethe energyis moving only in the radial direction).The flow ofenergy per second is the product of the intensity and the area;hence, I,9r:

1r9..

Becausethe areas9, and g, are proportional to the squareof their radii, we get I./I, : 9 r19. : (r,lrrl2. Moreover,it follows from eq. (2.106)that E is proportional to 1 and hence [,11,: ErlE,= (r,lrrl2.

(2.108)

Thus, geometricalspreadingcausesthe intensity and the energydensity of sphericalwavesto decreasernverselyas the squareof the distancefrom the source (Newman, 1973).This is calledsphericaldivergence. For a plane wave,the raysdo not divergeand hence the intensity of a plane wave is constant.Figure 2.24 could representa cross-sectionof a cylindrical wave, that is, a wavegeneratedby a very long linear source, arcs 9, and 9. being cylindrical wavefronts.Because the arcsare proportional to the radii, cylindricaldivercausesthe intensity to vary inverselyas the ra.qer?ce dius. Thus, we can write Irl I, : ErlE, : (rrlrr)-,

(2.l 0e)

nherem : 0, I, or 2 accordingas the waveis plane, e rl i n d r i c a lo. r s p h e r i c a l . Ratios of intensity,energy,or power are usually expressedin decibels,the value in dB being 10 log,oof the intensity, energy,or power ratio. Becausethese \ ary as the squareof the amplitude,dB is also given rs 20 log,oof the amplitude ratio. The natural log of rhe amplitude ratio (in nepers)is also used(seeprob. e m2 . 1 7 ) .

The foregoing assumesconstant velocity, whereas velocity usuallyincreaseswith depth, producingmore rapid spreading.A factor of Vlt is often used (99.8), where Z, is the stacking velocity ($5.4.1).The term "spherical divergence"is still used in this situation eventhough wavefrontsmay not be spherical. 2.7.2 Absorption (a) General. In the precedingsection,we considered variationsof the energydistribution as a function of geometry.Implicit in the discussionwas the assumption that none of the wave energy was transformed into other forms. In reality,as the wavemotion passes through the medium, the elastic energy associated with the wavemotion is graduallyabsorbedby the medium, reappearingultimatelyin the form of heat.This processis called ab.sorption and is responiiblefor the eventualcompletedisappearanceof the wavemotion (seealso 96.5).Toksozand Johnston(1981)summarize much of the literatureregardingabsorption. The measurementof absorption is very difficult, mainly becauseit is not easy to isolate absorption from other effectsmaking up attenuation(see$6.5.2). Moreoveq absorption varies with frequency,so that it is not clear how laboratory measurementsapply to seismicwavetravel in the earth. (b) Expressionsfor absorption. The decreaseofamplitude due to absorption appearsto be exponential with distancefor elasticwavesin rocks.Thus, we can write for the decreasein amplitude becauseof absorption A:

A o e\ , ,

(2.110)

whereI atd Aoarevaluesof the amplitudesof a plane wavefront at two pojhts a distance x apart, and 1 is the absorptioncofficient. Other measuresof absorptionare basedon the decreasein amplitude with time; to relatetheseto n. we assumea cyclic waveform: A : Aoe t" cos 2rvt,

(2.ttt)

and make measurements at a fixed location;ft is called

THEORY OF SEISMIC WAVES

60 lhe dampingfactor. The logarithmicdecrement(log dec) E is defined by

u: 'n(u,no,,,XTJl'i;t1. ,**) etn) It can be expressedin terms of the damping factor as 6:

hT : hlv : 2Thla,

(2.r13)

where Z is the period; 6 is measuredin nepers.Quality factor Q can be definedas Q : 2rl(fraction of energylost per cycle)

:2r(ElLE,

(2.1t4)

where AZ'is energy loss. Becauseenergy is proporand AEIEo: tional to amplitudesquared,f, : Eos-znt 2h Lt. SettingA,t : T we getLEIE,: 2hT: 2E and Q:

dhT:

t/6.

(2.lls)

If n is the number of oscillationsfor the amplitude to decreaseby the factor e, then eh'r : e, n: llhT and Q:

(2.116)

nr.

Stilf another manner of expressingQ is Q : cot 0, where $ is the lossangle. During one period, a wave travelsone wavelength so that if the loss of energyis due to absorptiononly, ( f r o m e q s .( 2 . 1 1 0 ) f a c t o ri s h T : 1 \ the attenuation and (2.1Il)), and we can interrelate11,6, and Q. (2.117) Q: nln),: al2\V : d6. Absorptionin the form givenby eq.Q.ll0) appears naturally in solutionsof the type given in eq. (2.56)if we permit the elastic constantsto be complex numbers.Real elasticconstantvaluescorrespondto media without absorption and complex values imply exponential absorption.Complex valuesof tr and p result in complexvelocityvalues(seeeqs.(2.58)and (2.59)). lf the llV in eq. (2.56)is replacedwith llV + jl/
--

Aei.tnt,r*jr/o)-/l :

wherex, is the distanceto the source.The table shows that lossesby spreading are more important than lossesby absorptionfor low frequenciesand short distancesfrom the source.As the frequencyand distance increase,absorptionlossesincreaseand eventuallybecome dominant. The increasedabsorptionat higher frequenciesresults in changeof waveshapewith distance.Peg-leg multiples ($6.3.2b)and possibly other mechanisms also produce waveshapechanges.Figure 2.25 shows the energy decreasingwith distance and with frequency; the frequency-dependentattenuation is greaterthan expectedfrom absorptionalone. 2.7.4 Dispersion;group velocity VelocitiesV a, and B, which appearin $2.2and subsequent sections,are phasevelocitiesbecausethey are the distancestraveledper unit time by a point of constant phase(for example,a trough) of a simple wave suchas thosein eqs.(2.53)to (2.55).This is not necessarily the velocity with which a pulse travels,called the group velocity. For the wavetrain shown in fig. 2.26a,wecan determinethe group velocity U by drawing the envelopeof the pulse (the double curve ABC, AB'C) and measuringthe distancethat the envelope travelsin unit time. The phasevelocity Z is given approximatelyby the rate of advanceof a distinct "phase break," as indicated in figs. 2.26a and 2.26b, but to find V accurately, we should decomposethe pulseinto its frequencycomponentsby Fourier analysis(seeeq. (l 5. I I 3)) and measurethe speedof each component l(o)sirt't.

Ae-\rer6lrtv 4,

which agreeswith eq. (2.1l0). 2.7.3 Relativeimportctnce of absorptionand spreading To compare the loss by absorption with the loss of intensity by geometricalspreading(seeeq. (2.108)), we have calculated the lossesin going various distancesfrom a point 200 m from the sourceassuming r1 : 0.15 dB/\. The results shown in table 2.1 were calculatedusing the following relations:

Divergence

Absorption:

intensity ross indB:

l3i:il [;,,1n,

: 0.3(x/\): 0.3(n,- 200yI - 0.3u(x"- 200)12000, Spreading:

intensity ross indB:

13i::i:

,[;,,lor,

Fig. 2.25 Loss of amplitude as a function of one-way traveltime, based on measurements with a geophone clamped in "diborehole with the source at the surface. The curves labeled "divergence plus transmission loss" are calcuvergence" and lated from sonicJog data allowing for loss of energyin transmission through reflecting interfaces. The 20-, 40-, 60-Hz curves show attenuation at those frequencies.(Courtesy of SSC.)

61

EFFECTS OF THE MEDIUM ON WAVE PROPAGATION I

i

Table 2.1 Energy lossesby absorptionand spreading (n: 0.15dBlwavelength and V: 2.0 kmls) Distance from shotpoint (x,) Frequency (v) 1200m 2200m 4200m 8200m

t

I

lHz 3 l0 30 100 Spreading All Absorption

0.075dB 0.22 0.75 1 a

7.5 16

0.15dB 0.45 1.5 A <

15 2l

0.3dB 0.6dB 0.9 1.8 3 6 9 l8 30 60 26 32

If phasevelocity Zis the samefor all frequenciesin the pulse,the pulse shapedoes not change and U : tr{ Howeveqif the velocity varieswith frequency,the different componentstravel with different speeds,the pulse shapechanges,and U + V that is, the medium is dispersive. Considerthe two harmonic wavesshown in fig. 2.26c that travel at slightly different velociries, the higher-frequencyone being faster than the lowerfrequencyone. The "pulse" that resultsfrom their interferencechangeswaveshapeand travelsat a velocity different from either of them (greaterthan either in this instance).If we write l(or)srt.' -,)for a plane-wave componenttravelingalong the x-axis, r being r(to), in general,,4(o) varies slowly whereasthe phase(xx ort) varies rapidly. When adjacent frequencycomponents are added togethet the net result is usually approximatelyzero becauseof destructiveinterference. However,when the phase(rr - or/)variesslowly,constructiveinterferenceoccursin the vicinity of a point (r, t). The condition for this is d. :u:* Orlx(to)x-(')/l

d x ( o )- r . O(r)

Point (x, l) will move with velocity U where U : d:c/dt(see{i2.2.5).Hence,differentiationof the above equationgives

frequency,we have inversedispersionand the opposite is true (asin fig.2.26a). of most Dispersionof body wavesis a consequence theoriesproposedto accountfor absorption.Aki and Richards (1980: 170-2) show that the assumptionof constant Q, which most data indicateis the situation in solid earth materials,and no dispersionare inconsistentbecausethey lead to noncausality.Their argument follows. Starting with a plane-waveimpulse at x : 0 : I a n d u s i n ge q s .( 1 5 . 1 2 7a) n d ( 1 5 . 1 3 6 )w, e have $(x, 0 : 6(t xll4 <) e-)dnv. Adding attenuation corresponding to constant Q (seeeqs.(2.110)and (2117)),the right side becomes t6txt2vo; we must use the absolute value of (l) to eej@x/t/ avoid havingthe amplitude increasewith increasingx when
t6,

dt

- W aO dl Va( ar )l] l

The derivativeis small, so Ll-V+.dV:V+rd': do dv

v - )rd,v, , CIA

( 2 . 1l 8 ) where V to,v, tr, dVlda, dVldv, and d tr4d),are average values for the range of frequenciesmaking up the principal part of the pulse. (See problem 2.1 for a more elementaryderivationof eq. (2.I l8).) When Z decreases with frequency,we have normal dispersion,and V is larger than H where the envelope travels slower than the individual cycles;this is the usual casewith eround roll. When Z increaseswith

^'

||

znLJ^

+ :

'\t2t Qteju, e- u,\rilr da

*-.-'utn-,,'n,.,., 6r]

J

-a[J:

eau+iB)a, + s.rr+ia, d6} J.-

where A = xl2VQ, and B : (t - xlt).Integration gives

{(x,t) : (ll2r)2Al(A, + R) tl2nYp (xl2vQ)' + (t - xlnt

(2.ne)

This function is shown in fig.2.27; it has a maximum valueat t : xlV and is roughly symmetrical.Observations show insteada sharperrise near x/Zfollowed by a slowerdecay.The function of eq. (2.1l9) has a finite value for I < 0, x * 0, and is thereforenoncausal, showing that our assumptionsare inconsistent.We conclude, therefore, that absorption necessarilyrequires that V varieswith to, that is, dispersionmust exist. Dispersion is important for severalreasons,perhaps the most important being that the energy of a pulsetravelswith the velocity U (exceptwherethereis appreciableabsorption;seeBrillouin, 1960:98-100). Dispersionof seismicbody waveshas not beendefinitively observed over the wide range of frequencies from h.ertzto megahertz.Most rocks simply exhibit little vafiation of velocity with frequencyin the seismic freguencyrange.Ward and Hewitt (1977)found the same velocity at 35 Hz as at 55 Hz in a monofrequency well survey to about 800 m. Futterman (1962)showsthat the dispersionexpectedfor seismic body wavesis small for usual situations.Dispersionis, however,important in connectionwith surfacewaves (see$2.5)and channelwaves($13.3)as well as other phenomena.

:t"l,[,"t,]] r:r'l :l,h

u:d! :

I r f0

r) :

62

THEORY OF SEISMIC WAVES

I I

*l 9 t b l

"l I

u r o u p v e f o c i l yn:

= "

P h a s € v c l @:iftf' : v

---(c)-

Fig.2.26 Comparison of group and phase velocities.(a) Definition of group velocity U and phase velocity Z (b) arrival of a dispersive wave at different geophones; (c) two sine waves of slightly different frequency and velocity traveling from left to

right form beats; the envelope travels with the group velocity U - Ao/Arc and points of constant phase within the beats with t h e p h a s ev e l o c i t i e s V , : a , l r , a n d V . - o , / r c . ( f r o m G e r k e n s ,

2.7.5 ReJlection and refraction;Snell'slaw

downwarda distanceV. L,t. By drawingarcswith center A' and lengthsequal to V, Lt and Z, Al, and then drawing the tangentsfrom R to thesearcs,we locate the new wavefronts,RS and RI in the upper and lower media.The angleat S is a right angle and A' S : V, L,t : -B'R,'therefore,the trianglesA'B'R and l'SR are equal,with the result that the angleof incidence0, is equal to the angle of reflection 0l; this is Lhelaw of reflection.For the refractedwave,the angle at ?'is a right angleand we have

Whenevera waveencountersan abrupt changein the elasticproperties,as when it arrivesat a surfaceseparating two beds,part of the energyis refectedand remains in the samemedium as the original energy;the balanceof the energyis refractedinto the other medium with an abrupt changein the direction of propagation occurring at the interface.Reflectionand refraction are fundamental in exploration seismology and we shall discussthesein somedetail. We can derive the familiar laws of reflection and refraction using Huygens'principle.Considera plane wavefront AB tncident on a plane interface as in lig. 2.28 (if the wavefrontis curved,we merelytate I and 8 sufficientlyclosetogetherthat AB is a plarle to the requireddegreeofaccuracy; however,seealso $2.8.1); 1.8 occupiesthe position A'B' when I arrivesat the surface; at this instant, the energy at .B' still must travelthe distance.B'Rbeforearriving at the interface. If B'R : V, L.t, then At is the time interval between the arrival of the energyat A' and at R. By Huygens' principle, during time At, the energy that rcached A' will have traveledeither upward a distance V, Lt or

I 989:37).

V, L,t : I'R sin0, and

vrat : l'R sin0,, hence,

sin 0,

sin e"

Vt

V2

(2.r20)

Angle 0, is calledthe angleof reJractionand eq. (2.120) is the /aw of refraction, also known as Snell'slav,. The anglesare usuallymeasuredbetweenthe raypathsand a normal to the interface. but these ansles are the

DIFFRACTION

63 sin 0, :

VrlVr.

(2.12r)

For anglesof incidencegreaterthan 0-,it is impossible to satisfySnell\ Iaw (usingreal angles)becauiesin 0, cannot exceedunity and total refiectior occurs. ThiJ does not mean that l00o/oof the energy is reflected as P-waves,however,becauseconverted S_waves(see $3.1.1)and evanescentwaves(see $2.3.1)are gen_ erated. Noting the method used to derive eq. (2.57) in $2.3.1,we write Snell'slaw for the case0, > e, (seefig. 2.29) in the form F r g . 2 . 2 7 I l l u s t r a t i n ge q . ( 2 .I l 9 ) .

same as those between the interface and the wave_ tionts in isotropic media. The laws of reflectionand refractioncan be combinedin the singlestatement:at an interface,the quantityp : (sin 0,)lV hasthe same value for the incident, reflected,and reiractedwaves. The quantity p is called the raypathparumeter lt will be shownin $3.l.l that Snell'slaw alsoholdsfor wave conversionfrom p- to S-waves(and vice versa)uDon reflection or refraction. The generalized form of Snell'slaw (eq.(3.1))will be undersroodin futureref_ 3rencesto Snell'slaw. When the mediumconsistsof a numberof parallel reds,Snell'slaw requiresthat the quantityp havethe )ame value everywherefor all reflectedand refracted :al's resultingfrom a given initial ray. The loregoingderivationassumeda planar surface .rnd therefore specular reflection. If the surface in_ ;ludes bumps of height d, reffectedwavesfrom them .rill be aheadof those from the rest of the surfaceby lil Thesecan be neglectedwhere2dl>\
;. 1.28

Reflection and refraction ofa plane wave.

sin 0, : (VrlV) sin 0, : sin (jrr - j0) : c o s j O : c o s h0 : L /? : cos 0: : sin j0 : j sinh 0; hence,eq. (2.56)becomes *

:

,qe

(ozlrasinh0ejo(r/racosh0_,1.

(2.122)

If we take 0 negativein fig. 2.29,the only changeis in the sign ofthe first exponentialon the right_handside. Thus, just as in the case of eq. (2.5i--),evanescent wavescan existon both sidesofthe interfaceand their amplitudesdecreaseas we go awayfrom the interface. The rate of attenuation is proportional to sinh 0, which has its maximum value at the grazing angle, e, .:^ lT The introduction of imaginary angles to satisfy-Snell's law for angles e*.eeding the critical angle means that the reflectioncoefficient($3.2)will be complex and phaseshifts will occur (seeoroblem 3.6b) that will be complicatedfunctionsof the anele of incidence. Snell'slaw is very useful in determining raypaths and arrival times and in deriving reflectoi position from observedarrival times,but it doesnot sive infbr_ mation about the amplitudes of the reflJcted and transmittedwaves.This subjectis taken up in chao.3. 2.8 Diffraction 2.8.1 Basicformulas In discussingreflectionand refraction,we statedthat when an interfaceis curved we merelyhaveto selecta portion suflicientlysmall that it can be considereda plane. However, such a simplification is not always possible,lor example,when the radius of curvatureof an interfaceis lessthan a few wavelengthsor the re_ flector is terminated by a fault, pinchout, unconfor_ mity, etc. In such cases,the simple laws of reflection and refractionare no longer adequatebecausethe en_ ergyis dffiacted ralher than reflectedor refracted. Be_ causeseismicwavelengthsare often 100 m or more, many geologicfeaturesgive rise to diffractions. The mathematicaltreatmentof diffraction is com_ plex and we shall give only a brief summaryof a sim_ plified treatment due to Trorey (1970).fe shall as_ sume a coincident source and receiver (see Trorev. 1977,for the noncoincidentcase)and constantveloc_ ity. Using $ in place of r!, we take the Laplacetrans_ ficrmof the waveequation.eq. (2.30).obtaining

THEORY OF SEISMIC WAVES

64

assumea constantreflectioncoefficientover y, so that c in eq. (2.123)is constant.Clearly,llr : 0: O over the hemisphere,hencethe contribution to the integral oyer the hemisphereis zero. We can also set O : 0 over the portions of the plane z : h exceptfor the portion I whoseeffectwe wish to evaluate. We now substituteeC.Q.123) in eq. (2.124),noting that r in eq. (2.123)is now r,,in fig. 2.30abecausethe sourceis now at the imagepoint O'; hence 0 r_ 3 r : i : h . 611 i)z r r'

V-s:9L : Fig. 2.29

Imaginary angles of reflection and refraction.

V' 6 : \lv' )

: rro.rrr(r,* ,r),

a26lAf er V' O : (slV)'Q,

where O(x, y, z, s) is the the Laplace transfqrm of 66, y, z, t) (see515.3),and the double-headedarrow indicatesequivalencein different domains.Note that we are assumingthat $ and 6$l0t are zero at / : 0 for all x, y, z. The solution of this equation for a point sourceat the origin is q : @tr)e

",tv,

(2.123)

wherer is the distancefrom the sourceto the point of observation,and V is the wave velocity.(This can be verified by direct substitution,noting that 12: x2 * y2 + z2,6rlOx : xlr, etc.)In general,c'should include the Laplace transform of the input waveform at the source,but in effectwe havetaken the transformto be unity, that is, the sourceis cEO (eq. (15.180)).The results for other types of sourcescan be found by time-domain convolution (see eqs. (15.201) and ( I 5.202)). In a source-freeregion, the P-wavepotential function $ is given by eq.(2.42)with Y : 0; hence,we can get another expressionfor $ by taking the Laplace transform of eq. (2.42),the result being

" -{'*,[ni;-u#]- :[:,.]].' 4ne:jJ (2.r24) The factor e-v/r/arisesbecause$ in the integrand of eq.(2.42)is evaluatedat time t : to - r/4 whereasO is the transformof$(x, y, z, t) (seeeq. (15.187)). 2.8.2Dffiaction effectof part of a planereflector We shallnow calculatethe diffraciion effectof an area 9 that is part of a plane reflectorz : h (seefig. 2.30a), both sourceand detectorbeing at the origin. We enclose the origin with a hemisphereof infinite radius with center(0, 0, ft), the baseof which is the planez : ft. In order to apply eq. (2.124),we replacethe source with its image at (0, 0, 2h), thus making the hemispherea source-freeregion.We ignore absorptionand

a!!,) : -h.

!.

0r1 6z rn' dr1 13' d O _ d O 6 r n_ _ r r . , ^ , , ( l s\/_r\ + d1 dro d1 Vl\ r, I r0 \rn

where we have set r,, : r afler differentiating. The result is

:,rll".*,0+ ).)av e tzs) 2,,e This surfaceintegral can be transformedinto a contour integral as follows. In fig. 2.30b,the elementof area is p dp d0 in polar coordinates;becauser' : p2 * h', p dp : , dr and, therefore,

2re: ,rlJ,",.,(:,+ ',)a,aee t26) If we integratethe first term by partswith respectto 4 we obtain l"

' ",'''

J,, rt

e ^'"1" se 2"tv' 6, = -2r1,,- l'2 or'

J., vr2

Substitutingin eq. (2.126),we get

'",/v - (ttrlte '*,'lde. 4tru:,n$ltn:rc (2.121) If the z-axisdoesnot cut y, we can get r : ( for points on the boundary,giving Q : -(chl4nldttlt,t. 2r€l/d0.

Q.128)

J

where I traversesthe boundary of I in the counterclockwisedirection. Ifthe z-axiscutsI (seefig. 2.30c),rt: h constant in eq. (2.127),so that Q : (ct2hlet,nrv- lshl4Trdrtl('1. 2' €//d0. J

(2.r2e)

whereI again traversesthe boundary in the counterclockwisedirection. If I includesthe entire xy-plane,

DIFFRACTION

65

(c)

; 1.30 Diffraction effect of a plane area g. (After Trorey, .-rl.) (a) Calculationusing a s u r f a c ei n t e g r a l ; ( b ) c a l c u l a t i o n .::rs a line integral where the origin is not over the area; (c) .ulation when the origin is over the area; (d) evaluating the : integral when ( is multivalued.

, . *,e have a continuousplane reflector,t : - and ': rntegralvanishes. The first tepm in this equation .\ representsthe s_implerefl&tion from this plane 'J the integral representsthe diffracted*uu.. Co-_ ::rion of eqs. (2.128) and (2.129)shows that the ::lcted wave is given by the same expressionin :] CASES.

\n lmportant point to note is that both the reflec_ r .rrrddiffraction terms in eqs. (2.128) and (2.129) -- Jerivedfrom the integral in eq.(2.125),wherethe .:,srationis carried over the entire surface.When we : :.tvSood think of reflectionand diffraction as oc_

curnng at a point or along a line, we are greatly sim_ plifying the actual phenomena.In fact, both reflec_ tions and diffractionsare the resultantsofenergy that return from all parts of the surface.From this point of view a reflectionis merelya specialtype of diffraction, a point ofview that has interestingpractical applications(see99.12.2). 2.8.3 Time-domainsolution/br dffiaction We now obtain the time-domain solution of eqs. (2.128) and (2.129).The inversetransform of the re-

THEORY OF SEISMIC WAVES

66 flection term on the right-hand side of eq. (2.129) givesthe impulse (cl2h)6(t - 2hln, that is, a repetition of the input at the sourceafter a delay of 2hlV which is the two-waytraveltimefrom the sourceto the plane surface,with the amplitude falling off inversely as the distance.Thus, the reflection has the same waveshapeas the source.The diffraction terms can be found as follows:We write t : 2ElV which is the twoway traveltimefrom the sourceto the variablepoint on the boundary.Equation (2.128)now becomes

o:

z(,!!'t)A!9\t) (t,+t,-t)(t,-f)tt2 (4chynlnLat)

(t7 + ti- t)(t, :0

(1>

(2.132)

(t
whereI : 2ElV t,: 2rlV and t, -- 2yolV. #f"r\,: "]uf?#o,(2,30) This valueof $(t) givesthe diffractedwaverecorded

We must pay careful attention to the limits of integration because(, and hencel, is in generala multivalued function of 0; for example,when 0 : 0. in fig. 2.30d, € can have any of the values {-, {,, or (,. To avoiddifficulties,the integralis calculatedas the point ofintegration goesfrom A to B (0 from 0, to 0r), then from 8 to C, C to D, and finally D to A, the proper valuesof { (and l) being usedalong each segmentof the path. Along a given portion of the path, say,betweenI : /r and t : t2(t2) /,), we have

o:

fact B and D are aI infinity, so that 0 increasesfrom -lrr to +]n as the point of integration,4 traversesthe botndary in a clockwise direction. The result (see problem2.19)is

hl:,';.(::)d/: f-

d(0e-"dr,

Comparing the right-hand integralwith we concludethat eq. (15-178a), 0(/):0 : (chlrlnt' )(d0ldt) :o

at point P(0, 0, 0) as the result of an impulse,c 6(l), appliedat the samepoint. If the input is cg(t)instead of c 6(t), O will have the factor G(s)and the response becomes0(r) * g(r) (seeeq. (15.195). Equation(2.132)givesthe diffraction effectwhether P is offthe plane,as in fig. 2.31,or over the plane; becausey,,changessign as P passesover the edge,the diffractedwaveundergoesa l80o phaseshift as P passesover the edge.Moreover,if we write D for the value of d for the diffractedwaveobservedwhen P is infinitesimally close to the edge and to the left of it, the total effect observedwhen P is the samedistanceto the right of the edgewill be R 4 R beingthe value of the reflectionterm in eq. (2.129).BecauseS(l) is continuous,

(2.131)

(tt.).

The derivative,d0/dl, is finite exceptwhen { is constant, suchas whereI is boundedby an arc ofa circle with its center at the origin for which cased/ : 0. In this specialcase,eq. (2.130)gives

R

D

or

( 2 r.3 3 )

Thus, the maximum amplitudeof the diffraction from a half-planeis half the amplitudeof the reflectedwave (as observedfar from the edge). Figure 2.32 shows what is expected from a half-plane based on eq. (2.132).As the edgeof the reflectoris approached,the diffractiongains in amplitudewhereas(R - D) decreases in amplitudeuntil at the edgeD : :R and the

Q : (chlnVrti)e 'o(0, - 0i), where 0, and 0, fix the end points of the circular arc, and t,,is the two-waytraveltimeto the arc. The inverse transform is g : (chlnv,ti)(e,- Oi)E(/- lo). When d0/dl is finite, we can get the time-domarn solution for the diffractedwaveby dividing the boundary of the arcaI so that the two-way traveltimet betweenP and the fioundary is a single-valuedfunction of 0 in eachpart, calculatingS for eachpart from eq. (2.131),then summingthe various 0's to get $ for the diffracted wave. 2.8.4 Dilfraction ffict oJ a halJ-plane A simple illustration of the method is the calculation of the diffraction effectin the important caseof a horizontal half-planeat a depth ft, the edgebeing parallel to the x-axis and a distanceyo from it. By referringto fig. 2.31, the edge of the half-plane is BD, where in

Fig. 2.31

Calculation of the diffraction from a half-plane

Surface

I / = 1 . 6k m / s

1.0

. , , i; : l-l*t': j

[iii] H[+--tr-tl[,1i

l--i-?-i]: t- i-T ti-l--t-f-r : ; i i - ; ( . t l: : t : f i i l l i . i ; i ] I j i ' a-: ili | -, ii - * i - - il : . , - - + f T -l , ' * j I i ; i i I : i i i i l' i i I I i I i I i ; I

i i I ; I i i , ; i rr , i I i i l , i I I I l , i _i , i i I ! i i i I i I I I I-i---ri-1i-- ;-*{- _f++*$-,=1-;

r l. ;* I i i I ; I t I i I l- l. i i, I , i l- i' , : :

E

tr

I.J

:-

i i, 1I ;r i i , i , i i i . i ; l , r i i , i I i i ' iI i i i i . i i ' iil-' ii .i iI - LI iI l+i,1i*l=i=-:1. -'. +'+ ts:r,-l-l-+

ilI i i l j i i ; l , ;I r : i i i l ; i ' ir ; I t i i i

;

i,t j,l-ir-irj i, ir- l-t i t ; i, ,J+i=l-i,=-i, ;j i j,i i'i-l i i i : : , 1i ,ii I I I

1.6

iIii

i I i'i ;

iii

,-- +11 j jT-i-,ii-'i--i l+ ,--;l-i i r ,i , L i i l-ri i i i ; - l 'I i l i , I I i , _ , . 1 I - r - i i " ] i

(b)

F'ig. 2.32 Seismic response of a half-plane. (After Trorey, 1 9 7 0 . ) ( a )M o d e l a n d ( b ) c o m p u t e ds e i s m i cr e c o r df o r c o i n c i d e n t

, l i

a

sourcesand geophones.The arrowhead indicates the location of the edge of the half-plane.

l l l , tl J

| = to + 2Ll t=to+2Lt

t:to+2Lt

Fig. 2.33

Diffracted wavefronts for'a laulted bed.

THEORY OF SEISMIC WAVES

68 sum is jR. The phasereversalofthe diffraction before the edge is reached (the backward branch of the diffraction) from that beyond the edge (the forward branch\is evidentin fi'g.2.32.

2.8.5 Using Huygens'principle to constructdffiacted wavefronts The surface integral in eq. (2.125) shows that the diffraction effectat a point is the sum ofeffectsarising from the entire diffracting surface.This suggeststhe use of Huygens' principle to construct diffracted wavefronts,and this is the casefor points more than a few wavelengthsaway from the diffracting source. Figure 2.33 illustratesthis construction for a faulted reflector.We assumea plane wavefront lB incident normally on the faulted bed CO, the position of the wavefront when it reachesthe surfaceof the bed at I : 1,,being COD. At t : t(i + Al, the portion to the right of O has advancedto position Gl1, whereasthe portion to the left of O has been reflectedand has reached position Efl We might have constructed wavefrontsEF and GH by selectinga largenumber of centers in CO and OD and drawing arcs of length VL,t; EF and GH would then be determined by the envelopesof thesearcs. However,for the portion Ef, there would be no centersto the right of O to define the envelope,whereasfor portion GH there would be no centers to the left of O to define the envelope. Thus, O marks the transition point betweencenters that give rise to the upward-travelingwavefront .E^F and centersthat give rise to the downward-traveling wavefront GH; arc FPG with center O is the diffracted wavefront originating at O and connecting the two wavefronts,EF and GH. The diffracted wavefront also extends into the geometrical shadow area GN and into region FM. The characteristicsof diffractionsin various sttuafurther in $6.3.1. tions are discussed

Problems 2.1 (a) If o,, is the only nonzero normal stress,that is, if o,,, : c,, : 0, useHooke'slaw,eq. (2.I 3), to show that strain Err : E-2,and verify eq. (2.20) for Poisson'sratio. (b) By adding the three equationsin part (a) for o,,, op, and o,,, deriveeq. (2.19)for Young'smodulus. (c) PressureI is equivalentto stresseso,, : or," : o", : -9. Substituting these relationshipsinto eq. (2.13),derive eq. (2.21)for the bulk modulus. 2.2 The entries in table 2.2 expressthe quantities at the headsof the columnsin terms of the pairs of elastic constantsor velocitiesat the left ends of the rows. The first three entriesin the ninth row are eqs.(2.19) to (2.21). Starting with these and eqs. (2.58) and (2.59),derivethe other relationsin the table.

2.3 (a) Firing an air gun ($7.4.3)in water createsa pressuretransient a small distance away with peak pressureof 5 atmospheres(5 x 105Pa). If the compressibilityof water is 4.5 x l0-'o/Pa,what is the peak energydensity? (b) If the samewaveis generatedin rock with \ : p : 3 x l0'0 Pa, what is the peak energydensity?Assume a symmetricalP-wavewith e,, : €rr: e,,, tij: 0 fot i*j. 2,4 To illustrate the interrelationshipand magnitude of the elasticconstants,completeTable2.3. Note that thesevaluesapply to specificspecimens,the valuesfor rocks range considerably,especiallyas porosity and pressurechange. 2.5 (a) Verify that S : .l(x - Vt) and * : g(x + Vt) in eqs. (2.46) and (2.47) are solutionsof the onedimensionalwaveequation,eq. (2.45).(Hint: Let ( = x - Vt and show that

ar!_ dfa(_d.f=r, 0x

d(

d( dx

etc.) (b) Verify that rp : J'(x + my * nz - Vt) + g((x + my * nz + Vt) in eq. (2.48)is a solutionof the planewaveequation,eq. (2.30). (c) Using the sametechnique,show that * : (llr)f(r - Vt) + (llr)SQ + Vt) in eq. (2.52)satisfiesthe wave eq. (2.50). equationin sphericalcoordinates, 2.6 (a\ Show that the waveequation,eq. (2.30),can be written in cylindrical coordinates(x : r cos 0, y : r sin 0, and : : ;; seefig. 2.34a)as a' {,

la{,

612

r 6r

la'q, f

I A:rl,

n' +-

T

12 602

622

V2 Af

(b) Verify that eq. (2.49)isthe waveequationin spherical coordinatesby substitutingthe following coordinatetransformation(seefig. 2.34b)into eq. (2.30): x:rsin0cos$,

0' I : ;:'J,?:''

(For an easiersolution,seeproblem 15.8.) 2.7 A pulse composed of two different frequency components,ou * Ao, can be representedby factors involving the sum and differencesof the frequencies. If they have equal amplitudes,we can write for the two components .4 cos (rcrx- art), I cos (r,x - o,l), ttl, - o)0 Ao, ru : 2nllto : where
S

j--s 9 A

> EA_

s

s

s

*I r lTt { l T | iF F F G + F rl'

I

s i

r,'

! 1v1

- -^ * iR r .

I

.eis _ln €l_i _1& JE- -l* _]! >;* _];

qr5

F!

ra 9ca

S h

N j

r

",x ;.= ?9 d

\

^

^

{i

.

t

i

.

- + - S c - r o - ' ' ^ + -8. -t -Y

O

>

^

-'

- -

d

4.

l

'E

(

a

F1

.J 5.<

-

..l

.{

b +

s

= ot

A E l .

I

t l 6.\ t 6

Hl^

=

X

1.1 |

r l Y l *

q R

l.\

<

-v

\O

-

r

l

Y

E

i

^

. + 6

b * l

6lO.t

I

h a]

l -

I

6

d

T

2.

11a:h r f xr*'lJ ilp l'i

N a l 1 I l..l H.l

O

\

I

I

I l

t

a i : b

l6l I

^= x - 7

l o 6

d E-r

r

<'

?le :ai f

I rt'l

-r.l lr'r I

l-

N

t l -

ll.l

-

q

q

C ctN

rt

,4

.t\ I c

e

^-!

a< c.l c.t

(.)

-

' Y

g H,.l

€ qi

-

' )

a kf

.

f

i

5

F

a rJ

j

a r

i

\

jl-

:

(

g

-T:-

;'

1T

5r< -f? 2

e,t a z 9 'j

_

g

.--

S

.: _

-1;--

& +-.ca

f**ir€=-- E z a ii

g

THEORY OF SEISMIC WAVES

70

Fig.2.34 Coordinate systems.(a) Cylindrical coordinates; (b) spherical coordinates.

(c) Show that the envelopemoveswith the group velocity tl where I- r = L a : d ' : Ar dr

v-xdv=v+tdv dt'r d\

(seefig. 2.26). 2.8 The magnitudesof period I frequencyv, wavelength \, and angular wavenumberr are important in practical situations.Calculate l1 \, and r ,for the situationsshownin table2.4. 2.9 (a) Show that the potential function ( : V(0 + 6yldz) - V' 1k of eq. (2.63)requiresthat A and 0- be solutionsof the P- and S-waveequations,eqs. (2.28) and (2.29), respectively.(I/rn l.' Recall that $ and 1 are solutionsof the waveequationand calculateV ' ( and V x ( u s i n ge q s .( 1 5 . 1 2a) n d ( 1 5 . 1 3 ) . ) (b) Show that the dilatation and rotation (eqs.(2.65)) can be derivedfrom ( in eq. (2.64)using eq. (15.14) and the resultsofproblem 15.7. 2.10 Justifyon physicalgroundsthe following boundary conditionsfor different combinationsof media in contact at an interface: (a) Solid-fluid: normal stressand displacementare continuous,tangential stressin the solid vanishesat the interface. (b) Solid vacuum: normal and tangential stressesin the solid vanish at the interface. (c) Fluid-fluid: normal stressesand displacements are continuous. (d) Fluid vacuum:normal stressin the fluid vanishes at the interface. 2.11 Using eqs.(2.8),(2.9),(2.12),(2.t3), (2.t\, and (2.64),show that the boundary conditions at the xyplane separatingtwo semiinfinitesolids require,for a wave in the rz-plane, the continuity of the following potential functions:

(d, - x*). (6, + x,). IV' $ + 2p(6""- x,), p(20,,+ x,,- x),

where the subscripts denote partial differentiation (theseare respectivelythe normal and tangentialdisplacements,normal and tangentialstresses). 2.12 A sourceof seismicwavesproducesa step displacementon a sphericalcavity of radius r0 enclosing the sourceof the form stepo(/):0(t<0), :kft>0). Show from eq.(2.77)that the displacementis given by

,r:10,, - 1). ^,, + rl

Is the motion oscillatory?What is the final (permanent) displacementat distancer? 2.13 Show that for harmonic wavesof the form $ : (Alr) cosa(rlV - l), the two termsin eq. (2.71),which decayat differentrates,are of equalimportanceat distance r : |tl2r. 2.14 Equations(2.86) and (2.87)for a Rayleighwave are valid at the surfacez : 0 for o : ll4. (a) Show that at depthsz * 0,the expressionsfor the u and w are x- and z-displacements r(x _ v^t), + 0.577e0.3e3s)sin u: rA(_e 0.8a8Kz 0 3e3s) 0.848K:+ : 1.468e w rl(-0.848e x cosK (x - 4t). (b) What are the valuesof u. w. and 0 when z : ll2r? When z : l/rc? (c) Is the motion retrogradefor all valuesof z? (Hint: Note that for the motion to changedirection,the amplitude of either u or )r must passthrough zero.) (d) What are the valuesof (, the Rayleigh-wavevelocity, when o : 0.4 and when o : 0.2? What are the correspondingvaluesofthe constantsin part (aX 2.15 Assumethreegeophonesso orientedthat one records only the vertical componentof a seismicwave, anotherrecordsonly the horizontal componentin the direction of the source,and the third only the hori-

REFERENCES

7l

Table 2.3 Example of magnitudesof elastic constantsand velocities Water Young'smodulus,E (x 10'Pa) Bulk modulus,k (x 10, Pa) Rigidity modulus, t'. (x 10' Pa) Lam6's\ constant(X 10' Pa) Poisson'sratio, o Density,p (g/cm,) P-wave velocity, o (km/s) S-wavevelocity, B (km/t

Stiffmud

Sandstone

Limestone

Granite

16

54

50

0.34 1.9

0.25 2.5

0.20 2.'7

2.1 0.5 1.0 1.5

0.43 1.5 1.6

Table 2.4 Magnitudesof T X, and x o (km/s)

Forv:15H2

r(s) Weathering(min.) Weathering(avg.) Water Poorly consolidatedsands shales at 0.75km Tertiary clasticsat 3.00 km Porouslimestone Denselimestone Salt Anhydrite

r (m)

Forv:60H2 K ( m ' )

7(s)

r (m)

K (m-')

0.1 0.5 1.5 2.0 J.J

+-)

5.5 4.0

6.1

zontal component at right anglesto this. Assume a simplewaveshapeand draw the responseof the three geophonesfor the lollowingcases: (a) A P-wave traveling directly from the source to the geophones. (b) A P-wavereflectedfrom a deephorizon. (c) An S-wavegeneratedby reflection of a P-wave at an interface. (d) A Rayleighwavegeneratedby the source. (e) A Love wave. Compare the relativemagnitudesof the components for short and long offsets. 2.16 (a) A tube wavehas a velocity of 1.05km/s. The fluid in the boreholehas a bulk modulus of 2.I 5 x 1O' Pa and density1.20g/cm3.The wall rock haso :0.25 and p : 2.5 glcm3 . Calculatep and o for the wall rock. (b) Repeat fot Vr: 1.20 km/s and 1.30 km/s. What do you concludeabout the accuracyof this method for determiningp? 2,17 The natural logarithm of the ratio of amplitudes is measuredin nepers.Show that I neper : 8.686dB. 2.18 A refraction seismicwaveletassumedto be essentiallyharmonic with frequency40 Hz is found to have amplitudesof 5.00 and 4.57 mm on traces2.50 and 3.00 km from the source.Assuminga velocity of 3.20 km/s, constant subsurfaceconditions,and ideal recording conditions,what is the ratio of the amplitudes on a given trace of the first and fourth cycles?

What percentageof the energy is lost over three cycles?What is the value of /r? 2.19 Using eq. (2.131),the generalequationfor the diffraction from a plane surface,verify eq. (2.132)for the diffraction effectof a half-plane.

Referenccs Aki, K., and P. G. Richards. 1980. Quantitatiw Seismology; Theory and Methods. San Francisco: W H. Freeman. Birch, F. 1966. Compressibility; elastic constants. ln Handbook of Physical Constants, S. P Clark, ed., GSA Memoir 97 Blake, F C. 1952.Spherical wave propagation in solid media. I Acoust. Soc. Amer.. 24: 211-15. Braddick, H. J. J. 1965. Vibrations, Waves,and Diffractions New York: McGraw-Hill. Brillouin, L. 1960. Wave Propagation and Group Velocity. New York: Academic Press. Bush, L, and S. Crampin. 1987.Observations of EDA and PTL anisotropy in shear-wave YSP, Expanded Abstracts, 57th Annual International Meeting of the Society of Exploration Geophysicists, pp.646 59. Tulsa: Society ofExploration Geophysicists. Cheng, C. H., and M. N. Toksoz. 1981. Elastic wave propagation in fluid-filled borehole and svnthetic acoustic loss. Geophysics, 462 1042 53. Crampin, S. 1981. A review ofwave propagation in anisotropic and cracked elastic media. Waye Motion, 3: 343-91.

72 Crampin,S.,R. McGonigle,and D. Bamford.1980.Estimating crack parametersfrom observationsof P-wavevelocity antso45: 345-60. ttopy. Geophysics, Dobrin, M. B. 1951.Dispersion in seismicwaves.Geophysics, 16:63-80. J. A. McDonald, Ebrom,D. A., R. H. Tatham,K. K' Sekharan, and G. H. F. Gardner.1990.Dispersionand anisotropyin laminated versusfractured media: An experimentalcomparison' ExpandedAbstracts,60th Annual InternationalMeetingof the pp. 1416-19.Tulsa:SociSoiietyof ExplorationGeophysicisls, ety of Exploration Geophysicists. Ewing,W. M., W. S.Jardetzky,and F. Press 1957.Elastic Waves in LayeredMedia.New York: McGraw-Hill' Futterman,W. I. 1962.Dispersivebody waves.J. GeophysRes', 67:5279-91. Gerkens,J. C. d'Arnaud. 1989.Foundationof ExplorationGeop/ryslcsAmsterdam:Elsevter. Grant. F. S.,and G. F. West. 1965'InterpretationTheoryin Applied GeophysicsNew York: McGraw-Hill. Hardage,B. A. 1985. VerticalSeismicProfling, Part A' PrinPress. ciptes,2ded. London:Geophysical Howell, B. 1959. Intoduction to Geophysics.New York: McGraw-Hill. Hsieh, J. S. 1975. Principlesof ThermodynanicsNew York: McGraw-Hill. New York:CambridgeUniversityPress' Lamb,H. 1960.Statics. Landau, L. D., and E. M. Lifshitz. 1986.Theoryof Elasticity, 3d ed. Oxford: PergamonPress. Lapedes,D. N., ed. 1978,McGraw-Hill Dictionary of Physics and Mathematics.New York: McGraw-Hill.. Logan,J. D. 1987.AppliedMathemalicsNew Yorkr John Wiley' Love. A. E. H. 1911.Some Problemsof GeodynamicsNew York: Dover. Love,A. E.H. 1944.A Treatiseon the MathematicalTheoryoJ' Elasticity.New York: Dover. polarizationsfor anMacBeth,C. 1990.Inversionof shear-wave isotropy using three-componentoffset YSPs. ExpandedAbMeetingofthe Societyof Ex' innual International st.::acls,60th ploration Geophysicists,pp. 1404-6 Tulsa: Society of Exploration GeoPhYsicists. Newman, P. 1973,Divergenceeffectsin a layeredearth Geo' plrysdcs, 38: 481-8. Postma,G. W. 1955.Wavepropagationin a stratifiedmedium' 20: 780-806. Geophysics,

THEORY

OF SEISMIC WAVES

Press.F.. and R. Siever.1978.Earth,2d ed. San Francisco: W. H. Freeman. Saada,A. S. 1974.Elasticity: Theoryand Applications.Oxford: PergamonPress. A. 1975.SeismicWaves.Moskow: MIR. Savarensky, M., and F. Muir. 1989.A calculusfor finely layered Schoenberg, 54: 581 9. media.Geophyslcs, Scholte,J. C. 1947. The rangeof existenceof Rayleighand Stoneley waves. Royal Astron. Soc. Monthly Notices Geophys Snpp,Ser.A, 106:416-28. S6gonzac,P. D., and J. Laherr6re.1959.Application of the conin Northern tinuous velocity log to anisotropymeasurements Prosp.,7z202-17. Geophys. Sahara:Resultsand consequences. in wells. Stoep,P M. 1966.Velocityanisotropymeasurements 900 16. Geophysics,3l: Stoneley,R. 1924.Elasticwavesat the surfaceof separationof two solids.Proc. Roy.Soc.(London),A-106:416-28. implicationsof aeolotropy R. 1949.The seismological Stoneley, in continental structures.Monthly Notices, Roy.Astron. Soc. Geophys.Supp., 5: 343-53. Toksoz,M. N., and D. H. Johnston.1981.SeismicWaveAttenuReprintSeries2. Tulsa:Societyof Exation,SEGGeophysical plorationGeophysicists. Trorey, A. W. 1970.A simple theory for seismicdiffractions. Geophysics, 35:.762-84. Trorey,A. W. 1977. Diffractionsfor arbitrary source receiverlo42':| 177-82. cations.Geophysics, Uhrig, L. F., and F. A. van Melle. 1955.Velocityanisotropyin 20: 774-9. stratifiedmedia.Geophysics, borehole Ward,R. W.,and M. R. Hewitt. 1977.Monofrequency 42t | 1t745. traveltimeswvey.Geophysics, and White. J. E. 1965.SeismicWaves- Radiation,Transmission, New York: McGraw-Hill. A t tenuation. Sound Applicationof Seismic White, J. E. 1983.IJnderground Waves.Amsterdam:Elsevier. Winterstein,D. F. 1990.Velocity anisotropyterminologyfor 55: 1070-88. geophysicists. Geophysics, poWinterstein,D. F., and M. A. Meadows.1990.Shear-wave larizationsand subsurfacestressdirectionsat Lost Hills Field. ExpandedAbstracts,60th Annual InternationalMeeting of the pp. l43l-4. Tulsa:Society Societyof ExplorationGeophysic,rrs of ExplorationGeophysicists.

3

Partitioning at an interface

Overview The partitioning of energyat interfacesis the central phenomenonof seismicexploration.Boundary conditions permit calculatinghow wave energy is divided among reflectedand transmittedwaves.The most easily understoodapproach,in terms of displacements, yields Zoeppritz' equations,and the calculation in terms of potentialsyields Knott's equations.Both approachesare given here,eventhough they accomplish the samepurpose,becauseof the importanceof partitioning. For the simple but important caseof normal incidence($3.2),which is commonlyassumedin most reflection work, Zoeppritz' (or Knott's) equations reduce to the familiar equationfor the normal reflection coefficient.This equationstatesthat the reflectionamplitude, comparedwith the incident amplitude,varies directly as the changein acousticimpedance(the product of velocity and density).Examplesof reflection coefficientmagnitudesare given. Reflectionat nonnormal incidence($3.3)leads to wave conversion and amplitude changes,especially near the critical angle.The Zoeppritz equations,while exact,do not givea feelingfor how amplitudesdepend on the various factors involved. Severalapproxlmations havebeenmade in an effort to achievean equation form that givesmore insight into the changesexpected for various situations. In particulaq the variation of amplitudeswith angle (93.4),more commonly expressedas amplitude variation with offset (AVO), has becomeimportant in the last few yearsas an indicator of hydrocarbongas. Refraction methods are usually basedon a simple head-waveconcept ($3.5) (although the waves lnvolved are, in fact, much more complex); the simple conceptyieldscorrect traveltimesand providesa basis for the interpretationof head-wavetraveltimes.

tions are the amplitudesof the wavesgenerated.When both media are solids, there are four equations resulting from the boundaryconditions,so that we must havefour variables.A P-wave(or an S-wave)incident on an interfaceseparatingtwo solidsmust in general generatereflectedand refractedS-wavesas well as reflected and refractedP-waves.Thus, for an incident P-wave,as shown in fig. 3.1, we havereflectedand refracted P-wavesat angles0, and 0, and reflectedand refracted S-waves at angles 6, and 6r. The waves whosemodeschangeat an interface(the reflectedand refractedS-wavesin the foregoingexample)are called convertedwaves. Note that S-waveshave 2 degreesof freedom,and motion perpendicularto the plane containing the incident waveand the normal to the interfaceis not involved in conversion from P- to S-wavesnor vice versa.Wherethe interfaceis horizontal,this is equivalent to sayingthat incident P-wavescan generatereflected and refracted P- and SZ-wavesbut not S.F1Waves,that incident SZ-wavescan generateP- and SZ-waves,but that incident Sl1-wavesgenerateonly reffected!a0d refractedS11-waves. Snell's law can be extended to cover converted waves.In fi5. 2.28, we change0i to E,, the angle of reflectionof the convertedS-wave.Writing o, and B, for the P- and S-wavevelocities,we have S i n0 , : B ' N A ' R : d t L t l A ' R , sin 6, : A'SIA'R: 9, L^tlA'R, hence.

3.1.1General The boundary conditions describedin $2.4.4lead to rather complex relationsfor reflectionand refraction at an interface.The nature of the two media fixesthe densitiesand elasticconstantsand thus the velocities. The angles of reflection and refraction are fixed in terms of the velocities,as will be shown. The only variablesremaining to satisfy the boundary condit-)

sin0,

Pr

0l

Following the same procedure for the refracted S-wave,we arrive at the generalform of Snell'slaw:

stn0r : I! j, : srn0: : An9,a :, r ' 0r

3.1 Application of boundary conditions

s i nE ,

g,

c2

(3.1)

9,

Note that p is the componentof the slownessof each ray in fig. 3.1 parallel to the interface.We conclude, therefore,that waveconversion,reflection,and transmissiondo not changethe componentof the slowness parallelto the interface. 3.1.2. Zoeppritz'equations Theseequations(Zoeppritz, 1919)determinethe amplitudesofthe reflectedand refractedwavesat a plane

PARTITIONING AT AN INTERFACE

74

- r 4 o c o s g t + A l cos 0r - B, sin 6,

- -A2 cos 0, - B, sin 6r,

(3.3)

,40sin 0, * ,4, sin 0 r + B r cos Er : '42sin 0: - -8, cos

(3.4)

By usingeqs.(2.8),(2.9),(2.13),and (2.14),the last two boundary conditions become

Pl, trl, rll, dl, Pl P 2 , ) \ 2 , P 1 ,d z , P 2

+ d --,ldz)+ 2p",(dw,ldz) tr,(drz,/dx : tr.(Da,/dx+ dwrldz)+ 2p.r(dwrldz), p.r(durldz+ dru,/Ex) : p,r(du.ld:+ du,,/Dx). Because

Fig. 3.1 Waves generatedat a solid solid interface by an incident P-wave.Displacement polarities assumedin $3.1.2 are indicated.

interfacefor an incident P-wave.We write A,', A,, 42, 8,, and ^8,for the displacementamplitudesof the incident, reflected,and refractedP-waves'and for the reflected and refracted S-waves(seefig. 3'l), the plus signs indicating the positive directions of displacements. We consider wavestraveling in the xz-plane, the interfacebeing the xy-plane' We can write for the incident wave(comparewith eq. (2.56))

:

f0ejo(rsinsl

: c o s f l l ) / r i :l

f09ro,r'(\

then [ I ' , s i n 0 , + ( \ r + 2 p , ) c o s0 , c o t 0 , ] ( A o + A ) - ( 2 p , c o s6 , ) . 8 , : [\, sin 0" + (\, * 2p,)cos0, cot 0,]1, + (2p, cos6,)8,,

/coroll.

where we have omitted the factor e-j'' becausewe shall not be differentiatingwith respectto time and the factor will cancelout in the boundary conditions' We can now write for the various wavesin fig' 3'1, rlro: loer'(', \rt:

Ap)@{l

qr/ : Brer'{r,

{r, :

l , e' r d ( r ,

I

l(3.2a1 r!' = f.er-(i. i

where = p(x - z cot 0,), f1r.ro, ( , : p ( x + z c o t 0 , ) , { r : p ( x - z cote,), (', : p(x + z cot E,),\lr: p(x - z c o t E j J

6

(the minus signsmean that the wavepropagationis in the direction of the negativez-axis). Next we resolvethe displacementsU', rfi into componentsalong the x- and z-axes.This gives I At zlo sin 0r ej-ao l, - A o c o s0 r e j ' { o* l , - I,

sin 0r ej'{r + sin 0r ej-\zc o s0 , e p r rc o s0 , e j ' { : -

Br cos 6' er-ai' Bzcos62er'{2, - 8 ,s i n 6 ' e r d i ' B , s i n6 ' e r r i '

The boundary conditions require that when z wr:

tr,(lo sin 0t + At sin 0, * .8, cos b') + (\r + 2p,) (.40cos0, cot 0, * l , c o s0 , c o t 0 , - 8 , c o s6 , ) : lrr(Arsin 0, - B, cosbr) + (\, + 2pr) (Arcose2cot 02+ B2cos6r);

fogjot{r+z:)/'r

{,,:

ut : uz: wr: ||z:

Dtj/dx: s.p*i, Erf,/dx : jtoprf,, : jorp(+cot6,)rfj, d,!,ldz : jop( + cot 0,){,, : we can drop the factor jtop and considerthat 0/3r +cot get the +cot for now We 6,' or 0, I and 6li)z: normal stresscondition

w2'

: I12t l'11

o",1,: o""ly

or

fr ^,* z*,)- 2p,sin' o,( A o +

aino

L - t[,^. *

sin0.

+ (2pzcos E,)-8,.

and(3.1), By usingeqs.(2.58),(2.59), /p,oi 2p,Plsin' 0,\ I-'. 1(Ao+A,)-(2P,BlcosE')B' \ P o ' l - 2p,Bisin' O,\ - - -lAr+Qpr}' rcos62)82. : l ' /o,ol :

\

P

*

,

1

Hence, prg?(l ?9in S?)(,1^ '+

At)-

2o'B'(sinE' tot-QJ3,

Pqt

: 0,

r, * 2d$T.ee$t4.

o.,1,: o'-12'

At z :0, all ofthe exponentialfactors reduceto er-pso that the exponentialfactors can be omitted when formulating the boundary conditions. The first two boundary conditions give

2p,) - 2p. sin'o.

A ) - ( 2 p , c o s6 , ) 8 ,

Finally, (Ao + A)Zt cos 26, - B,W, sin 26, : ArZrcos 262+ B2W2sin 2Er,

(3.5)

\PPLICATION OF BOUNDARY CONDITIONS ;:,eft Zi: p,c, and W: p,F,;Z, and W, are called - - .trtsticimpedances. The last boundary condition gives ll-rll(-Ao * l,)cos er + -Brcos 6, cot 6,] + [(- Ao * l,)cos 0, - B, sin 6,]] : lLz\t Azcos0, * B, cos6, cot Er) * (- Arcos02- ,8,sin 6r)), ..:i

75

equivalent to multiplication by I and by -r cot 0, and -r cot 6,, respectively, and the common factor ej.p'is omitted after the differentiation.Problem 2.ll gives the expressionsfor the normal and tangential displacementsand stressesin terms of derivativesof S and 1. The first boundary condition requiresthe continuity of normal displacementsat the interface,that is,

l).

* i l - A o + A ) 2 c o s 0 r + B , ( c o s6 , c o t 6 , - s i n 6 , ) ) = lL.l-2Azcos e2 + Br(cos6, cot E2- sin 6r)). 'p,\llp\) le Ao + A,)2 cos0, sin 6, * B, cos 26,1 = b'9llPP)?2Ar cos 0, sin 6, + B, cos 26r), . (g,la,)W,(-Ao * l,)sin 291+ Wpt cos 26r : - ( B r l a r ) W r A r s i n 2 0 r + W 2 B 2 c o2s b r . ( 3 . 6 ) Collecting together the results of applying the :.:.rndary conditions, we have the four Zoepprtiz :: Jations:

i

- A)Zt cos 26, - B,W, sin 26, : ArZ, cos 2E2+ B2Wzsin 26r,

- 1 + A ) ( g , l a , ) w , s i n2 0 , + B , w , c o s 2 6 , = - A.(S2|(}2)W, sin 20. + B2W2cos 26,.

(3.3) (3.4)

(3.6)

- 3 Knott'sequqtions r-,-.-.It(1899)was the first to derive equationsgiving : rmplitudesofreflectedand refractedwavesgener::-r at a plane interface.He used the displacement :' :3ntial functions$ and 1 of eq. (2.64)in the form: \ t e d i u m( l ) :

\ t e d i u m( 2 ) : Q2:.dtei'tz, yr: fi.si-1t,

z:o

(- du + d)cot 0, - %, : - Nz cot 0, - 9)r. The next condition is that the sheardisplacementsbe equal:

: (3*. (:*.:+) #),

- : o

(.{o + .{r) * 9J,cot 6, : d, - 9J,cot Er. The continuity of normal stressrequiresthat

\ V,6-r-,- lrTt- d-r'l ) x dz \dr-

I

be continuous;thus, (3 5)

Thus, for a P-waveof given amplitude lo incident , :he angle0, on a plane interfaceieparating two ml- " - : rrith given valuesof p, pr,o, and B, Snell'slaw de:irlrnes the angles 0, and 6,, whereas Zoeppritz' :: -.rrionsfix the reflectedand refractedamplitudesl, --,: B. Similarequationscan be derivedfor an inci:::.: .S-wave. For a fluid medium, B, : 0 becauseonly j -.irres a r ep r o p a g a t e d .

q, : s/oer.
:(::-*),

hence.

.:rs becomes

.{ + l,)cos 0, - .8, sin 6, - -A2 cos 02 - B, sin 6r, 1 l , ) s i n 0 , * 8 , c o sE , : lz sin 0, - B, cos 62,

(*-*)

\r(.do + "e/,Xl + cot2Or) + 2p",[(.N,u * .d,)cot20,- !Z},cot 6,] : \,d.( I * cot20,)+ 2yt"r(d2cot20, * (Z,cot D,). By usingeqs.(2.58),(2.59),and (3.1),this becomes p,(cot26,- l)(.d, * d,) - 2p",9),cot6, : pr(cot26,- l)dr* 2p";frrcot6,. Continuityof the tangentialstressmeansthat

-33) *(,#1,.33 is continuous;thus, IL,[2(*d,, + "d,)cot0r + g]r(cotrb,- l)l : Fr[-2d, cot e, + %,(cot26,- l)]. If we substitutea, : cot 0i, br : cot 6p and c,: bi - l, the precedingequationsbecome -a,do + a,d, - '!J, : -a,d, - ,fr., (3.1) .ilo+dt+ bllJt:.ilr-

br%.,

(3.g)

llrcrdo * p,c,.d,- 2p"rb r9), IL'czd, * 2p".b;%r, (3.9) -2p"rardoI 2p.,a,d,* lt"rc%, : -2p"radr* p"rcr,frr. (3.10) (Note that .d, and 9J,in theseequationsare the amplitudes of the potential functions $ and 1, not of the

PARTITIONING AT AN INTERFACE

76

(Volume of cylinder) x .E : (ar cos 0'Xp,to '*rlzai) : o')' ltro,too.d3(cot

equadisplacementsas in eqs. (3'3) to (3'6))' These. of equations subsituting by alsobe obtained il;t;" (3'6)' (3'3) to eqs' (3.13) into the form of eq. 3.I .4. Distributionof energY property' If Knott's equations have a very interestin^g third first.and we multipiy correspondingsidesof the second ofthe sides equation; and alsocorresponding products' and fourth equations,and then add these is the result

Becausesimilar expressionsmust hold for the energy have carried away by the other waves.we seethat we to ob(3' l.l by l :pr'ro in eq. term each onfy to mutiipiy tain tire distri6ution of energy among the various

+ (p, cot Er)9)i+ (prcot9r)dl (p, cot Or)-d? ( p , + c o t6 , ) % i : 1 P , c o t 0 , ) s 4 3 ( 3 ' ll ) relate Becausethe first and third Knott equations two the on stress and displacement to the normal fourth and second the whereas interface the of sides the t.iu,. to the tangential displacement and stress' area' per unit energy of pioJ""tt havethJ dimensions (3' I I ) F.l- tnit we make the correct surmise that eq' revarious givesthe distribution of energyamong the we this' f,ected and refracted waves' To demonstrate pothe terms,of in (2'105) r-epeatthe derivationof eq' becauseit tential function O (X doei not enter here we have "o.r"..n, S-waves).For the incident P-wave' E* for the kinetic energy per unit volume

uu+:1,,[(*)'. (Y)'l

waves. 3.2 Partitionlng

at normal incidence

Zoepprtiz'equationsreduceto a very simpleform for normal incidence.The curveschangeslowly for small angelsof incidence(say,up to l5'), so the resultsfor normal incidencehavewide application.For a P-wave at normal incidence,there are no tangential stresses , , : B . : 0 a n d e q s '( 3 ' 3 ) a n d d i s p l a c e m e n thse; n c eB to (3.6)reduceto A, * Ar: Ao, Z,A, - ZrAr: -ZrAn. The solution of theseequationsis A , 0 , p ,- c r P r- 2 2 - Z t - 5 2 22 A'o: ",0, * o,P' Z, + z, - L(\nZ): l{Aa/o + L'PlP), \

P:

r' : : A , : Au

2o,pr d z P z* c r P '

22, _.

( 3 .l 4 )

(3.15)

22+ Zt

coeffiEquations(3.14)and (3'15)give the reflection Equation 7l cient R and the transmissioncoemcient of a sequenceof isoi:.i+l tfto*t that the amplitudes in the log of changes of reiord a lated reflectionsare in seismic taken viewpoint the impendances, acoustic reenergy of tog manufacture($5.4.5).The fractions reE, and E^ by given n."",.0 and trans;itted are reflection called sometimes ;;;;Jt i*hich are also and transmissionenergycoemcients):

:l'[(*,)'.(3;*,)'l take 0l6x: Usingeqs.(3.2)and notingthat wemu3t -0o-rcoser)/ct'and6tdt= -jo' we have :ri,'ata):' forz:0,

?+: ;,'[,*'r'.''

^

u,prazAl C t , P , t l l:f r

* [t':,ttt'1,,'{'}.,*..,n01,,'' ,. the exTaking the maximum of the real part, we find is, pressiin for the energydensity 4 that E : lp,.i o(.4Ju,)'-

(3'12)

Comparisonwith eq. (2'105)showsthat do:

(a,lo)Ao,

( 3 .I 3 )

in the where ,4^ is the amplitude of the displacement of ProPagation. direction interifr. ".t..gy biought up to a unit area of the be the will P-wave incident the by time p.t rrnii fu"" energyin a cylinder of length o' of unit.cross-sectlon and inclined at an i-"utut.a parallel to the interface), that is' surface, the to normal angle O,toihe

,

('r..'r,)=

- o . r P r . u ' 2: A i 4 Z ' Z '

: Rr. (3.16)

Z" -'

,)r'l'),1': 2',r'' a,p,r,r2,4i

t3'll\

1' Note that eqs' (3'16) and Obviously,ER + Er: Z, are in-terchanged; i:.r2i"..'""ihanged if .1, an! not depend upon does partition hence, the energy -iontuint wave' When incident the *fri.tt -.alutransmltZ - l Z , : l . R : E o : 0 . a n da l l t h ee n e r g yi s : ",.'i' p' and p, that require ito,"itt"t thi! does not zero approaches contrast : "". As the impedance ", unity; R approaches and zero Iapproaches oi inntirv, is from unity' ifru., ,tt.iu*hei the impedancecontrast the stronger the reflectedenergy' Table 3.1 showshow the reflectedenergyvariesfor wlthln impedancecontrastssuch as may be expected contrasts the earth. Becauseboth densityand velocity only ".. r*"ff for most of the interfacesencountered'

PARTITIONING AT NONNORMAL INCIDENCE

ll

Table 3.1 Energy reflectedat interfacebetweentwo media First medium

Secondmedium

Interface

Velocity

Density

Velocity

Density

ztlz2

Sandstoneon limestone

2.0 3.0 2.1 4.3 t.5 1.5 1.5 0.5 2.4 2.4 2.2

1 1

3.0 2.0

2.4

1 ^

z. -)

2.4 1.0 1.0 1.0 1.5

4.5 1.5 3.0 0.36 2.0 2.5 2.2 2.5

2.4 2.4 2.0 2.5 0.0012 2.0

0.6'7 1.5 0.93 0.97 0.50 0.20 3800 0.19 0.96 1.39 0.69

Limestoneon sandstone Shallowinterface Deep interface "Soft" oceanbottom "Hard" oceanbotom Surfaceof ocean(from below) Baseof weathering Shaleover water sand Shaleover gas sand Gas sand over water sand

1 A

z.J z-J

1.8

) A

z-)

1.8 L. -)

F

0.2 -0.2 0.045 0.022 0.33 0.67 -0.9994 0.68 0.02 0.16 0.18

0.040 0.040 0.0021 0.000s 0 . 1I 0.44 0.9988 0.47 0.0004 0.027 0.034

All velocitiesin km/s, densitiesin g/cmr;the minus signsindicate lg0" phasereversal.

a small portion ol the energyis reflectedat any one interface; this is illustrated by the first four lines in table 3.1. The "sandstone-on-limestone" interfaceis about as large a contrast as is apt to be encountered, whereasthe "shallowinterface"and "deepinterface" figuresare much more typical of most interfacesrn the earth;hence,usuallyappreciablylessthan l,Z,of the energyis reflectedat any interface.The maJorexceptionsinvolvethe bottom and surfaceof the ocean and the surface and base of the weathering (see {5.3.2).A much larger proportion of the energycan be reflectedfrom these,and hencethey are especially lmportant in the generationof multiple reflections (96.3.2)and other phenomenawith which we shall deal later. Note that although the energyfractions E o and E, do not dependon which side of an interlacethe wave is incident,this is not true of the reflectedamplitude s/, becauseinterchanging Z, and Z, in eq. (3.14) changesthe sign of the ratio drld,, A negativevalue of "d, means that the reflectedwave is 180. out-ofphase with the incident wave; thus. for an incident wave sln cos to/ the reflectedwaveis s/, cos (ot + n). In table 3.1 , phasereversaloccurs for the situatrons wherc Z, exceedsZ.

33 Partitioning at nonnormal incidence Turning now to the generalcasewhere the angle of incidenceis not necessarily0', fig. 3.2 showsenergy partition as functionsof the angleof incidencefor certain valuesof parameters.Many curveswould be required to show the variations of energy partitioning as a function of incident angle becauseof the many parametersthat can be varied:incidentP-, SH-, or SVwave,P-wavevelocity ratio, densityratio, and S-wave

velocitiesin eachmedium (or the equivalentof defining Poisson'sratio for eachmedium). Figure 3.2a shows the partitioning of energy as a function of the angle of incidencewhen a P-waveis incident in the high-velocitymedium for a P-wavevelocity ratio a,la, : 0.5, a densityratio prlp, : 0.8, o, : 0.30,and o, : 0.25.For small incidentangles, all of the energyis in the reflectedor transmittedPwaves,E^r, and Er* respectively, and hencethere are essentiallyno S-waves.As the incident angle increases,some of the energy goes into reflectedand transmittedS-waves,d*. and .Er.r,respectively, mostly at the expenseof the reflectedP-wave.Note that at intermediateanglesof incidence,the reflectedS-wave carries more energy than the reflectedP-wave.Such convertedwaves(wavesresultingfrom the conversion of P-wavesto S-wavesor vice versa at an interface) are sometlmes recorded at long offsets where they are evidencedby alignmentsthat disappearas one tries to follow them to shorter offsets(see(R)"" in fig. 6.28b).As grazingincidenceis approached,the energy of the reflected P-wave increasesuntil at grazing incidenceall of the energyis in the reflected P-wave. The oppositesituation is shown in fig. 3.2.b,where a . l a , : 2 . 0 , p , l p , : 0 . 5 , o , : 0 . 3 0 ,a n d 0 2 : 0 . 2 5 . BecauseZ, : Z, the P-wavereflectioncoefficientis essentiallyzero for small incident angles.As the incident angle increases,S-waveenergyincreases. As the critical anglefor P-wavesis approached,the transmitted P-waveenergyfalls rapidly to zero and no transmitted P-waveexistsfor larger incident angles.Also, as the critical angle for P-wavesis approached,both reflected P-wave and reflected S-wave become very strong; such a buildup in reflectionstrengthnear the critical angle is called v,itle-anglerefection Sometimes it is possibleto make useof this phenomenonto map reflectorsusing long offsetswherethey cannot be fol-

PARTITIONING AT AN INTERFACE

78

1.0

1.0

7

.7

(,

o . E. u z u .

E U z U

o < 4

.1

.1

0

30

40

50

60

70

ANGLEOF]NCIDENCE (b)

1 0 4.0

3.0

(,

t.0

t U z u o U F

t I U z U 7 o uJ F o uJ

.9 .6 7 6

o u

J

J L

U t (L (L

uJ c

2 .1

,l 0

0

30

4C

50

60

70

ANGLEOF INCIDENCE (c)

A N G L EO F I N C I D E N C E (d)

F i g . 3 . 2 P a r t i t i o n i n g o f e n e r g y b e l w e e n t r a n s m i t t e da n d r e flected waves as a function of angle of incidence for the case of an incident P-wave. (From Tooley, Spencer,and Sagoci, 1965, e x c e p t ( c ) f r o m D e n h a m a n d P a l m e i r a , 1 9 8 4 . )( a ) C a s e w h e r e t h e v e l o c i t yi n t h c i n c i d e n tm e d i u m i s l a r g e r ;o , / u , - 0 . 5 , p . / p , 0 . 8 . o , - 0 . 3 , a n d o . - 0 . 2 5 . ( b ) C a s ew h e r e t h e v e l o c i t y i n t h e

i n c i d e n t m e d i u m i s s m a l l e r ;a , l a , : 2 . 0 , p . / p , - 0 . 5 , o , : 0 . 3 , and rr, - 0.25. (c) Fraction of energy reflected as a P-wave for v a r i o u s P - w a v ev e l o c i t y r a t i o s : p . / p , - 1 . 0 a n d o r : c r 1 : 0 . 2 5 . (d) Fraction of energy reflectedas a P-wave lor various density r a t i o s a n d o . / o , - 1 . 5a n d o , : o . - 0 . 2 5 .

lowedat short offsets(Meissner,1967).As the critical angle for S-waves is approached, the transmitted S-wavefalls to zero. If we had not had a densitycontrastbut otherwrse the valueshad been as indicatedin fig. 3.2b, there would havebeen a reflectedP-waveat small incident angleswhosefractional energywould havedecreased slightly as the incident angleincreased. Figure 3.2c showsthe P-wavereflectioncoemcient for various P-wavevelocity ratios when p, : p, and or : o: : 0.25.The reflectedenergyis zero for a velocity ratio of I (no impedancecontrast)and increases both as the ratio becomeslarger than I and as it becomes smaller than L The two peaks for ct,/ct,) I occur at the critical anglesfor P- and S-waves,respectively.Figure 3.2d showsthe energyof the reflectedPwave for various density contrastswhen a,/ct, : l'5 a n do , : o , : 0 . 2 5 . Koefoed (1962)gives 100tablesof the longitudinal and transversereflectionand transmissioncoefficients and the phaseshiftsfor anglesgreaterthan the critical angle(see$2.7.5)for incident longitudinal waves.

3.4 Variation of amplitude with angle (AVA) eqs.(3.3)to (3.6),conThe four Zoeppritzequations, tain four unknowns, At, Bp A., and -Br. Dividing through by ln, we can solvefor the four reflectionand transmissioncoemcientsR, : A,l Au, R" : BrlAu, rule T, : A,l An,and Q : BrlA6,usingeitherCramer's (eq. (15.3b))or matrices(eq. (15.22)).Aki and Richards (1980:chap. 5) derivedexpressionsfor thesefor P- and S-wavesincident on each of the five combinations of solid, fluid, and vacuum half-spaces.The expressionsare, however,quite complex. In the following, we assumethat Ap/p, Act/c, and AB/p are all small, p, cr, and B being averages(note that Ao/o neednot be small;seeproblem 3.10).These conditionsare satisfiedfor almost all sedimentarysituationswhereRo < 0.2 (R0is the reflectioncoefficient at normal incidence);in this case,the changesin the raypath direction are small. To show this we let 0, : e, + 40, 0 : (0, + gr)12: 0,;then sin 0, : sin (0, + Ae; : tin 0, cos A0 * cos0, sin A0 : sin 0, * A0 cos0,.

VARIATION OF AMPLITUDE WITH ANGLE (AVA) By using Snell'slaw: sin Orlsin0, = 1 + A0 cot 0 - crlo, - (ar + Act)/ct'- I + Ac/q, so A0 : (Ac/ct)tan0. Aki and Richards (1980:eq. (5.44))expand the terms in the exact expressionsfor the reflectionand transmissioncoefficientsfor a P-wave incident on a solid-solid interface,assumingthat the squaresand products of differentials are sufficiently small that they can be dropped. For the reflectedand transmitted P-waves,they get

-.(l],,,',; R,::[, +.i-",(*) -'(9,''.'(T)],

(3,e)

^ J : ^ Jj a+, ( , _ , : ^ ) 1, Substitutinginto eq. (3.18)and dividing by R,,,we have R , , l R , , : | + P s i n 2 0+ Q ( t a n : o- s i n : o ) , whereR,,: (Act/cr+ L,plp)|2from eq. (3.14),

(^ ) '

:

Ao/o

- zorl , -

I

Ao

n"tf--r'

(3.20)

I

Ao,Ap ,,Ap/p T

cr

I

p

T

Aa/o

The last term in eq. (3.20) is almost alwayspositive and thus increasesR" at very largeanglesofincidence (> 30'), but it is small for the anglesusuallyinvolved

(3'21)

Hilterman (private communication) rewrites eq. (3.20)in the form r

l t:

'l

R o = R n l ' - o f P,li n , e l(*.r 0- o" ) .' s i n 2 o \"/ [ J

* ^,o;

Theseequationsare valid when (a) Act/o, AB/B, and Ip/p are small(hence,A0 and AE are also small),(b) e < 80'if ct"( ct,,or (c) 0 < l0o if cr,) ct,.Hilterman (privatecommunication)believes condition(c) can be relaxedto 0 < 0, - 10" if o, ) (I,, where0, is the criticalangle.The form of the equationsallowsone to seethe separateeffectsof changesin density and Por S-wavevelocities,which are difficult to seefrom the exactexpressions. Equation(3.18)is frequentlyused to find the amplitude variation with offset (AVO) for reflectedP-waves. Shuey(1985)decidedthat Poisson's ratio was the elasticconstantmost directly relatedto the variation of R,, with 0, and thereforeusedeq. (2.60)to replace B with o. Takingthe log of eq. (2.60)and differentiatins. we have

+ Q)(l l - o

R"/Ro-l+Pg' '

(3,8)

r,:' ;(^f). (l*"u ') (:)

, : f n- o '

in reflectionseismology. The middle term in eq. (3.20) governsRp at intermediate angles;this term can have either sign and it is mainly the factor Ao/Ro that determines whether Rp decreasesor increaseswith 0. R. can changesign with increasing0 if the middle term in eq. (3.20) has polarity opposite to that of Ro. In some young clastic sections,Ap/p - -Act/o, which causescalculationdifficulties.For 0 < 30', eq. (3.20) can be further simplified(Shuey,1985:612) to

[,"",,

- . (l)' ,,",r]. 1322a1

where most of the dependenceon o is in the middle term. The relativecontributionsof the three terms in eq.(3.22a)are shownin fig. 3.3 for one situation.Hilterman further approximateseq. (3.22a)as R" = Rncos20+ 2.25Ao sin' 0.

(3.22b)

Ostrander (1984) applied theseresultsto practical cases (fig. 3.4). The reflection coefficient becomes more negative with increasing incident angles (fig. 3.4b)and vice versa(fig. 3.4c).For practicalreflection cases,three possibleresultsexist: l. For little changein Poisson'sratio (fig. 3.4a), the amplitudedecreases with increasingincident angle regardlessof the polarity of the reflectioncoefficient. For (a) a positivereflectioncoefficientand an increasein Poisson'sratio (which is apt to be true for a gas/watercontact or the baseof a gas sand embeddedin shale),or (b) a negative reflection coefficient and decreasein Poisson'sratio (which is apt to be true for the top of a gas sand embeddedin shale),the amplitude increaseswith incident angle. -). F-or(a) a positivereflectioncoemcientand a decreasein Poisson'sratio, or (b) a negative reflectioncoemcientand an increasein Poisson'sratio, the amplitudedecreases with incident angle at first and then the waveformreversespolarity and the amplitude increases with oppositepolarity. This is apt to be true for high-impedancereservorrs. Where a porous sandstoneencased in shale is water-saturated,Poisson'sratio is apt to be slightly smaller for sand than for shale, but when the pore spaceis filled with gas,Poisson'sratio for sandstoneis apt to be much smallerthan for shale.Consequently, the variation of amplitude with angle of incidence (AVA) is often regardedas a hydrocarbon indicator

Offset g

Offset

Offset

2.O

2.5

3.0

3.5

(a)

(b)

Fig. 3.3 Contribution of the three terms in eq. (3 22a) as a function of offset, for data from a well ollshorc Vermillion Pari s h " L o u i s i a n a .( a ) , ( b ) , a n d ( c ) r e f e r t o t h e f i r s t . s e c o h d 'a n d

(c)

third terms. respectively.(Courtesy ol Geophysioal Development Co.)

z

2

= 9

:==:I-:l

o

A N G T CO ' I N C I D € N C-I

z

o 2 o

(a) Fig. 3.4 Variation of a P-wave reflection coefficient with angle ofincidenceF . o r c u r v e sl , c t r l o , : p r l p , : 1 . 2 5 ; f o r 2 , l . l l ; f o r 3, 1.0; lor 4, 0.9; and for 5, 0.8. (From Ostrander, 1984.)(a) No change in Poisson'sratio at the interface (solid curves, or : o: :

0.3; dashed, or - o: : 0.2). (b) Decreasing Poisson's ratio ( s o l i d ,o , : 0 . 4 , o . - 0 . 1 ; d a s h e d ,o ' = 0 ' 3 , o z : 0 . 1 ) . ( c ) I n creasing Poisson'sratio (solid, o, : 0.1, o, : 0.4; dashed. o, 0 . 1 .o " : 0 . 2 ) .

HEAD WAVES

I

81

($10.8).A common situation in young clastic sediments (the "bright-spot" case,$10.8)is for a sand to have an acousticimpedancenearly the same as surrounding shale when liquid-filled but a much lower acoustic impedancewhen gas-filled.The result is a strong negativereflection marking the top of a gas sand and a strong positive reflection at the base (or at the fluid contact betweenthe gas- and liquid-filled portions); both reflectionsthen increasein amplitude with incident angle, as shown in fig. 3.5. However, changesin parametervaluescan alter theseresponses. Poisson'sratio can changewhere no reservoiris pres-

gf-

x' _--_4 ,

P

z g I

O

cofiPUt€o o.o OBSTRVED GAs SANO

lt lt ul

o o z 9

V2Lt

(b)

o.2

o.l

F

o

UI ll. lrJ E o.o l-

-/-.t' -t+*-

- - corr PUIED €+c oBs€RvED W A T T RS A N D **o--;-t*--F'u-ct-=.oTi

|000

231?

3624

4936

0t = 59'26'

5756

oFFSET (FEET) Fig. 3.5 Variation of amplitude with offset for a gas sand and a water sand. (From Yu, 1985.)

ent and also whereonly a little gas is present,so that an AVO anomaly does not necessaryindicatea com_ mercial reservoir.The measurementof AVO is also lraught with measurementand processingdifficulties (see Castagnaand Backus, 1993; Allen and peddv.

r993).

3.5 Head waves In refraction seismology,we make use of wavesthat havebeenrefractedat the critical angle(figs.3.6aand 3.6b); thesewavesare calledheadwaves,conical waves, or merely "refractions." The name ',conical waves" arisesbecausethe wavefrontsobtainedwhen we rotate fig. 3.6b about the vertical axis OL are conical surfaces generatedby wavefronts such as RO Head waveswerefirst postulatedby Mohorovi6i6in 1909to explain earthquakeobservations. In fig. 3.6a, we see a p-wave incident on the refracting horizon at the critical angle 0,. After refraction, it travels along the interface in the lower medium. This producesan oscillatorymotion parallel to and immediatelybelow the interface(as shown by the double-headedarrow just below the interface). Becauserelative motion betweenthe two media is not possible,the upper medium is forcedto move in phase with the lower medium. The disturbancein the upper medium travels along the interface with the same ve_ locity Vrasthe refractedwavejust below the interface.

Fig. 3.6 Head waves.(a) Motion at the interface; (b) wavefront emerging from relractor at the critical angle; (c) changes rn beam width upon refraction.

Let us assumethat thesedisturbancesrepresentedby the arrows reach point p in fig. 3.6b at time l. According to Huygens'principle,p then becomesa cen_ ter from which a wavespreadsout into the upper medium. After a further time interval Ar, this wavehas a radius of ( At whlle the wave moving along the refractor has reached Q, PQbeingequal to Il [t. Draw_ ing the tangent from Q to the arc of radius l( A,t, we obtain the wavefront RQ. Hence, the passageof the refracted wave along the interface in the lower me_ dium generatesa plane wave traveling upward in the upper medium at the angle 0, where

sin0 : t(AtlryLt : V,lU Thus, we seethat 0 : 0. so that the two inclined portions of the path are symmetricallydisposedwith respectto the normal to the refractor. In $3.1.4,we showedthat the energycarried by a wave was proportional to the cross-sectionof the beam; however,the beam width of the wave from M to Q in fig. 3.6b is zero, so head wavesshould have zero energy density and therefore should not exist. However,head wavesdo exist and frequently are very strong.The apparentconflict betweentheory and observationis due to the assumptionthat the sourceis at infinity so that the incident wave is plane. For a sourceat a finite distancefrom the interface,the incident wave is spherical and the situation is changed dramatically. The propagationof sphericalwavesin layeredme-

PARTITIONING AT AN INTERFACE

82

summarize the mathematics involved and Bortfeld (1962a,1962b)givessolutionsfor specialcases. The numbersand types of head wavespredictedby Cagniard'sresultsdependupon the relativevaluesof the velocities.Figure 3.7 showsthe head wavesfor an incident P-wave for two cases:(a) P, ) o, and (b) o, ) 9, > 9,. In the first case,five head wavesexist, the maximum possiblenumber; four are in the upper medium. one in the lower. In the secondcase,there are two head wavesin the upper medium and one in the lower. Problems 3.1 Derive the following results: (a) The displacementsof a free surfacefor an incident P-waveof amplitude lu are ulAo: lzlQn + n)l(la sin 0 * cos E)e:'o' 'r, \\tlAo: t-zlQn + n)l(n cos 0 + sin E) si'@' tt' wherem : (9/o)tan 26 and 4 : (a/B)cos26/sin 20. (Hint: The displacementsof a free surfaceare not restricted,so eqs.(3.3)and (3.4)haveno meaning'Set B r i n e q s .( 3 . 5 )a n d ( 3 . 6 )a n d e x p r e s rs, la n d Ar:0: w in termsof A,l AnandBrlAo.) (b) For normal incidenceon a free surface, ulAn: 0,

w lA , ' : - 2

(z: 0).

(c) At the free surfaceof a solid, where0 : 45", cr : 3 km/s, B/a : l/{i; then u l A o: 1 ' 7 9 3 '

wlAu: -l'035'

(d) At the surfaceof the ocean, ulAo: 0' (b) Fig. 3.7 Head waves. (After Cagniard, 1962.) (a) At an interface where o, ( pr; (b) at an interface where ct, > 9, > 9'.

dia has been discussedby many writers. Sommerfeld (1909)dealt with the propagationof electromagnettc wavesgeneratedby a sourceat an interfaceand with a sourceabovethe interface(Sommerfeld,1949:237 46). Joosand Teltow (1939)showedthat Sommerfeld's resultsappliedto elasticwaves.Jeffreys(1926)was the first to show clearly that the waveequationpredicted the existenceof head waves.Ewing, Jardetzky,and Press(1957:$3.3)used Sommerfeld'sresultsto develop the theory ofhead waves. The most completeaccountof headwavesis that of Cagniard (1962). He assumed a source giving a displacementof (l/r;srt*--') and usedLasteady-state place transform theory ($15.3)to obtain solutionsof the waveequationin terms of complexintegrals;convolution ($9.2)then givessolutionsfor other types of inputs.The mathematicsinvolvedis very complexand we evaluationof the integralsis difficult; nevertheless, regard the problem of the existenceof head wavesas resolved.Grant and West(1965:$6'3)and Dix (1954)

w l A o : - 2 c o s0 '

per(an incident ,SF1-component 3.2 For an ,SF/-wave pendicularto the paperin fig. 3.I ), write the boundary conditionsand find the amplitudesof all reflectedand refractedwaves.The absenceof P-wavesis important in S-wavestudies. 3.3 (a) Derive Knott's equationsand Zoepprtiz' equa' tions for a P-waveincident on a liquid solid interface when the incident wave is (i) in the liquid and (ii) in the solid. (b) Calculatethe amplitudeof the reflectedand transmitted P- and S-waveswhere an incident P-wave strikesthe interfacefrom a water layer (o : 1.5 km/s, B : 0, p = 1.0 g/cm3)at 20'when the seafloor is (i) "soft" (a : 2.0 km/s, : l'0 km/s, p : 2.0 glcm3): P "hard" (o : 4.0 km/s, : 2.5 km/s, p: 2.5 and (ii) B g/cm3). (c) Repeatpart (b) for an angleof incidenceof 30'. 3.4 Derive the Zoeppritz equations for an incident SZ-waveand (b) an incident Sf/-wave. 3.5 Show that the maximum amplitudeof an incident waveand its reflectionat the surfaceof the oceanoccurs at the depth \/(4 cos 0), where 0 is the angle of incidence,by expressingpressureI in the form used in eq. (3.2)and applyingappropriateboundary conditions.

i

REFERENCES

83

Surfaceofearth, S V:0.60

Geophone

km/s, p : 1.45g/cmr,thickness: 10 m, absorptioncoef. = 0.45 dB/\ Base of near-surfacelayer, A Source

V : 2.40 km/s, p : 2.35 glcml,thickness: 600 m, absorptioncoef. = 0.30 dB/tr Interface B V:3.20

kn/s, p : 2.68 glcm3,thickness: 800 m, absorptioncoef. = 0.25 dB/\ lnterface C V : 3.40 km/s, p -- 2.70 glcm3

Fig. 3.8

A layered model,

3.6 (a) Using eq. (2.56)to representa planewaveincident on a plane interface,show that a complex coefficientof reflection,R: a * jb, a, + b2< l, R being definedby eq. (3.14),correspondsto a reductionin amplitude by the factor (a2+ 6z7rtz and an advancein phaseby tan | (bla). (b) Show that an imaginaryangleof refraction,0, (see S2.7.5)in eqs.(3.3)to (3.6)leadsto a complexvalue of R, and henceto phaseshifts. 3.7 Calculate the reflection and transmission coefficients,R and 7 of eqs. (3.14) and (3.15),for a sandstone-shale interfacefor the following: ( a ) V " " : 2 . 4 3 , V " h : 2 . 0 2 k m / s ,p , " : 2 . 0 8 , a n d p . , : 2.23 glcm3; ( b ) V , " : 3 . 3 5 ,V " h : 3 . 1 4k m / s ,p " .: 2 . 2 1 ,a n d p , , : 2.52 glcm3 (c) What are the correspondingvaluesin nepersand in decibels? 3.8 Assumehorizontal layering,as shown in fig. 3.8, and a sourcejust below interfacel. (a) Calculate(ignoring absorptionand divergence)the relative amplitudes and energy densitiesfor the primary reflectionsfrom B and C and the multiples(see $6.3.2)BSA, BAB, and BSB (wherethe lettersdenote the interfacesinvolved).Compare traveltimes,amplitudes,and energydensitiesof thesefive events. (b) Recalculatefor l5- and75-Hz wavesallowins for absorption. (c) Recalculate amplitudes for the l5-Hz wave allowing also for divergence.Normalize valuesby letting the divergenceeffectsof reflection.Bbe unity. (d) Summarizeyour conclusionsregarding(i) the relative importanceof multiples versusprimariesand (ii) the relativeimportanceof differentattenuationmechanlsms. 3.9 Show that when angles in the Zoeppritz equations,eqs.(3.3)to (3.6),are small(so that the squares and productsare negligible), eqs.(3.14)and (3.15)are stillvalid and

B,: ]WL!+a2,, Ao (Iryt+ Wr)(Z,* Zr)' B, : ,2W9 12{ Ao (Wt + Wr)(Z,t Zr)' 4:2,02-2r0,, r:WFr_W2. 3.10 In 93.4,we statedthat Aolo is not necessarily smallwhenAa/ct,AB/B,andAp/pareall small;verify thisstatement. (Hint: Useeq.(2.60).) 3.11 How wouldyou recalibrate the scaleto change a plot showingamplitudevariationwith offset(AVO) into a plot of amplitudevariationwith angle(AVA)? Whatwill betheeffectif velocityincreases with depth? Referenecs Aki, K., and P. G. Richards. 1980. Quantitative Seismology: Theory and Methods, Yol. l. San Francisco: W H. Freeman. Allen, J. L., and C. P. Peddy. 1993. Amplitude Variation with Ofsetes: Gulf Coast Slndres. Tulsa: Society of Exploration Geophysicists. Bortfeld, R. 1962a.Exact solution ofthe reflection and refraction ofarbitrary sphericalcompressionalwavesat liquid-liquid interfacesand at solid-solid interfaceswith equal sh-earvelocities and equal densities.Geophys.prosp.. l0z i5 67. Bortfeld, R. 1962b.Reflection and refraction of soherical compressional wavesat arbitrary plane interfaces.Geophy.r.prosp., l0:517-38. Cagniard, L.1962. Reflection and Refraction of progressive Seismic Waves, E. A. Flynn and C. H. Dix, trans. New york: McGraw-Hill. Costagna, J. P., and M. M. Backus. 1993. Oflset-DependentReTheory and Practice oJ AVO,lniiysis. Tulsa: Society fectivity of Exploration Geophysicists. Denham, L. R., and R. A. R. Palmeira. 1984.Discussion on reflection and transmission of plane compressional waves. Geophvsics. 49l.2195.

84 Dix, C. H. 1954.The methodof Cagniardin seismicpulseproblems. Geop hysi cs, 19: 722-38. Ewing,W. M., W. S.Jardetzky,and F. Press.1957,Elastic Waves in LayeredMedic. New York: McGraw-Hill. Grant, F. S., and G. F. West. 1965.InterpretationTheoryin ApNew York: McGraw-Hill. plied Geophysics. Jeffreys,H. 1926.On compressionalwavesin two superposed layers.Proc Camb.Phil. Soc.,22:,472-81. Joos,G., and J. Teltow.1939.2,lr Deutung der Knallwellenausbreitung an der Trennschichtzweier Medien. Physik Z.' 40: 289-93. Knott, C. G. 1899.Reflexionand refractionof elasticwaveswith Phil Mag.,8:64 97. applications: seismological Koefoed,O. 1962.Reflectionand transmissioncoefficientsfor Prosp.,l0:. 304-5I . planelongitudinalincident waves.Geophys. Meissner,R. 1967.Exploringdeepinterfacesby seismicwideProsp.,15: 598-617. Geophys. anglemeasurements.

PARTITIONING AT AN INTERFACE Ostrander,W. J. 1984.Plane-wavereflectioncoefficientsfor gas sands at nonnormal angles of incidence. Geophysics,49" 163748. Shuey,R. T. 1985.A simplificationof the Zoeppritz equations. 50: 609-14. Geophysics, Sommerfeld,A. 1909.Uber die Ausbreitungder Wellen in der drahtlosenTelegraphie.Ann. Phys.,28: 665J 36. Sommerfeld,A. 1949.Partial DifferentialEquationsin Physics. New York: AcademicPress. and H. F. Sagoci.1965.Reflection Tooley,R. D., T. W.,Spencer, and transmissionof plane compressionalwaves.Geophysics, 30: 552-70. Yu, G. 1985. Offset-amplitudevariation and controlled50t 2697-708. amplitudeprocessing.Geophysics, Zoeppritz,K. 1919.Uber reflexionund durchgangseismischer VII WelGn durch Unstetigkerlsfliischen.Uber Erdbebenwellen det Wissenschaften B, Nachrichtender Ki)niglichenGesellschaft Math. Phys.,Kl: 57-84. zu Gdttingen,

4

Geometry of seismic wayes

Overview This chapter uses a geometrical-opticsapproach to derive the basic relationshipsbetweentraveltimeand the locations of reflecting/refractinginterfaces,most structuralinterpretationrelieson such an approach. The accurateinterpretation of reflection data re_ quires a knowledgeof the velocity at all points along the reflectionpaths. However,even if we had such a detailed knowledge of the velocity, the calculations would be tedious;often we assumea simple distribu_ tion of velocitythat is closeenoughto give useable results.The simplestassumption,which is made in \4.1, is that the velocityis constantbetweenthe surfaceand the reflectingbed.Although this assumption is rarely even approximatelytrue, it leads to simple formulasthat giveanswersthat are within the requiied accuracyin many instances. The basicproblem in reflectionseismicsurveyingis to determinethe positionof a bed that givesrisero a reflectionon a seismicrecord. In genelal.this is a problem in three dimensions.However,the dip is of_ ten very gentle and the direction of profiling is fre_ quentlynearlyalong eitherthe directionof dip or the directionof strike.In such cases,a two-dimensional solution is generally used. The arrival time-versusoffsetrelationfor a plane reflectorand constantvelocity is hyperbolic.The distanceto the reflectorcan be found from the reflection arrival time at the source point if the velocity is known. The variation of arrival time as a geophoneis moved away from the source, called normal moveout, providesthe most important criterion for identifying reflectionsand a method of determining velocity. The dip is found from differ_ encesin arrival times of a reflectionat different loca_ tions after correctionfor normal moveout:dio moveout is relatedto dip and alsoto the angleof approach of wavefrontsat the surfaceand to apparentvelocity. Reflectiordip and strike can be found from the componentsof dip moveout at the intersectionof seismic lines.Reflectingpoints move updip as source-receiver offset increases,so that the traces in a commonmidpoint gather do not have common reflectins points. Section4.2 dealswith reflectionraypathswherevelocity changesvertically;this resultsin changesin ravpath direction.One solutionin somesituaiionsis io use equivalentaveragevelocity.For parallel velocity layers,the slope of the traveltimecurve gives(in the 6)

limit as x -+ 0) the root-mean-square(rms) velocity. Vertical velocity is often expressedas a function of arrival time or depth. Where velocity is linear with depth, wavefrontsare sphericaland raypathsare arcs of circles,facts that can be usedin graphicalplotting ol depth sections. Section 4.3 concerns the geometry of head-wave pathsas usedin refractionexploration.In most cases. we assumea seriesof beds,each having a consranr velocity, the velocity increasingor *. go to deeper beds,and then we deriveformulas relatingtraveltime, offset,depth, dip, and velocities.The casesconsidered include a single horizontal refractor, several horizontal refractors,and a singledipping refractor. Velocities can be found from slopesof the traveltimeversus-offsetcurves, depths from the intercepts of projections to the source point, and dip from the differencesin depth at two sourcelocations.Wherevelocity increaseswith depth, diving waveseventually return to the surfaceeven where reflectionis not in_ volved. Refraction paths in the case of a linear rn_ creasein oveiburdenvelocity are also considered.

4.1 Reflection paths for constant velocity 4.1.I Horizontalreflectot normal moveout The simplesttwo-dimensionalproblem is that of zero dip illustrated in the lower part of fig. 4.1. The reflecting bed, AB, is at a depth I below sourceS. Energy leavingS along the direction SC will be reflected in such a direction that the angle of reflectionequals the angleofincidence. Although the reflectedray CR can be determined by laying off an angle equal to a at C, it is easierto luseimagepoinr { which is locatedon the samenormal to the reflectoras S and as far below the bed as S is above.If we join I to C and prolong the straight line to R, CR is the reflectedray (becauseCD is parallel to S{ making all the anglesmarked a equal). Denoting the averagevelocity by ( traveltimeI for the reflected wave is (SC + CR)IV. However, SC : C1, so that 1R is equal in length to the actual path, SCR. Therefore, I : INV and in terms of x, the source-to-geophone distance(ofset), we can write

Ltzlz:yzq47z,

(4.1)

GEOMETRY OF SEISMIC WAVES

86

v2t2l4h2_x2l4h2:1.

(4.2)

Thus, the traveltimecurve is a hyperbola,as shown in the upper part offig. 4.1. The geophoneat R will also record the direct wave, which travelsalong the path SR. BecauseSR is always lessthan SC + CR, the direct wave arrivesfirst. The traveltimeis /, : xlV and the traveltime curves are the straightlines OM and ON passingthrough the orig i n w i t h s l o p e so f t l / Z When distancex becomesvery large,the difference betweenSR and ,SC + CR becomessmall, and the reflection traveltime approaches the direct-wave traveltimeasymptotically. The location of the reflectingbed is determinedby measuring /,,, the traveltime for a geophone at the sourcepoint.Settingx : 0 in eq. (4.1)' we seethat

I : lvt,. Equation (4.l) can be written f - x2lv2 + 4h2lv2: x2lV2 * ti.

(4.3)

(4.4)

If we plot 12againstx2 (insteadof / versusx, as in fig. 4.1), we obtain a straightline of slope l/22 and intercept ri. This forms the basis of a well-known " schemefor determining V the X1 - T2 method"; this will be describedin $5.4.4a. We can solveeq. (a.l ) for t, the traveltimemeasured on the seismicrecord. Generally 2ft is appreciably largerthan x, so that we can usea binomial expansion ( $ 1 5 . 1 . 4 ca)s f o l l o w s : + (xl2h)' ltt' : /0[l + (xlVt)2lt'' t : Qhlnl (4.5) : /0u + t{xtVtJ, -'U{xlVto'So + . . .1. lf t. t., x,, and x, are two traveltimesand offsets,we haveto the first approximation Lt : t, - t, = (xl - x])l2Vrt,,.

(4.6)

In the special case where one geophone is at the sourcepoint, At is known as the normal moveout (NMO), which we shall denoteby Lt,*o. Lt rro - x2l2V2tu: x2l4Vh'

(4.7)

At times,we retain anotherterm in the expansion(see alsoproblem4.lc): Ltfiro:

x2l2v2t,,- xalSvatl : (v2l2V' t)[l - (xl4h)' ].

(4.8)

From eq. (4.7), we note that the normal moveout increasesas the squareof the offsetx, inverselyas the squareof the velocity,and inverselyas the first power of the traveltime(or depth - seeeq. (4.3)).Thus, reflection curvatureincreasesrapidly as we go to more distant geophones;at the sametime, the curvaturebecomesprogressivelylesswith increasingrecord time. The concept of normal moveout is extremelyimportant. It is the principal criterion by which we de-

\

l-) I

Fig. 4. I

Traveltime curve for a horizontal

reflector

cide whetheran eventobservedon a seismicrecord is a reflectionor not. If the normal moveoutdiffersfrom the valuegivenby eq.(4.7)by more than the allowable experimentalerror, we are not justified in treating the eventas a reflection.One of the most important quantities in seismicinterpretationis the changein arrival time causedby dip; to find this quantity,we must eliminate normal moveout.Normal moveoutmust also be "stacking" (adding together) eliminated before common-midpoint records (see $8.3.3).Finally, eq. (4.7) can be used to find Z by measuringx, /n, and Lt *rr,,, this forms the basis of the Z-Aln method of findingvelocity(see$5.4.4b)and alsoof velocityanalysis ($9.7).Brown (1969) discussesrefinementsto handle dip and long offset. 4.1.2 Dipping reflector;dip moveout When the bed is dipping in the direction of the profile, we havethe situationshownin fig.4.2, { beingthe dip, and /r the distancenormal to the bed. To draw the raypath for the reflectionarriving at geophoneR, we join imagepoint 1to R by a straight line, cutting the bed at C. The path is then SCR, and t is equal to (SC + CR)IV;becauseSC + CR : 1R, applicationof the cosinelaw to triangle S1Rgives V2f :

JR2

: x2 + 4h' - 4hxcos(ln + i) : x2 * 4h2+ 4hx sin l.

(4.e)

REFLECTION PATHS FOR CONSTANT VELOCITY

87

On completingthe squares,we obtain

- (x + 2h sin {), (2h"* €F (2h cosO' V2t2

= l.

Thus, as before,the traveltimecurve is a hyperbola, but the axis of symmetry is now the line x : -2h x sin { instead of the l-axis. This means that r has different values for geophonessymmetricallyplaced on opposite sidesof the sourcepoint,unlike the case for zero dip. Settingx equalto 0 in eq. (4.9)givesthe samevalue for ft as in eq. (4.3); note, however,that h is not measured vertically as it was in the earlier result. We call points C, C' , C" in fi5. 4.2, where the anglesof incidenceand reflectionare equal,reflectingpoinls.(These are sometimecalled "depth points," but this term is also usedfor the point on the surfacemidwaybetween sourceand receiver;we call the latter a midpoint, and to avoid confusion we shall avoid the term "depth point.") The updip displacementof reflectingpoints compared to midpoints for dipping reflectorsis important in migrating data (99.10.2)and in the common-midpointmethod(98.3.3). To obtain the dip, (, we solve for I in eq. (4.9) by assumingthat 2h is greaterthan x and expandingas in the derivationof eq. (4.5).Then

'!(,*"'* 111''" g) v\ 4h, )

,."\(, *

x' + 4ftr sint).

th '

(4.10)

I

using only the first term of the expansion.The srmplestmethodof finding t is from the differencein traveltimes for two geophonesequally distant from and on oppositesidesof the source.Letting .r in fig. 4.2 havethe values*Ax for the downdip geophoneand -Ar for the updip geophoneand denotingthe equivalent traveltimesby l, and /,, we get

''" ,], r - r. * (Ar)';f,o" [r r. - r.

+

(44'-

8h2

[r at,,: t,-

4h A x s," ,]

/A" 'in €\ I. : ,.

\

o

)-

2a'x

nsrnt'

Dip { is given by

sin{-:r(,4*)

(4.r1)

The quantity L,tolAxis calledrhedip moveout.(Note that dimensionally, dip moveout is time/distance, whereas normal moveout is time. Note also that

Fig. 4.2

Traveltime curve for a dipping reflector

DMO or dip-moveoutprocessingt$9.10.21 involves different concepts.)For small angles,{ is approximatelyequalto sin (, so that the dip is directly proportional to A/, under thesecircumstances. To obtain the dip as accuratelyas possible,we use as large a value of Ax as the data quality permits; for symmetrical spreads(S8.3.1), we measuredip moveoutbetweenthe geophonegroups at the opposite ends of the spread, Ax then beinghalf the spreadlength. Dip moveout can also be measuredby the time difference between 1,,at different sourcepoints.As shownin fig. 4.3, A/u : /,,,- /,,,and

sin{ : I;, (u. U*),

(4.12)

whereAx is the distancebetweensourcepoints.When we measuredip on a recordsection($8.8.3), Ax is the distancebetweenany two convenientpoints. It should be noted that normal moveout was eliminated in the derivationof eq. (4.ll). The terms in (Ax)' that disappearedin the subtractionrepresentthe normal moveout. Figure 4.4 illustratesdiagrammaticallythe relation betweennormal moveout and dip moveout.Diagram (l) representsa reflection from a dipping bed; the alignment is curved and unsymmetrical about the sourcepoint. Diagram (B) shows what would have been observedif the bed had been horizontal; the alignment is curved symmetricallyabout the source position owing to the normal moveout. The latter rangesfrom 0 to 13 ms (l millisecond: 10-3s = I ms, the unit of time commonly usedin seismicwork) at an offset of 400 m. Diagram (O was obtained by subtractingthe normal moveoutsshown in (B) from the arrival timesin (A).The resultingalignmentshows

88

GEOMETRY OF SEISMIC WAVES

tr.,: \/sin a:

2rk",

(4.13b)

where L" is the apparentwavelength,and r"l2r the apparentwavenumber. Equation(4.13a)is somewhatsimilar to eqs.(4.I I ) and (a.l2), but it has a different significance,becauseit givesthe direction of travel of a planewaveas it reachesthe spread,Vbeingthe velocity between C and the surface. In eqs. (4.11) and (4.12), V is the averagevelocity (54.2.2)down to the reflector,and ( is the angle of dip. Becausesin o can be very small, the apparent velocity \ (and L,) can be very large, and for energyapproachingvertically,

4: *'

Fig.4.3 Geometry involvedin dip moveout measured between sourcepolnts or on recordsections. the effect of dip alone; it is straight and has a time differencebetweenthe outside curves of l0 ms, that is, A/, : l0 ms when Ax : 400 m. Thus, we find that t h e d i p i s 2 5 0 0 ( 1 0x l 0 - Y 8 0 0 ): 0 . 0 3 1r a d : . 1 . 8 ' . The method of normal-moveoutremovalillustrated in fig. 4.4 was usedto demonstratethe differencebetween normal moveout and dip moveout. If we require only the dip moveout, Lto, we merely subtract the traveltimesfor the two outsidegeophonesin (l). Frequently,we do not have a symmetricalspread and we find the dip moveoutby removingthe effectof normal moveout. As an example, refer to fig. 4.4, curve (D), which shows a reflection observedon a spreadextendingfrom x : -133 m to n : +400 m. L e t / , ,: 1 . 2 2 5s , t r : 1 . 2 2 3s , t , : 1 . 2 4 2s , a n d Z : 2800 mis. From eq. (4.7), we get for Atrro at offsets of 133and 400 m, respectively, the values I ms and 8 ms (rounded off to the nearestmillisecond because this is usually the precision of measurementon seismic records).Subtractingthesevalues,we obtain for the correctedarrivaltimes /, : 1.222and t,: 1.234: hence,the dip moveoutis l2l(53312)ms/m. The correspondingdip is ( : 2800(12x l0 V533) : 0.063 rad : 3.6'. An alternativeto the precedingmethod is to usethe arrival times at 133m and x : +133 m, thus obtaining a symmetricalspread and eliminating the needfor calculatingnormal moveout.However,doing this would decreasethe effectivespread length from 533 m to 266 m and thereby reduce the accuracyof the ratio (L,tolL,x). The apparentvelocity V. of a wavefront is the ratio of the distance(Ax) betweentwo points on a surface (usually,the surfaceof the ground) to the difference in arrival times (Al) for the same event at the two points. It is given by V , : L , x l L t : V J s i na ,

(a.l3a)

where o is the angle of approach(fig. 4.5); o is sometimes calledapparentdip. We can divide this equation by the frequency to give

4.1.3 Cross-dip When the profile is at an appreciableangle to the direction of dip, the determinationof the latter becomes a three-dimensionalproblem and we usethe methods of solid analytical geometry.In fig. 4.6, we take the xy-planeas horizontal with the z-axisextendingvertically downward. Line OP of length ft is perpendicular to a dipping planebed that outcrops(that is, intersects the xy-plane)along line Mll if extendedsufficiently. We write 0r, 0,, 0, for the anglesbetweenOP and the x-, y-, and z-axes,and (., m, n for the direction cosines of OP The angle E between MN and the xaxis is the direction of strike of the bed while 0, : {, the angle of dip. The path of a reflectedwave arriving at geophone R on the x-axiscan be found using imagepoint L The line joining 1 to R cuts the reflectorat Q; hence,OQR is the path. BecauseOQ : QI,line 1R is equalto Zl, t beingthe traveltimefor the geophoneat R. The coordinatesof 1 and R are respectively(2h(, 2hm,2hn) and (x, 0, 0); hence,we have p1, : : : :

(IR)2 (x - zhe)' + (0 - 2hm)' + (0 - 2hn)' x2 + 4h2(C,+ m2 i n2)* 4hh x2 + 4h'- 4hk,

because(2 * m2 + n2: I (problem15.9a). When x : 0. we obtain the samerelation between h and tuas in eq. (4.3).Proceedingas in the derivation of eq. (4.I 0), we get for the approximatevalue of /, /:ro(l+

x) - 4h(x\

8n, )

By subtracting the arrival times at two geophones locatedon the r-axis at "r : +Ar, we find

at, : t,,,((L,xlh) :2( d:cos0,

A,xlV.

- j"ffit;)

(4.14)

REFLECTION PATHS FOR CONSTANT VELOCITY

89

+400 m +383 +267 +200 + 133 +67 ,o

0 -67 -

IJJ

- 200 -26'l - 333 -400 m

Fig. 4.4 Relation between normal moveout and dip moveout. For curves (A), (B), and (Q, /" : 1.000s and Z : 2500 m/s. For

curve(D), to: 1.225s, t, = 1.22!s, t.: m/s. /is the averagevelocity.

1-242s, and I/ = 2800

(x*mylnz:h. Settingz : 0 givesthe equationofthe line ofintersection of the reflectorand the surface;this strikeline has the equation (x*mY:ft' The interceptsof this line on the x- and /-axes are hl( and hlm. Referring to fig. 4.7, we find that tan ,= : Fig. 4.5

Finding the angle of approach of a wave

: q{.14{)

If we also have a spread along the y-axis (crossspread), we get

,??: cos,r: !;, (a;;),

(4.r 5)

whereAr, is the time difference("cross-dip")between geophonesa distance 2 L,y apart and symmetrical about the source.Because ,4 : cos € : tl _ (C2+ m2)lr2 sin € :

(l

-

4z|D :

C hlm : ,,. nt( m

(C2 + m2)tt2

: ln [(o*)'* f1")'1"' (416) 2 [\Ax/ \Ayl I The componentsof dip moveout, Lt.lLx and Lt,,lL'y, are also called apparentdips. To find the strike E, we start from the equation of a plane (that is, the reflector)that has a perpendicular from the origin of length ft and direction cosines({, m, n), namely(seeproblem 15.9b),

(L,t,lA,y)

(4.r1)

Consider the case where the profile lines are not perpendicular,for example,where they are in the r, and r, directions of fig. 4.8a and the dip is in the ro direction. We expressthe dip moveout as the vector (drldx)ro: AO; the componentof dip moveouton the line in the r, directionis thus (dtldx)ro' r, : (dlldx)cos 9 : OB (seeproblem 4.2a).The converseproblem of finding the total dip moveout from measurementsof the componentsof dip moveout OB and OC can be done graphically,as shown in fig. 4.8b (seealso problem 4.2b), or mathematicallyas follows.We take one profile along the x-axis and the other along the y'-axis at an angle a to the x-axis. By taking the length of a symmetricalspread along the y'-axis as 2Ay', the coordinatesof the ends of the spread(relativeto the r-, y-axes)are * Ly' cos a, +Ay' sin o. Then V2ir*: (2h( t- Ay' cos c)2 + (2hm + Ay' sin a)2 + (2hn)2 : (Ly'), + 4h2-r 4hLy'((. cos ct + zt sin a).

GEOMETRY OF SEISMIC WAVES

90

path for a dipFig. 4.6 Three-dimensional view oi a reflection ping bed.

The dip moveoutalong this line, At'lAy', is Lt' ILY' : 2(( cosa * m sin o)/ Z

Fig.4.7

D e t e r m i n a t i o no f s t r i k e

(4' 18)

Becausect is known, { can be found from AllAr and m f r o m e q .( 4 . 1 8 ) . 4.t .4 Reflet'tionpointsfor ffiet receivers When the sourceand receiverare coincident and the velocity is constant,the locus of a reflectingpoint R for constant traveltimeas the dip ( varies is a circle (fig. 4.9a).However,when the sourceand receiverare ofset by'-x, the locus is an ellipse(fig' a'9b) with the sourceand geophoneat the foci; this follows from the : Vt is a definition oi an ellipsebecauseSR + RG given by is I traveltime the for constant.The equation (4.9), namelY, eq. (Vt)' : 4h2+ 4s2+ 8ftssin {' Expressingthis in terms of the depth at the midpoint M,h':/r*ssin{: (4'19) (Vt)' : 4(h')' + 4s2cos2{

Apparent dip 20 ms/km

(comparewith problem4.3); Levin (1971)writesthis equation: t' : 4(h')' 1V2* 4s2lV]ro,

z

2hn'

2' -l

s+h(

::

\eo

\e6-

where V*ro= Vlcos(; thus, hro> V n ng.'i."lU,the reflectionpoint R has moved updip by fR": AL. To determine AI, we find the coordinut.t 1"0,zo)and 6r, z,l of points P and. R-.Because ( ur it'putiit"l to 51' xo is s h'(' and zois.h'n' and sourcethe 2s and S'I, n being direction cosines of joing.optto'* distance.lf (x' z) is a point on the line ing l and G we must have

2' - ! :21!J]

\a

9.20)

: 1r,

hn

where k is a parameterthat fixes the location of the point (x, z) albng,IG To get k' we use the fact that IG

\-d

\a-

-

\ --

\-io

\'

- - - - t\

Fig. 4.8 Determining dip and strike from nonperpendicular O ob"r".uations. (a) Relaiion between the point of observation (b) ExO) from updip (l always is and the reflecting point,4 ample of a graPhical solution.

VERTICAL VELOCITY GRADIENT AND RAYPATH CURVATURE cuts the reflector at R, so (x, z) must satisfy the equation ofthe reflectingplane (seeproblem 15.9b): -(x*nz:h.

9l

changesfrom one velocity function to another do not necessarilyimpose a seriousburden upon the interpreter.

Substitutingthe previousvaluesfor (x, z), we get -(l2s - k(hc + s)l + n(khn) : ft,

4.2.2 Equivalentaveragevelocity

that is, k: (h + 2k)l(h + (s) : (h + 2h)lh'. Therefore, the coordinatesof R are (seeproblem 4. I l). x, - 2s - k(h( + s) : 2s - (h( + s)(lr + 2k)lh' : xo - (n2s2lh' (4.21a) , and k(hn) : (h + 2(s)(hn)lh': zu - (,2ns2lh'.

z,:

(4.21b) Finally, (AL)'

("0 -

:

x,), * (zo - z,)2 (s2lh')2((2n4 + (4n2)

:

(s2lh')2((2n2),

:

and 67 : (s' lh')sin{ cos g : (s,l2h')sin2f. G'2Ic\ If we wish to stackdata elementsthat havethe common reflectionpoint R, we have to stack updip from the midpoint by the distanceAx, where Ax : Atlcos 6 : (srlft,)sin{.

(4.22a)

The updip offsetAx changesthe zero-offsettime by A,t : 2 A,x sin llV : 2(s2lh,Z)sinr(. (4.22b) The DMO (dip moveout)correction(99.10.2) accommodatesthis updip movementof the reflectingpoint as offsetincreases. 4.2 Vertical velocity gradient and raypath curvature 4.2.1 Efec't of velocityvariation The assumptionof constant velocity is not valid in general,the velocity usually changingas we go from one point to another.In petroleumexploration,we are usually dealing with more or less flat-tying bedding and the changesin seismicvelocity as we move horizontally are for the most part small, being the result of slowchangesin densityand elasticpropertiesof the beds.Thesehorizontal variationsare generallymuch lessrapid than the variationsin the vertical direction wherewe are going from bed to bed with consequent lithological changesand increasingpressurewith increasingdepth. Becausethe horizontal changesare gradual, they can often be taken into account by dividing the surveyarea into smallerareaswithin each ofwhich the horizontal variationscan be ignoredand the same vertical velocity distribution used. Such areasare often large enough to include severalstructures of the size of interest in oil exploration so that

Vertical variations in velocity can be taken into account in various ways.One of the simplestis to use a modification of the constant-velocitymodel. We assumethat the actual sectionexistingbetweenthe surface and a certain reflectinghorizon can be replaced with an equivalentsinglelayer of constantvelocity 7 equal to the averagevelocity betweenthe surfaceand the reflectinghorizon; 7 is the equivalentaveragevelocity. This velocity is usually given as a function of depth (or of tu,which is nearly the sameexceptwhen the dip is large).Thus, the sectionis assigneda different constant velocity for each of the reflectorsbelow it. Despitethis inconsistency, the method is usefuland is extensivelyapplied.The variation ofthe averagevelocity with /,, is found using one of the methods describedin $5.4.For the observedvaluesof the arrival time to,we selectthe averagevelocityTcorresponding to this reflector; using the values of lo, the dip moveout, LtulL,x, and \ we calculatethe depth ft and the dip ( usingeqs.(4.3)and (4.I 1).

4.2.3 Velocitylayering When the velocityis constant,eq. (4.l) showsthat a graph of tr versusrr is a straight line with slopellV2. Ifthe velocityvariesin the verticaldirection,raypaths will bendas requiredby Snell'slaw (eq.(3.1)).A commonly used method to take into account vertical velocity variationsis to replacethe actualvelocitydistribution with a number of horizontal layersof different velocities, the velocity being constant within each layer. We can approximate any vertical velocity changesas closelyas desiredby using enough layers. A graphical method using a wavefront chart can be usedto find the depth and dip ofa reflectinginterface; the preparation and use of these charts will be disc u s s e di n 9 8 . 8 . 3 . In effect,we replaceactual raypathswith a seriesof line segmentsthat are straight within each layer but undergoabrupt changesin direction at the boundaries between layers. Larger portions of travelpaths are spent in the higher-velocity layers as the sourcegeophonedistanceincreases. The resultis that a graph of /2versus,r' is slightlycurved,as shownin fig. 4. l0b. Dix (1955) showedthat eq. (4.4) can still be used exceptthat the slope ofthe :r2 t2curve at x : 0 yields the inverseof the rms velocity squared,llV,2^".Weapproximate the x2-t2 curve by the straight line

t' :x2lV!^"+tfi; hence,

dtldx : xlVl^, t.

(4.23)

G E O M E T R Y O F S E I S M I C WAVES

92

- S + S +

S,G

s

G

ilrvr

(a) Fig. 4.9 Loci of reflection points for various dips. (a) Coincident source and geophone; (b) geophone offset from source.

The angleof approach,l' is given by

sini, : n,'dx + : :L V,'*,t

g'24)

using eq. (4.23). Also, writing A/, for the vertical traveltime through the ith bed, and, becausex ts small. we have

jx : : =

Ax, * Ax, : hrtan i, + h2tan v, Lt, sin l, * v, Lt.,sin i, i,lV, (4 Lt, + Vl L,tr)sin + Ltr)(x.lvh"t) (V1 Lt, Vl

from eq. (4.24) (note that x cancels here becausewe haveasiumedit to be small).Becauset - 2(L't, + Alr)' we get

i ,,o,. i't

'Vr2m s:

i

e

It

This equationcan be generalizedfor r horizontal beds (Dix, 1955),giving lz

v2

-

v 2

-+ v2

r.,

(4.2s)

(4.26)

Shah and Levin (1973)give higher-orderapproximations necessaryto get more accuracy for large values of x. Fig. 4.10 Derivation of the formula for x2-t2velocity in twolayermedium.(a) Reflectionpath. (b) xt-l2 curve(the curvature The reciprocalof the slope of the dashedline is exaggerated). (tangentto the curve at x : 0) givesthe rms velocity'The bestfit stiaight line for someportion of the curve is what is often *ru.r.tid; the slopeof this line (showndashed)is the reciprocal "stackingvelocity" of the squareof the % ($5.a.aa);it depends on the portion beingfit.

4.2.4 Effect ofvariable velocity on raypath direction Changesin the direction of rays at interfaces are determiied by Snell'slaw (eq. (3.1)).For planar parallel layering (fig. a.ll), the angle of emergencefrom a tayer equatsthe angle ofentry into the next layer and

V E R T I C A L V E L O C I T Y G R A D I E N T A N D RAYPATH CURVATURE the raypath parameter p : (sin i)lV : (sin i)lVo : ArlAx (seeeq. (4.13a))specifiesray direction,'thai is, p is constant along any ray and is fixed by the direction in which the ray left the source.Note that l/Z is the slownessandp is the component of slownessparallel to the interface,hence,the componentof slowness parallel to the interface is constant for each ray. In earthquakestudies,it is often assumedthat the earth is divided into concentric sphericalshells(layers) of constant velocity, as in fig. 4.12. In this case, the angleofentry into a layeris not equalto the angle of exit from that layer, that is, i, * it. However. becauseOP : r, sin r, : r. sin i!. uiing Snells law shows that (r, sin ir)lVr: (r, sin r.)/2,. Thus, in this case, direction can be specifiedby a raypath parameterp,.. p' : (t"sin i")|V".

4

dz

V:

I t a ni d z , I

I

t:

l

lz

l" I

PVdz

- lPvz'1trz' l. [t

-u'1"':o'""' t : l"'.o l, ;'1."r,'-oi,,r, ),"

: j f"o, ro- cost),

,:tl'

a ),"u(l -

(4.31)

. : l r -n f + ( l u-

u2)tt2 a

[l

ll'

u,1trzjl,

:;,'lffi(H3;1 :''"

tan ! i. 2 l tt,|r):

(4.32)

P:[l+(x')213/2lx',

(4.28)

,/ cos I g

where

z

J .z . o r , - - '

hence, -r ' :

.:

The parametric equations(4.31) and (4.34) give the coordinates x and z, the parameter i being related to the one-waytraveltimer by eq. (4.32)or (4.33). The raypath given by eqs. (4.31) and (4.34) is a circle; this can be shown by calculatingthe radius of curvature p, which turns out to be a constant:

V(z),

FZ

x:

V:Vo*az, where Voisthe velocity at the horizontal datum plane, Z is the velocity at a depth z below the datum plane, and a is a constant whosevalue is generallybetween 0.3/sand 1.3/s. If we introducea new variableu : pV : sin r, then du : p dV : pa dz, and we can solvefor x and t as follows (p is the raypath parameter):

i : 2 tan t(e. tan jil, (4.33) : z (V V)la: (sin i sin io)lpa. (4.34)

In the limit when n becomesinfinite, we get

dz

Sometimeswe can expressZas a continuousfunction of z and integrateeqs.(4.29)and (4.30).One caseof considerableimportanceis that of a linear increaseof velocity with depth, namely,

4 = 1Q),

A,x, : Az, tan i,, Lz' At,: \cos i,'

-:t3fl/.

4.2.5 Linear increaseof velocity with depth

hence,

: sin ro _ , V o r '

sini_sinro:, V V o r ' dx dt -

(4.30)

- (Pv7'1"'

BecauseZ is a function of z, eqs.(4.29) and (4.30) furnish two integral equationsrelatingx and r to the depth z. Theseequationscan be solvedby numerical methods when we have a table of values of Z at various depths.

(4.27)

At times,the assumptionis made that the velocity varies in a systematiccontinuousmanner and there_ fore can be representedby a velocity function. The actual.velocity usually varies extremelyrapidly over short intervals,as shown by sonic logs (see $5.a.3); however,if we integratethesechangesover distances of a wavelengthor so (30-100 m), we obtain a func_ tion that is generallysmoothexceptfor discontinuities at marked lithological changes.If the velocity discon_ tinuities are small, we are often able to representthe velocity distribution with sufficient accuracy by a smoothvelocityfunction. The path of a wavetraveling in such a medium is then determinedby two integral equations. To derivethe equations,we assumethat the medium is divided into a large number of thin bedsin each of which the velocity is constant;on letting the number of beds go to infinity, the thicknessof each bed becomes infinitesimal and the velocity distribution be_ comes a continuous function of depth. Referring to fig. 4. I l, we havefor the nth bed srl_L

,:|;,

93

(4.2e)

, :d x tan i, using eqs.(4.31) and (4.34), i: ,, d2x d. .di dl x : ;.: ;.(tanr). : sec,i oz. o, dz dz : pa sec2i,using eq. (4.34). f

94

GEOMETRY OF SEISMIC WAVES

s

F-'"--l l

\

Hence,

l

( I + tan2i)r/2 | "f I = : l lV^\ .:constant. pasec't pa \4/slnro

l

\

r

t

t

l

- - - F - J \

Fig.4.1I

l

Raypathwherevelocityvarieswith depth.

Figure 4.13 shows a ray leaving the source at the angle lo. The center, O of the circular ray lies above the surface a distance p sin io, that is, Vola. Because this is independentof io,the centersof all rays lie on the samehorizontal line. This line is locatedwherethe velocity would be zero if the velocity function were extrapolatedup into the air (becausez : -Vola at this elevation). To determinethe shapeof the wavefront,we make use of fig. 4. 14.The raypathsSl and S.Bare circular arcs with centersO, and O, respectively.If we continue the arcs upwardsto meet the vertical through S at point S', line O, O, bisectsS'S at right angles.Next, we selectany point C on the downward extensionof S'S and draw the tangentsto the two arcs, CA and C.B.From plane geometry,we know that the squareof the length of a tangent to a circle from an external point (fiorexample,CA' ) is equal to the product of the two segmentsof any chord drawn from the samepoint (CS. C,S'in fig.4.14).Using both circles,we seethat CS' CS' : CA2: CB2, hence, CA : C,B.Thus, a circle with center C and radius R : C,4 cuts the two raypathsat right angles. Because,Sl and S.Bcan be any raypaths and a wavefront is a surfacethat meetsall rays at right angles, the circle with center C must be the wavefront that

Fig. 4.12 Seismicray in a sphericallylayeredearth with construction to show the geometricsignificanceof the ray parameter.

Fig. 4.13

Circular ray leaving the source at the angle L.

Fig. 4.14 Construction of wavefronts and raypaths for linear increase of velocity.

REFRACTION PATHS passesthrough A and,B.Even though arc SA is longer than S4 the greater path length is exactly compensatedfor by the higher velocity at the greaterdepth of raypath sl. We can draw the wavefront for any value of r if we can obtain the valuesof 11 and R in fig. 4.14. Thus, the quantities .F1and R are equal to the values of ; and x for a ray that has i : ]r at time r, that is, SD in 'crf the diagram. Substitution i: jn in eqs. (4.31), (4.33),and (4.34)yields tan)io: s '', sin lo : sechct, cos r0 : tanh at, H : (llpa)(l - sin io) : (Vola)l(llsini) - 1l : (Z/a)(coshat - l), (4.35) R : (llpa)cos io: (Vola)cotio : (Vola)sinhat. (4.35) Equation showsthat the centerof the wavefront moves downward and the radius becomeslarger as time increases. Field measurementsyield valuesof the arrival time at the source/oand angleofapproach ArlAx. Because the ray that returns to the sourcepointmust have encountereda reflectinghorizon normal to the raypath and retracedits path back to the point of origin, the dip is equal to angle i, at time t : t"tn.Thus,to locate the segmentof reflectinghorizon cbrrespondingto a set of valuesof t,, and AllAx, we make the following calculations: (a) , : )ro,

95 of the overlying bed so that it never carries a head w a v e( s e e$ l 1 . 2 ) . 4.3.2 Singlehorizontal refractor For the caseof a singlehorizontal refractinghorizon, we can readily derivea formula expressingthe arrival time in terms of the offset, the depth, and the velocities. In fig. 4.15, the lower part shows a horizontal plane refractor separatingtwo beds of velocities Z, and V2,whereVr) V,.For a geophoneat R, the path of the refractedwave is OMPR, 0. being the critical angle.The traveltime/ can be written

t :

OM

+

MP , P R 1-

Vt

vt

V2

: \- f! !?"s. + v2

_x T V2

MP v2

-f

^OM vl

z

2h "Z, cos0,

2h l r r- 4 r i n e'I \ tr/,cos 0 \ V,

x

2hcos0

V2

Vl

(4.36)

wherewe haveusedthe relation sin 0. = V,IV.inthe last step.This equationcan also be wiitten t : (xl Vr) -t t,,

(4.37)

where t,:

(2hcos0,)/2,,

(4.38)

(b)io:,'"-'(n.i,i), (c) j, : 2 tan-t(s"' tan

;,0), (d) H :(V/a)(cosh at * l), (e) R: (Vola)sinhat. With thesevalues,we find C lay off the radius R at the angle i,, and draw the reflectingsegmentperpendicular to the radius, as shown at the point A in fig. 4.14.This method is easilyadaptedto a simpleplotting machine (Daly, 1948) or to wavefront charts (Agocs,1950). Refractionstudiesinvolvinglinear increaseofoverburden velocity are discussedin 94.3.6. 43 Refraction paths 1.3.1General Refraction seismology involves the study of head waves($3.5)using primarily first arrivals,the equivalent of first breaksin reflectionseismology(see,however,secondarrivals,$11.2).For a head wave to be generated,the velocity below an interface must be higherthan that aboveit; accordingly,we shallassume rn the following sectionsthat the velocity increases downwardmonotonically.However,this is not always thecase,and problemssometimesresultfrom a hidden thlind) zone,a layer whosevelocity is lower than that

h :

) V , t , l c o s0 , .

Obviously,the head wave will not be observedat offsetslessthan the critical distance,OQ in fig. 4.15, writing x' for the critical distance, x' : OQ: 2htan0.: 2htan[sin | (V1lVr)) : 2h[(VrlV,)2 -

lfi/2.

(4.39)

The relation betweenx'lh and V.lV, is shown in fig. 4.16.As the ratio VrlV,increases, x'decreases.When VrlV, equals1.4,x' is equal to 2h. As a rule of thumb, offsetsshould be greaterthan twice the depth to the refractor to observerefractionswithout undue interferencefrom shallowerhead waves. Equations(4.36)and (4.37)representa straighrline of sfope llV, and intercepttime tr This is illustrated in fig. 4.15, where OMQ, OMP'R', OMPR, and OMP'R' are a seriesof refractionpathsand DWS the corresponding time-distance curve. Note that this straight-lineequationdoesnot havephysicalmeaning for offsetslessthan r' becausethe refractedwavedoes not exist for such values of x,' nevertheless, we can project the line back to the time axis to find r,. The problem to be solved usually is to find the depth h and the two velocities V, and Vr. The slope of the direct-wavetime-distancecurve is the reciprocal of V, and the same measurement for the refraction eventgives V,We can then calculatethe critical angle

96

GEOMETRY OF SEISMIC WAVES

0. from the relation 0" : sin-t (VtlV), and use the intercepttime, t,, to calculateft from eq. (4.38). In fig.4.15, the time-distancecurvesfor the reflection from the interface AP" and for the direct path are representedby the hyperbola CDE and the straight line OF, respectively.Becausethe path OMQ can be regarded either as a reflection or as the beginning of the refracted wave,the reflection and refraction timedistancecurves must coincide at x : x', that is, at point D. Moreover,differentiatingeq. (4.1) to obtain the slope of the reflection time-distance curve at x : x', we find

fa,l:["]

l a " J" =.,|,q i l , -'

:L(_o2

part of the refraction path is traversedat velocity V, so that as x increases,eventually the refraction wave will overtakethe direct wave. In fig. 4.15, these two traveltimes are equal at the point W lf the offset correspondingto W is r., we have

Vt

We see. therefore. that the reflection and refraction curves have the same slope at D, and, consequently, the refraction curve is tangent to the reflection curve atx:x'. Comparing reflected and refracted waves from the samehorizon arriving at the samegeophone,we note that the refraction arrival time is always lessthan the reflection arrival time (except at D). The intercept time /, for the refraction is lessthan the arrival time /o for the reflectionat the sourcepointbecause t, : (2hlVr)cos0,,

to: 2hlV,;

hence,/, < lo. Starting at the point Q, we seethat the direct wave arrivesaheadof the reflectedand refractedwavesbecauseits path is the shortestof the three. However,

2h

Vt

V2

z

cosu.,

=:('';,'') o+,,,,,, =t(ffi,1'"

\

V2

Y

, r : t l- ,L,)I *' r,

(4.40)

v,\oM + M el

= l s l n 0 : -l

Y

This relation is sometimesused to find ft from measurementsof the velocities and the crossoverdistance x.. However,usually we can determinetr more accurately than x. and henceeq. (4.38) providesa better method of determiningh. The relation between4ift and VrlV, is shown in fig. 4.16. 4.3.3 Severalhorizontal refractors Whereall layersare horizontal,eq. (4.36)an be generalized to cover the caseof more than one refracting horizon. Considerthe situation in fig. 4.17, wherewe have three layersof velocities,V,, V, and Zr. Whenever Vr) V,we havethe refraction path OMPRand corresponding time-distance curve I4/g just as we had in fig.4.15.If Vr) Vr) V,,travelby a refraction path in V, will eventually overtake the refraction in V, The refraction paths such as OM' M" P" P' R' are fixed by Snell'slaw: sin 0, v.l

: s,tl&: v2

where 0., is the critical angle for the lower horizon and 0, is lessthan the critical anglefor the upper horizon. The expressionfor the traveltimecurve SZis obtained as before:

'| lo'

t : :

OM' + R'P' M'M' + P'P" + - + vl v2

M'P' v3

2h, 2h, * Z, cos 0, Vrcos 0,, x - 2h. tan0. - 2h tan 0

V. x Vj

2h,

l.

+------...........---ll

Z, cos 0", \

-

ftri"e",)

2h, (t * rin e,) Z, cos0, \ ,4

: - ^+ A

M

P

'

y

z

P

P

"

Fig. 4.15 Relation between reflection and reliaction raypaths and traveltime curves.

L/

- - - - 2 c o s 0 . -+ 2h 2h + ,, + vl v2

cos0,:

+

t

(4.4r)

97

REFRACTION PATHS

curve to permit it to be analyzed correctly. We can find all of the velocities(hence,the angles0, also) by measuringthe slopes of the various sectionsof the time-distancecurve and then get the thicknessesof the layers from the intercepts

&.""

(4.43) '.: t-#,.. ('.- Z'^'';,'u')

:/depth,

' z

4.3.4 Effect of refractor dip

cr,-. St/car lce/depth)

0

l

3

5

V z /V t

Fig.4.16 Relation between critical distance x', crossoverdisrance x", and velocity contrast.

The simple situations on which eqs. (4.36) to (4.43) are basedare frequently not valid. One of the most serious defects is the neglect of dip because dip changesthe refraction time-distance curve drastically. The lower part of fig. 4. I 8 showsa vertical dip section through a refracting horizon. Let t be the traveltime for the refraction path OMPO'. Then, we have I

I I I I v,l o

I

o

OM + O'P MP T Vt V2 (ho+h,)tan9, _ hd+ h, *OQZ, cos 0. V2 : x c o s '(* u cho, s+0h. .. . (4.44)

_

V2

Vl

If we place the source at O and a detector &t O' , we are "shooting downdip." In this case,it is convenient to have I in terms of the distance from the source to the refractor ho; hence, we eliminate fr, using the relation h,:

h o* x s i n ( .

Writing t, for the downdip traveltime, we obtain to: (xlVr)cos( + (x/()cos 0. sin | + (2hulV,)cos0, : (xlV,)sin (e" + €) + (2holV,)cos0, :(xlV,)sin(0.+0+t,d,

(4.4s)

where tro: (2holVr)cos 0,. Fig. 4.17 Raypaths and traveltime curves for two horizontal refractors.

Thus, the time-distance curve for this refraction is also a straightline whoseslopeis the reciprocalof the velocity just below the refracting horizon and whose interceptis the sum of terms of the form 2h,cos 0rlV,, each layer above the refracting horizon contributing one term. We can generalizefor n layers:

,: ur,*z

where 0, : sin-'(V,lV^). This equation can be used to find the velocitiesand thicknessesof each of a series of horizontal refractinglayers,eachof constantvelocity higher than any of the layers above it, provided each layer contributes enough of the time-distance

)

The result for shooting in the updip direction is similarly obtained by eliminating ftr:

t,: (xlV,)sin(0. - {) + r,", where

t,,: (2hJV,j)cos 0".

) )

(4.461

)

Note that the downdip traveltime from O to O' is equal to the updip traveltime from O' to o,' this sourcepoint to sourcepoint traveltime is called the reciprocal time and is denoted by l" The concept that traveltimealong a path is the sameregardlessof the direction of travel is an example of theprinciple of reciprocity. Theseequations can be expressedin the sameform as eq. (4.37): to: (xlV) * t,o,

(4.47)

(xt\) + t,,,

(4.48)

t,:

GEOMETRY OF SEISMIC WAVES

98

Raypaths and traveltime curves for a dipping re-

Fig. 4.18 fiactor.

where V :

Vtlsin(0. + €),

(e. - 0. V,: V,lsin (4.49)

Voand V,are apparentvelocitiesand are given by the reciprocalsof the slopesof the time-distancecurves' For reversedprofiles,suchas shown in fig.4.18' eq. (4.49) canbe solvedfor the dip ( and the critical angle 0, (and hencefor the refractor velocity Z'): l sin r(VtlV,)1. o, : j [sin t(VtlV,l+ ] ,' o . r ' , -r( t(v € : I lsin l l v l ) s i n v t l v " ) 1 .) The distancesto the refractor, h, and h,,, can then be found from the interceptsusing eqs.$.a\ and (a.46). Equation (4.49) can be simplified where { is small enough that we can approximateby letting cos ( - I and sin € : €. With this simplificationeq. (4'49) becomes VJVr: sin (0. + t) = sin 0. + ( cos 0.' VlV,: sin (0. - O : sin 0. ( cos 0,; hence,

and assumingthat { is small enoughthat higher powers of ( are negligible: ' V , , : ( V , l s i no , X c o s{ + c o t 0 , s i n { ) : Vr(l - t cot 0,)' V , : V r ( l + t c o t0 , ) ; hence,

v,: :(v,t+ v,)

(4.s2)

4.3.5 Diving waves It is obviousthat raypathswill eventuallyreturn to the surface wheneverthe velocity increaseswith depth. The wavestraveling by such raypaths are called diving wayes.Symmetry shows that for horizontal velocity layering,the angleof emergenceis io(fig. 4.19a);at the : deepest point on the raypath (h-), i : 90" and p point on llV^, that is,p is the slownessat the deepest the iaypath for a diving ray. We can rewrite eqs.(4.29) and (4.30)for this situation as

,:2fr-

(4.53a)

s i n0 , : ( V t l v r \ : : V t I l l l v d l + 0 l V , l j , so that

u v,-;l (U V )+ (rl 4) 1.

( 4.s1)

An even simpler approximate formula for Vt (although slightly less accurate) can be obtained by applying the binomial theorem($15.1.4c)to eq. (4.49)

dz- .lo(4.53b) ', -' vrzllt 1pv(:)1't''2' )n the doubling factor of 2 arisingbecauseof the raypath from h^back to 0. If x, / measurementsare available for diving ways from a common source and if the velocity hai increasedwith depth monotonically,then

99

REFRACTION PATHS eqs.(4.53)can be solvednumericallyfor V(z). For a linear increaseof velocity,eqs.(4.53)become (seeproblem 4.20a) x : (2Vola)cotio : QVolal)sinh(atl2), (4.54a) (4'54b) t : (2la) ln [cot (iol2)]' and the maximum depth of penetrationis h^: (VJa)fcosh(atl2)- ll.

(4.55)

For the case of concentric spherical layering (fig' 4.19b),eqs.(4.53)become(seeproblem 4.20b)

o:'r",-##w,,*' (456a) ' : 2 1 " v t r l, R,_h^

dr

where A is the angle subtended at the center of the Earth by raypath SO and R" is the radius of the Earth' Richter (1958:app. VI) givesa numericalsolution for Z(r) in eqs.(4.56).Using LDEF in fig. 4.19b' we can write eq. (4.27)as

I I lh^ I I

: u'/uo?r'.rr., i)lVo: VndtJR"UO p' : R"(sin At the deepestPoint, P' : r^l\, p' : r^lV^:

w,.,

(4.s6b)

lp'V(r)lrl' ltt2'

so

(R1zr)sinlo : Et/bAor. (4.58)

If we plot a curve of traveltimeI versusangular distance A for an earthquakeevent at various stations, the slope, Ltl6L, givesr ^l V-. With diving waves,two velocity situationsrequire specialattention.A velocity gradientin a layer that is substantiallyhigher than the gradient abovethe bed, as in fig. 4.2ba,causesa very sharp increasein raypath curvature and a folding back of the time-distance curve. Such a triplication of branchesof the timedistancecurve (fig. 4.20c)is usuallydifficult to seebecausethe later branchesbecomelost in the later cycles of the earlier arrivals. Unless all branchesare correctly recognized,errors will be made in solving the time distance observationsfor Z(z)' The other situationleadingto errorsin determining Z(z) is that of a velocity inversion(fig' a'20b)' The inversionmay producea gap in the time-distancecurve' as in fig. i.iOc, anA this gap may not be recognized becausediffraction tends to fill it in' Although fig' 4.20 shows situations for planar layers,similar (but more complex) situations occur with concentric sphericallayersin earthquakestudies'

(b) (a) PlaFig. 4.19 Raypaths for increasein velocity with depth nar velocity layering; (b) concentric velocity layering'

|

-

I.sN

L I ' tNP

J' - ' 'PR

:2tr*+(x-2MI\\lV^' Noting that (sin i)lVo : (sin 0.)/ V, : llZ-, we find from eq. (4.32) I . /tan ]0.\

trr: i," \45/

:t"'"1(r^r-r,--rr,,)("-urn'')l

:j["",n-'(A *'n'(A], 4.3.6Linear increasein velocityabovea refractor The caseof a high-velocitylayer overlainby a layerin which the velocity increaseslinearly with depth (fig' 4.21) is of considerablepractical importance'The relation betweent and x for a horizontal refractor can be found as follows:

'x : where use has been made of the identity cosh get (4.31)' we l)'"]. From eq. ln[x + (x' MN : (l/pa)(cos io - cos 0.)

: (rlpa){lt- (VJV,,)'l'''- tt - (4lV-)'\'''}'

GEOMETRY OF SEISMIC WAVES

100

t i o n a t t o : 2 . 3 5 8 s , g i v e nt h a t t h e v e l o c i t yv : Z ' S O km/s. (b) Typical errors in t,V tomight be 0'6 m, 0'2 km/s, unOj -s. Calculate the corresponding errors in A't*ro approximately.What do you conclude about the accuracy of A/".o calculations? (c) Show itiat ttre more accurateNMO equation,eq' (4.8),can be written Lt|uro- L,trro(l - LtNMJ2t), in terms of the first-ordervalue of Lt NMogiven by eq' (4.7). Taking into account the errors in x, V /n,when is this equationuseful? 4.2 (a) Show that the quantity dtldx can be considered as a vector or component of a vector according as d/ correspondsto the total dip or component of dip. of fig' 1b; Using fig. 4.22, verify that the constructio-n (Hint: Ex(4'18)' eq' as results +.SUgiu.t th. tu-. pressf , m, and OC in terms of OA.) i.: Stto* that the equation for a dipping reflection, eq. (4.9),becomes (Vt)z : (2x cos $' + 4h7 determlne Fig. 4.20 Velocity situationsmaking it difficult to gradient' iiri"-Airtun"" curves (a) Region with large velocity (c) Time-distancecurvesfor inversion' velocity with n"gion ift (a) (solid linel and (b) (dashedline); diffractionswill ;ii;;; complicatethesecurves.

(Gardner, 1947),where ft is replacedby ft., the slant depth at the midpoint betweenthe sourceand receiver (seefig. 4.23).

V,=l'o+oh, 1J:^

in velocityin the with a linearincrease Fig.4.21 Refraction upperlayer.

Fig. 4.22 Combiningdip components'

Substituting in the first expressionfor t gives (4.59) t : (xth) + to, where /o : intercept time:

| (V^l%)- "olq_:V_ry!^Dl to= " -(2la){fcosh + ll - (V'lV^)')'''I'(4'60) (% l4)'l''' tl The slope of the head-wavetraveltime curve gives V-' A curve is plotted of to againsth,(or V) for given valuesof Zoa\d o, arrdh,and V,are read from this curve for particular measurementsof r0'

( I I I I I

s' Problems 4.1 6\ Calculate the normal moveout A/r.o for geophones 600 and 1200 m from the source for a reflec-

i,' A'

Fig. 4 23

Derivation

of X2-72

relation

for a dipping

bed'

PROBLEMS

l0l

+.4 (a) Using the dip-moveoutequation,eq. (4.11), and the resultsof problem 4.3, verify the following result (due to Favreaccordingto Dix, 1955): tan(:

t l t 2 A B -t i ) t t 2 '

where{ : dip, I : tru - tr", tu": traveltimebetween sourceA and receiverB, tn: traveltimeat sourceS tseefig. 4.23.y. (b) Using eq. (a.9), show that sin { :

- t2")l\h,r. Vr(t?rn

(c) Under what condition is the result in part (b) the sameas eq. (4.I l) and also consistentwith part (a)? 4.5 The expressionfor dip in terms of dip'moveout, eq. (4.l1), involves the approximation of dropping higher-orderterms in the quadraticexpansionusedto get eq. (4.10).What is the effecton eq. (4.11)if an additional term is carried in this expansion?What is the percentagechangein dip? 4.6 In fig. 10.5b,the reflectiontime at the top is 1.0s and the depth 1500m, the reflectiontime at the bottom is 1.4s, the interval velocitybetweeneventsI and 2 is 3300m/s,and the tracespacingis 100m. Calculate the depth and dip ofthe three reflectors. 4.7 A well encountersa horizon at a depth of 3 km with a dip of7". Sourcesare located200 m updip from the well with a geophoneat depthsof L0 to 2.6 km at intervals of 400 m. Plot the raypathsand traveltime curvesfor the primary reflectionfrom the 3-km horizon and its first multipleat the surface.AssumeZ: 3.0 km/s. (Hint: Seefig. 6.33.) 4.8 The numbered ticks at the top of fig. 8.5 are 1 km apart. (a) Select two fairly steeply dipping reflections,assume velocities(fairly high in this area), and determine the approximatedips. (b) Figure 8.5 is a migrated section; by what horizontal distancesare the reflecting points for these eventsdisplaced,that is, how far did they migrate? (Hint: See$8.8.3and fig. 8.30.) 4.9 @) SourcesB and C are respectively600 m north and 500 m east of sourcel. Traveltimesat A, B, and C for a certain reflectionare to: 1.750,1.825,and 1.796s. What are the dip and strike of the horizon, Z being 3.25 km/s? (b) What are the changesin dip and strike if line AC has the bearingN80'E? a.l0 (a) Two intersectingseismicspreadshave bearings N10'E and N140'E. If the first spreadshowsan event at to : 1.160s with dip moveout of 56 ms/km and the same event on the secondspread has a dip moveout of 32 ms/km, find the true dip, depth, and strike, assumingthat (i) both dips are down to the southand west,and (ii) dip on the first spreadis down to the south and the other is down to the southeast. The averagevelocity is 3 km/s. rb) Calculatethe position of the reflectingpoint (migratedposition) for each spreadin (i) as if the cross rnformationhad not beenavailableand eachhad been rssumedto indicatetotal moveoutl comDarewith the

resultsof part (a). Would the errors be more serious or less seriousif the calculationswere made for the usual situation where the velocity increaseswith depth? 4.11 Verify the derivation of the expressionfor k in $4.1.4and of eq. (4.21a). 4.12 Given the velocity-depthdata shownin fig. 5.19, what problems would you expect using simple functional-fit relationsin the different areas? 4.13 (a) CalculateV and V,^"down to each of the interfacesin table 4. l. Why do they differ (give a geometrical explanation)? (b) Pfot V and V,_.versusdepth and versustraveltime and determinethe best-fit straight lines for the four cases.What are the main problemsin approximating data with functional fits? Table 4.1 Layeredmodel Depth (km)

Velocity (knts)

0 1.00 1.00-2.50 2.50-2.80 2.80-4.80

2.00 3.00 6.00 4.00

4.14 (a) Assumingflat bedding,calculatedepthscorresponding to t0 : 1.0,2.0,2.1 , a n d 3 . 1 s u s i n gt h e velocity functions for V and Z.*. determinedin problems4. l3a and 4. | 3b.What errorsare introducedrelative to the depthsgiven in table 4. l? (b) Using the velocitydata in table4. l, tracea nonvertical ray through the various layersand find the arrival times and reflectingpoints offlat reflectorsat eachof the interfaces. 4.15 (a) Repeatthe calculationsof problem 4.14a assumingdip moveout of 104ms/km and find the dip in eachcase. (b) Tracerays assumingthe velocity is constantat the vafuesof V and V,^"calculatedin problem 4. 13. Find the arrival times and reflectingpoints of reflectorsat eachof the interfaces. 4.16 Figure 7.45showspart of a seismicrecordwhere the geophonegroup spacingis 50 m, the offsetto the near groups being 50 m and that of the far groups 600 m. (a) What is the velocity of the first-breaks? (b) Assumingthat the sourceis below the baseof the LVL and that the LVL velocity is 500 m/s, how thick is the LVL? (c) Arrival times at the sourcepoint for two reflections are given as 0.415 and 0.778 s; what are the average velocitiesto thesereflectors? (d) For these reflections, the arrival-time differences betweenthe far traces in opposite directions from the sourcepoint are given as +0.005 for both reflections. What are the dips of thesereflectors? (e) What is the dominant frequency of these reflections (approximately)?

GEOMETRY OF SEISMIC WAVES

102

t

frr =300m

,/r = 3 km/s

ft: =300m

/z = l'5 km/s

Y1 = 3 kmls

Iz. = 6 km/s

Vt = 6kmls (a)

(b)

Fig.4.24 Two different geologic sections that give the same refraction time distance curves.

4.17 (a) Giventhe velocityfunction V : |.60 + 0.60: km/s (z in km), find the depth, dip, and offset of the point of reflection when In : 4.420 s and AllAr : 0. 155 s/km. What interpretation would you give of the result? (b) If the ray continued without reflection,when and where would it emerge?What moveout would be observedat the recording spread?Calculate the maximum depth of penetration. 4.18 (a) Show that the two geologicalsectionsillustrated in flg. 4.24 produce the same time-distance curves. (b) What would be the apparent depth to the lower interfacesin figs. 4.24a and 4.24b if 4 : 3.15 km/s insteadof 6 km/s? 4.19 Figure 4.25 showsa refraction profile recorded as a ship firing an air gun moved away from a sonobuoy. Identify the direct wave through the water and disuseits traveltimesto give the source-to-sonobuoy tances(assume1.5km/s as the velocity in water). (a) Identify distinctivehead-wavearrivals,determine their velocities,intercept times,and depths of the refractors assumingflat bedding and no velocity inversions. (b) What is the water depth? Identify multiples and explain their probable travel paths. (The data in the upper right corner result from paging ($8.6.3)and actually belong below the bottom of the record.) a.20 (a) Verify eqs.(4.54) and (4.55).(Hint: Use eqs. ( 4 . 3 1 )t o ( 4 . 3 4 ) . ) (b) Deriveeqs.(4.56).(Hint: ln fig. 4.19b,M.BC gives (V 6t)' : (6r)' + (r EA)' ; using eq. (4.27), show that O' : Qllz1, (6A/6r);eliminating first 61, then 64, and integratinggiveseqs.(4.56a)and (4.56b). 4.21 If the velocity function in problem 4.17 applies above a horizontal refractor at a depth of 2.40 km, where the refractor velocity is 4.25 km/s, plot the traveltime-distance curve. 4.22 Given that situations (a) through (h) in fig. 4.26 involvethe sametwo rock types,draw the appropriate time distancecurves.Diagram (c) showstwo casesfor dip in oppositedirections.In figs.(i) and O, the velocity in the lower medium varies laterally accordingto the density of the shading. 4.23 Barton (1929) discussesshooting into a geo-

phone placed in a borehole(fi9. a.27) as a meansof determiningwhere the bottom of the borehole is located. (a) Given that A, B, D, and.E are equidistant from lA in the cardinal directions and assumingstraight-line travel paths at the velocity V and that the traveltimes from D and -E are equal, derive expressionsfor CC' and CW in fig. 4.27ain terms of the traveltimesfrom A and B, tA(.,ar'd tB(.,. (b) What are the valuesof tAC,and trr. for V : 2.500 kmls, AW : BW : CC' : 1000m, CW -- 200 m? (c) How sensitiveis the method, that is, rvhat are L,(CC')lL,tnr.,,and L(CW)IL/,..,?For the specificsituation in part (b), how much changeis there in WC and CC' per milliseconderror in /r.,? (d) Modify the assumptionsin part (b) by taking the velocity as 1.5 km/s for the first 500 m and 3.5 km/s for the lower 500 m. What are the actual traveltimes now and how would thesebe interpretedassumingthe straight-pathassumptionin part (a)? 4.24 SourcesA and B arelocatedat the endsof a225m spreadof l6 geophones.Using the data in table4.2, find the velocities,dip, and depth to the refractor.

Table 4.2 Refractionprofle r, (m)

/1 (ms)

/8 (ms)

rr (m)

0 l5 30 45 60 75 90 105 t20 r35 150 165 180 195 2t0 225

0 10 21 30 4l

98 92 87 8l

225 210 195 180 165 150 135 120 105 90 75 60

\tl

I I

59 65 70

65 60 52

t)

40

'78 81 85 89 94 98

43 )t

31 2l l0 0

A< JU

l5 0

F

Fig. 4.25 Sonobuoy refraction profile in Baffin Bay. Source was a 1000-in.rair gun. (Courtesy of Fairfield Industries.)

,F]

(e)

A

Fig.4.26 Time-distance curves for various twoJayer configuraions. This figure is adapted from Barton (1929) in the first publication in English on the seismic method The part above O of each diagram provides space for a curve of arrival time versus distance foi the model shown in cross-sectionbelow O' In each

te , ,

t t t

^-

- -- - - -, - ! ^

W

-2-----'

€t I

t I I

v (a)

c a w

\

lt l.'

(bl Fig. 4.27 Mapping a crooked borehole by measuring traveltlmes to a geophone at C' in the borehole (From Barton' 1929.) (a) Plan view; (b) vertical section AWB'

(t)

(/)

case,the velocity in the cross-hatchedportion is higher than that above. Part (a) has been completed to show what is expected. In (c), two alternatives are given so two sets of curves should be drawn. In (i) and (), refractor velocity varies horizontally and is proportional to the shading density.

REFERENCES References of veAgocs,W. B. 1950.Comparisonchartsfor linearincrease 15::22636. locity with depth.Geophysics, Barton,D. C. 1929.The seismicmethodof mappinggeologic pp. 572 624.New York: structure.In Geophysical Prospecting, AmericanInstituteof Mining and MetallurgicalEngineers. Brown, R. J. S. 1969.Normal-moveoutand velocityrelations for flat and dipping bedsand for long offsets.Geopftlsics, 34: 1 8 09 s . Daly,J. W 1948.An instrumentfor plotting reflectiondata on the assumptionofa linearincrease ofvelocity.Geophysics, 13: 153 t.

105 Dix. C. H. 1955.Seismicvelocitiesfrom surfacemeasurements. 68-86. Geophysics,20: Gardner.L.W. 1947.Verticalvelocitiesfrom reflectionshoot12: 221-8. ing. Geophysics, Levin,F. K. 1971.Apparentvelocityfrom dippinginterfacere36: 510-16. flections.Geophysics, Richter, C. F. 1958. ElementarySeismology.San Franctsco: W. H. Freeman. Shah,P.M., and F. K. Levin. 1973.Grosspropertiesof time 38:643 56. distancecvves. Geophysics,

t

Seismicvelocity

()verview Knowledgeof velocity valuesis essentialin determining the depth, dip, and horizontal location of reflectors and refractors, in determining whether certain things like head wavesand velocity distortionsoccur, and in ascertainingthe natureofrocks and their interstitial fluids from velocity measurements. We developa heuristic appreciationof the factors that affect seismicvelocity from a conceptualmodel of a sedimentary rock. F. Gassmann,M. A. Biot, and J. Geertsmadevelopeda model for a fluid-filled porous rock, and G. H. P. Gardner, L. W. Gardnet and A. R. Gregory hypothesizedthat microcracksin nonporous rocks lower velocity.Fracturing also generally lowersvelocity. Lithology is the most obvious factor we would expect to control velocity.However,velocity rangesare so broad and there is so much overlap that velocity alonedoesnot providea good basisfor distinguishing lithology.Sand velocities,for example,can be smaller or larger than shalevelocities,and the sameis true tbr densities; both velocityand densityplayimportant rrles in seismicreflectivity. Porosity appearsto be the most important single factor in determininga rock'svelocity,and the dependenceof porosityon depth of burial and pressurereli'tionships makes velocity sensitiveto these factors also.Velocity is generallyloweredwhen gas or oil replaceswater as the interstitial fluid, sometimesby so nruchthat amplitude anomaliesresult from hydrocarf,on accumulations. The near-surfacelayer of the earth usually differs markedlyfrom the remainderof the earth in velocity lnd some other properties. This makes the near\urtacelow-velocitylayer(LVL) especially important; !rur determinationsof depths,attitudes,and continuLn of deeper eventsare affectedas reflectionspass through this layer.In arctic areas,a zone of permanentlyfrozenearth,permafrost,distortsdeeperevents lecauseof an exceptionallyhigh velocity.Fluid pres.ure that exceedsthat of a column of fluid extending :!r the surface("normal" pressure)lowersseismicve;.rci1yt11ti.is usedto predict abnormal pressures.Gas just belowthe sea rrdratesthat form in the sediments ir-rorin deepwater also producevelocity changes. Velocity terminology is often misusedand causes rruch confusion.Section5.4.1 attemptsto clarify the rrecisemeaningof average,root-mean-square, stack-

107

ing, interval, Dix, phase,group, apparent,and other velocity terms. Seismicvelocity is measuredin boreholesby sonic logs ($5.4.3)(and by vertical seismic profiling discussedin $13.4).Velocityis also measuredby surface seismicdata becauseof the dependenceof normal moveout on velocity.The reflection-coemcient equation can be usedto obtain velocity information from amplitudes.a form of inversion.

5.1 Model of a sedimentary rock 5.1.I A pack oJ uniformspheres Seismicvelocityas givenby eqs.(2.58)and (2.59)relates to a homogeneousmedium, but sedimentary rocksare far from homogeneous. Theseequationscan be written, for solid media, cr2: (tr + 2p")lp,

B' : t-rlp,

and for fluid media, c t ,: \ / p ,

g.:0;

hence,in general,

y : lKto),,2,

(5.1)

where K is the effectiveelastic parameter.Thus, the dependence of Z upon the elasticconstantsand density appearsto be straightforward.In fact, the situation is much more complicatedbecauseK and p are interrelated,both dependingto a greateror lesserdegree upon the material and structureof the rock, the lithology, porosity,interstitial fluids, pressure,depth, cementation,degreeof compaction,and so on. The most notable inhomogeneityof sedimentaryrocks is that they are porous, containing fluid-fllled spaces within them. Porosity is simply the pore volume per unit volume.Wang and Nur (1992b)discusstheorres relatingseismicvelocity to the compositionof rocks, The simplest rock model consists of identical spheresarrangedin a cubicpattern(fig.5.la) with the matrix subjectedto a compressivepressureL If the radius of the spheresis R, the force ,F pressingtwo adjacentspherestogetheris the total force acting on a layer of n X n spheres(that is, (2Rn)' 2|)divided by the number of spheres(!t'), or F : 4R29. This force causesa point ofcontactto becomea circleofcontact

SEISMICVELOCITY

108 of radius r and the centersto move closertogethera distances (seefigs. 5.lb and 5.1c),r and s being related to R, I and the elastic constants4 o of the spheresby Hertz' equations (see Timoshenko and Goodier. 1951:3'72-7): r : [3(l - o2)RFl4E|t3, (5.2) t s : [9(l - o')'F'l2RE']n. I When a P-wavepasses,I changesby A9, resulting i n c h a n g e sL F : 4 R ' L 9 a n d A s : - 2 R e , w h e r e e t s the strain in the direction of F (seefig. 5.ld). Thus, the effectiveelasticmodulus K is given by Ag : 1. .(: - '

I2R -s --l

L

!g:

AJ

r AF : | 3E'9 1'''

2Rl,

Lgtl-o')'l

on differentiatingeq. (5.2).The averagedensity is the weight of a spheredivided by the volume of the circumscribedcube,that is, p :71ttO3p)/(2R)3 : t/orP, p being the density of the material of the spheres. Thus, we get for the P-wavevelocity, ("n,.,

Fig. 5.1 Effects ofcompression on a cubic packing ofspheres. (After White, 1965.)(a) Cubic packing; (b) force causescenters to move closer together; (c) force causes point contact to become circular area of contact; (d) effect of change in force.

V."r,.: (Kl-p)tt': I81E2gl(l - o2)2?rrpslr/6. (5 . 3 ) Gassmann(195l) calculatedthe velocityfor a hexagonal packing of identical spheres(fig. 5.2) under a pressureproduced by the weight of a thickness: of overlyingspheres;he obtained for a vertical ray Vn,-: fl28E' gzl(l - o2)2rr2p)fi/o, (5.4) where g is the accelerationof gravity. BecauseI is nearlyproportionalto:, eqs.(5.3)and (5.4)give the samevariation of velocitywith depth. Faust (1953) found an empirical formula for velocity in terms of depth of burial z and formation resistivityR', that is consistentwith eqs.(5.3)and (5.4): 4:

900(zR')"o,

/5 5\

Vrbeing in m/.1,: in m, and R' in ().m. However,the deviations of individual measurementswere very large, indicating the presenceof other factors that havenot beentaken into account. Random packsof well-sortedparticleshaveporosities in the range of 45 50Vo,but under pressure,the particles deform at the contacts,and as a result the density increasesand the porosity decreases(Sheriff, 1911;theelasticconstantsalsochange see$5.2.5). on the model 5.1.2 Expectationsba,ged What velocity relationshipsmight we expectbasedon the foregoingmodel of a rock? Clearly,porosity will be an important factor in velocity becauseit should alTectboth the effectiveelasticityand the density,and indeed it is often said that porosity is the most important factor in determiningthe velocity of a sedimentary rock. The contactareabetweenspheresis not proportional to the pressureforcing them together,so we may expectthat the pressuredependenceofvelocity will not be linear but will diminish with increasing

pressure(or with depthof burial).Fluid filling the intersticesin a rock may be expectedto resistthe effects of the overburden,that is, the overburdenweight tendsto squeeze out the porositywhereasthe interstitial fluid tends to preservethe porosity. Thus, the effectivepressureon a rock will be the differencebetweenthe overburdenpressureand the fluid pressure, the di/firential pressure.If the pore spaceis connected to the surface,the fluid pressureshould be that of a column of porefluid extendingto the surfacewhereas the overburdenpressureis the weight of the overlying rocks. Where this is true, the pressureis said to be normal. However,if the pore fluid cannot escapeto allow the grain-to-graincontactsto adjustto normal pressure,then some of the overburdenweight will be supportedby the interstitial fluid and we will havean overpressured situation. An overpressuredrock will "feel" the same differential pressureas it would at some shallowerdepth, where it would have a lower velocity, and, hence, we expect overpressuringto lower the velocity. We would not expectthe deformationof a rock under high pressuresto be elastic.Hence,if a rock from which the porosity has beensqueezedout by depth of burial should be uplifted, we would not expectporosity to return, except for a small amount becauseof some remaining elasticity. The porosity of a rock might be expectedto dependon the maximum stresses it has endured since formation, that is, porosity may dependon the maximum depth of burial rather than on the presentdepth. Gas as a formation fluid is much more compressible than a liquid and hencegas in the pore spaceshould lower the velocity much more than oil or water. In fact, gas is so compressiblethat the presenceof just a

MODEL OF A SEDIMENTARY ROCK

109

t r )

W ! z

.V

\

l

7

\

.

(c)

\

@)

Fig. 5.2 Close packing of uniform spheres.(a) Cubic packing (as in fig. 5. la), an arrangement that is not gravitationally stable. (b) Hexagonal packing, gravitationally stable and the densest packing possible. (c) First layer of a hexagonal stack, showrng two classesof sites (.4 and B), adjacent sites of which cannot both be occupied at the same time (for example, the two dashed

locations). (d) Second layer of spheresshowing how occupyrng some I and some B sites leavesextra spacein between.(e) Hexagonal stack with left side occupying I sites and right portion B sites; the consequence is increased porosity. The random choice of,4 and B sites leads to a completely random pack after a few layers.

small amount of gasshould lower the effectiveelasticity nearly as much as a large amount, and hencewe expectthe effectof gas on velocity to be very nonlinear.Gas in the pore spacewould affectdensityas well as effectiveelasticity; if we gradually introduce gas into the pore space,the first small amount of gas should have a large effect on the numerator of eq. (5.1), but additional gas will have much less effect, whereasthe effecton the density term in the denominator will be linear with the amount of gas.Thus, as the amount of gasis graduallyincreased,we expectat first a sharpdecreasein velocityand thereaftera gradual steady increase in velocity. The near-surface weatheringlayerbeinggenerallyabovethe water table, we expect it to have exceptionallylow velocity. BecauseS-wavesdo not travel through fluids,the nature of the pore fluid should have little effect on S-waves

compared to that on P-waves;however,it will still havea minor influencebecauseof its effecton the density. By changingP-wavevelocity much more than Swave velocity, the presenceof gas will change the effectivevalue of Poisson'sratio and hence change amplitude-versus-offset relationships. Cementationand pressure-induced recrystalization would be expectedto decreaseporosity. Very few of the things that might happento rocks increaseporosity (seefig. 5.3a).Hence,generally,we may expectporosity to decrease(and velocity to increase)with increasein depth of burial (fig. 5.3b),cementation,age, as sorting becomespooret and so on. The major failure in expectationsis that an increase in densityusuallydoesnot lower velocity,as might be expectedfrom eq. (5.1). Phenomonathat changethe densityusuallychangethe effectiveelasticitymore, so

SEISMIC VELOCITY

ll0

0.0s

suming that relative motion between the fluid and rock is negligible.We shall follow the account given by White (1983:57-63). The rock is assumedto be a porous skeleton or framework with the pore fluid moving in unison with the rock so that there are no viscousenergylosses.To distinguishvarious componentsof the system,we use the following notation for the bulk moduli: kt.,k., k*, and k refer respectivelyto the fluid filling the pore space,the material comprising the skeleton,average values for the skeleton plus empty spaces,and the fluid-filled skeleton.We use $ for the porosity and C: llk for the incompressibility.We assumethat the saturatedrock is isotropic and that the fluid has no effecton the shearmodulus, so p : p*. The average density is simply the volume-weightedaverage:

Averageporosity,P 0 . 2 0 0 . 3 00 . 4 0 0.10

0

) ^ lz

x o

l0

/ 3 '9/

J

st

P:0P/+ (1 0)P,.

We considera cube of the saturatedrock and apply an incremental pressure A0. We assume that the pores are interconnectedso that the fluid pressureis that applied to the pore openingson the cube faces (however,no fluid enters or leavesthe cube because there is no fluid motion relativeto the rock). We write for the total pressure

l5

-. (a)PorGity (%) ? p 3

(5.6)

0

A0 : A9* + Lgt.

r08'

(5.7)

From the definition of dilatation, we can write eq' (2.18)in the form -LY|V : C 49.

6

; d

o o

Thus, the pressureA9r changesthe fluid volume by - LlrtlY : the ma6C, tW,. But A0, also compresses terial of the skeleton,so -A{,/Y : (l - d)C" AE. Finally, A9* compresses the skeleton so that -AT./1/ : C, A0*. Adding these three effects,we find for the total volume change

( - L Y | V ) : [ 6 C , +( 1 - 6 ) C "A )E + c. ag*.

Fig. 5.3 Factors affecting porosity. (a) Porosity in a clastic rock decreaseswith depth of burial (compaction), cementation, and "limpoorer sorting, but is essentiallyunchanged by uplift. The it-of-porosity" line refers to normally pressured situations and ignores possible secondary porosity. (After Zieglar and Spotts. 1978.) (b) Porosity depth curves. (From Atkins and McBride, 1992; reprinted with permission.)

rltoaa SAtrlt lllrllxc raal (tlw tlctoclactll

o

E

!o that the explicit densityterm in the denominatorgives the wrong implication. Biot, Geertsmaequations 5.1.3 Gassmann, To obtain a useful formula for the velocity of a fluidfilled porous rock, the effectsofporosity and the pore fluid must be taken into account. Gassmann(1951) derivedexpressionsfor the effectivebulk modulus as-

(5.8)

oattto

.ttot

lllrll

rclotllY

l.ta

oaitlll

l.l

(tturr

DrY tlrtrt lr tlrlttio ilcrclrcllxo

lttlr lt ttc'c rilllrlD)

r o o ' m s a m c n f r

Pr$sure(P.s.i.) Fig. 5.4 Effect of microcracks on velocity of gabbro. (From Gardner, Gardner, and Gregory, 1974.)

I

Velocitg(km/s) 2 3 4

I

I

Alfuvium, * Drg sand, Veaihering

c-r H-?

l-2H l-

7r

t,5-t I

F-4-.|

4 Hud

F3a

Glacial

-21

Shale

Smd, Sandstone

Lirnestone

Dolomile

Arrhgdrile,

r Ggpsum A Selt

6rrnite

l-1 FS

4-l

l-5 + Fig. 5.5 P-wave velocities lor various lithologies. Data from ( l ) G r a n t a n d W e s t ( 1 9 6 5 ) ;( 2 ) Kearey and Brooks ( 1 9 8 a()3 ;)

Lindseth (1979);(4) Mares ( 1984);(5) Sharma ( 1976);(6) Sheriff a n d G e l d a r t ( 1 9 8 3 ) ;a n d ( 7 ) W a t e r s( 1 9 8 7 ) .

SEISMIC VELOCITY

tt2

- l) in the nuAdding and subtracting (k*lk,)(k*lk" merator, we get

k:k*+

Fig. 5.6 Histogram ofvelocity values tabulated in Birch (1942) for different lithologies. (From Grant and West, 1965.)

3 . 3t rt

x

Best-fit lines Sands

tr 3.lH

Shales

2.9LL Oc

r E N

d

-

-SUd_ltne

V)

Fig. 5.7 Portion of SP- and velocity logs for a well in the U.S. Gulf Coast. The SP-values distinguish sands from shales. (After Sheriff. 1978.)

(t-k*lk,)2

60lkf - vk") + (l/k,xl k*lk") ( 5I. 1 )

Thus, k equals k* for the skeletonplus a term that dependsin part upon the fluid filling the pores. BeM : k + 4p"13(seeeq. (2.58)and table2.2),we ca:erse can add 4p,13to both sidesand get (t - k*lk")2 M:M** -

+(ykf uk")+ (uk")(r u.,o(1,4

Becausea2 : Mlp, the P-wavevelocity dependsupon the fluid bulk modulus and the porosity as well as the rock properties.On the other hand, the fluid influencesP only through the density (seeeq. (5'6)). One might expect the coupling betweenthe rock skeletonand the fluid to be greaterat low frequencies; Gassman's equation is therefore called the lowfrequencysolution. Biot (1956)assumedthat the fluid "highcould flow through the pore spacesto give a frequency solution"; this introduced the additional factorsof fluid viscosityand matrix permeability.Biot alsodefinedthe low-frequencyrangeof the applicability of the Gassmannequationas v < 0.1(e$l2rrrP,),

(5.13)

wheree is fluid viscosity,and r is matrix permeability. Geertsmaand Smit (1961) derived an equation We obtain another equation for - LYIY'by considering that A9* producesa relativevolume changein the skeletonplus pores equal to C* A9* and A0, results in a relative change in the skeleton material C" Ag, (the volume changeof the fluid is taken care of in the'term C* A9*). Adding, we find that

( - L 1 fl 1 f ) : C , A E + C * A g *

VEI-oclTY (lus)

(5.9)

Equations(5.8) and (5.9) are now solvedfor A0, and A0* with the result

- C) Lgr: e^1'r'1r)G"- C*)/[oC*(C" +c, (c, - c*)1, Ag*: (-A1//D0(C"- C)tl+C*(c,- c/) + c.(c, - c*)1. Adding and using eq. (5.7),we havefor the effective bulk modulus

As

t(c,-_91(E lr : "k,__r_ c -Lyfv oc*(c, - C,)+ c"(c,- c*)' o(r/k"- !tk) + (rtk"-rtk*) _ d ( l / k * )l (t k , - l t k , l + ( l l k , x l l k , - l l k * l '

14000 fr ONE WELL

97 WELLS

(a)

(bl

(s.l0)

Multiplying numeratorand denominatorby k* gives

- uk) + (k*lk"- l)l/to(l/fr"- rlkf) k : tk*601k" + (l/k,xk*/k,- l)1.

Fig. 5.8 Shale velocities in the Ship Shoal region, ofshore Louisiana. (From Hilterman, 1990.) (a) A velocity analysis in one well; (b) histograms showing velocities in 97 wells throughout the area.

EXPERIMENTAL DATA ON VELOCITY from the Biot equationswhere wavelengthis greater than pore size: u -

{[(:. ?)+

il-c,/c),

IiJ," (l -0-C./C)C.+6gJpJ (5.l4)

This equationgivesvaluessimilar to eq. (5.12).These equationsfit experimentaldata reasonablywell considering how many variablesare usually not known precisely.

5.1.4 Model of a nonporousrock The foregoingsectionsbasicallyexplain observedvelocity variationsas attributablemainly to changesin the porosityand the fluid filling the porosity.However, nonporousrocks also show variation of velocity with pressureand other parameters. Gardner, Gardner, and Gregory (1974) hypothesized that nonporous rocks have minute voids (,.microcracks")that result in loweringthe velocity.Generally, rocks are composedof many minerals that have different temperature coefficientsof expansion, so that a temperaturechange will create stressesand open up microcracks.To testtheir hypothesis,they determined the velocity-pressureresponsefor a gabbro with only 1.77oporosity and then heatedthe gabbro to 750'C and cooled it, after which they repeatedthe velocity-pressuremeasurement(fig. 5.4). The loweringof the velocity pressure curveis presumedto be due to the creation of new microcracks.After being subjectedto pressure,the samplereturnedto a higher velocity when the pressurewas lowered;presumablv. the pressurehealedsome of the microcracks.probably,repetition of the pressurecyclewould heal more mrcrocracksand elevatethe velocity-pressurecurve still more, approachingmore closely the preheating curve. One might also expect the heat-treatmentinduced stressesto gradually dissipatewith time so that the curve would climb gradually. The inclusion of fluid in microcracks greatly increasesthe P-wavevelocity,but leavesthe S-wavevelocity nearlyunchanged(Nur and Simmons,1969).

5.2 Experimental data on velocity 5.2.1 General Velocity can be determinedfrom measurements (a) in situ (see95.4)or (b) on samplesin a laboratory.press (1966)listsmeasurements of both types.Care has to be taken that measurementson samplesare not distorted by changesin the sample conditions; many early measurementsgave misleadingvalues because they were made on desiccatedor otherwise altered samples.Gregory (1977) discusseslaboratory measurementsand givesa number of referencesfrom outsidethe usualgeophysicalliterature.Reportsofvelocrty measurementsin the literatureare numerous,and

ll3 in the following sections,we cite only those believed to be representativeand that give insight into the interrelationshipof factors. The usual way to determinethe effectsof various factors is to observewhat happenswhen we let them vary one at a time; we then assumethat when more than one factor changes,the effectwill be the sameas if the effects changed sequentially.However, the factors are not independent;thus, for example,changes in external(overburden)pressure(or depth of burial) are apt to changethe interstitial-fluidpressure,the porosity, and the density.Also ordinary descriptionsof rocks often ignore the facts that they have various structures and are heterogeneousin composition. Thus, interpretation of experimentaldata regarding the parameters governing rock velocity becomes difficult and the data in the literatureinvolveappreclable scatter. Despitethe central role that velocity plays in interpretation and the fact that it is often the principal source of uncertainty,much of the literature (Press, 1966;Robieet al., 1966;Christensen,1989;Nur and Wang, 1989)ignoresthe factorsaffectingvelocity,and others give such broad rangesthat the data are not very useful.

5.2.2 Efect of lithology Lithology is probably the most obvious factor affecting velocity and some of the data from the literature are summarizedin fig. 5.5. The most impressiveaspectsof this figure are the rangesof values(somemeasurementsextend beyond the rangesshown) and the tremendousoverlapof the valuesfiordiffering lithologies.Thesesuggestthat velocityis not a good criterion for determining lithology except in a general sense. High velocity for sedimentaryrocks generally indicates carbonatesand low-velocity sands or shales, whereasintermediatevelocitycan indicateeither.The broad rangesfor each ofthe lithologiesillustratethat many other variablesare involved,especiallyporosity and age.The Grant and West (1965)histograms(fig. 5.6) of the data from Birch (1942) also show broad rangesand overlap,as do the data tabulatedby press (1966)and Christensen (1989). Velocity measurementsare sometimesused to discriminate between sandstoneand shale in areas of clastic deposition.Sandsand shalesin boreholesare usuallyidentifiedon the basisof self-potential(SP)or gamma-raylogs; fig. 5.7 showspart of SP and sonic logs in a well that was part of a large study involving many wells in the U.S. Gulf Coast region. Regression analysisfound a differencebetweenthe best-fitvelocity lines for the sand data and the shaledata. but the scatterofindividual valuesexceedsthe differencesbetweenthe best-fitlines.Statisticalpredictions,such as of the overallsand/shaleratio. sometimesare satisfactory when based on reasonably good local data,

6000 El roooorrooo I S eooo-zooo li3llttooo'lzooo Q rcn -aw Z.l'tzow'ttooo - tlooo @ aooo-eooo f| raooo

-rooootrt.* $l sooo

sandand shalevelocitiesand denstFig. 5.9 Maps of average for offshoreLoutii.! i.t,ft. depthintervaifrom7000to 8000ft mapsand the ,iunu. fn. coastlineshowsnear the top of these

leasesystems' block markingsshow the Louisianaoflshore Corp ) (a) SandveDevelopment Geophysical tie of ii"*,.rv (d) shaledenstty' iocitvl (bi shalevelocityl(c) sanddensity;

I Density1gm/cci -1e6 ffi), zzc-zc. I $ ree-zor Q zt -zze Qtzoz -zto Zl zsg-zqs @t zro -zt, D zls -zsz $f ztz -zzl J zsz*

SEISMIC VELOCITY

116 whereaspredictionsfor specificsamplesare little better than guesses. Hilterman (1990) found much variation in shale properties,presumablybecauseof variationsin grain sizeand cementation.The velocitiesof shalesseenin one well (fig. 5.8a)almost trackedhistogramsof average shalevelocitiesfrom 97 wells within the area (fig. 5.8b), both showing fairly broad ranges of values. Shalemembersare more continuousthan sandsand the strongestand most continuousreflectorsare often caused by shale-shalerather than shale-sandcontrasts. In studies of U.S. Gulf Coast wells, Hilterman found that curves of velocitiesand densitiesagainst depth for sandsversusshalesvary considerablyfrom area to area; this variability is illustratedin the maps of fig. 5.9. Becauseof the variability, Hilterman prefers to base synthetic seismogramstudies on edited data from nearby wells rather than to usegeneralized values. Figure 5.10 showsthe dependenceof reflectioncoefficientson the density and velocity differencesbetween sandsand shalesaccording to his stu{ies. In very young sediments,sand-shaleacousticimpedance contrastsare causedmainly by density,rather than velocity, differences,but in older and more deeplyburied sediments,velocity differencesdominate.In the PlioPleistocene, a shale-to-sand reflectionis generallynegative,but for the Lower Miocene,it becomespositive at greaterdepths.Where the densityand velocitycontributions have opposite polarity and roughly equal magnitude,reflectionsare very weak (as in overpressuredUpper Miocenesection,fig. 5.10b). Hilterman also found (fig. 5.1I ) that Poisson'sratio o decreaseswith increaseof velocity for both sands and shales,clean sandshaving appreciablysmallero valuesthan shales.This impliesthat water-filledsands may show an increasein amplitudewith offset.The o contrastbetweensandsand shalesbecomessmalleras the clay content of sandsincreases. Sonic logs ($5.4.3)and density logs (Telford, Geldart, and Sheriff,1990:$l1.7.2 and I1.8.3)often result in poor synthetic seismogrammatches to observed data in the Gulf Coast area. Hilterman believesthat sonic logs indicate sand velocitiesthat are too high becausethey measurean invaded-zonevelocity that exceedsthe velocity of uninvadedsand.The editing of sonic-logdata for syntheticseismogrammanufacture ($6.2.l) attempts to correct for this. The use of deep induction logs (which depend on porosity, like the soniclog; seeTelford et al., loc. cit:652-4) to give the acousticimpedancefor syntheticseismogrammanufacture often resultsin better matchesto actual seismic records. Sandstonesoften contain appreciableclay filling the pore spaces,and clay content is the next most important factor (after porosity) in determiningvelocities.Han, Nut and Morgan (1986)saythat the reduction of P-wavevelocity when pores are clay-filledis about 30Voof that when fluid-filled,and the factor for

S-wavevelocity is about 40Vo. A graph of S-wavevelocity for different lithologies showsspreadscomparableto those for P-wavesexcept that the data are much sparser.By cross-plottingPwaveslownessagainstS-waveslowness,Pickett(1963) found that the domains of different lithologies separated (fig. 5.12a)but some authors quote valueswell outside the indicated ranges (Hamilton, l97l). The ratio of P- to S-wavevelocities(B/c) is thus to some extent indicativeof lithology,as illustratedalso in fig. 5.12b,and S-wavesurveyinghasbeenemployedto determine lithology.Hamilton summarizesthis usageof B/ctdata; he notesthat there is generalagreementthat B/c < 0.5 for unconsolidatedsands,but he notes that consolidatedrocks do not alwayshaveB/o > 0.5. The data for shalesare still very sparse;Hamilton (loc. cit.) quotesvaluesrangingfrom 0.08to 0.36,but some authorsbelievethat the rangefor shalesoverlapsthose of other lithologies to such an extent that lithology identification by B/a measurementis no longer as promisingas once thought. The P- to S-wavevelocity as both porosratio in sandstonesgenerallydecreases ity and clay content increase(Han et al., 1986).

5.2.3EffectoJ density The densityof a rock is simply a volume-weightedaverage of the densitiesof the rock constituents.The densities of the mineralsthat constitutemostsedimentary rocks (table 5.1) encompassa relativelynarrow rangeof abour +7Vo(halite excepted).The major reason why rocks vary in density p is becausethey vary in porosity (seeeq. (5.6)).Histogramsof density (fig. 5.13)resemblethoseof seismicvelocity(fig. 5.6).The densitiesof igneousand metamorphicrocks are generally higher than thoseof sedimentaryrocks because they havelow porosity. Seismic velocity appears to be proportional to mean atomicweight(Birch, l96l), determinedby dividing the molecularweight by the number of atoms. This is shownin fig. 5.14.Most of the relativelyabundant minerals have mean atomic weights around 20 (table 5.1). Metallic ores generallyhave higher mean atomic weights,for example,30.4for ilmenite,31.9for hematite,and 33.1for magnetite. Gardner et al. (1974)graphedvelocity againstdensity (fig. 5.15) and found that the major sedimentary lithologiesdefineda relativelynarrow swathacrossthe graph. The principal exceptionsare the evaporites (anhydrite, gypsum, salt) and carbonaceousrocks (coal, peat, lignite). They determined an empirical equation relating velocity and density, often called Gardner'srule: P

:

avtt4,

( 5 .l 5 )

wheredensity p is in g/cm3,a : 0.31 when velocity I/ is in m/s anda:0.23 when Zis in ftls.This equation is often used to obtain density values in synthetic seismosram constructionor in inversion.

EXPERIMENTAL DATA ON VELOCITY

-0.02

whereAt is the specifictransit time (slowness),Alrand Al- the specifictransit times of the pore fluid and rock matrix, respectively. In terms of velocity I{ this equation is

Relloclion Coollicl.nl

00

0.02

tt7

0.04

I

- d ) = o, l v ' r y+ ( l .

o ( o -

to

(a) n0ll0cllon Coorlicionl

- 0 02

0.0

0.02

004

o o O r

o o

SVelocily

(b) F l c l l o c l i o nC o o l l i c i o n l

-o.o2 -o.04 o -l-_ _ _ t _ _ _ _ _ l

o O c o s

(s.l 6b)

Equations(5.16),the time-average equations,were developedby Wyllie, Gregory,and Gardner (1958)(see fig. 5.16).Howevequnlike eq. (5.6),which is rigorous, eqs.(5.16)are statisticaland empirical.They make no allowancefor the structureof a rock matrix, the connectivity of pore spaces,cementation,or past history, all of which might be expected to affect velocity. Equations(5.16)are usedextensivelyin well-loginterpretation,often with values(table 5.2) for L,t, and Lt,, (or ( and V^) that are empirically determined to give the bestfit over a rangeof interestrather than the actual slowness(or velocity) values,and the fit may be poor outside the intended range, for example, for poorly consolidatedhigh-porositysediments. It should be noted that the interstitial water in shalesis mostly bound water rather than free water in pore spaces;nevertheless, the volume fraction occupied by this water is usually treatedas porosity. Equations(5.| 6) are sometimesgeneralizedby adding terms for the volume fractions occupiedby other constituents.For example,Han et al. (1986) found that adding terms for clay content reducedthe scatter from 6.6 to 2.8Vofor P-wavevelocityand from 10.3to 5.lVo for S-wavevelocity. However,they also found that they could fit velocity measurementsbetter than thoseof slowness;their equationsare

o 0

-o.o4

I

V

0.o

o : 5.59- 6.930* 2.18C1-t2.l%o)kmls ,, ',", : 1 8 . 3- 2 . 2 1 6- ' 7 . 2 C k f t l s , ] B : 3.52 4.910 1.89(-r4.3vo\kmls ,, ,,0, : f 1 . 5- l 6 . l d - 6 . 2 C k f t l s , ]

Oensilya

0

lo

Table 5.1 Density of representativesedimentaryrock minerals(after Robieet al., 1966)

(cl Fig. 5.10 Sand-shale reflection coefficients at normal incidence attributed to differencesin density and velocity values of sand and shale, Gulf of Mexico Tertiary. The solid curves are for normal pressures and the dashed ones for overpressured conditions. (Courtesy of the Geophysical Development Corp.) (a) Pliocene and Pleistocene;(b) Upper Miocene; (c) Lower Miocene.

5.2.4Efect of porosity As previously stated,porosity is often the most important factor in determining a rock's velocity. An equationanalogousto eq. (5.6)is often used:

Ar = d Ar,* (t - d) Ar,,,

( 5 .l 6 a )

Mineral

Formula

Calcite Dolomite Anhydrite Flalite Quartz (ct) Albite Orthoclase Kaolinite Muscovite

CaC0. CaMg(CO.), CaSO. NaCl

Density (g/cmt)

2.71 2.8'7 2.96 2.16 2.68 sior 2.62 NaAlSi.O, 2.55 KAlSi3Os 2.60 Al,si,os(oH)o KAlr(AlSi.O,uXOH), 2 . 8 3

Mean atomic weight

20.0 18.4 22.'7 29.2 20.0 20.2 21,4 15.2 19.0

Many natural minerals vary in composition and hence tn density.Kaolinite and muscoviteare includedas representative of clay minerals.

SEISMICVELOCITY

118

V E L O C I T Y( f t / s )

V E L O C I T Y( f t / s l

(b)

(a) Coast Fig. 5. I I P-wave velocity versusPoisson'sratio for Gulf shale ,uid, und shales.Triangles indicate sand values, circles

Corp ) (a) values. (Courtesy of the Geophysical Development (shaly) sands' Shalesand clean sands; (b) shalesand dirty

Table 5.2 Maftix velocities commonly used in sonic-log interpretatbn V,,,

Unconsolidated sand Sandstone Shale Limestone Dolomite Anhydrite Salt Gypsum Granite Casing

km/s

kft/s

ps/m

ps/ft

<5.2 5 . 59 . 0 1 . 84 . 9 6 . 47 . 0 7.0 6.1

< 17.0 1 8 . 01 9 . 5 6.0 16.0 21.0 23.0 23.0 20.0 15.0 18.0 20.0 17.5

> 193 1 8 2o r 1 6 7 20s 550 l 56-143 139 164 2\8,220 182 164 187

>58.8 )).) or) l.u 62.5 t6'7 47.6 43.5 43.5 50.0 66.76T.0 55.6 50.0 57.0

4.O

5.5 6.1 57.4

whereporosity $ and clay content C are volume fractlons. 5.2.5 ElJbctsof depth of buriul and pressure with increasingdepth of Porosity generallydecreases and hencevelocity inpressure) (oioverburden burial creaseswith depth. The elasticconstantsalso depend on the pressurebecauseof the structure of sedimentary roiks, which are not homogeneousas elasticity theory assumes. The rocks of the LouisianaGulf Coastare generally relativelyundisturbedclastic rocks whose conditions are similar to the rock model describedin $5'l'l' Gregory (1977) gives velocity versusdepth data for Gulf Coast sandsand shalesunder normal pressure

conditions(fig. 5.17).The useof a ll4 exponenlglves a betterfit than the l/6 exponentof eq' (5'5)' The pore spacesin rocksare filled with a fluid under u p..r.ur., which is usually different.from that resuiting from the weight of the overlying rocks; the effecti'e pressureon the granular matrix is the difference between the overburden and fluid pressures' Normal fluid pressureis that of a column of fluid extending to the surface.Where formation fluids are overorissured,the differentialpressurebecomesthat appiopriate to a shallower depth and the velocity t.nOs to be that of the shallowerdepth ($5'3'4)'Laboratory measurements(fig. 5.18) show that velocity is .rr.niiully constant when the overburdenand fluid p..rr,r... u.. changed,provided the differentialpresiure remainsconstant.Abnormal fluid pressurecon-

119

EXPERIMENTAL DATA ON VELOCITY

'

400

'/ ./

./ ^

6

N --

-Y

€

3 250

\

/

E .r'

o

,'

,,',

r, .r'^^^r(rr. .-to"'rt'

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,'

,''

1&

,,

,''

/

-j,, , '.r,

,'

.r{1t r}}'l,r(t

t', l t

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{;,

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,'

, l ' - ,t .r-'-

,,(Fri',:p"'," ,'/n)"

,'

. ,?: .n(

/

t .h9o'

P'wavc v.loci(y, d(kft/3) t4 18 t6

/

,1'ot ,'

A'Lt't2'

t2

,/

,' ," l;**,t'',,",r'(':." .-

J

ut

7-

,{' d"'7',' y'''' y''/ ;i'y'' /

,"' .""'1","",'("',"t'

-,t

>

F O

I2a+.

20

066

F i g . 5 . 1 4 V e l o c i t ya t l0 kilobars versusdensity for silicatesand oxides. The numbers refer to mean atomic weishts. (From B i r c h .1 9 6 1 . )

3 060

i rr" I

s ?

@ 050

8 50

40

Velocity (kft/s) 12 15

60

P.wrv€ vclocily. a(km/s)

(b)

2.8

Fig. 5.12 Relation between S- and P-wave velocities (/, and (,) for various lithologies. (a) Cross-plot of laboratory measurements (after Pickett, 1963). (b) Use of S- and P-wave velocity r a t i o ( p / c r )a s a n i n d i c a t o r o f l i t h o l o g y ( f r o m S h e r i f f ,1 9 8 9 13 8 8 ) .

l

t

t

t

1 ^

d

2.O

t

Sdl ond olluvio

3.0 4.0 5 Velocity(km/s)

Sondstonas Shola5

Fig. 5. I 5 P-wave velocity density relationship lor different lit h o l o g i e s( t h e s c a l ei s l o g , l o g ) .T h e d o t t e d l i n e s h o w se q . ( 5 . 1 5 ) and the dashed lines show constant acoustic impedance ( k g / s . m r x 1 0 6 ) .A f t e r G a r d n e r e t a l . 1 9 7 4 : a n d M e c k e l a n d Nath. 1977.)

Limasiona

l

l

l

r

\l

a

tE

!

\

I

t

I

I

o

s5

I I

I I

\

z

I I

r.4

1.6

'.8

2.O

2.2

2.4

2.6

2.a

3.0

Densig (g/cc) Fig. 5.13 Histogram of density values tabulated in Birch (1942) for different lithologies. (From Grant and West, 1965.)

stitutesa severehazard in drilling wells and one use is in predictingsuch of seismicvelocitymeasurements z o n e s( s e e$ 5 . 3 . 4 ) . The variation of velocity with depth, often referred to as the velocity function (54.2.4),is frequentlya reasonablysystematicincreaseas we go to greaterdepths. Velocity versusdepth relationshipsfor severalareas are shownin fig. 5.19. Gardner et al. ( l9l 4: 175 -6) state:"With increasing depth the velocity increasespartly becausethe pressure increasesand partly becausecementationoccurs at the grain-to-grain contacts. Cementation is the more important factor." Their graph for sands is

SEISMICVELOCITY

120

\

\

x \

20

r 1

,

\i" o '

t:

\^" e' o ,\^-p:' -^ o\6 p ^' d":o \ \ \r., o o

o o, l0

o,

^

'

o Sandstone . Limestone

^

r!.

"\:

^

"\f

, \';:' {q.,:.,

Velocity(kft/s) I2 I0

14

4.O 4.5

3.5

o

' '16'

,' :'(rl

5.0

V e l o c i t y( k m / s )

Fig. 5.16 Velocity-porosity relationship. The horizontal scale is linear in transit time (l/l'). The dashed line is the time-average e q . ( 5 . 1 6 b )f o r V - : 5 . 9 4 k m / s ( 1 9 . 5k f t / s ) a n d V , : 1 . 6 2 k m / s (5.32 kft/s). (After Wyllie et al., 1958.) Velocity (km/s) t.0

4.5

l0

l2

14

Velocity(kft/s)

l0

with depth for in-situ sedimentsis more rapid until they becomeconsolidated;below about 6000 ft they roughly follow the time-averageequation. Some authors interpret data as showing a simpler curve (e.g., Faust, l95l), that is, they regard consolidationas a more continuousprocess. An exampleof the variation with depth of ct, B, and p/o in a predominantly sand-shalesection is shown in fig. 5.2L Gardner and Harris (I968) considervalues of p/a < 0.5 as indicating water-saturatedunconsolidated sand. 5.2.6 Effectsofage, frequency,and temperature An early form (Faust,l95l) of eq. (5'5) includedthe ageof the rock as a factor in determiningvelocity'In fig. 5.22,each data point is the averageof many values.Older rocks generallyhavehigher velocitiesthan younger rocks, but most geophysicistsagreethat age is merelya measureof the net effectof many geologic processes, that is, olcierrocks havehad longer time to and so be subjectedto cementation,tectonicstresses, on, which decreaseporosity.The history ofrocks varies so much in time and spacethat the time factor must be only approximate. Time-dependentstrain may play some part, but how large a part is not known. Experimentaldata generallysupportthe conclusion that dispersion(variation of velocity with frequency) is small over the range from hertz to megahertz.We expect velocity to changewith frequencybecauseof absorption(52.7.2),the manner of changedepending on the absorptionmechanism.Nur and Wang (1989:

t2

PRESSURE(MPa) r4 .:

; lo 6

lq

".."{--=:--Etffi lL1999o

t;

l8

'/ 20

,r' ^ -d-- o------o--

'.9 2X 24

':ry-o

/--o--o------o--

i A

o

e : iE

o--

'o'-

{E

F C)

ul

Fig. 5.17 Velocity depth relationship for Gulf Coast sands and shales. Best-fit quadratic curves are also shown. The step graph shows data ior offshore Venezuela.where conditions are similar. The dotted curve shows averagevelocity versus depth for the Venezuelandata. (Data from Gregory, 1977.)

shown in fig. 5.20; the curves for Ottawa sand and glassbeadssubjectedto pressureindicate what they believewould happen if sands were buried without consolidationor cementation.The increasein velocity

aP'rno^ a

o J

28

-o--

--o---o -o---o---o-

1*

6 o J

u.l

-O---O---O--O---O aP.am O<:O-:o---o---o!P.&) -o ---o!P' tqD - -1.- - - o

EXTERNAL PRESSURE (P,s.i.) Fig. 5.18 P-wave velocity for two sandstonecores versus confining pressure where the differential pressure is held constanl at the indicated values (in psi). (After Hicks and Berry, 1956.)

EXPERIMENTAL DATA ON VELOCITY 318-19)discussdispersionin rocks.Thereis somedispersionin liquid-saturatedrocks,but not in dry rocks, from which Spencer(1981)concludesthat dispefsion is due to fluid movement along pore surfaces.Han (1987) observed some dispersion, which decreases with increasingporosity and increaseswith clay content; dispersionalso decreaseswith pressure.Wang (1988)found that dispersionincreaseswith fluid viscosity,but it is still small in rocks saturatedwith light oil. Murphy (1985)found that crincreasedby 157obetween 2 and 200 kHz. Velocity appearsto vary slightly with temperature (fig. 5.23),decreasingby 5 to 67a for an increaseof l00oC. However,velocity in heavy crude oil and tar varies considerablywith temperatureand the same appliesto rocks saturatedwith them (fig. 5.2q (Wang and Nur, 1988).This forms the basisfor the monitoring of enhanced oil-recovery programs based on thermal stimulation.that is. steamand fire floods.

I2l

Averagevelocity(km/s)

3

a

9

o

Fig. 5.19 Velocity depth relationships for selected wells. (a) Data from Gulf of Alaska Cost-B2 well. offshore U.S. East Coast, wells in Tyler (#l ) and Dewitt (#2) counties in TexasGulf Coast, Illinois Basin, and Permian Basin. (b) Data from Sacramento Valley, Yolo Co., Calif.; Central Valley, Calif. (from Stulken, l94l ); Wind River Basin, Fremont Co., Wyo.; Williston Basin, Divide Co., N.D.; and the Java Sea. (c) Average velocity to various depths for the data in parts (a) and (b).

t0

The velocity in water-saturatedrocks increases markedlyas temperatureis loweredthrough the freezing point (fig. 5.25).As the temperaturedrops,the liquid in the larger pores freezesfirst, the salinity of the liquid controlling the freezing curve. At a slightly lower temperature,the liquid in the smaller pores freezes. 5.2.7E/fectof interstitialfuid

(kftA)

E

q

Porousrocks are almost alwayssaturatedwith fluids, generallysalt water,the poresin oil and gasreservoirs being filled with varying amounts of water, oil, and gas.The replacementof water by oil or gas changes the bulk density and the elasticconstants,and hence also the P-wavevelocity and the reflectioncoefficient. Thesechangesare sometimessufficientto indicatethe presenceof gas or oil. Horizontal variationsin reflection amplitude,velocity,frequency,and other factors are sometimesimportant indicatorsof oil and gasaccumulations(see$10.8).The low velocitieswhen gas fills the pore spaceat least partially explain the low velocities observed in the weathered (LVL) layer ($5.3.2)and why its lower boundary is so often the water table. The nature of the interstitial fluid does not change the shear modulus appreciably and hence ,S-waveve-

SEISMICVELOCITY

122

everyamplitude anomaly was associatedwith a commercialgasor oil field. Domenico (1974),applyingthe Geertsmaequation (5.14), showedthat only a small amount of gas in the pore spaceproduceda large decreaseof velocity (fig. 5.27a) and a large change in reflectivity(fig. 5.27b).The Geertsmaformula allows for the fluid compressibilityas well as the densityand the elasticmoduli of the matrix material. Domenico (1976, 1917)partially verified the theoretical results with laboratory experiments.

(knVs) VELOCITY ,2.5 3,

PosTEoctu s^l{osANosHAuS '{'ztrl ztto

e . .T

E

5.2.8 Summaryof factors afecting velocity

I F (L UJ

Tatham and McCormack (1991) summarizedthe effectsof various parametersin fig. 5.28, including two parametersthat we did not discuss:pore shape and anisotropy.If the poresare approximatedby ellipsoids,the aspectratio is the smallestaxis divided by the largest.The aspectratio of circular poresis unity. whereas small aspect ratios denote very elongate pores.Velocity decreases when pores are elongate.In transverselyisotropic media, the velocity dependson the direction of particle motion with respectto the axis of symmetry(u2.6.2).

!

; F

GUtfcl^sl S^llDS BRINE

G

u, o

a

\

l0t 6

7

. E 9 1 VELOCITY (kfiis)

J3 n

0

53 Apptication of yelocity concepts 5.3.1Intoduction

Fig. 5.20 Velocity-depth for an unconsolidatedsand pack and for actual sands. The dash dot curve shows curve shapes indioated by some authors.

(km/s) VELOCITY

o,!

oa

g!r

F '. 6

vEloclrY (km/s)

tt oa,

-

o'

o't!

ot

Fig. 5.21 Velocity depth for a well in Chambers Co., Texas. Note the decrease in the ratio of P- to S-wave velocities in shales.(After Lash, 1980; and Tatham and McCormack, 1991.)

locity changesonly slightly (mainly becausethe density changes).The ratio of P- to S-wavevelocity (a/9) has beenusedas a method of distinguishingthe fluid filling the pore space(fig. 5.26and $l3.l.l). in locating hydrocarbonsby Some early successes increasedreflectionamplitudeled to expectationsthat

An understandingof the factors affecting velocity helps us forseethe kinds of velocity variationsto expect in an area and hencethe velocity distortions to expectin seismicdata ($10.5).In the U.S.Gulf Coast area, a younger section is encounteredat a given depth as one goesseawardbecauseofthe seawardregional dip. Lithologic changesare relativelysmall and the maximum pressuresto which the rocks havebeen subjectedare the existing pressures,which depend mainly on depth, not age.Hence,the velocity(fig. 5.9) does not vary greatly from area to area, and lateral velocity changesare relatively small, although still large enough to modify structures significantly.In contrast, areas subject to recent structural deformation and uplift, suchas many mountainousareas,exhibit rapid lateralvariationsofvelocity. In suchareas, many of the rocks have been subjectedto greater stressesby burial to greater depths than at present, and the result is rapid lateral changesin velocity that profoundly affect seismicinterpretation. Empirical data suggestthat the maximum depth to which a rock has beenburied is a measureof the rrreversible effect on porosity and is therefore an lmportant parameterin determiningporosity. In short, porosity is often determined principally by the existing differential pressureand the maximum depth of burial. The irreversible change in porosity (and consequently in velocity)with depth of burial has beenused to determinethe maximum depth at which a section formerly lay. If the velocity depth relationshipfor a

o z

'"')'"'"t^'n

0 0 IJ o \r

',fru*,i::;;:;:

ld IJ I

o o 9 1 L

U 0

J l!

O ORDOVICIAN . DEVONIAN O MISSISSIPPIAN 'PENNSYLVANIAN . O O 6

r! r

1

" T E R T T A R (yp o s r - e o c e N e ) u

r

2

3

4

5

DEPTH Fig.5.22

PERMIAN J U R A S S I C- T R I A S S I C CRETACEOUS EOCENE

5

7

IN

IOOO

8

9 r E-L

I

V e l o c i t y v e r s u sa g e a n d d e p t h o f b u r i a l . ( F r o m F a u s t , 1 9 5 1 . )

-'-

4.4

2.7 -r-o-

-o-

_c_ _ _ _ -p _6 o_ _Mr p€u_ _ e l _ -

.- -"- - - - -6ro-YlU---.o--

L

isr_rr{s{'_r__+

--r-f

2.5

_rl:Yii

o +.t

=

--!-

-.-

)

1-r--r-

!t-

t--_-_r

- - --l 5- M- P- a- - - o - -

__,r\

r 1_ 6 Mpp

'

-

-'-

-a\-

'a-a-

-

6 MPa

a-r-l

4.0 100 (oC) Temperature

0

100 (oC) Temperature

Fig. 5.23 Velocity in brine-saturatedBerea sandstoneas a function of temperature and pressure.(After Timur, 1977.)

27oiJ 4600 2600

a 9a4dt '6

^2500

{

€zroo '8 raoo

€.* o

o

.E.*

)m o

I P3800 E o

o

wnh 1ooIFt Otl

3600

s

€zrm o 2oo0 r9@

z

o

{

p

6

0

8

0

t

m

t

2

0

Temperature1.0C )

(?) Fig. 5.24 Velocity in Berea sandstone saturated with 10. ApI oil as a function of temperature. (From Wang and Nur, 1992a.)

wtth looApt otl 4

0

6

0

0

Temperature 19C ) (b) (a) P-wave velocity and (b) S-wave velocity.

SEISMICVELOCITY

124

l0

$H o a

t

u

U

H

c -

=

E

E

3 g

E

& :e 5

,ng unH a E0E X rob

$F

nE T[ n

Fig. 5.25 P-wave velocity in Berea sandstone as the temperatuie passesthrough the freezing point. (From Timur' 1968')

0.05 h 0 . 1 0 .9 0.15 o 0.20 U.ZJ

0.30 o "- "5- "0- 0L

0

0r0

020 Porosity0

030

-6

'o c

040

Fig. 5.26 Relation ofS- and P-wavevelocities and porosity for gas- and water-saturated rock. (From Sheriff, 1989; with data from Gregory, 1976.)

given lithology can be establishedin an area not subjected to uplift, the maximum depth of burial may be approximatedfrom the observedvelocity-depth relationship and hence the amount of the uplift can be inferred. In fig. 5.29, the shaleand limestoneregression lines (curvesA and Q representmeasurements "pure" shalesand limestonesthat are believedto on be at their maximum depth of burial. Curve 4 which is obtained from thesecurvesby interpolation,is the predictedcurve basedon the relativeamountsof shale and limestone actually present and assuming the rocks to be at their maximum depth of burial. The displacementin depth required to fit this curve to the actual observationsis presumedto indicate the amount of uplift that has occurred. This technique can sometimesbe usedto determineif rocks haveever been buried deeply enough to acquire the high temperatures required for hydrocarbon generation (see $ 1 0 . 1l ). . 5.3.2 The weatheredor low-velocitylayer Seismicvelocitiesthat are lower than the velocity in water usuallyimply that gas(air or methaneresulting from the decomposition of vegetation)fills at least some of the pore space(Watkins,Walters,and Godson. 1972).Such low velocitiesare usually seenonly

on land near the surfacein a zonecalledthe weathered layer or the low-velocitylayer, often abbreviatedLVL. This layeqwhich is usually4 to 50 m thick, is characterized by seismicvelocitiesthat are not only low (usually between250 and 1000m/s), but at times highly variable. Frequently,the base of the LVL coincides roughly with the water table, indicating that the lowvelocity layer correspondsto the aeratedzone above the water-saturatedzone, but this is not always the case. In areas of seasonalfluctuation of the water table,leachingand redepositionof mineralsmay produce the effect of double-weatheringlayers.Doubleweatheringeffectssometimesresult from a perched water table or changesat the baseof glacial drift that is at a different depth than the water table. In sandy desert areas where there may be no definite water table,the LVL may gradecontinuouslyinto sediments with normal velocity. In subarctic areas, muskeg swampis mushywith low velocity in summerand frozen with high velocityin winter (seealso $5.3.3).In other areas,the nature of the low-velocity layer and the problems associatedwith it changeconsiderably "weathering"as used with season.Obviously,the term "weatherdiffersfrom the geologist's by geophysicists ing," which denotesthe disintegrationof rocks under the influenceof the elements. The importanceof the low-velocitylayeris fivefold: (l) the absorption of seismicenergy is high in this zone:(2\ the low velocity and the rapid changesin velocity have a disproportionatelylarge effect on traveltimes;(3) becauseof the low velocity,wavelengths are short and hence much smaller featuresproduce significantscatteringand other noise;(4) the marked velocity changeat the baseof the LVL sharply bends seismicrays so that their travel through the LVL is nearly vertical regardlessof their direction of travel beneath the LVL; and (5) the very high-impedance contrast at the baseof the LVL makesit an excellent reflectot important in multiple reflections and in mode conversion.Becauseof the first factot records where the sourcesare within this layer often are of poor quality; shots in boreholesare often placed 5 to l0 m below the LVL. Methods of investigatingthe low-velocity layer are discussedin $8.5 and methods of correctingfor its effectsin $8.8.2. In someareaswherethereis significantcompaction with depth within the low-velocitylayeq the increase in the velocity Z with depth z approximates V :

aztln,

( 5 .I 8 )

where a afid n are empirically derived constants. Blondeauand Swartzdevelopedthe Blondeaumethod for determining the vertical traveltime to a datum whenthe velocityobeyseq.(5.18)(Duska,1963;Musgrave and Bratton, 1967). If the first-break timedistancecurve is nearly a straight line when plotted on log-log paper,the method is applicablefor making weatheringcorrections($8.8.2).The line's slope gives r. The calculationprocedureis discussedin problem

H y d r o c a r b oSna t u r a t i o n

t.0

H y d r o c a r b oSna t u r a t l o n

0.5

1.0

t0

3.0 -

-1oloI -

<:^3050m

8 o

2.5

n-/

-lggi-

I

6

E

1 8 3 0m

:

=

2 . 0I :

g

o

3

0.5

6

- - 0.2

.= ': o o o G

o

"----98--"

-18J0, -_ ---r-:-:;1-=,i,e.

- -k,]*i'i

t.5 6 f0 n r

t.0

a

r.0

0.5

0.5

WalolSaturetlon

W a t o rS s l u l a l l o n

(a)

(b)

Fig. 5.27 Effect of gas/water or oil/water saturation on velocity. Solid curves are for gas, dashed for oil. (After Domenico,

1974.)(a) P-wavevelocityversussaturation,and (b) reflection coefficientfor gas/oilsandsoverlainby shale.

ROCKPROPERTY LITXOLOGY

POROSITY

POREFLUIOIYPE

PORESNAPE

OEPTHOF SURIAL, coNsoLr0lTr0r lNo aP

TEXPERATURE

AMSOTROPY

x ao e

:

]

aq t> I

olspEcl

n^l,o''o

--oEPIH+

Fig. 5.28 Summary of effects of different rock properties on P- and S-wave velocities and their ratios. (After Tatham and McCormack. 1991.)

T€YPER TUNE+

t -

---

o mnzrtn

n

SEISMICVELOCITY

t26

0.5 Presentdepth 800 m ^

1.0

g

o

1.5

2.5

r.s 2 2.s

ro i"*nro,*,n,1,

Fig. 5.29 Finding maximum depth of burial from velocity. (From Jankowsky, 1970.)

8.21. This method has been applied in glacial-drift areas. 5.3.3 Permalrost The temperature of near-surfacerocks is usually about the mean annualtemperaturefor the location, and in arctic and some subarcticareasthis temperature is below the freezingpoint. Seismicvelocity generally increasesmarkedlywhen the pore fluid in a rock freezes(fig. 5.25).In muskegareaswhere the nearsurfacematerial is essentiallyswampwhen not frozen and rich in undecayedvegetation,the velocitymay increasefrom 1.8km/s or lowerto 3.0 to 3.8km/s upon velocfreezing.Timur (1968)reportsBereasandstone ity changingfrom 3.9 to 5.2 km/s, Spergenlimestone from 4.4 to 5.7 km/s, and black shalefrom 3.6 to 3.9 km/s upon freezing.The amount by which the velocity changedwas roughly proportional to the porosity. The portion ofthe sectionthat is frozenyear-round is calledpermafrost.There is usually a layer aboveit that thaws in the summerand the generalincreaseof temperaturewith depth imposesa lower limit. Permafrost thicknessvariesfrom tensof centimetersto a kilometer.Where it is very thick, the velocity near lts basemay decreasewith depth gradually until velocities are normal for the rock type. Where the permafrost is relativelythin, the decreasein velocity at its basemay be fairly abrupt. A body of water on the surfaceusually does not freezedeeperthan a few metersand the water insulatesthe sedimentslying belowit from the cold so that permafrost is often absent under water bodies.The lateralchangefrom normal velocitiesunder lakesand

rivers to high permafrostvelocitieson adjacent land areascan be very abrupt and can producethe appearance of major fictitious structuresdeeperin the section. Permafrostis found on the continental shelf of the Arctic Ocean where the presentseafloor was exposedduring a period of loweredsealevel(Hofer and Yarga, 1912).The thermal conductivityof earth materials is so low that thermal equilibrium has not yet beenreestablished. Whereasrefraction at the baseof the low-velocity layer tends to make raypathstraversethe layer more nearly vertically,thereby simplifying correctionsfor the layer,refraction at a permafrost boundary makes raypathsmore oblique and increasesthe travel within prothe permafrost,complicatinganalysis/correction cedures.The effect is greater with long-offsettraces, which generallyhave a larger horizontal component of travel.Also, the baseof permafrostis often gradational and not necessarilyhorizontal.The result is that our ability to correct for permafrosteffectsis often poor. To complicatethe problem, we usuallycannot determine accuratelypermafrost thickness and velocityvariations. Another phenomenonassociatedwith permafrost is /rost breaks(ice breaks)that result from crackingof the ice outwardfrom the source(fig. 5.30).Thesesudden energyreleasesoccur abruptly at various times after the sourceactivationand involveappreciableenergy release,so that their effectis that of repeatederratic extra sourcesthat may obscurereflectionsfrom the primary shot. Frostbreaksare lesslikely to occur so one may haveto use as the sourceenergydecreases, smallersourcesthan otherwisedesirableand increase the amount of stacking to compensate(Rackets, 1 9 7l ) .

detet'tion 5.3.4 Abnormul-pressure "Normal" pressure for rocksis the situationwherethe pressureof the fluid in the rock's pore spaceis that of a hydrostatichead equal to the depth of burial. If the density of the fluid is p,, normal fluid pressure0, : p,;, where z is the depth. Drillers often speak of the gradient,d9,ld: = p, whichis about l0 kPa/ pressure m or 0.45psi/ft for p,: 1.04g/cmrtgradientsbetween "normal")' 0.48and 0.43psi/ft are usuallyregardedas The pressuregradient due to the rock overburdenis about d9,,,/d::22.5 kPa/mor 1.0psi/ft (for p,,,: 2'3 g/cmr).The effectivestresson a rock (as discussedin 55.2.5) is due to the differential pressuregradient L,(dgldz) : dT,,ftz - dT,Mz : 12.5kPa/m or 0.55 psi/ft. Abnormal or overpressuresituations (subnormal pressuresare also occasionallyencountered)result from a sealingof formationsas they are buried so that the formation liquid cannot escapeto allow the formation to compact under the increasedoverburden pressure(Plumley,1980).In effect,part of the weight of the overburdenis transferredfrom the rock matrix

r 0

3 2

""'u

:*;,,r:--;i:

5 5

Fig. 5.30

Seismicrecords from the Arctic showing frost breaks. (Courtesy ofpetrocanada.)

I

SEISMICVELOCITY

t28 to the fluid in the pore spaces.Consequently,the rock "feels" that it is under a differentialpressureappropriate to someshallowerdepth and the velocity of the rock is that of the shallowerdepth (Dutta, 1987). The deeper portions of many depositional sequencesinvolve fine-grained sedimentswhere the permeability was not sufficient to allow the interstitial water to escapeduring compaction and abnormal pressuresarecommon.This is especiallytrue in young Tertiary basinswheredepositionhasbeenfairly rapid, suchas the U.S.Gulf Coast,the Niger and Mackenzre deltas,and along the continentalslopesin many areas. Abnormal-pressureformationsmay behaveasviscous fluids lacking shear strengthand becomeinvolved in diapiric flow (seefig. 10.23)or becomeweak detachment zonesfor faulting (Gretener,1979). Sonic and induction well logs (Telford, Geldart, and Sheriff, 1990:$l L2.6) in a boreholeencountering high pressure(fig. 5.31)often show a fairly abrupt onsetof high pressure;it is sometimesmore gradual.The high pressurein this well shifts the trendline toward higher transit times (lower velocity);the magnitudeof this shift is a measureof the overpressure(Reynolds, 1970.1973). Reflectionswithin or below an abnormally pressured sectionmay permit calculatinginterval velocities by velocityanalysis($9.7.3)(Keyseret al., l99l); curvesfor two situationsshowing overpressuringare shown in fig. 5.32.Analysesfor overpressureare usually plotted as transit time (Al : ln versusdepth to make them more comparableto sonic logs. Analysis often includespredicting the mud weight that will be required when drilling (fig. 5.32b). Mud weight of about 9.2 pounds/gallonproducesnormal pressurein a wellbore. The tendencyin picking velocity analyses($9.7)is to honor only velocity data that show a monotonic increaseof stackingvelocity with depth, so that conventionalinterpretationsare apt to ignore velocity inversions, which often indicate abnormal-pressure zones.Thus, data are apt to be stackedwith too high a velocity,causingdeteriorationof reflectionswithin such zones,and often the zonesappear rather dead. Multiples ($6.3.2),of course, are also usually evidenced by low stacking velocities and hence may make the interpretation of abnormal-pressurezones difficult. The predictionof abnormalpressureis of considerable importancein drilling plans to minimize the possibilitiesof blowoutsand other drilling problemssuch "gas cut," "heaving" shale,and "bridging," indias cated in fig. 5.31. Abnormal-pressurezonesare also of importancein predictingreservesbecausegas reservoirs within them can contain exceptionallylarge amounts of gas for the reservoirvolume. When searchingfor abnormal pressures,velocity surveys ($5.4.2)are usually run with smaller increments than when the objectiveis primarily to determine the stackingvelocity.Aud (1976)contendsthat velocity-scanincrementsshould be 50 ft/s and time

increments10 ms; he also arguesthat data should be at least l2-fold with offset sufficientto give at least 100 ms of normal moveout. He often picks events spacedonly 100ms apart. The techniqueof averaging data for a number of adjacent midpoints generally but also to poorer leadsto lessnoisein measurements, identification of velocity values with specificevents. Becauseof the uncertaintiesin the valuesdetermined from any singlevelocity analysis,weightedaveraging of the resultsof severaladjacentanalysesimprovesreliability. 5.3.5 Gas-hydrate fficts Reflectionsthat cut acrossthe beddingare sometimes seenon deep-waterseismicdata a short distancebelow the seafloor, as in fig. 5.33.Theseare often attributed to gas hydrates,icelikecrystallinelatticesof water molecules in which gas molecules are trapped physically.Theseare stableunder the temperatureand pressureconditions found just below the seafloor in deep water.They apparentlycan form where the gas concentrationexceedsthat necessaryto saturatethe sedinterstitialwater.The velocityin methane-hydrate imentsis about 2.0 to 2.2 km/s (Tucholke,Bryan, and "base ofgas-hydratereflection" usuEwing, 1977).A ally is roughly conformableto the sea floor and the depth of the reflection beneath the sea floor correspondsroughly to the limit of stability of methanehydrate (fig. 5.34).The reflectionis thus interpretedas marking the interface between hydrate and gas trapped by the overlyinghydrate.The gas trapped in this way may somedaybe an energyresource.

5.4 Measurement of velocity 5.4.I Velocityterminology "velocMany adjectivesare usedprecedingthe word ity," the speedwith which a wave travels.Velocity in media dependson the proportions and heterogeneous distributionsof the media and the direction of a wave (and amplitude for very large disturbances).Terms such as averagevelocity ,/ (the distance traveled divided by the time required to traversethe path) depend on the raypath. Likewise,the root-meansquare (rms) velocity Z.-. refers to a specificraypath; rms velocities are typically a few percent larger than correspondingaveragevelocities.

v

,r,to, f' :'o n,

,|.ot

v : ^ ,:

1,",',0' (5.re)

J,.,

Although averageand rms velocityhavemeaningonly with respectto a particular path, a vertical path and horizontally layeredmedia having velocities\, thicknessesAz,, and traveltimesthrough the layersA/, are

(9

B T < f

9

sol{lc ps/lt ,

@NDUCNVITY mmhor

F

or ? 9 3W = -9.t

s ssgpggEEf;gE

DRILLING HISTORY REMARKS

9.t TOP OF AENORTAL P R E S S U R EZ O N E \ cul,ft.ovh! .icl.--fAlgoo.

lrl trt lr

= -

Sl.*r"o *",to,. rtollhi Cin.Clr'Crc.-7?tC

q lrl

Holr h.ovrd'dgCAlp|rWo.h.tt oar ond

o

faSorafac. l.l.! Oor celtlne I no haarlrll

z 9 (J lrj

o UJ J

l5.S Gd el.6b.lda haaoacrlELroDhe [t? eor cll 15,7Gc cul- Flo|. rnotl rltaoo rlttr 9|n, ott. 13.9Cc cll . chlalda lrtaora

o J F

z

Gorsul.hoL haovdrlort 73'lalwnr. .agohad3lw 16.0Cor cut Gorq/l,b?ldearrloalurdl monl Gorcalrtltg.r l6.JGorol.l.lde6 I tlo io{DL O$ olftcDlne G6 cul 0or cul 0o. qi,hola bdne h ra|url 0aqi,palldba LD.l?ttt

Fig. 5.31 Effectof abnormalpressure9n-t-oli"and induction (cJnductivity;logsof an offshoreU S' Guli Coastwell' (From MacGregor,1965;reprintedwith permission')

nl|6

130

SEISMICVELOCITY

100

1 / V( p s / m ) 2 0 0 4 0 0 8 0 01 2 0 0

1 / V( p s / m ) 2oo 300 I 4 L B S , /G A L ,

16Lgs./ GAL. E

4

:

LBS./GAL

c L 5 6 a) o E 2

d o

-

a

o t o

PR€OICTEO PORE PRESSURE LB./GAL.

-, ^E : E CL

roo

t?.5 t7.6

o

4

o

rs2

60 80100

2@

80 loo 1 / V ( p s il l ) (b)

1 / V( p s / f t ) (a)

Fig. 5.32 Lowering of seismicvelocity as indicator of overpress u r e .( a ) A I t e r R e y n o l d s ,1 9 7 3 ;( b ) a f t e r P c n n e b a k e r1, 9 6 u .

often implied, that is, depth divided by the traveltime verticallyto that depth (straightraypaths).Then, by replacingthe integralswith sums,we obtain

\ v. tr. I 4., - - 7 - 7

/-

!

rz: ,t,

| Y,',,,= \. ^Eo', )o',' /' ,/

:

,-^.

().lU)

l.II,

the former is eq. (5.24),the latter eq. (4.26).Stacking velocity ( (also called NMO velocity)is the velocity value determinedby velocityanalysis(seeeq. (5.23) and $9.7);these measurementsare usually made by finding the best-fithyperbolato data that are not perfectly hyperbolic.For isotropichorizontal layers,( : Z.-" in the limit as the source-receiverdistanceapproacheszero(seefig. 4.10). Interval velocity( is the averagevelocity over some interval of the travel path. The interval may be the distanceover which a soniclog measuresor the interval betweenparallel horizontal reflections.The latter is also called the Dix velocity given by eq. (5.25), which can be written as

,r,-vit,-1J,, I, -

(s.21)

t, ,

where V, is the rms velocity and l, is the zero-offset

arrival time correspondingto the rth reflection. Becausewavefrontsare surfacesof constantphase, perpendiculardistancesbetweenthem yield pha,sevektcitl:. Apparentvebcity refersto the apparent speed of a givenphasein a particulardirection,usuallythe spread direction; thus, apparent velocity is greater than instantaneous velocityby the secantofthe angle betweenthe direction ofwave travel and the direction in which the apparentvelocity is measured.Groupvelocity wasdiscussed in {2.6.2and 2.7.4. 5.4.2 Conventional well ,\urveys The most direct methods of determiningvelocity require the use of a deep borehole.Three types of well surveys are used: the "conventional" method of "shooting a well," a variation called vertical seismic profiling(VSP),describedin 913.4,and soniclogging, describedin the next seclion. Shootinga wel/ consistsof suspendinga geophone or hydrophonein the well by means of a cable and recordingthe time requiredfor energyto travel from a sourcenear the wellheaddown to the geophone(see fig. 5.35).Air guns in the mud pit or in the water for marine wells are often used as energy sources.The geophone is specially constructedto withstand im-

MEASUREMENT OF VELOCITY

t3l

Baseof gashydrate

Fig. 5.33 Seismic line on the Blake Outer Ridge, offshore U.S. southeast coast, showing base of gas-hyilrate reflection. Water

mersion under the high temperatures and pressures encountered in deepwells.A mechanicalarm presses the geophoneagainstthe boreholewall to assurecou_ pling. The cablehas a threefoldrole: it supports the geophone,it servesto measurethe depth ofthe geo, phone,and it carrieselectricalconduciorsthat brins the geophoneoutput to the surface,where it is rel corded.Sourcesare locate<J at points near the well_ head.The geophoneis movedbetweenactivationsof the sourceso that the resultsare a set of traveltimes from the surface down to various depths.The geo_ phone depths are chosento include the most rm_ portant geologicalmarkers,such as tops of forma_ trons and unconformities,and also intermediate locationsso that the interval betweensuccessive mea_ surements is smallenoughto givereasonable accuracy (often200 m apart). Resultsof a typical well surveyare shown irr fig. _ 5.36.The verticaltraveltime,t, to the depth,;, is ob_ tainedby multiplyingthe observedtime by the factor zl(22+ r:1trtto correctfor the slantdistance.The aver_ age velocity betweenthe surfaceand the depth : is then given by the ratio :/1. Figure 5.36showstire aver_ age velocity Z and the vertical traveltimeI plotted as functionsof :. If we subtractthe depthsand times for two sources,we find the interval velocity \, the aver_ age velocity in the interval 2,,,- 2,, by means of the formula V,:

t,,, -

t ,,

(s.22)

Time measurementsoften havean accuracyof only 2 ms and consequentlyinterval velocity valuesare not as accurateas traveltime_depthdata. Shootinga well givesthe averagevelocitywith good accuracy.It is, however,expensivebecausethe cost in_ cludesnot only the one-half to one day's time of the seismiccrew,but also the cost of standbytime for the well (which often far exceedsthe seismiicost).poten_

depthis from 3000to 3600m. (FromShipleyer al.. 1979.)

tial damageto the well is anotherfactor that discour_ agesshootingwells;while the surveyis beingrun, the well must stand without a drill stem in the hole and henceis vulnerableto cave-in,blowout,or other seri_ ous damage.A further disadvantagein new explora_ tlon areasis that seismicsurveysare often completed beforethe first well is drilled. 5.4.3 Velocitl;(sonit) loggmg The continuous-velocity surveymakesuseof one or two pulsegenerators and two or four detectors. all lo_ cated in a singleunit called a soncle,which is lowered into the well. Figure 5.37a shows the borehole_ compensated sonic-logging sonde developed by Schlumberger. It consistsof two sourcesof seismic pulses,S, and S", and four detectors,R, to Ro, the

| 000

10.0

500

5.0

10.0

-

Hyd rate s t ab l c

q

I zoo

:tt

E

!

u

r.n

g e

100

I O ; Hydrirte not stabie


x o ^ .

E

o t.o E

q

u) o

x

0.2 l0

20

T e n r p e r a t u r(e" C ) Fig. 5.34 Limit of stability of melhane hydrate in water con_ taining 3.5% NaCl. The horizontal distance between the dotted vertical line (indicating sea-fioor temperature of 3.C) and the stability-limit line is roughly proportional to the thicknessof the gas-hydratezone. (After Tucholke eI al.. 1977.\

SEISMICVELOCITY

t32

l-_,____-.1 sr

Fig. 5.35

s2

s3

Shooting a well for velocitity.

"span" distancesfrom R, to R, and from R, to Ro "boreholeoften being6l cm (2 ft). The spacingof the (4 the "longm ft), on is 1.22 compensated"sonde spaced"sonde2.44m (8 ft). With the longer spacing, there is greaterlikelihood that the velocity measured will be that of unaltered formation. The velocity is found by measuring the traveltime differencefor a pulse traveling from S, to R, and Ro, similarly for a pulsegoing from S" to R. and R,, then taking the average of the differences.The sondeis run in boreholes filled with drilling mud, which has a seismicvelocity of roughly 1500m/s; however,the first energyarrivals are the P-wavesthat have traveled in the rock surrounding the borehole.Errors arising from variations in boreholesizeor mud-cakethicknessnear the transmitters are effectivelyeliminated by measuring the differencein arrival time betweentwo receivers;errors resulting from such variations near the receiversare reducedby averagingthe resultsfrom the two pairs of receivers.The sonic log (fig. 5.37b) showsas a function of depth the transit time divided by the span(expressedin ps/ft), the result being the reciprocalofthe P-wavevelocity in the formation. The traveltimeinterval betweensonic-logreceivers is measuredby a device that automatically registers the arrival of the signal at each of the two receivers and measuresthe time interval betweenthe two. Becausethe signalat the receiveris not a sharppulsebut insteadis a wavetrain,the detectoris actuatedby the first peak (or trough) that exceedsa certain threshold value. At times, the detector is not actuatedby the samepeak (or trough) at the two receivers,and hence

the incrementof traveltimewill be in error.This effect, called cycleskry, usually can be detectedand allowed for becausethe error is exactly equal to the known interval between successivecycles in the pulse (Kokesh and Blizard, 1959). The accuracy of sonic-log values is often rather between poor, as evidencedby frequentdisagreements regularand long-spacedlogs.This fact shouldbe realwho are inclinedto believesonic ized by geophysicists, and distrust seismicdata when faced with disagreements. Sonic logs often suffer from inadequate penetration, hole caving,alterationswith time after drilling the borehole, and other factors. The boreholecompensatedsonic log generallydoes a good job of compensatingfor sondetilt and minor hole irregularities.Errors becauseofcycle skip can usuallybe recognized and compensated. Measurements may be affectedby invasionof the boreholefluid into formations so that the velocity measuredis not that of the unaltered formation. Long-spacedtools help ensure penetration beyond the invaded zone; they use a different schemefor compensationfor sondetilt and hole-sizeirregularities.Dispersion,becauseof differencesbetweenthe P-wavevelocity at the frequencies of soniclogs (ordinarily 20 kHz) and seismicfrequencies,usuallyis quite small, probably lessthan l7o. The instantaneousvelocity fluctuates rapidly in many formations,as seenin fig. 5.37b.While the velocity distribution, if consideredin detail, is an extremely irregular function, the wavelengthsused in seismicexplorationare so long (generallygreaterthan 30 m) that the rapid fluctuationsare not significantin determiningthe path of waves. The soniclog is automaticallyintegratedto give total traveltime,which is then shown on the log as a function of depth by meansof ticks at intervals of I ms. There is a tendencyfor small systematicerrors to

{2

J

Velocity (D and lnterval Velocitv (4) (km/s) F i g . 5 . 3 6 P l o t o f a w e l l - v e l o c i t ys u r v e y .

MEASUREMENT OF VELOCITY

t% T-TF

--rt E Bl

Itrl ? I jt---r_

I

4

'6 I I

* __

r"-,lr(e)

o) Fig..5.37 Sonic logging. (a) Borehole-compensated logging sonde (courtesy of Schlumberger). (b) BHC Acoustilog survey in a dolomite-shale sequence. The left curve is g"--i_.oy .._ sponse (courtesy of Atlas Wireline Services Division of West_ ern Atlas).

133 accumulate in the integrated result. Check shots (an abbreviated"conventional" well survey ($5.4.2))can be made at the baseand top of the sonic log (and occasionallyin between)so that the effectof the cumulative error can be reducedby distributingthe difference in a linear manner.The sondemay includea well seismometer of the type usedin shooting a well to facilitate taking check shots. Sonic logs are used for porosity determinationbecauseporosity appearsto be the dominant factor in seismicvelocity.Although soniclogsare of greatvalue to the geophysicist,they are usually not run with the geophysicalusesin mind and henceoften do not produce all the information that the geophysicistwants. For example,checkshotsare not necessaryfor porosity determination and, therefore,are often omitted; the log usually does not cover the entire hole depth and sonic-logdata are rarely availablefor the shallow part ofthe borehole.Thus, the sonic-logdata are usually incomplete so that using such logs for velocity control involvesassumptionswherethe data are missing. Of course,head-wavetravelis not achievedwhere the formation has lower velocity than the borehole mud, which is apt to be the casein the upper part of the borehole.Depending on the use to which soniclog data are to be put, editing and correctingmay be needed. Editing involves detailed comparisons of different logs to locateimprobablevaluesand replacing them with valuesbelievedto be more reasonable. Editing is often essentialif good synthetic seismograms (96.2.1)are to be made from log data. Missing shallow data are especiallya problem in preparing syntheticseismogramsfrom sonic-logdata. An array-sonictool (fig. 5.38)employingdownhole digitizing is now coming into use. Compared with a conventionaltool employingtwo transmittersand two receivers,it containseight additional receiversspaced 6 in. (15 cm) apart that record full waveforms(fig. 5.39a). The high-frequencyP-wave traveling in the formation arrives first, usually with very low amplitude,to be followedby a higher-frequency, strongerSwaveand then a lower-frequency, very high-amplitude Stoneleywave. The Stoneleywave travels along the boreholewall and sometimesprovidesindicationsof fractures and permeability changes. Arrivals are picked by computer to yield plots of the p-, S-, and Stoneley-wavetraveltimes(fig. 5.39b). Sometimesa "Poisson's ratio log" (fig. 5.a0)is also computed.This information is proving useful in studies to evaluate and optimize production from reservoirs. Severalother variations of acoustic borehole logs are usedsometimesthough they havelittle application in seismicwork. The amplitude of S-wavesis sometimesdisplayedon a "fracture-finderlog," high amplitude indicating the absenceof fractures.The p-wave amplitude when the sondeis inside the casins is displayed in the "cement-bond log"; the amplitude is high when the casingis hangingfreely becausethe energy is trapped in the steel casing, and low when bonded to the formation by cement. The full wave-

134

SEISMICVELOCITY V, : Vlcos I for constant velocity and reflector dip {, V" : V,^, for horizontal velocity layering and reflectors. V"- V,^,lcos( for dipping but parallelvelocity layeringand reflectors.

Mud At Measurem€nl Soction

Logging JRec€iver I Section 3.9 fr

To winch

'r

fl t","r",o Cartridse

Lf I l-Gamma RaY (optional) Sonic -f-j DigiiizingI I Canridge| |

fl

soni.

Eighi Wid€band Coramic Receiv€rs

Two Cersmic Roceivers Sonic Logging Sonde

I l-Logging | | Receivet

l-..1secrion | | sonic

Two Ceramic Transminers

ll-Logging I I Sonde

In the generalcase,thereis no simplerelationshipbetween I{ and V,^,. The stacking velocity I{ is used in correctingCMP data beforestacking(henceits name) even where its relation to the velocity distribution is very complicatedor not known. We sometimesusethe equivalentaveragevelocityV $4.2.21for depth determination(seeeq. (5.20)):

L4 o',

H

V:i=t,

d L-c"rio"t V

(oPtionat)

(s.24)

;

Ilr,

(a)

Fig. 5.38 Schematic of an array sonic-logging sonde. (From Charnock.1990.)

form is sometimesdisplayed,and this log is also used to measurethe quality of bonding the casingto the formation. 5.4.4 Velocity from traveIt ime-ffi et measurements (a) X2-72 method. The arrival time of reflectedenergy dependsnot only on the reflectiondepth and velocity abovethe reflector,but also on offset distance. Several methods (including the velocity-analysis methodsthat will be describedin $9.7)utilize this dependenceon offsetas a meansof measuringthe velocity. Two classicalmethods,X2-72 and Z A?Tare central to surface velocity-measurementmethods,even though both methodshavefallen into disuse. The X2 T2 methodis basedon eq. (4.4).We write tr:

xrlV?+ ti.

623)

When we plot t2 as a function of x' (fig.5.41), if the velocity is constant,we get a straightline whoseslope is llv! and whoseintercept is rfi,from which we can determine the corresponding depth. If velocity changesare not extreme,the x2-t2 curve can be approximatedby a straightline. This is equivalentto fitting a portion ofthe curve offig.4.lOb by a straight line. The slope of this line givesthe stackingvelocity, (, the velocityassumedin CMP stacking($8.3.3). When we havehorizontal velocity layeringand horizontal reflectors, ( is nearly the sameas the rms velocity Z,-. in eq. (4.26). Under different circumstances, V, : V for constant velocity and horizontal reflectot

This equation also assumeshorizontal velocity layering (and vertical raypath). When the regular seismicprofile does not have a sufficientlylargerangeof x-valuesto enableus to find the velocity with the accuracyrequired,speciallongoffset profiles can be acquired(Dix, 1955),but, with the longer offsetsused in CMP work, they are rarely required.An X2 T2 surveycan givevelocitiesaccurate within a few percent. Once velocitieshave been determinedto two successiveparallel reflectorsusing eq. (4.26),the interval velocity can be found from the Dix equation.Writing V, for the rms velocity to the rth reflector and V, for the rms velocity to the reflector above it, eq. (4.26) gives

a t+i v 2 L t " : v , L i o , , .

fri t,,:2v: l

1

:

'

1

/Vi A,t,= vi, / l

l

;

l

tr,.

l

Subtractingand dividing both sidesby Al, then gives the Dix velocity(seeeq. (5.21)):

v:: ( vii tt,- vll

LtJ tLr ,. ( s .2s )

Note that this equatiJnimpliestt'ut tf,. travelpathsto the (n- 1)th and rth reflectorsare essentiallyidentical except for the additional travel betweenthe two reflectors. When the two reflectorsare not parallelor when the offset is large, this condition is not satisfiedand the Dix equationmay give meaninglessresults. (b) T-AT method. The Z-AT methodis basedupon eq. (4.'7),which can be written in the form (2t,, Lt,)tt2

(s.26)

With symmetricalspreadsLl,can be calculatedfrom the arrival timesof a reflectioneventat the source(/o) and at the outside geophonegroups, l, and ro. Dip

MEASUREMENT OF VELOCITY

135

(a)

(b)

F'ig. 5.39 Array sonic logging.(From Charnock, 1990.)(a) Waveformrecordedby the upper eight receiversin fig. 5.38

showing the P-, S-, and Stoneley-wavearrivals. (b) Log of the velocities of the three waves.

moveout is eliminatedby averagingthe moveoutson t h e o p p o s i t es i d e so f t h e s o u r c ep o i n t :

or (4.56).However,the completex-l curveis often not availableso that this method is rarely usedexceptto determinethe velocity in the Earth's mantle and core whereother methodscannot be used.

N,:

:lQ, /u)+ Qr t,,)l - tu. : ){t, + tr)

(s.2't)

The valuesof Al, given by this equationare subject to largeerrors,mainly becauseof uncertaintiesin the near-surfacecorrections.To get useful results,large numbers of measurementsmust be averagedin the hope that weatheringvariationsand other uncertainties will be sufficiently reduced (Swan and Becker, 1952).

5.4.5. Measurementsbasedon reflectionamplitude In concept at least, with amplitude preservedin recording and processing,acousticimpedancechanges are proportional to seismicamplitudes.The reflection coefficientequation at normal incidenceis given by e q .( 3 . 1 4 ) : ^

(c) Best-Jit approaches. Most velocity determination is done in data processing,which is discussedin $9.7.Thesemethodsare basedon either( 1)findingthe hyperbolathat bestfits coherenteventsassumedto be primary reflectionswithin somegiven spaceand time window, or (2) finding which stacking velocity producesthe "best" stackedsection.Suchmeasurements are generallysufficientlyaccuratefor stackingbut not alwaysfor the lithologic conclusionssometimesdrawn from them. (d) Measurementson diving waves. Where the velocity increasesmonotonically with depth and velocity layeringis parallelto the surface,x-t datacan be used to determinethe velocity as a function of depth for diving waves($43.5)by solutionof eq. (a.53),(a.54),

-

2,.,- Z -lA(ln4-!k,q, Z,t+2,

(5.28)

whereZ,and Z,*, are acousticimpedanceson opposite sidesof the interface giving rise to a reflection, h,(ln 4 is the changein the logarithm of the acoustic impedance,and ,4 is the amplitudeof the reflection;k is called the scaler,a constant dependingon system gain. Applying this relationshipto real data, we assume that the amplitude is not affectedby noise or the shapeof the seismicwavelet.Although we cannot achievetheseconditions,we can sometimesget close enoughto get usefulacousticimpedancedata. Using eq. (5.28)to give reflectioncoemcientsfrom impedancedata is an example of a direct problem, whereasobtaining impedanceinformation from the is an inverseproblemand an amplitudemeasurements example of inversion.Trace inversion is preceded by

136

SEISMICVELOCITY

I

I

Poissonratio

I

Fig. 5.41 Seismicgather plotted on -r2-t2scale.(From Waters, 1987:214; reprinted by permission of John Wiley & Sons, Inc.)

tay: z( t) l: l,oy I.o''" Fig. 5.40 Acoustic impedance and Poisson'sratio logs. Shown to the right is an interpretation ofthe lithology and fluid content for a Middle East carbonate reservoir.(From Charnock, 1990.)

o'' J,ooat Z(t\

processingto removeas much of the noiseas possible and the effectsof the embeddedwavelet.If we also have knowledgeof density,we can solve for velocity rather than acousticimpedance. Equation (5.28)can be solvedfor the acousticimpedance below an interface in terms of that above the interface:

/.,:

| + :kAt 2. . | - ')kA,

(s.29)

If the density and velocity,p, and Vp respectively, are known for the shallowestlayeq then Z, : prV, is known and Zrcan be found from amplitude,4, of the first reflection,23 can be found from l, and so on. The resultingrecursivelyderivedvalue ofthe acoustic impedanceas a function of arrival time or depth is called a synthetic acoustic-impedancelog or seismic log. A more common way for solving for Z as a function of time Z(t) or of depth Z(z) is to integrateeq. (5.28):

tnZ(t)- tnz(0) (5.30)

= z(o)e*p ltf,atrl04.

Neither the scaler k nor the constant of integration Z(0) can be obtainedfrom the seismicdata alone.Becausewe havedata only within a limited passband,we "low-frequency usually let Z(0) representthe missing value; it is the starting in addition to components" usually made time-dependentrather than constant. The contributions of low frequenciesare evident in fig. 5.42. Velocity determinedfrom normal moveout ($9.7)can sometimesgive part of the missinginformation (perhapsvaluesas low as 7 Hz if we can calculate for intervals as small as 150 ms); howevet moveoutderivedinterval velocity is often unreliable,especially when calculatedfor small time intervals.Betterresults are achievedby using logs in a nearby well wherethe geology is expectedto be similar. Missing high frequenciessimply limit the amount of detail derivable. A seismiclog is often expressedas a velocity time series,that is, as a syntheticsonic log. This requires either knowledgeof the density distribution or an assumption about the relation betweenthe velocity and the density.Commonly,we assumeeitherthat the density is constant, in which case it drops out, or that

MEASUREMENT OF VELOCITY

t37 10,000

<250

<EO <60

<40H2

'0+

-250

(d

10.000

< 6 l'lz (b)

Fig. 5.42 Sonic log and amplitude within various frequency bands. (From Lindseth, 1979.)(a) Sonic log with various high-

cut filters. (b) Sonic log (left) decomposed into portions above and below 6 Hz.

Gardner'srule, eq. (5.15),holds. In the latter case, we have

5.15),we should not expectquantitativeagreement. In particular,evaporites(anhydrite,gypsum,salt) and carbonaceoussediments(coal, lignite) depart considerably from the velocity density relation for other sedimentarylithologies. Synthetic acoustic-impedance(or sonic) logs emphasizethe acousticimpedance(or velocity) of beds rather than the contrastsat the interfaces,making it easierto relateto rock properties.Using seismiclogs, we often see featuresthat we would otherwisemiss

Z:p(V)V-avst4,

t' V(t): V(0)expl(4kl5t a,1. (5.3| ) ),AOt which is the sameform as eq. (5.30)exceptthat a new constant,4k/5, replacesk. Becausenot all lithologies follow the same velocity-density relation (see fig.

SEISMICVELOCITY

138 even though the syntheticlogs merely representa rearrangementof the seismicinformation rather than new information. The principal limitation with syntheticlogs is the assumption of a linear relation between reflectivity and amplitude, which in effect assumesa noise-free seismicrecord (showing the effect of primary reflections only) plus recording and processingthat have preservedamplitudesfaithfully.The successfulmanufactureof seismiclogs requiresexcellentreflectionrecords. The additional limitations involving the determining of V(0) and k are minimized where the application is to interpolatevelocity information between wells or to extrapolatevelocity information in the immediatevicinity of well control. Syntheticsonic logs constitute a powerful tool for locating stratigraphic changes,porosity changesand hydrocarbon accumulation under these circumstances(Lindseth, 1919). synthetic-log displayis shownin A black-and-white fig. 5.43.There is one synthetic-logtracefor eachseismic trace input. The horizontal scalefor eachtrace is linear in transit time (the reciprocal of velocity); the transit-time scaleis shown for one trace (dark curve at the left). This trace showsthe usual increaseof velocity with depth.An actualsoniclog hasbeensuperimposed at the right and another synthetic-logtrace (dark curves) for comparison. Synthetic sonic-log tracesdo not show the detail of soniclogs becausethe seismictrace is deficientin high frequencies,synthetic logs often cannot follow sharp velocity changesfaithfully, and they often drift becausethe low-frequency "constant" is incomplete.They also often show periodicity becausethe embeddedwavelethas not been completelydeconvolvec. It is difficult to read transit-time valuesfrom synthetic sonic-logdisplayssuchas shownin fig. 5.43and velocityor transit timesare usuallycolor-encodedand superimposed(plate I ). 5.4.6 Other sourceso/'velocityinformution In addition to determining velocity from measurements in boreholes,traveltime offsets,or amplitudes as discussedin the preceding sections,velocitydependent processingpotentially provides another sourceof velocity information. One may vary the velocity in processingin order to maximize the coherof the resultingpicture.One ence(56.1)or consistency example would be to vary the velocity distribution (migration velocity) so that the most coherent migratedstructure($9.12)results.Mills et al. (1993)built a velocity model where the velocity changedat migratedreflectionslayer by layer.After determiningthe velocity in shallowerlayers,they varied the velocity in the next layerin a trial-and-errormannerto determine which would match most closely the moveouts observedon selectedgathers. Velocity information is potentially given by other processessuch as generalinversionmethods ($13.9).

However,the techniquesto accomplishthis on a practical basishavenot yet beenworked out. Somevelocity information can also be obtained from types of measurementsthat do not dependon reflectiontravelpaths such as head-wavevelocity or surface-wave dispersion. 5.5 Uses of velocity data Velocity information is used in many processingand interpretationsituations,and in most of these,the accuracy of the velocity data is not as good as is needed. "kinds" of Table 5.3 lists someof the usesof different velocity; this list is basedon Al-Chalabi (1979). Problems "limit-of5.1 What physical fact determines the porosity" line in fig. 5.3a?What is implied for measurementsthat fall to the right of this line? 5.2 Figures5.12aand 5.12b arebasedon differentexperimentaldata. Show the compatibility or incompatibility of thesefigures. 5.3 (a) Assumethat sandstoneis composedonly of grains of quartz, limestoneonly of grains of calcite, and shaleof equalquantitiesof kaoliniteand muscovite. For sandstone,limestone,and shale saturated with salt water (p : 1.03g/cmr),what porositiesare impliedby the densitiesshownin fig. 5.l3? (b) What velocitieswould be expectedfor thesevalues accordingto Gardner'srule (eq. (5.15))?Where do thesevaluesplot on fig. 5.5? (c) From the graph oi fig. 5.3b,what densitieswould you expectat 7500ft and how do thesecomparewith figs.5.9cand 5.9d? 5.4 Assumethat the velocityin calciteis 6.86 km/s and in quartz5.85km/s.What velocitiesshouldbe expectedfor 10, 20, and 30% porosityin (a) limestone composed only of calcite; (b) sandstonecomposed only of quartz?Wheredo thesevaluesplot on figs.5.5 a n d5 . 1 6 ? 5.5 (a) Why do the velocity-depthcurvesfor the various areasshown in flgs. 5.17 and 5.19 depart from each other?Incorporateyour knowledgeof the geology ol the variousareasin your answer. (b) Plot the shaleand limestonevaluesfrom fig. 5.29 for depthsof 1000and 2000m on fig. 5.17.How do they compare? 5.6 (a) Assumea subsidingarea without uplift activity. A shale is normally pressureduntil it reachesa depth of burial of 1400m, at which point it becomes cut off from fluid communication,that is, interstitial fluid can no longer escape.If it is found at a depth of 2000 m, what velocity and what fluid pressurewould you expect?If at a dePth of 3000m? (b) Assumea shaleburied to 3000m and then uplifted to 2000 m, being normally pressuredall the time. What velocity and fluid pressurewould you expect? (c) Assume the shale in part (a) is buried to 3000 m and then uplifted to 2000m, without fluid communi-

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the heavy trace at the left is indicated at the top. At the right, one heavy trace shows a synthetic log along with an actual sonic log for comparison. (Courtesy of Technica.)

SEISMICVELOCITY

140 Table5.3 Usesof velocitydata Velocity

Main uses

Precision requirements

Stacking velocity

Stacking of seismicsections Preliminary migration processrng rms velocity estimation

Modest to low Modest to low Dependent on situation

rms velocity

Estimation of migration velocitY Interval-velocity estimation Average-velocityestimation

Generally modest Dependenton situation Dependenton situation

Interval velocity

Gross lithologic and stratigraphic studies Generalinterpretationpurposes Age estimation Detection of abnormal pressure Ray tracing Migration processing Average-velocityestimation

High to modest Modest to low High to modest High to modest Dependenton situation Generallymodest Dependenton situation

Averagevelocity

Depth conversion Generalinterpretationpurposes

Generally modest Modest to low

Precisionrequirements: high = 0.1 to 1.0% modest:lto5'k low > 5%

After Al-Chalabi( 1979). What velocity and fluid prescation beingestablished. you What if uplifted to 1000m? expect? surewould 5.7 By comparingfigs. 5.19and 5.36,what can you deduceabout the nature of the rocks in the well for fig. 5.36? 5.8 What shalevelocitiesare consistentwith the oil-sand data shown in fig. 5.27b?(Determine velocities for two valuesof water saturationfor eachof the three depths, neglecting differences between sand and shaledensities.) 5.9 (a) Figure 5.17 shows velocity versusdepth for normally pressured shales. How do the velocities "top abnormal shownin fig. 5.31aboveand below the pressure"comparewith the curve?What depth would correspondto normal pressurefor the overpressured shale?What porosity would you expect for the overpressuredshale? (b) Plot the velocitiesfor l00%owater saturationfrom fig.5.27on fig. 5.17.How do they compare? 5.10 Assumethat raypathshavean angleof approach of l0', 20', and 30'in the subweatheringwith a velocity of 2400m/s. (a) For a weatheredlayer l0 m thick with a velocity of 500m/s, how do traveltimesthrough the weathering cpmparewith that for a verticallytravelingray?What is the horizontal component of the raypath in the weathering? (b) For permafrost100-mthick with a velocityof 3600 m/s, answerthe questionsin part (a). 5.11 In the early days of refraction exploration for salt domes, sketcheswere drawn indicating that the angle of approachto the surfaceshould have a large with threehorizontal component,but measurements component seismographsshowed very little hori-

zontal component and controversy arose therefore over whether the travelpathscould be as drawn. Explain the apparentdiscrepancybasedon your concept of the actual earth. 5.12 (a) Assume six horizontal layers each 300 m thick and having a constantvelocity within eachlayer (fig. 5.44a),the successive layershaving velocitiesof 1 . 5 ,L 8 , 2 . 1 , 2 . 4 , 2 . ,7a n d3 . 0k m / s .R a y - t r a c teh r o u g h the model to determine offset distancesand arrival times for rays that make an angle of incidencewith the baseof the 3.0 km/s layer (at A) of 0, 10, 20, and 30o.Calculatestackingvelocity from eachpair ofvalues (six calculations)and compare with the average and rms velocities. (b) Repeatassumingthe layersdip 20', as shown in fig. 5.44bfor reflectingpoint I (c) By trial and error, shift the reflectingpoint updip to achievecommon midpoints. (d) Repeatby modifying the model to that shown in fig. 5.44cfor reflectingpoint C. (e) Shift the reflectingpoint C as required to achieve common midpoints. 5.13 Velocity analysis usually results in a plot of stacking velocity against travel time. Bauer (private "quick-look" method of communication) devised a determiningthe interval velocity,assuminghorizontal layering and that the stacking velocity is averagevelocity. The method is shown in fig. 5.45. A box is formed by the two picks between which the interval velocity is to be picked; the diagonal that does not contain the two pick points, when extendedto the velocity axis,givesthe interval velocity. (a) Provethat the method is valid and discussits limitations.

REFERENCES

141 Z , = 1 . 5k m / s V ^ = t A Vt = 2.1

Vt=24 Vs= 2.7 V" = 3.0

k) Fig.5.44 Models of 300-m-thick layers(measured perpendicu Iar to the bedding), each ofconstant velocity.

Fig. 5.45 "Quick-look" interval-velocitydetermination.

(b) This method is useful in seeingthe influence of measurement errors; discuss the sensitivity of interval-velocitycalculationsto (i) error in picking ve_ locity values from this graph; (ii) .r.o. ln picking times;(iii) picking eventsvery closetogether,and (ivj picking eachevent late. 5.14 Figure 5.46showsdata from a well-velocitvsur_ vey tabulatedon a standardcalculationform. (a) Plot time, averagevelocity, and interval velocity versusdepth graphsusing a sea-leveldatum. (b) How much error in average-velocity and interval_ velocity values would result from (i) tlme_ measurement errors of I ms and (ii) depth_ measurementerrors of I m? (c) Determine Vo and a for a velocity-functionfit to thesedata assumingthe functionalform V : Vn* az, where V is the interval velocity and z is depth. 5.15 Analysis of an X2 T2 surveygivesthi resultsin table 5.4. Calculatethe interval velocities. 5.16 Determinethe velocity by the X1-7, method us-

b)

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ing the data givenin table5.5, l, beingfor a horizontal reflector and t" is for a reflector dipping l0o toward the source. 5.17 (a) Given that the trace spacingin fig. 9.46b is 50 m, determinethe stackingvelocity,depth, and dip at approximately 0.5, L0, l.5, 2.0,2.3. and2.4 s. (b) What problems or ambiguities do you have in picking theseevents? (c) How much uncertainty is there in your ability to pick times and how much uncertaintydoesthis intro_ duce into the velocity,depth, and dip calculations? 5.18 (a) In fig. 9.24d,pick stackingvelocity versus trme pairs and calculateinterval velocities;the analv_ s i si s f o r S P 1 0 0 . (b) What can you tell about the lithology from this? (c) If the section that is present in the syncline but missingover the anticlineconsistsof young, poorly consolidatedrocks, what valueswould you "^p..t u velocity function at Sp 45 to show? (d) Note the downdip thinning of the section from about 0.75to 1.25s at the left end of the section;sus_ gestthe explanation. 5.19 (a) Determine velocity versus depth from fig. I1.8. The direct wave travelsthrough the water and can be usedto give source-receiver distances.Assume horizontal reflectors. (b) Determine the apparent velocitiesof refractors and correlaterefractionwith reflectionevents. 5.20 Given orthogonal dip and strike seismiclines; will velocityanalysesat the line intersectionsyield the samevalues? 5.21 (a) Becausea velocityanalysisfor a causalwave_ let ($15.5.6a) is not made on the waveletonset.how will this affect the stacking-velocityvalue? Table 5.4 X2 T2 survevresults

I 2 3

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143

Table 5.5 X2-72 data .x (km)

l, (s)

l, (s)

r (km)

1, (s)

/B (s)

-t (km)

l, (s)

/u (s)

0.0 0 .I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.855 0.856 0.858 0.864 0.868 0.814 0.882 0.892 0.904 0.906 0.930 0.945 0.950 0.9'79

0.906 0.902 0.898 0.898 0.899 0.902 0.903 0.909 0.916 0.922 0.932 0.943 0.950 0.965

1.4 1.5 1.6 t.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.' 7

1.005 1.017 1.037 r.068 1.081 1.105 Lll8 l.l5l 1.166 1.203 1.237 1.255 1.283 1.304

0.977 0.991 1.004 1.019 1.037 1.058 1.066 r.083 1.102 1.121 1.t27 158 t]1 195

2.8 2.9 3.0

1.330 1.360 1.404 1.432 1.457 |.487 1.513 1.548 1.580 1.610 t.649 t.674 t.708

1.202 |.234 1.253 t.2'72 |.296 r.304 |.334 1. 3 5 6 |.37' 7 t.40'7 1.415 1.438 1.459

l.l

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(b) What will be the effectof NMO srretch($9.7.3)? (c) lf a datum (98.8.2)is usedthat is appreciablyremoved from the surface,what effectwill this have'l REFERENCES A l - C h a l a b i . M . 1 9 7 9 .V e l o c i t y d e t e r m i n a t i o nf i o m s e i s m r cr e flection data: ln DcwloptnL'nt.sin Gcopltsicul E.rplorution M t ' t l t o d . s ,V o l . l . A . A . F i t c h . e d . , p n . I 6 8 . A m s t e r d a m : E l s e ri c r . A t k i n s , J . 8 . . a n d E . F ' .M c B r i d e . 1 9 9 2 .P o r o s i t t 'a n d p a c k i n g o l ' Holocene river, dune. and beach sands: Blll AAPG. 76: -il9 55. A u d , B . W . 1 9 7 6 .H i s t o r y o f a b n o r m a l p r e s s u r ed c t c r m i n a t i o r r {'rom seismic dattr. OTC Pr(print.\. paper 2611. Dallas, TX: C)ffshoreTechnology Conftrencc. B i o t . M . A . 1 9 5 6 .T h e o r y o f ' p r o p a g a t i o n o f ' e l a s t i c w a v e s r n llurd-saturated porous solid. J. Acou.tt. Sor'.Appl. plt.:.. 26: t82 5 Birch. F-. 1942, Hundhook o/ Ph.ysitul (onstunt,s,CSA Special P a p e r - j 6 . G e o l o g i c a lS o c i c t yo f A m e r i c a . B i r c h . F , .1 9 6 1 .T h e v e l o c i t y o l ' c o m p r e s s i o n aw l a v e si n r o c k s t o f () kilobars. J. ()aophys.Rc,r.,66: 2199 2223. Charnock. G.. cd. 1990. IIidtlle Eust Wt'l! Et'uluution Rtt,ien. Ridgefield. CT: SchlumbergerTechnical Services.

J . l

3.2 3.3 3.4 3.5 3.6 3.1 3.8 3.9 4.0

F a u s t , L . Y 1 9 5 1 .S e i s m i cv e l o c i l y a s a f u n c t i o n o l d e p t h a n d geologic time. Gaphrsit.r, l6: 271 tl|. F a u s t .L . Y 1 9 5 3 .A v e l o c i t yf u n c t i o n i n c l u d i n gl i t h o l o g i cv a r i a tion. Gtttph.r'sicr;l8:271 88. Gardner. G. H. Fl. L. W. Gardner, and A. R. Gregory. 1974. F o r m a t i o n v e l o c i t y a n d d e n s i t y T h e d i a g n o s t i cb a s i c sf b r stratigraphic traps. Gurph.r..sir'.r 39: 770 80. G a r d n e r ,G . H . F l , a n d M . H . H a r r i s . 1 9 6 8 .V e l o c i t ya n d a t r e n u ation crf elastic waves in solids. ?'rzllsaclions ol the gtlt Attnuul Log S.t'nyltiutrt, Paper M. G a s s m a n .F l | 9 5 1. E l a s t i cw a v c s t h r o u g h a p a c k i n g o f ' s p h e r e s . Geoplt.t':;ic.s, l6: 673 85. G e e r t s m a .J . ,a n d D . C . S m i t . 1 9 6 1 .S o m ea s p c c t so f e l a s t r cw a v e propagation in fluid saturated ptrrtrus solitJs.Gtryln,.rrcs,26: 1 2 6 98 1 . Grant, Fl S., and G. F. West. 1965.lnterpretution T'ltttr) in AppIitd Gaoplt.t',siti. Ncw York: McGraw-Hill. G r c g o r y .A . R . 1 9 7 6 .F l u i d s a t u r a t i o nc f f e c t so n d y n a m i c e l a s t r c p r o p e r t i e so f s e d i m e n t a r yr o c k s . G c r p l r | s l r ' .41l : U 9 5 9 2 1 . G r e g o r y , A . R . 1 9 7 7 .A s p c c t s o f r o c k p h y s i c sf r o m l a b o r a t o r y a n d l o g d a t a t h a t a r e i m p o r t a n t t o s e i s m i ci n t e r p r c t a t i o n .I n Sei.snir Strutigrupht' Applitutiort,sto Iltlroturhon E.rphrur i o n , ( ' . F . . P a y t o n ,c d . , p p . l 5 4 6 . A A P G M e m o i r 2 6 . T u l s a : American Association of PetroleurnGeology.

C h r i s t c n s e n .N . L 1 9 8 9 . S e i s m i cv e l o c i t i e s .l n P r u t t i c u l H u n t l hutk ol Plt.t'.siLulPropertic,sol Rot.k,;unt! Mint'ruls, R. S. Carmic h a e l .e d . . p p . 4 2 9 5 4 6 . B o c a R a r o n . F L : C R C p r e s s .

Gretcner. P E. 1979.PorL,Pre.ssurc: Funduntntuls, Generul Runifrtutiots, und Intpliuttion,s lir Structurul Geology, AAPG Educ a t i o n C o u r s e N o t e S c r i e s4 . T u l s a : A m e r i c a n A s s o c i a t i o no f Petroleum Ceology.

D i x . C . H . 1 9 5 5 .S e i s m i cv e l o c i t i e sf r o m s u r l a c em e a s u r e m c n l s . Gaoph.t'sits, 20: 68 86.

H a m i l t o n , E . L . 1 9 7 1 .E l a s t i c p r o p e r t i e so f m a r i n e s e d i m e n t s . J. Geoplt. Re.s., 76:,-579 604.

Domenico. S. N. 1974. Ellcct of water saturation on seisrnicreflcctivity of sand rescrvoirs encased in shale. Geopht.sic,s, 39l. 759 69.

Han, D. 1987. Efi'ectsof porosity and clay content on acoustic p r o p c r t i e so f s a n d s t o n e sa n d u n c o n s o l i d a t e ds e d i m e n t sP . h.D. thesis.Stanford University.

Domenico.S. N. 1976 Effect of brine gas mixture on velocity i n a n u n c o n s o l i d a t e ds a n d r e s e r v o i r :G e o p h r s i t s , 4 l :8 8 2 9 z l .

H a n , D . , A . N u r . a n d D . M o r g a n . 1 9 8 6 .E f f e c t so f ' p o r o s i t y a n d clay content or.l wave velocities in sandstones.Geopfu'.sics, 5l: 2097 2t07.

D o m e n i c o .S . N . 1 9 7 7 .E l a s t i cp r o p e r t i e so f ' u n c o n s o l i d a t e dp o r o u s s a n d r e s e r v o i r sG . a t p h . t , s i c s , 4 21 :3 3 9 6 8 . D u s k a . L . 1 9 6 3 .A r a p i d c u r v e d - p a t h m e t h o d l b r w e a t h c r i n g and drift correction. Geopltr.sit.s, 28:925 47. D u t t a , N . C . . e d . 1 9 8 7 .G e o p r e s . ; u r e S:E G G e o p h y s i c sR c p n n l S e r i e s7 . T u l s a :S o c i c t y o 1 ' E x p l o r a t i o nG e o p h y s i c i s t s .

H i c k s , W G . . a n d J . E . B e r r y . 1 9 5 6 .A p p l i c a t i o n o l ' c o n r i n u o u s velocity logs in determination of fluid saturation of reservorr rocks. Geopll,,rlc.s, 2lt 739 54. Hrlterman. F 1990. Is AVO the seisrnicsignature of lithology? A case history of Ship Shoal South Addition . The Leudin,q E d g e , 9 ( 6 ) :1 5 2 2 .

SEISMIC VELOCITY

t44

deducerockgoperSheriff,R. E. 1977.Using seismicdata to "ii"t.inTet'oleum Geologv'Vol l ' G D' Hobit iitipments son,ed., pp. ial tl+- London:AppliedScience' Exploration R. E. 1978.A First Coursein Geophysical Sheriff, " Boston: International Human Resources i"i"i"iirir*/ion. DevelopmentCorP. to Geophysirul Melftod'sEnglewoodCliffs' NJ: Kearev, P., and M Brooks lgS4' An Introduction Sheriff,R. E' 1989.Geophysical E-xolotar'ion.Oxford: Blackwell Scientific' PrenticeHall' Ro-driguez'1991' Seismologl" W., L. K. Johnston, R' Reeses,and G Kevser. ''ii;;;;il'.;i.Jtti"" (sepSheriff. -vnt. R. E., and L' P Geldart lgS3' Exptoration fiom surface seismic' wotld oil Z. N.* york: CambridgeUniversityPress' tembir): 115*24 Shaub' K' J' Shiolev.T. H., M. K. Houston, R T Buffier'^F^J' "i,i['ri,iin"",'i. Geometrical factors tn Kokesh, F P., and R. B' Blizard lg5g w. r-oaa.and J L' worzel' lgTg Seismicevion conttsonic logging. Geophysics,24z64 76' i..... i". tla.spread possiblegashydrate^horizons AAPG' 6322204 13' rises'-Brz/l and converted waves and reflections tfop.t n.niui Multiple 1980. C. C. Lash, "[tta 45: 1373-l4l l' in fluid UV " deep vertical wave test Geophysics' J. W. 1981.Stressrelaxationat low frequencies Spencer, of disperston' modulus stratifor and process A Attenuation logs rocks: sonic saturated R. O 1979.Synthetic Lindseth, -giapttlc'interpretation' Re.r.,88: 1803-12' Gettphvsits' 4'4: 3' 26' J. GeoPhYs. San southeastern of reservotr Stulken,E. J. 1941.Seismicvelocitiesin the "j""q"it J. R. l965 Quantitative determination MacGregor, ";;;;;; 6:327-55' I l' 1502 Geophysics' 492 AAPG' California of Bull v"ff"v logs' f."- conductivitv of velocitiesobSwan,B. G.. and A. Becker'1952'Comparison ",uin.J Mares'S.1984.Introt]uctiontoAppliedGeoph,-sics.Dordrecht: and well velocitysurveysGeoanalysis ielta-time Uy Reidel. 11:57585. physics, considerations L. D., and A. K' Nath lgTT Geologic Meckel, ^r"t StrutSel'rrnjr ln R. H., and M. D' McCormack'l99l' Multicomponent Tatham, ^-S-;;;;;i;gy tit"i,g*pttic modeling and interpretation Expbrution' C E Payin PetroleumExploration'Tulsa: Societyof Explot'iHytltocurbon A.ppli.,ttiuns ,igrupnvl 2 6 T u l s a :A m e r i c a n A s M e m o i r GeoPhYsicists. ration A A P G r a , p p . ' + t z i;":,eJ., Applied sociation of Petroleum GeologY' W. M., L. P Geldart,and R. F'' SherilT1990 Telford, ^ Press' University Barton' A Cambridge C York: and dtiiytir: ,2ci ed. New Mills, "i;;: G. F'.,M. A. Brzostowski, S Ridgway' pre-stack depth ; ".r".i,v model building technique for S., and J N Goodier' lg5l TheoryoJ Elastk'ity' Timoshenko, m i g r a t i o n . F i r s tB t e a k , l l : 4 3 5 4 3 ' McGraw-Hill' York: New ed. 2d velocities:Theory verwavesln porousmeMurphy, W. F 1985.So-nicand ultrasonic Timur,A. 1968.Velocityof compressional 8' 33:584 95' *r.,.exoeriment. Geophys' Res Lett'' l2: 85 dia ai permafrosttemperaturesGeophysi<s' Practicalapptication lg6T and Bratton' H R of Murgruu", A. W., and dependence Timur,A. 1977.Temperature .compressional Relruc.tion ProsySu o' 42: of Blondeau weatherlng solution ln Sel'rnrl< Gertphysits' rocks in shearwavevelocities Society of Expecting,A. W Musgrave. ed , pp' 231 46 Tulsa: GasB. 8., G. M Bryan, and J I Ewing .l9TT Tucholke, ploration GeoPhYSicists' from the Western ""tit-p'ofiler ,n hortzon, hydrate !11a velocity anisotNur, A., and G. Srmmons lg6g Stress-induoed IriorthAtlantic.Butl.AAPG,61:698'707' J Geophys Res" T4z ropy in rocks An experimental study' hydrocarWang,Z. 1988.Wavevelocitiesin hydrocarbonsand 66 7. to EOR monitoring' applications with rocks, bonlsaturated antl Aco,ustit'Velotities Nur, A. M.,and Z' Wang' 1989' Sel'slai( Ph.D.thesis,StanfordUniversrty' Series l0 Tulsa: Sociin ReservoirRrr*s, Geophysics Reprint on waveveWang,2., and A. Nur' 1988'Effectof temperature ety of Exploration Geophysicists' with heavyhydrocarbonsSPE sandstones and sands in locities depth' magnitude Pennebaker,E. S. 1968' Seismicdata indicate Eng.,3(l): 158-64' Reserv. 73 8' ol abnormal pressures liltrld Oil' 166(7): in seismic Wang, 'Hl:rv;i2.. and A. Nur' \992a' Aspectsof rockphysics their applicaG. R. 1963. Acoustic character logs and Pickett, ';;; ReservoirGeophvsits'R. Sheriff' ed ' ln i"*"iil"tt* 15:.650 67' ; formation evaluation J' Petrol' Tech' pp. )S! :10. Tulsa:Societyof ExplorationGeophysicists' Survcy of fluid.preSsure: in andac
experiencein the Hofer. H., and W Varga. 1972' Seismogeologic 605-19' Beaufort Sea. Geophysics,37: of some^factors "Jankowsky, W. 1970. Empirical investigation rocks Geophvs' carbonate in ve-locities wave ^ri""ii"g Lf"ttic P r o r p .1 8 : 1 0 3 l 8

Sharma' P V. 1986' Geophysical dam: Elsevter.

i'iii""".r ?i""t Gri'v oi cuurotnia'Butt' AAPc' 62:

6

Characteristicsof seismicevents

Oreniew --: basictask of interpretingseismicrecordsis that ' .--lecting those eventson the record that represent : -.:tarv reflections(or refractions),translatint the ar_ - .. rimesfor these into depthsand dips. uid rnup: .s rhe reflecting(refracting)horizons.In addition, '-: :nterpretermust be alert to eventsfeaturessuch :r -iidr1_E€S in amplitudeor characterthat mav vield , --ibleinformationaboutothertypesof events,such ., :rultiplereflections and diffractions. The character_ ,: ,. of eventsare the subjectof this chapter. Tie leaturesthat allow one to recognizeand iden_ - :. in event coherence, amplitudestandout,charac_ ': . jrp. anJ normal moveout- are discussed in 66.l. l.i:rences in arrival time becauseof offsetprovide .- :speciallyusefulmethodof distinguishing between -.::,-rions,refractions, multiples,and other types of : :__-\. P:rt of the downgoing wavetrain from a seismic ' -:.'e is reflectedat eachinterfacewherethe acoustic -.::dance changes. Syntheticseismograms showing "-: :eflections expectedfrom a model aid in under_ .::iJlns seismicreflections. The strongestreflections -: -:llr result from unconformitiesor sisnificant -:::tSes in lithology,but reflectionsalso occ*urfrom - :..r Iithologicchanges. Most reflectionsare the in--:::rence compositesfrom a number of closely .:..-ed interfaces.Occasionalreflectionsare caused : . :mpedance changesassociated with phasechanges : iuid contactsrather than beddingcontacts.Re:::::.rns occur not only becauseof energyreturned ':':u the reflecting point, but becauseofenergy re:-=ed liom the entire Fresnel-zonearea. Reflection .::.:iltude.phase,and overallappearance changebe-: -:3 of reflectorcurvature.Where the center of cur:i-ire of a synclinalreflectoris below the observing :.:1. a buriedfocusoccursand a reversebranchap:.:.s. the reversebranch having convex-upwardcur.:-re rather than the concave-upwardcurvature of ,-.::eflectorcausingit. Section6.3 takes up the characteristics of several -.::ieS of nonprimary reflectionevents.Diffractions -.:'.i rlor€ normal moveout than reflectionsand are :-:red eventson stackedunmigratedsections.They :t"r certain relationshipsto the reflectionsfrom re-:,-rr-rrsu'hosetermination often generatesthem. The ,::st ofa diffraction givesthe location ofthe diffract145

ing point for simple velocity situations;this point is usefulin locating beddingterminationssuch as occur at faults and salt-domeflanks. Diffractions also provide a mechanismfor getting seismicenergyinto regions that cannot be reachedon a geometrical-optics basis. Multiples are classifiedas long-pathif they show as separateeventsor short-path if their effect is merely that of changingreflectionwaveshape.peg-legmultiples are important factors in changingthe waveshape and removing high frequencieswith increasingtraveltime. Long-path multiples confuse interpretation unlessthey are recognizedfor what they are.The characteristicsof surfacewavesconcludeQ6.3. Resolution refers to the ability to distinguish between adjacentfeatures.Vertical resolutionconcerns the minimum separationbetweeninterfacesfor them to show as separatereflectors;the resolvablelimit is about a quarter wavelength.The interpretation of beds thinner than a quarter wavelengthhas to be based on amplitude rather than time-interval measurements.Wherea bed is subdividedinto severalvery thin beds,the reflectionamplitude is a measureof the net, rather than gross, thickness of the bed. Horizontal resolutiondependson a numberof factors.The migration process($9.12)attemptsto increasethe horizontal resolution. The attenuationof reflectionswith traveltime and frequencyis discussed in 96.5. The shapeof the seismicwaveletchangeswith traveltime becauseof absorptionand peg-legmultiples, and also becauseof filtering actions in recording and processing.Minimum-phasewaveletsare distinguishedfrom zero-phaseones,the Ricker waveletbeing the most commonly assumedzero-phasewavelet. Distinction is made betweencoherentand incoherent noise,and repeatableand ambientnoise.A discussion ofthe attenuationofnoise concludesthis chapter. 6.1 Distinguishing features of events Recognitionand identification of seismiceventsare basedupon five characteristics: (a) coherence,(b) amplitude standout,(c) characteq(d) dip moveout, and (e) normal moveout. The first of thesecharacteristics, coherence. similarity in appearance from traceto trace (seefie.6.1). is by far the most important in recognizingan event.

146

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

Amplitudc standout

(a)

(b)

(c)

Phasc brcak

Fig. 6.2 lllustrating spatial aliasing. When traces are spaced t o o f a r a p a r t , t h e r e i s a m b i g u i t y i n t h e i r m e a n i n g .l s ( b ) o r ( c ) the correct interpretation of (a)'J

Fig. 6.1.

Characteristic of seismicevents.

When a wave reachesa spread,it tends to produce approximatelythe same effect on each geophone.If the waveis strongenoughto overrideother energyarriving at the sametime, the traceswill look more or lessalike during the interval when this wave is arnving. Coherenceis a necessary conditionfor the recognition of any event. Recognitionof coherencesetsa maximum to the tracespacing.Most eventsextendfor severalcycles,so coherencewill involve ambiguity if the tracesareseparated by more than halfthe shortest present(see.fig. apparentwavelength 6.2).The spatial samplingof seismictracesis subjectto the sampling just as any samplingis. theorem($9.2.2d) Amplitudestundoutrefersto an increasein amplitude such as resultsfrom the arrival of coherentenergy;it is not alwaysvery marked,especiallyif AGC (see$7.6.3)is usedin recording.Coherence and amplitude standouttell us whetheror not a strongseismic eventis present,but they saynothing about the type of event. Character(or signature)refers to a distinctive appearanceof the waveformthat identifiesa particular event.It involvesprimarily the shapeof the envelope, the number of cyclesthat show amplitude standout, the dominant frequency, and irregularities in the phaseresultingfrom the interferencebetweencomponentsof the event.The reflections observedon seismic sectionsusually result from the interferencebetween component reflectionsfrom a closelyspacedseriesof interfaces,and the appearanceof a reflection,that is, its character,dependsupon the spacingand magnitude of the individual acousticimpedancecontrasts. Usually, these are relativelyconstant over moderate distancesand so the reflection exhibits coherence. Charactermay help identily reflectionevents.Reflections are usuallyfairly short eventswith little ringing and their frequency components are usually in the range l5 60 Hz, exceptfor shallowhigh-frequencyreflections and very deep reflections,which may have considerableenergyevenbelow this range. Moveoutrefersto a systematicdifferencefrom trace

f; I

to tracein the arrival time of an event:it is the most distinctive criterion lor identifying the nature of events.We distinguishbetweendip moveout,systematic changesin arrival time becauseof dip, and normal moveout, systematicchanges with sourcegeophonedistance;thesehave been discussedfor reflectionsin $4.1.1and 4. 1.2.It is fairly easyto separate dip and normal moveoutswith split spreads,but not with end-on spreads. With planar reflectors, dip moveoutproducesa nearlylinearalignment,whereas normal moveoutis characterized by alignmentcurvature, but reflectorcurvatureor velocitycomplications can obscure this distinction. Reflection normal moveoutsmust fall within certair.r limits setby the velocity distribution.Reflectionsoften have small dip moveoutsbut occasionallythey have large dip moveouts(as with fault-planereflections). Eventsother than primary reflectionscan also exhibit linear and curved alignments,as will be seenin lollowing sections. One very powerfultechniquefor distinguishing between reflections,difiractions, reflected refiactions, and multiplesis to displaythe data (sometimesafter sorting into CMP gathers)after correctingfor (a) weatheringand elevation(static correclior.s,because the correctionis the samefor all arrival times on a given trace;see$8.8.2)and (b) normal moveout(dynamic correction).Such correctedrecords can be made in data processing.Providedthe correct normal moveout has been removed,reflectionsappear as straightlines,whereasdiffractionsand multiplesstill havesomecurvature(fig. 6.3) (becausetheir normal moveoutsare largerthan thoseof primary reflections) and refractions and other lormerly straight alignments haveinversecurvature. 6.2 Reflections 6.2.I Synthetic seismogram.s It is generallyassumedthat the waveletreflectedfrom (or transmitted through) an acousticimpedancediscontinuity has the same waveshapeas the incident wavelet.and that a seismictrace recordsthe succession of such waveletsand thus the successionof

:LECTIONS

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Offset (m)

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'F

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(b) Aftcr normal movcout corractions

- -: Types of events on a seismic record. ldentities of 650 m/s; b : refrac: rre as follows: a - direct wave. /: ,: base of weathering, V,, : 1640 m/s; 1' - refraction from :i'fractor. V^ - 4920 mlsl, d = reflection liom the refractor I - 1640m/s; r, : reflection from a flat reflector, v - lglO ' = reflection from a flat refleclor, T = ZIOO m/s; g : t...r liom a dipping reflector,V = Zttt: m/s: /r : multiple of - multiple of e; .l : ground roll, VR- 5'75 m/s; k - arr

wave, V - 330 m/s; / - reflected refraction from in-line disruption of'refractor (: nt - reflectedrefraction from broadside disruption of refractor in t: After proper normal-moveout correction. the primary reflections are straight. Processing usually involvessetting to zero all valuesearlier than some "mute schedulc," here indicated by the dashed line; this is called a front-end mute. Consequently,the data that might otherwise appear in the upper-right triangles are usually not seen.

'l I I i

- - r - : i t i c i m p e d a n c ed i s c o n t i n u i t i e(sl i g . 6 . 4 ) . T h i s - :rept. that a seismictrace is simply the superposi:: trf individualreflections, is basicto the convolumodel(a mathematical :-..r1 calculationthat enables -- :.. determinethe effectof a filter on a signal;see - I I t. the reflectionprocessherebeingconsidered as . ilter." Carrying out the calculationsfor an as-:'.ed distributionof physicalpropertiesresultsin a ::)tt,tiL.yei.smogram, one of the commoner types of . i . r r dm o d e l i n g( $ 10 . 4 . 4 ) . The most common simplificationinvolvesa one- :-,:nsionalsyntheticseismogram whereit is assumed -:: ra\paths are vertical and interf-aces are hori:-,:.r1. Reflectionand transmissioncoeflrcientsare - . l b r n o r m a l i n c i d e n c e( e q s .( 3 . 1 4 )a n d ( 3 . 1 5 ) ) . l, -iactions and other wave modes are usually ig::d. althoughmultiplesmay be included.Most o1-::.. the acousticimpedancevaluesrequiredby these :--i-ttiousare obtained from borehole logs. Often, :.'. \onic logs are availableand densityis eitheras.-::ed to be constantor to bear somerelationto ve' - : : \ ( s u c ha s e q . ( 5 . 1 5 ) ) S . m a l l v e l o c i t yv a r i a t i o n s ,:: .rtienlumpedtogetherinto largerstepsto reduce -: numberof interfacesto be considered. and samr .::sis usuallyon a regulartraveltimeinterval(rather -.-:r on a regular depth interval, as with logs).

Amplitude-changingfactors other than those involving reflectioncoefficientsareoften ignored.The downgoing seismicwaveshapeis assumed,often a Ricker

: f I

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(d

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Fig. 6.4 Relation between reflections and acoustic impedance changescausing them. (a) Acoustic impedance changes,and (b) resulting reflections drawn for a minimum-phase wavelet.

148

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

wavelet($6.6.2),but sometimesthe waveshapeusedis the embedded wavelet (determined by wavelet processing,$9.5.9)of the actual data to which the synthetic seismogramis to be compared. The major use of syntheticseismogramsis to compare them with actual seismicdata in order to identify reflectionswith particular interfaces,so that mapscan be made on the bedsof particular interest.This procedure is also used to distinguish primary reflections from multiples and other events.Seismicsectionsoften involvetime or phaseshifts (includingpolarity reversals)of unknown magnitude(which are sometimes time-dependent),so that the ability to match synthetics to actual data adds considerableconfidenceto an interpretation. Figure 6.5 illustrates the concept of synthetic seismogram manufacture. The reflectivity model, shown as a stick graph ("stickogram") portraying the magnitude and polarity of discretereflection coefficients,is often derived from edited log data. Editing involvescorrectionfor sonic-logcycleskips and other borehole log errors; editing may involve .changing valuesso that they are consistentwith the complete ensemble of logsrecordedin a borehole.Conductivity or other logs that show changesin rock propertiesin a borehole are sometimesused instead of sonic and density logs.The reflectivitymodel is then convolved with a wavelet,comparedwith the actual seismicdata, and the difference("error trace") usedto modify the model (and sometimesthe wavelet);the procedureis iterated until the match is judged to be sufficiently close.As with all typesof modeling,the resultis not unique.becauseit is alwavsoossiblethat somediffer-

ent model could producean equally good match. Figure 6.6 showsa primaries-onlysyntheticseismogram matched to actual seismicdata. Although the match here is consideredgood, a multitude of differencesexist,perhapsbecausethe model or the assumed waveletwas not exactly correct, multiples or density variations were not allowed for, the match is to a common-midpoint sectionrather than one involving only vertical travel and horizontal bedding,errors in the acquisitionor processing,or for other reasons. The effectsof multiplesare sometimesincorporated as successive modificationsof the propagatingwaveform (Vetter, 1981),although often they are ignored also. Consider two adjacent interfacessuch that the one-way traveltime between them is the sampling interval, A, the reflection coemcientsfor a downgoing wavebeing R, and R,*,. If w(t),is the downgoing waveform approachingthe interface R,, the upgoing waveform reflectedby R, will be u(t),: R,w(t),.Neglectingthe very slight loss on transmissionthrough R,, the downgoingwave at R,*, will be w(l + A), plus the peg-legmultiples generatedbetweenR. and R,*,, that is. : lr(/ + A), - R,R,*,w(/ + (R,R,*,)2w(/ + 5A)r

_*11)'

the second term on the right being the one-bounce peg-legmultiple, the next the two-bounce,and so on. Thus, we can include the effectsof multiplesby modifying the downgoingwaveformat each step.We need to likewisemodify the upgoing (reflection)waveform for the peg-legmultiplesthat it generates. Where the match between synthetics and actual

T r a c em o d e l i n g

R

Adjust

Ir,todel trace

model

Fig. 6.5 Procedure fbr manufacturing synthetic seismogram. "Trace modeling" consists of convolving the reflectivity model representing the geologic data with a wavelet that is often extracted from the data to be matched. The difference between the

model trace and a trace from the actual data (the error trace) is often used to modify the model. (From Stommel and Graul,

r978.)

ri

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i ati I i \ \ \\\\\\\r\\ir..1.1 Fig. 6.6 Synthetic seismogram(right half) compared with actual seismicdata (left half). The synthetic seismogrammanufacture used a sonic los from a well on the seismicline and embed-

.(i\'.\'.(.(.,.!.\'.\\\t.i''r\\r'.rr\itt,

ded wavelets extracted from successivetraces of the actual data densities being calculated from Gardner's rule (eq. (5.15)) (Courtesy of Grant Geophysical.)

150 data is good, the reflectingsequencecan be modified according to stratigraphicchangesthat might occur, so that the effectsof suchchangeson the seismicdata can be ascertained.For example,we might assume that a sandunit shales-outin a facieschange,that the filI in a stream channel differs from that in the adjacent unerodedformation, that the formation subcropping under an unconformity has changed,or that a small reef grew.Syntheticsthen give the interpretera better idea of what to look for in order to locate the hypothesizedfacieschange,channel,subcrop'or reef. This usage of syntheticsprovides one of the main methodsfor the stratigraphicinterpretationof seismic d a t a( $ 1 0 . 7 ) . The model for manufacturingone-dimensionalsynthetic seismogramsmay actually be two- or threedimensional.A model might simulatethe changesexpected along a seismicline connectingtwo or more wells where the changesbetween the wells are explainedon the basisof facieschanges,unconformities, or faults. The synthetic seismogramfrom such a model is still consideredone-dimensional,howeveq unlessnonverticaltravel paths are considered. The synthetic seismogramconcept is generalized further in $10.4.4' and discussed 6.2.2Nature of reflections Reflectinginterfacesin sedimentaryrocks usually occur much closertogetherthan the seisrnicwavelength, and so the observed waveshapeis the interference compositeof a number of componentreflectionsand (as stated in $6.1) interferenceis largely responsible for reflectioncharacter.Thus, the appearanceof a reflection dependsupon the spacingand magnitudeof the component acoustic impedance contrasts. The compositenature of a reflection is illustrated by fig. 6.7. Although the waveshapeassumedin this classic illustration is poorly selected,the point is made clear that observed reflection events do not correspond one-to-onewith lithologic changesand that identifying individual cycleswith formation tops basedon traveltimemay lead to errors. It is often assumedthat acousticimpedancecontrasts coincide with major lithologic boundariesand indeed this is often true. However,the lithology may changewithout a changein acousticimpedanceand the acousticimpedancemay changewithout any major changein lithology.Hardage(1985)'in a vertical seismicprofiling study ($13.4),identified somereflections with the acousticimpedancecontraststhat cause them (fig. 6.8). Note that reflectionsB and D occur within fairly massive shale units where the shales differ somewhatin their properties,as evidencedby well logs.ln anotherarea,an abundanceofreflections on a seismicsection(fig. 6.9) was interpretedas indicating appreciableinterbedding of sand and shale (low stacking velocity values suggesteda siliciclastic section); howeveq a subsequentstratigraphic hole found hardly any sand; changesin the shalewere re-

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

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Fig. 6.7 Portion of an East Texas synthetic seismogram.(After Vail, Todd, and Sangree, 197'7:113;reprinted with permission') (a) Depth in feet, (b) interval velocities blocked from sonic log data. (c) reflection coefficient spikesbased on a one-to-one relation betweenacoustic impedance and interval velocity,(d) wavelets corresponding to the spikes in (c), and (e) composite of the waveletsin (d).

sponsiblefor the reflections.To be sure, such situa"exceptionsthat prove the rule" and tions are the most reflectionscorrespond to distinctive lithologic changes. A facies boundarymarks a changein the lithologic or paleontologicalcharacteristicsof contemporaneous sediments,whereasa time boundarymarks what was at one time the surface of the solid earth. Although it is commonly stated that facies and time boundariescrosseach other,they usuallycoincidelocally and their apparentcrossingis the result of the inadequatesampling on which faciesboundariesare one expectslithoBecause usuallybased(see$10.7.3). logic changesto be associatedwith facieschanges,one might expect reflections to follow the way facies boundaries are drawn. Howeveq overwhelmingevidence indicates that reflectionscoincide with time boundaries (except in occasional unusual clrcumstances). Reflectionsare often associatedwith unconformities.and often the bestand strongestreflectionscome from unconformity surfaces.The nature of an unconformity reflection,however,changesas the properttes ofthe rocks aboveor below the unconformity change. Unconformitiesare often associatedwith angularities betweenthem and subcroppingreflectionsor onlapping or downlapping reflections;this tends to make unconformity reflectionsmore obvious becausean observer'sattention is drawn to pattern irregularities. Unconformities have special importance in seismic studies($10.3.6). stratigraphic In many areas,especiallyareas of Tertiary clastic deposition,the nature of depositionchangeslaterally along time lines and consequentlyreflectionschange

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B, C, and D correspond with interfacesat the depths A', B', C,, and D' on the well logs. (b) Detail of responseon different logs at interfaces A' and C'.

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

152

1.0 at,

r! = =

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Fig. 6.9 Seismic line in an area of clastic deposition showing abundant reflectionsfrom a section that is almost entirely shale.

their character and are discontinuous.Commonly, one can map a given reflection only over a limited "phantoming," area; this leads to the practicecalled which is discussedin $8.8.3.On the other hand, in some areas, formations extend over very large distancesso that individual reflectionscan be followed over great distances. Although most reflections mark unconformities and/or time surfaces,fluid contactswithin porouspermeable rocks provide acoustic impedancecontrasts that cut acrossthe bedding.Theseare responsiblefor flat spots,one of the most important hydrocarbonaccumulationindicators($10.8).Figure 6.10 showssuch a reflection.Acoustic impedancecontrasts(hencereflections also) can be causedby chemical or phase changes,such as hydrate reflections(fig. 5.33) and ofopal (seefig. 6.ll); changesdue to recrystalization such situations,however,are relativelyrare.

(Courtesy of Teledyne Exploration Company.)

Fig. 6.,10 Portion of a West Ekofisk, Norwegian North Sea, seismic section showing a discordant horizontal reflection attributed to a gas liquid contact. The vertical lines indicate the limit of porosity based on the seismicdata; I and ,B have been interpreted as the top and base of the reservoir. (From d'Heur, 1992: 952; reprinted with permission.)

6.2,3Fresnelzones When we use rays to representwavetravel,the implication is that a reflectionoccursat the reflectionpoint. However,a reflectionis made up of energyreturning from a fairly largearea ofthe reflector.A Fresnelzone is the area from which reflected energy arriving at a detectorhas phasesdiffering by no more than a half-

t .

cycle; thus, this energy interferesmore or less constructively. Considera sourceand a coincident detector,S, as in fig. 6.12.SPuis perpendicularto a reflectingplane, and R,, Rr, . . . are such that the distancesSP', SP. SP,, differ by \/4. Generally,h, >> R, >> tr';

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il il lll lt l ll

lll

F= u "; .g E 6 :9 c

t

se

lll

.- sE qI

t t

-

o

ilp =E E

l l ! :

| ; l l c

!

t t

I [

lll

A

F o 6 9 h

a d

t

a , ^ ' '

L

o. o

i l

^ d

--

il

LI]

EE

t t

i ' -

l l

d ^

6

A

r vt

.<

'

o

6

U-o^F

3 H; = " : :< q o

N

6

n < < ! = . -

o x t

- E g c o S N a A q

a

Y -^ q v €

.

!

:

:i x

9 " e n

o€ > 6

. as 3 a ed,

=_ o

e

.dg; abr- 9 llrUtu

154

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S Becausetnand t,*, differ by jT contributions from successive zonesare alternatelyplus and minus.Thus, the effectat S is an alternatingseries,$, : S, - S, + S. - So + . . ., whereS,*, is a positivequantity given by ,4 times the factor in the square bracketsin eq. (6.4). BecauseS, decreasesas r increases,the series convergesand we may write (Wood, 196l: 34)

O': js' + (is'- s' + is') + ( : s -, E + l . s ,+). . ' .

(6.5a)

A graph of $, as a function of the radius is shown in fig. 6.13. In the foregoing equation, the terms in parenthesesare approximatelyzero; hence, Fig. 6.12

$r:

Geometry of Fresnel zones.

Srl2,

(6.5b)

that is, the major contribution to the reflectedsignal comesfrom the first Fresnelzone (the adjective"first" is frequentlydropped).The radius of S, is

hence,

R, - (n\hnl2)t2, h,S - rrlrhol2,

whereAS is the area of each of the annular rings. We shall calculate the energy returning to S from the (n + I )th ring. If we apply eq. (2.129)to a circle with the origin over the center(fig. 6.12),h becomesftoand ( becomesh,, and the integration with respectto 0 merelyinvolvesmultiplying by 2r. If we now calculate the Laplacetransform O(s)for two circlesof radii R, and R,*, and subtract the secondfrom the first, we obtain the effectof the (n * l)th zone:

: ),n"()."',"^r, o(^t) rc.z) +,",,,,*,,,). This solution correspondsto an impulsesourcec b(t) (see $2.8.1).Taking the inverse transform (see eqs. ( 1 5 . 1 8 0a) n d ( 1 5 . 1 8 7 ) w ) , e h a v ei n t h e t i m e d o m a i n

- t.)+at: lra,fl,6(t

*,u(,

- t. -

Rr: (llhol2)tn: 1Vl2)(tlv)t/2,

,u';

)

)l

(6.6a)

where/znis the depth, I the arrival time, Zthe average velocity,and u the frequency.The outer portion of S, makesrelativelylittle net contribution to the final result,as is seenin fig. 6.13,and hencean ffictive Fresnel zone is sometimes taken as a smaller radius : Rlt[i. The first Fresnelzone is often taken as a measure of the horizontal resolution of unmigrated seismic data. For a depth of 3 km and velocity of 3 km/s (t : 2 s), the Fresnel-zoneradius rangesfrom 300 to 470 m for frequenciesof 50 to 20 Hz (seefig. 6.14). Figure6.15 showsthe responseof small segmentsof a reflectingsurfaceat a depth of 1500m for a 30-m dominant wavelengthfor which the Fresnel-zoneradius is 150 m. When the reflector dimensions are somewhatsmallerthan the Fresnelzone,the response is essentiallythat of a diffracting point.

(6.3)

where r, : 2h,14 t,*, : (2h, + l\12)lv : t, + Tl2, T being the period. If the input at the sourcehad beenI cos rol instead of an impulse,eq. (6.2)would havethe additional factor Asl(s2* r,r' )(seeeq. (15.183))and, upon taking the inversetransform,eq. (15.187)would give

drr

- t^) 60 : )n"'41lth:)step(t r,)cosa(t - (l//rl*,)step(t- t, - ]r).or a(t - t, - irll - r,) : )lr,,l11ttt,,\step(r

- t, - Lr)\coso(r - t,) + lll(h, + lr,ylstep(r - t,) - thol2hl)Alstep(t ,

l-ld

First zone

l

2nd

t

r 3rd 4thi5th

Rdius R

- t, - ,T)lcoso(t - t,). + (l - |tl2h")step(t

Whenl>{t.+';r1, $(t) : A[(holh)(l \/4ft,)]cosa(t t,) = Afholh]lcosa(t - t").

(6.4)

Fig. 6.13 As the radius ofa circular reflector increases,the amplitude begins to build up slowly, then rapidly, tapering off toward the edge of the first Fresnel zone. Interference causes the amplitude to oscillate as successivedestructiveand constructive zones are added. Obliquity causes the oscillation to dampen more raoidlv than shown.

REFLECTIONS

r55

2.WAY TI M E

VELOCITY

FREOUENCY

-20 E

Fig. 6.14 Nomogram for determining Fresnel-zone radii. A straight line connecting the two-way time and the frequency intersectsthe central line at the same point as a straight line con-

necting the averagevelocity and the radius of the zone. For example, a 20-Hz reflection at 2.0 s and a velocity of 3.0 km/s has a Fresnel-zoneradius of 470 m.

The foregoing discussionassumesa point source, for which the travel paths from sourceback to detector differ by a half-cyclefor successive Fresnelzones. Fresnelzonesare sometimesspecifiedwith respectto a plane incident waverather than a sphericalwave,in which casethe half-cycledifferencesbetweensuccessive Fresnelzoneshave to be accommodatedentirely in the reflector-to-detector portion of the travel path. This resultsin an enlargementof the Fresnelzone,the radiusin this casebeing

mic traces whose Fresnelzonesinclude the portion. We wish the amplitude of a migrated trace to be proportional to the reflectivity,and we do this by summing (actually or effectively)traces around the reflecting point; clearly,if we wish to obtain the correct reflectivity,we must include all observationswithin the Fresnel-zoneradius.We note that migrating seismic line data effectivelycollapsesthe Fresnelzone in the in-line direction, but does not shrink its sizeperpendicularto the seismicline.

R , : ( \ f t n ) r , 2= ) V Q , I r ) ' , ,

(6.6b)

The amplitude of a reflectionwill be diminishedif the reflectivityanywherewithin the Fresnelzoneis diminished.For example,if we should be recording immediatelyover a largevertical step,half of the Fresnel zone would not contribute to a reflection,and the reflection amplitudewould only be one-halfof that over the reflectorremotefrom the step(seeeq. (2.133)).Because seismictraces are spacedmuch more closely than the dimensionsof the Fresnel zone, a specific portion of a reflectorwill contribute to all of the seis-

6.2.4 Effectsof reflector curvature Geometricalfocusingas a result of curvatureof a reflector affectsthe amplitude of a reflection. In a constant-velocitymedium, the wavegenerated by a point sourceis sphericalwith radius of curvature equal to the source-to-wavefrontdistance.We shall assumethat such a wave encountersa sphericalreflecting horizon centereddirectly below the source and with radius po.We can then apply the well-known formula of geometrical optics for reflections from

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

156

filllllllIlIl

!!!!!!!\\\!!LLLr

il[il1 tILrtLIt tl ))

(r) Fig. 6.15 Reflection from strips of various widths. (After Neidell and Poggiagliolmi, 1977.) (a\ Cross-section of model;

spacing of vertical lines equals the Fresnel-zonediameter; seismicsection.

curved mirrors:

Consider a cone of energyfrom the sourcethat is reflectedfrom a sphericalcap,MN (fig. 6.16b);the reflected energy is spread over a larger area at the surface for the anticline than for the plane, and over a smaller area for the gentlesyncline.Thus, reflections should appear stronger over gentle synclines and weakerover anticlines. When a synclinehas a radius of curvaturelessthan its depth, p. is positive and p is negative,and the energy passesthrough a focusbelow the surface(seefig. 6.17a):this is a buried-focussituation. Obviously,the likelihood of a buried focus increaseswith reflector depth. A reflectioninvolvinga buried focusis calleda reversebranch.The senseof traverseof the reverse branch is reversedfrom the usual,that is, as the source point travels from left to right, the reflecting point travelsfrom right to left. Where the sourceand geophoneare not coincident (that is, for offset traces),the reflectedwavemay focus even where the reflector'scenter of curvature is not below the surface,as in fig. 6.17b.Thus, long-offset traces may involve buried-focus effects even where short-offsettracesdo not. Common-midpoint stacking where short- and long-offsettracesare combined after normal-moveout correction usually does not allow for this situation correctly. When the curvatureof a synclineis not constant,as in fig. 6.17d, reflectionsmay be obtained from more than one part ofthe reflector,and the reflectedenergy has multiple branches,most commonly three. The two deeper reflections in fig.6.18 involve multiple branches;eachshowsbranchesfrom eachflank ofthe synclineand a reversebranch from the curvedbottom of the syncline.Three-dimensionalmultiple-branch effectsare discussedin $6.2.5. Just as light can be focusedby passingthrough a lens,seismicwavescan also be focusedby curved ve-

llu+llv:21R, wherea and v are the object and imagedistances,respectively,and R is the radius of the mirror. Positive R correspondsto a concavemirror and negativev to a virtual image.The object and image distancesare equivalentto the radii of curvatureof the incidentand reflectedwaves,pj and p,, the radius of curvature of the reflectingsurfacep,,being positive for a syncline, negativefor an anticline.The formula now becomes llp,+ Ilp,: 21p,.

(6.7)

If the distanceto the reflectoris ft, p, : h for a point source,and p,:hp,/(2h-p,,).

(6.8)

When the reflectedwavereachesthe surface,its radius of curvaturep is p:

h)l(2h- p,,).

p,- h:2h(p"-

(6.9)

We can expressthis in terms of curvature,the reciprocal of the radiusof curvature,in a normalizedform as n p

\r'l

l,,i;; ) -

t -

(6.10)

Figure6.16ais a graphofeq. (6.10).Reflectorcurvature of +6 correspondsto a point diffractor,and zero curvature(hlps :0) correspondsto a plane reflector for which the convex-upward curvature of the wavefront producesnormal moveout. When the reflector center of curvatureis at the surface(p" : h1,the reflected energy concentratesto a point. Curvature greaterthan this producesa buried-focusas discussed in what follows.Figure 6.16b showsthe imagepoints for a diffracting point, anticline,plane and syncline, and fig. 6.17 showsthe buried-focuscase.

REFLECTIONS

t5l

Concaveupward curyature of reflection wavefront at surface Centerof curyatureof reflector at the surface Increasinganticlinal curyature of reflcctor

Increasing synclinal curyature of !eflector

n/o

-r";":7diffraction / Convex upward curyature of reflection wave front at surface

l1

\\

\tl Planer D.

'^tl \\,^ I I

Diffracting pointd = lr)

, j \l l

,^l I

I

(b)

Fig. 6.16 Effect ofchanges in reflector curvature. (a) Normalized curvature of wavefront at the surface. hlo. as a function of reflector curvature, hlp,J, for a point source. The letters d, a, r and s refer to diffracting point, anticline, plane reflector, and syncline, respectively. (b) Effect on wavefront curvature as reflector changes from anticline to syncline. Respective wavefronts for d, a, r, and s are D, A, R, and S; image points 1o, In, Io, and 1y, and radii ofcurvature po, pp p& and py.

locity surfaces that result in seismic rays being bent by refraction;such situationsare often very comDlex. Curvatureat the baseof the weatheringcin be especially important becauseof the largevelocity contrast usuallyassociatedwith this surface.Variationsin oermafrost thickness and gas accumulation. .un ilro causefocusingeffects(fig. 6.19). Fermat'sprinciple explainsthat a wavewill take that raypathfor which the traveltimeis stationarywith respect to minor variations of the raypath, that is, for which the changein the traveltimefor an incremental changein raypathis zero.For most situationsthe raypath involvesthe minimum traveltime,that is, travel over any neighboringpath will take longer;henceFer_ mat's principle is often called theprinciple of least ttme or the brachistochroneprinciple. Snell'slaw, Huygens' principle, and many other laws of geometricaloptics can be derivedfrom this principle (seeproblem 6.3). An incident wavefrontapproachingthe reflectorin a buried-focussituation (fig. 6.17e)encountersthe reflector beforethe wavereachesthe reflectingpoint R, which satisfiesSnell'slaw Contributions to the reflection from the region surrounding R will thus arnve earlier than the reflectionfrom R itself,that is, the reflection point involvesa maximum in a Fermat,sprinciple sense,thus contrastingwith the more usual situation where the reflection point involves minimum traveltime.The fact that reflectioncontributionsfrom the region surroundingthe reflectionpoint arrive earlier manifestsitself as a changein the waveshapeof the reversebranch reflectioncompared with normal branches.With wavefrontsthat passthrough a focus (as for the reversebranch,),this phaseshift is n if the reflector is sphericalor ir if it is cylindrical. A irr phaseshift can be seenby comparing the waveshape of the reversebranch lor the lower event in fig. 6.lg with other events;the reflectorshere are cyliridrical. Such a phase shift is rarely useful in identifying buried-focusevents,but it will affect calculationsof reflector depth where picking is done systematically on the same phase, for example, always picking troughs,and it affectscommon-midpointstackine. 6.2.5 Three-dimensionalelfects In $4.1.2,we definedthe reflectingpoint as the point at which the angle of incidenceequaledthe angle of reflection. Seismic data are usually mapped at reflecting points. A line connectingreflectingpoints is called the subsurfacetrace. There is a subsurfacetrace for each reflector.Where there is a componentof dip perpendicularto the seismicline, the reflectinspoint liesto the sideof the seismicline ratherthan belowit. Such cross-dipeffectsare often ignored, usually becausethe cross-dipis not measured,and sometimes such neglectleadsto seriouserrors.In $6.3.1,for example, we assumethat the diffracting point is in the vertical plane containing the seismicline; if insteadit is from the truncation of a reflectorby a fault that is not perpendicular to the line, then the diffractine

E ot

(a) TC

, l

(c)

Fig. 6. l7 Focusing caused by reflector curvature. (a) Raypaths peipendicular to and equally spacedalong a curved surface Veiocity is constant throughout section. (b) Raypaths through a focal region when center of curvature C is below surface (c) Long-offset tracesfrom sourcesS, and S' passthrough foci even where short-offset traces S" do not. (d) Reflectionsfrom several points on a reflector when curvature changes.(e) Incident waveiront of radius p, impinging on reflector of radius p. when P')Ps

(e)

- l

EVENTSOTHER THAN PRIMARY REFLECTIONS

159

I

0.100 I

0.200

I0.3m I

0.400 I

b soo I

I 0.500 I 0.7m

{

0.Em

t-

Fig. 6. I 8 Reflection from cylindrically curved reflectors(whioh rrc planar outside the curved region). For all three reflectors, .he radius ofcurvature - 1000 m and V - 2000 m/s. Depths to

the bottom ofthe synclinesare 800, 1200, and 1600 m, respectively. The traces are 100 m apart with coincident sources and receivers.(Courtesy of Chevron.)

rrrint may move along the fault as sourceand/or re--eilersmove and the curvature of the diffraction on :he seismicrecordwill be lessthan that givenby eqs. 6.ll)and(6.13). In $6.2.4.we examined reflector curvature effects .,rd in fig. 6.18 showedmultiple-brancheffects.If a .eismic line crossesa synclineother than at right .ngles (line BB' in fig. 6.20b),the reflectionbranches may come lrom oppositesidesof the seisrnicline and \he \engt\r of \\re reriersebranc\ ma1 be sttetc\red,out and thus show smaller curvature(comparefig. 6.20d with fig. 6.20c). In the extreme situation where the seismicline is parallel to the axis of the syncline,the multiple branchesappear as parallel horizontal reflectors(fig. 6.20e).Wherethe synclineis plunging,the different brancheswill not be parallel. The Fresnel-zoneconcept of $6.2.3replacesa reflecting "point" with a reflecting"area," the area of

the first Fresnelzone. Featuresto the side of the reflecting point but within the reflectingarea will produce effectson the seismicline, as shown in fig.6.21. Hilterman (1970)showedthat such effectscan make structuresappearto be appreciablylargerin areathan they actually are (fig. 6.22) when mapping 2-D data.

63 Events other than primary reftections 6 . 3 . 1D f f i a c t i o n s Diffraction phenomenawere discussedin $2.8,where it was shown that the reflection from a half-plane and the diffraction from its edgeare continuousand indistinguishableon the basis of character.Diffractions from edgesthat are perpendicularto the seismicline, however,exhibit distinctive moveout. In figs. 6.23a and 6.23b for sourcepositions abovethe diffracting

C H A R A C T E R I S T I C SO F S E i S M I C E V E N T S

r60

negative,so the diffraction moveout can be either greateror smallerthan given by eq. (6.11).If we consection,as shown sidera coincidentsource-geophone in fig. 6.23d,the diffraction traveltimecurve will be lr:

- to * 4 a,tn. (2lv)(x' + h2)\t2

(6.13)

This is the sort of sectioncommon-midpointstacking seeks to simulate. However, a diffraction on a common-midpointgather(fig. 6.23e)has a traveltime curve /::

-.

-i -:'.

(a) Low Fig. 6.19 Fbcusing produced by velocity variations deeper from raypaths focusing gas accumulation velocity in a reflection. (b) High-velocity wedge producing a focus'

edges,the diffraction traveltimecurve (for a commonsource gather with the source over the diffracting point) for h >> x is given bY * (x2 * h2)1t21 tu: (ll)[h : 2hlV + x2l2Vh : t,,4 2 L,t,,

( 6 . 1l )

whereAt, : x2l4Vh,the normal moveout for a reflection (seeeq. (a.7)). However,when the sourceis not vertically over the diffracting point (fig. 6.23c)' the diffraction traveltime curve is given by + [(r - a)2+ h')\''] tr: (llV){(h2 + a2)tt2 : (2hlv) + (a2l2Vh)+ (x - a)' l2Vh : to * (2a2- 2ax + x' )l2Vh : to + 2 Lt, + ab - x)lVh; 6.12) thus, the diffraction moveout varieswith the location of the source.The last term can be either positive or

Un{h'

+ (xt2 * blz|rz+ IE + (xl2 - b)' )lt' ], ( 6 .l 4 )

which differsfrom that given by eq' (6.13).Obviously, diffractions will be attenuatedby common-midpoint stacking. Note that the normal moveout applied in processingusually is that for a deeperreflection(that is, for largervelocity) becausethe diffraction time t,,is larger than for the reflection terminating at the diffracting point. The earliestarrival time on a diffraction curve is for the trace that is recordeddirectly over the diffracting point (exceptfor unusual velocity-distributionsituations), but the diffraction will not necessarilyhaveits peak amplitude on this trace. Consider three half-planesat the same depth but with differentdips (fig. 6.24).Thediffractionsfor each of thesecrest at the location of the edgeof the halfplane, have the same curvature, are tangent to the reflections.and the maximum amplitudesof the diffractions occur at this point of tangency.Thus, the diffraction crest locates the diffracting point, the diffractioncurvaturedependson the depth and the velocity abovethe diffracting point, and the amplitude distribution along the diffraction dependson the attitude of the half-Plane. A reflectorthat is bent sharply,as shownin fig' 6'25' could be thought of as the superpositionof two dipping half-planes,each terminating at the bend point' Thus, the two diffraction curveswould coincidein arrival time and would add together constructivelyin the region betweenthe respectivereflections'In fig' 6.25b,the reflectorto the right of x : 2.1 km gives : 2' l km rise to P'^Band the reflectorto the left of x givesAP; diffraction fills in the gap PP' and makes the seismicevent continuous without a sharp break in slope. As another exampleof diffraction effects,consider the reflection from a reflector with a hole in it' as shownin fig.6.26. Diffraction tendsto fill in the hole' Figure 6.27 shows the location and amplitude of wavJmotion a short time after a plane wavefronthas passedby the point of a wedgethat is a perfectreflecior. The reflectedwavefront has an associateddiffraction BAC andthe portion of the wavefront that missed the reflectingwedgealso has an associateddiffraction FDE.The portion of the diffraction DErepresentsenergy reachinginto the shadowzone hidden from the incident wavefront by the reflector.Diffraction pro-

EVENTS OTHER THAN PRIMARY REFLECTIONS

r6l A'

a Centerof curvature

(a)

t : x x xx xx

g

')l o

N : ts

€€

i i x

C l r )<

€

€

B,

; : i: 9-

:

i t II '

l

: l

:l ; l ! l : l

\

I ,-l

: l

il : l

it

I

I

I

I

I

.ll - /

I

t\

I

C,

\) B'

ft\

(e)

Fig. 6.20 Buried-focus effectson lines at different angles to the strike of a syncline. (a) Syncline cross-section;(b) contour map of reflector showing subsurlace traces (dashed line) of BB' and

(dotted lines) of CC'; (c) arrival times along line AA'; (d) arrival times along BB'; and (e) arrival times along CC'.

vides a mechanismfor getting seismicenergyinto regionsthat cannot be reachedon the basisofgeometrical optics. The downgoing diffraction FDE in fig. 6.27 might be subsequentlyreflectedby another reflectorto give a reflected diffraction, or an upcoming reflection could be diffractedto give a diffractedreflection,and so on. The diffraction curvature for such compound eventswill differ from that of simplediffraction events that havethe samearrival time. Additional compound eventsare shownin fig. 6.55. Figure 6.28showsa vertical step.The top of stepI will generatediffraction, D,, in fig. 6.28b.The bottom of step B will also generatea diffraction, Di, but the portion of this diffraction to the right of B will travel partially at a lower velocity than the reflectionto the left ofthe step. The reflectionfrom the left portion of the baseof the model will arrive earlier than the reflectionfrom the right portion becausemore of its travel path is at the higher velocity.Furthermore,each of thesere-

flection segmentswill have an associateddiffraction appearingto come from C even though the baseof the model is continuous;such diffractions are called phantomdiflrut'tionsand they result from lateral velocity changesin the sectionabovethe reflector.Consider a geophonejust to the right of a point directly over the stepin fig. 6.28a.The travel path for a reflection from the base of the model is shown by the dashedpath, but some energy will travel the dotted path and arrive earlier than the reflection;the latter producesthe phantom diffraction (diffraction energy doesnot necessarilyobey the rules of geometricaloptics). Phantom diffractions are occasionallyseenon seismicrecords,especiallyin areasof thrusting where some travel paths pass through thrust plates and othersdo not. 6.3.2 Multiples (a) Distinction betweentypes. Multiples are events that have undergonemore than one reflection. Be-

162

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

I

DePth= 1 5 0 0 m

,-,". n JL

1200m

-

n

Line A

(a) Box

Box

0.9

tttttiitttt

'(((((

I

l.t

1.3 (c)

Fig. 6.21 Modeled lines acrossa box structure. Trace spacing. 85 m, Fresnel-zoneradius 280 m for 30 Hz. (Courtesy of Geo-

quest.) (a) Model, (b) subsurface trace of line I above the top of the box, (c) line B above the plane I 50 m beyond the box.

Fig. 6.22 Grid of 2-D seismiclines across a physical model of irregularly shaped gas field (solid curve). Dotted outline shows seismic interpretation; failure to allow for 3-D Fresnel-zone effects led to 407u overestimate of field size. (After Hilterman, 1982.)

causethe amplitudeof multiplesis proportionalto the product of the reflectioncoefficientsfor each of the reflectorsinvolved and becauseR is very small for most interfaces,only the largestimpedancecontrasts will generatemultiples that are strong enough to be recognizedas distinctiveevents. We distinguish between two classesof multiples, which we call long-path and short-path.A long-path multiple is one whose travel path is long compared with primary reflectionsfrom the same deep interfaces,and hencelong-path multiples appear as separate eventson a seismicrecord.A short-pathmultiple, on the other hand, arrivesso soon after the associated primary reflectionfrom the samedeepinterfacethat it interfereswith and adds tail to the primary reflection; hence,its effect is that of changingwaveshaperather than producinga separateevent.Possibleraypathsfor

EVENTS OTHER THAN PRIMARY REFLECTIONS

163

^(, A/.

F-.--*l

l*t-*l

Fig. 6.23 Diffraction traveltime curves. (a and b) Commonsource arrivals lbr diffraction and reflection; Ar,,is the reflection normal moveout; (c) dillraction for source offset from diffract-

rng point; (d) source and geophone coincident at different locations; (e) midpoint not over diffracting point.

thesetwo classes are showninfig.6.29.

pulsebecomesmodified as a resultof passingthrough a sequenceof interfaces,and their frequencyspectra in fig. 6.30cshow the lossof high frequencywith time. Figure 6.30d showsthat the averageattenuationdue to peg-legmultiples is equivalent roughly to 1\ : 0.085+ 0.055dB (see92.7.2b). This is the samesiruation that wasdiscussedin relationto syntheticseismograms(Q6.2.l ). Ghostsare the special type of multiple illustrated in fig. 6.29.The energytravelingdownward from the sourcehas superimposedupon it energythat initially traveledupward and was then reflecteddownward at the surfaceor baseof the low-velocitylayer (LVL) in land surveysor at the surfaceof the water in manne surveys.A 180'phaseshift,equivalentto halfa wavelength, occurs at the additional reflection,and hence the effectivepath differencebetweenthe direct wave

(b) Short-pathmultiples. Short-path multiples that havebeenreflectedsuccessively from the top and base of thin reflectors(fig. 6.30a)on their way to or from the principal reflectinginterfacewith which they are associated(often called peg-leg multiples) are lmportant in determining the waveformsof the events recorded on a seismogram.These peg-legmultiples delay part of the energy and thereforelengthen the wavelet.The strongerpeg-legmultiplesoften havethe samepolarity as the primary becausesuccessive large impedancecontraststend to be in oppositedirections (otherwise,successive largechangesin velocity would causethe velocity to exceedits allowablerange).Pegleg multiples effectivelylower the signal frequencyas time increases.Figure 6.30b showshow a simple im-

t64

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

/ > > t <

l '

,

(a)

I

I

,

)

1

(b)

II

--t

t

/ >.s .

I

II (c)

Fig. 6.24 Reflectionsand diffractions from half-planes terminating at the arrows (coincidenl sourcesand geophones).(Courtesy of Chevron.) (a) Termination at the downdip end of the

half-plane; (b) the flat half-plane; (c) termination at the updip end of the half-plane.

and the ghost is )\12 + 2D,, where Q is the source depth below the reflectorproducingthe ghost.Similar ghosting results from buried detectorsor a marine streamertowed at depth Q. The interferencebetween the ghost and the primary dependson the fraction of a wavelengthrepresentedby the differencein effective path length. Becausethe seismicwaveletis made up of a range of frequencies,the interferenceeffect will vary for the different components.Thus, the overall effect on the waveletshapewill vary as Q is varied. Relativelysmallchangesin Q can resultin largevariations in reflectioncharacter,creatingseriousproblems for the interpreter.Thereforethe depth below the base of the weatheringor the surfaceof the water is maintained as nearly constantas possible. Ghosts affect directivity as well as waveshape.Figure 6.31 showsa point sourceat a depth 4 : c\; if

then the the sourceemits the wave A cos (rr effectat P is 0 " : I cos (rr, - <,1/)- I cos (rr, - /"dt) I cos(rr - at - rcl cos 0) -l cos (rr - <,ll* rc\ cos 0). Expandingand noting that r\ : 2n, we get mately - ol) sin (2rrccos 0) *r: 2A sin (rr : 2A sin (2nc cos 0) cos (rr - at -

(6.l s) Figure 6.53 is a graph of rJr"as a function of 0. For r large,the total wavemotion lagsthe original wavemotion by 90o and has an amplitude that dependson 0. Becausea waveletcontainsa spectrumoffrequencies, different componentswill add differently at various

EVENTS OTHER THAN PRIMARY REFLECTIONS

0.6

0.9

165

' ) l

(b) Fig. 6.25 Reflections and diffractions from a sharply bent ref l e c t o r .D i p s a r e 3 l ' a n d l 1 ' t o t h e l e f t a n d r i g h t o f x = 2 . 1 k m .

anglesof 0, resultingin changesin waveshape. Ghosts are especiallyimportant in marine surveys becausethe surface of the water is almost a perfect reflectorand consequentlythe ghost interferencewill

't< t.tot o

Fig. 6.26

(1?;'ll-

Effect of a hole in a reflector.(Courtesy of Chevron.)

(Courtesy of Chevron.) (a) Model; (b) reflections and diffractions (dashedcurve).

be strong.If Q is small in comparisonwith the dominant wavelengths,appreciablesignalcancellationwill occur. At depths of l0 to 15 m, interferenceis constructive for frequenciesof 40 to 25 Hz, which is in the usual seismicrange.The sameeffect occurs with the upcoming signal from the reflectorswe wish to map. Hence,marine sourcesand marine detectorsare often operatedat suchdepths. A particularly troublesometype of multiple producesthe coherentnoiseknown as singing(alsocalled ringing or water reverberation),which is frequently encounteredin marine work (and occasionallyon land). This is due to multiple reflectionsin the water layer. The largereflectioncoefficientsat the top and bottom of this layer result in considerableenergy being reflected back and forth repeatedly, the reverberating energy being reinforcedperiodically by reflectedenergy. Depending upon the water depth, certain frequenciesare enhanced,and the record looks very sinusoidalas a result (seefig. 6.32). Not only is the picking of reflectionsdifficult, but measuredtravel-

r66

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

o

\

\ \q. \", '.I

?^\o-

2a\

\ e*\

'r(,

I I

I I I \

Fig. 6.27

Diffractions from a perfectly reflecting wedge. (Courtesy of Chevron.)

times and dip moveoutswill probablybe in error.This type of noise and its attenuation are discussedin

se.2.4. (c) Long-path multiples. The strongest long-path multiples involve reflectionsat the surface, the sea floor, or (on land) the baseof the low-velocity layer (LVL, also called weatheredlayer; seetable 3.1 and $5.3.2),wherethe reflectioncoefficientis very largebecauseof the large acoustic-impedance contrast. Becausethis type of multiple involvesat leasttwo reflections at depth, its amplitude dependsmainly on the magnitudeof reflectioncoefficientsat depth, and multiplesof this type will be observedas distinctiveevents when thesecoefficientsare abnormally high. Because R in eq. (3.14)may be as largeas 0.7 at the baseof the LVL and perhaps 0.2 for the strongestinterfacesat depth,the maximum effectiveR for suchmultipleswill be of the order of 0.2 x 0.7 x 0.2 : 0.03.This value

is in the rangeof typical reflectioncoefficientsso that such multiples may have sufficientenergyto be confused with primary events.The principal situation where weakerlong-path multiplesmay be observable is where primary energyis nearly absentat the time of arrival of the multiple energy.It is important that multiples be recognizedas such so that they will not be interpretedas reflectionsfrom deeperhorizons. Becausevelocity generally increaseswith depth, multiples usually exhibit more normal moveout than primary reflectionswith the same traveltime.This is the basisof the attenuationof multiples in commonmidpoint processing, which will be discussed in 59.10.4. However,the differencein normal moveout is often not large enough to identify multiples. The attenuationof multiples is also the primary objective of predictivedeconvolution(99.5.7). The effectof dip on multiples that involve the surface or the base of the LVL can be seenbv tracins

EVENTS OTHER THAN PRIMARY REFLECTIONS

I

t67

Low velocity

or

(b)

(

o

r

l

1

z

o

P

l

P

U m l | r r I s UI i ' i I t R, L

l

S ,Rr

R)ps

I

I a l

z |lf

c

,,F

fl

l

iw.i

{ tc,td

F li.lilotor

Fig. 6.28 Effects of a step. (From Angona, 1960.) (a) Model; (b) split-spread record with the source over the step: P - direct wave;S : surfacewave; Rr_and R, : reflectionsto left and right of step, respectively; Ro: reflections from the base of the Short-path multiplcs

d

E

-

! u ? A ; =

-E o

3r z e

9

.

fi'

It

fiq* h ldt.. model; D, and D1 : diffractions from top and base of the step; D'i : phantom diffractions that continue R, beyond the center; (R)p. : converted wave; and M : multiple.

Long-path mulliples ? .Y3

,-

-

^-6 Eiq

sa

?9E d E

,,e

- =

e=

bE;

=.E e

Fig.6.29 Typesof multiples.

raysusingthe method of imagesassumingthe velocity is constant.In fig. 6.33,we trace a multiple arriving at symmetricallydisposedgeophones,G, and Gro.The first imagepoint, 1,, is on the perpendicularfrom S to AB as far below .4.8as S is above.We next draw the perpendicularfrom 1, to the surface of the ground wherethe secondreflectionoccursand place Irasfar abovethe surfaceas 1r is below. Finally, we locate I on the perpendicularto AB as far below it as I is above. We can now draw the rays from the source S to the geophones(working backwardsfrom the geophones).The dip moveout is the differencebetween the path lengths IrGroandIrGr;it is about double that

of the primary (I,Gro - I,G,).The multiple at the source will appear to come from 1r, which is updip from 1,, the image point for the primary, and 1rS is slightly lessthan twice 1,S. Hence,we can seethat if the reflector dips, the multiple involves a different portion of the reflector than the primary and has a traveltime slightly less than double the traveltime of the primary. The latter fact makes identifying multiples by merely doubling the arrival time of the primary imprecisewheneverappreciabledip is present. The arrival time of the multiple will be approximately equal to that of a primary reflectionfrom a bed at the depth of 1,. If the actual dip at 1, is not double that at

168

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

(r)

After 0.7 s

Altcr 1 . 4s

(a)

Aftcr 2 . 7s

After 5 . 5s

(c)

40

60 80 (Hz) Frequency

Fig. 6.30 Changes in waveshapeproduced by peg-leg multiples.(a) Schematicdiagram showing peg-legmultiples adding to the wavetrain. (b) Waveshapesafter different traveltimes for an actual layered section (alter O'Doherty and Anstey, l97l). (c) Frequency spectra of waveshapesshown in part (b). (d) Histo-

/N t \ ctrl I

Fig. 6.31 Directivityof sourceplusghost.S : source,1 : image(effectivesourceof the ghost),and P = observingpoint.

0.12 0.16 0.08 A t t e n u a t i o n , 4(Id B )

gram of attenuation coefficientsdue to peg-leg multiples based o n 3 l w e l l s i n d i f f e r e n t b a s i n s ;1 : a t t e n u a t i o nc o n s t a n t a s i n e q . ( 2 . 1 1 0 ) ( e x c e p t f o r t h e a t t e n u a t i o n m e c h a n i s m ) ,a n d \ : wavelength (after Schoenbergerand Levin, 1978).

AB (and one would not in generalexpectsucha dip), then the multiple will appear to have anomalousdip. If the multiple should be misidentifiedas a primary, one might incorrectly postulatean unconformity or updip thinning, which might lead to erroneousgeologic conclusions. In deep-watermarine surveysmultiples of the sea floor may be so strong as to virtually obliterateprimary reflections.Furthermore,the rangeof reflection anglesmay be so great that the effectivereflectioncoefficient varies widely for different offset distances. Figure 6.34 showsthe buildup of amplitude observed near the critical angleand how this occurson different tracesfor successive multiples.Predictivedeconvolution ($9.5.7)is generallyineffectivein attenuatingsuch multiplesbecauseit assumesthat, for any given trace, the reflection coefficientsat the reflectors involved are constant.However,if sea-floordip is small,the reflection anglesfor the simple reflection at offset x will be

EVENTS OTHER THAN PRIMARY REFLECTIONS

r69

jlf'#rl]tililllf [T, nllil,4H l',"1s",>1.1) I I I I I I I l,t'rli'l',1',)1'!,".P.)

lll'fJTrfrrTi it Iirtt.]il,t,t DI t'r D'l???t:I?),)t;rt)t-)) D)I) j'-'-i-.f f ,rfft

, t t t l i r | : , ' , \ _

Df"F )D4"n.,""x*-j,'_ll

_I.i-i_l _ ; . i i i I : i.,J; . . . .D ' ru|;.)..1,.).t l . rr i ; f , ; ; . t ; , i , ,: : , . : ; ; ; i : ! l l l l l i lr "t>rt't'',u'>a

E F

, iiili;i.,,x,riir,ii,,f3:::::l|HI::IA

tr|jrir ;i:::::iir* :;:::::[iir _; jtii, I f::ir l;il:;3:::::::xJ ||||||l:";""n'FD'tsh

!::Fff

rtr, ttr))) t r, rrDr)f rr>rr D))rrrrrrfm) _ll_Jf. i.i" ,:"Dr'!.>D)r!.t.t }},lt b)ilrrDr D)Df rr., ) r , , .,., I I i1 --llJ,:?ffD)'t"li.)rt.)tr-r,r'.rx'ri>)' r',r,.r..rrDiil..l j jrr")';1 ..,,i.r.r>;,p-, r i Ip I I ;,r ; I I I I r"r,r^".---l;,f

2 0-lqi It nn,'.n^.r.. r t J-;i-:i*l*

a.:i:r;r'::':1::i'-i

rjr; lxii;::j--+ ; fi l?lli,*Hry rrrr

>,,.,.!i.ili;l:;;:llli: _Ji.l'_i"l-'-l_:mr --;;.'ri':;)r{..} 6i j,iir j ir i,r *Dr I >rt 6'-t}A -

\.,

{

}t!;,8.}tt

).t

( I }, r, f ,rrt

"

7.,,,.'

,1ii; irr' r r'*r or*rr; ipir .+ii ;,3::l:ffit;

+i::ififiHr f*:: :*llil:sr ' jill,iii*l: **ifr n:riii ;:l :;; 1 1:L :l :::,,i i;l;';;.H;l'ffi ::Ii:1' ::X:l 1,,,':::ll' i-l'ilr:;1.:

li{f il'+1+r-';7;r;i#r.5;n-iiiffi ; 5*;lr i',,_: -br:l ':: ') r'i"lijJJ;;, 'fllt iii

jii::

?.,J::::,F;:: I::lt iliH;j:

ii.i:i;ill[. ill:l:i:j :.j ri::::ll:i,'^l::

-;q!,rr)..sr.;:.{1.,

,o-ii':;l :::fi-i ..;::i:-:::l l:i:i:: l::,:.-'*e.-b"E'd

\a)

Fig.6.32 Seismic recordshowing (Courtesy singing. of HaliburtonGeophysical (a)Fieldrecord. Services.) and(b)thesame the sameas for the first multiple at offset2r (fig. t5.35) and for the next multiple at offset 3x, and so on, so that the primary on one trace sometimescan be used to predict and compensatefor the multiple seenon anothertraceat anotheroffset.This providesthe basis for radial multiplesuppression (99.5.13). 6.3.3Refractions The onsetof headwavesis often followedby a number of parallel alignments,that is, they seemto involve a

after singing has been removed by deconvolution processing

($e.2.4).

long wavetrain consisting of severalcycles.As the offset distance increases,the number of cycles increasesand the peak energy shifts to later in the wavetrain, an effect called shingling (fig. 6.36a). The amount of shinglingis greaterwhen the refractor is of limited thickness.Becauseof this shift of energy,it is often impossibleto pick the onset time required for applicationofhead-waveequationssuchas eqs.(4.36) to (4.48).Most timing of head wavesis done on later peaksand troughs,and a correction is applied to obtain the onset time. This processoften givessatisfac-

l.

,i

,li\

/ t'l \

r ,

1 ' /

/

\ t t

Offsct (ftl

v h Fig. 6.33 Constructing raypath ol a multiple from a dipping bedwherevelocity is constant. -7

i

J Ji-t {

F -

( '

+

o o

E F

I

i

+

ilt

t t

il

t- t t

I

ill

Fig. 6.34 Change of amplitude with offset for sea-floor multi: olJs. offshore easiern Canada. Trace spacing 100 m and offset of first trace : 425 m. The amplitude buildup occurs near the critical angle (seeproblem 6.13). (Courtesy of Chevron')

EVENTSOTHER THAN PRIMARY REFLECTIONS

t7l

I

I

I I

Fig. 6.35 Relation between offset and angle of reflection for pnmary and multiple reflections from a flat reflector.

tory resultseventhough absorptionand other mecha_ nisms shift the frequency spectrum lower with increasingdistanceso that latercyclesdo not perfectly parallelthe waveonset. Several mechanismscontribute to the shingling effect(figs.6.36bto 6.36d).Someof the enersvthat peelsoff the refractorcan be reflectedat bedsparallel to the refractor and returned to the refractor at the criticafangle(suchas M andM, infig.6.36b)to form delayedhead waves.Multiple reflectionsof this type can peal off the refractor continuouslv.and for anv parallelreflectorthat is significantin ireating multiples,they tend to add in phase(the head waveconsequent to a reflectionat M havingthe samedistanceto travef as one reflectingat M,). The result is to steal energyfrom the front ofthe headwaveand add cvcles at thetailend. Wavesthat bounce repeatedly(fig. 6.36c)in layers within the refractoralso add tail to the refractionwavetrains.Diving wavesresultingfrom a velocitygradient in the high-velocityrefractor (fig. 6.36d)havethe similar effectof adding tails. A velocitygradientin the refractorconsiderablystrengthensa headwave,as shownin fig. 6.37. As the refractor becomesthinner,destructiveinterferencebetweenthe headwaveand the reflectionfrom the baseof the refractor also weakensthe head wave, as shown in fig. 6.38.Poisson'sratio also has an effect on head-waveamplitude. Note that the head wave sometimes is phase-shifted, as in fig. 6.38. Refractions(headwaves)are not usuallya problem on reflection records.They are generallyof low frequency, have straight alignments (prior to normal_ moveout correction),and are attenuatedby stacking. Head wavesare only observedwhere the offset eiceedsthe critical distanceand, as shown in fig. 4.16 and by eq. (4.39),the criticaldistanceis lessthan the refractor depth only for V,IV, > 2.24. Velocity contrasts of this magnitude are possiblebelow the base of the weathering,for example,where carbonatesor evaporitesare overlain by sandsand shales,but usually head wavesfrom deeperrefractorsdo not appear on enough traces to make their moveout useful in identifyingthem, and they often disappearin the muting of the first-breakregion (the upper-right triangular regionof fig. 6.3).

(b)

fTt-_l

t

\ (d)

Fig. 6.36 Mechanisms that lengthen the refraction waverrain. (a) Section showing head-wave shingling. (Courtesy of Geophysical Development Co.) (b) Reflections of head waves from parallel reflectors above the refractor. (c) Repeated reflections within the refractor. (d) Velocity gradient in the refractor.

6.3.4 ReflectedreJract ions Wherea refractoris terminated,suchas shownin figs. 6.39a and 6.39b,the head wavewill be reflectedbackward. It may appear on the later portion of a reflectron record some distance from the actual refractor termination.When the refractor termination is nearly perpendicularto the seismicline, the reflectedhead wavewill havea nearly straightalignmentwith an apparent velocity approximatelythe negativeof the refractor velocity.The head wavewill be reflectedeven though the law of reflectionis not satisfiedat the refractor termination.The refractor terminationmay be either againstlower- or higher-impedance material so that the reflectedhead wavemay havepolarity either oppositeto or the sameas the head wave.Where the refractoris massive,reflectionsas in fig. 6.39cmay appear much like the reflectedheadwave(figs.6.39aand 6.39b). Where the refractor termination is off to the side of the line (fig. 6.39d),the event may have some curvature(pseudo-normalmoveout)acrossthe record ( s e ep r o b l e m6 . 1 5 ) .

I I

CHARACTERISTICS OF SEISMIC EVENTS

172

OFFSET

6

o'. ui

=

5

l !

,

, , r , r

, l , r , r r r l t r r r l

(a) OFFSET

G 0.4 IrJ

= F

Fig. 6.37 Strengthening ofa head wave by a velocity gradient' Reflection normal moveout has been removed so that head waves curve upwards. (Courtesy of Geophysical Development Co.) (a) Velocity step from 2000 to 4000 m/s (b) Velocity step

6.3.5 Surfac'ewaves Surface waves(ground roll) are usually present on reflection records.For the most part, theseare Rayleigh waveswith velocitiesranging from 100 to 1000m/s or so. Ground-roll frequenciesare usually lower than thoseof reflectionsand refractions,often with the energy concentratedbelow l0 Hz. Ground-roll alignments are straight,just as in the caseof refractions, but they havemuch lower apparentvelocities.The envelopeof ground roll builds up and decaysslowly and often ground roll includesmany cycles.Surface-wave energygenerallyis high enoughevenin the reflection band to overrideall but the strongestreflections;however,becauseof the low velocity,different geophone groups are affected at different times so that only a few groups are affected at any one time. Sometimes there is more than one ground-roll wavetrain,each with a different velocity.Occasionally,where surface wavesare exceptionallystrong,in-line offsetsare used to permit recordingdesiredreflectionsbeforethe surface wavesreach the spread.

irom 2000 to 3000 m/s, and then gradient increasingthe velocity to 4000 m/s. (The lower event from 0.65 to 0.74 s is a converted wave. Note also a phase shift of the wide-angle reflection from that near normal incidence.)

effectscan be attenuatedby the useof Surface-wave arrays($8.3.5to 8.3.9and problem8.6)'by frequency filtering (ground roll can be seenon the 0-6-Hz and slightly on the 6-12-Hz panels of f,g. 9.20)' and by apparent-velocityfiltering (seefig. 9.38)' 6.4 Resolution 6.4.1General Resolutionrefersto the minimum separationbetween two featuressuch that we can tell that there are two featuresrather than only one' With respectto seismic waves,we may think of (a) how far apart (in space or time) two interfacesmust be to show as separate reflectors(verticalresolution)or (b) how far apart two featuresinvolvinga singleinterfacemust be separated to show as separatefeatures(horizontalresolution)' (The word "resolution" is often usedlooselyto denote the ability to tell that a featureis present.) Clearly, the ability to see and distinguishfeatures dependson the signalinoiseratio and the knowledge

RESOLUTION

T I J

jlllil llf|ll

OA

G ul

= F

o.6

ffifffll OFFSET

'r!' j

o.4 3 ul F

Fig. 6.38 Effect of refractor thickness on head wave. Head wave loses amplitude because of destructive interference with reflection from baseof refractor when the refractor is thin. Normal moveout for reflection at top ofrefractor has been removed

so that the head wave and the reflection from the bottom of the refractor curve upwards. (Courtesy of Geophysical Development Co.) (a) Refractor 1.5 wavelengthsthick, and (b) refractor 3/4 wavelength thick.

and experienceof the interpreter. Where a correct model is used for interpretation,it is possibleto exceedconventionalresolutionlimits,that is,if we know a priori exactlywhat we are looking for in very good data, then subtledifferencescan be usedto locateand identifyit. If seismicwaveletswereextremelysharp,resolution would not be a problem. Howeveqreal seismicwavelets involve a limited range of frequenciesand hence haveappreciablebreadth (see96.6.t ).

are two waveswhen the arrival of the seconqwave causesa perceptiblechangein the appearancecf the first wave. Rayleigh(Jenkinsand White, t957:300)delinedthe resolvable(separable)limit as being when the two events are separatedby a half-cycle so interference effectsare minimized. Ricker (1953b) and Widess (1973) used slightly different criteria, which resulted in slightlysmallerresolvablelimits. Kalweit and Wood ( 1 9 8 2 d) i s c u s rse s o l u t i o n criteria. For a boxcarfrequencyspectrum(seeeq. (15.123)), the waveletshapeis that of a sinc function. The Rayleigh criterion is equivalent to a width of approximately 213u,,, whereu, is the upper frequencylimit of the boxcar (seeproblem 6.l8). Thus, we must record higher frequenciesif we are to achievehigher resolution (Sheriff.1977).

6.4.2 Verticalresolution Let us first consider resolution in the vertical direction. For two horizontal reflectorsa distanceL,z apart, the deeperreflectionlags behind the shallowerby the fraction 2 A:/1,of a wavelength.We can tell that there

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

t74

as the signal/noiselevel and the experienceof the interpreterin similar studies.Thesethree examplessuggestthat the Rayleighdefinition of resolvablelimit is reasonable. 6.4.3 Tuningand thin-bedeffects

(b)

(c)

Fig. 6.39 Reflected refraotions. (a c) Refractions reflected fiom f aults or salt domes. (d) Isometric drawing of refractions reflectedfrom the termination of the reliactor to the side of thc sprcad; paths are shown from source Sto geophonesG, and G'; dashed lines indicate head-wavetravcl in thc refractor'

As an illustrationof verticalresolution,fig' 6.40b showsthe effectof a wedgewhosevelocity is intermediate betweenthat aboveand below it. The waveshape clearly indicatesmore than one reflectorwhen the wedge thicknessexceedsl\14 (12 ms). Figure 6.40c showsa wedgewith a velocity different from that of the surrounding material. The waveshapeis nearly constantbelow a thicknessof \/4, wherethe amplitude is at a maximum becauseof constructiveinterference(tuning;see$6.4.3).Note that the wedgestill produces a significant reflection when it is appreciably thinner than the resolvablelimit, and a bed only tr/20 to \/30 in thicknessmay be detectablealthough its thicknesscannot be determined from the waveshape.For wedge thicknessless than tr'/4,the waveshape is the derivativeof that for a single interface ( s e e$ 1 5 . 1 . 7 ) . Similar resolution considerationsapply to structural features.Figure 6.41showsa seriesof faults with varying amounts of throw, the fault being quite evident when the throw is \/4 or larger. Obviously,the ability to resolvedependson other factors also, such

When a bed embeddedin a medium of differentproperties is 1/4 wavelength(\/4) in thickness,the reflections from the top and baseof the bed interfereconstructively (as in fig. 6.40c) and the amplitude increases,an effect calledtuning.Tuning is important in the analysisof hydrocarbon reservoirsand other thin-bed interpretation situations.A thin bed is defined as a situation where the aggregatethicknessof is lessthan \/4' bedsunderconsideration Figure6.42 showstiming and amplitudemeasurementsfor the thining-wedgesituationsshown in fig. 6.40.Where V.) V.> /,, as infig.6.42a,the destructive interferenceat tr/4 producesan amplitude minithe amplitudegivesapmum. For largerthicknesses, proximately the correct reflectivity of the interfaces of the sidelobes with successive althoughinterference time Peak-to-peak oscillation. some causes wavelet greaterthan \/4 give for thicknesses measurements although the approximatelythe correct thicknesses produceminor errors. sidelobes successive The more common situationencounteredin reser($14.4)is shown in fig. 6.42b,where voir geophysics at l/4 V.: V, * V..Then constructiveinterf-erence maxlmum. amplitude produces tuning a wavelength Trough-to-peaktime measuremelltsgive approxifor thicknesses mately the correct grossthicknesses sidelobes greaterthan \/4 (although the successive produceminor errors)but no informationfor thicknesseslessthan \/4. Thin-bedthicknessinformation below can be obtainedfrom amplitudemeasurements 3\/16 thickness.The amplitude thicknessgraph is nearlylinearbelowabout\/8, but the amplitudeis relto thicknessin the tuning vicinity. ativelyinsensitive waveletsand mapsthe peak If one useszero-phase and troughthat indicatethe top and baseofthe wedge in fig. 6.42b,the arrival times give correctvaluesin the thick-bedsituationbut not in the thin-bedregion. The observedpeak and troughcan comeno closertogetherthan \/4, so that for a thin bed, they effectively push each other apart, giving arrival times that are too early fbr the top and too late fbr the bottom of t h e b e d . T h i s o b s e r v a t i o ni s i m p o r t a n ti n r e s e r v o i r studies. geophysics Meckeland Nath (1977)calculatedthat, for sands embeddedin shale,the amplitudewould dependon the net sandpresentprovidedthat the thicknessofthe is lessthan \/4. Mahradi (1983)verientiresequence fied this using physicalmodels(fig. 6.a3).For gross thicknessesless than tr/4, waveshapesare the same and amplitudes(fig. 6.43f)lie on the samecurveas in fig. 6.42b,whereasfor grossthicknessesgreaterthan changeand amplitudesno longer lie tr./4,waveshapes for fig' 6'43f (Note: The measurements on this curve.

INFLECTIOT\I POINTS 2ro

RESOLVED

RAYLEIGH'S CRITERIOf\I

UNRE9LVED

+ DECREASIIJG IMACESEPARATIO\I (a) TWO -

26

24

22

W A Y L A Y E B T H I C K N E S S( M I L L I S E C O N D S ) 20

18

16

14

1t

1n

Q

c

A

1

A

.n

E UJ

=

100

a--

TWO - W A V L A Y E R T H I C K N E S S( M I L L T S E C O N O S )

26 24 22 2 0 1 8 1 6 1 L ' t 2

ln

a

A

a

s uJ 1On F-

(c) Fig. 6.40 Rcflectionsillustrating vcrtical resolution. Zero_ phasc sinc wavelets; thickness of tr/4 corresponds to 12 ms. ( A f t e r K a l w e i t a n d W o o d , l 9 8 l : 1 0 . 1 89 . ; 1 a yl i l u s t r i . r t i n rse s o l u t i o n , ( b ) r e f l e c t i o n sl i o m s i n g l ei n r e r l l c e t U p p e r r e t l e c t i o n ya n d

w e d g e o fi n t e r m e d i a t e v e l o c i t y ( V , >V . > V , o r V , < V . < V , ) , and (c) the same as part (b) except the wedge is en-rbeddedin a medium of different velocity (V, - V, * V.l.

+r

rll

I

E

E

a

F

)l

+r

+r

I

I

I

I

'({

'{

I

( I

|l

I

F-is.6.41

Reflection from a faulted reflector, with the fault throw indicated as fractions of the dominant wavelength.

@ 6

U z

z

Y

F 6 F O 2 z

o E 3

' oF

c o

SU ti

<

t

2

s

I 7^ o 1

!

Eq

E E

o

4

gJ

> =

;

] o ' F

TWO - TVAV TRUC TXICKN€SS lxtLltsEcoa{osl

(a) Fig. 6.42 Amplitude and timing measurements for wedges shown in Iig. 6.40. The interferencemechanism is shown above the diagrams. Zero-phase sinc wavelets; the horizontal dashed (dotted) lines indicate the amplitude and traveltime as if inter-

TVO-WAY TRUE THICXNESS

(rrLLrsEcoiosl

(b) lerence is not involved. (After Kalweit and Wood, 1982: 1043.) (a) Case where Z, > V, ) V, and (b) where V.: V, * V..

a

ATTENUATION weremade at the center of eachportion in figs. 6.43a to 6.43e to avoid distortions becauseof diffractions from the discontinuities.) 6.4.4 H orizontalresolution The Fresnelzone (96.2.3)is often taken as limiting horizontal resolution on unmigratedseismicdata although other factors such as signal/noiseratio, trace spacing(sampling),three-dimensionaleffects,and so on, also affect how far apart featureshave to be to be distinguishedas separatefeatures.Note in fig. 6.15 that there is little evidenceof reflector shape(that is, that the reflectorsare flat) when they are lessthan one Fresnelzone wide. Resolutionon migratedsectionsis difficult to quantify becauseit dependson many factors,especiallythe presence of noise.Migration(S9.13) can be thoughtof ascollapsingthe Fresnelzones,and hencethe Fresnelzone sizecannot be usedas a criterion for horizontal resolution on migrated sections.Ordinary migration collapsesthe Fresnelzone only in the direction of the migration so that (unlessthree-dimensional migration is performedon 3-D data) correction is not made for contributions perpendicularto the line. One of the most important factors is the quality of the unmrgratedsection;migration rearrangesthe noise as well as reflections,creating what is sometimescalled migrationnoise. Actual migration is performed on sampled data (sampledspatially,that is, at discretegeophonelocations, as well as at discretetime intervals).Spatial (99.1.2b; aliasingconsiderations seealsofig. 6.2)limit the angle of approach, which in turn limits the amount of dip that can be migrated. The sampling theorem dictates that at least two samplesper apparentwavelengthmust be obtainedin order to recognizefeatures,even with perfect data. Thus, for example,to recognizea stream channelon a horizonslice(fig. 12.16)generallyrequiresbin srzes no largerthan 1/3or l/4 the channelwidth. Horizontal uncertaintyalwaysexceedsvertical uncertainty, often by a factor of at least 2. Schneider (1978)gives an exampleshowing that 5Vovelocity error smearsthe position of a discontinuityover a horizontal distanceequal to 5Voof the depth; local velocitiesare usuallynot known betterthan this. 6.5 Attenuation 6.5.I Atrenuutiunme(huni.w.t The amplitudesof eventson a seismicrecord depend upon a multitude of factors (fig. 6.aq. Someof these factors(for example,recording/processing) are within our control. The effectsof otherscan be estimatedand then compensatedfor. Still other factors affect data with about the sametraveltimesin about the sameway and thus do not introduce significant trace-to-trace differences,the main factor on which interpretational decisionsare based.

t77 Divergenceis usuallythe major factor causingtimedependentamplitudechanges(see$2.7.3).The energy spreadsout so that the wavedecreases in strengthbut the total energyin the wavefielddoes not change.If the medium were homogeneous,the amplitude weakening would be inverselyproportional to distance,or Zt,' however,becausevelocity generallyincreaseswith depth, raypath curvaturemakes the wave spread out more and thus makesthe decreasein amplitudelarger. Newman (1973) showed that, for parallel layering, the amplitude decreasedepends approximately on llV?^,t, and Hardage(1985)showedthat this facror is appropriatefor observeddata (fig. 6.45). Absorption (52.7.2)causei wave energyto disappear by converting it to heat. However,like dispersion, most of the factors affecting the amplitude of wavesas they travel through the earth (partitioning at interfaces[chap.3], interferencewith other wavessuch as peg-legmultiples [96.3.2b],and diffraction or scattering) redistributethe waveenergyrather than cause it to disappear.Sometimescompensationfor these various factors is approximatedby multiplying by an empirical exponentialfactor. In general,seismicamplitude decreasesexponentially with time,as shownin fig.2.25.Higherfrequencies are attenuatedmore than lower frequenciesso that the spectrum of a seismicwaveletchangeswith time (fig. 6.46).Hauge (1981)studiedcumulativeattenuationin a largelyclasticsection(fig. 6.47)for VSP data.Spencer(1985)concludesthat attenuationmeasurements are not promisingas a diagnosticof lithology becauseof the intrinsic scatterproducedby pegleg multiple interference. Unlike most of the effectsin flg. 6.44, which are generallyunderstood,the basicmechanismsby which elastic-waveenergy is transformedinto heat are not clearlyunderstood.Toksozand Johnston( I 98I ) summarizedthe stateof our knowledgeabout attenuation and absorption.Variousabsorptionmechanisms have beenproposed(White, 1965,1966)but none appears adequate.Internal friction in the form of sliding friction (or stickingand sliding)and viscouslossesin the interstitial fluids are probably the most important mechanisms, the latterbeingmore importantin highpermeability rocks. Other effects,probably of minor significancein general,are the loss when part of the heat generatedduring the compressivepart of the waveis conductedaway,piezoelectricand thermoelectric effects,and the energyusedto createnew surfaces (of importance only near the source). Many of the postulated mechanisms predict that, in solids, p should depend upon frequency;howeveq Q appears to be independentof frequency(that is, 11is directly proportionalto frequency;seeeq. (2.1l7)). In liquids, Q is inverselyproportional to frequency.The loss mechanismin rocks must be regardedas an unsolved problem (Aki and Richards,1980:156-7, 169-70). Often, no distinction is made between "attenuation" and "absorption."Becauseof difficultiesin measuringabsorptionand also becausethe quantity of in-

(a)

Net

0.0r4

lh€knessl Totalthickness:

0.02s

0.v2

0.056

0.070

0.052

0.093

0.r32

0.172

wavdengfih

1.0

-

Not thrckness: Totalthicknessl

0.043

0.086

0.129

0.172

0.166

0.289

0.412

0.215 0,535

wavdenglh

(e) Fig. 6.43 Reflections from interbedded lithologies Net and grossthicknessesare given in terms of the dominant wavelength' iFrom Sheriff, 1985, after Mahradi. 1983.) (a) Reflections from plates of varying thicknessesmeasuredas fractions of the domtnant wavelength. (b and c) Reflections where lithologies alter-

nate. (d and e) Reflections from beds of different thicknesses' (f) Graph of amplitudes versus net thicknesses,with asterisks indicating points for which gross thicknessesare greater than r/4; A, daia from part (a); O (b)r r (c); X (d); V (e); O, A' f' * from other models.

Nct

thicknoss:

0.157

Totelthickness:

0.2

0.3

(d) Net thicknessl

Totalthicknoss:

0.8

0.9

uJ

Q.;

F@ = o-

0.08

0.16

0.24

N E T T H I c K N E S (SX ) (D

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

180 Geophone Superimposed notse

snsitivity

Instrument balance

Table 6.1 Absorptionconstants for rocks

L f Peg-leg multiples

from thin reflectors

ariation of reflection angle

D(dB): rr

a

and coupling

20-200 70-130 20-'10 50-200 135 190 5-50 200-400 75 300

Sedimentary rocks Sandstone Shale Limestone Chalk Dolomite Rocks with gas in Pore sPace Metamorphic rocks Igneousrocks

0 . 1 60 . 0 2 0.04 0.02 0.16,0.05 0.06-0.02 0.02 0.02 0.630.06 0.02-0.01 0 . 0 40 . 0 1

(FromSheriff'1975) amplitude' affecting Fig.6.44 Factors terest is usually the net decreasein wave amplitude, measurementsare often made of total attenuation without regardto its causeand the resultsusedto determinea valueof 1 in e9.Q.ll0) (seefig' 2'25)' Although this may be a usefulmethod of treatingattenuation, it has no proper mathematicalbasisbecausethe attenuationdue to partitioning, peg-legmultiples,and so on is not a continuousfunction of distance,as requiredby eq. (2.1l0). s 6.5.2 Absorp t ion measurement Attenuation is due to both absorptionand a number of more or less predictablefactors as describedin Q6.5.1.In the laboratory, measurementsare usually made of absorptionbecausethe other factors can be calculated(at least approximately);however,laboratory measurementsare invariably made at high frethe quencies (becauseof scaling requirem^ents ^for 384) 1990: Sheriff, and Geldart, Telford, model; see and so have doubtful significanceunder actual field conditions (becauseabsorption 11increaseslinearly with frequency;seewhat follows)' In field measurementsof absorption,the effectsof partitioning and other significantfactors must be aliowed for to obtain meaningful absorption values' Difficulties in achievingthis have resultedin wide divergencein absorptionmeasurements'Measurements of ibsorption havebeensummarizedby Attewell and namana (t966), Bradleyand Fort (1966),and Toksoz a n d J o h n s t o n( 1 9 8 1 ) . : Attewell and Ramanafound a best-fitvalue of 1 authors, 26 from values of average the 0.2 dB/km for and Waters(1987:33) givesa fairly extensivetable of tn Q values.The ranges of values are summarized one-half to be appear 6Ut. O.t. Q valuesfor S-waves to one-thirdthosefor P-waves.Tullos and Reid ( 1969) report measurementsin the first 3 m of Gulf Coast : 0'24) but seiiments of n : 13 dB/wavelength(Q m (Q : 300 next 0.15 to 0.36 dB/wavelengthfor the Pierre in the measurements 20 to 9). Often-quoted = 0'39 dB/km (1958) were al. et McDonal T by shale : 3'3dB/km f o r P - w a v e (sQ : 3 5 , E : 0 ' 9 ) a n d 1 formation in massive for S-waves;the Pierreshaleis a velocity P-wave a with m thick 1200 Colorado about

50 UJ

o z

UJ I E UJ

620 -) I OE lrl

ro E o t!

o o r 6 U

o o F

o

UJ u a UJ

z =

l

o.I

t

j

o.2 0.3

o N E - W A YT I M E( S ) Fig. 6.45 Gain needed to overcome spherical divergence for VSP data. The slope of the line on the log log plot is nearly { i . 1 n r o m H a r d a g e , 1 9 8 5 :1 7 3 . )

of 2330 m|s. Q is generallyindependentof amplitude for strains less than l0 a, which coversvirtually all situations. seismic-wave Experimentalevidencesuggeststhat the absorption coemcient 1 is approximately proportional to frequency,that is, r1tris roughly constantfor a particular rock. Such an increaseof absorptionwith frequency ($6.5.1)providesone mechanismfor the observedloss of high frequenciesand the changeofwaveshapewith distance. Peg-leg multiples ($6.3.2b) and possibly changes.In other phenomenaalsoproducewaveshape interbeddedsections,the loss in amplitudebecauseof peg-legmultiple effects(fig. 6.30d)appearsto be comDarableto that due to absorption.

SHAPE OF THE SEISMIC WAVELET

I

E

-20 E

Frequency, Hz Fig.6.46 Change in wavelet spectrum from a VSP study. Both curves are normalized with respect to the highest amplitude. ( F r o m B a l c h a n d L e e , 1 9 8 4 :1 6 . )

6.6 Shape of the seismic wavelet 6.6.I De.sircdwavcletcharacteristit's An interpreterwould like to have seismicsections show simpleone-to-onerelationsto interfacesin the earth and as much detail as possible(maximum resolution), that is, sectionswhere beddingcontactsare sharply imaged at their correct locations, with no noise to confuse matters. To achieve short, sharp events requires a broad spectrum with good highfrequencycontent. To show it at the correct location requiresmigration and knowledgeof the embedded waveshape. To showthe contrastsat interfaces,amplitude valuesmust be faithfully preserved. If we think of a seismicwaveletas resulting from the superpositionof many harmonic wavesof different frequenciesand amplitudes (Fourier synthesis concept), we see that cosine waveswith zero phase shift will have maximum constructiveinterferenceat I : 0, thus producing the maximum possibleamplitude there.At certain other valuesof l. the waveswill add up to give smaller peak amplitudes but the broader the band of frequenciesincluded,the farther one has to go from r : 0 for theseto achieveappreciable amplitude. Higher frequenciesin the bandwidth are also necessaryto producea sharp peak. Thus, the desired waveshapeis best achievedwith a narrow zero-phasewavelet(fig. 6.48a)with minimal sidelobes to interferewith other events. Figure 6.48showshow the waveshapechangeswith the bandwidth characteristics.Note the increased

l8l magnitudeof the central peak comparedto any other half cycle and the increasedsharpnessof the central peak as the bandwidth widens (figs. 6.48a to 6.48e). Waveletsdo not changevery much as bandwidthsincreasebeyond about 2.5 octaves.Waveletsbecome leggy as the bandwidth slopesbecome steeper(figs. 6.48fto 6.48h).Two waveletshavingthe samespectral shape and number ol octavesbandwidth but whose spectraare displacedfrom each other along the frequency scale have the same waveshapes(exceptfor time scaling);the one with the lower frequenciesis simply broader in the time domain. Waveletshaving the samespectralshapeand bandwidth measuredin hertz rather than octaveshavethe sameenvelopebut differing number of cycleswithin the envelope(figs. 6.48i and 6.48j). In acquisition,we try to achieve higher frequenciesand broader bandwidths,but absorption and other mechanismsusually limit energy aboveabout 60 Hz. Most of the natural mechanismsthat affect the shapeof real wavelets($9.2.3)are minimum-phaseor nearly so (see Sherwood and Trorey, 1965). A minimum-phase wavelet(915.5.6a) is causal(that is, it is zero for negativetimes)and has the energyconcentrated in the early part ol the wavelet.Real wavelets are also causaland the first detectablepeak or trough is alwaysdelayedfrom the onsetof the waveletso that the picking and timing of arrival timesare alwayslate. Furthermore,as arrival times increase,the increased attenuationof the higher frequenciescausesthe spectrum to shift toward the low frequencies,so wavelets build up more slowly,and the delaysbetweenreflecCorrectcomtion onsetsand their detectionincrease. pensationfor delaysis very dilicult to achieve. The embeddedwavelet($9.2.3)after processingis sometimesapproximatelyminimum-phase,but often has a nearly constant-phasespectrum.Most displays in 1994attemptto achievezero-phasewavelets(whose phase spectra are identically zero and that are not causal).Antisymmetricwavelets(whosephasespectra are identically 90o; see fig. 6.48b) are also encountered frequently. The SEG standardpolarlly convention(fig. 6.49)for minimum-phasewaveletsis that, for a positivereflection (a reflectionfrom an interfacewherethe acoustic impedance increases),the waveform begins with a downkick, representedby negativenumbers,this has a historical basisand is almost universallyagreedto. For a zero-phasepositive reflection,the waveletcentral point of symmetryis a peak representedby positive numbers;a minority usethe oppositeconvention. Displays sometimesshow the opposite of the forepolarity). going (SEG negativepolarity or reverse 6.6.2Ricker v:avelet The embeddedwavelet($9.2.3)is often convertedto a zero-phase equivalent in processing ($9.5.9 and 15.5.6d).The embeddedwaveletis made symmetrical and the time scaleis shifted(but not alwavscorrectlv)

N

I E!

DOWNHOLEPHONE

!

z I F

) zul F F

ut

0.4 0.3 0.2

0.0

F

5 l = l o

' .';r.-.:.tJ. jai'1""'t*"r'v'''' .4, ;:..1. .

0.1

{.il

i

TI

f tl

tlt

tI

.l ,0

rl

IIIT

I

t.t

4000 DEPTH (ff)

Fig. 6.47

Cumulative attenuation as a function of depth Silt-sand intervals are shaded. (From Hauge, 1981

Time(ms)

o Frequcnc'y (Hz)

(a)

.,.i-=-l--:

Fr6quency (Hz)

Froquencf (Hz)

(b)

(c)

o

-l

AilA ^T . tF"^l/\lll\ ? " _F , , , V , ,V, ,1

Er

, Time(ms)

Timc (ms)

::l

Froqu.ncy(Hz)

FrequenryHz)

(0,

G),

Frequonc-y(Hz)

Fig. 6.48 Impulses filtered with various bandpasses(After Yilmaz. 1987: 23 4.) (a to e) Changing bandwidth by increastng the high-frequency cutoff; bandwidths are approximately 1, 1.5'

(h)

Frequonay (Hz)

o

Froquoncy (Hz)

.0)

2,2.3, and 2.6 octaves. (f to h) Changing filter slopes; slopes j) are approximately 120, 60, and 24 dB/octave' (i and Shifting frehigher to interval frequency same bandwidth containing the quencies.

NOISE

183

.".'r -..! + | Normal polarity

Rwerue polarity

(a)

Nomal polarity

t

ll

Reverse polarity

l

(

r

rr r

t

l

l

r

r

'

I

l

(

)

l

l

t fl

(bt

I

l

Fig. 6.49 Standard polarity. (a) For a positive reflection. a minimum-phase wavelet begins with a downkick, and (b) the center of a zero-phasewavelet is a peak.

so that the waveletcenter indicatesthe arrival time. Conversionto a zero-phaseequivalentdoes not solve problemswith time-varianteffects. The most common zero-phasewaveletis the Ricker v'avelet(Ricker,1940,1944,1953a), expressed in the time domain (fig. 6.50a)as f(t)

:

(l

-

2n2v2rt2)e

-6vrt)2,

(6.16)

or in the frequencydomain (fig. 6.50b)as F(v) : (2l1ln)@)1v' ,)et't'ut2,f(u) : 0,

(6.n)

where/(l) ++ F(u), and v, is the peak frequency(see problem 6.21). The distance between flanking side lobesin the time domain, To (fie.6.50a),is 7,,: {l6trttv,,. Also, Q : T,,,1ll 6.7 Noise 6.7.I Typeso.fseismicnorse The reliability of seismicmapping is strongly dependent upon the quality of the records.The quality of seismicdata variestremendouslyfrom areaswhereexcellent reflections (or refractions) are obtained to areasin which the most modern equipment,complex field techniques,and sophisticateddata processingdo not yield usabledata (often called NR areas,that is, "no reflections"). areasof In betweentheseextremes lie most areasin which usefulresultsare obtained.but

the quantity and quality of the data could be improvedwith beneficialresults. We use the term signalto denote any event on the seismicrecord from which we wish to obtain information. Everything else rs noise, including coherent eventsthat interfere with the observationand mearatio, abbrevisurementof signals.The signal-to-noise ated S/N is the ratio of the signal in a specifiedportion of the record to the total noise in the same portion. Poor records result wheneverthe signal-tonoiseratio is small;just how small is to someextenta subjectivejudgment. Nevertheless,when S/N is less than unity, the record quality is usuallymarginal and further. deterioratesrapidly as the ratio decreases Seismicnoisemay be either(a) coherentor (b) incoherent. Coherentnoise can be followed acrossat least a few traces;incoherentnoiseis dissimilaron all traces, and we cannot predict what a trace will be like from a knowledgeofnearby traces.The differencebetween coherent and incoherent noise is often a matter of scale and if we had geophonesmore closely spaced incoherentnoisewould be seenascoherent.Nevertheless,incoherentnoiseis definedwith respectto the records being usedwithout regardfor what closerspacing might reveal. Incoherentnoiseis often referredto as randomnoise (spatially random), which implies not only nonpredictability but also certain statisticalproperties;more often than not the noise is not truly random. (It should be noted that spatial randomnessand time randomnessmay be independent;the usual seismic trace is apt to be random in time becausewe do not know when a reflectionwill occur on the basisof what the trace has shown previously,with the exceptionof multiples. Coherentnoiseis sometimessubdividedinto (a) energy that travelsessentiallyhorizontally,and (b) energy that reachesthe spread more or less vertically. Another important distinctionis between(a) noise that is repeatable, and (b) noisethat is not; in other words.whetherthe samenoiseis observedat the same time on the same trace when the sourceis repeated. The threeproperties-coherence,traveldirection,and repeatability- form the basisof most methodsof improving record quality. Coherent noise includessurfacewaves,reflections, or reflectedrefractions from near-surfacestructures suchas fault planesor buried streamchannels,refractions carried by high-velocitystringers,multiples,and so on (Olhovich,1964).All of the precedingexcept multiplestravel essentiallyhorizontally and all are repeatableon successive sourceactivations. Incoherent noise, which is spatially random and also repeatable,is due to scatteringfrom near-surface irregularitiesand inhomogeneitiessuch as boulders and small-scalefaulting; such noise sourcesare so small and so near the spreadthat the outputs of two geophoneswill only be the samewhen the geophones are placedalmost sideby side.Nonrepeatablerandom noisemay be due to wind shakinga geophoneor caus-

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

184 t

r

l

o

l

r

'

equivalentto frequencyfiltering (fig. 6'51)' In this operation, low-frequencycomponentsmainly interfere constructively,whereas high-frequencycomponents tend to interferedestructively.This type of summing is apt to occur in the ground mixing of geophones within arrays or sourcesin sourcearrays,but it also occursin vertical stackingand other typesof stacking in processing.

l

l

6.7.3 Methodsof'attenuatingnoise

(b) Fig. 6.50 Ricker wavelet. (a) Time-domain representationand (b) frequency-domain representatton.

ing the roots of trees to move. generatingseismic waves,stonesejectedby a shot and falling back to the earth near a geophone,a personwalking near a geophone.and so on. 6.7.2 Stut'kingt()ull(nuul('rundomnoi'se If we add severalrandom noisestogether,therewill be somecancellationbecausesomewill be out of phase with others.Assumethat we have n geophones,each of which is respondingto coherentsignalS but has on it' A measurement randomnoiseN, superimposed -r,will then be r,:

S + N,'

The averageis our best estimateof the signal and we identify the standard deviation o with the noise, so that

s=;:llx.

N=o,

o,:12{r,- ")': )>*; The signal-to-noiseratio, S/,{ is thus

s:i: N

o

t (llntt' )(\l{i)tt1

n"-x

(rlf ),/l

(6.18)

As n becomeslarge,o approachesa limit that depends on the statisticaipropertiesof the noise; hence' for random noise, the signal-to-noiseratio varies as /'rlr for rularge. Sumniing a number of identical traceswhere there are small random timing differencesamong them ls

Becausethere are many types of noise,various noiseattenuatingmethods are employed.All are basedon noiseand of the differencesbetweenpropertiesof the"signal" is somesignal.Inasmuchas the nature of and signal both properties of what subjectiveand the attenuation noise known, not completely nolse are cannot be comPletelYobjective. Noise attenuationbeginswith the field recording' To the extent that noise has appreciableenergyoutsidethe principalfrequencyrangeofthe signal,it can be attenuatedby limiting the frequenciesrecorded' Very low-frequencycomponents(suchas high-energy ,u.iu". wavesrich in low frequencies)may be filtered out during the initial recordingprovided the low frequenciesare suliciently separatedfrom the reflection fiequencies.However, if the spectrum of the noise ou.ilupt the signal spectrum,then frequencyfiltering is of limited value in improving record quality' The dynamic rangeof field instrumentstoday is usually sufficiently wide that often the only low-frequency filtering used in the field is that resulting from the limited low-frequencyresponseof the geophones' filteringemLikewise,often the only high-frequency ployed is that required to prevent aliasingin digitizing. Cancellationof random noisedoes not place any restrictionson geophonelocations(exceptthat they cannot be so closetogetherthat the noiseis no longer spatially random). If we connect together, for exzrmple,16 geophonesthat are spacedfar enoughapart thai the noise is spatially random but still close enoughtogetherthat reflectedenergytravelingalmost in phaseat all 16 geophones' verticallyis essentially the sum of the l6 outputswill havea signal-to-noise ratio four times greaterthan the output when the geophonesare placedsideby side.If, on the other hand' we are attenuatingcoherentnoise and the 16 geophones are spread evenly over one wavelengthof a coherent-notsewavetrain (for example' ground roll)' then the coherent noise will be greatly reduced (see considerations $8.3.6 and problem 8.6b). Similar sources' multiple of arrays of use the to apply 'The contributionof noisecoming from the sideof the line hasgenerallybeenunderestimated'We cannot deal properiy with data arriving from off to the side of thi line unlesswe recorddata to the sideof the line' Areal arrays ($S.3.8)are sometimesused effectively for attenuatinghigh-angleoff-the-line noises'Major

P R OB L E M S

185 versally used is very effectivein attenuating several kinds of noise.The summationtracescompriseenergy from several sourcesusing different geophone and sourcelocations.The field techniquewill be discussed in $8.3.3and the processing(which is usually done in a processingcenterrather than in the field) in $9.10.4. A number of other noise-attenuatingtechniques (such as apparent-velocityfiltering) are also applied in processingand describedin chap. 9. In fact, most of the operationsdone in seismicprocessinghave the attenuationofnoise as their principal objective.Their application has the advantageof trial and error and subjectivejudgment is usually a factor in deciding which processesto employ and which parametersto vary.

F'ig. 6.51 Filter effect of timing errors in stacking. The numbers on the curves are standard deviations of the timing differc n c e sa m o n g t h e t r a c e ss t a c k e d .

noise attenuationresults from 3-D recording and processing. Noise can also be attenuatedby adding together traces recordedat different times or different places or both. This forms the basisof severalstackingtechniques,includingverticalstacking,common-midpoint stacking,uphole stacking,and severalmore complicated methods.The gain in record quality often is largebecauseof a reductionin the levelof both random and coherent noise. Provided the static and NMO correctionsare accuratelymade, signal-tonoiseimprovements for randomnoiseshouldbe about 5 (or l4 dB) for 24-fold stacking. Vcrticalstacking involvescombining together several recordsfor which both the sourceand geophone locationsremainthe same.It is extensively usedwith weak surface energy sources and many marine sources(see$7.2.4and 7.4).Verticalstackingusually impliesthat no trace-to-trace correctionsare applied, but that correspondingtraceson separaterecordsare n.rerelyadded to each other. The effect, therefore,rs essentially the sameas usingmultiplesourcessimultaneously.In difficult areas,both multiple sourcesand vertical stackingmay be used.In actual practice,the surfacesourceis moved somewhat(3 to l0 m) between successive recordings.Up to 20 or more separate recordsmay be vertically stacked,but the stacking of many recordsbecomesexpensiveboth in field time and in processing,whereasthe incrementalimprovementbecomessmall after the first few. Vertical in subsestackingis oftendonein the field,sometimes quent processing.Marine vertical stacking rarely involvesmore than four recordsbecause,at normal ship speeds,the ship movesso far during the recordingthat the data are smearedwhen stacked;smearingmeans that the changesin the reflectingpoints affect the arrival times so much that the signal may be adversely affectedby summing (the effect is similar to using a very fargegeophoneor sourcearray). The common-midpointmethod that is almost uni-

Problems 6.1 In table 6.2, classifydifferent types of eventsand noiseon the basisof commonlyobservedcharacteristics. 6.2 A salt dome is roughly a vertical circular cylinder with a flat top of radius400 m at a depth of 3.2 km. If the averagevelocity abovethe top is 3.8 km/s, what is the minimum frequencythat will givea recognizable reflectionfrom the dome? 6.3 (a) Use Fermat'sprincipleof leasttime to derive the law of reflection($2.7.5).(Hint: Expressthe traveltime for the reflectionSMR in fi9.6.52in termsof the variabler, then set dtld-r equal to zero.) (b) Repeatpart (a) for the refractedpath SMQ. (c) Repeatparts (a) and (b) for reflectedand refracted convertedS-waves, thus verifyingeq. (3.1). 6.4 Redrawfig. 6. l6b for a planewaveincidenton the reflector,and explain the significanceof the changes that this makes. 6.5 (a) Show that the slope of the diffraction event with sourceS, in fig. 6.23bapproaches+ llV for large x. (Hint: Expandthe expression in eq.(6.I I ) for r >> h.) (b) What is the slopeof the asymptotefor fig. 6.23d? 6.6 Assume that fig. 6.34 shows relative amplitudes correctly (divergencehaving been allowed for). The water depth is 420 m and the velocity below the sea floor 2590m/s. (a) lf the reflectioncoefficientis maximum at the critical angle,on what traceswould you expectthe maximum amplitude for the first, second,third and fourth multiples? (b) What should be the ratio of the amplitude of the successive multipleson the short-offsettrace?How do thesecalculationscompare with observations?What unaccounted-forfactors affectthis comparison? 6.7 (a) Given that 0 ( c < f I in eq. (6.15)for the directivity resultingfrom ghosting,discussthe conditions under which the amplitude of rf, is zero. (b) For a source below the base of the low-velocity layeq compare the amplitude and energy of ghosts generatedat the baseof the low-velocitylayer and at

C H A R A C T E R I S T I C SO F S E I S M I C E V E N T S

186 of events Table6.2 Characteristics

:0

g

E

E a

s^ i

b c

i

\

t

o:

9

E

=

F

,

r

:u

6

F

.

:

e

i

E

e ?

:

=.

g s s g €E = : i ! ! a

"

: 9 &E -i

-=

-

-

p E

9

j

Ed : ; t g p ; , . E Z = 9 + at .

e . ! 4 + d d

8 zy C E

E u f : t b: f -

E

. g

&

t

E

&

g

F

i

2

r

e

F A

c

c

2 F =E E+

-

*: gc Ea

H

;o , E

u

F F g . e r - e ' 3 ' . a E H - e= ; * zE := zg z$ Eo of i =

E + s E a

=

8

a

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principle. lawfromFermat's Fig.6.52 DerivingSnell's the surfaceof the ground, given that V, : 1.9 km/s, V, : 0.40 km/s, and that the densitiesjust below and within the LVL are 2.0 and 1.6g/cm3,respectively. (c) Assume that the LVL is jl in thicknessand that 1I : 0.6 dB for the LVL; now what are the ratios of the ghost amplitudesand energies? 6.8 An air gun is fired at a depth of l0 m. The Waveform includesfrequenciesin the range 10-80 Hz' the amplitudesof the l0- and 80-Hz componentsbeing the samenear the source.Comparethe amplitudesof thesecomponentsfor the waveletplus ghostat considerabledistancefrom the source in the directions 0o,

30', 60', and 90'to the vertical. 6.9 Show that eq. (6.15)givesthe directivity diagrams shownin fig. 6.53. 6.10 A multiple reflectionis producedby a horizontal bed at a depth of 1.100km, the averagevelocity being 2.95km/s. A primary reflectionfrom a depth of 3.250 km coincideswith the multiple. (a) By how much do arrival times differ at points 200, 400. 800. and 1000m from the source? (b) If the shallowbed dips l0', how much do the arrival times at 400 and 800 m change?What is the apparent dip of the multiple? 6.ll A primary and a multiple each arrive at 0.600 s at x : 0; the stackingvelocitiesfor them are 1800and Calculateand plot the residual 1500m/s, respectively. NMO for offsetsof 300 n, wheren : l, 2, . . ., after application of the NMO correction for the primary velocity.What is the shortestoffsetthat will give good multiple suppressionfor a waveletwith a 50-msdominant period? 6.12 Pautsch( 1927)showedthat a horizontal or vertical interface could give identical first-arrival curves (fig. 6.5a).Add secondaryarrivals and reflectionsto thesediagramsto show how they can distinguishthe two cases. 6.13 An unlabeledeventcan be seenin fig. 6.38bwith the projectedarrival at x :0 at about 0.60 s; identify it. 6.14 Draw arrival-time curves for the five events shownin fig. 6.55.

PROBLEMS

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6.15 (a) A horizontal refractor at a depth of 1.20km is being mapped along a N S line. The overburden velocity is 2.50 km/s and the refractor velocity 4.00 km/s. The refractor is terminatedby a linear vertical fault 3.50 km from the sourcepoint.Determine the traveltimecurves when the fault strikes (i) E W (ii) N-S, (iii) N30'w. (b) Repeat for the E-W fault for a refractor that dips 10'to the north with the sourcepointto the south. (c) What effectswill the manner of terminatingthe refractor have,that is, how will the amplitude of the reflectedrefractiondependon the dip of the terminating fault? (d) Most commonly a refractor will terminateagainst rock of lower acoustic impedance,but the opposite situation can also happen. What differenceswill this make? (e) Extend the proflle for case(a), part (i), an appreciable distance beyond the fault so as to plot the diffraction from the refractor termination. Assume uniform 2.50-km/smaterial beyond the refractor termination. 6.16 (a) Determine the traveltimecurves for the refraction SMNPQR and the refraction multiple SMNTUWPQR in fig. 6.56. (b) Determinethe traveltimecurveswhen both refractor and reflector dip 8' down to the left, the depths shown in fig. 6.56 now being the slant distancesperpendicularto the interfaceat S.

(c) What happenswhen the reflectordips 3'to the left and the refractor 5' to the left? 6.17 Explain why wavesin fig. 6.40binterferedestructively and in fig. 6.40cconstructivelywhen the wedge thicknessis j\. 6.18 (a) A wavelethas a flat frequencyspectrumfrom 0 to u,, abovewhich the spectrumis zero; show that the Rayleighcriterion givesa resolvablelimit t,, where t, : 0.'7151v,.(Hint: Transform a boxcar spectrum box,,,(u) to the time domain and find the location of the first trough.) (b) Show that the value of t,for a waveletwith a flat spectrumextendingfrom u. to nvr(that is, la octaves wide, where n : 2'") is given by the solution to the equation nx cosnx - sin n,t - -rcos r * sin x : 0. w h e r ex : 2 n v r t , . (c) Solvethe equationin part (b) for m : 3,2, and 1.5 and comparethe relation betweent, and m. (d) Noting that part (a) involvesan infinite number of octaves,how many octaves'bandwidth are required to give nearly the sameresolution? 6.19 (a) Approximately what are the dominant frequenciesfor reflectionsin fig. 8.5 arriving at the right sideof the sectionat about 0.6" 1.2,and 1.8s? (b) If the velocitiesat those reflectorsare 2000, 3000, and 5000 m/s, respectively,what are the resolvable limits (tuning thicknesses)?

EVENTS CHARACTERISTICS OF SEISMIC

188

R€flected diffraction Fig. 6.55

a horst' Reflectionsand diffractions involving

( N 8 ' 3 ' l l Ji s r e 6 . 2 0 D e n h a m ' sh i g h - f r e q u e n cl yi m i t to the lated both to the loss oi high frequenciesand Recsystem' recording the 4"""-i" range ($7.6.1)of of 0'15 absorption bv loss ;;;i;;i;ii.nliwlttr'a

lossbeandhigh-frequencv iiiliiSi.i.zu), spreading. 6'30' fig' in as illustrated multiples ;;;;';l peeliee

recording fate S+ OS asltte dynamic range of the 1 system. "i.ii verify (^l Using the result e az 41 (116\ttze-o-l4d' eq' from follows that eq. (6.l7i for a Ricker wavelet ( 6 .I 6 ) . (b) Show that u. is the peak of the frequencyspectrum. that (c) Show that TJT*: ./: (r.. fig. 6.50) and Tov,: t[6tn. represent 6.22 Selectrandom numbersbetweeni9 to : four Sum 2' S signal a noit. 1{, and add to each stanthe mean' the determine + and N' uut".t of S * noise)' darJdeviation o, and the ratio signal/(signal mean the how Reoeatfor 8, 16, and 32 values'Note lncreases' ofvalues number "onuarga,toward S as the on io* "?pp.ouches a limiting value (which depends see noise; the $15'2'12)' of properties ifr. ttutitii"ul converges and how the ratio signal/lsignal+ noise). given in is numbers l. (A table of random ;;;; app.c.)

References Seismologv" Aki, -i;";;;K., and P. G. Richards 1980' Quantitaive W H' oni urtho,lr, Vols I and II' San Francisco: Freeman. and its applicaAngona, -t'i-o'n F. A. 1960.Two-dimensional modeling 82' to'seis-ic problems' Geophysics'2* 468 Wave attenuation and Attewell, P. 8., and Y. V Ramana lg66 ';;;;i in rocks Geophvsics' frequencv of functions "t i.i.til. 31: 1049 56. Vetticat Seismic Profiling' Balch. A. H., and M. W Lee 1984'

Fig. 6.56

Multiply-reflected refractton'

Boston: lnternaTet'hnique, Applit'utions, untl Cuse Hislorics' tional iluman ResourcesDevelopment Corp' in rocks' In J. J., and A. N. Fort' 1966' Internal friction Bradley, ";;;-;;;;;'"7'inv,ira

constants,S P' clark' ed'' pp l75 e3' q7. Co: GeologicalSocietyoiAmertca' Boulder, dsa vl-"i. NorwayCentralGraben' d'Heur,M. 1992.WestEkofiskField' AAPC,75:.946-68' Bult. Sea. North A" Principles'2d Pro'filing: Hardage,B. A. t985. VertitalSeismit' Press' ed. London:GeoPhYsical from vertical of attenuation Hause.P S. 1981.Measurements 1546' 46: seismicprofiles.Geophysics, seismicmodeling' Hilterman, F. J. 1970.Three-dimensional 37' 1020 35: Geophysics, from threeHilterman, F. J. lgs2 Interpretive 199s911 ^Ji-"ntionut 47: 784-808' modeling'Geophysics' lg85 Depositional Hubbard,R. J.,J. Pape,and D G Roberts tectonicand seouencemapplng as a techniqueto establish. and eviluate hvdrocarbonpotential .;ffi ;;;;it;Ti;'"i*o't ln ser'rmicsy-atlsrufit II' ;';?;r;;; "B;;; ion,in.n'ot margin' eds'' pp 79-91' AAPG Woolv-erton' G. D. and ci. n of PetroleumGeoloAssociation Amencan Memoir 39.Tulsa: glsts. oJ'Optic's' Jenkins,F. A., and H. F White' 1957'Fundamentals New York:McGraw-Hill' resolution R. S., and L. C Wood 1982'The limits of Kalweit, 'oi 41:421 39' t"t6-ptrasewaveletsGeophysics' R' L Sengbush' McDonal, F. J., F. A. Angona, R' L Mills'

REFERENCES

R. G. Van Nostrand, and J. E. White. 1958. Attenuation of shear and compressional waves in Pierre Shale. Geophysics, 23:421 39. Mahradi, 1983. Physical modeling studies of thin beds. M.Sc. thesis,University of Houston. Meckel, L. D., and A. K. Nath. 1977. Geologic considerations for stratigraphic modeling and interpretation. ln Seismic Stratigraphy Applications to Hydrocarbon Exploration, C. E. Payton, ed., pp. 417-38, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Neidell, N. S., and F. Poggiagliolmi. 1977. Stratigraphic modeling and interpretation. ln Seismic Stratigraphy - Applications to Hydrocarbon Exploration, C. E. Payton, ed., pp. 389 416, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Newman, P. 1973. Divergence effects in a layered earIh. Geopftyslcs,38:481-8. O'Doherty, R. F, and N. A. Anstey. 1971.Reflectionson amplitudes. Geophys.Prosp., 19: 430-58. Olhovich, V A. 19fl. The causesof noise in seismic reflection and refraction work. Geophysics,29: 1015*30. Pautsch, E. 1927. Methods of Applied Geophysics. Houston: Minor Printing Co. Ricker, N. 1940. The form and nature of seismic waves and the structure of seismograms.Geophysics,5:348 66.

189 tigraphy - Applications to Hydrocarbon Exploration, C. E. Payton, ed., pp. 3-14, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Sheriff, R. E. 1985. Aspects of seismic resolution. ln Seismic Stratigraphy 11, O. R. Berg and D. G. Woolverton, eds., pp. I 10, AAPG Memoir 39. Tulsa: American Association of Petroleum Geologists. Sherwood, I W. C., and A. W. Trorey. 1965. Minimum-phase and related properties of the response of a horizontallystratified absorptive earth to plane acoustic waves. Geophysics, 3 0 : 1 9 17 . Spencer,T. W. 1985. Measurement and interpretation of seismic attenuation. ln Developments in Geophysical Exploration Methods 6, A. A. Fitch, ed., pp. 73 110. Amsterdam: Elsevier. Stommel, H. 8., and M. Graul. 1978. Current trends in geophysics. In Indonesian Petroleum Association Proceedings, Jakarta. Indonesia. Telford, W. M., L. P. Geldart, and R. E. Sheriff. 1990. Applied Geophysics,2d ed. Cambridge: Cambridge University Press. Toksoz, M. N., and D. H. Johnston. 1981.SersmicWaveAttenuatlon, Geophysical Reprint Series2. Tulsa: Society of Exploration Geophysicists. Tullos, F. N.. and A. C. Reid. 1969. Seismicattenuation of Gulf Coast sediments. Geophysics,34: 516 28.

Ricker, N. 1953a.The form and laws of propagation of seismic wavelets. Geophysics,18: l0 40.

Vail, P R., R. G. Todd, and J. B. Sangree. 1977. Chronostratigraphic significanceof seismicreflections.ln SeismicStrutigraphy Applications to Hydrocarbon Exploration, C. E. Payton, ed., pp. 99 I16, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists.

Ricker, N. 1953b.Wavelet contraction, wavelet expansion, and the control of seismicresolution. Geophysics,18:769 92.

Vetter, W J. 1981. Fbrward-generatedsynthetic seismogram for equal-delay layered models. Geophys Pro,sp.,29:. 163-73

Schneider,W. A. 1978.Integral formulation for migration in rwo and three dimensions. Geophysics,43:49,76.

Waters, K. H. 1987. Refection Sei,smology, 3d ed. New York: John Wiley.

Schoenberger,M., and F K. Levin. 1978. Apparent attenuarion due to intrabed multiples. Geophysics,43:730 7.

White, J. E. 1965. Seismic Waves- Radiation, Transmission, and A ttenuation. New York: McGraw-Hill.

Sheriff, R. E. 1975.Factors affecting seismicamplitudes. Geopi. Prosp., 23,. 125-38.

White, J. 1966.Static friction as a source of seismicattenuation. Geophysics,3l: 333 9.

Sheriff, R. E. 1976. Inferring stratigraphy from seismic data. Bull. AAPG, ffi:528 42.

Widess, M. B. 1973. How thin is a thin bed? Geophysics,38: I 176-80.

Sheriff, R. E. 1977. Limitations on resolution of seismic reflections and geologic detail derivable from them. ln Seismic Stra-

Wood, R. 1961. Phy.sical Oplics. New York: Dover.

Ricker, N. 1944. Wavelet functions and their polvnomials. Geophysics,9:314 23.

Yilmaz, O. 1987. Seismic Data Processing.Tulsa: Society of Exploration Geophysicists.

7

Equipment

Oveniew Becausethe objectivesof seismicsurveysvarv tremen_ dously,from surveyshavingvery shallowobjlctives to those having deep objectivesin difficult areas, the scaleof seismicequipmentvaries.However,the concepts are generallythe sameand we will not discuss all the variations.Pieuchot(1984)along with Evenden and Stone (1971)provide the principal referencesfor seismicequipment. Of first-orderimportanceis determiningwheredata are acquired (47.1).To locate the traces to be com_ bined in common-midpointstacking,surveyinghas to be more accuratethan formerly required.Land work today often employs electronic distance measurins and Global PositioningSystem(GpS) measu..-.ntrl Locating positions at sea where there are no landmarks dependsmainly on radiopositioningand satellite observations,with reliance on the Global positioning Systemincreasing. Although a multitude of energysourceshave been used at times, most large-scaleland work useseither explosives or vibrators(57.2.2and 7.3.1).The choice of sourceis usually basedon economics.Sourceslocated below the weatheringlayer usually produceless noisethan thoseon the surface,but the cost ofdrilling holesis so high in many areasthat explosivesare not used. Transporting large surface sources prevents their usein someareas;fortunately,drilling is usually easywhere the surfaceis too soft to support surface sources. Air guns ($7.4.3)are used almost exclusivelyfor large marine surveys.The bubble effect(g7.4.2)is often the limiting constraint for sourcesimmersed in water.Severalother types of marine sourcesare also used,especiallyfor shallowsurveys. Seismicwavesare detectedalmostexclusivelyby velocity geophones(97.5.1)on land and pressurehydrophones($7.5.3)at sea. Exceptionsoccur in some marsh and transition-zonework. The characteristics of data recorded by geophonesand submergedhydrophones differ, and matching records where both are employedare discussedin $7.5.5.The use of very largenumbersof recordingchannelshas led to digitization near the geophonesand hydrophones,as discussedin 97.5.2and 7.5.4. Digital versusanalog representationof data is discussedin $7.6.4.Most recording today is digital, but portions of recording systemscontinue to be analog. l9l

The analog display of data (97.6.6)is important in communicatingwith an interpreter.Color displaysare increasinglyusedto facilitateinterpretation. 7.1Determinlng location 7.1.1Land surveying Most types of conventionalsurvey instrumentshave been used in seismicsurveying.plane table and alidadewereextensivelyusedinto the 1960s,and transittheodolitesand chaining continue to be used today. Successivesource points are often measuredwith a wire of appropriatelength,a theodolite(fig. 7.l ) being used to measureelevations,to keep the line straight, and to surveyin side features.The theodolite is used to turn both horizontal and verticalanglesand to give rangesby stadiameasurements. Electromagnetic distance measuring equipment 69 7.2), often called EDM, is widely used today. EDM and GPS (97.1.5)instrumentsare often usedto run in a base survey and to tie to benchmarksand wellheadsevenwherenot usedfor all sourceand seophone locations. Most EDMs use laser diodes that emit a pulse of about 0.9 pm length; the pulse is reflectedfiom the rod, and the round-trip time is measuredto give distancesto about 5 parts in 106(Spradley,1984).Horizontal and vertical anglescan be measuredto about 30 minutes. Laser EDMs have a maximum ranse of about 2 km. For long ranges,a microwave EDM transmits a radio beam toward the "rod," which either reflectsor retransmitsthe beam back to the EDM, where the phases of the reflected and transmitted waves are compared.If the two-way range is (n + k)\, r being an integer and k a fraction, k is measuredto about l/100 wavelength(tr - l0 m). The ambiguity as to the value of n is resolvedby making measurements at several frequencies.A computer calculatesthe range, which can then be read directly. Becausebeaming 10-m wavelengthsrequireslarge antennas,the l0-m wave modulates a high carrier frequency (10 to 35 GHz) that can be beamed. EDM instruments can be equipped with digital readouts giving ranges and vertical and horizontal angles,and the data are recorded on a floppy disk, which becomesthe surveyor's"notebook." It can then

EQUIPMENT

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the time differencebetweentwo signals,which givesa differencein distances;(3) a differencein frequency (4) becauseof Doppler shift, which gives a velocity; and accelerometers; oriented of means by acceleration (5) direction with respectto north, usuallydetermined with a gyrocompass.With velocity and acceleration position is determinedby integration' measurJments, can also be classifiedaccording systems Navigation -way locations are determined: (l) piloting' to the whereinlocation is determinedwith respectto known locations; (2) deadreckoning,wheteinlocations are determined with respectto a known starting point and based known course;una (:) celestialmeasurements on the altitudes of the Sun or stars at known times; with respectto navigationsatellitescan measurements in be included this class. Navigation/positioningsystemshavebeenundergoing rapid technologicaldevelopmentin recent years una tttit continuesin 1994.Prior to the development of atomic clocks,one could not dependon systemsat differentlocationsto be operatingon the sametiming system;the availabilityof relativelycheap' highly accurateatomic clocks has changedthis' Systemerrorssuchas minor changesin velocity becauie of meteorologicor ionosphericuncertaintiesor minor perturbations in a satellite'sorbit limit accuracy; theseerrorscan be removedto a largeextent by translocation,which involvesmeasurementoI appar-

be read into a personalcomputer for automateddata reductton. 7.1.2 Marine Positioning Marine seismicnavigation involves two aspects:(a) placing the ships at a desired position and (b) deie.minl.tg the actual location afterwardsso that the data can be mapped properly.ln assessingthe accuracy of a navigationmethod, we distinguishbetween absoluteand ielative accuracy.Absolute accuracyis important in tying marine surveysto land surveysand in returning to a certain point later, for example' to locatea well. Relativeaccuracyis important primarily to ensureaccuraterelativelocationsof midpoints' Ab-t70 m are usually sufficient' solute accuraciesof whereasrelativeaccuraciesof i5 m are desirable'The accuraciesobtained in a survey (which are usually very difficult to assess)depend upon the systemand equipmentused,the configurationof reference(shore) stations,the position of the mobile stationswith respectto the referencestations,variationsin the propagation of radio waves,instrumentmalfunctioning'opJrator error, and so on. Systemscapable of giving adequateaccuracy under good conditions may not realizesuch accuracyunlessconsiderablecare is exercised at all times (Sheriff 1974).Spradley(1984)discussesseismicmarine positioning in detail' Navigation systemsgenerallymeasuresome of the followiig: (1) tire time betweentransmissionand re(2) ceipt of"a signal, which gives a distance(range);

is replactng a Fig.7.2 Field survey instruments The surveyor theodolite and electronic an with hand left his in ....iu.. CFS up to 5 km with distance meter. The EDM measuresdistances of l(3 mm * 2 parts per million)' the theodolite u".*u"y of arc' -"ur".at horizontal and vertical angles to 3 seconds (Courtesy of Leica Inc.)

I DETERMINING LOCATION ent position changesat a fixed station and using these to correct positionsof mobile stations.Tianslocatron is now usedwith systemsof different types. Redundancy of positioning is highly desirable. Where position is determined simultaneously by different systems,the redundancyprovidesbackup in eventof failuresand a neededcheckon possiblemalfunctioning. The overdeterminationof position also providesan assessment ofthe accuracyactually being achieved.The history of marine geophysicalsurveying includesmany instancesof work lost in whole or in part becausepositions could not be relocated.Although the Global PositioningSystem(g7.1.5)is rapidly rising in prominence,other types of systemsare still usedand henceare describedin the following sectlons. Location uncertainty can be reduced significantly by analysisof the data afterwards.Data acquiredlater in a surveymay be usedto reducethe uncertaintyof data acquiredearlier.Much positioninguncertaintyis systematic,and analysisof the entire body of data may clarify the nature of errorsand permit correcting for them. 7.1.3 Radiopositioning Radiopositioning(or radionavigation)systemsare generallyclassifiedaccordingto their frequencies.In high-frequencyradio systemssuchas shoran, -qeneral, radar, Autotape, and Hydrodist use frequenciesof 3 to 9 GHz (wavelengths of 3 to l0 cm) and achieve good accuracy,but their travel is line-of-siehtand so their range is limited to about 40 km (dep-endrng on antennaheights).Medium-frequency systemssuchas Hi Fix, Decca,Toran, and Raydistusefrequenciesof 1.5to 3 MHz (wavelengths of 100to 200m); they can bend with the Earth's curvature and their range is about 150km. Low-frequency systemssuchas LoranC usefrequenciesof the order of 100kHz (wavelength ol 3 km) and have a rangeof 2000 km; the global Omegasystemoperatesat the very low frequencies ol l0 to l5 kHz (wavelengths 20 ro 30 km). Radiopositioningmethodscan also be classifieddepending upon the type of measurement:(a) systems that measurethe time required for a radio-frequency pulseto travel betweena mobile station and a shore station(examplesare radar,shoran,the rho-rho mode ..f Loran-C), and (b) systemsthat measurethe differ:nce in traveltime (or phase)of signalsfrom two or nore shorestations(theseincludeRaydistrM,LoracrM, Decca,Pulse-SrM, Hi-fixrM,ToranrM,ANATM,ArgorM, :re phasemode of Loran, and Omega).Angular mea:Llrementsare not ordinarily usedin radiopositioning recausedirection cannot be determinedwith enough :recision with antennasof reasonablesize. Radar and shoran are similar in pinctple. Radar jepends upon the reflectionof pulsesby a target, the jistanceto the targetbeingequalto one-halftheprod:.-r of the two-way traveltime of the reflectedpulse =nd the velocity of the radio waves.Shorar,rdiffers

193 from radar in that the target is a shore station that receivesthe pulse and rebroadcastsit with increased power so that the return pulseis strong.Two or more shorestationsare usedand the position of the mobile station is found by swingingarcs as in fig. 7.3. Radar and shoran are high-frequencysystems,radar frequenciesbeing in the range 3000 to 10,000 MHz, shoran in the range 225 to 400 MHz. Because such high frequenciesare refractedonly very slightly by the atmospherethesemethodsare basicallyline-ofsight.With normal antennaheightsof about 30 m, the range for shoran is roughly 80 km. If the shore stations can be located on hills adjacent to the sea, greater rangescan be obtained. By using very sensitive equipment(directionalantennasand preamplifiers),rangesof250 km can be obtained;this variation is calledextended-range, or XR, shoran.The extension ofrange beyondthe line-of-sightappearsto be due to refraction,diffraction, and scatteringfrom the troposphere.In sometropical or subtropicalregions,strong temperaturegradientsin the atmosphererefract the radio wavesso that rangesof 300 km or more can be obtained. The distance between the ship station and each shorestationis normallymeasuredwithin -r25 m (+ 0.2 ps), sometimes within -r5 m. The error in location dependsmainly upon the anglebetweenlinesjoining the shore stationsto the mobile station, as shown in fig.7.3;anglesbetween30oand l50o are usuallyconsideredacceptable. Severaldevicesutilize the same principles as shoran, but usethe higher radar frequencies;they ,,interrogate" a small transponder, a devicethat emits a signal immediately upon receipt of the interrogating signal.Theseinclude RPSTM,MinirangerrM,and TrisponderrM,which use frequenciesaround 9500 MHz; AutotapelM and HydrodistrM,which use frequencies around 3000 MHz; and SyledisrM,which operatesat 450 MHz. The effective range of these devices is strictly line-of-sight,but they are extremelyportable. Their accuracyis often excellent,of the order of 5 m. Loran-C involves the broadcast of a coded se-

Fig. 7.3 Effect of station angle on errors in range-rangeposition determination. 0 : station angle, I : mobile station, .B and C: shore stations. Point ,4 can be anywhere within the "parallelogram" formed by the four arcs. (Note: Range errors are not to scale.)

EQUIPMENT

194 quenceofpulses offrequency 100kHz, the broadcast times being controlled very accurately by atomic clocks.ThJstability of relativelycheapatomic clocks makesit feasibleto carry sucha standardon a seismic ship, so that the instant of signal transmissioncan be determined and hence the range to the transmitter' Such range determinationis called the rho mode, or the rho-rho or rho-rho-rhomodes if rangesare determined to one, two, or three transmitters.Despite the 3-km wavelength,rangescan be determinedto 20 to 100m. Long travel paths may be involved, however'so that minor variationsin the speedof the radio waves because of variations in the conductivity of the ground or moisture in the atmospherecan introduce iizeablepropagationerrors(seeproblem 7'l)' To minimize suih irrors, the systemshould be calibrated in the local area. The shipboardatomic clock may drift slowly,so that the drift has to be checkedevery few days. If two shorestationssimultaneouslybroadcasta radio pulse or coded sequenceof pulses'a mobile station can measurethe differencein arrival times and so find the differencein distancesto the two shore stations. The locus of points with constantdifferencein distancefrom two shore stations(,4 and B infig' 7'4' for example)is a hyperbola with foci at the two stations; thus, a singlemeasurementdeterminesa hyperbola PQ passingthrough the location of the mobile station, R. If the differencein arrival times for a second pair of stations(B and C) is measured'the mobile stationis locatedalso on the hyperbola VW andhence at the intersectionof the two hyperbolas' This principle forms the basis of the phasecomparisonmodes of Loran and Omega,long-range radionavigationsystemsmaintainedby the U'S' Government. Omega is a worldwide system,but its long wavelength(23 to 30 km) and seasonaland diurnal variationsin the ionosphereprecludeachievingaccuracy greater than about I km. Loran-C is available over much of the northern hemisphere,especiallyin American and Europeanwaters.With care,the accuracy of Loran-C phase comparisonsmay be nearly that of its rho-rho mode.Deccais anothersystemgenerally comparableto Loran-C, used mainly in Western Europe. Medium-frequency radiopositioning systems can broadcastcontinuous waves(CW) from severalstations, locations being determinedby comparison of phase.Phase-comparisonsystemsused in seismic exploration generallyoperatein the frequencyband l'5 io 4.0 MHz and haverangesup to 650 km' Referring again to fi,g.7.4,shore stationsA and B transmit tyn"htonout steady continuous sinusoidal signals that are exactly in phase 7t U' !\e midpoint oi baseline AB. A mobile station with a phaseat comparisonmeter will show zero phase.difference MN bisector perpendicular points on the M andat all lf MP : ]f ttre phase-comparisonmeter indicates zero phasedifferenceat P also; if the mobile station -ou"i u*uy from P in such a direction that the phase

difference remains constant' it traces out the hyperbola PQ. In general, a point R moving in such a way that (n = 0, -rl,'r2, t3, ' ' ') RA - RB : nlt, tracesout the family of hyperbolasshown in the diagram. The zo\e between two adjacent zero-phasedifferencehyperbolasis called a lane'lf we start from a known point and maintain a continuousrecord of the phasedifference,we know in which lane the mobile station is located at any given time' By using a secondpair of stations(one of which can be located at the samepoint as one of the first pair of stations) transmitting a differentfrequency,we obtain a.second family of hyperbolas,henceanother hyperboliccoordinate of the mobile station' The accuracyof location with increasinglane width as we go farther decreases from the basestations,alsoas the angleofintersection of the hyperbolasdecreases.Location accuracyis of the ord#of 30 to 100m. If the continuouscount of lanes is lost, however,one could be considerablyoff metersgive only the location as the phase-difference indicatewhich lane' not do and lane i within oosition ih, f..qu.n"y can be changedperiodicallywith consequent changes in the phase at a given position' which can be usedto identify the lane' Atomic clocks are alsousedwith medium-frequencysystemsto allow devices' their useas rang€-measuring Translocationcan be used to improve accuracy this by removingthe effectof propagation.variations; a fixed at in observations variations using involves station to correct those made simultaneouslyat a mobile station.With translocationand at leastone redundant measurement' I to 3 m accuracy can be achieved(Musser,1992)' 7.1.4 Transitsatellitepositioning Shipscan determinetheir location from Transit satellitei in polar orbits 1075km above the Earth' Each satellitetakes about 107 minutes to circle the Earth' being in sight for about 18 minutes horizon to horizon.-Eachiatellite transmitscontinuouswavesof frequencies 150 and 400 MHz. The frequenciesmeasured by a receiver are Doppler-shifted (S7'l '6) becauseof the relativemotion of the satellitewith respect to the ship. Because radiowaves travel at the speed of light ( I/), relativity affects the Doppler-shift equation (seeeq. (7.2)): \f the velocitiesof the source and observerare respectively Vrand Vo, the observed frequencyuo is

+ vo- vs\ v o : v sl v\ v - h * v r l ' where vs is the source frequency. The difference between tlie ship's and satellite'slongitudes and latitudes at closestapproach (seefig' 7.5) are calculatedfrom the Dopplei sttitts.nte satellitetransmitsinformation that gives its location evety 2 minutes' A small computeion the ship combinesthis information with the

195

D E T E R M I N I N GL O C A T I O N

Fig. 7.4

Hyperbolic coordinates for radionavigation system based on measurement of time differences.

Doppler-shift measurements and the speed and courseof the ship to give the ship'slocation. Each satellitecan be observedon four or more orbits each day (exceptnear the equator); hence,with four to five satellites,twenty or more determinations are possibleeach day. Howeveq the satellitesare not uniformly spacedand do not havepreciselythe same orbital period so that sometimesmore than one satellite is visible, whereasat other times, severalhours intervene without any satellite being visible. About "passes"result in two-thirds of the satisfactoryfixes or determinationsof position. Satellitefixes may be accurateto within + 50 m, providedthe ship'svelocity is accuratelyknown (Spradley,1976).The principal disadvantageof Transit satellitenavigation is that it givesno information about position during the interval betweenfixes. Commonly, Doppler sonar ($7.1.6),gyrocompass, and satellite navigation are combined (Kronberger and Frye, l97l), or elseradionavigation($7.1.3)and satellitenavigation.The satellitegivesthe periodicupdating information neededto maintain Doppler-sonar accuracy or to remove ambiguities or propagationerror effectsfrom radionavigation, whereasthe Doppler sonar along with the gyrocompassand/or radio systemsgive the velocity information neededfor an accuratesatellitefix. Translocationcan be usedto imDroveaccuracvof satellitelocation determination.

7.1.5GlobalPositioningSystem(GPS) The Global Positioning,or Navstar, System(Dixon, 1992)consistsof twenty-oneto twenty-four satellites at elevationsof 22 200 km; it is operatedby the U.S. Governmentand it permits determinationof latitude, longitude, and elevationby trilateration. The system (with 25 satellitesin 1994)is extensivelyusedfor geophysicalpositioningin the marine environmentand is also used to set basestationson land. Four satellites are to be equally spacedin each of six orbital planes that make 55oangleswith the Earth'sequatorialplane (fig. 7.6). Each satelliteorbits the Earth in about 12 hours. Each satellite includes four atomic clocks, which provide an ultrastabletiming system.Orbital perturbations are observed by stations in Diego Garcia, Hawaii, Kwajelein, and Ascension,and a facility at Colorado Springs, Colorado, keeps the satellitesat their proper locations and their timing systemssynchronized. Each satellite broadcastson carrier frequenciesof 1575.42(Ll) and 1227.6(L2) MHz (and another for military use). The 50-Hz information is superimposedon the carriersby biphasephaseshifting using *90o to indicatea one and -90'to indicate a zero. The superimposedinformation includes a "handover word," which permits synchronizinga us"almanac" inforer's time with a satellite'stime. and

EQUIPMENT

196

, )DK

////rr/////)

For high angleof maximum Time of closest approach Fig. 7.5

Position fix from transit navigation satellite.(After Sheriff, 1990:259.)

mation, which givesthe satellite'sposition for the next l8 days, correction factors for troposphericanomalies,and other information. Broadcastsfrom the individual satellitesare distinguishablebecausetheir almanac information begins at different time intervals after the handover word. Two broadcast codes are used,a P-codefor military use permitting greateraccuracy than the civilian-usercode. Severalreceivertypes(Burns,1992;seealsofig.7.2) are available,includinghand-heldreceivers.The user's position is determinedby the simultaneoussolution of range information from four satellites.A receiver must searchamong the transmissionsfrom the satellites visibleat any time for the signalsthat will give the best position information. Three satellitesand the userform a tetrahedron,and the most accuratetrilateration resultswhen the tetrahedron'svolume is maximized;observationof the fourth satelliteis requiredto resolvedifferencesbetweenthe satelliteand user time systems.If the userknows a priori which satelliteswill give the best information, the receiver need search only for the signalsfrom them; this speedsup determination of a location.

rl

GPS permitsother measurementmodesin addition to the systemjust described.The phasedifferencebetween the satellitesignalsand the receiverreference can be used to give the differencein coordinatesof successivestations.The Doppler-frequencyshift can be measuredto give locationsin the samemanner as with Transit satellites($7.1.4).The useof two frequencies permits correction for refraction in the Earth's ionosphereand atmosphere.Translocation(dffirential GPS),in which readingsat a fixed secondreceiver up to 500km awayare utilized,can be usedto remove short-term satellite perturbations. Differential GPS geophysicalusagecan achieve2 to 5 m accuracy(Jensen.1992:Musser,1992). GPS readingsgive coordinateswith respectto the satellitecoordinatesystemand haveto be transformed into local coordinates.The accuraciesachieveddepend on the modes of use,that is, the measurement duration, whether the receiveris static or in motion, whetherlocationsare requiredin real time or later for postprocessing,whether absolute locations or only relativelocationsare required,and especiallywhether translocation is used. The U.S. Government warns

DETERMINING LOCATION

t97

Fig. 7.6 Global Positioning System schematic showing satel_ lites in orbit about the Earth. (Courtesy of Wild Heerbrugg Ltd.)

that GPS may be degradedfor securityreasonsto give locationaccuracyof only 50 to 100m. 7.1.6At'oustit'and inertialpositioning Acousticor sonarpositioningusesboth sonar range and frequency-shiftmeasurements.High-frequency acoustictransducers,also called pingers,are used in severalways;they emit acoustic(sonar)pulsesin the kilohertz range that can be detectedby other transducersto give the distancesbetweenthem. They are sometimesincorporatedin the source-detection system ({7.1.7)to locatethe sourcesand streamerswith respectto the ship and to eachother. For surveysof restrictedareas(fig. 7.7a),anchored pingertransponders can be used.The ship or sonde to be locatedtransmitsa sonar pulseand the transpondersemit coded responseswhen they sensethe interrogatingpulse.Most systemsmeasurethe twoway traveltime,though some also measurethe phase differenceat severalsensorson the ship to determine the direction to the transponder(much as moveout givesthe apparentdirection of a seismicray). Four or more transpondersmight be set I to 6 km apart where water depths are 20 to 500 m. The range is improved if the transpondersare 5 to l0 m abovethe seafloor. Recoverabletranspondershaving lifetimesof 5 years are available. The locationsof anchoredtranspondershaveto be verified, not only becauseof uncertaintiesin transponderlocations,but also becauseof local velocity and propagation variations. Verification is usually done by criss-crossing over the area while using some other navigation system.The transponderlocations should also be verifiedperiodicallybecauseanchored

/ l /, / l t I .i' ,\/ I t

t

t'

\

I

I I

rl 1{

I

a

9.\

\ Fig. 7.7 Sonar navigation. (a) Measuring sonar range to bottom-set transducers (from Ingham, 1975; reprinted by permission of John Wiley & Sons Ltd.). (b) Doppler-sonar navrgat i o n ( f r o m S h e r i f f ,1 9 9 0 : 9 0 ) .

transponders sometimes move, especially during storms. Acoustic transponderspermit relative positioning of +5 m, whereasabsoluteaccuracydepends mainly on the method used to position the transponders. Dopplersonaris a dead-reckomrzg system,that is, it determinesposition with respectto a starting point by measuringand integratingthe ship'svelocity.The ship'svelocity is measuredby projectingsonar beams againstthe oceanfloor in four directionsfrom the shio (fig. 7.lb). Thesebeamsare reflectedback to the ship but their frequenciesundergoa Doppler shift because of motion of the ship with respectto the oceanfloor.

I

EQUIPMENT

198 The frequency shift in each beam thus gives the component of the ship's velocity in that direction. The Doppler effect relates to the compressionof wavefronts aheadof a moving sourceor as seenby a moving observer.If Z is the velocity in the medium and tr{ thi componentof a ship'svelocity in the direction of the acouitic beam, the wavelengthtransmitted will be (V - V")lv,but a stationary observerwould seeit as Vlvo; hencevo : v,Vl(V - V").If an observerwith componentof velocity Zois moving toward a stationary source,he would observevn: v,(V + V)lV For : a moving sourceand observer,we have vo v"(V r -moving ship the Vo)t(V O. When the observeris on V,, and becomes /o point reflection, of toward the

vo:v,(v+v)l(v-v,).

(7.2)

The fore and aft measurementsare averagedto mtnlmize the effectsof pitching motion of the ship, and to minimize the starboardand port measurements effectsof rolling motion. The four beamsoften actually look in directions45" to the ship'scourse,which givesimprovedsensitivity,rather than inJine and perpendicular to the ship'scourse.Thesemeasurements can be resolvedto give the ship'svelocity (in conjunction with direction information from a gyrocompass), and the velocity can be integratedto give the ship's position. Small errors in velocity measurementaccumulate in the integration,resultingin position uncertainty of the order of 100 m/hr. Accuracy has to be maintainedby periodicupdates,that is, periodicdeterminations of location by independentmeasurements' In deepwater,scatterof the sonar beamsby inhomogeneitiesin the water dominates and the Doppler shifts give a measureof the velocity with respectto the water ratherthan the oceanfloor, resultingin considerableloss of accuracy.A 300-kHz Doppler-sonar "see" bottom shallowerthan about systemcan usually 200 m, whereasa 150-kHz systemcan seeto depths of 400 to 500 m. Inertial navigationcan be accomplishedby measuring accelerationin orthogonal directions,integrating once to get velocity and a secondtime to get location are relativeto a known startingpoint. Accelerometers usually locatedon a stableplatform that is kept horizontal by a levelingfeedbacksystemand whosedirection in spaceis maintainedby a gyro feedbacksystem' Periodicfixesfrom an independentnavigationsystem minimize the accumulationof systematicerror' The uncertainty with inertial systemsin geophysicaluse increasesat a rate of about 200 m/hr.

7.1.7Locatingthestreamer A seismicship usually tows a long streamerthat may extendfor 5 km or more behind the ship.Even though the location of the ship is known' the streamercan drift by appreciable amounts. There is usually a tail buoy (fig. 7.8) at the end of the streameron which a radar reflector is mounted; the direction to this re-

Fig. 7.8 Tail buoy containing a radar corner reflector'.a Syledis raiiopositioning receiver,and a transmitter for radioing information back to the seismicship. (Courtesy of Compagnie Gen6rale de G6oPhYsique.)

flector can be measuredwith the ship's radar' However,it is often impossiblein rough seas,to distinguish the tail buoy reflectionfrom water-wavebackscatter, particularly when the tail buoy is in the wavetroughs' A radio or GPS receivercan be mounted on the tail buoy so that its location is known in the radiopositioning or GPS systembeing used to locate the seismic ship. Magnetic compasses,typically 8 to 12 in a 5-km streamer.are included within the streamer(frg. 1.9a) (Proffit, l99l: 30). The readingsfrom theseare digitized and sentback to the ship.High-frequencywaterbreak detectorsare also includedwithin the streamer; thesemeasurethe channelwave($13.3),which travels in the water layer at the speedof sound in water (fig. 13.19),and thus give the distancefrom the seismic pingersare sometimesincorsource.High-frequency porated in the system,particularly when more than one sourceor more than one streamerare employed (fig. 7.9b), to locate the sourcesand streamerswith respectto the ship and to eachother,as in 3-D seismic acquisition(see$12.1).The information from the various sensorsis reducedby computeralgorithmsto give the midpoint bins into which the data fall.

7.2 Impulsive land energr souroes 7.2.1The desiredsource An ideal seismicsourcewould generatea wavethat ( I ) containsenoughenergythat it can be detectedreadily after traveling greatdistances,(2) has short duration so that closelyspacedinterfacescan be resolved,(3) is repeatable,and (4) does not createnoise that will

IMPULSIVE LAND ENERGY SOURCES

t99 MagneticHeadingCompasses

Magnetic Gyro l-leadingCompass Compass System

Tailbuoy

LengthMeasuringUnit PrecisionDirectionFindingSystemwith OpticalBearingControlSyst€m (a)

contradict; increasingthe energylengthensa wavelet and consequentlydecreasesthe resolution. Vertical stackingprovidesan alternativeway of increasingthe energywithout lengtheningthe waveletif the sourceis repeatable.The problemsof interferencewith nearby sources conceptually can be solved either in processing,if the sourcecharacteristicsare known, or by sourcedesign. 7.2.2 Explosivesourcesin borehole,s

_

PORI

_

STAREOARD

-

CROSSCABLE

(b) Fig. 7.9 Locating the streamer. (a) Sensors for locating the streamer with respect to the ship and the seismicline (courtesy of Prakla-Seismos).(b) Acoustic system of pinger transducers for locating sources(G, and G.) and streamers(C', and C'.)with respectto pingers P and S, which provide a baselinewith respect to the ship. Separatecross-cablepingers at several locatrons rn the streamersmeasure the streamer separation (such as the distance C,C) (courtesy of Western Atlas International).

interfere with the detection of reflections.A scaling law for sources(Ziolkowski, 1980)indicatesthat increasingthe energyincreasesboth the waveletamplitude and the time scale, but does not change the waveshapeif the sourceenvironmentis the same(fig. 7.10):This law statesthat ru"(t): awuftla),

(7.3)

where wr(l) is the wavelet amplitude generated by source A, and the energy of source I is ct3times that of sourcel. Thus, requirements(1) and (2) tend to

(a) Drilling. When dynamiteis beingusedas the energy source,holesare drilled so that the explosivecan be placed below the low-velocitylayer.The holes are usuallyabout 8 to l0 cm in diameterand 6 to 30 m in depth, although depths of 80 m or more are used occasionally.Normally, the holesare drilled with a rotary drill, usuallymounted on a truck bed, but sometimes on a tractor or amphibiousvehiclefor working in difficult areas.Somelight drills can be divided into units small enoughso that they can be carried.Augers are used occasionally.In work in soft marshes,holes are sometimesjetted down with a hydraulic pump. Typical rotary-drilling equipment is shown in figs. 1 . l l a n d7 . 1 2 . Rotary drilling is accomplishedwith a drill bit at the bottom of a drill pipe, the top of which is turned so as to turn the bit. Fluid is pumped down through the drill pipe, passesout through the bit, and returns to the surfacein the annular region around the drill pipe. The functions of the drilling fluid are to bring the cuttingsto the surface,to cool the bit, and to plaster the drill hole to preventthe walls from caving and formation fluids from flowing into the hole. The most common drilling fluid is mud, which consistsof a fine suspensionof bentonite,lime, and/or barite in water. Sometimeswater alone is used; sometimesair is the circulating fluid. Drag bits are usedmost commonly in soft formations;thesetear out piecesof the earth. Hard rock is usually drilled with roller bits or cone bits, which cause pieces of rock to chip off becauseof the pressureexertedby the bit teeth. In

EQUIPMENT

200

areas of exceptionallyhard rock, diamond drill bits are used.

3 (b) Fig. 7. I 0 Signature of two air guns where the second has eight times the energy of the first. The polarity is negative.SEGstand a r d . ( F r o m Z i o l k o w s k i , 1 9 8 4 :l 3 )

(b) Explosive sources. Explosives were the sole source of energy used in seismic exploration until weight dropping was introduced about 1954.Explosives continue to be an important seismic energy sourcein land work. Two types ofexplosiveshavebeenusedprincipally: gelatin dynamite and ammonium nitrate. The former is a mixture of nitroglycerin and nitrocotton (which form an explosivegelatin)and an inert material that binds the mixture together and that can be used to "strength" of the explosive'Ammonium nivary the trate is cheaperand lessdangerousbecauseit is more difficult to detonatethan gelatin dynamites.Ammonium nitrate and NCN (nitrocarbonitrite) are the dominant explosivesused today (in such forms as are alsoused NitramonrM).Other typesof explosives occasionally. Explosivesare packagedin tins or in tubesof cardboard or plastic about 5 cm in diameter that usually The contain 1 to l0 pounds(0.5to 5 kg) of explosive. tubesand tins areconstructedso that they can be eas-

ii"i'-iS'

''--l P Fig. 7.11 Shothole drilling. (a) Mayhew 1000 drilling rig (courtesy of Gardner-Denver). (b) Light drill that can be moved by helicopter (courtesy of PraklaSeismos). (a)

-ry .,

IMPULSIVE LAND ENERGY SOURCES ily joined togetherend to end (fig. 7.13a)to obtain various quantities of explosives.Ammonium nitrate sometimesis used in bulk form, the desiredquantity being mixed with fuel oil and poured directly into a dry shothole. The velocityof detonation(that is, the velocitywith which the explosiontravelsawayfrom the point of initiation in an extendedbody of explosive)is high for the explosivesused in seismicwork, around 6000 to 7000 m/s; consequently,the seismicpulsesgenerated have very steepfronts in comparisonwith other energy sources.This high concentrationofenergy is desirablefrom the point of view of seismic-wave analysis,but detrimentalfrom the viewpointof damageto nearbystructures. The U.S.Bureauof Mines, Bulletin656,.,Blasting vibrations and their effectson structures."statesthat it is the velocityof ground motion rather than displacement or accelerationthat correlatesbest with damage.Damageis minimal if peak velocitydoesnot exceed12 cm/s but a "safe criterion" is 5 cm/s. This translatesinto an empiricalrule of ,r : k mr/2,where i' : 50 lor x in feet and m in pounds,or fr : 23 for ,t in metersandm in kilograms.The InternationalAssociationof GeophysicalContractorssetsthe following

201 minimum distances: Pipe lines Telephonelines Railroad tracks Electriclines Transmissionlines Oilwells Water wells,cisterns, masonrybuildings

200ft (60m) 40 fr (r2 m) 100ft (30m) 80 ft (24m) 200ft (60m) 200ft (60m) 300ft (90m)

Electricblastingcapsare usedto initiatean explosion. Theseconsistof small metal cylinders,roughly 0.6cm in diameterand 4 cm long (seefig. 7.13b).They contain a resistancewire embedded in a powder chargethat deflagratesreadily.By meansof two wires issuingfrom the end of the cap, a large current is passedthroughthe resistance wire, and the heatgeneratedtherebyinitiatesthe deflagration ofthe powder, which causesthe explosionof an adjacentexplosive in the cap.The cap has previouslybeenplacedinside one of the explosivecharges,so that the explosionof the cap detonatesthe entirecharge. Primersaregenerallynecessary in settingoffthe explosion in ammonium nitrate explosives. Theseare cansofmore powerlulexplosives that are usedas one of the elementsin making up the total charge.A cap is insertedinto a "well" in the end of thecan of primer to set it off. (c) Placementand firing of explosives. Charges are usualfypusheddown a boreholewith loudingpolcsbecausethe density of the chargemay be slightly less than that of the boreholemud. The depth to the top of the chargeis measuredby how many loadingpole lengthsare required to push the chargedown the borehole."Wings" fastenedto the chargeexpandand dig into the boreholewall whenthe chargemovesupward, preventingit from risingin the borehole. The currentthat causesthe blastingcap to explode is derivedfrom a blaster; this is basicallya devicefor charginga capacitorto a high voltageby meansof eitherbatteriesor a hand-operated generatorand then dischargingthe capacitorthrough the cap at the desired time. Incorporatedin the blasteris a devicethat generates an electricalpulseat the instantthat the explosionbegins.This time-breakpulsefixesthe instant of the explosion,t : 0. The time-breakpulseis transmittedto the recordingequipmentby a ielephoneline or radio, where it is recordedalong with the seismic data. The efficiencyof an explosionis increasedwhen it is confinedand under pressure.A boreholeis usually filled with water or mud to "tamp" the explosion. When the hole will not hold water.the boreholeis usually filled with sandor looseearth. The explosiongenerateswastegasesthat ejectthe fluid, rocks,and other debrisfrom the borehole(holebloy,l:when this debris falls back to the earth,it causesnoiseon nearbygeo-

EQUIPMENT

202 Crownblock

Swivcl (mud is injcctcd into ccntcr of Kcllcy and drillstcm)

Draw-works (for pulling drillstcm from thc borcholc) Rotary tablc (turns Kcllcy and hcnc€ drillstem and bit) Pull.down PullcY(Puts Prcssure on thc drillstcm and blt) mud pit (collccts mud -Portablc rcturning up thc annular sPacc around drillstcm) Mud-rcturn hosc

Kcllcy

Transfcr csc

Mud flows back to surfacc through annular sPace b€twccndrillstcm and holc wall, carrYingthc cutiingswith it

(FromSheriff,1990:93') Fig.7.12 Rotarydrill schematic.

phones.Thereis often a time delaybetweenthe explo"catcher" is iion and the hole blow. A devicecalled a sometimesplacedover the top of the boreholeto prevent the capwiresflying into the air and possiblyfouling nearby power lines. (d) Effects of charge size and depth on reflections. We now give somegeneralrules on the use of chargesin boreholesbased on experience'the exact ptoc.sset surrounding an explosion still being unknown. Shotsbelow the water tableare almost alwaysmuch more effectivethan those above.Clay is a better shot medium than sand, with more effectiveenergytransmission.The boreholeis more apt to be reloadableif the medium is clay; detonationpacksthe surrounding "pot" that is less apt to collapse'A clay to form a small charge is sometimesused to enlargethe space available("pot the hole") for large chargeswhere required. Caliche and limestone are often poor shot media. Hole conditions,especiallychargedepth,may affect the characterof the seismicwavelet,but the changes are usually not extreme when the charge size is

Kcllcy scrcwed into drill stcm

Drill stcm

Bit scrcwcdonto drill stcm (mud cmcrgcs through bit)

roughly the same.Charge depth affectsnear-surface and noisegeneratton. ghosting The fiequenciesof the seismicwavesfrom small explosive chargesis generallyhigher than those from iarger charges.Likewise,severalsmall chargesseparatIA Uy a iew metersand detonatedsimultaneously generatl higher-frequencywavesthan-a concentrated ihutg. containingthe sameamount of explosive' Borehole conditions are different for subsequent shots. The first shot is apt to fracture the borehole walls, force gasesand borehole fluid into the surrounding formations,and otherwisechangethe immediate environment.Thesechangesoften causea few millisecondsdelay in the seismicwave generatedby "hole-fatigue"correctionmay subsequentshots,and a times match' The holereflection that be required so (in clays)' Subsenegative be can correction fatigue ouJnt thot. in a sand medium may be less effective generatorsof seismic energy' and a shothole may "deaden." even (e) Directional charges. Severaltechniquesare used at times to concentratethe energytravelingdownward from an explosion.The detonatingfront in an explo-

l0_1 SOURCES IMPULSIVE LAND ENERGY the seismicwave siveusually travelsmuch faster.than orlglnatlng wave ,n iit.-tot-u,ion, so that the seismic lagstehind explosive-charge top of a long ;;.;,h. evenwhen the ii,. *uu. from the bottom of the charge is the usual (which ; e.tonated.at the top ;i;; vedetonattng *Jttt"al. Explosiveswith low effective long in made are these but locity are sometlmesused, are that are difficult to load' Delav ulits iili6l;i;;;t explosometimesused betweenseveralconcentrated the formation to ,i". "ftu.g.t to allow the wave in may consistof these front; the explosive ;;il;;tih betweenthe delay fixed a introduce ari"t "ipt t*hich the time and shock initiatesthem ii#,ft.i.i"nating detonatwound ii,." ift.tl*r"., "*plodt) or helically has to..travela i"*""tJ it. ,hat tile detonatingfront imExpendible 7'5c)' problem (see iois., airtunce)

Fig. 7.13

detonatewhen Dactblastershavealsobeenused;these

iil

ex;;;;;i"ut.a uv theshockwavefromanother

plosion. '' (seefig' 7 '14a)of CtnriO., the effect at a point P explosion of a linear source of th.-;;;1t"";ous : aI' and strength o per.unit.length' o i;;g,h;tf per unit of length per beine the energy recelved at P form Xrr.t-lng harmonic waves-of the ;;;""il;;. wave a for P at !i"' - lree eq. (2.56)),'theresultant : is g..r.ru,.a at all Points of MN at r 0 h(!):

II

Oej(Kr-ot) d-

7zs+

oe-r'l I I r

a\12

ej.'d-.

zo-a)tl2

cap' joined end to end' and (b) seismicblasttng Pont') (a) Cans of Nitramon Seismic explosrves.(courtesy of Du

EQUIPMENT

204

T I

alvl 2

T-

alul2

L

(b)

(a) Geometry lor deFic.. 7.14 Continuous linear source plot for various values (b) directivity and i.."-i"itg directivity. of a.

For r,,))

atr,

r : r o + Q - z ) s i n 0 n :ro(l-sin2Ou)*zsin0n ro cosroo* : sin 0n.

Thus, fd^/l

h(t\ : oe

,.tlfri J:1

g tr(r9 coszoe+: sinon) da\/2

sin(n'asin 0n) : od\er((1)'" na sin 0^ -

6'4\gi{"ro

'') SinC(na sin 0,,),

where sinc . : (l/r) sin 'x. lf the linear sourcewere P concentrate
The length of the sourceis apt to be one-wavelength (a : or less(rz' l), and occasionallytwo wavelengtfs varfor response 2). Figure 7.14bshowsthe directivity is achieved directivity little that Note lou. u'utu.sof a. when a is small. Shapedchargesare sometimesused to concentratethe energytravelingvertically,but they ineffective. are generallY ,l,itnough fi.g. 7.14 implies a distributed vertical source. tlie derivation also applies to a distributed horizontal source. surJucesources 7.2.3Lurgeimpul.sive Before magnetic tape recording (in the early 1950s) provided the ability to build up the effectivesource in.rgy by vertical stacking, explosivesprovided al-oriih. only seismicenergysource'A number of alternativeimpulsivesourcesweredeveloped,mostly in the 1960s,but most havenow disappeared'The earliestnondynamitesource,the thumperor weightdropper (fig. 7.1!; d.op. a heavy weight (-3 tonnes) from ab"out3 m; it ii still usedin some desertareastoday' Among the sourcesthat havedisappearedaregasguns (such is Dinoseis)that usedan explosivegas mixture in an expandablechamberheld againstthe surfaceby the weight of the transport vehicle' X laid air gun (fig. 7.16),consistingof a pan conused'The taining waterind an air gun, is sometimes. weight' vehicle's the by ground the pan iield against work marine in used those to gun is similar air ttre water the into (see$7.4.3).The suddenreleaseof air .*punat the pan diaphragm and imparts an impulse inio the eartl. The unit is caught before it can fall back onto the surface,thus preventing a secondary lmpact. F,xplosivechargeson or near the surfaceare somehabitation' In air times used in areas remote from "flashless"explosives small 1950), (Poulter, shooting (so they do not start grassfires)are placedon sticks I (fig' 7'17); to 1.5m high distributedin a sourcearray (such cord detonating with are usuallyconnected they -PrimacordrM) very A explosions' the initiate to as ear damage can^ that generated is wave air strong wave air the of drumi of anyonenearby'The impact on the surfacegeneratesthe seismicwave' Explosivedeionating cord is sometimesburied 0'3 to I m in the earth or laid in shallowwater' A vibratory plough (fig.7.18)can be usedto bury the cord; upio l00L can be usedfor a shot'The speedofdetonation of the cord, about 6'5 km/s, determinesthe number of capsrequiredto detonatethe entire length of cord within the desiredtime interval' Long cords that hred from one end have directional properties dip prevailing the where situations in can be utilized laid is sometimes cord Detonating is known. direction on the water bottom as a sourcein shallowwater and in marine surveys(Aquaseis)and sometimesis used source' energy other waYsas an quite The energy from most surface sources is vertiusually are weak. an<Jso a number of records

IMPULSIVE LAND ENERGY SOURCES

205 cally stackedtogether($6.7.3)to build up the effective strength. The sources usually generateappreciable ground roll and the sources are moved a small distance(3 to 5 m) betweensourceactivationsto attenuate the ground roll in the vertical stackine. Although the foregoing are primaiily surface sources,gas guns, air guns (Brede et al., 1970),and other devicesare sometimesused in boreholes,especially in soft marsh where there is little risk of beins unableto recoverthe equipmentfrom the borehole. The nature of the surfaceaffectsthe waveshapethat surfacesourcesgenerate.Figure7.19showsthe spectra from land air guns on two types of surface.

7.2.4 Small surfacesources

Fig. 7.15 Weight-dropunit. (a) Weightdropping;(b) impres_ sionsin the sandleft by the impacts(from Sheriff,t9g9).

tl!!

Plt!

Ati out

Modifled versions of the major energy sourcesare used as sourcesfor engineering,ground water, and other surveys that do not require large energy,but small sourcessuchas thoselistedin table7.1 are often more cost effective.Systemsthat sum the effectsof many impacts(seeSosiein $7.3.2)are also used.Gravitational accelerationof weight-drop devicescan be supplementedby other means(enhancedweight drop). Enhancedweight-dropdevicesare also usedto generate S-wavesby impact againsta baseplateat an angle to the vertical. This generatesboth P- and ,S-waves; the azimuth of the deviceis then rotated l80o so that the generatedS-waveshaveoppositepolarity, and the secondrecord is subtractedfrom the first, thus adding the S-waveeffectsand subtracting the P-wave effects (fie. 13.3).

otAPxnAoI

Dt^tHt^oI

(a)

(b)

F i g . 7 . 1 6 L a n d a i r - g u n s c h e m a t i c(Courtesy of Bolt Technology Corp.) (a) Armed and (b) fired

tilrxo OUDA

EQUIPMENT

206

Severalof the sourceslisted in table 7.1 were compared by Miller et al. (1986).The relativeenergyfor singleimpactsis shown in fig.7.20.They found about a 48-dB energy decreasebetween 110 and 1000 Hz. The strongersourcesproducedgood reflectioncoherence from sand--claycontacts80 to 200 m deep with 340-Hz low-cut filters. 7.3 Nonimpulsiveenergysources 7.3.I Vibroseis (a) Introduction. Unlike most energy sources that try to deliverenergyto the ground in the shortesttime possible,the Vibroseissourcedeliversenergyinto the ground for severalseconds.A control signal causesa vibrator (usuallyhydraulic)to exert variablepressure on a steel base plate pressedagainst the ground by the weight of the vehicle(fig. 1.21b).The pressureI generallyvariesaccordingto the relation Fig. 7.17 Air shooting. Charges on small wooden poles are connected by Primacord ready for detonating.

Fig. 7.18 Plough for planting detonating cord The cord feeds down through pipes behind blades that vibrate as they are pu-

9(t) = 11,1sin 2nl[uo + (duldr)r]'

(7'5)

2/t to I m below the surface' illed forward, planting the cord (Courtesv of Primacord Services.)

NONIMPULSIVE ENERGY SOURCES

201

Table 7.1 Small seismicrlurces Source

Silencedrifle BetsySeisgunrM Buffalo gun

Sledge hammer Brutus

Bean BagrM SoursilerM Elasticwave generatorrM HydrapulserM Dynasource Primary SourcerM

Hydraulic hammerrM YumatsuImpactorrM

Prezoelectric

Spark PakrM

Descriptionof source Projectiles Modified rifle with silencingdevice,fired into ground or water-filled hole 8-gaugeshotgunon portablebase Firing rod dropped down l-m iron pipe ro detonate shell below qround surlace

Cost ($)

s00 5000 Shotgunshellfired into dirt roadbed (alsoa downhole source) 4-cm hole augered1 m into ground; gun placedin hole; hole filled wtih water

Impactors(weightdrops) 7.3-kghammer striking steelplate of roughly Plate set in dirt roadbedwith l-2 equivalentweight hammerblows 136-kgweight raised 1 m by gasoline-poweredPlateset in dirt roadbedby several hydraulicpump and dropped onto steel weight drops plate; trailer-mounted 136-kgweight in soft bag dropped about 3 m No site preparation 200-kgweight drop ofabout I m Both P- and ,S-wavegenerator I l4-kg weight drop acceleratedby elastic Plate set in dirt roadbedby several bands;resetby electricwinch; trailerweight drops mounted Weight drop augmentedby compressedgas No site preparation Vacuum-assisted 45-kg weight drop; gasoline- Plateset in dirt roadbedby several powered,trailer-mounted weight drops Compressed-airaccelerated,136-kgweight Plate set in dirt roadbed by several drop; truck-mounted;with hydraulic weight drops; both P- and S-wave positioningcontrol generator Hydraulically acceleratedweight drop onto No site preparation baseplate Hydraulicallyaccelerated200-kgweight drop No site preparation Electrical Piezoceramictransducerstack with highvoltagedischarge;l-kV power supply vehicletransported Pulsegeneratedby discharging5-20 kJ of energybetweenelectrodesin salt water; operatesoff l-kW generator

Explosive l0 cm of 200-grainPrimacordinsidetubing I m below surface;detonatedby standard blaster Erplosives Ammonium nitrate and nitromethane explosivemixed on site;detonatedby seismicblastingcap POD (propane-oxygen Sleeveis driven againstbottom of borehole detonator) bv exolosion \lini-Primacord

r)mnipulse'r'\'{

Comments

Seefig. 13.3

Tradenames: BetsySeisgun Bean Bag Soursile Elasticwavegenerator Hydrapulse

5000-1 5,000 < 500

< 500 5000-I 5.000

500015,000

500015,000 > 15.000

lmpact plateseton ground

5000-I 5,000

50-cmhole dug in ground; garbage bag placedin hole and filled with salt water

> 15,000

Small tubing pounded I m into ground; Primacordpushedto bottom and tamped with sand l0-cm hole augeredI m into ground; explosiveand cap placedat bottom; hole packed

< 500

< 500

Both P- and S-wavesenerator Mapco DevelopmentalGeophysics Geom6chanique Bison Instruments CMI

Primary source Hydraulic hammer Yumatsuimpactor Spark Pak Omnipulse

Shear-waveTechnology Prakla-Seismos JapexGeoscienceInstitute GeomarinesSystems Bolt Technology

EQUIPMENT

208

oiston to chambersin the center of a large reacting mass. The chamber is divided into two parts into which oil under high pressureis alternatelypumped or exhausted,so that the massoscillatesup and down' The force causingthe steelmass to oscillateis equal and oppositeto the force exertedby the baseplateon the ground. The vehicleweight acts as a hold-down weight to pressthe vibrator firmly againstthe ground; airbags and springs decouple the oscillating system from the vehicleso that it is not unduly shaken'The hold-down weight must exceedthe maximum upward instantaneousthrust to prevent the baseplatefrom leaving the ground; peak thrusts are sometimesas much is l6 tonnes.For more detailedaccountsof vior Geyer(1989)' bratordesign,seeWaters(1987:62ff.)

E

*5 i

0

Frlqumcy (H.)

Fig. 7. l9 Effect of surface on spectra of wave resulting from a land air gun. (Courtesy of Bolt Technology.) (a) Concrete surface, and (b) clay surface.

(duldl) being either positive (upsweep)or negatrve (downsweep) and constantin the usual caseofa linear "sweep."The amplitude,4(t)is usuallyconstantexcept during the initial and final 0.2 s or so when it increases from zero and decreasesto zero (see fig' 9'8)' The '7 sweepusuallylastsfor to 35 s with a frequencyvarying from about 12 to 60 Hz or vice versa. Baaunt. reflectionsoccur at intervalsmuch smaller than the sweeptime, the seismicrecordis the superposition of many wavetrainsand the field recordsare uninterpretable even to the experienced.Subsequent to resolvethe (seeS9.3'4)is necessary data processing with (cross-correlation processing the data; in effect, the sweep)compresseseach returning wavetraininto a short wavelet,thus removing much of the overlap (seefig. 9.9). The cross-correlationrequiresthat frequ.n.i.t do not repeatduring a sweep;ifthey did, the would find more than one match for cross-correlation the repeatedsuccessionof frequencies,creating fictious events. ( b ) Vibrators. A schematicdiagram of a vibrator is shown in fig.7.21a. The baseplate is connectedby a

(c) Fietd technique. Miller and Pursey(1956)calculated the distribution of energyfrom a P-wavevibrator in a semi-infiniteearth to be 1o/nP-wave(concentrated in the downward direction), 26% S-wave (beamedat about 30'to the vertical),and61oksurface wave. The near-surfacelayer causesmost of the 'Senergy' waveenergyto be convertedinto surface-wave ensurface-wave of proportion Becauseof the large are arrays receiver and source large ergy, relatively generallyused. Active geophonegroups usually are not locatednear the vibrators,so that there is an appreciablesourcepointgap (often exceeding300m)' By using only groups offsetfrom the vibrator source,the range of amplitudes recorded is much smaller; this simplifiesthe detectionof deepreflectionsthat otherwise would be lost when superimposedon the wavetrains of shallow ones.However,the large sourceoffsetand source-receiverarraysalso limit the recording of shallowreflections.Severalvibrators (usuallythree or four) and vertical stackingare usedin addition to the sweeplength to build up the seismicenergy'The sourcesare moved betweensweepsto attenuatethe ground roll in stacking. Ideally, the input to the ground is a copy of the pressureapplied -and to the steelplate,but in fact heating' compaction of the surface material irushing, (pressuresare as high as 200 kg/cm2)result in the input to the ground varying nonlinearlywith the presiure exertedby the vibrator. This introducesharmonics not presentin the original input signal'The second harmonic is apt to be troublesomebecauseit can correlatewith the sweepto superimposea spuriouscorrelation ghoston the record.Correlationghostproblems can be eliminatedby using upsweepsso that the ghost arrivesbeforezero time, or by using very long downsweepsso that the ghost doesnot arrive until after the region of interest. It is generallyeasier to lock the phlaseas a downsweepstarts but upsweepsare much iasier on the equipment,so that there is no clear-cut preferencebetweenupsweepsand downsweeps'Standard practiceis to move the vibrator a couple of mesweeps' ters betweensuccesslve Vibroseissourcesproduce low energydensity:as a result,they can be usedin citiesand other areaswhere

Relative energy

to' SiI enced 30-06Rifle. l0 kJ SparkPak /. J tg hamer l2 9a. 8uff.l 8 9a. Setsy S i l e n c e d5 0 - c a l. l0 9..

Euffal

l-o*-cut

lli ni -primacord 8ru tus []astic lave Generator

oynasource (vAfo 8 9a. Euffa Prihary ll4

g Them€r

Fig. 7.20 Relative energies for sources listed in table 7.1. Shadedbars representreflections;open bars, ground roll. (From Miller et al..1986.)

hold do*n mass

truseplate -'-i,-t*f,,tfg

F t g . 7 . 2 l V i b r a t o r s .( a ) B a s i cp r i n c i p l eo f a v i b r a t o r .( b ) p h o t o o f a v i b r a t o r ( c o u r t e s yo f C o n o c o ) .

1rr,.i

(a)

4 t

.. il

1'?, .

t,t.

t

* 'i'*

u r.j

f ^r

t ,rA *r"i

ffiI:

;"d#

E Q UI P M E N T

210 exDlosivesand other sourceswould causeextensive damage(Mossman,Heim, and Dalton, 1973)'Vibrosei is now usedfor over half of land seismicexploration. The large weight of vibrator vehiclesrestrictstheir use in some areis. Air-cushion (hovercraft)vehicles can be used acrosstidal flats and other areaswhere ground-pressurerestrictionsapply; to make the vehiIl. into a seismicvibrator, the air flow is modulated by the sweeP. ( tl) Phasecontrol. The signal/noiseratio (S/AI)is improvedmore by usingseveralvibratorssimultaneously ittan Uy vertical stacking. With four vibrators, the noiseis imS/N with respectto nonsource-generated succesfour stacking whereas provedby a factor of4, '{4 : 2 beby only S/N the improves ,iu. ,*..pt causethe noise will be different on the four sweeps' (S/N also variesas {f, where ?"is the sweeplength') However. the different vibrators must be phaselockedtogether. The earth acts as a spring to baseplatemotlon, creating a resonantsystemwhosepropertiesdepend on both the baseplatearea and the soil characteristics' systemsare usedto keepthe inPhase-compensating signal'Eachvijectedsignils lockedonto a reference bratorhasits own digitallycontrolledsweepgenerator that is triggeredby a signal sent from the recording truck; thii-procedurepreventsa noisy radio signal from affecting the sweeps.The output of a strongis integratedto givea baseplate motion accelerometer velocity,which is then sent to a phasecomparator the output whereit is comparedagainstthe reference; constant keeps that phase shifter a is usedto control the phase differencebetween the referenceand the baseplatevelocitY. ( e) Nonlinear and pseudo-runrlomsweeps' The useof nonlinear sweepsis equivalent to filtering the data (Goupillaud,1976).The natural attenuationof high frequenciesin a wavelet'spassagethrough.the earth actsas a filter and is a major factor in limiting resolution. Becausethe Vibroseismethod givesus control over the wavelet'sfrequencycontent, we may use a rQ increasethe high-frequencyenergy nonlinearsv)eep input to compensatefor absorptionlosses'Usually'we cannot significantlydecreasethe time spent sweeping low frequenciesbecausethe total bandwidth needsto be maintained, so increased time spent sweeping higher frequenciesmeanslonger sweeps'With a linear sweep,eq. (7.5) showsthat it takesdouble the time to sweepa high-frequencyoctavethan it takesto sweep twice the next lower octave (becauseit encompasses insweeps nonlinear hence as many frequencies), of effect The sweeps' linear over time creaser;cording using a nonlinear sweep in improving the highfrequencycontent and consequentlythe resolutionis shownin flg.7.22. Goupillaud (op. cit.) also showed that if a linear

sweepis divided into segmentsand the segmentsrearranged,the correlationprocessto extract the information from the recordeddata will not changeprovided there are no discontinuities between the segments.He proposeda random arrangementof the A selments that he called a pseudo-random. .sweep' sweep a with correlate not will sweep pseudo-random using a different sequence'Thus, the use of pseudorand-omsweepsmakesit possibleto record with several sources (at different locations) simultaneously and be able to separatethe data in processing'Other coding schemesare also used. It is possibleto offset vibrators from each end of a cable and thus double the amount of data acquiredin a givenrecordingtime' Ground-roll amplitude is also lower with pseudorandom sweepsbecausethe low frequenciesthat contribute most fo ground roll are distributedthroughout the sweep. 7.3.2Sosie An impactor such as those shown in fig' 7 '23 can be (up usedas a sourcefor shallowpenetrationsurveys to I s) usingthe SosielMmethod(Barbierand Viallix' strikesthe ground 5 to 1973).The impactor(v'hackar) is madefor about recording l0 timesper secondand a The imimpacts) 1800 to 900 (therefore. 3 minute.s pact timescan be consideredrandom for seismicfreouencies.A sensoron the baseplaterecordsthe moment of eachimpact for use in correlation'Random repetitivefiring of other small sources,suchas small Viporchoc'" units (see$7.4.4)in marinesurveys,can alsobe usedas Sosiesources. With the Sosiemethod,the output of a geophone groupis addedinto its summationregisteragaineach ii*.-u n.* impulseis appliedto the earth' Because many impulsesoccurwithin the recordinglength'the output of eachgroup is belngaddedto its summatlon register simultaneouslyat many places' Sourceenergy adds in phase with respectto the ge-nerated times,but addsrandomlyat otherttmes arrival f,top., (seeproblem7.6). 7.3.3Cltoit'eoJ lund sour(es The choiceof seismicsourceis almost alwaysan economic one.The source(or drill) is usuallythe largest' heaviestpieceof equipmentthat a seismiccrew requires.Hence,if the sourcecan reach its locattons' other acquisition operations probably can reach theirs.Explosivesin boreholesare probably preferred exceptwheredrilling is difficult by most geophysicists o. .*p.niiu., wherelegalrestrictionsexist,or proximity to trabitationsor itructures preclude the use of .hurg., of sufficientsize. Some of the large surface ,ou.i.. are so heavy that they require large vehicles for their transportand thus accessbecomesthe major factor. Most equipmentcan be mounted on a varlety ofplatforms (suchas boatsor barges,trackedvehicles' marsh buggies,and very shallow-draftair boats pro-

\1.\RINE EQUIPMENT

-

ll lmprovement in resolution resulting from use of non_ : -:,: \'ibroseis s w e e p .L e f t h a l f , n o n l i n e a r l 0 - t o - 9 0 _ H zs w e e p ;

- .ld by an air propellor)for usein differenttypesof :::rin. includingmarsh, swamp,and shallow-water --:.rs.The local availabilityof equipmentis apt to be , ::.'iding factor on small work projects. -J

\larine equipment -: . Getrcrul -' : shipsusedfor marineseismicacquisitionare usu_ _ .rrge.30 to 95 m in lengthand 5 to 20 m in beam. - . ship,as shownin fig.7.24,drawsup to 6 m. * Fical - :hip carriesenoughfuel,water,and othersupplies rerate at seafor 30 to 60 daysand accommodates -' : . 60 people.Replenishment of suppliesand per_ . ':-:l erchanges can be carriedout at seaso that oo_ :-:r.,-rfiS can continuesteadily.Thereis usuallya plat-::: ..rnwhich a helicopter can land. The ship is apt :.rke its own fresh water by reverseosmosis.It is -:-::allr equippedwith a numberof radio and navi--' . i aids for communicationand positioning,rn_ . -:.:tg a weatherfacsimilereceiver.The ship'sspeed - : .rcquiringdata is usually5 to 6 knots (2.5 to 3 -. : 'i 9 to I I km/hr), but the ship can make about ' . :..rs (28 km/hr) when not acquiringdata.A high -:-r:3 trf automationis employedto sustainefficient . - : : . J L r U SO D e r a t l o n s .

right hali linear l3-to-50-Hz sweep.(From Andrew.

Figure 7.25 showsviews aboard seismicships.The severalkilometers of streamersare stored on larse reels(fig. 7.25a) from which the streamercan be 6d over the stern into the water.Arrays of air guns are sometimesmounted on paravanesthat are suspended lrom the deck ceiling when not in use(fig. 7.25b).The rnstrumentroom (fig. 7.25c)often appearsmuch as the control room in any largecomputing facility. As on land, early marine seismicacquisition used explosivesas sourcesalmost exclusively. Althoueh the efficiencyof an explosionin water incieaseswii-hwater depth, it was soon discoveredthat the wastegases oscillatedin the water (bubbleoscillations),each oscillation effectivelyproducing a new seismicrecord superimposedon the earlierrecord so that the resultins recordmixture could not be interpreted.To ou...o-i this problem, the practicewas to suspendthe charee from a floating balloon within about 2 m of the suiface, allowing the waste gasesto vent to the atmospherebeforethe bubblecould collapseand generate a secondseismicwave. Safety required that only one capped explosive could be aboard a ship at any time; this meant that shotscould not be fired at short time intervals,so that the common-midpoint method could not be used effectivelyin the marine environment.Oncenonexolo-

{

F

'

r. ?i; "

i.-q; f"+.3 t'-ti

the base plates indicate the times of impacts tCourtesy of Fig.7.23 Two impactors used as mini-Sosie source. Geophones on Wacker-Werke.)

outer paravanesand directly from the ship) and two sources Fig. 1.24 A seismicshrp towing three streamers(from the two Geophysical ) (Courtesy ofWestern (from thc inner paravanes).

[.

MARINE EQUIPMENT

2t3 comes zero when the bubble expansionreducesthe gas pressureto the hydrostaticpressure.However,by then, the surroundingwater has acquired maximum outward velocity and so continuesto move outward while decelerating(becausethe net force is now directed inward). Eventually,the water comes to rest and the net inward forcecausesthe bubbleto collapse. The water convergesinto the limited bubble volume, thus rapidly compressingthe gasto high pressure;this implosionis associatedwith a very rapid increasein the pressure,which starts a new bubble expansion. This processcontinuesas the bubble rises,some energy being dissipatedon each oscillation.Eventually, the bubblebreaksthrough the surfaceofthe wateqthe gasesvent to the atmosphere,and the cycleis broken. Seismicwaveswill be generatedby the high pressures associatedwith eachbubblecollapse. The period of the bubbleoscillation(seefig. 7.31) dependson the energystored in the bubble and can

Tlme (s) 0.1

o.2

0.5

E - r o ct

.50

(a)

1807 Fig. 7.25 Views on a seismic ship. ((a and b) Courtesy of Prakla-Seismos, (c) oourtesy of Seismograph Service.) (a) Streamer reels for two streamers,(b) air-gun subarray, and (c) rnstrument room.

trst)

q CL o 5 6 q o G

1

40

37

sive marine sourcesbecameavailable,the use of explosivesas marine sourcesdisappearedquickly (fig. t.2t).. o

7.4.2Bubbleffict The bubble ffict (Krameq Peterson,and Walters, 1968)just describedappliesto any sourcethat injects high-pressuregasesinto the water; it is illustrated in fig.7 .26.As long as the pressurein the gasbubbleexceedsthe hydrostaticpressureof the surroundingwater. the net force acceleratesthe water outward. The net force decreasesas the bubble exDandsand be-

o o o

Fig.7.26 Bubbleeffect.(From Krameret al., 1968.)(a) Depth of bubbleasfunctionof time,(b) bubbleradius,(c) bubblepressure,and (d) velocityofwater adjacentto the bubble.

214 be brought within the seismicrangefor small sources so that the oscillationcontributesto the effectiveness of the source.The bubble energyis roughly proportional to the maximum volume of the bubble as it startsto implode. The bubble effect is important in determining the of almost all marine sources,even sourcewaveshapes for thosedesignedexpresslyto minimize the effectby dissipatingthe bubble energyafter the first collapse. The waveshapegeneratedby a small air gun is shown in fig. 7.28a. 7.4.3Air guns The most widely usedlarge marine energysourceis the air gun (fig. 7.27),a devicethat dischargesair under very high pressureinto the water (Giles, 1968; 1972). Schulze-Gattermann, Pressures up to l0 000 psi (70 MPa) are used,although 2000psi (14 MPa) is most common.The air gun shownin fig. 7.27a is in the armed position, ready for firing. Chambers A and ,8 are filled with highpressureair that enteredA al the top left and passed "shuttle."The into ,Bthroughan axial openingin the latteris held in the closedpositionby the air pressure (becauseflange C is larger than flangeD, resultingin a net downward force). To fire the gun, the solenoid air at the top opensa valvethat allowshigh-pressure to reach the undersideof flange C This producesan upward lorce that is large enough to overcomethe force holding the shuttlein the closedposition,and consequentlythe shuttleopensrapidly.This allowsthe high-pressure air in the lower chamberto rush out through lour ports into the water.The bubbleof highpressureair then oscillatesin the samemanner as a bubble of waste gasesresulting from an exploston. However,becausethe energyis smaller,the oscillating frequencyis in the seismicrangeand thereforehas the effect of lengtheningthe original pulse (rather than generatingnew pulsesas with dynamite).The upward motion of the shuttle is arrestedbefore it strikesthe top of chamberI becausethe upward force falls off rapidly as the air entersthe water and the downward force of the air in the upper chamber increases.The shuttle then returns to the armed position, and the lower chamber again fills with air. The explosivereleaseof the air occursin I to 4 ms whereasthe entire dischargecycle requires25 to 40 ms. Other types of air guns operate in essentiallythe same way. The "sleevegun" (Harrison and Giacoma, 1984)opensthe gun chambermore quickly so that the air is released more rapidly,resultingin increasedpeak pressureand strength. seismic-wave The lower chamber in an air gun may be divided into two parts connectedby a small orifice (Mayne and Quay, l97l), which resultsin a delayeddischarge of the air in the innermost chamber.The flow of this air into the bubble continuesfor some time after the initial discharge,retarding the violent collapseof the bubble and diminishinethe subsequentbubbleeffect.

EQUIPMENT The waveformemittedby a singleair gun, shownin fig. 7.28a,oscillatesbecauseof the bubbleeffect.The delayedghost reflection from the water surface has opposite polarity and comparableamplitude to that from the gun itself; it is primarily responsiblefor the secondhalf-cycleof the waveletshape.For the same air pressure,the energyoutput of an air gun is proportional to the product of volume and pressure.Air-gun sizeis usually taken as the volume of its lower chamber and gun sizesgenerallyrangefrom l0 to 2000in.3 (0.16to 33 liters).Usually,severalguns (often six or seven)ofdifferent sizesare usedtogetherin subarrays such as shown in fi1.7.28c.Gun dischargesare timed so that their initial impulsesinterfere constructively, but their subsequentbubble pulsesinterferedestructively; the wavelet from a subarray is shown in fig. 7.28b. The action of an air gun is affectedby nearby guns unlessthe gun separationis greaterthan a wavelength, that is, greaterthan l0 m for 150Hz. Howeveqinteraction can be calculatedand allowed for (Ziolkowski et al., 1982),so that elementscan be spacedmore closely. Hydrophoneslocatedabout I m from eachgun can measurethe pressurefield, and this permits calculation of the compositefar-field signature.Monitoring the pressurefield permits decisionsto be made should conditionschange,for example,if a gun should fail.

7.4.4Inplodersand other marinesources Severaltypes of sourcescreatevoids in the water into which the water rushesimplosively.With the FlexichocrM,an adjustable-volumechamber is evacuated and the walls of the chamberare held fixed by a mechanical restraint;upon removal of the restraint, the hydrostaticpressuresuddenlycollapsesthe chamber. Air is then pumped into the chamberto expandit and to securethe restraint for the next activation.A flexichoc signatureis shown in fig. 7.30f. With the HydroseinrM,two platesare suddenlydriven apart by a pneumaticpiston to createa void betweenthem. With the BoomerrM,a heavy surge of electrical current through a coil on one plate createseddy currentsthat force the platesapart. Theseimploders are generally weak sourcesbut are suitable for profiling applicat i o n s( $ 8 . 6 . 3 ) . The water gun obtainsenergyfrom compressedair but doesnot inject the air into the water.Releaseof a shuttle(fig. 7.29a)forcesthe water in the lower chamber out through the ports into the surroundingwater at high velocity. The moving slugs of water create near-vacuumsbehind them, and the implosion as the surroundingwater rushinginto the voids providesthe main seismicenergy.Bubbleoscillationis minimal becausethere are no gasesto oscillate.A water-gunsignature is shown in fig. 7.30g. Electric arcs (sparkers)utilize the discharge of a large capacitor to create a spark betweentwo elec-

I I

High-pressure att

High-pressure.atr

Port i

a

Pon

H igh-pressure atr

(cJ

Fig.7.27 Air gun (Courtesy ofBolt Associates.)(a) Charged and ready for firing, (b) firing, and (c) photograph ofan air gun.

o,

L

o

o

o

o-

o.

t-

01

0.2 03 Time(s)

0lt

0 2 0.3 0 1 0.5 0.6 07 Time ( s )

u)

(b)

(a) 18 4h

SEA SURFACE

HYOROPIIONES

GUNS

VOLUMES c rNcHEs)

(c)

Fig.7.28 Waveform signaturesgeneratedby air guns. ((a and b ) F r o m Z i o l k o w s k i . 1 9 8 4 :9 , 1 0 , l 3 ; ( c ) f r o m P a r k e se t a l . , 1 9 8 4 : 106.) (a) Single 0.8-L air gun at a distancc of I m; pressure of 1 3 5b a r s ,g u n d e p t h o f 7 . 5 m . ( b ) A r r a y o f s e v e n a i r g u n s ;p o l a r -

(a) Fig. 7.29 Water gun. (Courtesy of Seismograph Service.) (a) Compressed air forces piston down, ejecting water from lower chamber at high velocity. (b) Because of its high velocity, the

ity is negativeSEG standard. (c) Geometry ofan air-gun subarray; two to four subarrays separated by l0 to 50 m are often used simultaneously.

(b) water continues moving outward, creating a void behind it. (c) Water rushes into the void, producing a sharp implosion

MARINEEQUIPMENT

217 . Severalother types of sourcesare used for very shallow penetration. These include high-powered piezoelectrictransducersmade of barium titanate, lead zirconate,or other materialsthat chansetheir dimensionswhen subjectedto an electricfieldl Frequencies of 2 to l0 kHz can be generatedat about 100W achieving20 to 100m penetration. Other kinds of marine sourcesdesignedto avoid the bubble effect have nearly disappearedfrom use.One was the sleeveexploder,or AquapulserM,which used the explosionof a propane-oxygenmixture in a closed flexible chamber with the waste gasesvented to the atmosphere.Another was the VaporchocrM,or steam gun, which injected a bubble of superheatedsteam into the water. The FlexotirrMinvolved detonatinga small explosiveat the centerof a steelcageso that the bubble oscillationwas damped as the wa=ter flowed in and out of openingsin the cage.The MaxipulserM recordedthe bubbleoscillationand tried to removeit in subsequentprocessing. A marineversionof the Vibroseisemployingseveral sourceunits hasbeenused;however,becausea seismic boat travelsan appreciabledistanceduring the time of a Vibroseissweep,undesirable"smearing" of reflection points results.

7.4.5Choiceof marinesour('es

Fig. 7.30 Far-field waveshapesgenerated by marine seismic s o u r c e s .( a ) S i n g l e 1 2 0 - i n . ra i r g u n , ( b ) a r r a y o f a i r g u n s o f different sizes.(c) sleeveexploder, (d) VaporchocrM,(e) MaxipulserM,(f) FlexichocrM,(g) water gun, and (h) 5-kJ sparker. Curves show waveshape features, but amplitudes are not to scale. B indicates bubble effects,1 implosion. parts (a) and (g) a r e f r o m M c Q u i l l i n , B a c o n , a n d B a r c l a y( 1 9 7 9 ) ;( b ) , ( c ) , a n d ( e ) from Wood et al. (1978); (d) from F'arriol er al. (1970); (f) from manufacturer's literature; and (h) from Kramer, peterson, and Walter(1968).

trodes in the water, thus vaporizing the water and effectivelycausinga small explosion.The penetration of a 5-kJ sparker is generallyless than 300 m, but sparkerarrays deliveringas much as 200 kJ at 50 to 2000 Hz achievepenetrationof 1000m. A variation of the sparker,WassprM,involvesconnectingthe elec_ trodesby a thin wire that is vaporizedin theiischarge; this increases the bubble duration and its low_ frequency content, and also permits operations in fresh water,where conductivity is too low to permit reliablesparkeroperation.

The marine-sourcecharacteristicsmost sought after arehigh peakpressure and low secondary oscillations. These can be determined empirically by firing the sourcein deepwater (so that water-bottomreflections will not confusethe results)and observingthe waveform with a calibratedhydrophone75 to 100m below the source.Although the experimentseemseasy,its implementationis difficult. It is not easy to keep the hydrophoneat the desireddistancebelow the source when the ship is moving, and statictestconditionsare apt to be different from operationalconditions.The source waveforms (signatures) for several energy sourcesare shown in fig. 7.30. Rayleigh(191'/),while studyingthe soundsemited by oscillatingsteambubbles,relatedbubblefrequency to bubble radius, pressure,and fluid density; and Willis (1941), while studying underwaterexplosions expressedthe relationshipin terms of sourceenergy (the Rayleigh-Willisformula): T :

36pt/2gt| st6EU3

(7.6\

where ?"is the period of bubbleoscillationin seconds, p is the fluid densityin g/cm3,Oois the absolutehydrostatlc pressurein pascals(N/mr), and E is the energy in joules.If we assumea densityof 1.024g/cm3for sea water and replaceOowith h + 10,whereft is the depth in meters(10 m is I atmosphere),the formula becomes r : 0 . 0 1 7 E t . \ (+h l 0 ) 5 , 6 (7.7) Figure 7.31showsthe energiesofvarious sourcesver-

EQUIPMENT

218

Equivalent pounds of 60% dynamite at 9 m depth

0.001

i

E

rn -

|

E p

' 'utnt lutt"

**iyffi'W*\sP

= o

U

I

0.1

0 . 0r

O I 00 ft Aquaseis (linear explosive)

o

1 0 0 f t G a s s p{ l t n e a rg a se x P l o d e r )

100

';4 I 0r

Sparkers

| I 04

Enersvin foot Pounds I 06

I 05

Energy in joules

Fig. 7.31

Energy-frequency relationships for marine sources at 9-m depth. (From Kramer et al . 1968.)

sus dominant frequency.In general,largeenergytnvolveslow frequencyand vice versa.

nals of the coil. The geophoneoutput for horizontal motion is essentiallyzero becausethe coil is suspendedin sucha way that it staysnearly fixed relative to the magnetduring horizontalmotion.

7.5 Detectors 7.5.1TheoryoJgeophones (a) General. Seismicenergy arriving at the surface (frequentlyreofthe ground is detectedby gettphones phone,s, or jags) or detectors, ferredto as seismometers, by hydrophones.Although many types of geophones have been used in the past, modern geophonesfor land work are almost entirely of the moving-coilelectromagnetictype. Hydrophonesfor marshand marine in boreholes) work (and sometimesfor measurements are generallyof the piezoelectrictype; these will be in Q7.5.3. discussed The moving-coil electromagnetic geophone is shownschematicallyin fig. 1.32a;and fig. 7.32bshows a cutawaymodel.The schematicdiagramshowsa permanentmagnetin the form of a cylinder into which a circular slot has beencut, the slot separatingthe central South Polefrom the outer annular North Pole.A coil consistingof a large number of turns of very fine wire is suspendedcentrally in the slot by means of light leaf springs. The geophone is placed on the ground in firm contact with it in an upright position. When the ground movesvertically,the magnetmoves with it, but the coil, becauseof its inertia, tends to stay fixed. The relative motion betweenthe coil and magneticfield generatesa voltagebetweenthe termi-

(h) Equationsofmotion. The theory ofgeophones has been discussedin severalplaces (Scherbatskoy and Neufeld,1937;Washburn,1937;Dennison,1953; Lamer,1970;Pieuchot,1984).We let : : displacementof the surface : displacementof geoPhone :, : displacementof geophonecoil relative to the Permanentmagnet m, r, n : mass, radius, number of turns of the coil I : current in the coil r mechanicaldamping factor; r(dz,ldt)beingthe dampingforce spring constant : flLz, where a force/stretchesthe sPringbY A: H : strength of permanent magnetic field K : 2rrnH K i : force on coil due to current R ,L : total resistanceand inductanceof the coil plus externalcircuit A geophonecoil in motion is acted upon by three forces:the restoringforce of the springs,the force of friction, and the force resulting from the interaction of the permanent magnetic field with the magnetic

DETECTORS

2t9

Fig. 7.32 Moving-coil electromagnetic geophone. (a) Sche_ m a t i c a n d ( b ) c u t a w a yo f d i g i t a l - g r a d eg e o p h o n e .( C o u r t e s y ot Geo Space.)

field of the current. The first two are retarding(nega_ tive) forces,and the last is positive.Newton,ssecond law of motion gives -.s-- '

d, d"\ * *-, ' : *"(\dd' r: ' - dr df l'

(7'8)

Faraday'slaw of induction relates:, to i.. emf inducedin coil : -d0 : -d$ d;, . dl dz. dt : -2rrnHdt' = dt

it (seeproblem7.9),giving

.H':ffi]i (ito) (;.fr):', l',:.

The term involving(d/dl) representsdamping,r/m giving the mechanicaldamping and K2lmR the elecro_ magneticdamping.If the dampingwerezero,the sys_ tem would be simple harmonic with natural Jiequen(.y u,,,where

,":1.,: (r;)ffi"

,,d=, dt

( 7 . 1l )

= Ri + L:it, When the damping is not zero,we write where$ is the flux throughthe coil. Solvingfor:,, d-dr

||

fl.*r0."9'*r;i:(f:;

dr\

. . l R +r -t d" 't l1' .

where

K\

Differentiatingeq. (7.8) and substitutingfor d:,/dl givesthe geophoneequationof motion;

'#**')!,. (T), .3:j* F.';)fl1.1l1-r A3d13

(7.e)

For a geophoneoutput to be independentof fre_ quency,Z : 0 (becauseinductive reactancedepends on frequency).Although this cannot be achieved,we assumethat Z is sufficientlysmall that we can neglect

2h<'t,,:

t

m

K

+

(7.12)

2

mR'

i beingthe dampingfactor of eq. (2.1Il). This is the equation for damped simple harmonic motion, and the solution is given in standard texts (potter and Goldberg, 1987:43-l\. (c) Transientresponse. The transientsolution is obtained by setting the right side of eq. (7.12)equal to zerc. Let us assumethat i : 0, dildt : un at t : 0; then the solution has the following form, depending

EQUIPMENT

220 on the value of ft: For h> I (overdamPed), i -- {uJlao(h' - l)"'l}e io'r sinh ftr,ot(h' l)tt2l:' (7.13) For ft : I (critically damped), i : uote -o';

verted it (if n is odd); becausethesechangesdo not vary with o, the waveshapeis unchanged' If a geophoneis subjectedto a harmonic displacethe responseto an ment (Lamer, 1970:80-1, discusses input impulse) such that the velocity d-ldt : V,, cos o/, then V

(7.14)

Thesesolutionsare shown in fig. 7.33 in terms of the resonant period d; they are transient solutions becausei eventuallybecomeszeroowing to the exponential factor. For ft > l, the current starts to build up becauseof the sinh factor, but then decreasesas the exponentialfactor beginsto dominate.When ft < l, the output is a dampedsinewave.Fot h : l, the critically dampedcase,the output just fails to be oscillapeaks occur at intervals tory. When li < l, successive 7,,: 2nlon(l - h2)tt2,

(7 '16)

peaksis and the ratio of successive (1 '17) i'|i" " : exp f2rh(l - h2)tt2l' The logarithmicdecrementE (seeeq. (2.1l2)) in nepers(seeproblem2.17)is givenbY b : ln (i,,|i,,) : 2nh(l - h2)tt2' (7.18) we can obtain ft when ft < I by measuring6. (d) Responseto a driving force. Imagine a mass suspendedby a rubber band from a hand (usingan illustration from Pieuchot,1984).Moving the hand slowly but harmonicallymovesthe suspendedmassin unison with it. As the frequencyof the hand motion increases (maintainingconstantamplitude),motion of the mass will lag behind. As the frequencyincreasesstill furtheE a point will be reached where the suspended mass and the hand are 90o out of phase;this is the naturalfrequencyu,,(seeeq. (7.1l)) ofthe systemand the massis now moving with maximum amplitude.As the hand moves still faster, the motion of the suspended mass decreasesand becomesfarther out-ofphase with motion of the hand; eventually,at very high frequencies,the phasedifferencebecomesl80o and the amplitude of the motion of the mass approacheszero.Thus, we expectamplitude and phase responsesas shownin fig.7 .34.To havethe amplitude of the massrelativelyindependentof the frequencyof the hand, two things are necessary:the natural frequencymust be well below the frequenciesof interest and the phaseshift must vary linearly with frequency. To show the latter, we start with the wave A cos t'.l/ and add to its phasethe quantity (ka + nr), where/c is a constant, and n is integral; the result is the wave +l cos o(l + k). Thus, adding the linear phaseshift has delayedthe waveby k time units and possiblyin-

z

& :' .ov*:,,, *' 0 "--- ----

For /z < I (underdamped),

i : {ilJ[to.(l - lrzttrzf]e"o'sin [rrro/(l h2)tt2]. ( 7 .I 5 )

d

dt2

f-!= -,'n,,cos .,/,

dl3

and eq. (7.l2) becomes drl dl2

dt ^, ,. : + ln@^" d l + (|)if "

- a 2 K V , ,cos (l)l R

(1.1e)

The solution of this equationis made up of two parts: a transientsolutiongivenby eqs.(7.13)to (7.15)and a solution representingthe forced motion of the geophoneresultingfrom the motion of the ground (Potter and Goldberg,1987:61-2).The latteris i:

(V,,lZ)cos (
(1.20)

where 7: tanl:

(R.ir' ulK,ljx[l- (o/o,,)212 + (2ftolrou)rlr/:,] ( 2 h a l a , , ) l l ( a o l o n )l'l-.

J (7.21\

Thus, the amplitudeof i for a givengeophonedepends upon Zo,t'rlon, R, K, and ft. When o) J @, Z -+ NK and the amplitude of i becomesi* : VoKlR. One of the most important factors of merit of a geophoneis the output voltageper unit velocity of the case.We can define the geophonesensitivity,f (also called the geophonetransductionconstant),by the relation amplitudeof glltput voltage (1.22) f- : amplitudeof geophonevelocity Assuming the geophoneis connectedto an amplifier with essentiallyinfinite input impedance (the usual case),the output voltage is the voltage across R., the shunt resistance.Using eqs.(7.20) and (7.21\' we get f : R,(VolZ)lVo:R I Z K(R"/R)/(o/orn),

(7.23)

where : f (l"i,li"r.,,)

0, l. l , '

an,

when to : 0, when
For practicalpurposes,the geophonesensitivityis determinedlargelyby K and h, that is, by the radius and number of turns of the coil, by the magnetic field strength, and by the damping. Modern geophones havesensitivitiesof about 0.7 V/cm/s. Curvesof f are shown in fig.7.35a for various values of /r. When /r : 0, the output becomesinfinite at

D E TE C T O R S

221

o

g

v.z

Fig. 7.33

Free oscillation of a geophone as a function of damping factor, /r.

qt !t

=

a

E

0 1800 Et

g €) .4

f

'gtto

v Fig. 7.34 Amplitude and phase for movement of a mass on a spnng.

the natural frequency;obviously,this is merelya theo_ retical result becausezero damping can never be achieved.As I increases,the outpuipeak decreases in magnitude and moves toward highir frequencies. When ft - 0.7, the peak disappearsind the ianse of

flat responsehas its maximum extent. As i increases beyond this value, the low-frequencyresponsefalls off. The generallyacceptedchoice oi j\o/o of critical damping for geophonesthus results in more or less optimu.moperatingconditions with respectto amplitude distortion in the geophone output. Obvrouslv. t h e d a m p i n go f a g e o p h o n ei s a k e y f a c t o r i n d e _ termining its performance.The damping factor (ex_ pressedby h in eq.(7.12))can be increisedby winding the coil on a metal "former" so that eddy currentsin_ ducedin the former by motion of the coil will oppose the motion; h can be increasedto about 0.3 by this means.The damping is usuallyfurther increasedby a resistancein parallel with the coil (insidethe case). The output of a geophoneis shifted in phasewith r€spectto the input, as shownin fig. 7.35b.The phase shift "y (seeeq. (7.21))will changeihe waveshape(ex_ cept.fora linear phaseshift; seethe preceding;,that is, produce phasedistortion, becausethe seisriic signal comprisesa rangeof frequencies. F i g u r e 7 . 3 5 as h o w st h a t f o r h : 0 . 7 , t h e d i s t o r _ tionlesssignalband extendsfrom about l.2tonupward; hence,the lower the natural frequency,the wider the distortionlessband. The natural frequency of geo_ phonesemployedin petroleumexploraiion(u,,)rs usu_ ally 7 to 28 Hz for reffectionwork and 4.5 Hz for re_ fraction. The decreaseofsensitivity below the natural frequency(fig.7.35a)often providis the lower limrt to the passbandto be recorded. (e) Other aspects. Geophone coils are often divided into two parts that are wound in oppositedirections and so wired that signalsdue to motion add, whereas those that result from stray electricalpickup in the

EQUIPMENT

222 (a\

lo

\ E o

h=0.3 h = 0.5. \

B w

!n=r.o.

o o f l

r r

/t T /

I

h=2.4

r R e l a t i v ef r e q u e n c y( o / @ o )

(b\

,>Y

lt

h = 0.5/

-L) 4

1t 4

,b I :^

o,/ 0

'

^

W

-ln

4

0.1

1.0 R e l a t i vler e q u e n c(yo / @ o)

F i g . 7 . 3 5 T h e d e p e n d e n c eo f g e o p h o n e r e s p o n s e so n d a m p i n g f a c t o r / r . ( A f t e r D e n n i s o n , 1 9 5 3 . )( a ) S e n s i t i v i t ya n d ( b ) p h a s e response.

coils cancel; this feature is called hum-but'king.Geophones also have spurious resonancesbecauseof modesof motion other than that intended,but these usually occur at frequenciesabove the seismicpassband. An entire geophone group is considered to be equivalentto a singlegeophonelocated at the center of the group. Becausethey are generally used in groups,six to nine geophonesusuallyare permanently connectedin stringswith 3 to 6 m of wire separating clipsat the end ofthe adjacentphones.Spring-loaded string provide solid connectionsto the cable,yet are

easy to connectand disconnect.Theseclips usually are geometricallypolarized (for example,one side is wide and the other narrow) to prevent connecting them backwards.Whereasthe damping of each geophone will be affectedby the presenceof the other geophonesbecauseof the changein resistanceof the circuit, the equationof motion for an array will be the same as for a single geophoneexceptwith modified parameters.If there are n parallel branches,each of which contains n geophonesin series,the resistance and dampingwill be the sameas for a singlegeophone (Lamer,1970:$3.7).

DETECTORS

223

We haveassumedthat the geophonefbllowsexactly the motion of the surfaceof the ground, but the geo_ phoneis not rigidly fastenedto the ground,so the geo_ phone-groundcoupling also affectsthe response.We expect the coupling to affect the responsejust as source-groundcouplingdoes(fig. 7.19).Lamer (loc. cit.: 88) statesthat coupling dependson the factor Mlprt, where M ts the geophonemass (inclu
Fig 7.36 Battery-powereddigitizing unit that filters, amplifies, a n d d i g i t i z e ss i x d a t a c h a n n e l sw i t h 2 4 - b i t a c c u r a c y (. C o u r t e s y o 1 'l n p u t / O u t p u t . l n c . )

storesthe informationtemporarilyuntil called upon to transmit it. Radio links are also sometimescom_ bined with conventionalcable-linkedgeophonesto spanrivers,canyons,and otherobstacles acrosswhich cablescannot be laid.

II I

t I

7.5.3Hydrophones Hydrophone,s, or marine pressuregeophones,are usually of the piezoelecrric type(Whitfill,l9i 0). Synthetic piezoelectric materials,suchas bariumzirconate.barium titanate,or lead mataniobate,are generallyused. A sheet of piezoelectricmaterial developsa voltage differencebetweenopposite faceswhen subiectedto mechanicalbending.Thin electroplating on th.r. surfacesallows electricalconnectionto be made so that this voltagecan be measured.Disc hydrophones1fig. 7.38a)are essentially two circularplatesof piezoelectric ceramic mounted on the ends of a hollow brass cylinder. Electrical connectionsare made so that if both bend inward, as they would in responseto an increaseofpressureoutsidethe unit, the inducedvoltagesadd, whereasif the platesbend in the samedirection, as they would in responseto acceleration,they cancel (fig. 7.38b).This feature is called acceleration canceling.Cylindrical hydrophones(fig. 7.38c)are es-

I l

l

I !

EQUIPMENT

a1A

Fig.7.37 OpseisrMtelemetry system. The system is capable of' recording up to 4 lines with 2 spreads/lineand 1016 channels/ spread. (Courtesy of L. Denham.) (a) Remote telemetry unit serving 4 geophone groups. The seismictraces are digitized and

stored in the unit's memory until instructed to transmit them. (b) Program unit of the central recording system. Communication between central and remote units is by horizontally polarized RF waves.

Electrical connection

Eflectof acceleration to lelt

Effect of pressure lncrease \

Piezoelectric disc

(r)

(a)

ui----------

o\ f

>--1

\

Piezoelectticceramic

I

' B r a s sc a P

(c) Fig. 7.38 Hydrophones. (a) Disc hydrophone, (b) accelerationcanceling feature of a disc hydrophone, and (c) cylindrical hydrophone.

sentially thin hollow piezoelectricceramic cylinders closedat the endsby brasscaps.A changein pressure outside the cylinder induces stressesin the ceramic and hencea voltagedifferencebetweenthe insideand outsideof the cylinder. The sensitivityof eachgeophoneelement(75 to 250 pV/Pa) is small so that threeto fifty elementsare usually combined in seriesto make up a hydrophone group; theseare distributed over 3 to 50 m. Natural

frequenciesare of the order of l}a Hz, well outsidethe seismicrange. Piezoelectrichydrophoneshave high impedance,so an impedance-matchingtransformer may be included with each group. Sometimes,charge amplifiersare usedinsteadof transformers.Digitizers may also be locatednear the hydrophonesso that the signal sent to the ship will not be degradedby transmission-linelosses. Hydrophonesrespondto changesin pressure,that

DETECTORS

is, to the accelerationof the fluid medium (seeeq. (2.67)).Moreover,the pressureand pressuregradienl are proportional to the time derivativeof a seismic wave(seeeq. (2.68));this meansthat for a harmonic wave of amplitude l, the hydrophoneoutput is pro_ portionalto jaA, that is,the output doubles(increases 6 dB) for each octave.Hence, the responseof a hydrophonediffers from that of a velocity geophoneby the factor j resulting in a 90" phase difference(see Al5.l.5),and a riseof 6 dB/octave(dueto r,r).Because pressureis nondirectional.a hydrophone'soutput is independentof wavedirection, whereasreversingthe direction of wave travel inverts a geophone'soutput ( s e e$ 7 . 5 . 5a n d 9 . 5 . 4 ) . 7.5.4Streamers Hydrophonesare mounted in a long .streamertowed behindthe seismicship at a depth between10 and 20 m. A diagramof a streameris shownin fie. 7.39.As of 1994,most streamershad 96 to 500 chinnels and wereup to 6000m in length. Streamersare constructedin sections25 to 75 m in length. Groups are usually 6.5 to 50 m in length,containingup to 35 uniformly spacedhydrophones.Data are digitizednearthe hydrophones with up to l2 .channelsper digitizer. Transmissionto the ship is often by fiber-optic cable at data rates of about 7 megabits/second. Transmissionin digital lorm elimir.rates distortion produced by leakageand transmission-lineeffects. The hydrophonesand other sensors,connecting wires,and stressmembers(to take the strain of towing) are placedinsidea neoprenetube 7.5 to 9.0 cm in diameter(fig. 7.a0).The tube is then filled with sufficient lighter-than-water liquid to make the streamerneutrallybuoyant,that is, so that the average densityofthe tubeand contentsequalsthat ofthe sea water.A lead-in sectionperhaps100m long is left betweenthe ship'sstern and the streamerproper. A depressorparavanepulls the streamerdown to the operatingdepth and a compliant sectionpreventsshocks causedby wave action on the ship from affectingthe active sections.Sometimesdead sectionsare inserted betweenactivesections.The last group is followed by anothercompliant sectionand a tail buoy (fig. 7.8) that floats on the surface.The tail buoy with a radar reflector is used to locate the farthest groups in the streamer,the direction from the ship to this buoy being determinedby visual or radar sighting. The tail buoy may also include equipment to locate it in the navigationsystembeing used.The tail buoy helps retrieve the streamerif it should be broken accidentally. Depth controllers(one is shown in fig. 7.40b)are fastened to the streamerat 5 to 12 places.These sense the hydrostaticpressureand tilt vanesso that the flow of water over them raisesor lowersthe streamerto the proper depth; they are ineffectivewhen the streamer is not in motion. The depth that the controllersseek to maintain can be controlledby a signalsentthrough

225 the streamer;thus, the depth can be changedto accommodatechangesin water depth or to allow a ship to pass over the streamer. When not in use, the streameris stored on a large motor-driven reel (fig. 7.25a)on the stern of the ship. The streamer contains several(perhaps l0) magnetlc compassesand water-breakdetectors(which respond to high frequenciescarried through the water from the seismicsourceor pinger sources)to help locatethe streamerduring recording(see$7.1.7).Depth detectorsverify the streamerdepth. A marine detectionsystempicks up noisesof severalkinds(Bedenbecker, Johnston,and Neitzel,1970): (l) ambientnoisedue to waveaction,shipping,marine life, and so on; (2) locally caused waterborne noisesuchas that causedby the turbulencegenerated by motion of the lead-in cable, depressorparavane, depth controllers, and tail buoy through the water, and energyradiatedfrom the ship becauseof propellers,motors,and other machinery;and (3) mechanically inducednoisetravelingin the streamersuch as resultsfrom cablestrumming,tail-buoyjerking, and so on. Usually, (3) is dominant except in rough weatherwhen ( I ) dominates.Towingnoiseis reduced by (a) making the streamersystemas smoothas possibleand keepingdepth controllersand other deviations from a smooth streamerat least 3 m from the nearesthydrophone,(b) usinga lead-insectionto increasethe distancebetweenthe ship and the nearest hydrophonegroup, and (c) using compliant and stretchsectionswith nylon rather than steeltensile members to reduce energy transmitted along the streamer.A separate,small, short streameris sometimes usedto record short-offsettracesbecausethere is usuallyan appreciable distancefrom the ship to the n e a r e sgt r o u pi n t h e m a i ns t r e a m e r . 7.5.5 Mutching hydrophoneand geophoneret:ords Becausehydrophonesrespond to pressurechanges and geophonesrespond to particle velocity, records recordedwith hydrophonesdo not match those recorded with geophones.The mismatchis most often observedin transition-zonework, whereland linesrecorded with velocity phonesare extendedinto marsh or shallowwater wherehydrophonesare usedat shallow depths.The mismatchcommonly showsas an apparentphaseshift of90. (fig. 7.41c). Reflectionfrom the surfaceofthe earth reversesthe polarity; this makes an upcoming positive reflection into a negativedowngoingreflectionor an upcoming compressioninto a downgoing rarefaction.A buried velocityphonewill seethe downgoingghost as having the samepolarity as the upgoing reflection(reversed in polarity twice, once by reflectionand once because it is traveling in the opposite direction), whereasa buried pressurephone,havingno senseas to the direction of wave travel, will seeit as having the opposite polarity.Where the ghost reflectionfollows soon after the primary reflection,the effectis similar to applying

i

I 226

E Q UI P M E N T Cablcrccl on stcrn of ship

Teil buoy wllh reder rcflcctot

Towin3bridlc Lcad-in t€ction

Dcprcssorparavanc

Dcpthconrrollcr

Comphanttcclion to rsoletcrlrcrmcr from shocktfrom ship

Dcad rccnon

LiYCscctroncontr|nrnt | 20- | 0Ohydrophoncsin | 2.5- | 00 mctcrslcntth. Croup I

Croup 2

7

L , as l B r o u p

Dcpih controll€ron dcadscclion

F i g . 7 . 3 9 M a r i n e s t r e a m e r (. F ' r o mS h e r i f l 1 9 9 0 . )

a derivative operator (as was seen in the reflection from the thin wedgeof fig. 6.40c;seealso gl5.l.7). Figure7.41showsa minimum-phase waveletrecorded with a pressurehydrophoneon the seafloor at various depths.A ghostfollowsthe eventdelayedby the twoway traveltimein the water,making the waveshape nearly the same as the derivativeof the waveshape from a surlacegeophone.The differencebetweenthe responseof hydrophonesand submergedgeophones can be utilizedto attenuateshostinsand reverberat i o n s( { 9 . 5 . 4 ) . 7.6 Recording 7.6.I Amplifier requircment.s A seismicamplifiermust amplify signalsovera range of frequencies and amplitudes. The liequencyrangeis relativelysmall and designingto cover this rangeis comparatively simple.Howeveqhistorically,the large rangeof amplitudes(ftg. 7.aD has presentedsenous designproblems. The lower limit of signal amplitudeis set by the noiselevelof the amplifier,around 0.2 pV for modern amplifiers;signalsweakerthan this are lost in system noise.The upperlimit, fixedby the geophones, is generally a few tenths of a volt. If we set this at 400 mV, the signafrange(or dynamicronge\thal the amplifier must acceptis 126 dB. Verticalstackingfurther increasesthis range.The newestseismicamplifiersuse 24 bits (23 magnitudebits plus one sign bir; 97.6.4) and havea dynamicrangeof 140dB. Accuracy of recording is also important. If we require 0.1'2,accuracy,we must have four significant figures,or l0 bits ($7.6.4), because2r0: 1000,and we requirea gain of 60 dB to achievethis accuracy.If we hope to recoversignalsbelow the noiselevelin subsequent processing,still larger rangesare required.An amplifierwith a fixed gain of -r dB and dynamicrange of y dB could amplify without distortion signalsas strong as (-t, - x) dB, whereassignals as weak as (60 - x) dB would be recordedwith the requisiteaccuracy.Thus, x should be small to recordwithout distortion, but this limits the accuracy;hence,fixed gain often is not desirable,and historically provision has beenmade to vary the gain during the recordingpe-

riod. Modern 24-bit instantaneousfloating-point (lFP) amplifiers have sufficientdynamic range to overcomethis problem.

7.6.2 Retordingin.\truments As statedin $7.6.I, recordinginstrumentsolten have to compressthe signalrangeas well as to recordvery weak signals.An amplifier is also usedto filter the geophoneoutput to enhancethe signal relativeto noise.Discussions of seismicamplifiersare given by E v e n d e na n d S t o n e( 1 9 7 1 )a n d P i e u c h o(t1 9 8 4 ) . Early recordingresultedin paperfield recordswith dynamic rangesof 20 to 26 dB (Pieuchot.loc. cit.); subsequentinterpretationhad to be done on these. Becausethe instrumentationcould not handle the broaddynamicrangeof signalsactuallyencountered. automaticgain control (AGC) was usedalmost universallyto compressthe signalrange.The adventof reproduciblerecordingin the early 1950sexpanded the rangeto 50 or 60 dB, but this rangewasstill inadequateand AGC continuedto be used.In order to obtain betteramplitudeinformation,effortsweremade to recordthe gain.Thesesometimes took the fbrm of gangedguin, wherethe gain of all channelswas made gain, where the gain the same,or of preprogrammed was determinedas a functionof recordtime beforea record was made.Thesetechniquescontinuedto be used with early digital recording,which had gain rangesof 80 to 90 dB. Digital instrumentsgenerally multiplexthe data beforedigitizingso that only a single channelis digitized.Today,digitizing is sulliciently rapid that one doesnot haveto havethe gain rangein the propergeneralmagnitudeaheadof time. Modern seismicamplifiersgenerallyemploy solidstatecircuitry, which allows them to be very compact and rugged.Although they are usuallycarried in a recording truck or other vehicle,they are smalland light enoughthat they can be hand-carriedwherenecessary

|f.e.1.a3). Today,most recordinginstrumentsoutput the data in digital format. However,we first describeanaloginstruments (which are still used somewhat)because they illustratebetter the functions that amplifiersare called on to perform.

R E C O R D IN G

227

F i g . 7 . 4 1 ) S c i s m i cs t r e a m e r .( a ) E l e m e n t s o f s t r e a m e r :a , p l a s t i c s p a c e r sc o n n e c t e d b y s t r e s sm e m b e r sb ; b u n d l e o f c o n d u c t o r s ( to carry data to the ship;rl hydrophone sensor.(b) Streamer in the water with d e p t h c o n t r o l l e r ,r , ( c o u r t e s yo f S e i t m i cE n g i n e e r i n g ) .( c ) C o n n e c t i n g s t r e a m e r s e c tions together (courtesy of praklaSeismos). 7.6. 3 Analog

recortling

A block diagramof an analogamplifieris shown in fig. 7.44;the numberand arrangementof circuit ele_ ments vary, of course.The cable from the geophones may be connectedto a balancecircuit that permitsadjustingthe impedanceto groundso as to minimizethe coupling with nearby power lines, thus reducing pickup of noiseat the power-linefrequency(high-tine pickup). The next circuit elementusually is a filter to attenuatethe low frequenciesthat arise from strong ground roll and that otherwisemight overdrive the first amplificationstageand introduce distortion. Amplifiers are multistageand havevery high maximum gain,usuallyof the orderof 105( | 00 dB), some-

timesas much as l0? (140dB); 100dB meanstnar an input of 5 pV amplitudeappearsin the output with an amplitudeof 0.5 V. Lower amplificationcan be obtained by means of a multiposition master garn switch. The gain is varied during the recordinginterval, starting with low amplificationduring the arrival of strong signalsat the early part of the record and ending with the high gain value fixed by the master gain setting.This variation of gain with time (signal compression)can be accomplishedwith automutrc gain (volume)control, usually abbreviatedAGC or AVC. This is accomplishedby a negativefeedback loop, a circuit that measuresthe averageoutput signal levelover a short interval and adiuststhe sain to keeo

EQUIPMENT

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the output more or lessconstantregardlessof the input level. lf the time betweena changeof amplitude and the consequentchangeof gain is too short, the output amplitude will be nearly constant and reflection eventswill not stand out; if the time is too long, subsequentreflectionswill not stand out. In either case.information will be lost. The use of AGC was standardprior to the 1960s,and AGC is still used, especiallyin making displays.A gain trace is plotted on the recordshownin fig.7.45. It is important in making corrections ficr nearsurfaceeffectsthat we be able to observeclearly the Jirst-breaks,the first arrivals ofenergy at the different geophones.(For a geophonenear the source,the first arrival travels approximatelyalong the straight line from the sourceto the geophone;for a distant geoohone.the first arrival is a head waverefractedat the

fbr a minimum-phase wave. (b) Amplitude spectra fbr a geophone and a hydrophone at 1.5-m depth. (c) Phase spectra for part (b).

baseof the low-velocity layer seethe discussionof weatheringcorrections, $8.8.2. If we allow the AGC to determinethe gain prior to the first arrivals,the low input level(which is entirely noise) will result in very high gain; the output will then be noiseamplifiedto the point whereit becomes difficult to observethe exact instant of arrival of the first-breaks.This problem is solved by using initial .suppre.s,sion or pre,\uppress ion. A high-frequency oscillator signal(about 3 kHz) is fed into the AGC circuit, which reactsby reducingthe gain so that the noise is barely perceptible;the high-frequencysignalis subsequentlyremovedby filtering so that it doesnot appear in the output. With the reduced gain, the relatively strong first-breaksstand out clearly.As soon as the first-breakshaveall beenrecorded,the oscillator signal is removed,usually by a relay triggeredby one of

RECORDING

E o E f

:

o. E

Tlme

(s)

Fig. 7.42 Amplitude of signals from short-offset (solid line) and long-offset (dotted line) groups. (From pieuchot, l9g4: 55.)

the first-breaks.Thereafter,the AGC adjuststhe gain in accordancewith the seismicsignal level (seethe gain trace in fig.7.45). Seismicamplifiersare intendedto reproducethe input with a minimumof distortion,and hencethe gain (without filters) should be constantfor the entire frequencyspectrumof interest.For reflectionwork, this rangeis about l0 to 100 Hz, and for refractionwork, the range is about 3 to 50 Hz; most amplifiers have flat responsefor frequenciesfrom about 3 to 200 Hz or more. Frequency.filtering refers to the discrimination againstcertain frequenciesrelativeto others.Seismic amplifiershavea number of filter circuits that permit us to reducethe rangeof frequenciesthat the amplifier passes.Although details vary, most permit the selection of the upper and the lower limits of the passband. Often it is possibleto selectalso the sharpnessof the tuto// (the rate at which the gain decreases as we leave the passband).Figure 7.46 shows typical filter responsecurves.The curves are specifiedby thetr cutof that is, the frequencyvaluesat which the fiequencies, gain has dropped by 3 dB (30% of amplitude, 50%of power);the curve labeled"Out" is the responsecurve of the amplifier without filters. Seismicamplifiersmay includecircuitry for mixing or compositingthat is, combiningtwo or more signals to give a singleoutput. Mixing in effectincreasesthe sizeof the geophonegroup and is sometimesusedto attenuate low-frequency surface waves. The commonest form, called 50% mixing, is the addition equallyofthe signalsfrom adjacentgeophonegroups. \{agnetic tape recordingnow has virtually eliminated the need to mix during recordingbecausewe can alriaysmix in playback. The time-breaksignaloften is superimposedon one ..f the amplifier outputs (on trace 3 in fig. 7.45),where :r appearsas a sharp pulsethat marks the point I : 0

for the record. When explosivesare being used, the output of an upholegeophone (a geophoneplacednear the top of the shothole)is also superimposedon one of the outputs (on trace 4 in fig. 7.45); the interval betweenthe time-breakand the uphole geophonesignal is called the upholetime (t"); it measuresthe vertical traveltime from the shot to the surface and is important in correctingfor near-surfaceeffects($8.8.2). For the first 30 years or so of seismicexploration, the outputs of the ampliflerswererecordeddirectly on photographicpaper by meansof a camera.However, about 1952,recording on magnetic tape began, and today it is nearly universal(seefig. l.2l). The feature that originally led to widespreaduse of magneticrecording was the ability to record in the field with a minimum of filtering, automaticgain control, mixing, and so on, and then introduce the optimum amounts of theseon playback.Later,a more important advantageturned out to be the ability to producerecordsections (see$8.8.3),which provedto be powerful aids in interpretation.However,magnetictape recordingdid not developits full potential until the introduction of digital techniquesduring the 1960s. Analog magnetictape recordersusually had heads for recording26 to 50 channelsin parallel.In the early years,direct recordingwas used;the output from the amplifier went directly to the recording head, the intensity of magnetizationof the tape being proportional to the current in the recordinghead and hence proportional also to the signal strength.Lateq direct recordingwasdisplacedby frequencymodulation and pulse-widthmodulation techniquesbecausetheseare more noise-freeand can accepta wider rangeof signal strengths.Today almost all recordingis digital. 7.6.4 Digital representation Digital recording was first introduced into seismic work early in the 1960sand by 1975was almost universal (seefig. 1.21).Whereasanalog devicesrepresent the signal by a voltage (or other quantity) that variescontinuouslywith time, digital recordir?g represents the signal by a seriesof numbers that denote valuesat regular intervals,usually 2 or 4 ms. Digital recordingis capableof higher fidelity than analog recording and permits numericalprocessingof the data without distortion. Howeverthe beginning(geophone response)and end (display) of the recording process continue to be analog. Beforedescribingdigital recording,we shall discuss digital representations.Although we could build equipmentto handledata usingthe scaleof 10,which forms the basisof our ordinary arithmetic,it is more practical to operateon the binarv scaleof 2. The binary scaleusesonly two digits, 0 and l; hence,only two different conditions are required to representbinary numbers,for example,a switchopenedor closed. Binary arithmetic operationsare much like decimal ones.The decimal number 20873is a shorthandway of sayingthat the quantity is equal to 3 units plus 7 X

230

EQUIPMENT

(a) Fig.7.43 Portable field recording systems.(a) Recorder lor 24b i t l 0 l 6 - c h a n n e l d i s t r i b u t e d s y s t e m .( C o u r t e s y o f H a l i b u r t o n Geophysical Services.) (b) Portable 24-channel signal-

enhancement seismograph incorporating liquid-crystal display ( t o p ) a n d t h e r m a l p r i n t e r .( C o u r t e s yo f E G & G G e o m e t r i c s . )

l0 pfus 8 x 10' plus 0 x l0r plus 2 x 101.Similarly, the binarynumberl0l l0l I is equalto l unit plus I x 2 p l u s 0 X 2 , p l u sI X 2 r p l u sI x 2 a p l u s 0x 2 s p l u s I X 26,which is the sameas the decimalnumber91. We can usepositiveand negativesquarepulsesto represent I and 0 or representthem in other ways.Each pulserepresenting I or 0 is calleda bit, and the series of bits that givethe valueof a quantityis calleda word.

digitization is done near the geophonesor hydrophones($7.5.2)in small units that handleonly a few channelseach. An analogmonitor ret'ordis made of the data written on the tape.The tapevaluesare read by read-afierwrite heads,demultiplexedto sort them into their respectivechannels,an AGC is usually applied and often filtering,and the resultingvoltagesare fed to the camera.This is often only an 8-bit procedurebecause of the very limited dynamic range (about 20 dB) of a paper display. Binary gain c'ontrol,wherethe gain was changedin stepsof 2, was usedin digital amplifiersbeginningin 1968.Samplingnear a zero crossingof the seismic trace could producedistortion if accommodatingthe big gain changerequiredby the amplitudeof the next sampletook too much time. In the early 1970s, responseshad become fast enough to adjust the amplitude without prior constraints and instantaneous foating-point (IFP) amplifiers (fig. 7.48)cameinto use.Theseconsistof a series of amplifier stagesall with the samefixed gain. Gates connect the outputs of the amplifier stageswith the systemoutput, only one gateat a time being open. If the output amplitude is above (or below) a preset range, the open gate is closed and an adjacent gate openedto decrease(increase)the gain. Early IFP amplifiersusedsevenamplifierseachwith a gain of 4 and werecalledquaternary-gain amplifiers.Many IFP amplifiers use 15 or more stages,each with a gain of 2, and today 24-bit (signplus 23 gain stages)systemsare coming into use.

7 6.5 Digitul instruments The stagesof a digital amplifier are, of course,analog prior to the digitizing unit. Thesesometimesinclude 669.7.a7) a lineflter to reduceradio-frequencynoise and a notchflter to removeexcessive high-linepicked up by the cableswherethe data are not digitizednear the detectors.A preamplffierincreasesthe gain by a constant amount before multiplexing and digitizing. A low-cutfilter is usuallyprovidedto supplementgeophone filtering in removing excessiveground-roll effects.An aliasfilter is always included to prevent aliasing (see $9.2.2c).A multiplexer connects the different channelssequentiallyto the digitizer; from here onward there is only one channel to the digital tape recorder. The digitizer (or analog-to-digital (AID) converter)holds signal voltagesin a sampleand-holdcircuit while the voltagesare comparedwith standardsin order to measuretheir values.The measurementoutput may be in the form of a gain word plus another value. The data are thenformatted (arrangedin the proper manner) and written onto magnetictape.Today,many systemsare distributedso that

MixinR from orlier c h an n c l s

Fig. 7.44

Block diagram of analog seismicamplifier

232

EQUIPMENT

Head-chsk pulse z

Time break Uphole break

brcak, tracc 5 break, trace 4

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Seismicrecord (playback). Courtesy of Chevron.)

Compact l2-24-channel signal-enhancement seismic recordersare availablefrom severalmanufacturers; one is shown in fig. 7.43b. Signal enhancemenl meansthe ability to vertically stack a number of individual records.Theserecordersinclude 12-15-bitdigitizers and storedata on floppy disks.Their frequency rangeis 3 to 5000Hz and they can sampledata at 0.5 ms. They display data both on graphic displaysand

hard copy using thermal printers.They weigh l5 to 25 kg and severalare expandableup to 120 channels. Some also include some processingcapability facilitating refraction interpretation or application of NMO correctionsto reflectiondata. Simplifiedtimersare sometimesusedin engineering refractionwork; they measureonly the traveltimesof the first arrivalsand displaythem in digital form.

RECORDING

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Tape transport lread-after-writg

al-to-analog((D/A; converter

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Fig.7 .47

Block diagram of digital recording system.

7.6.6Display The data recorded on magnetic tape must be pre_ sentedin visual form for monitoring and interpreta_ tron.The classiccameraconsistsof (l) a seriesof gal_ vanometers, one for each geophone group, that transformsthe electricalsignalsinto intensespots of light moving in accordancewith the signals,1i1a Ae_ vice for recording accurate time marks, and (3) a

means.for recordingthe positionsof the light spotson a moving pieceof paper.Historically,this was iccom_ plished mainly by photographic methods. More widely used today are electrostaticcameraswherern the light spot producesan electricchargeimage and printing powder adheresto the paper whereverit is charged.This dry-write processusesordinary paper, which is cheaperthan photographic paper and also

EQUIPMENT

(From (IFP)amplifier. floating-point Fig.7.48 Instantaneous 1984:177.) Pieuchot,

dispenseswith liquid developer-fixersolutions.Some plotters, especiallythose in fixed installations,are of the raster type wherein a matrix of very fine dots is usedto createthe image;a very fine beam of light (often from a laser)is sweptacrossthe paper,the beam being turned on and off very rapidly to produce the dots. With a raster plotter, the information from the variouschannelsis formattedin a microcomputerand are no longer used.lnk-jet individualgalvanometers plotters,whererasterdots are sprayedonto the paper, are sometimesused,especiallyto producecolor plots. Each individual graph representingthe averagemotion of a group of geophonesis called a truce. A simple graph of amplitude against arrival time is called a wiggly trace mode of display (fig. 7.49c). Where part of the area under a wiggly trace curve ls blacked in, the display is called variable-areu(fig. 7.49b);usuallypositivevalues(peaks)are blackedin. Sometimesthe light intensityis variedinsteadof the mode light-spot position to produce vuriable-densi4: ,f:9.1.a9il. Modes are also sometimessuperimposed (figs.7.49a and7.49e). Conventional black-and-white, variable-area/ wiggle-tracedisplayshavethreeseriousshortcomings: (a) black peaks and white troughs look so different that an interpreteris biasedtoward the peaks,neglecting the information in the troughs, and it is very difficult to comparethe relativeamplitudesof a peak and adjacent trough becausethey look so different; (b) high-amplitude peaks are clipped so that their magnitudesare lost; and (c) horizontal positions are carried sidewaysby the trace excursions.A color disBy play (e.g.,Plate2) correctsfor theseshortcomings. seeingadjacentpeaks and troughs with equal clarity, reflectionsfrom the top and bottom of a reservoircan often be recognizedbecausetheir amplitudesvary in unison. The mode of displayand parameterchoicesgreatly affectwhat an interpreterseesin the data. Among display parametersare horizontal and vertical display scalesand trace spacing;width, amplitude, and clip level (maximum amplitude that can be plotted) of wiggly traces; degree of blackness,bias (minimum value, which will be blacked in) and clip level of variable-areatraces;and so on. Usually,the effective

t

=w

[Jlilt]illllill F i g . 7 . 4 9 M o d e s o f d i s p l a y i n gs e i s m i cd a t a . ( C o u r t e s yo f G e o Space.)(a) Wiggle superimposed on variable area, (b) variable area, (c) wiggle. (d) variable density, and (e) wiggle superimposed on variable density.

vertical scale(time scale)is greaterthan the horizontal scale,that is, sectionsare horizontallycompressed.The vertical scaleis, of course,variablewith depth when time is plotted linearly,as is usuallythe case.Scaleratios of approximately1:l are most helpful when making a structural interpretation, but considerable vertical exaggeration is often used for a stratigraphicinterpretation.Color is sometimes superimposedon sectionsto display additional information. Problems 7.1 The velocity of radio waveshas the following values (km/s) over various terrains: normal sea water, 299,610: fresh water, 299,250; normal farmland, 299,400; dry sand, 299,900l-mountainous terrain, 298,800.Ifrange calculationsare basedon travel over normal seawater,what are the errors in rangeper kilometerof path over the various terrains?

PROBLEMS l3)

7.2 If the error in Shorantime measurements is -r 0.I p.s,what is the sizeof the parallelogramof error in fig. 7.3when (a) 0 : 30oand (b) 0 : 150.?Takerhe veloc_ ity of radiowavesas 3 X lOskm/s. 7.3 A satelliteis in a stableorbit around the Earth when the gravitationalforce(mg)puiling it earthward equalsthe centrifugal forcemV2lR, where.s is the ac_ celerationof gravity,m and V the satelliteimass and velocity, respectively,and R the radius of its orbit about the centerof the Earth. (a) Determinethe accelerationof gravity at the orbit -Earth, of a Transit satellite1070km abovethe know_ ing that g at the surfaceof the Earth is 9.gl m/s, and that the gravitational force varies inversely as the squareof the distancebetweenthe centersof gravity of the masses. (b) What is the satellite'svelocity if its orbit is stable? (c) How long doesit take for one orbit? (d) How far away is the satellitewhen it first emerses overthe horizon? (e) What is the maximumtime of visibilityon a single satellite pass?(Assume the radius of the Earth is 6 3 7 0k m . ) 7.4 Sieck and Self (1977)summarize,,acoustrcsvs_ tems,"as shown in table7.2. For each of thesecalcu_ latethe following: (a) The wavelengths. (b).Thepenetrationgivenby Denham,srule (gg.3.l l) and reconcilewith the statedpurposes. (c) Trade literature claims 30-cm resolution with im_ plodersand 2-5-m resolutionwith sparkers.How do these figures compare with the resolvablelimit ({i6.4.2)? (Note that absorptionin wateris very small, so that effectively absorptiondoesnot beginuntit the seafloor is reached.) 7.5 (a) An explosioninitiatedat the top of a column of explosivesof length atr. travelsdown the column with velocity 4. By comparison with the same amount of explosiveconcentrated at the centerof the column and explodedinstantaneously at the same time as the column,showthat the array response .Fis s i n c .F: [ n a ( s i n0 t ) V , l V , ) ] , \being the velocityin the rock,and 0,,the sameas ln fig. 7.14a.Under what circumstances boesthis result reduceto that of eq.(7.4)? (b) Calculate,Ffora column l0 m long giventhat : \,

= 2 . 1 k m / s ,a n d 0 o: 0 o ,3 0 o , ! 9 ^ , V , : 5 . 5 k m / s ,4 60.. 900. (c) If the column in (b) is replacedby six charges, each 60 cm lo.ng,equallyspacedto give a total len;th of l0 m, the chargesbeing connectedby spiralsof-detonat_ ing cord -with velocity of detonation 6.2 km/s, what length of detonating cord must be used berween adjacent charges to achieve maximum directivity downward? (d) What are the relativeamplitudes(approximately) of the waves generatedby the explosivesin (c) at angleso,, : 0o, 30o,60o,and 90owhen L- : 40 m? 7.6.Imaginean impulsivesourcestriking the ground at trmes nA apart, where r is a random number be_ tweenl0 and 20,and A is the samplinginterval.Given reflectionswith amplitude+5 at O,7Z at 5A, -l at l3A, +3 at 29L, +t at 33A, and -2 ar 42A^. add the reflection sequenceas would be done with Sosiere_ cording($7.3.2) for 10,20, and 30 impulsesto seehow the signalbuildsup as the multiplicityincreases. 7..7 How much energyis released(approximately)by thg lir eun arrayin fig.7.28cwhenthe initial pressure is 2000psi (14 MPa). (Energyreleased: work done by the expandinggas : J g d Z; Assumethat the changeis adiabatic,that is, gV' o : constant.the final pressureis 2 atmospheres,and that the guns are far e n o u g ha p a r tt h a t t h e yd o n o t i n t e r a c t . 7.8 The dominant period of a marine seismicwave_ shapeis often determinedby the sourcedepth, that is, by_thesecondhalf-cyclebeing reinforcedby the ghost reflectedat the surface.Assumingthat thij is true for the sourcesignaturesshown in fig. 7.30, determine their depths. 7.9 If we wish to take it into accountapproximately the small term L d3/dl3 in eq. (7.9) (still neglecting other terms involving L), show that for a harmonic wave, lt can be included in the term involvins /l in e q .( 7 . 1 2 ) . 7.10 A 96-channel streamerwith 25-mgroupshasthe hydrophonesspaceduniformly throughout its length. The lead-in and compliant sectionstogether are 200 m in length and the tail sectionand buoy connection are 150 m. Assumea ship'sspeedof 5.g knots (3.0 m/s) and a current perpendicularto the direction of traversewith a speedof 1.9knots. (a) What are the perpendicularand in-line compo_ nentsofthe distanceto the farthestactivegroup with

Table 7.2 Acoustic.systems System Fathometers Water-columnbubbledetectors Side-scansonar Tuned transducers Imploders Sparker

Frequency(kHz)

r2 80 2

l f

38 250 3.5-7.0 0 . 85 . 0 0.04-0. 15

Purpose To map water bottom To locatebubbleclusters,fish, etc. To map bottom irregularitres To penetrate30 m To penetrate120m and find gas-charged zones To map to 1000m

236 respectto the traversedirection? (b) Ifthe velocityto a reflector2.00km belowthe ship is 3.00km/s and if the reflectordips 20'perpendicular to the traversedirection: (i) By how much will the arrival time be changed for the far trace? (ii) If this should be attributed to a change in velocity rather than cross-dip,what velocity would it imply? (c) Assume that the amount of streamerfeathering (drift of the streamerto one side) is ascertainedby radar sighting on the tail buoy with an accuracyof only -r3o' (i) How much uncertaintywill this produce in locating the far group? (ii) How much change in arrival time will be associatedwith this uncertainty? (d) Over what distancewill the midpoint tracesthat are to be stackedwhen making a CMP stack be distributed? 7.11 Use figs. 7.35 and 7.46 to determine the filter equivalentto a geophonewith v0: 10 Hz and h : 0.7 feedinginto an amplifier with a l0-70-Hz bandpass filter and a 4-ms aliasfilter. 7.12 Figure 7.50illustratesfilter characteristics. Evaluate the importance of (a) low-frequency cut, (b) high-frequencycut, (c) bandwidth,and (d ) filter slope on; (i) time delay to a point that could be timed reliably;(ii) apparentpolarity;and (iii) ringing.The conclusionscan be generalizedfor filters of other design types. 7.13 Figure 7.51 showswaveshapechangesproduced by the analog filtering in modern digital instruments. What can you concludeabout the effectson picking? 7.14 Express the numbers 19 and l0 as binary numbers. (a) Add the binary numberstogetherand convertthe sum to a decimalnumber. (b) Multiply the two binary numbersand convert to decimal. (Note that mathematicaloperationsare carried out in binary arithmetic in the sameway as in decimalarithmetic.) 7.15 Assume a 96-channelseismicsystemrecording with 2-ms samplingand 25-sVibroseisrecords.What is the data rate (samples/second) and the number of bits/record?How doesthe data rate comparewith the capacityof a 9-track magnetictape moving ar a 6250How many bits bytes/inchrate, using 4 bytes/sample? of memory are requiredto store one channelof data? What is the effectof the headerand ancillarv information, and parity bits? References Andrew, J. A. 1985.The art and scienceofinterpreting stratigraphy from seismic data. ln Seismic Exploration of the Rocky Mountain Region,R. R. Gries and R. C. Dyer, eds.,pp. 95 104. Denver: Rocky Mountain Association of Geologists and the Denver Geophysical Society.

EQUIPMENT

bottom cable. Paper read at the 59th Society of Exploration G e o p h y s i c i s t sA n n u a l M e e t i n g . Barr, F. J., R. N. Wright, W L. Abriel, J. I. Sanders.,S. E. Obkirchner. and B. A. Womack. 1989. A dual-sensor bottomcable 3-D survey in the Gulf of Mexico. Paper read at the 59th Society of Exploration GeophysicistsAnnual Meeting. Bedenbecker,J. W., R. C. Johnston, and E. B. Neitzel. 1970. Electroacoustic characteristicsof marine streamers. Geophvsics,35: 1054-72. Brede, E. C., R. C. Johnston, L. R. Sullivan, and H. L. Viger. 1970. A pneumatic seismic energy source for shallow-water/ marsh areas. Geophys Prosp, 18: 581-99. Burns, R. F.1992. GPS receivers- A directory. Sea Technologl' (March):13-18. Dennison, A. T. 1953.The design ofelectromagnetic geophones. Geophys.Prosp., l: 3-28. Dixon, R. C. 1992. Global positioning system. ln Entyclopedia of Earth System Science,W A. Nierenberg, ed., pp. 395 407. New York: Academic Press Evenden, B. S., and D. R. Stone. 1971. Seismit Prospetting In' struments, Vol. 2: Instrument Perlormante and Testing. Berlin: Gebruder-Borntraeger. F a r r i o l , R . , D . M i c h o n , R . M u n i z , a n d P S t a r o n . 1 9 7 0 .S t u d y and comparison of marine seismic source signatures. Paper read at the 40th Society of Exploration GeophysicistsAnnual Meeting. Geyer, R. L. 1989. Vibroseis,Geophysics Reprint Series.Tulsa: Society of Exploration Geophysicists. Giles, B. F 1968. Pneumatic acoustic energy source. Geophys. Prosp.,16z21 53. Goupillaud, P. L. 1976.Signal design in the Vibroseis technique. Geophysics,4l: l29l 1304. H a r r i s o n , E . R . , a n d L . M . G i a c o m a . 1 9 8 4 .A n e w g e n e r a t i o n air gun. Paper read at the 54th Society of Exploration CeophysicistsAnnual Meeting, Atlanta. Ingham, A. 1975.Sea Surve.ying.New York: John Wiley. Jensen.M. H. B. 1992.GPS in offshore oil and gas exploration. 4. The Leuding Edga, ll(ll):30 Kramer, F S., R. A. Peterson, and W C. Walters, eds. 1968. Seismit Energy Sources- 1968 Handhutk Pasadena:Bendix United Geophysical. Kronberger, F. P, and D. W Frye. 1971. Positioning of manne surveyswith an integrated satellitenavigation system. Geophys. Prosp.,19:487 500. Lamer, A. 1970. Couplage sol-geophone. Geophvs.Prcsp., 18: 300-l 9. Mayne, W H., and R. G. Quay. 1971.Seismicsignaturesof large l162-73. air guns. Geophysics,36:. McQuillin, R., M. Bacon, and W Barclay. 1979.An Introdut'tion to Seismic Interprelation. Houston: Gulf Publishing Co Miller, G. F., and H. Pursey. 1956. The field and radiation impedance of mechanical radiators on the free surface of a semiinfinite isotropic solid. Proc Royal Soc., A-2232321. Miller, R. O., S. E. Pullan, J. S. Waldner, and F. P Haeni. 1986. Field comparison of shallow seismic sources. Geophysit's,5l: 2067-92.

Barbier, M. G., and J. R. Viallix. 1973. Sosie - A new tool for marine seismology.Geophysics,3S:673 83.

Mossman, R. W., G. E. Heim, and F. E. Dalton. 1973.Vibroseis applications to engineeringwork in an urban area. Geophysit's, 38: 489-99.

Barr, F J., and J. I. Sanders. 1989.Attenuation ofwater-column reverberationsusing pressureand velocity detectorsin a water-

Musser, D. D. 1992. GPS/DGPS in offshore navigation. positioning. Sea Technology(March): 61-6.

R E F E RE N C E S

O-60 Hr

6-60 Hr

18-6OHr

d-l-

6-36 Hr

6-50 Hr

6-@ Ht

;

l-

'tf_

*t-

t----

Fig. 7.50 Impulse responsesof minimum-phase filters. The respectiverows differ in filter slopesand the columns in passbands

(spccified by 3-dB points). (Courtesy of Grant-Norpac.) Efl'cct o f ( a ) l o w - c u t f i l t e r i n g a n d ( b ) h i g h - c u tl i l t e r i n g . Potler. M. C.. and J. L. Coldberg. 1987.Muthcmutirul Methotl,r. E n g l e w o o dC l i f l s , N . J . : P r e n t i c eH a l l .

0.2

I

P o u l t e r ,T . C . 1 9 5 0 .T h e P o u l t e rs e i s m i cm e t h o d o l - g e o p h y s i c a l e x p l o r a t i o n .G e o p h . r , s i cl 5s :. l 8 l 2 0 7 . P r o l f i t , J . M . 1 9 9 1 .A h i s t o r y o f i n n o v a t i o n i n m a r i n e s e i s m i c data acquisition. Thc Lcutling Edgc, l0(3):24 30 R a y l c i g h ,L o r d . I 9 l 7 . O n t h e p r e s s u r ed e v e l o p e di n a l i q u i d d u r ing tlre colfapsc of'a sphcrical cavity. Phil. Mug., 34: 94 8. S c h e r b a t s k o yS, . A . . a n d J . N e u l ' e l d . 1 9 3 7 . F u n d a n r e n t a lr e l a tiorrs in seismometry.Gutph.t'.ricl; 2: 188 212. S c h u l z e - G a t t e r m a nR . . 1 9 7 2 .P h y s i c a la s p e c t so f t h e a i r p u l s e r as a seismiccnergy source. Gaoph,t,,s. Pntsp.,20; 155 92. Sherifl. R. F.. 1974.Navigation requiremenls fbr geophysicalexploralion. Gcophys.Pntsp.. 22: 526 33. Sherifl, R. E,. l9ll9. Gutph.rsitul Methut,s. Englewood Cliffs, N . J . : P r e n t i c eH a l l . SherifT.R. E. I990. Entvktpcdit Dittionur.t of E.rplorution Gu> p / r l s l r ' . r3 d e d . T u l s a : S o c i e t yo f E x p l o r a t i o n G e o p h y s i c i s t s . S i e c k . H . C . , a n d G . W S e l l . 1 9 7 7 .A n a l y s i s o l ' h i g h - r e s o l u t i o n seismicdata. ln ^Str.rnrrlStftttigfttphl Applitutiort.sto Htdroturhon E.rpktrutiott, C. E. Payton. ed., pp. 351 86, AAPG M e m o i r 2 6 . T u l s a :A m e r i c a n A s s o c i a t i o no f P e t r o l e u mG c o l o gists. S p r a d l e y ,H . L . . 1 9 7 6 .A n a l y s i so f p o s i t i o n a c c u r a c i e sl r o m s a t ellite systenrs I, 1976 update. ln 1976 Olfslnre Tet'hrutlog.r' Conferenre Prt'prints, paper 2462. Dallas: Offshore Technology Conf'erence.

F i g . 7 . 5 1 F a r - f i e l d a i r - g u n s i g n a t u r e st h r o u g h v a r i o u s i n s r r u m e n t f i l t e r s .( a ) N o e x t r a f i l t e r i n g ;( b ) o u t 1 2 4 H z , l 8 d B / o c t a v e : ( c ) o u t 1 2 4 H 2 , 7 2 d B l o c t a v e ;( d ) o u r 6 2 H L l B d B / o c r a v e ;( c , 8 l 2 4 H z w i t h s l o p e so f l 8 a n d 7 2 d B / o c t a v eo n l o w - a n d h i g h f r e q u e n c ys i d e s ,r e s p e c t i v e l y(;f ) l 8 1 2 4 H z w i t h l 8 a n d 7 2 d B / octave slopestand (g) 8 62 Hz with 36 and 72 dB/octave slopes. Timing marks are l0 ms apart.

Parkes, G.. A. Ziolkowski, L. Hatton, and T. Haugland. 1984. The signatureofan air gun array: Computation from near-field measurements including interactions Practical considerations. Geophl sic.r,49: 105- I I . Pieuchot, M. 1984. Handbook oJ Geophv.sical Explorution, hl. 2 : Sei.smitInstrumentatiotL London: Geophvsical Press.

urul Nuvigution./or Geophl,sicul Spradley.H. L., 1984. Surve.t'irtg E.rplorutiort. Boston: International Human Resources Development Corp. W a s h b u r n ,H . W . 1 9 3 7 .E x p e r i m e n t a ld e t e r m i n a t i o no f t h e t r a n sient characteristicsof seismograph apparatus. Gt'opltl'sit.t,2: 213 52. Waters, K. H. 1987. Refettion Seisnnlog.t',3d ed. New York: John Wiley. Whitfill. W. A. 1970.The seismicstreamer in the marine seismic system. In 1970 Oll.shore Technology Con/erence Preprint,s, paper 1238. Dallas: Offshore Technology Conference. W i l l i s , H . F r .1 9 4 1 . U n d e r w a t e r e x p l o s i o n s T i m e i n l e r v a l b e tween successiveexplosiolts.British Report, WA-47: 21. Wood, L. C., R. C. Heiser,S. Treitel, and P L. Riley. 1978.The debubbling of marine source signatures. Geophysits, 13: 715 29.

238 Ziolkowski, A. 1980. Source array scaling for wavelet deconvolution. Geopl1,s.Prosp.,28: 902 18. Ziolkowski, A. 1984.The Delft airgun experiment. Firsr Break, 2(6):9 18.

EQUIPMENT

Ziolkowski, A.. G. Parkes, L. Hatton, and T. Haugland. 1982. The signature ofan air gun array: Computation from near-field measurementsincluding interactions. Geophysit.s, 47: l4l3 21.

8

Reflection field methods

Overview

Refraction data acquisitionis discussedseparately in chap. ll, 3-D acquisition in chap. 12, and S-wave,verticalseismicprofiling, and crossholeacquisition in chap.13.

Fieldmethodsfor the acquisitionof seismicreflection data vary considerably, dependingon whether the area is land or marine,on the natureof the geologic problem,and on the accessibility of the area.One of the most important aspectsin cor.rtrolling data costs 8.1 Basic considerations is avoidingdelayssuchas whensomephasesof opera8.1.I Data acquisition tions have to wait on other phasesbeforework can begin. High-qr-rality field work is essentialbecause Virtuallyall seismicacquisitiontodayis perlbrmedby nothingdone subsequently geophysicalcontractors, either for oil- or gascan remedydefectsin the b a s i cd a t a . E v a n s( 1 9 8 9 )a n d P r i t c h e t t( 1 9 9 0 )d e a l basislor subsecompanyclientsor on a speculative with fieldtechniques. quent sale.The latter probablyconstitutes20 to 25"1, The organizations of field crewswho acquireseisof U.S.acquisitionsas of 1994.Acquisitionmethods mic data and proceduresfor carryingout surveysare havebecomefairly standardized and,contraryto eard e s c r i b e dT. h e c o m m o n - m i d p o i n(tC M P ) m e t h o di s lier beliefs,clients generallyno longer believethat the field n.rethodusedalmostexclusively today.Usutheir own fieldmethodsprovidea significantcompetially,onewantsdatato be acquiredin the samemanner tive edge over their competitor's.Speculativedata along straightlir.resso that observedchangesin the costsconsiderablylessbccausecostsare distributed data may bc ascribedto geologicratherthan acquisiover severalclients,and lower unit costspermit acquiringmoredata.Wheretractscomeup for competition changes.Practicalconstraintsthat restrictacquisitior.r are discussed. tive bidding,companiesotien t-eelthat they have to An array of- geophonesusually l-eedseach data buy most availabledata to avoid the possibilitythat char-rnel; sourcearrays are also often used. Arrays their competitionhasan advantage. haverespclnse characteristics that dependon the specThe clatathat resultfiom wclrkdonelbr a soleclient trum and velocityof a waveand the directionfiom belongto the client,who can usethe data exclusively, which it comes;thesepropertiesare usedto attenuate trade the data fbr data owr.redby others,or sell the surveysbelongto the certaintypesof noise.The selectionof field paramedata.The data lrom speculative ters dependson both geologicobjectivesand noisc contractorwho paid for them or to the groupof comparriesthat subsidizedthe acquisition.The terms of conditions.Specialsituationsand objectivessomeas to timesrequirespecialtechniques, suchas undershootsaleusuallyplacerestrictionson the purchasers ing, crooked-linc,extendedresolution.and uphole who is permittedto seeand usethe data and also resurveys. strictionson luture salesby the data owners.In some Marine surveysacquiredata at a very fast rateand countries,data go into the public domain after some high hourly cost. fundamentalfacts that distinguish specifiedperiodol time. marine operations.Shallow water and obstructions Most acquisitionin the United Statesin 1994is on sometimescontrol acquisition.Specialmethodsmay a turnkeybasis,wherepaymentis on a per data-unrt be requiredin the transitionregionnear a coastline, basisratherthan a time-requiredbasis. composingthe surf zone,beach,and lagoonalareas ir.rlar.rd from the beach;in this zone.environmentsgen8.1.2 Crev'organi:ution erally changerapidly. Correctionshave to be made lor elevationand Seismiccrewsdiffer greatly in size,ranging lrom two weatheringvariationsto preventthem from influencor three people for a shallow land survey for engiing (distortingand sometimescompletelyobscuring) neeringobjectivesto more than a hundred peoplefor the reflectiondata on which interpretation is based. surveysin jungle areaswheremany men are required The correctionscalculatedby the field crew are the to cut trails and bring in supplies.Consequently,the Addifirst,and oftenthe most important.corrections. organizationof the crew varies,but thqse shown in for land crews. made fig. 8.1 are representative tional (or residual)correctionsare subsequently A supervisoqor porty chia/, usuallya professional in data processing. t-19

REFLECTION FIELD METHODS

240

8.l. 3 Environmentaland safety considerations

Supcr\i\or('r Pril)'ichrcl

t_jnc

Fig. 8.1

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sh(rrer

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( lhlt/gconhonc

S e i s m i cc r e w o r g a n i z a t i o n .

geophysicist,has the overall responsibilityfor a field crew.He is often assistedby an administratoror office manager, especially when many personnel are lnvolved. A party manageris usuallyresponsiblefor field operations.His main responsibilityis to obtain maximum production and adequatequality at reasonable cost.Other field personnelreport to him; he also hires field helpers.He is responsiblefor safety,equipment maintenance,maintaining adequatesupplies,paying bills, and operationof the field camp whererequired. The surveyorhas the responsibilityof locating survey points in their proper places.As the advanceman on the ground, he anticipatesdifficultiesand problems that the survey will encounterand seeksto avoid or resolvethem. This involvesinvestigatingalternattves so that the surveyobjectivesmay be achievedat minimum cost. He determinesthe best accessroutes for subsequentunits. He may be assistedby a permit man, who contacts land owners and tenants and secures permissions to conductthe survey.He is alsoassisted by rotlmenwho help with measurements.In areasof difficult access,he may also supervisebrush cutters and bulldozeroperatorswho clear the way. The observeris usuallynext after the party manager in field authority. He is responsiblefor the actual field layoutsand data acquisition,includingoperationof by a junior obthe instruments.He is usuallyassisted lay out the cable who hustlers of acrew serverand .iug and geophones. Other membersof a field crew vary dependingon the nature ofthe survey.A crew may haveone to four drillers,occasionallymore, and assistantsto help drill and haul water for the drilling operations,or two to perhaps five operators of surface source units (see for detonat$7.2.3and 7.3.1).A shooteris responsible ing explosivesat the proper time and for cleaningup the shothole area afterwards.Cooks and mechanics may be includedwhereoperationsare performedout of field camps. A marine seismiccrew usually consistsof a party manager,chief observeror instrumentengineer,three or four junior observers,two navigationengineers,a chief mechanic.and three or more mechanics'This is supplementedby the ship'screw of captain,mates,engineers,deck hands,and cook and mess/cabinattendants.

A seismiccrew is not only responsiblefor any damagesfrom its operations,but also for environmental considerationsand safety.Crewshavean obligationto minimize the environmentalimpact of their operations, which should be plannedand executedin sucha way as to minimize changesin the land (International Associationof GeophysicalContractors, 1993).This has not always been the situation; in some areas ln former times, the trails that seismiccrewscut (often "open up the coun5 m wide) were apt to be usedto try," but attitudes have changed and seismicoperations today should be as unobtrusiveas the work permits. Some of the tracks made years ago through forests,swamps,tundra, and desertsare still evident and are cited as argumentsagainstfuture work. New tracks should be of minimal width and the land should be restoredas nearly as possibleto its former condition. Crew and public safety also should be continually on the minds of all crew members.Crewsshould hold safetymeetingsperiodicallyto remind crew members of safety concerns.The safety manualspublishedby the International Association of GeophysicalContractors(l99la, l99lb) shouldbe reviewedby all crew membersperiodically.Every accidentshould be studied so that the causesmay be rectifiedin order to avoid similar accidentsin the future. Any outdoor work involvesdangersfrom falls,cuts,infections,insectbites, and poisonousplants,and any work with machinery involveshazards.Often, the greatesthazardsofall are relatedto the operationof vehicles. Public relations is another concern of field work. Courtesy calls should be made on those apt to be affectedby the field work or influential in informing the public, evenwhereone has no obligation to do so. Where field work is uncommon, an education program may be requiredto inform the public as to what seismicoperationsinvolve.

8.I .4 Condut'toJ a.fieldsurveY Most seismicfield crewstoday are operatedby contractorswho acquirethe data for client oil companies. Usually,the processbeginswith a bid requestsentout by a client. Ifexperiencein the surveyarea is lacking' prospectivecontractorsscout the area, often accompaniedby client personnel,to form opinionsabout the equipmentrequiredand problemslikely to be encountered.Thesemattersand any anticipatedconflictsare discussed.The contractor estlmatescosts, suggests and preparesa bid. The modificationof specifications, client evaluatesbids from the variouscontractorsand selectsthe bestbid. The client and winning contractor meet to resolvechangesin the specificationsthat may havedevelopedin the courseof the bidding and a contract for the work results. The contractor beginsequipment preparation and sendsan advancegroup to the field to arrange for

FIELD OPERATIONS FOR LAND SURVEYS officespaceand personnelaccommodations,commu_ nications permits, supply, storage,and repair facili_ ties,initiatespermitting operations,and reiruits local labor. A client representativemay participate in some of theseactivities.Once permitting is complete enough, survey layout is establishedand the main body of the contractor'sequipmentand personnelar_ rive. After some field experimentation,the survey proper gets underway. The data are preparedfor processingand periodi_ cally transferredto the processingcenter.The survey resultsare also regularlytransmittedto the client reo_ resentative.Unexpectedproblemswill inevitablyha;e arisenduring the surveythat will have to be resolved betweenthe party chief and the client representative. Thereprobablywill be modificationsor extensions to the program. Once the field work is concluded,the crew and equipment will be reassignedto the next prolect,and the party chiefwill prepareand presenta Iinal report to the client.

6.2 Field operations for land surveys 8.2.1 Theprogram The program of work is usuallydictatedby the clients, but the conduct of the work is the contractor'srespon_ sibility.Acquisition proceduresare often developedin meetingsbetweencompany and contractor geophysicists.A representativeof the client company 1..Ui.O_ dog") may be attachedto the field crew while the work is beingdone to monitor the work and alter the pro_ gram in the light of results.Speculative work is done in much the samemannerexceptthat the ..client"is the samecompany. Beforebeginninga survey,the questionshould be asked,"Is it probablethat the proposedwork will pro_ vide the requiredinformation?"Good practice(As_ n i c h a n d D u n l a p .1 9 5 9 )i s r o . . s h o o rt h e p r o g r a mJ n paper" before beginning the survey,estimatingwhat the data are likely to show,anticipatingproblemsthat may occur,asking what alternativesare availableand how data might be obtained that will distinsuish be_ tweenalternativeinterpretal.ions. Data migration(99.12)may requirethat linesbe locatedelsewherethan directly on top offeatures in order to measurecritical aspectsof a structure.Crestal areasmay be so extensivelyfaulted that lines across them may be nondefinitive. The structures being soughtmay be beyond seismicresolvingpower.Lines may cross features such as faults so obliquelv that their evidences are not readilyinterpretable. Lick of crosscontrol may result in featureslocatedbelow the seismicline being confusedby featuresto the side of the line. Near-surfacevariationsalong a proposedline may be so largethat the data are difficult to interpret. uhereasmoving the seismicline a short distancemay rmprovedata quality. Obstructionsalong a proposed line may increasedifficulties unnecessarilv.whereas moving the line slightly may achievethe sime obiec-

24r tives at reducedcost. Where the dip is considerable, merely running a seismicline to a wellheadmay not tie the seismicdata to the well data. Lines may not extendsufficientlybeyondfaults and other featuresto establishthe existenceof such features unambisu_ ously or to determinefault displacements. In geneial, linesshouldextendwith full coveragebeyondthe area of interestto a distanceequal to the target depth. 8.2.2 Permitting Once the seismicprogram has beendecided,it is usu_ ally desirable(or necessary)to meet with the owners and/or leasorsof the land to be traversed.permission to enter landsto carry out a surveymay involvea pay_ ment, sometimesregardedas advancepayment ..for damagesthat may be incurred." Even where surface holders do not have the right to prevent entry, it is advantageousto explain the nature of impending op_ erations.Of course,a seismiccrew is responsiblefor damagesresulting from their actions whither or nor permissionis requiredto carry out the survey. 8.2.3 Laying out the line

Once the preliminary operations have been comp_ leted, the surveycrew lays out the lines.This is often done by a transit-and-chainsurvey that determines the positionsand elevationsof both the sourcepoints and the centersof geophonegroups.The chain is of_ ten a wire equalin lengthto the geophonegroup interval. Successivegroup centersare laid out along the line using this chain, eachcenterbeing marked in a conspicuousmanner,commonly by meansof brightly colored plastic ribbon calledflagging. The transit is usedto keepthe line straight and to obtain the eleva_ tion ofeach group centerby sightingon a rod carried by the lead chainman. The survey may be tied to points that have been surveyedin with higher preci_ sion, perhapsby useof electromagnetic distancemeasurements(g7.l.l) or GPS (97.1.5),to avoidaccumu_ lating errors,and side shotsare made to relatenearby structures,streams,roads, fences,and other features to the Iine location.Radiopositioning systems(97.1.3) are sometimesusedfor horizontal control, especially in marsh and shallow-waterareas where eGvation control can be obtained from the water level. A surveyor'sfield notes should be sufficientlvcom_ pletethat anothersurveyorcan accuratelV.."onr,.rr", the surveylrom them.With elecrromagnetic surveying equipment, measurementsand survey notes may be recordedon magnetictapes or floppy disks that can be input into a personalcomputer after the day'sfield work. The computerthen reducesthe survey data, adclosureerrors,and plots updatedmaps daily. .1usts One of the surveyor'sresponsibilitiesis to plan accessroutes for the units that follow. In areas of difficult terrain or heavy vegetation,trail-building or trail-clearingcrewsmay be required.Theseare often under the direct supervisionof the surveyor.

242 8.2.4Field procedures When the energysourceis explosives,the surveyoris followed by shotholedrillers. Dependingon the number and depth of holes required and the easeof drilling, a seismiccrew may have from 1 to 10 drilling crews. Whenever conditions permit, the drills are truck-mounted. Water trucks are often required to supply the drills with water for drilling. In areas of rough terrain, the drills may be mounted on tractors or portable drilling equipment may be used. In swampy areas,the drills are often mounted on amphibious vehicles.In desertareas,air insteadof water or mud may be usedas the circulatingmedium.Where there is hard rock at the surface,percussiondrilling is occasionallyused;the drill tool is repeatedlydropped onto the rock to break it up. Usually,the drilling crew places the explosivein the holes before leaving the site. Drilling is often a major part of data-acquisition costs. When surface-energysourcesare used, there is of courseno shotholedrilling. The sources,often consisting of four to five truck-mounted units, move into position and await instructions from the recording crew Despitethe fact that no explosivesare "shot" and "shotpoint" are involved,terms such as "vibrator point" is used still sometimesused;often with Vibroseis. The recordingcrewcan be dividedinto threeunits: (l) the sourceunit responsible for positioningand activating the surface-energysources or for loading (2) the jug (when required)and firing the explosives; hustlerswho lay out the cables,place the geophones in their proper locations, and connect them to the pick up the geophonesand cables,and subsequently cables;and (3) the recordingunit that doesthe actual recordingof the signals. After the cablesand geophonesare laid out and tested,the observerchecksthat all geophonesare connected,that the amplifiersand other units of the recording systemare properly adjusted,and that everything is ready for a recording.Finally, he signalsthe sourceunits via radio or connectingwire (telephone) to activatethe sourcesor to fire the explosive. When all is ready for a shot (if explosivesare being used),the shooterarms his blaster,the deviceusedto set off the explosive,by a safety switching arrangement, and advisesthe observerthat he is ready.The "arm" button that causesa observerthen pressesan "tone" to be transmittedto the shooterand startsthe recording system.A signal sent from the recording equipment actually fires the shot. The blaster then transmits back to the recording equipment the shot instant (time-break). When a seismiccrew usessurface-energysources, the source units move into place and a signal from the recorderactivatesthe sourcesso that the energyis introduced into the ground at the proper time. The energyfrom eachsurfacesourceis usuallysmall compared to the energy from a dynamite explosion, so

REFLECTION FIELD METHODS that many recordsare made for eachsourcepointand subsequentlyvertically stacked($6.7.2)to make a single record.Severalsourceunits generallyare usedand theseusually advancea few metersbetweenthe component "subshots"that will be combinedto make one profile.It is not uncommonto usethreeor four source trucks and to combine 20 or so componentsubshots. After the data are recorded,the observerstudiesa monitor recordto seethat the recordis free of obvious defects.The monitor record is not usedfor interpretation, but may be usedto determineweatheringcorrecin $8.8.2.When finishedwith the retions,discussed cording at one source location, a roll-along sv,itch connectsthe proper elementsfor the next record and the sourcecrew moveson. Sometimesthis roll-along duty is performedby the instrumentsoftware.A computer doessomeof the checkingand recording. With the standard singlefold recording method (98.3.2)usedbeforereproduciblerecording,interpretation had to be done on the paper recordsobtained in the field, and considerableeffort was made to get would examthe bestrecordspossible.A geophysicist ine each record immediatelyafter it was acquired to decideon changesin recordingconditions.He would vary explosivesizeand depth,field layout,and instrument settingsin an effort to improve the record. Several shotsweregenerallytakenin eachborehole,drills sometimesstandingby to redrill a hole that might be lost. The high production and high efficiencyneededin order to achievelow cost per kilometer have altered field procedures.With common-midpointrecording, sourcepoints are closetogether,usually25 to 100m (75 to 300 ft) apart comparedwith 400 to 600 m for singlefold recording. The redundancy of coverage on any individualrecord,so lessensthe dependence that occasionalmissedrecordscan be tolerated.Also. the broad dynamic rangeof digital recording has removedmost of the needto tailor instrumentsettings to particularlocal conditionsand for filteringin the profield.The goal offield recording(and subsequent cessing)is generallyto haveconditionsthe samefor everyelement,so that changesin the data may be attributed to geologicchangesrather than changesin the field conditions. dictatethat the recordingoperCostconsiderations ation must not wait on other units.Shotholesmay be drilled for the entire line beforerecordingevenbegins so that the recorder never waits on the drills. Extra cablesand geophonesare laid out and checkedin advance.The roll-along switch makesit possiblefor the recording unit to be located physically at a place different from where it is locatedelectrically.The recording unit connectsto the cable at any convenient location, for example,the intersectionof the seismic line and a road. The roll-along switch is adjustedso that the proper geophonesare connected.The time betweensourceactivationsmay be only a few minutes and the recordingtruck may move only once or twice during the day.The shootingunit often walks the line

FIELD LAYOUTS becauseit needsno equipmentexceptthe blaster,and perhapsshovelsto fill in the shothole after the shot. The recordingunit does not have to traversethe line and so is subjectto lessabuse.Damagesare reduced becauseless equipment moves along the line. Thus, other benefitsaccrue besidesincreasedefficiencyof recording. Severalpoints should be noted in the foregoingdiscussion.Field operationsrequiremoving a seriesof units through the area being surveyed,and balance has to be achievedso that the units do not delayeach other, especiallyso that the recording unit is not delayed.Extra drills or layout personnelor overtimeare usuallyadded to achievethe requiredbalance.Crews often work irregular hours, working long days sometimes to make up for time lost becauseof weather.A variety of transport vehiclesare used: trucks where possible,marsh and swampbuggieswherethe ground is soft, tractors in light forests,boats,jack-up barges, air boats,helicopters,and so on. Generally,the energy sourceunits(drills,vibrators,and so on) arethe heaviestunitsand determinethe transportmethod.In some areas,operationsare completelyportable,everything, includingsmall drills being carried on men'sbacks. Transport often representsan important part of a crew'scost and determineshow much production can be achieved. Completerecordsshould be kept so that yearslater it will be possibleto determinefield conditionswithout ambiguity.Most of the routinereportingis done by computerlogging,but the field crew should specificallynote anythingunusual.The most important recordsare generallythose of the surveyorand observer,but drillersand other unitsshouldalso submit completereports.All reports should include the date and time of day and should be written as eventshappen ratherthan at the end ofthe day.The daily reports should include tape-reel numbers collated with sourcepoint numbers, specification of source and spread configurations,notes about deviations from surveyedpositions,information about all recordings, including repeats,all record settings,size of charge and depth to its top and bottom, any facts that affect the validity ofdata suchas electricalleakage,changes in surfacematerial,excessive noise,reasonsfor delays in the work. and so on.

E3 Field layouts 8.3.1Spreadtypes By spread,we mean the relativelocationsof the source and the centersof the geophonegroupsusedto record the reflectedenergy.Severalspreadtypesare shownrn fig. 8.2 and thereare many variationsof these.ln splitdip recording,the sourceis at the center of a line of regularlyspacedgeophonegroups;for example,if 120 groups are being recorded,the sourcewould be midway betweengroups 60 and 61. Howeveqthe source usuallygeneratesconsiderablenoise.and an adiacent

243 2a

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L. C r o s ss D r e o d Fig. 8.2 Types of reflection spreads.The symbols 0 and x representsourceand geophone-groupcenter locations, respectively.

geophonegroup often yields only a noisy and unuseable trace. The geophonegroups nearestthe source thus are often not used,producinga gapin the regular geophone-groupspacing.The sourcepointgap may be only a singlestation or a number of stations(resulting in a gapped split) with near-traceoffsets of 100 to 700m. Asymmetricalspreadsare common today.A common spread is the end-on,where the source is at one end of regularly spacedgeophone groups. This arrangement also often involves in-line offset of the source.Occasionally,the sourceis offset 500 to 1000 m perpendicularto the seismicline to permit the recording of appreciabledata before the arrival of exceptionallystrongground roll; suchspreadsare called broadside spreads and both broadside-T and broadside-Lspreadsare used, the former having the sourceoppositethe spreadcenterand the latter opposite one end of the spread. With cross-spreads,Iwo

244

REFLECTION FIELD METHODS

linesofgeophonesare laid out roughly at right angles so that three-dimensionaldip information may UeoU_ tained. Additional spreadarrangementsused in 3_D recordingare discussed in gl2.l.2 and 12.1.3. 8.3.2 Singlefold recording Virtually all routine seismicwork consistsof conttnu_ ousprofling, that is, the sourcesand geophonegroups are arranged so that there are no gaps in the data other than thosedue to the discretesamplingbecause of the geophone-groupinterval. prior to tte 1960s, each reflectingpoint was sampledonly once to yield singJefoldrecording.An exceptionwas that the points at the ends of a record (tie points) sometimeswere sampledagain with the adjacent record. Various ar_ rangementsof sourcesand geophonegroups are em_ ployed to achievethis. Singlefolarecordingis in con_ trast to common-midpoint recording where each reflectingpoint is sampledmore than o=nce. Continuous-coveragesplit-dip recording is illus_ trated by fig. 8.3a. Sourcesare laid out at Lsular in_ tervalsalong the line of profiling.often 400 to S+Oapart.A seismiccablethat is two sourceintervalslong is used.Provision is made to connectgroups of geol phones (for example,24 groups) at regulai intervals along the cable(called thegroup intervill. Thus, wrth sourcepoints400 m apart, 24 groups are distributed along 800 m of cablemaking the group centersabout 35 m apart. With the cablestretchedfiom point 0, to point O., sourcepointO, is used;this givessubsurface control (for flat dip) betweenA andB. The portion of cable^betweenO, and O, is then moved bitween O, Ooand sourcepointO. is used;this givessubsur_ 1nd face coveragebetweenB and C. The travel path for the last group from-sourcepointO, is the reversedpath for the first group from sourcepointO, so that the subsur_ face coverageis continuous along the line. The geo_ phone location at the sourceis often not recorded. 8.3.3 Common-midpoint method C,ommon-midpoint (C M p) or ..roll-along" recording (Mayne, 1962,1967)is illustratedin fig. il.4a. We have evenlyspacedgeophonegroups,which we shall num_ ber by their sequencealong the seismicline rather than by the trace that they representon the seismic record.Geophonegroups I to 24 are connectedto the amplifier inputs in the recording truck and sourceI is used. By assuminga horizontal reflector,this gives suhsurfacecoveragefrom a to g. Geophone groups 3 to 26 are then connectedto the amplifier infuts, the being made by meansof the roll_alongswitch :^h^"1g.: ($8.2.4)rather than by physicallymoving thJseismic cable.SourceB is then used,giving subsurfacecover_ agefrom b to i. SourceCis now uJedwith geophones 5 to 28, giving coveragefrom c to r, and ,J on dorvn the seismicline. Note that the reflectingpoint for the energyfrom sourceI into geophoneg.oup 2l is point

Fig. 8.3 Spreads to give continuous subsurface coverage.(a) . Symmetrical split spread where half of the spread is moved for_ ward lor successivesource locations. (b) End-on spread where sources are located at each end before the entire sbread is advanced; the source at O, will complete coveragefrom B to C

I, which is also the reflectingpoint for the energyfrom B into geophonegroup 19, from C into 17, from D into 15, from .t inro 13, and from Finto il. After removal of normal moveout, thesesix tracescan be combined (stacked)together in a subsequentdata_ processingoperation. ln this situation, the reflecting g9i1t/is sampledsix times and the coverageis called "6-fold" recording (sometimescalled 600%). Obvi_ ously,the multiplicity tapersoffar the endsof the line. Most present-dayrecordingusesat least l2_fold mul_ tiplicity, 24- and 4S-fold are common, and at times multiplicity exceeds500. To help keep track of the many tracesinvolved in CMP acquisition,stackingcharts are used (Morgan, 1970).A surface sracking charr (fig. g.4b) has geo_ phone location g as one coordinateand sourcelocation s as the other,that is, the trace observedat s from so-urce s is indicatedby the location (g s). A vaiiation of this chart, a subsurfacestacking chart (fig. g.4c), has the trace plorted ar [(g + s)12,sl. Occasionally,one of the regularlyspacedlocations will not^bea su-itableplace for u ,ouri. (perhapsbe_ causeof risk of damageto nearby buildingg and ir_ regularly spacedsourcepoints(or geophoie groups) -used, will be used. Thus, if point .E couid not be a sourcemight be located at E, insteadand then eeo_ phone group l4 (insteadof l3) would receivethe"en_ ergy reflected at / Figure g.4b shows the surface stackingchart when ,8, is used insteadof E Note in fig. 8.4bhow the six tracesthat havethe common mid_ point/line up along a diagonal;points along the op_ positediagonalhavea common offset,whereis poinis on a horizontalline havethe samesource.and points along a verticalline representtracesfrom a common geophonegroup. Stackingchartsare usefulin makine static and NMO corrections and ensuring that the

FIELD LAYOUTS

245 midpoint gather. The wavefield could aiso be represented by the samples at the same time for different locations,or as time s/lces(seealso $12.3).A threedimensionalrepresentationof data on a sinsle seismic -commonline, oriented in common-midpoint and offset directions, is sometimescalled ofset space. 8.3.4 Practical constraintsand specialmethods

",'i,

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tracesare stackedproperly. Geophonessamplethe seismicwavefieldat discrete locations,just as digitizing samplesa seismictrace in time (97.6.5).This spatial sampling obeys the sampling rules (59.2.2),and inadequatespatial sampling producesaliasing(that is, createsfalsedip alignments; seefig. 6.2)just as inadequatetemporal samplingcreatesfalsefrequencies. Each x on the stackingchart in fig. 8.4b represents an observedseismictrace that extendsin time. The data can be examinedin different directions,as indicated by the dashed lines; this proves useful in the study of noises,such as near-surfaceirreeularities. ghosts.multiples, and convertedwaves.Di-splaysol' the data in different directionsare called goihri, 1o, sometimesdomains);thus, a field record is a commonsource gather, but we can also make a commongeophonegather, common-ofset gather, or common-

(a) Gaps in coverage. As stated earlier, a common goal of field work is to have everythingthe same at each point along a line, so that an interpretercan attribute a changein the data to a changein the geology rather than changingfield conditions.However,uniformity is rare in land recordingbecauseaccessis restrictedat somelocations,perhapsbecauseofnearby wells or habitations. Wherecertainsitescannot be occupiedor wherethe sourceeffort has to be decreased,extra sourcelocations may be usedto compensateat leastpartially for decreasedmultiplicity or weakersource.The effort at nearby locationsmay be increased,the seismicsource offsetto the side,the line direction changedslightly,a dog-leg(jog) introduced in the seismicline, or some other effortsmade to partially compensate.Clear notation of the field changesshould be included in the field recordsand subsequentlytransferredto the seismic sectionsto alert an interpreterto the changes.Recording condition changesoften show on stackedsections by changesin the first-break pattern (seefig. 8.5). The ends of seismiclines produce differencesin multiplicity and data quality (fie. 10.3).To maintain multiplicity closerto the end of the line, extra source locations may be used with land recordins. Where end-on shooting is being used with the actiie spread preceding the source down the line (..pushine the spread"). the source units may proceed through the activespreadregion,which is held constant,when the end of the line is reached("shooting through the spread"). (b) Effect of direction of shooting. The direction in which a surveyis carried out can affectthe data quality. Dangerfield(1992)showslines run acrossan area where gas leaking from a reservoircausesdistortion (fig. 8.6); by comparison,lines run tangential to the gas area show a remarkable improvement in data quality. O'Connel, Kohli, and Amos fl992) show differencesin the quality of vertical sectionsfrom marine 3-D data volumes(fig. 8.7),wherethe acquisition directions differ by 90". A gather from an east-west line involves raypaths having different amounts of travel in the north-south salt body. (c) Undershooting. Long in-line or perpendicular offsetsare sometimesused where one cannot record over a desired region, perhaps becauseof structures, river levees,canyons, cliffs, permit problems, and so on. This technique is called undershootins.Under-

tfi

246

REFLECTION FIELD METHODS

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=s+*:#;3 =€E': o Fig. 8.5 Section in Ardmore Basin, Oklahoma. Lack of.access results in the V-shapedgaps seenat the top ofthe section. Data quality often deterioratesin such regions.ih" lin" shows a oosr-

shooting is used in the marine environment by two boats.thattravel parallelto eachother (fig. g.g) to ob_ tain data under platforms.Undershootingis uiro ur._ ful_whereraypathsare so distortedby shallowfeatures of limited extent that sensecannot be made of deeo events,as might be the situation in mapping under_ neath a salt dome,reef,gas leakingfrom-f deip reser_ voir,_orlocal region of very irregular topography or weathering. (d ) Crookedline methods. Becausemany interpreta_ tion criteria,suchas changesin dip rate,becomemore difficult to usewhen line directionchanges,effortsare made to keep lines straight. However,sometimesac_ cessand/or structuralcomplicationsmake it impossi_ ble to locate lines in desiredlocations.The fieid re_ cording may be done in the same way as CMp surveying, except that the line is allowed to bend (Lindsey, 1991),and the departuresfrom regularity are accommodatedin subsequentprocessing.The cor_ rect source-to-geophone distances(as opposedto dis_ tancesmeasuredalong the line) must be calculatedso that the proper amounts of normal moveout can be applied and the correct midpoints actually deter_ mined. Usually, a best-fit straight line (or ieries of straight-linesegments)is drawn through the midpoint plot (fig. 8.9), rectangularbins are c;nstructed, and those traces whose midpoints fall within a bin are stackedtogether.The bins are often perpendicularto the final line, but sometimesbins are oiiented rn the strike direction. The lateral extent of a bin mav be

zxtu

tive flower structure. (From Harding, Gregory, and Stephens, l 9 8 3 .)

made smalleras the expecteddip increases. Becausethe actual midpoint locations are distrib_ uted over an area,they contain information about dip perpendicularto the line and in effectproducea serles of cross-spreads, from which the true dip can be re_ solved.Lines are sometimesrun crookedintentionally to glvecross-dipinformation.

Fig. 8.6 Raypaths for seismic lines across (A) and tangential (B) to a gas-obscuredarea.

FIELD LAYOUTS

247 8,3.6 Unifurm linear arrays (a) Responseto harmonic waves. Arrays are used to discriminate between waves arriving in the vertical and horizontal directions.They are uniform andlinear when the elementsare spacedat equal intervalsalong the seismicline, or areal whenthe elementsare distributed over an area.The responseofan array is usually ilfustrated by the array response,defined as the ratio of the amplitude of the output of the array to that of the samenumber of elementsconcentratedat one location. Figure8.10showsan arrayof n identicalgeophones spacedat intervals Ax. We assumethat a plane harmonic wave with angle of approacho arrives at the left-hand geophoneat time I and that the geophone output is I sin tot. The wave arrives at the rth geophoneat time I * rAt, whereAr: (Ax sin ct)iZ; the output of the rth geophoneis I sin o(t - rL,t1: 1 sin (ol - r1), where"yis the phasedifferencebetween geophones,that is, successive : : 2ttv(A,r sin a)l V: (2nAxl\) sin ct ooAt f : 2rL,xllt,,, where L,, : L/sin o is the apparentwavelengthin eq. ( 4 . 1 3 b )T. h e o u t p u to f t h e a r r a yo f r p h o n e si s

I

n

l

h u ) : L , 4 s i n ( o -r r 1 1 ) |:o : I [sin (jr1)Ain (lr)] sin [t,ll j(,? l)^y] Fig. 8.7 The same east west line extracted from two migrated 3-D surveys, where the acquisiton lines were oriented respectively east west (upper) and north.south (lower). Data are better on the latter survey becauseraypaths did not have to penetrate the north south salt body (shaded). (From O'Connell. Kohli. and Amos. 1993.)

8.3.5Array concepts The term array rcferseither to the pattern of a group ofgeophonesthat feeda singlechannelor to a distribution of sourcesthat are fired simultaneously. It also includesthe nearby locationsof sourcesfor which the resultsare combinedby vertical stacking.A waveapproaching the surface in the vertical direction will affect each geophoneof an array simultaneously,so that the outputs of the geophoneswill combine constructively;on the other hand, a wave traveling horizontally will affect the various geophonesat different times, so that there will be somedestructiveinterference. Similarly, the wavestraveling vertically downward from an affay of sourcesfired simultaneously will add constructivelywhen they arrive at the geophones,whereasthe wavestravelinghorizontallyaway from the sourcearray will arrive at a geophonewith different phasesand will be partially canceled.Thus, arrays provide a means of discriminating between wavesarriving from different directions.

(seeproblem l5.l2c). The array output thus lagsbehind that of the first geophone;for n odd, the lag is that ofthe centralgeophone;for r even,it is the mean of those of the two central geophones.The array responseF dependson both n and 1: F : [ a m p l i t u d eo f h ( t ) l n A l : l s i n( j n 1 ) / [ ns i n ( ] 1 ) l l : lsin [(nnAx sin ct)/tr]{nsin [(nAr sin ct)/\]]l : lsin [rn(Ax/\)sin ct]/{n sin [n(Ax/\) sin o]]l ( 8 . 1)

S E AL E V E L

REFTECTOR

FIECTOR

Fig. 8.8 Use of two boats to obtain data underneath a marine platform.

248 REFLECTION FIELD METHODS

F : t , I - - - t - . 1 .

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with eq. (7.4), which givesthe response for !com93re a continuous array). Array responseis often plotted using . as abscissa \,, V. : Vlsin a (: uppu..ni;;i;;;;*. L"::,. .q. (:)3y. : L,rtl,x: isin ol/V, dip mo.veout lee1,rent ano so on, other quantitiesremainingfixed.or using the dimensionlessabscissa,Ax/),,,(see fig. g.l la). The graph usually consistsof a series'ol.i^irnu iroU"rt separatedby small values.For_A: : \,,, F : l,'giving the first alias lobe,and beyond.ttris. ttre'enti.epattern repeats. The lobes between the principal _iin tob, = (o 0) and the alias lobe ur. .ull"d iA" toi"rr. f* uniform spacing,the position "f th; fi;J';;;; o. tir. width of the principai lobe, aepenas ."'/lr, *rri"r,

has cross-dashes.showingthe output trace spacing. The solid rectanglesshow the bins (areasof midpoints combin'eain stacklng) to make a single outpu.t trace, one rectangle for prolectrng perpendicular to the line, the other for pro.leciing along srruc_ t u r a l s ( r i k e .( C o u r t e s yo f G r a n t _ N o r o a c . l

is one qeophonespacinggreaterthan the distancebe_ tween^theend geophones,(n _ 1) L.x; n Ax is called Ihe efective array length. For nonuniform arrays, the effectivearray length is raken ". th. ;;il^A.r of a uniform array whose principal lobe h?s ,h. ,urn. width at F : 0.7. The region between tt " poini, *t ".. the responseis down bt3 dB, that is, r.fr'.i. r = O.Z,

illil:9,th.l,on(t

resion(sometimes tt",e r.iectregion

to rhe 6 dB points, that is, F : Ir.:el'xed.wrth..r.:p:.t_ y.)i occasronally. it is definedby the nulls that separate the side lobes from the main jobe and-tfre principut aliaslobe). The nulls in fig. g.I I occur when the effectivearray , length is _anintegral number of waveterryths;wave peaks and troughs are then sampled .quujly io tfrut they cancelin the sum. An exception ,o ltiir-i".u., u, the alias lobe, where the wavelength equals,h-",pu"_ ol ttrg individual geophones;"""f. g."pi"i" tt.n lng records the sameamplitude so that utt iaa'in pt ur.. Ann1.91t wavelengthor apparentvelocity is olten , tne vanableto be studiedand array diagramsare of_ ten plotted with a linear wavelengit ,"i1., as in ng 8.12ainsteadof th'ereciprocalsca"le, ". ir'ng. b.f f ", and with a logarithmicverticalscalein aeciUeis. array responsecan also be plotted in polar form, as in fie.

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L

Apparentdip (ms/n,) (c)

Fig. 8. I I Responseof arrays to a 30-Hz signal. The alternatrve scales shown in part (a) apply to all three arrays. The effective length of the array controls the width of the main lobe. and the element spacing controls the location of the secondary (alias) peak. Weighting increasesthe attenuation in the rejeci region. The dashed curves indicate the array responseto a tell-shaped

Irequency spectrum peaked at 30 Hz with a width of 30 Hz. (Courtesy ofChevron.) (a) Five inJine geophonesspaced l0 m apart; (b) five geophonesspaced l0 m apart and weighted l, 2, 3,2, 1 (or nine geophonesdistributed among the five locations according to these weights); (c) nine geophones spaced 5.5 m apart.

2s0

REFLECTION FIELD METHODS

0 5 F

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60

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Apparentwavelenglh(m) (a)

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arguesthat, with common-midpointstacking,the entire spreadconstitutesthe effectivearray length, that is, all the tracesin a common-midpointgatherare lnvolvedin attenuatingground roll, air waves,and other noises.The stack array is a uniform linear array involving the entire common-midpointgather.This can be achieved in a number of ways with geophones spreaduniformly over the entire geophonegroup interval: (a) with a split spreadhaving sourceslocated midway between group centers and source spacing equalto the geophone-groupinterval,(b) with an endon spreadhavingsourcespacingequal to half the geophone group interval, and (c) in other ways. The NMO correction,sourcepointgaps,and minor variations of ground-roll propertiesalong the line usually of the stack array sigdo not lessenthe effectiveness nificantly. 8.3.7 Weighted(tapered)aruays

-(b)Fig. 8.12 Array directivity plots for five inline geophones spaced l0 m apart for 30-Hz signal.(a) Plot that is linear in apparent wavelength; (b) polar plot for velocity of 1.5 km/s; s o l i d c u r v e i s f o r 5 0 - m s p a c i n g( w i t h n u l l s a t I | . 5 ' , 2 4 ' , 3 7 ' , a n d 53") and dashed curve for l0-m spacing (null at 90").

8.12b.In this case,the radiusvectorgivesthe valueof ,Fas a function of the anglect. The case of a continuous sourcewas discussedin \7.2.2e for a vertical source,but the situation is the samefor a horizontal sourceexceptfor a 90orotation of fig.7.14. ( b) Response to transients. Actual seismic wavetrains are almost alwaysrelativelyshort transientsinvolving a spectrum of wavelengths (frequencies) rather than a single harmonic wave as usually assumedby array theory.The effectof changingapparent wavelengthis to stretchor compressthe array diagram. A transient wavelet can be thought of as a superpositionof different apparentwavelengthcomponents(the Fourieranalysisconcept,$15.2),eachof which would produceits array responsewith its peak amplitudeequal to the amplitude of the Fourier component, and the effectivetotal responsewould be the sum of these.This describesthe convolutionoperation ($9.2.1),and the array responseto a transientis obtained simply by convolving the harmonic array responsewith the wavelet spectrum. The effectiveresponsefor a bell-shapedspectrum is shown by the dashedlinesin fig. 8.1L Effectiverejectionis generally poorer (exceptin the alias-loberegion)than the rejection for a harmonic wave. (c) The stack array, The width ofan array reject region is proportionalto the arraylength.Anstey(1986)

Arrays where different numbers of elementsare located at the successivepositions are called tapered arrays. Compared with a linear array with the same overall array length, the main lobe and principal alias "reject lobes are broadened,but the responsein the generally array length The effective smaller. region" is is lessthan the actualarraylength.Figure8.1lb shows of a 1,2,3,2, 1 array(the numbersindithe response cating the number of elementsbunchedat successive locations).Taperingcan also be accomplishedby varying the outputs of the individual geophonesor by varying the spacingof the geophones.Arrays are also sometimesweightedat the endsof the array to attenuate long-wavelengthevents. Tapered arrays also result from combinations of sourceand receiverarrays,wherethe effectivearray is the result of convolving($9.2.1)the sourcearray with the receiverarray. The Vibroseisarrangementillustratedin fig. 8.13providesan example. 8.3.8Areal arrays The principal application of linear arrays is in discriminating againstcoherentnoise traveling more or less in a vertical plane through the array. Coherent noisetravelingoutsidethis planecan be attenuatedby an areal array (Parr and Mayne, 1955;Burg, 1964). Someareal arraysare shownin fig. 8.14.The effective array in a given direction can be found by projecting the geophonepositions onto a line in that direction; thus for the diamond array of fig.8.14a,the effective array in the in-line direction is that ofa tapcredarray , hereas 1 , 2 , 3 , 2 , I w i t h e l e m e nst p a c i n gL ' x : a l " , l 2w at 45o to the line the effectivearray is 3, 3, 3 (or the sameas a three-elementuniform array) with L,x : a. Where sourcesare locatedat different azimuths,as in land 3-D surveying, the differencesin array responsewith direction affect the componentswithin a bin differently and thus introduce undesired differencesamong the bins. An array such as the windmill

FIELD LAYOUTS 33m

--l t*

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F-300m-_-i Fig. 8.13 An arrangement used with surface sources such as Vibroseis.Four units 33 m apart lollow one another from left to right down the seismic line, operating simultaneousjy at loca_ tions spaced 16.5m apart. The positionsof the vibiators fbr successivesweepsare displaced vertically to avoid overlap. Re_ cords at six successivelocations are summed (vertically stacked) to make an output field record. The four central geophone groups (each a linear group 100 m long) are not used for each output record (becauseof vibrator truck noise). The recordins

array of fig. 8.149whoseresponseis nearlythe same in all directionsis suitablefor this situation.

8.3.9 Practicalconstraintson arrays Response diagramssuchas thosein figs.8.I I and g.| 2 apply equally to arrays of geophonesand arrays of sources.They also apply to the summing of tracesin vertical stackingor other types of summing, such as is done in data processing.Theoretically,we get the sameresultsby using I sourceand l6 geophonesas by using I geophoneand 16 sourcesspacedin the same manner and activated simultaneously.However, we use multiple geophonesmuch more than multiple sourcesbecausethe cost is usually less.In difficult areas,both multiple sourcesand multiple geophones are usedat the sametime. With most surfacesources. two to four units are used.The recordsfrom several successivesource locations not very far from each other are often summedto make an anay sum (vertical stack)and a sizeableeffectivesourcearray may be achieved in this way (fig. 8.13). Array summrng achlevesgreaterattenuationofrandom noisethan us_ ing simultaneousmultiple sources. The cancelingofcoherent noiseby using geophone and sourcearrays presentsa more challengingarray designproblem than does the cancelationof random noise. In the caseof random noise, the locations of the elementsof the array are unimportant provided

connectlons are advanced one group after source locations 6 and 12. The source locations used for one output record are shown by solid triangles. (a) Successivelocations alone the line of source units and active geophone groupsi 1b) effective array from combining the source and geophone arrays (the result of convolving them); numbers indicate the number of sweeoscon_ t r i b u t i n g t o e a c h p o r t i o n , t h e I o c a t i o n b e i n g r h a t f o r s o u r c el o _ c a t i o n sl 3 t o 1 8 .

no two are so closethat the noiseis identicalfor both. For coherentnoise,the size,spacing,and orientation of the array must be selectedon the basisof the prop_ erties of the noise to be canceled (Schoenberger, 1970).If the noise is a long sinusoidalwavetrain,an array consistingof n elementsspacedalong the direc_ tion of travel of the wave at intervalsof Lln, whereL is the apparent wavelength,will provide cancelation (seeproblem 8.6b). However,actual noise often con_ sistsofseveraltypesarriving from differentdirections, each type comprising a range of wavelengths;more_ over, the nature of the noise may changefrom point to point along the line. One sometimesresortsto areal arraysin areasof severenoiseproblems(althoughthe in-line distribution of elementsis almost always the most important aspect).Numerousarticleshavebeen wntten on the subjectof arrays;McKay (1954)shows examplesof the improvement in record quality for different arrays. In addition to the difficultiesin defining the noise wavelengthsto be attenuated, actual field layouts rarely correspond with their theoretical desien (see fig. 8.15 and Newman and Mahoney, 1973).tri.uru.ing the locations of the individual geophonesis not practicable.In heavy brush, one may have to detour when laying out successive geophones,and often one cannot seeone geophonefrom another so that even the orientation of lines of geophonescan be very irregular. In rough topography, maintaining an array design might require that geophonesbe at different

252

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(b) X-array;(c) rectangular 3 X 3 diamond; array;(d) crow's foot array;(e)odd-armstar;(f) herring-bone array;(g) windmill array.

elevations; this may produce far worse effects than those that the array is intended to eliminate. Similar problems arise where the conditions for planting the geophonesvary within a group (Lamer, 1970),perhapsas a result of loosesand,mucky soil, or scattered rock outcrops. The best rules for array design are often (l) to determine the maximum size that can be permitted without discriminating against eventswith the maximum anticipateddip and (2) to distribute as many geophonesas field economywill permit more or lessuniformly over an area a little lessthan the maximum sizepermitted,maintaining all geophoneplants and elevationsas nearly constant as possibleeven if this requiresseveredistortion of the layout (seealso $8.4.2).

Arrays may also be of value in refraction work (Laster and Linville, 1968). 8.3. l 0 Spatial samplingrequirements A successfulseismicsurvey should be designedwith the objectivesin mind and with some knowledgeof the geology.The subsurfacesamplinginterval should be small enough to avoid aliasing during processing and interpretation. The sampling theorem (59.2.2) statesthat signalsshould be sampledat leasttwice per wavelength.The highestfrequencyofinterest u-,., the velocity,and the maximum dip (hence,the maximum angle of approach) fix the shortest apparent wavelength and thus determinethe maximum permissible

SELECTIONOF FIELD PARAMETERS sub^surface spacing. The limiting value of the sub_ surfacespacing,D_.,, is, therefor!, D-* S (Ir")^J2 = tr_r(2 sin c_.") = (Vlv^ )l(2 sin o-u*) = (Vlv^^,)l[2V(At/Lx)^^*] < 10001[2v^^*(ArlAx)*"_], g.2a) eqs. (4.13b), (2.4), and (4.t3a)), 1.f,r:i ,t*":rsively wnereAmin rs the minimum wavelength,(L-)_,is the minimum apparentwavelengrh irieq. 1as ,-". t+.ii6), the maximum frequency,o** the maximum angte of approach,.lino_". : V(Atl[i)^_*, with the maximum apparent dip moveout, (L,tlAx,j*., given in _ittir."_ onds per unit distance.tt is prucieni io attow u-*urgin of safety becauseit is difficult to O.t.rrnin. ,-._ unO (Lt/Lx)*^^ exactly,and hence we often,p""ifil'tirr"" samples per shortest wavelength (S.ol;n, 'rsst), that is, D < 1000/[3u_.*(A/Ax)_.*].

(8.2b) r"mpli ng intervals computed according to ,t *r::?.,.. rne precedrngdesign^considerations generallyraige trom 10 to 100 m. Geological consiraint, 1fo. "*_ ample, the preknowledgethit there are no iarge Oips; can permit relaxing the spatial aliasing constraint. However,most data are migrated una _i!.ution algo_ rithms create noise where spatial ,urnpiinr'i, mua"_ quate;this may providethe limiting.onrt.uini. Intelli_ gent interpolation (g9.11.2)can bi used tolelax the spatial aliasingconstraint as far as rnig.ution is con_ cerned.

8.3.I I Extendedresolution Although conventionalgeophones and recordingsys_ j..: 1g usualtyadequatefor recordingup t; 125Hz '1*t (and higher) and normal alias filteis i"t "ut sharply above l/4A, where A is the sampling *t") p"._ mit recordins up to 2.50Hz fo. t; ;;;piing, tr,. bandwidth of most reflectionru.u.y, i, oiir?or, ,0 to 60 Hz. Becauseboth vertical una loriroritJ .esolu_ tions (96.4)are limited by the high_fr;;;;;;"r_p"_ must expand the passband upward ro ::,ilt, Y: acnteve hrgher resolution. Techniquesfor doing this are sometimescalled extendedresolution. limitations are usually due to (,,,1!_"-,llg.n-Uequency r, umttatlonsrn the source,(2) processes within the earth that discriminateagainst high fr"qu"ncies, (:t conditions at or near the surfaci, ln"tuaing o.ruy effects,and occasionally(4) recorAine inri.urn?n,r. Surface sourcesare often limited"with ..ro"", ,o high frequenciesbecauseof mechanicaf unj "Jupring as well as high near-surfaceutt.*uion lllbl.Tr fin comparison with a source in a borehole) resutting y: passesthrougtr the weathere
,s? sorption (52.7.2)and pegJegmultiples (g6.3.2b). The practical limiting factor is the amplituie of useful high-frequency reflection energy ""_p"r.a to the noise level. Denham (l9Sl) gives tf,. "_pi.l"ut io._ mula u_u*: l50lt, wherev-"* is the maximim useable frequency,and t is the two_waytime. AccorOing to this, the upper limit should be greater than iO H, fo, eventsshallowerthan t : 2.5,. The processes within the earth..arelargely beyond ou. .ontroi und set the utilmate tlmrt to the resolutionthat can be achieved. greatestsinglecauseof high_frequ"n.y tom i, ^The often ground.mixing owing to tie use'oi-g.opt on. arrays (and also source arrays); random tilme shifts with a standarddeviationof i ms u.. .quiuui.nt to u 62-Hz^filter(fig. 6.5I). Thus, although ul.uyr'p.ouiA" one of the best waysof attenuatings-urface i"uu., unO ambient noise,alternativemethois should fe useOf high resolutionis desired.Low_frequen.f nt.ring una burying the phones should b. "onrid.r"d (it e uest .e_ sults.are obtained by burying the phones Ulrow ttre weatheredlayer, but burial by only tO to 50-cm often achievesdramatic improvement).Consideration must arso be grven to spatial aliasing($g.3.10), which may dictate group spacing as small as-tS to'iO _. Sorn"_ umes a hybnd spread is used, group intervals being

'oTF:ltu. Iong-offset groups lseip.oUtem t. i:;.

Il the geophoneresponseis not adequate, acceler_ ometers (which have a frequency ..rponr. increasing 6 dB/octave relative to. velocity g..pf,"""r-iSZ.S.:ll b. usedeven though rnaintenan"ecosts increase. :,"^n wnen filtering is used to attenuate severe .iow--cut ground roll, a passbandofat least 2 octavesshould be retained in order to achievegood wavelet ,h;;; Significant improvement in high_frequ.i"V .._ sponse may be obtainable in some areas without marked increasein acquisition costs, although pro_ cessingcosts may increase.Extending tfr" i..-qu"n.y spectrumupward usually improvesUottr stratt# anO deepresolution.

E.4 Selection of field perametens 8.4.1 Noise analysis Systematicinvestigationof coherent noise often be_ gins with shooting a noiseprofile (also called,a miuo_ spre.ad walkaway).This is a small_scaleprofile with -or a single geophone per the geophones being -trace, spacedascloselyas I to 3 m over a to*talspreal Iength of the order of 300 m or more. If the weathering or elevation is variable, corrections should U" _uO" fo, each trace.The correcteddata, often in the form ofa recordsection,suchas shownin fig. g.l6a, ur" r,uai"a to determine the nature of the cohlerent events-onthe records,the frequenciesand apparentvelocities ofthe coherent noise, windows betwein noise trains where reflection data would not be overridden by su-chnoise, and so on. Oncewe havesomeindications of the types of noisespresent,we can designarrays o, oit e. neta techniquesto attenuatethe noise ani then n.ta_t"rt

REFLECTION FIELD METHODS

254

l

I

Wavelength(m) 100

50

33

25

20

c ] l n E

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*,

20

30r 0

o.o2 Wavenumber (cYcles/m)

Fig. 8. I 5 The theoretical responseof a I 2-elementin-line array (heavier curve) and as it was actually laid out on the ground.

our techniquesto seeif the desiredeffectis achieved. The data from noise analysesare sometimestransformed (see$9.1.3and 15.2.3)into the frequency wavenumberdomain and displayedas u r plots (often called/ k plots),suchas shownin fig. 8.16b.Radial lineson sucha plot show a constantproportionality betweenv and rc.,,that is, constant apparentvelocity (by eq. (2.4)).Suchu-r plots are helpfulin seeingthe characteristicsof different types of events and determining the best ways of attenuatingthe portions regardedas noise-dominatedin order to improve the signal-to-noiseratio (for example, with arrays, frequencyfiltering,apparentvelocityfiltering($9.9),and soon). 8.4.2 Determiningfield parameters Field parametersshould be determined in a logical way (Anstey,1970;Sheriff,1978),although existing how equipment(numberof channelsand geophones, cablesand geophonesare wired, and so on) usually prejudicesdecisionsconsiderably. 1. The maximum offset.the distancefrom sourceto the farthestgroup, should be comparableto the depth of the deepestzone of interest.This usually resultsin large enough nordal-moveout differencesto distinguishprimary reflectionsfrom multiplesand other coherent noise, but offsetsshould not be so large that reflectioncoemcientschangeappreciably,that conversion to shearwavesbecomesserious,and that the approximationsof the CMP method becomeinvalid. If data quality in the deepestzone of interest is sufficiently good, the maximum offsetmay be increasedup to the value of the basementdepth. 2. The minimum offsetdistanceshould be no greater than the depth of the shallowestsection of interest. noise Getting sufficientlyfar from source-generated sometimesdictates a greaterdistance,but this may causea loss of usefulshallowdata.

3. The maximum array length is determinedby the minimum apparentvelocity of reflections.The minimum apparent velocity usually occurs at the maxrmum offset.The shortestapparentwavelength(highest frequency) at this minimum apparent velocity should be just within the main lobe of the array'sdirectivitypattern(fie. 8.11). 4. The minimum useful in-line geophone spacing within arrays is usually determined by the ambient noise.Ambrentnoise,sometimesby source-generated noise characteristicscan be determined experimentally by recordingindividual geophonesspaced0.5 to I m apart to determinethe minimum geophonespacing for which the noiseappearsto be still incoherent. This minimum spacingis often 2 to 5 m, the smaller value being where noise is mostly generatedlocally (such as noise caused by the wind blowing grass, shrubs,or trees),and the larger value where noise is mainly caused by distant sources (such as microseisms,surf noise, traffic noise, and so on). If more geophones are available, an areal distribution of phonesat this minimum spacingwill be more effective than crowding the phonesclosertogetherin-line. Areal arraysare rarely requiredto attenuatenoisecoming from the sideof the line. Air-waveattenuationmay requireclosergeophonespacing. 5. Group intervalshouldbe no more than doublethe desiredhorizontalresolution,thus providingsubsurfacespacingequalto the desiredresolution.However, the group interval should not exceedthe maximum permissiblearray lengthindicatedby rule 3. 6. The minimum numberof channelsrequiredis determined by the combination of spread length and group intervaldecisionsalreadyreached. 7. The minimum chargesizeor sourceeffort is determined by the ambient noise late on the record. Random noise should not affect repeatedrecords until after a depth below the deepestsectionof interest;if this is not the case.the source effort should be increased.Figure 8.17 showsthe differencebetween6and l2-fold multiplicity for a particular prospect;the 12-fold data are sufficientlysuperior to warrant the additional effort. However, charge size and source effort are more often too large rather than too small. It should be rememberedthat line length, line orientation, and line spacingare also field parameters. The deteriorationof data quality near the ends of a seismicline can be seenin fig. 10.3. Figure 8.18a shows another example (in addition to figs. 8.6 and 8.7) of differencesin data quality as a result of line orientation,and fig. 8.18billustrateshow line orientation and spacingmight affect the interpretationof a l a u l t e dd o m a ls t r u c t u r e . Figure 8.19 summarizeshow the field parameters dependon the natureof the explorationproblem.Special circumstancesmay require variations from customary guidelines.For example,mapping a zone of

SELECTIONOF FIELD PARAMETERS

255

Offset (m) h N

a O

F F

F i g . 8 . 1 6 N o i s e a n a l y s i so r w a l k a w a y .G e o p h o n e sa r e s p a c c d 1 . 5m a p a r t ; t h e o f f s e tt o f i r s t g e o p h o n ei s 4 2 5 m . I d e n t i f i c a t i o n of'cvents: 1890 m/s is the refiaction liom base of weathering; 510 and 630 m/s are ground-roll modes; 330 m/s is the air wave; and 3140 m/s is a refraction event. (After Sheriff, 1991.) (a) Noise analysis and (b) frequency wavenumber sketch. A horizontal slice of the data diagrammed in part (b) can be taken by frequency bandpassfiltering, a vertical slice by the use ofarrays, and a pie-shapedwedge by apparent-velocity filtering (see also fig 9.36c).

maximum interestmay require recording in an offset window where desired reflections are between the wavetrains following the first arrivals and those causedby surfacewaves. Although one is usuallyconstrainedsomewhatby the equipmentavailable,it may be possibleto usegeophones with higher natural frequencywhere ground roll is especiallystrong or accelerometers wherehighfrequency responseis inadequate.Field parameter choiceswith marine surveysare apt to be limited by t h ee q u i p m e nat v a i l a b l e .

8.4.3Field testing Sustainingefficientfield operationswith many channels and changesin surfaceconditions, as generally

a o q

Wavenumber. x (rycles/m)

o) occur on land, is challenging.Recordingequipment today includes considerable computing capacity, which is usedto carry out a variety of tests,keep records of what was done, and assistin planning field designchangesthat may be required during the survey.The systemcan generatea varietyofcontrol plots that monitor operationsas they proceed. The recordingequipmentexecutesand analyzesinstrument teststo verify that the instrumentsare functioning properly.It checksthe entire system,including geophonesand vibrators.Simplifiedprocessingcan be carried out, including transformationsbetweentime, frequency, and f k domains, autocorrelationsand crosscorrelations, frequencyand velocity filtering, deconvolution, velocity analysis, muting, amplitude scaling,refraction statics,stacking,and other operations.

256

Fig. 8.17 An example of the results of 6- and l2-fold multiplicity.

Recordingequipmentalso doesmuch of the bookkeeping required. It can plot and update stacking charts as data are acquired. This capability can be used to show how stacking would be affectedby rearrangedsource-receiverlocations,which may be required becauseof accessproblems. The recording equipment and portable workstationscan also be usedto generateplots to assistin field design.Softwareis availablefor the design and display of in-line and areal arrays, correlating Vibroseisrecords,performing spectralanalyses,calculating and displaying frequency-wavenumber(J-k) information, plotting changesof amplitudewith trme or offset,and other analysistools. This capabilitycan be especiallyhelpful while experimentingto determrne optimum recordingparameters. 8.5 Defining the near surface 8.5.1 Upholesurveys An upholesurveyis one of the bestmethodsof investigating the near surfaceand finding the thickness(Dr) and velocity (V*) of the low-velocitylayer (LVL) and the velocity of the subweathering layer (Vr). An uphole survey requires a borehole deeper than the base of the LVL. Usually shots are fired at various depths in the hole, as shown in fig. 8.20a,beginning at the bottom and continuinguntil the shot is just below the surface of the ground. Usually, a complete spread of geophonesand an uphole geophone are used. Arrival times are plotted against source depth for the uphole geophoneand for severaldistant geophones,including two spacedat least200 m apart, as shown in fig. 8.20b. Sometimesthe procedureis reversedwith a sourceon the surfacerecordedwith a

REFLECTION FIELD METHODS downhole cable containing a number of sensors spacedperhaps 3 m apart. The plot for the uphole geophonechangesabruptly wherethe sourceentersthe LVL; the slopeof the portion below the LVL gives V", the break in slope gives D,,, and the slope abovethe break gives Zr. For distant geophones,becausethe path length changesvery little, the traveltimeplot is almost vertical as long as the source is in the high-velocitymedium. However, when the source enters the LVL, there is an abrupt changein slopeand the traveltimeincreasesrapidly as the path length in the LVL increases.The refraction velocity at the base of the LVL (V) is obtained by dividing the time interval betweenthe vertical portions of the curves for two widely separatedgeophones (for example Ar,, in fig. 8.20a) into the distance betweenthe geophones.The refraction velocity Vr. may be different from the velocity given by the slope of the deeperportion of the uphole geophone curve, VH,becausethe highest-velocityportions ofthe subweatheringcarry the refraction first-breakenergy, whereaslower-velocityportions also contributeto the uphole measurement; however, often VH. - Va and thesedifferencesare ignored. The valuesof Vr- and Vrmay also differ becausethe time interval /,, is usually more accuratethan that measuredby the uphole geophone. 8.5.2 Near-surJacerefiaction The contact between the near-surface,low-velocity (LVL) layer ($5.3.2)and the underlying rocks (subweathering)is usually fairly sharp and has a largevelocity contrast.Consequently,the first energyto reach most geophonesis head-waveenergy involving the baseof the LVL. Refractionfirst-breakinterpretation ($l 1.3to I1.5) are suitablefor determining techniques the vertical traveltimethrough the LVL and thus for correcting reflection traveltimes.The interpretation problem is often framed in terms of determiningthe LVL thicknessfrom which the LVL traveltimecan be determined,but the LVL velocity may also vary significantly,so that measuringtraveltimesavoidshaving to decide whether observedchangesare caused by LVL thicknessor velocity changes.When using surface sources,theremay be someambiguity about partitioning traveltimesbetweenthose under the source and thoseunder the geophoneswherethe LVL correction varies,but the relativepoint-to-point corrections, which are more important, are apt to be correct. Errors in the assumptionstend to causeLVL correction errors to accumulatealong the line, creating overall time-shift errors, but these usually do not cause serious problems. The most serious errors of this kind often have a wavelength about the same as the spreadlength. The first-break pattern can suggestchangesin the LVL, as illustratedin fig. 8.21for buried sources.Subsequentprocessingalso often employs refraction statics analysis($9.6.4)to refine staticsdeterminations.

STRIKE LINE

(a)

survey Reconnaissance

(b) Fig. 8.18 Line orientation and spacing effects.(a) Intersecting strike and dip land lines; the small reefseen on the dip line is not

evident on the strike line; and (b) maps resulting from differenl layouts of lines over a faulted dome are mapped diflerently.

258

REFLECTION FIELD METHODS Flrod-Galn

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(b)

(a)

Fig. 8.19 Determination of survey parameters.(a) Aspects of the exploration problem that define field parameters,and (b) determination of parametersfrom a seismicrecord.

E.6 Marine methods ional murineoparations 8.6.I Convcnt

Dr

Upholc gcophonc

,l

I Fig. 8.20 Uphole survey.(a) Traveltime versusdepth ofsource and (b) vertical section showing raypaths.

L.

Conventionalmarine operationsinvolveone ship that tows both the sourcesand hydrophonestreamer.The ship has to be sufficientlylargeto tow so much equipment, and henceefficiencyin acquiringdata becomes of high priority. With so much towed behind it, the ship must maintain its forward motion so as not to lose control over the tow and tangle the towed elements. With such a long tow, changesin direction must occur gradually,and henceit takes appreciable time to changefrom one line to another(usuallymore than I hour). The tows also restrict operations to water depthsgreaterthan l0 m and to areasthat are relativelyfreeof obstructions.These,then, are the major factors governingmarine data acquisition.Today, most 2-D marine data acquisitionemploysshipsthat are largerthan actuallyrequiredbecausethey are used mainly for 3-D acquisition($12.1.2),which involves towing more equipmentthan requiredfor 2-D acquisition. When on station, the streameqsources,and other towed equipment are unreeledfrom the stern of the ship; this typically requires 4 hours or more, de-

\I.\RINE METHODS

259

Tu-

1

t

t

l

l

v

I '2

^

tt

.z

.z

To,,, \i---T-T-f-

.t

a

2

1

(d) Fig-8.21 . . F i r s t - b r e a kp a t t e r n sf i o m a b u r i e d s o u r c ef b r v a r i o u s L V L c o n d i t i o n s .( a ) ' . N o r m a l , ' p a t t e r n f i o m s o u r c es l i g h t l y b e _ low the base LVL, (b) from source deep within rhe subiveather-

pendingon difficultiesin ballastingthe streamer so that it is neutrally buoyant. Navig-ationof the ship dylilg recording is done by a navigation computer, which steersthe ship and, at the proper times, rssues "on-location" commandsto start the recordingsys_ tem and fire the sources.The helmsmankeepsoverall watch so that he can take over steeringof tire ship if somethingunforseenshould occur. The recording of the various sensorsis largely automatic,the chief function ofthe observers,navigators,and others beins to seethat everythingis functioning properly. Marine seismiclinesare usuallyfiolyiong to mini_ mize the fraction of time spent getting into"position. Minor failuresduring the acquisitionof a line of data, such as the failure of one or two hydrophonegroups or air.gunsin_an array or of a missed.ou.". loJatron, are toleratedbecauseof the hugecost in time required to remedy the situation. Normal operationsproceed at about 6 knots (l I km/hr or 3 m/s) on an around_ the-clockbasis.Hence,about 250 km of CVp Oata

(D rng, (c) from double-weathering layers, (d) source in the LVL, (e) low-velocity layer below the first high_speed refractor, and ( f ) s h o r t h i g h - v e l o c i t y" s r r i n g e r " w i t h i n o r n e a r r h e b a s eL V L .

could be recordedin a day if alt the time were spent acquiring data. Of course, this much production is neverachievedbecauseof the time spentgoing to the surveyarea,moving from line to line, repairingequip_ ment, waiting on good weather,and other faitors. A ship typically collects data about 60% of the time. With sourcesactivatedevery l0 to 15 s, sourceloca_ tions are spacedabout 30 to 45 m apart. Records (sometrmescalledsubshols)may be vertically stacked as the data are acquiredor recordedindividually for combininglater.At regularintervals,a few recordsare given a cursory examinationto detect fairlv obvrous defects,but detailedmonitoringis not possibleat the pace of operations.Nevertheless,the relativelycon_ stant water environmentsurroundingsourcesuna ny_ drophonesand the generalabsenceof a weathenne layer usuallyresult in uniformly good data quality. A variation occasionallyemployed is to tow the streamerat an angle with the water surfaceso that ghosting effects(96.3.2b)differ for the different hv_

t

I

260

drophone groups; these effects are then attenuated when the data are stacked.Another variation used to reduce ghost effectsis to tow two streamersat different depths.Sometimesa short ministreameris towed at shallower depth to record near-offset data, which may be missing with the regular streamersthat are towed deeper. A tremendousamount of data is generatedby a marine survey with data obtained from a large number of sensors.Essentially,all the data are recordedautomatically on magnetictape to be read and digested later by computer programs.A modern 3-D seismic ship, for example, towing multiple sources and streamerswith perhaps500 channelseach, generates a hugevolume of data with subshotsperhapsevery l5 to 20 s. With redundant navigation systems and source-streamerlocating systems,such as shown in fig.7.9, the volume of navigationdata may equal the volume of seismicdata. 8.6.2 Shallow-waterand obstructedoperations For operationin shallowwater.drag cablescarrying hydrophonesor gimbel-mountedgeophonescan be pulled along the sea floor. Cablescan be laid on the bottom in areas where obstaclesrestrict operations, the cablesbeingconnectedto recordingequipmenton an anchoredrecordingship or barge,with a separate mobile boat usedfor the source. 8.6.3 Pro.filingmethods Environmentaland engineeringobjectivesusuallydo not require appreciablepenetrationof seismicenergy and hencedo not requirethe strongsourcesnor elaborate detectionsystemsused in hydrocarbonstudies. We use the general term prcfiling for such methods. Profiling involvessmallerequipmentand doesnot require large,dedicatedships.Equipment is usually installedon boats of opportunity (leasedlocally). Profilingalmost alwaysinvolvesa trade-offbetween the use of small, high-frequencysourcesto achieve high resolution and larger sourcesto achievegreater penetration. Profiling covers a broad range of operational sophistication,ranging from the use of single detectors that simply record reflections from the sea floor and a thin veneerof sedimentsbelow it to the use of streamers employing the commonmidpoint method to achievepenetrationsof the order of 1000m. Thus, thereis no sharpdistinctionbetween profiling and full-scale marine operations.Profiling systemsare summarizedin table 7.2. Profiling systemswere entirely analog for many years.The simpler systemsemployeda singlesource and single hydrophone group whose output was merely amplified and plotted. One simple strip recorder useda wire wrappedaround a cylinderin a helical manner;the electricalcurrent from the detection systempassedthrough the wire and completedits circuit through a metal plate tangent to the cylinder.A

REFLECTION FIELD METHODS

record of the current was made on electrosensitivepaper passingslowly(in the x-direction)betweenthe cylinder and the plate. The source was fired when the wire touched the top of the paper,and the plot point movedin the y-direction acrossthe paperas the cylinder rotated.Reflectionswith arrival timesgreaterthan the cylinder rotation time were sometimeswritten on top of the recordingof shallowerdata, an effectcalled paging (seefig. 8.23). Today, both small air guns (57.4.3) and other sourcessuchas describedin $1.4.4 are used.Sea-floor multiples usually are quite strong, becausesystems with only a single detector cannot discriminatebetweeneventsbasedon their normal moveoutor directions of arrival, and data arriving after the sea-floor multiple are often so contaminatedthat they are unuseable.Data are sometimesrecordeddigitally so that they can be processed.The major amplitude loss in travel through the water is due to geometricaldivergence,there being little absorption and loss of high frequenciesuntil waves enter the sediments;towing systemsclose to the sea floor reduce lossesbecause of divergence. Profiling is extensivelyused in engineeringstudies to determine the nature of sediments(fig. 8.22), for example,to find sand layersthat can support structures erectedon pilings. It is usedto locate pipelines buried in the mud. An important applicationis in surveyingfor hazardsthat might causetrouble in drilling, such as foundation problems,gas seeps,shallow gas accumulations,buried channels,and faults.Becauseit is relativelyinexpensive,profiling is extensivelyused in oceanographicstudies(fig. 8.23). Profiler operationsoften use a number of different kinds of sensorssimultaneously(figs. 8.24 and 14.3), including magnetometersand sniffers(to sample the seawaterfor analysis).One profiling variation is sidescan sonat transducers in a towfish radiate short sonar pulsesthat scanthe seafloor to the sideofthe fish and record energythat is back-scatteredfrom the sea floor (fig. 8.25).The result is essentiallya map of the sea-floor relief showing reflectors such as rocks, ledges,metal objects,and sand ripples as dark areas, and depressionsand other featuresas light areas.The usefulcoverageis out to about three times the height of the towfish (often about 20 m abovethe seafloor), with a gap of no coverage underneath the towfish. Data can be corrected for divergence,slant range, and variations in boat speed. The synthetic-apertureconcept is sometimesused to sharpen the in-line footprint; the directivity of a radiation pattern is inversely proportional to the antenna size,and the successive imagesobtainedas the towfishmovesthrough the water can be combined to effectivelygive a source array and thus improvedirectivity. 8.7 Transition-zone operations A transition zone is a region where environments changerapidly.Usually,we try to minimizeacquisi-

i

D - A T AR E D U C T I O N

tion changesto avoid misinterpretingtheir effectsas geologicfeatures,but changesare necessaryin transition zones.The most common transition zone is the region near a coastlinewhere a survey area may in_ cludeshallowwater,surf zone,beachdunes,mud flats, lagoons, marsh, and swamp. These different environments are apt to require quite different kinds of equipment. Energy sourcesmay be those ordinarilv used on land (dynamite in boreholes and Vibrosels,for example) and at sea (air guns). Explosive detonating cord might be used where the water is not deep enoughfor the useofair guns.Velocity geophonesof both conventionalland types and gimble-mountedin bay cablesand hydrophonesin both marsh casesand streamersmight be usedin different parts of the area in conjunction with severalkinds of sources.Telemetry might be used to get information from the detec_ tors to the recordingunit. Means of moving about the region may include trucks, small boats, jack-up barges (to raise them abovewaveaction), rubber boats,pontoons,and amphibious craft such as hovercraftand marsh buggies, and parts of the area may haveto be portable.Loca_ tions and elevationsmight be determined partly by marine positioning and partly by conventionalland surveymethods.

8.8 Data reduction 8.8.1Fieldprocessing The first stepin either field or processingofficeis usually to check the inventory of data to make certain that all data are in hand. Checksfor data consisrencv may also be made. Field processingusually includesvertical stacking and sometimes other processing. Vertical stacking usuallyimpliescombining recordsthat wereobtained without any major movementof sourcesor detectors and without any differentialstaticor normal-moveout corrections made to them. Especially with surface sources, four to six source locations within the geophone-groupspacingdistanceare occupiedto give the effectofa sourcearray (as in fig. 8.13).Occasionally, vertical stackingalso includescombininerecords from sourcesat severaldepthsin a boreholea-ftertime shifting according to differencesin uphole time. Instrument software carries out testing and analyzing the test results. Field recordinginstrumentsoften havethe capability of doing appreciableprocessing,but usually field personneldo not havetime for doing anything except almost automatic routine processing.The correlation of Vibroseisrecords,the summationof Sosierecords, and demultiplexingto trace-sequential format are examplesof suchroutine processes done in the field. The processingcapabilitiesof field instrumentsare utilized during field experimentalperiods and sometimesat night when surveyresultsare neededquickly. Marine

261 operations may involve more extensive processing and interpretation. Field recordsof all kinds must be completelyidentified. "Observerbsheets"are usedto record field data such as (a) company and prospect names; (b) line, profile, and magnetic tape numbers; (c) information about sourceand activegeophonelocations;(d) number and spacingof geophonesand geophonegroups; (e) source pattern or size and depth of explosive charges;(f) instrument settings;(g) time of the recording; (h) history ofany processingdone; and especially (i) anything unusual, such as changesin source locationsbecauseof accessproblems.Computersmay automatically record much of this data. In earlier times, the appropriate information was written on the back of paper records immediately after recording. 8.8.2 Elevationand weatheringcorrections The first calculation of corrections for elevation and weatheringvariationsis carried out in the field. These field statics are based on survey, uphole, and firstbreak information, and are subsequentlyusedin processingas the first estimate.Marsden 0993) reviews staticscorrections. Variations in the elevationof the surfaceaffect traveltimes and it is necessaryto correct for such variations as well as for changesin the low-velocity layer (LVL). Usually, a referencedatum is selectedand corrections are calculated so that in effect the source and geophonesare locatedon the datum surface,it being assumedthat conditions are uniform with no LVL material below the datum level. Many methods exist for correcting for near-surface effects.Theseschemesare usually basedon (l) uphole times, (2) head waves from the base of the LVL (refraction statics),or (3) the smoothing of reflecrions. Methods (2) and (3) are employedin automaticstatics processing($9.6),often on an iterative basis; these generally involve some statistical smoothing. In the following we shall assume that V* and V* the velocitiesin the LVL and in the layerjust below it, respectively,are known, having been determined from an uphole survey or the refraction first-breaks. With explosivesin boreholesas the source,we also assume that the source is below the baseof the LVL; if this is not true, some modifications have to be made in the following equations(seeproblem 8.18).Similar modifications have to be made when surface sources are used. Figure 8.26 illustratesa method of obtaining the corrections for lo, the sourcearrival time. E, is the elevation of the datum, E" the elevation of the surface at the source, ,8" the elevation of a geophone group, D, the depth ofthe source below the surface, and /,, the uphole time, the time required for energyto travel vertically upward from the source to a geophone on the surface (in practice, the uphole geophone is within a couple of meters of the shothole).The deviation of reflection paths from the vertical is usually small

262

REFLECTION FIELD METHODS

6

7

Shotporntsabout 300 m apail Holoccnc

IrJnrgrc\\ron

0 channel barid^oned ii,\-t \.".'i"'"'"", -\'{, fill r clat t --'/ r'itr.- n,tr1\:-

l0

-__=+-_-\.\_-

Delta fringe deposrtsof c l a y s .s r l t sa n d s h e l l s

tu D i \ t f l b u l a r \ c h x n n e lc o m p l e x o l s a n d .s r l t a n d c l a r

Drslortrons produced by s h a l l o wv e l o c r t l v a r r a t r o n s

20

g

'

30 40 d 50

08^ !

E

toF

l4

':_-i

Fig.8.22

ii

Profiler record showing subbottom deposits.(After King, 1973.)

enoughthat we can regardthe pathsas vertical;therefore, the time required for the seismicwave to travel from the sourcedown to the datum is Al., where (8.3) Lt": (E, - D" - E,t)lVH. Similarly,the time for the waveto travel up from the datum to a geophoneon the surfaceat .Bis At*, where Ltr:

Lt"+ t,o.

(8 4)

The correction, A/0, for the traveltime at the source is then Aro: A/. + At":2L,t.+ t,, = 2(E" * D, - E,)IVH + t,h.

(8 5)

Subtractionof A/n from the arrival time /u is equivalent to placing the sourceand the geophonegroup at the sourcelocation on the datum plane,therebyeliminating the effectof the low-velocitylayer if the source is beneaththe LVL. At times,the sourcemay be so far below the datum plane that Ato will be negative. When eq. (4.11)is used to calculatedip, the dip moveoutmust be correctedfor elevationand weathering. The valuesneededin eq. (4.11) are the differences in arrival time at A' and C' in fig. 8.26.The correction to be applied, Al., called the differentialweathering correction,is the differencein arrival times at opposite endsofa split-dip spreadfor a reflectionfrom a horizontal bed. The raypathsfrom the source.B down to a horizontal bed and back to geophonesat A and C haveidentical traveltimesexceptfor the portions l',4 ar'd C'C from the datum to the surface.Assumingas

before that A'A and C'C are vertical, for At, we get the expression Ar,:(A/")6-(At"), : (Ar, + t,t)(.- (Lt, * t,n)n,

(8.6)

wherewe assumethat A and C are sourcelocationsso that the quantities in parenthesesare known. If we take the positive direction of dip to be down from A toward C then At, must be subtractedalgebraically from the observeddifferencein arrival times at I and C to obtain that for A' and C'. The followingcalculationillustratesthe effectof the correction.We take as datum a horizontal plane 200 m abovesealevel; V, ts 2075m/s. Data for three successivesources,A, B, and C suchas thosein fig. 8.26, are shownin table8.1. Let us supposethat a reflection on a split profile from source-Bgivesthe following data: to : 2.421 s, tu: 2.419s, and t. : 2.431s. Then, the corrected value of t,, is 2.421 - 0.074 : 2.347s, and the correctedAt,,is A t , t : 2 . 4 3 r - 2 . 4 1 9- ( - 0 . 0 0 9 ) : 0 . 0 1 2+ 0 . 0 0 9 : 2l ms. Supposethat (for anotherreflection)to: |.392 s, t n : 1.401s, and r.. : 1.395s; then the correctedvalueof lo is 1.318s and the correctedA/.,is

Ltu: 1.3nt- 1.401- (-0.009) : -0.006+ 0.009: *3 ms,

6 . 0s

: 1 . 0s .?

./.':

:txt';l: .;'.'r.:

:ll;.t'H

l:\lS.=

Fig. 8.23 Profiler record, offshore Japan. The water_botromre_ flection, l, with traveltime 1.0 to 2.0 i indicates water depths of 7 5 0 . t o 1 5 0 0m . T h e s h i p t r a v e l e dg . 5 k m b e t w e e n the 30_min marks at the top of the record. Most primary reflections are obsoured by the Iirst and second water_bottom multiples. A and

1L More than a kilometer of sedimentsare indicated by the re_ flections nea-rC, multiples of these appear paged back at D. E indicates a fault scarp on the ocean hoor.'fu.. diffractrons, probably from sea-floor reliefslightly offset to the line. Note the onlap and thinning above G (Courtesy ofTeledyne.)

Table8.1 Coruections to reflectiontraveltimes

Surfaceelevation,E, (m) Depth of source,D, (m) Uphole time, r,, (ms) Source-to-datumtime, Al, (ms) Datum-to-geophone,Ar" (ms) Correction time, At,,(ms) Differentialcorr., Ar,.(ms)

SourceC

Source,B

Source,4

248 l5 48 16 64 80

244 l3 44 l5 59 74 -9

2 5 71 20 | measured 53J

r 8I

7l

I

J

calculated

(Courtesyof Bureaud'EtudesIndustrielles Fig. 8.24 Profilingoperations employingseveral typesof sensors. et de Coop6ration de I'Institutdu P6trole.)

fiiii# (a) Fig. 8.25 Side-scansonar.(a) Schematicdiagram;(b) portion of a record showingreflectionslrom sea-floorrelief and gap underneaththe toufish (courtesyof CGG); (c) recorder,tow fish, and towing cable(courtesyof EG&G Marine Instruments).

t.

I

o1*; '

'lr -l 700

I

600

I

500

I

400

I

3-00

I

2oo

|

10o

|

0

I

1oo

I

200

r

3CO

I

400

I

5oo

I

600 dtslance

(b)

,

7oo n melerS

REFLECTION FIELD METHODS

266

Baseoi LVL

lL, D,-Ed)

Fig. 8.26

E.-D.-Ed

Calculation of weathering corrections.

Thus, the correction can changethe apparent direction of dip as well as the dip magnitude.Accuratecorrectionsare essential. Correctionsare often requiredfor geophonesin between source locations where uphole data are not available.The first-breakscan be used fior this purpose. In fig. 8.27, G is a geophoneintermediatebetween adjacent sourcesA and B for which we have first-break traveltimes.Let t no and t uo be the firstbreak times for pathsA'C'G and B'C"G. Almost always GC' and GC" are within 20' of the vertical and C'C" is thereforesmall. Thus, we can write the approximaterelation t A ( . +t B C : ( A ' B ' l V ) + 2 t w - @ B l V r ) + 2 t w , t, being the traveltimethrough the weatheredlayer at G. Thus. (8.7) tn:'.ltoo * t"o - UBI V)1. Subtracting/- from the arrival times in effect places the geophoneat the base of the LVL; to correct to datum, we must subtract the additional amount, (8, - E,t * Dw)lVH,whereE, is the mean elevationof the geophone grotrp,Dw being found by multiplying I, bY V*. Occasionally, special refraction profiles are acquired to obtain data for making correctionsfor intermediategeophones.Theseprofiles are usually of the standardtype usingsmall chargesplacednear the surfaceor a nondynamitesourceon the surface;theseare interpretedusing standardmethodssuchas the GRM ($l1.3.3)or Wyrobek's(see$11.4.4)to find the depth and traveltimeto the baseof the LVL. Alternatively, a sourcemay be placedjust below the LVL, as shown in fig. 8.28;in this event,we must modify eq. (4.36) becausethe sourceis at the baserather than the top of the upper layer.Thus, x-Dntan0

Dn

VH

Vrcos 0

I :

x

in between buried Datumcorrectionfor geophones ili;.t;3t

profile suchas that shownin fig. 8.28).The weathering velocity V*can be found by (l) measuringthe slope of a plot of the first-breaksfor geophonesnear the source(correctingdistancesfor obliquity), (2) dividing D. by tunfor a sourceplaced near the baseof the LVL, (3) an uphole survey,or (4) firing a cap at the surfaceand measuringthe velocity of the direct wave. 8.8.3 Picking refections and preparing cross-sections Although the following proceduresare those used prior to the era of computer-processed record sections, they are also used occasionallytoday, and the conceptsthat they involve continue to be central to today's techniques.When the record quality is poor, almost any alignmentmay be mistakenlyidentifiedas a reflection.The best criterion in such casesis often the geologicalpicture that results.If this picture does not make sense,we should reexaminethe geophysical "making sense" data with more skepticism.Naturally, does not mean that the result must fit our preconceived ideas,but rather that it must be geologically plausible. When the interpreter decidedthat an event was a "timed" it. When legitimatereflection,he marked and working with individual split-dip records,the arrival timesat the centerof the recordand at the two outside traces(or the differencebetweentheseoutside times, At.,),usually correctedfor weatheringand elevation, were recorded. Besidestiming reflections,the interpreterassigned

D.cos0 T

VH

Vw

(8.8) Most near-surfacecorrection methods require a knowledgeof Vn and sometimesof V* as well. The former can be determinedby (l) an uphole survey,(2) a specialrefractionsurvey,as previouslydescribed,or (3) analysisof the first-breaksfor distant geophone groups (becausetheseare equivalentto a refraction

Fig. 8.28

Refraction weathering profile.

DATA REDUCTION

261

a grade to each, for example, VG, G, p, ? (for very good, good, poor, questionable).These grades referred to the certainty that the event was a primary reflectionand the accuracyof measurementof the arrival times. Sometimesa two-letter grading system was usedto separatethe gradingofcertainty from the grading of timing accuracy. The processof identifying eventson a seismicrecord and selectingand timing the reflectionsis referred to as picking the records.Figure 7.45 showsa picked record. The next stageafter picking individual recordswas to prepare a composite representation,usually by plotting on a sheetof graph paper.The sourcelocations weremarked at the top on a horizontal line indicating the datum plane. The reflection events were plotted, and sometimesalso the surfaceelevations,the depth of the baseof the LVL, and the first-breakar_ rival times. Cross-sections are calledtime sectionsif the vertical scaleis linear with time or depthsectionsif linear with depth. A cross-sectionis unmigratedwhen reflections are plotted vertically below the source.A migrated .\ection(such as fig. 8.29) is one for which we assume that the seismicline is normal to strike so that the dip moveout indicatestrue dip, and we attempt to plot in their actual locationsthe segmentsof reflectinghorizons that producedthe recordedevents.The scaleon hand-migratedsectionswas usuallylinear with depth. Unmigrated sectionsgive a distorted picture of the subsurface,the distortion increasingwith the amount of dip; they are used for interpretationonly in areas

I r$

i

! i ! f:r

ofgentle dip or when an accuratestructuralpicture is not required. Figure 8.30 showsthe effect oi tact of migration on simple structures.The anticline at the left would appear on an unmigrated section as the dashedline R'SZ and the synclinewould appear as the dashedline T'UV'. Failure to migrate decreases the curvature of an anticline and incr-eases that of'a syncline.Failure to migrate also positionsa fault incorrectly. Severalmethods of migration have been used.The simplestis to assumea constantvelocity down to the reflectorand swing an arc whoseradius is half the arrival time romultiplied by the averagevelocity.The radius O,R in fig. 8.30makesan anglewith the vertical equal to the reflectordip (calculatedusing eq. (4.1l)) and a straight-linesegmentequal to half the spread length is drawn at R perpendicularto the radius lrangent to the arc) to representthe reflectingsegment(see also fig. L3b). This method of migratingpositionsrhe reflector segmentsincorrectly and gives them the wrong dip, comparedwith correctlyallowing for a velocity gradient (seeproblem 8.20e). Migration was sometimescarried out with simple plotting machines.Some of these assumeda functional form for the velocity,such as one describedby Rockwell(1967),which usedeq. (4.35)assuminga linear increaseof velocity. Probably the commonest method of handmigrating reflectionsuseda wavefrontchart, a graph showingwavefrontsand raypathsfor an assumedvertical distribution of velocity.Figure 8.31is a simplified version of such a chart. It shows the location of a

r t

Detum

o : l

tr* l

r*o l

i'.

\\ * 75oO-

*

4 "-

-!'

Fig. 8.29

Hand-migrated cross-section.(Courtesy of Chevron.)

Fig. 8.30

Effect of lack of misration

t_.

t

268 wavefront at different times after the source instant, the successivepositions being labeled with the two_ way traveltime. Raypaths are also shown; these are Fur1l Uv applying Sneil'slaw ut.u.f,lfrurree in veloc_ ity. Raypaths are labeled in terms of dif moveour, A,tolAx.Dip moveoutcan be measuredfroir corrected record sectionssuch as stackedsections, as the differ_ encein arrival time, Ar, betweenpoints a distanceAx apart. To plot a cross-section, thJwavefront chart of_ ten was placed under transparent graph paper with the chart origin at the appropriatesouice l,ccation. A reflection with certain values of /o and A/r/Ar was plotted by interpolating between til" *uu"flnt, unO rays (actual wavefront charts have more closely spacedwavefronts and rays than those shown in fig. that interpolation was -o.. u*u.ute) and 9:1-t_,t" qrawrng a straight line of length equal to half the spread.lengthtangent to the wavefroni at the point (lo, L,tol\,x).The reflectiondenotedUy ttre symlol _o_ in fig..8.31correspondsto /n = 2.350s unJlrrA" : f f O ms/km. With an asymmetricspread, we correct the mea_ sured moveout for the differencein normal moveout betweenthe two ends of the spread ;" ;;J the dip moveout, LtulL,x.With in_lineoffsetr, we find L,t,,/L,x by applying normal-moveout corrections (and cor_ recting the arrival time to find rnwhen the offsetsare large).On record sections.(see *f,^if.ff"*J, *..uy measurethe dip overany distanceover whichitre hori_ zon is dipping uniformlv. Identifying a reflectionseenon one record as repre_ sentingthe same interfaceas a reflection on anottr". record is calledcorrelation. Itwas basedpartly on sim_ ilarity of character and partly upon'rf"'ilp and agreementrn arrival times (time_tie).The travel paths for reflectionson the outside tru"", "iua;'u".nt pro_ fiIes that provide continuous subsurface "o"u"rug.uru_ ally were the same except that they were traverseOin opposite directions;hence,the traveltimes stroufAUe the.sameprovided that adequate*.urh;;;;;nd ele_ vation correctionshavebeenapplied.no. e*"a_pte,in fig. 8.3, trace I on the profile i.orn ,our" 7, 1r"i,t r'ath OrBO3) should have the ,u-. "o..."t.A t.uu_ eltime as trace24 on the profile f.o. ,ou."" O If the record quality is ioo poor to p;;;l; Jjntinu_ ous reflectionsor ifthe continuousreilections that are presentare not in the part of the sectionwhereinfor_ matron rs required, we draw phantoms, lines on the sectionso that they are parallel to uA;u""nl-aip sym_ bols that the interpreterconsidersuuiia. Wh"l. Outu conflict or are absent,the phantom is drawn in the manner that seemsmost reasonableto the interpreter on the basis of whateverevidence."y U. uruifuUf". Recordsectionscan be regarded as a form of time placing successive .."o.d, ,io" Uy !y ::::i"j llllned sloe. Kecord sectionsare usually corrected for eleva_ tion, weathering,and normal *u"out. if,.i alsptay a large amount of data in compact form. potentral A disadvantagewith record sectitns i. tt ut ttJv ur" .o

REFLECTION FIELD METHODS grap.hic that people who lack understanding of the significanceofthe data tend to attachincorrect mean_ ings to them, as though the section *"* u ..photo_ graph" of the rock formationsin the ground. Problems 8.1 Land cablesare built in sectionsthat are identical. The connectionsto the plugs at "u"t, ,nJof the sec_ tion effectively rotate the system Uy it" riumber of groups to be connectedto that section. Thus, if pins l, 2, and 3 at one end of the sectionare connectedto takeoutsfor_3groupsin a section,at thetlher end of the sectionthey will connectwith pins 4, 5, and 6 of the next section.There are more sets of wires than chann^els being usedat any one time, for example,per_ haps 96 independent pairs of wires for use witfr +g channels.However,occasionallyso many rectio.rsand geophonesare laid out that a distant g.oup ofphon", is connectedto the samechannel ur"u n"'ua". group; this mistake is called rollover.Sketch porsiUte a ar_ rangement for the connectionsin one section with takeoutsfor three channelsand explain t o* roilouer could appear on the seismicrecords. 8.2 Assumea reflector2.0 km beneath the midpoint and a dip of 20. with constant overburden-velocity; how much does the reflecting point move between membersof the common_midpointset for offsetsof 0, 0 . 5 , 1 . 0 ,1 . 5 ,a n d 2 . 0 k m ? E.3 (a) Draw surfaceand subsurfacestacking charts

splitspread whererhe,ou.". i.'rnia*uy Pj,i.^11-11"". Detweenthe geophone_group

centersand the centers ofgeophone groups l2ind l: ur.,"pu.ut.Jiy ttr.. times the normal geophone_group ,pu.ing. nrru_. u source interval double the geophon._grJup interval and that sourceand geophonetocatlonJUey'ond point A canno-tbe occupied.The source,however, au., _oua a location l, producing asymmetric spreads --shootlng (:1, throughthe spread..). (b) Assumethat geophonesbut not sourcescan be lo_ cated over a distanceof l0 geophone_group intervals centeredat point B Whaj is the effect & rnuttipti"ityt If the da-? quality deterioratemarkedly when multi_ plicity falls, what can be done to alleviaie the deterio_ ration? 8.4 For an airwavewith a velocityof 330 m/s and two geophonesseparatedby 5 m. ai what frequency is maximumattenuationachieved? 8.5 Reflectionsin the zone of interesthave apparent velocitiesaround 6.5 km/s, whereasthe veloJity .lust belowthe uniform LVL is 2.1 km/s.If we wish io avoiO cancelationbelow 80 Hz when using an array, what is the maximum inJine array length? 8.6 (a) Under what conditionsls the responseof a lin_ ear array ofn evenlyspacedgeophonesziro for a wave horizontally (such as ground roll)? lpu-"ling (b) If r geophonesare distributLd uniformly over one full wavelength,show that the response;, i : l,n. (c) What is the responseof the ariay in part (a) when

PROBLEMS

269 1 0 0 0m

I

i

Fig. 8.31

Simplified wavefront

chart fior horizontal velocity velocitv layering. li

arriveperpendicularto the line of geoiit"#:;tt M" is theresponse of thearrayin part lll rhe]vaves arriveat 45. to thefi". u,ia, "jd.,"(a)when fe] \eneatparr(d) forn : t6. d. / showthateqs.(g.I and(7.4) ) areconsistent.

rapering issometimes;ril;;il;ii,oou_ l;l li*r bllng elements

a mixingboxin rhen",9.,_:l(3) using-unequal spacrng Wh.atargumenrswould you use fbr or :l^:lT:",:each of rheseapproaches? :C:rl:t 8.9 Figure8.32showsifi" A*"ii"ity effectof a group lengthtypicalof end_on marine,f,oot'ing."gl,*iff change(a) with arrivar,irnrl'iif'"] l:1: :urves .rr", j.1:::::* and (c).for grearersracking u.fo.itvl r. it Deuerlo proceedin theupdip o, ao*fiAii'ii..r.r,onr:

at someL ,pu..d .r.,.;; "i;;;iilTiffiii::i:*J1'":%i1? 8.r0(a)rhetapered aniyyi,t,

i"j,';,'if :ii, ,, ,,

270 REFLECTION FIELD METHODS

Dip Fig. 8.32

40"

Response to€pparent

dip of a effecrive lengthof 50**l:A*,]:: ".li*r tapered array with u,n""il,il,,, ,,u.t,n, is l 5 km/s'offsetis -lo0m. lAlter suuiiuna si.n-'r, i|'iitr" ll +,[], l, l]. Usethis fact to sketchthe arrayresponse. (b) This approximar

il:rl#",.iiijJ'f,:i:.*'i;;'i""

arrav withunequauv spaced

';n":UiX ach ieved ; ;.. ;l,T l; as shown in g.33.ii:;:ft-:il#i

fie. How_coul"dii...'rr.irgr, .""n four elually spaced geophones, !:t"t be used to approxtmate a cosilne array more exactly than in part (a)? 8.1I (a) The noisetestshown.in g.34 fig. used36 geo_ phonesspacedl0 m apart and dyn"amite,t ot, uu.iou. in-line distances away. Th" ;;;;^;,"_),",nd,"u,., ground roll. What is-itsvelocity, ao_inuni.f..quency, and wavelength?what. ;;;ff'; - - t!"opr,on. fngti' group haveto attenuateit? (b) What are the velocity, dominant frequency,and wavelengthof other wavetrains? Witt tti leopfrone group lengthdetermined in part tuloii*uol ,r.rt (c) What causesthe alignment B,_;J""^^"-" 8.12 Assumethat vo to man objectives3 to 5 k.n d.;;;';;;;;,iJ-':'

j:LXi:"fff togentrerr'rslsurracJtfr i,",r.',li["f; Tj lO.O Dipsat objective depth

1) maybeup to 10".The velocityat the bise of th; jo;_;.i;;,;'iuy.. ,, z km/s,thatat objecrive aeptr,is + tnVs,"uniuiin. 0"."_ ment (8 km) probablyibout O f.rni"'fi". ,r.tu..sourceunitsand4g_chann.el recording.quiirn.n, u.. available. Both ground roil 1Vo_ 80"0 #;;;nd air_ wavesmay be problems,but ihe low_velocity Iayer (aboutl0 m thick with a veloci; ;;;ffi m/s) is probablylairly unilorm.The arei i, iririy "rlry u"A wr]l probabty to"acrrteve 3l*11" data ;rortqualitv.propoieb. ..;;i;; adequate fi.ld ;;th";; anoe*_ the basesior your proposals. gl{n 8.13Seismic fieldworkis usuallycarriedout in a unr_ .i"l"r l:iT "rl.lh. with group inrervatseverywhere the spreadeitherall on one,ta. i, l1T? ,u_rn.,_ nc.aboutrhesourcepoint. Thel"y";i;i;;;.torn, tenc to bedetermined bv whatequipment is aihaiJ or habit_rather than ihe natureof the problem Uy to be solved;. for example,the lengrh"I;J;;;;; strnss (severalgeophones for a sin*gle c-;t;.i;o".rnu_ nentlywiredtogerher) mayaiirar.'in.g.opn|"! i",.. val and the available equlpmentthe numberof chan_ nelsand thus the effectrve spreadlength.Sometrmes

Fig. 8.34

Walk-away noise rest. (From yilmaz, l9g7: 35.r

PROBLEMS

271 hrbrid spread ($8.3.11)arrangementsare used to SIGNA L make llller use of equipment.Assumethat you have r: ntLrrechannelsavailablethan the numbergiven o by fol_ 1.'rsingthe guidelinesof $g.4.2;what ciicumsrances might leadyou to usethe extrachannelsto (a) extend A the sp.read_lengrh beyondthat givenby guidjine E 1t.y; tbt fill in the spacebetweenthe sourceind 4 the mrni_ mum offset given by (2); (c) interleave additional Sroupssomewherein the middle of your spread;(d) Ial out a partial spread on the oihe. siie 20 30 ao 50 60 70 to 90 ot.the source-pointwherean end_onarrangementis being FREQUENCy used:and (e) lay out a short ".o., u.-? If you have almost but not quite enoughchannelsto usea solit arrangement comparedto an end_on,what arethe ad_ NOISE \anragesand disadvantages ofusing the split with: (f) Itl longergroup intervalsthan givenby guiaeline(5); (g) shorteningthe maximumoffset;or (li) increasing ts the minimum offset? 8.14 Signaland noisecharacteristics z determinedfiom 4 a prevlousdynamitesurveyare shownin fig. g.35. The principal objectiveis at 3000m with a staJking veloc_ rtl of 3000m/s. (a) What frequenciesshould be 20 30 ao 50 60 70 coveredbv a l i n e a r sweep? FRE9tTENCy (b) Iia downsweep of g s is used,at what time w i l l t h e correlationghostappear?Will it interferewith Fig. tt.35 Signal and noise spectra from the oba particular area. ( F r o m E v a n s .I 9 8 9 . ) lective'? (c) lf a singlevibrator sweepof g yields s an S/N ratio : 0.2 at the objectivedepth foi a 20_to_60_s sweep,how many sweepsper vibrator point will be requiredto give .S/N: 2.0? Table 8.2 Reftet'tiondatu (d) Assumethat recordingcontinuesfor an additional time of 6 s (listentime) beyondthe sweep Age Depth (m) time and V (m/s) 1 , ,( s ) V, (mls) that it takes l0 s for moveupbetweensweeppornts; Recent how long will be requiredfor four vibrators 300 2000 0.333 2000 to record Eocene 3000 one vibratorpoint? 3000 2.222 3140 Cretaceous 5000 3300 8.15 You wish to surveya 10 x 70 km 3.000 4137 area where anticlinalstructureswith their long axesnorth south are expected.the minimum size structureeconoml_ callyviablebeing2 km across.Maximumdip expected plot the results, shading positive values as in a is l5'and reflectorsare listedin tableg.2.The Recent variable-areadisplay. reflector will be useful in making staticscorrectlons. (b) Repeatfor noiserangingfrom * 20 *2}(signal/ to A r.roisetest gavea prestackS/.V : 0.5. noise: ]1. Whot can you concludeabout the (a) DetermineIine spacingand orientation. extrac_ t r o n o l s r g n a bl u r i e di n n o i s e ? (b) What_multipticityis requiredro give : S/N 3.0 (c) Sum the fivewaveformsin part (a) and alsoin part after stack? (b) to show how stackingenhances (c) What coherentsignal geometry should be used,that is, .spread versusrandomnoise. near- and far-channel offsets, spacing required to (d) Replacethe elementsin the wavelets in pans 1a; avoidaliasingfor l5 to 40 Hz, and minimum number and (b) with + I or - I as the valuesare positive of channelsrequirecl? or negatrve(sign-bit expression) and repeatpart (c). Ho* long will the surveyrequire,assuming 1d) 270 8.17 Uphole surveysin five different'lunrelateij km/month for 24-fold? areas give the uphole-timeversusdepth information (e) Answerpart (d) for 210 km/month in taUte for 4g_fold. 8.3. Explain the possiblevelocity layeringfor 8.16 (a) Take the waveletthat has valuesat each succes_ case.How reliably are velocitiesand depth"of s r v e4 - m si n t e r v a los f 0 , . . . , 0 , g , 7 , _ g , _ 6 , weath_ 0,4,2, errngdefined? 0, . . . , 0 (with l0 zerosat eachend)and add random 8.18 The correctionmethodsdiscussedin noise (a random number table is given in upp. $g.g.2as_ C; in sume that the sourceis below the baseof itre the rangefrom a l0 to - 10,that is, with sijnal/noise t_VL. What ch^anges are required in the equationsof this : l. Do this five times for different noise v-aluesand sectionif this is not the case?

P

272

REFLECTION FIELD METHODS

I ?

Fis. 8.36

E l e v at r on s

E l e v art o n s

Reflection first-breaks.

F i g . 8 . 3 7 I l l u s t r a t i n gB l o n d e a uw e a t h e r i n gc o r r e c t i o n .

8.19 Figure 8.36 showsthe first arrivals at geophone stations 100 m apart from shots 25 m deep at each end of the spread.(There are actually I I geophone stationswith the shotpointsbeingat the first and eleventh stations;however,the geophonegroup at each shotpointis not recordedbecauseof hole noise.)The uphole geophoneis recordedon the third trace from the right. The weatheringvelocity is 500 m/s. (a) Estimatethe subweatheringvelocity Vrby averaging the slopesof lines approximatingthe first-breaks. The valley midway betweenthe shotpointsproducesa changein the first-break slopes,as if two refractors are involved, which is not the case.How can we be sure of the latter? (Hint: Plot an elevationprofile as an aid in determiningthe best-fitline through the first breaksto give Vr.) (b) Determine the weatheringthicknessesat the two shotpointsfrom the uphole times.

(c) What correctionsA/,, should be applied to reflection timesat the two shotpointsfor a datum of ll00 m? (d) Calculatethe weatheringthicknessand the time correction for each geophonestation. (e) Plot correctedreflectionarrival times in an K T' plot and determinethe depth, dip, and averagevelocity to the reflector giving the reflection at 0.30 and 0 . 2 1s . 8.20 The arrival time of a reflection at the sourcepoint is 1.200s and the differencein arrival times at geophones1000m on oppositesidesofthe sourceis 0. I 50 s, near-surfacecorrectionshaving beenapplied. Assumethe line is perpendicularto the strike.Determine the reflectordepth, dip, and horizontal location with respectto the sourcepointassuming:(a) no dip moveout and the averagevelocity associatedwith a vertical traveltimeis V : 2630 m/s; (b) the observed dip moveout and I (c) straight-raytravel at the angle of approachand V,,: l830 m/s; (d) straight-raytravel at the local velocity abovethe reflector,3840m/s; (e) the migratedposition determinedfrom the wavefront c h a r to f f i g . 8 . 3 1 . 8.21 The Blondeau method of making weathering corrections starts with a curve of emergentarrival times versusoffsets,I versusx in fig. 8.37 plotted on log-log graph paper.The slope gives.8 : (l - lln),

REFERENCES

2'73

Table 8.3 Upholesurveysinfive areas Depth (m)

5 8 t0 t2 l5 18 2l 25 30 35 40 50

Area I (s)

Area .B(s)

0.012

0.01I

0.025

0.023 0.024

0.030 0.028 0.034 0.036 0.039 0.046 0.051

Area C (s)

0.020 0.024 0.027 0.031 0.034 0.036

0.032 0.035 0.037

0.039

0.044

0.044 0.048

wherelln is the exponentin eq. (5.1g).To find 1,,,the vertical traveltime to a given depth :_, we find 4 a tabulatedfunction of I (seeMusgraveand Bratton, 1967:233).Then, x : Fz_ so that we can get / from the x-l curve. Finally, t, : x/F. Verify this procedureby derivingthe following relations: (a) z : z_ sin,| wherei is the angleof incidencemea_ suredwith respectto the vertical at depth z. (b) x : F2,,,wheret, : 2nll'' sin,i di : function of n, henceof -Balso. (c) I : (Gla)(xlfla, where G : 2nlX,,sin,-ri dr. (d) dx/dr : xl Bt is the horizontai component of ap_ parent velocity at the point of emergence. (e) dxldl, the horizontal component of apparent ve_ locity of the wavefrontat any point of the irajectory, rs V,,: xlBt.

(r) r, : Ii- dztv: tr. References

Agnich,F.J.. and R. C. Dunlap.1959.Standards of perfor_ mance

Area D (s)

Area E (s)

0.012 0.016 0.018

0.008 0.020

0.022 0.030 0.033 0.035 0.039 0.042 0.047

0.030 0.031 0.032 0.036 0.043

Evans, B. 1989. Handbook for Seismic Data Acquisition in Oit Exploration. Perth, West Australia: Curtin University of Tech_ nology. Harding, T. P., R. F. Gregory, and L. H. Stephens.19g3. Con_ vergent wrench fault and positive flower siructure, Ardmore Basin, Oklahoma. In Seismic Expressions of Stuctural Sryles, A. W Bally, ed., AApG Studies in Geology 15. Tulsa: Ameri_ can Association of Petroleum Geologists. International Association of Geophysical Contractors. l99la. Land Geophysical Operations Sufety Manual,7th ed. Hous_ ton: IAGC. International Association of Geophysical Contractors. l99lb. Marine Geophysical Operations Safety Manual, 7th ed. Hous_ ton: IAGC. International Association of Geophysical Contractors. 1993. Environmental Guidelines for Wortdwide Geophysical Opera_ tron^r.Houston: IAGC. King, V L. 1973.Sea bed geology from sparker profiles, Vermillion Block 321, offshore Louisiana, 1973 Offshore Technology Conference Preprints, paper 1802. Dallas: Offshore Technology Conference. L-aqer, A. 1970. Couplage sul-geophone. Geophys. prosp., lg: 300-l 9. Laster, J., and A. F. Linville. 1968.Preferential excitation ofrefractive interfaces by use of a source array. Geophysics, 33: 49-64.

rn petroleum exploration. Geophl,sics, Zl: Sti Zq. Anstey, N. A. 1970. Seismicp_rospecting Instruments, Vol. L Signal Characteristics and InstiumenT iprri|..rLrr' n*ti", Gebruder-Borntraeger.

Lindsey,J. P 1991.Crooked lines and taboo places.The Leading Edge, l0(ll):74 7.

A_nstey,N. A. 1986. Whatever happened to ground roll? The Leading EdSe, 5(31: 40-5.

Marsden, D. 1993. Static corrections Edge, l2(l)t 43-9 and 12(21:115-20.

Brown,^A. R. l99l. Interpretation of 3_Dimensional Seismic Data,3d ed., AApG Memoir 42. Tulia: a_.ri.un,{rro.iutlon of Petroleum Geologists.

Mayne, W. H. 1962.Common-reflection-point horizontal datastacking techniques. Geophysics,27; 927 38.

Brown, A. R. 1992. Technologies of reservoir geophysics. In

Reservoir Geophl,sics, R. E. Shiritr,.a , pp.ii-uiiT"iiu, soct_

ety of Exploration Geophysicists.

Burg, J. B. 1964.Three-dimersionalfiltering with an array of seismometers. Geophysics, 29: 693 713. Dangerfield,J. A. 1992.Ekofisk.Fi-eld development:Making im_ agesof a gas-obscured reservoir In Reservo'ir Ce'"oiiii^, r.. e. Sheriff,ed., pp.98 109.Tulsa:Socieiy"iE"pi6.#"" c."_ physicists. D-enh.am, L. R.. 1981.Extendingthe resolutionof seismic re_ flection exploralion.J. Can. Soi.Expl. G";;i;;.

A review. The Leadinp

Mayne, W. H. 1967. Practical considerations in the use of common-reflection-point techniques. Geophysics,32: 225-9. McKay, A. E. 1954. Review of pattern shooting. Geophysics, 19:420 37. Morgan, N. A. 1970. Wavelet maps A new analysis tool for reflection seismograms. Geophysics,35: 447-60. Musgrave,A. W., and R. H. Bratton. 196'T.Practicalapplication of Blondeau weathering solution. ln Seismic Refraction Prospecting,A. W. Musgrave, ed., pp. 231-46. Tulsa: Society of Exploration Geophysicists. Newman, P., and J. T. Mahoney. 1973. Patterns of salt. Geophys.Prosp., 2l: 197-219.

With a pinch

214

REFLECTION

FIELD

METHODS

l

O'Connell.J. K.. M. Kohli, and S. Amos. 1993.Bullwinkle:A 58: | 67-76. unique3-D experiment.Geophysics,

M. 1970.Optimizationand implementationof Schoenberger, marine seismicarrays.Geophysics, 35: 1038 53.

Parr,J. O., and W. H. Mayne. 1955.A new methodof pattern 20: 539 64. shooting.Geophysics,

Explorution Sheriff, R. E. 1978.A First Coursein Geophysical and Intelpletat on. Boston: International Human Resources Development Corp.

Pritchett, W. C. 1990.Acquiring Better SeismicData. London: Chapmanand Hall. Rockwell,D. W. 1967.A generalwavefrontmethod. In Seismic Refraction Prospecting,A. W. Musgrave, ed., pp. 363-415. Tulsa:Societyof ExplorationGeophysicists. streamersysSavit,C. H., and L. E. Siems.1977.A 500-channel tem, 1977Offshore TechnologyConferencePreprints,paper Conference. 2833.Dallas:OffshoreTechnology

Dictionaryof ExplorationGeoSheriff,R. E. l99l. Encyclopedic physics,3d ed.Tulsa:Societyof ExplorationGeophysicists. Tulsa:Societyof ExYilmaz, O. 1987.SeismicData Processing. plorationGeophysicists.

9 I

Data processing

rl

ir

Overyiew The "digital revolution" changedseismicexploration beginningin the early 1960s,about 20 yearsbeforethe widespread use of CDs in recorded music. The changeswere of the same kind: much lessnoise, no deteriorationof signal by repeatedprocessing(playing), and the ability to reshapethe information content into more easilyunderstoodforms. Radar, one of the technologicaladvancesof World War II, was usedin detectingshipsand aircraft. However,noise frequently interfered with its application and considerabletheoreticalwork devotedto the detection of signalsin the presenceof noise led to the developmentof a new field of mathematics,information theory. At first, this theory was very difficult to understand becauseit was formulated in complex mathematicalexpressionsand employedan unfamiliar vocabulary.However,the developmentof digital computer technologyconsiderablysimplified the understandingof the basic concepts,and the number'of applicationshas expandedgreatly. Early in the 1950s,a researchgroup at the MassachusettsInstitute of Technologystudied the application of information theory to seismic exploration (Flinn, Robinson, and Treitel, 1967).These studies combined with the new digital technology changed seismicexplorationconsiderably.Today,most seismic data are recordedin digital form and subjectedto data processingbeforebeing interpreted. The basic conceptsare expressedin a number of books and papers(Lee, 1960;Robinson and Treitel, 1964, 1980;Silverman,1967;Anstey, 1970;Finetti, Nicolich,and Sancin,l97l;Kanasewich,1987).Probably the two best current books on seismicdata processing in general are Hatton, Worthington, and Makin (1986)and Yilmaz (1987). In chap. 15, we give the mathematical concepts most important in data processing,and in this chapter, we show how theseconceptsare applied without beingconcernedwith mathematicalproofs.This chapter discusses mainly functionsin digital form, whereas chap. 15 treatsboth continuousand digital functions. Equation numbersgive the nearly identical equations in chap.15. Usually, we think of seismicdata as the variation with time (measuredfrom the sourceinstant) of the amplitudes of various geophone outputs. When we take this viewpoint, we are thinking in the time domain, that is, time is the independentvariable.We also

275

sometimesfind it convenientto regarda seismicwave as the result of the superpositionof many sinusoidal waves differing in frequency, amplitude, and phase; the relative amplitudes and phasesare regardedas functions of frequency and we are thinking in thefrequencydomain. The frequency-domain approach is illustrated by electrical systemsthat are specifiedby their effectson the amplitudesand phasesof sinusoidal signals of different frequencies.For example, graphsof filter characteristics usuallyshow amplitude ratios or phaseshifts as ordinateswith frequencyas the abscissa. Three types of mathematicaloperationsconstitute the heart of most data processing:Fourier transforms, convolution, and correlation. Fourier transforms ($9.1.3)convert from the time domain to the frequency domain and vice versa, and they and other types of transformscan be used to convert into and out of other domains also. Sometimeswe transform in two dimensionsto other domains,such as thef-k ($9.9),r-p ($9.ll.l), or other domains.The essential aspectof transformsis that in principle no information is lost in the procedure,although in actual application, very minor degradationoccursbecauseofapproximations, truncation, and so on. Transforms provide alternateways of doing things that are sometimes advantageous. Convolution ($9.2) is the operation of replacing each elementof an input with a scaledoutput function; it is the mathematical equivalent to filtering. such as occurs naturally in the passageof seismic wavesthrough the earth, in passingelectricalsignals through circuits, and so on. The limitations on sampling and signal reconstitution,that there be no frequency components above half the sampling frequency, are explained using convolution concepts. Sometimesundesirablefiltering can be undone by deconvolution. Correlation ($9.3) is a method of measuring the similarity betweentwo data sets.A common application is determiningthe time shift that will maximize the similarity. Correlation is also the means for extracting short signalsof known waveshapefrom long wavetrains,as is usedin Vibroseisprocessing.If a data set is correlatedwith itself (autocorrelation),a measure of the repetition in the data is obtained. Phase($9.4)plays an especiallyimportant role in seismic exploration. The wavelet injected into the earth by most impulsive sourcesis nearly minimum

l

l

II

I

l

II

,l 'l

DATA PROCESSING

276 phasebut the waveletinjectedby the Vibroseisis far from minimum phase.A zero-phaseoutput provides the best resolution and hence is the desiredwavelet for interpretation, and the phase of seismicdata is changedin processing. The objectiveof most data processingis enhancing the signalwith respectto the noise($9.5).Deconvolution is probably the most important processthat aims to improve vertical resolution. Improvements resulting from discriminationon the basisof frequency are employed in various deconvolution techniques: deterministicinversefiltering,recursivefiltering,leastsquares(Wiener)filtering, waveletprocessing,and so on. Deconvolutionis, of course,not restrictedto just one dimension. Compensationfor near-surfacetime delaysis the objectiveofstatics corrections($9.6).The first approximation to staticsis generallybasedon measurements in the field, but this is usually supplementedby useof a surface-consistentstatics correction procedure. However, surface-consistentstatics can leave longwavelengtherrorsand refractionstaticsanalysisis often employedto removesucherrors. Normal moveout is the discriminant employedin velocity analysis ($9.7).Both the velocity spectrum and velocity panelsas usedin velocity analysisgenerally yield velocityinformation at locationsa kilometer or so apart. Horizontal-velocity analysisprovides a continuous examination of the stacking velocity for specificreflectinghorizons. The preservationof amplitudeis important because of its significancein interpretation($9.8).Amplitudes amvary for a number of reasons.Surface-consistent plitude analysis is similar in concept to surfaceconsistentstaticscorrection. Apparent-velocity (or 2-D, dip, or f-k) filtering (99.9)providesa very powerful tool for discriminating against coherent noise trains. Stacking techniques ($9.10)also aid considerablyin discriminatingagainst noises of various types. DMO processingremoves much of the reflection-pointsmear inherent in CMP stackingwhen reflectorsdip. DMO is sometimesapplied after NMO correctionsand sometimesbefore. Muting of noise trains can also be usedeffectivelyto discriminateagainstcoherentnoise trains. Weighted stackscan provide considerablymore discrimination against certain types of noises than simple CMP stacking. Specialprocessingtechniques($9.1l) such as slant (r p) stacking, intelligent interpolation, and automatic picking can be used under special circumstances.Attribute analysisand other methodsthat rearrange the information content of seismic data sometimesprovide viewpoints that give interpretive insight into featuresof various kinds. Migration, the repositioning of data elementsto representthe subsurfacelocation of features,is often the last major step in processing.Migration is generally basedon the explodingreflectormodel, wherereflectorsare replacedby distributedsourceswith mag-

nitude proportional to the reflectivity,the sourcesare exploded simultaneously,and the waves travel only upward (at t/z the velocity) to generatethe recorded section.Migration involvesusing the waveequationto backtrack the wavefieldto locate the reflectors.Most methodsemploy variationsof three methods:an integral solution (Kirchhoff or diffraction-stackmigration), a solution in the frequency domain (Stolt or Gadzagmigration), or a finite-differencesolution in the time domain. The major problem is how to handle (in fact, how to determine)velocity variation, especially laterally varying velocity. Depth migration is usedto take into account lateralvelocity variations. proceduresare describedin Typical data-processing are often made at workdecisions 13. Processing $9. stations,and to some degree,processingcan be tailored to the specialrequirementsof specificdata. Often practical compromiseshaveto be exercised. This chapterconcludeswith a brief introduction to generalizedinversion. 9.1 Transforms 9.I. 1 Integral transforms Fourier transformsare an exampleof a classof operations called integral transforms, which are used to transform a function into a relatedfunction of different variables.The transformis accomplishedby multi"kernel," which is a plying the original function by a variables, and then eliminatsets of function of both ing the first set ofvariables by integratingthe product with respectto thesevariablesbetweendefinitelimits. The kernel must be such that the processcan be reversedby integration using a different kernel (sometimes the reciprocalof the first) to obtain a function of the first variables. The original function and its transform are said to specifiedby the variables;thus, ifll) is be in domain^s in the time domain, its transform F(u) is in the frequency domain (note that the dimensionsof / and v are reciprocals). Although in general the most widely used transform is the Laplacetransform, the Fourier transform (including the special forms known as z-transforms and Hilbert transforms)is most important in seismic work. The most common transformsare listed in tab l e 9 . 1. A crucial factor regardingtransformsis that (theoretically)no information is lost in transforming.Thus, we can start with a waveform in the time domain, transform it into the frequencydomain, perform varlous operationson the transform, and then transform the result back to the time domain to obtain the original function with modifications equivalent to the frequency-domainoperations.For example,if we remove all frequenciesbelow 60 Hz in the frequencydomain, the time-domain result will be the original Transformsenableus waveletminusthesefrequencies. processing domain and part part in one of our to do in anothel taking advantageof the fact that somepro-

271

TRANSFORMS Table 9.1 Commontransformsusedin seismicwork Original domain

Transform type

Transform domain

Kernel

time (t) l-D space(x) 2-D space(x, /) time (r)

Fourier Fourier 2-D Fourier Laplace Radon (slant stack)

Frequency Wavenumber Wavenumber wavenumber s-domain 1-p

e!)2tu1

,\-I

cessescan be done more easilyin one domain than rn another.In actualpracticethe integralsmust be calculated by using seriesand a small amount of information is lost in transformingbecauseof the truncation of theseseriesand round-off errors,but we can make theseerrors as small as desiredby carrying the calculations far enough.

erjKr erj(K/+Kp!)

. 15.3) e "'(cf$ (cf.$9.1 .5)

a continuous variable. Equation numbers also show equivalentsin chap. 15.) g(/) in terms of cosineand Equation (9.I ) expresses sine curvesof amplitudesc, and b,, eq. (9.2) in terms of cosinecurvesof amplitude c, that havebeenphase shifted by 1,, and eq. (9.3) in terms of complexexponentials.All three forms are equivalent,that is, c^: ancos "y,+ b, sin ^y,,: (ql + b))tt' , (9.7,1s.103)

9.1.2Fourieranalysisand synthesis Fourier analysisin our context involvestransforming functions from the time domain to the frequency domain and Fourier synthesesthe inverseprocessof transformingfrom the frequencydomain to the time domain. This distinction is artificial, however,and analysisand synthesiscan be interchangedwithout making any diflerencein the result. "well-behaved"($15.2.1) If we have a reasonably periodic function g(l) of period ?' (that is, if g(l) repeatsitself everytime t increasesby Z), then the function can be representedbya Fourierseries,

(9.8) a , : d , + a , ; b , : j ( c , ,- o , ) ; ( 9 . 9 ;1 5 . l o 6 ) o--,: )(a,+jb"). Note that : )ro: iou: 0o averagevaluesofg(l). (9.10;15.101;15.103)

With seismicdata g(t) is usually a voltageor equivalent with a zero mean;in this case,eq. (9.10)means that there is no dc component (an: bn: co : ot : 0). Equations(9.1) and (9.2) show that g(r) can be regardedas the sum of an infinite number of harmontc (cosineand sine) wavesof frequenciesu, having amg0l : :a,,+ | ta,cos 2nv,/ + b. sin2rv,t), ( 9 . 1 1; 5 . 9 3 ) plitudes a,, b,, or c, and phases1,. Theseequations thus representthe analysisof g(t) into componenthar: 1r,,, * i .,.o, (Znv,r- t , ) , ( 9 . 2 ; 1 5 . 1 0 2 ) monic waves.

:

( 9 . 31; 5 . 1 0 5 )

._L_o.",**,,

9.1.3 Fourier tansforms As period ln becomeslarger,it takeslonger for g(l) to repeat; in the limit when I becomesinfinite, g(r) no longerrepeats.In this case,we get in placeofeqs. (9.3) a n d ( 9 . 6 )( s e e$ 1 5 . 2 . 3 )

where

t I

C ( I ): I G ( u ) e t 2 " ' ' d r , . t-* ( 9 . 1l ; l 5 . l o 9 )

(9.4:15.99,15.100) ,,:

- 1,) dI cos(2nv,,r 12trl |1rr,,,

: tan t(b,la,l lo = 0."y,

r--,: (tlT)f, t{t)e,,^",at

(n*01.

]

)

( 9 . 51; 5 . 1 0 3 ) (9.6;15.106)

(Subscriptsindicatediscretesets,as with v,, 4,, and so on, whereasfunctional notation, suchasg(l), indicates

G( v)=

f__t} e- j^"' at. (9.12;15.108)

The function G(v) is the Fourier transform of g(t) and g(r) is the inverseFourier transform of G(v). Using the symbol +> to denote equivalentexpressionsin different domains,we write g(t) * G(v) We also refer to g(r) and G(v)as a transformpair.

i '

I' t

278

DATA PROCESSING

Equations(9.I l) and (9.12)canbe written in several ways. In general, G(u) is complex: A ( v ) s : t t " t , ( 9 . 1 3 ;l 5 . l 1 3 a )

ey):

wherel(u) and 1(u) are real, and A(v) is also positive. We call A(v) the amplitude spectrum(often called the frequencyspectrum-see $15.2.l) and 1(v) the phase spectrumofg(l). Substitutionin eq. (9.11)gives c(/) :

A(y)slz*t+tti) dy.

(e.14)

I

(r,

A(v\ cosf2tvt * 1(u)]du. (9.15)

Because(u) is ""-0,*, and imaginary parts,

f+-

f+-

|

|

C(*. y)el.**2.ilr dK du,

J-J_-

( 9 ' 1 8l 5 ; 'l17) f , f . r) : r)e-"*'2n"dx dt, J _J _gk. (9.19;15.116)

the ll2tt factor entersbecauser : 2nllt, whereasu : 1/Z(see$2.1.1).When we regardx and u as the independent variables,we are in thef k domain.

we may separateit into real

9.1.5Radon(t-p) transforms The Radon transform, also known as a slant stack or projection, is a line integral of some property of the medium along a specifiedline (usuallya straightline). The transform can be defined in many ways, the following one being that given by Stewart (1991).The Radon transform of a property such as amplitude, g(x, y) along the line RS in fig. 9.1, is the integral .S

G{v):R(u)+jX(u);

se,6) * qd. 0) : | 8(r 0) ds.

( 9 . 1 6l;5 . 1 l 3 a ) when g(l) is real, R(v) -- X(v) :

lll2n

f

For actual waveforms,g(l) is real and hence from eq. (9.14)and problem l5.l2a, we get Sttt : '

glx. tl :

I

(s.20)

dn

Line RS has the equationx cos 0 + y sin 0 : f, so the integralcan be written g(l) cos 2tvt

l:

dt,

(9.t7;t5.tt2)

The integrals R(v) and - X(v) arecalled the cosineand sine transform,s,respectively.When g(l) is real, R(u) and X(v) are respectivelyeven (symmetrical, that is, R(u) : R(-y)) and odd (antisymmetrical,that is, X(v) : - X(-v)) functions.Wheng(r) is real and even, X(v) : 0; when g(r) is real and odd, R(u) : 0 (see p r o b l e m sl 5 . l 7 b t o 1 5 . 1 7 d ) . We have regardedthe Fourier transform as a transformation of the independentvariable, / e y. The variableshavereciprocaldimensions:/ is measuredin time units (for example,seconds),u in reciprocaltime units (for example,per second).The samemathematics clearly appliesto any other dimensionand its reciprocalsuchas distance(for example,meters)and reciprocal distance(for example,per meter).

c@' Y) * G((,0) ftf" : y ) 6 ( x c o s0 + y s i n 0 - Q d x d y , I l tt*, JnJR (9.21\ whereE(r): 0ifr t 0, E(r): I ifl : 0(seeg9.2.land 15.2.5);using the delta function ensuresthat the only contributionsto the integralsare at points on line RS w h e r e x c o s0 * y s i n 0 - ( : 0 . The Radon transform changes(maps) g(x, y) from the x-y domain onto a cylinder in the l-0 domain, the cylinder being given by the equations0 < 0 < 2r, 0 = (, < *- or onto the half-cylinder 0 < 0 < n, - * < ( < * o .

The Radon transform is closelyrelated to the r-p transform, the only differencebeing in the equation used to specify RS. For the r p transform, we define RS by the equation/ : r - px (that is, r : Usin0, p : cot 0) and eq. (9.21)becomes s(x, y) *

G{r. pl:

ftft - r) dx dy. | | r,", /) 6(-v+ px ,J n,l n

o.22\

9.1.4 MultidimensionalFourier transforms In $9.1.3, we describedthe Fourier transform as involving a change from one independentvariable to another,t <+ v.The transformationcan be appliedto any number of variables,but most often only two are involved; for example,a seismicsection is usually a plot of time versusdistance(for example,trace spacing). In this case,we can generalizeeqs. (9.11) and (9.12)to the two-dimensionaltransform:

The theory of the Radon transform will be developed further in $13.5on tomography;applicationsof the r-p transformwill be discussed in 89.1l.l. 9.1.6 Implementationof transforms In $9.1.I,we describedthe Fourier transform in terms of continuousvariablesand inteqralsfrom -- to *-.

CONVOLUTION

279

out energy from other traces to provide cancellation except where the migrated data lie. As another ex_ ample, we can think of a reflector as the aggregateof many closely spaceddiffracting points and the reflec_ tion as the interferencecompositeof their diffractions. An alias(S9.2.2)resultswhen not enoughcomponents are includedto producethe requireddestructiveinter_ ference. 9.2 Convolution 9.2.1 The convolutionoperation Fig. 9.1

lllustrating the Radon transform

In practice,we deal with a finite number of discrete samplesand the operationis done with a fast Fourier transform, which requires2k samples,where k is an rnteger;we add (pad) a string of zerosto satisfythis condition. A long string of zerosaddedat the end of our waveform also helps avoid wraparound, the aliasingthat occursbecauseour wavefoimreDeatsev_ ery 2k samples.Fourier transforms do not perform w e l l a t a b r u p td i s c o n t i n u i t i e( s9 1 5 . 2 . 7 T ) .h e i w o n u _ mericalintegrations(as in eqs.(9.11)and (9.12))re_ quired to transform from the time domain to the fre_ quency domain and back again involve roundoff errors, but thesecan be made as small as desiredbv using sufficientterms (at the cost of increasedcom_ puter time). The Fourier transform involvesfitting cosiirecurves to our waveform, so we must include a sufficiently long portion in our analysisif the frequency_domain representationis to have meaning.A time seriesex_ tendingfrom 0 to n - | consistsof only n independent piecesof information, and its representationin tt. frequency domain, therefore,can have only n independent values.The time domain involvesonly real values,whereasvaluesare complex in the frequency domain, that is, eachfrequencyrequirestwo indepen_ dent values (real and imaginary, or amplitude and phase)to expressit. Hence,there can be independent valuesfor only nl2 frequencies. Transformsprovideequivalentwaysof representtng _ the same information content in differeni domains. We sometimesrefer to transform operationsas map_ ping from one domain to anotheqand we utilize mao_ 'in ping in a number of different ways. For example, additionto the transformslistedin $9.1.1,we maDbe_ tween the domains listed in table9.2. Another way of looking at Fourier synthesisof a seismicwaveletis that eachcomponent,being infinite in length, spreadsits energyover all of space,-butthat other componentsinterferedestructiveiywith it ex_ cept where the waveletis located.This conceproro_ videsa usefulway of thinking in a numberoi siiua_ tions. For example, DMO ($9.10.2) and migration algorithms($9.13)spreadthe energyofeach trice out over the operator shape,and we rely on the spread_

Convolution is the time-domain operation of replac_ ing eachelementof an input function with an output function scaledaccordingto the magnitudeof the in_ put element, and then superimposingthe outputs. Most systemswith which we deal are linear and time_ invariant, or nearly so (g15.4).The output of a linear systemis directly proportional to the input and is in_ dependentof the time when the input occurred,that is, it is time-invariant.Let us assumethat we feed into the systemdata sampledat regular intervals,A, for example,a digital seismictrace.The output of the sys_ tem can be calculated if we know the impulseresponse of the system,that is, the responseof the systemwhen the input is a unit impulse(915.2.5).We wiite rhe unit impulseE, or E(r),dependingon whetherwe are deal_ ing with sampleddata or continuous functions.The impulse responseof the systemwill be zero Drior to / : 0 and then havethe valuesl,,l,fr, . . at succes_ sive samplingintervals.We representihe processdia_ gramaticallythus: 6, -+ system - .f, : l,"fr,"f,,fr, . . .1. Writing 6, , for a unit impulse that occurs at t : n rather than aI t : 0, we can illustratelinear and time_ invariant systemsas fiollows: Linear:

kE,--r system-s kf,: Ikfo,kft,kf, Time-invariant: 6, , +

s y s t e m- . f , - , :

't. ,1,

[0,0,0,. .,0,fr,-f,,-fr,. .] -;;et cs

In the last bracketon the right, the first output differ_ ent from zero is foand occurs at the instant t : nL, (which we write as t : n by counting in units of A). Obviously, any input that consistsof a seriesof sampledvaluescan be representedby a seriesof unit impulsesmultiplied by appropriateamplitudefactors. We can then use the abovetwo propertiesto find the output for eachimpulse,and by superimposingthese, we get the output for the arbitrarv inout. We shall illustrate convolution by considering the output for a filter whose impulse response is / [f,,,"f,,f,]: Il, -1, jJ. Wnen the inputg, is [go,g,, &] : tl, j,-jJ, *. apply to rhe input the seriesof impulses[E,,ja, ,, -ja, ,] (the last two subscriptsmean_

280

DATA PROCESSING

Table9.2 Mappingbetween domains Offset-CMPtraces Common-midpointdomain Unmigrateddomain Location time section(or map) Offset-time (gather)

eNMO+ eDMO= eMigration= eT D conversion+ eVelocity analysis=

ing that the impulsesare delayedby one and two sampling intervals,respectively)and obtain the following outputs: E ,_ + [ 1 . _ 1 .

this case,the summationbecomesan integration:

h(t\: fut +g(tt= l- frr,ru - r) dr.

]1.

( 9 . 2 7 ; 1 5 . 1 4155; . 1 9 6 )

16,, - fo,j, _j,N, - j 6 , -_ ,t oo, , _ j , ; , _ j 1 Summing,we find the output

,j tl ,_ j ,_ ;,;,_]r [6 ,+ ;6 ,_-, j a ,_-r Convolution is illustratedin fig. 9.2. This operationis equivalentto replacingeachelementof the one set by an appropriatelyscaledversion of the other set and then summingelementsthat occur at the sametimes. If we call the output fr, and denote the operation of taking the convolution by an asterisk,we can express this as h,:.f,*8,

:

Common source receivertraces Common reflection-pointdomain Migrated domain Location-depth section(or map) Stacking-velocity time at zero offset

\"rrr, *

: l.f&r"fug, + f,En,fug. + -f,g, +.frgo,. . .1. (9.23:t5.211) (We usethe sum sign to mean summationover all appropriate valuesof the summation index.) Note that we would haveobtained the sameresult if we had input / into a filter whose impulse responseis g,; in other words,convolutionis commutative:

The convolution theorem states that the Fourier transform of the convolutionof two functionsis equal to the product of the transforms of the individual functions;we can statethe theoremas follows: J, *- F(v) : lF(v)l s:tto, 9,, , G{v) : lG(u)le.,rct,t, .f, * 9,., F(v)G(v): {lf'(y)l.rrrt,t}{lG(u)lsjrrr,r;, . ' l F ( u ) l l G ( u )gl j l r r { , t + r * t , t l

( 9 . 2 81; 5 . 1 4 5 )

wherelf'(u)l and lG(v)lare the amplitude spectra,and T1(u)and %(v) the phasespectra.This meansthat if two sets of data are convolvedin the time domain, the effectin the frequencyciomainis to multiply their amplitude spectra and to add their phase spectra. Equation (9.28) thus provides an alternativeway to carry out a convolution operation: (1) transform the function to be convolved,say,g,, and the convolution (or filter) operatorf, into the frequencydomain; (2) multiply their amplitude spectra at each frequency value and add their phasespectra;and (3) transform the result back into the time domain. Outpul

h, : .f,* g, : g,* f, : Z.frg,-o= Lgo.f,_r. e.24) Whereaswe have been expressingconvolution as an operation on sampleddata, we can also convolvea sampleset with a continuousfunction:

h(t\ : f, * g(r)=

2frsQ - k).

(e.2s\

Each term in the summation representsthe function g(t) displacedand scaled(displacedto the right k units and multipliedby -fo).A specialcaseof eq. (9.25)is that of convolvinga continuousfunction g(/) with a unit impulselocatedat t : n:

6,_,* g(r):

)u-_,rtr

* k) : g(t - n) (9.26)

(because6o_,is zero exceptfor k : n, where 6o , : l). Hence,convolvingg(t) with a time-shiftedunit impulsedisplacesthe function by the sameamount and in the samedirection as the unit impulseis displaced. One can alsoconvolvetwo continuousfunctions.In

Fig. 9.2

Filtering as an example of convolution

CONVOLUTION

281 ,!

I

lr h

\01

I 1 2 5H z

0

* convot!eo wilh conlb (r,)

(b)

I

N l-oooas

? am = 250H2 Alias rpectra -

t' y r el d \

)

rcl(l\

,I

Alirs spectra

I I I l

, t,fl'il'\,,,,rnr..,

* r25

0

t ('onvolred rrrth \tnc

/

0 i r nul ri p lcr d by r boxcar

v A

\tt)

,rl ryqJL$/q^_

t F o o o 4s I yr el d s I I

'tolds

(.) e

_\r_'.1 2 5H t

bacl'

I v

Fig. 9.3 Sampling and reconstituting a waveform. The left column shows a time-domain sequence,and the right shows the equivalent frequency-domain sequence.

Becauseof certain symmetrypropertiesof the Fourier transform, the reciprocalrelationshipalso holds, that is, multiplication in the time domain is equivalent to convolutionin the frequencydomain (see$15.2.8): 2nfg,, , F(v) * G(v). (9.29;15.146) 9.2.2 Sampling,interpolating,and aliasing (a) Sampling. In analog-to-digital conversion, we replacethe continuous signal with a seriesof values at fixed intervals.It would appear that we are losing information by discardingthe data betweenthe sampling instants. The transform relationship in eqs. (9.28)and (9.29) can be usedto understandsampling and the situationsin which information is not lost (see a l s o$ 1 5 . 5 . 1 ) . We make use of the comb or samplingfunction; Ihis consistsof an infinite set of regularly spacedunit impulses(fig. 9.3b).The transformof a comb is also a comb: comb(l) * k, comb(u), ( 9 . 3 0 1; 5 . 1 5 5 ) where k, : 2rlL,, A being the sampling interval (see$15.2.6). If the comb in the time domainhas elements every 4 ms, the transform has elementsevery l(0.004 s) : 250 Hz. We shall also make use of the boxcar (fig. 9.3d), a function that has unity value betweenthe values+vo and is zero everywhereelse.The transform of a boxcar,boxr"^(v),is a sincfunction:

til 2t'n', b o x!vt\ . ( v ) . , k . s i n c 2 r v , , t: k ', lTlv^l

( 9 . 3 11; 5 . 1 5 2 ) where k, dependsupon the area of the boxcar (see e q .( 1 5 . 1 2 3 ) ) . Figure9.3ashowsa continuousfunction y(l) and its transformY(u): tQ) "

Y1',

The amplitude spectrum, lI(v)|, is symmetric about zero for real functions, negativefrequencieshaving the same amplitude values as positive frequencres. (Negativefrequenciesresult from the use of Euler's formula when we combine Fourier series in sinecosineform into the complexexponentialform.) The sampleddata that representy(t) can be found by multiplying the continuousfunction by the comb (hencethe name "sampling function"). If we are sampling every 4 ms, we use a comb with elementsevery 4 ms. Accordingto eqs.(9.29)and (9.30), 2n comb(r)Y(t)'' k.,comb(u)x Y(v). Convolution is equivalentto replacingeach data element (eachimpulsein comb (v) in this instance)with the other function, I'(v) (seeeq. (9.23)).This is illustrated in fig. 9.3c. Note that the frequencyspectrum of the sampled function differs from the spectrum of the continuousfunction in this exampleby the repetition of the spectrum.

I

a

282

DATA PROCESSING

We can recover the transform of the original func_ tion by multiplying the transform of tni sampled function by a boxcar.The equivalenttime_domainop_ eration (seeeqs.(9.28)and (9.31))is to convolvetle sampledata with the sinc function; as shown in fig. 9.3e,this restoresthe original function in everydetail. The sincfunction thus providesthe precise,,operator" for interpolatingbetweensampleuulu.r. (b) Interpolating. Interpolating between sampleval_ ues is necessaryin many processes,for example,to time shift a trace by lessthan a multiple of the sam_ pling interval for static or normal-moveout correc_ tion. In the simplestform, we might try to obtain a value in betweentwo samplesby linear interpolation; fig. 9.4, which showsthe amplitude spectrum'of a lin_ ear operatot indicatesthat this is unsatisfactory for frequenciesgreater than a sixth of the samplingfre_ quency(about 40 Hz for the 250 Hz sampling in this case).An 8-point truncatedsinc function, un i-n,..po_ lator of reasonablelength, gives good results up to 3/8 of the sampling frequency(about 95 FIz), which is.adequatefo-rmost operations.Interpolating is also drscussed in g9.lI .2 and t5.t.j (seeeq. (15.74t\. (c) Aliasing; ympling theorem. When the complete sinc function is usedfor interpolating,no information whatsoeveris lost in the processof samplingand in_ terpolating. However,if the continuousiunc-tionhad had_aspectrum (showndashedin fig.9.3a),which in_ cluded frequencycomponentshighei than'i 25 Hz (in this example),then the time-domiin multiplication by the samplingfunction would have producedan over_ lap of frequencyspectraand no longer would we be able to recoverthe original spectrumfrom the spec_ trum of the sampleddata nor could we recover the original waveform. Whether the original waveform is recoverable depends, therefore, upon whether the original waveform contains frequencieshigher than half of the samplingfrequency. The relationshipsdemonstratedin the foregoingare 1.0

0.0 0

25

50 ZS Frequency(Hz)

100

125=V"

Fig. 9.4 Amplitude spectra of interpolators for the midpoint between two samplesof data sampledevery 4 ms. The solid line is the-linearinterpolator; the dashed line is ihe g_point truncated sinc function. (From Hatton, Worthington, und Mukin, 19g6,

3 8).

summarized by the sampling theorem;no information is lost by regular sampling provided that the sampling frequency is greater than twice the highest frequency component in the waveform being sampled (seealso $15.5.1).This is equivalentto sayingthat there must be more than two samplesp., iy"L for the highest frequency.The sampling theorem thus determines the minimum sampling we can use. Becausethis mrni_ mum sampling allows complete recovery of the wave_ form, we can further concludethat notiring is gained by using a finer sampling. Thus, sampling"ratesof 2 and 4 ms permit us to record faithfully p.o-uid"dnon. of the signal spectrumlies above250;;d 125Hz, re_ spectively.In actual practice, the limits are half to two-thirds of these frequenciesbecauseof analog aliasingfilters (seewhat follows). Half the sampling frequencyis called the Nyquist frequency,u", that is, vu : ll2\'

(9.32)

Any frequency present in the signal that is greater than the Nyquist frequencyby thJ amount Au will be indistinguishablefrom the lower frequencyv* _ Av. In fig. 9.5, we seethat a samplingrate of 4 ms'(that is, 250 samplesper second)will allow perfect recording of a 75-Hz signal, but 175- and ZSb_Hzsignalswill appear as (that is, will alias as) 75 and O tti 1OUz is the same as a direct current), respectively.Aiias sig_ nals that fall within the frequency band in which we are primarily interestedwill appear to be legitimate sisngl_s.To avoid this, alias fiitirs (see ng. i.+Ay are used before sampling to remove frequeicy compo_ nents higher than the Nyquist frequency.ihi, rnurt be done beforesamplingbecauseafierwardsthe alias signals cannot be recognized.Actual analog alias filters often haveslopesof j2 dBloctaveand cutofffre_ quenciesone- to two-thirds of an octave below the Nyquist frequency in order to prevent aliasing (see problem 9.5). Digital filters can be made with almost arbitrarily steepslopes;someinstrumentssamplewith a.frequencyhigh enough that there is little .n.rgy to aliasand then apply a very steepdigital aliasfilter be_ fore frequency. -resamplingto the desired *-fting Aliasing is an inherent property of all syitemsthat sample, whether the sampling is done in time, in space,or rn any other domain such as frequencv.For example, trace-to-tracemiscorrelation becausethe trace spacing is larger than half the apparent wave_ length constitutes aliasing (spatial iliasing; see $.9.2.2d).Alias filtering must be done beforesarnpled data are resampledat a lower rate, as is often done in processingto reduce processingtime. The term "aliasing" is also usedto denoteother types of errors in sampling_,for example,errors becauseihe sampling is not regular and errors in correlating sedimentary cycleswell-to-wellwherethe well spacingis too large. Signalswith frequencieshigher than the Nyquist ^ frequencycan be recovereduniquelyifall the frequen_ cies are known to lie within a narrow passband(see problem 9.4).Thus,4-mssamplingwould be adequate

CONVOLUTION

Fig. 9.5 Sampling and aliasing. Different lrequenciessampled at 4-ms intervals (250 times per second). (a) 75_Hz signal, (b)

to definesignalsuniquelyif the only frequenciespres_ ent were,for example,betweenI and L 1 kHz. (d) Spatial sampling. Spatial sampling occurs when we usegeophonesto samplea waveat differentpoints in space.Figure 9.5 representsa harmonic wave recorded at a fixed point as a function of time and the samplingis done at fixed time intervalsAt. However, the figure could equally well representwave motion observedalong a line ofgeophonesat a given instant of time, the samplingbeing done at fixed intervalsof distanceAx. In time sampling,we take l/Ar samples per unit time, whereasin space sampling, we take l/Ax samplesper unit distance.In time sampling,we had a Nyquist frequency,u" of eq. (9.32),which gave the maximum number of wavesper two unit time in_ tervals, and aliasing occurred for components that have frequencieshigher than u". In spacesampling, we havea Nyquist wavenumbeq r*l2t: rrl2r : llX*:

1126*,

(e.33)

which gives the maximum number of waves Der two unit lengths.Aliasing occurs when a component has more wavesper unit length than this, that is, when K..) K" (or l, < \r). Figure 6.2c shows an alias arlgnment.

175-Hz signalyieldsthesamesample values as75 Hz,and(c) 250-Hzsignalyieldssamples of constant value(0 Hz).

9.2.3 Filtering by the earth We can think of the earth as a filter of seismicenergy. We might considerthe waveresultingfrom an explosion as an impulsek8n,that is, the wavemotion at the sourceof the explosionis zero both beforeand after the explosion and differs from zero only in an extremelyshort interval (essentiallyat t : 0),and during this infinitesimal interval, the motion is very large. Ideally,the signalthat we record would be simply k6, convolvedwith the impulseresponseof the earth (assumingthe earth is a linear system-see915.4).The result would be zero except for sharp pulsescorrespondingto the arrivals ofdifferent reflections.Ifthis were so, we could easily determine from the recorded data the complete solution to the seismicproblem. However,in practice,we get back not only primary reflections,but also multiples, diffractions, surface waves, scattered waves from near-surface irregularities, reflected refractions, and so on, all modified by filtering becauseof absorptionand other causes,and with random noisealwayssuperimposed. We can regard the recorded seismicwave as the result of successive convolutionsof the initial impulse E, with a number of impulse responsesrepresenting various factors modifying the wave during its passage

284 DATA PROCESSING through the earth. The major zonesaffecting the sig_ nal are (a) the zone near the sourcewhere the stresses and absorption. of energy (especially at higher frequencies)often-are' "it.._.; *. .. . y.it" s, for the impulseresponseof this zone; (b) the sequenceof reffectorswith imputse re_ sponsee,: this is the signalthar seismic re_ flection work is intendedto find; (c) the near-surfacezone, which has a dispro_ portionate effectin modifying the signal;its impulseresponsers n.; (d) additionalmodifying effects becauseol-ab_ sorption, wave conversion, multiples and diffractions,and so on; the combination of theseis representedby p,. Combining theseeffects,we obtain for the recorded waveform g,, g , : k E , * J , * € , * f r , * p , : ( k 6 ,* s t * nt*O,rirlr: ,,, This equation expressesthe convolutional model, that ls, a selsmlctracecan be thought of as a seriesof con_ volutions.The convolutionalirodel i. ""ni.uito .ort data processing. Equation(9.34)is at.o w.lilen gt:et*li,r

(e.3s)

*h:t.,y,,includes (k,6, * s, * n, * p,): w, is calledthe emoe.dded (o_requivalent)wavelet;it is the waveletthat woutd be reflectedfrom a singleisolated interface.The convolutional model often includesadditive noise r, (usually,but not necessarily, assumedto be ranOom;. The convolutionalmodel oi a noisy ,.i*i. i*.. i, 8,:Q,*wt+rt.

(e.36)

Whenwe usea Vibroseissource,the input to the earthis.along wavetrainr,,and the resultiit seismic tracegj is 8::v,*sl*e,*fi,*p,

(e.37)

(wherewe write si rather than s, becausethe filtering processesnear the vibrator may be different from those.nearan impulsivesourceowing to the different magnitudeof the stressesinvolved). 9.2.4 Waterreverberation end deconvolution Let us examinethe effect of multiples resulting from

the rop and bouo; oi u-*ui"i ruy.. 1gfleltion.11 (Backus, 1959). WewrirenA for the.ounJ_irip ,.uu_

eltime from top to bottom and back, n being ui, inr._ ger. We assumethat the reflection coefficieits at the surfaceand bottom of the water tuy., ur. ,u"n tfrut the ratio of the reflectedto incidentamplituaes are _ f and +R, respectively,the minus sign aenotlnfpnase reversalat the water_air interfacel We assufre also that the amplitude of a. wave returning directly to a hydrophoneafter reflectionat a certain lorizon iwith_ out a "bounce" round trip betweentop and U"tto_

of the water layer) is unity and that its traveltime is l. A wave that is reflected at the same t o.iron unO suffersa bounce within the water layer .ith.. U.fo.. or after its travel down to the reflecior will arnve at time / + nA with the amplitude _R. Because there are two raypathswith the sametraveltimefor a single_ bounce wave, one that bounced before traveling downwardand one that bouncedafter returning from depth, we havein effecta wavearriving at time t + nA with the amplitude -2R. There will*be three waves that suffertwo bounces:one that bouncestwice before going downward to the reflector, one that bounces twice upon return to the surface, and one that bouncesonce before and once after its travel down_ wards;eachof theseare of amplitudeR2, so that their tl a wave of amplitude 3R, arriving at time I * :"T 2nA. Continuing thus, we seethat a hytlrophone will delT.t s_u_c:essive signalsof amplitude, f ,'_Zn, :nr, -4R3, 5R4,. . . , arriving at iniervals of nA. We can thereforewrite the impulseresponseof the water layer Ior vanous water depths,z : ),nVL,,where Z is ihe velocityin water:

l; : tl, -2R, 3R,,-4Rr, 5R4,. . .l, (,?: t).1 : [ , 0 , - 2 R , 0 , 3 R , , 0- 4 , R., . . .], (n:41 : [ , 0 , 0 ,- 2 R , 0 , 0 , 3 R , , . . . ] , (n:,) (e.38) and so on. Thus, the water layer acts as a filter, If we transform this to the frequency domain, we find a largepeak (the sizeof the p"ut i*.ruring *itf, in-creasing R) at the frequencyll2nL, andat multiples of this frequency.Theseare ih" f.equencies that are reinforcedat this water.depth(that ir, tfr. f..qu.n"i., for which interferenceis constructive). The result of passinga wavetrainthrough a water fuy.. i, ,h. ,u_. as multiplyingthe amplitudespectrum of the wave_ rorm wrthout the water layer by the spectrumof the lmpulseresponseof the water layer.Wheneverthe re_ flection coefficientis large (and Lence n is largl) anO U2nL (or one of its harmoni"cs)ties .t1:, i..?,u.n.y wrtnln the sersmicspectrum, the seismic record will appear very sinusoidalwith hardly any variation rn amplitude throughout the.recording p!.ioJ ng. 6.32).Becauseof the overridingor"iifurionr, ir.. ii *U -' U. difficult to interpret the prima{, .eflectlons.' A filter i, having the property that .f,*i,:6, (e.3e) is called the inverseflterof f,.If we passthe reverbera_ tory output from the hydrophonesthrough the inverse water-layerfilter (in a data_processing ce-nter),we will remove the effect of the water_layerfilter. T(e inu..se of the water-layerfilter is a simple filter lthe Backus filter) with only three nonzeroterms:

i : [,2R, Rr, (n: l)l : [ , 0 , 2 R , 0 ,R , ] , {n: 2ll : ! , 0 , 0 , 2 R ,0 , 0 ,R 1 {n: 3)l

(e.40)

CORRELATION and so on (seeproblem9.laand$15.5.5).Figure6.32b showsthe result of applying such a filter to the data shownin fig.6.32a. The processof convolvingwith an inversefilter is calleddeconvolution and is one of the most important operationsin seismicdata processing(Middleton and Whittlesey,1968).Whereaswe haveillustrateddeconvolution as removing the singing effect of a water layer,we could also deconvolvefor other filters whose effectswe wish to remove if we know enough about the filtersand the signal(see99.5and 15.7). 9.2.5 Multidimensional convolution In the foregoingsections,we have assumedonly one independent variable, time. We have assumedthat eachseismictraceis beingprocessedby itself and that the only data availableare the successionof samples of that trace,the actual situation for many processes. There is, however,no need to restrict ourselvesto a successionof time samplesfor only one trace, and someprocessesinvolve convolution with two or even more variables. The convolutionoperation,eq. (9.23), can be written lor two variables./ and n as h , , , : / ' , , ,* g , n = L L J ' r , , , g , r , , , , k D' (9.411 ; 5.164) This equationstatesthat the convolution result is the superpositionof valuesat nearby times and locations (if r and n, respectivelyindicate time and location) after each has beenweightedby the filterl,, . As with one-dimensional convolution,the operationis commutative.It is also possibleto carry out the operation by transforming1,, and g,,, usinga two-dimensional Fburiertransform(see89.1.4, 15.2.4),multiplyingthe two-dimensionalamplitude spectraand adding the phasespectra,and then transformingback to give/r,,,. If t is time and r is distance,then the transformed domain is the frequency-wavenumber domain. 93 Correlation 9.3.I Cro.ss-r'orrelation The cros.g-correlation function is a measureof the similarity betweentwo data sets.One data setis displaced varying amountsrelativeto the other and corresponding valuesof the two setsare multiplied togetherand the products summed to give the value of crosscorrelation. Wherever the two sets are nearly the same,the productswill usually be positiveand hence the cross-correlation is large;whereverthe setsare unlike, some of the products will be positive and some negativeand hencethe sum will be small. If the crosscorrelation function should have a large negative value,it meansthat the two data setswould be similar if one were inverted (that is, they are similar except that they are out of phase).The two data setsmight be dissimilarwhen lined up in one fashion and yet be similar when one set is shifted with respect to the

285 other; thus, the cross-correlationis a function of the relativeshift betweenthe sets.By convention,we call a shift positiveif it involvesmoving the secondfunction to the left with respectto the first function. We expressthe cross-correlationof two data sets,Jr, and 1,,,as

d-(") : ItOr.",

(9.42;15.147)

wheret is the displacementof y, relativeto x,. (Note that $-(r) is a data setrather than a continuousfunction, becausex and y are data sets.)Let us illustrate cross-correlationby correlating the two functions, -r,: il, -1, jJand-y,: I, j,-:f,shown in fig.9.6. Diagram (c) showsthe two functions in their normal positions.Diagram (a) showsy, shifted two units to the right; correspondingcoordinatesare multiplied and summed as shown below the diagram to give 0,,(-24). Diagrams(a) to (e) showy, shiftedby varying amountswhile (/) showsthe graphof 9,,(r). The cross-correlationhas its maximum value (the functions are most similar) when y, is shifted one unit to the left (r : +A). Obviously we get the sameresults if we shift n, one spaceto the right. In other words, 0.,(t) : d,.(-r).

(9.43:15.15'7)

The similarity betweeneq. (9.42)and the convolution eq. (9.23)should be noted. We may rewriteeq. (9.42)in the florm 0.,(r): S,.(-r) :}yoxr,:Ltrx n o, : y,* x ( 9 . 4 4;15.158) y . . i .: .* Hence, cross-correlationcan be performed by reversingthe first data set and convolving. If two data setsare cross-correlated in the time domain, the effect in the frequencydomain is the same as multiplying the complex spectrum of the second data set by the conjugateof the complex spectrumof the first set. Becauseforming the complex conjugate involvesonly reversingthe sign of the phase,crosscorrelationis equivalentto multiplying the amplitude spectraand subtractingthe phasespectra.In mathematical terms, : lX(v)lsrr,t't, x,<+ X(v) : lI(u)l srr'r"t, Y, <+ Y(v) : lX(u)lgrr,t't, x , <+ X(v) 0 . , ( t ) e + X ( v ) Y ( v ) : l x ( u ) l l ) z ( u ) rl et r , t ' rt ' t ' t t . (9.45; 15.141) We note that changingthe sign of a phasespectrumls equivalentto reversingthe trace in the time domarn. Anstey (1964) gives a particularly clear explanation of correlation. 9.3.2Autocorrelation The specialcasewhere a data set is being correlated with itself is called autocorrelation.In this case,eq. (9.42)becomes

0..(t) : Ir*r**,.

(e.46)

286

DATA PROCESSING (b) Shift of - I

(a) Shift of -2

(c) Shift of 0

- aI

0*ye2)= 0 + 0 + =+l

0*y(-l)=0-l - - 4

(d) Shift of + I

(e) Shift of +2

0*y(+l)=0+l+l =+l Fig. 9.6

r l

d*r(O)= I

- 2l

l 4

-+l

(/) Graph of Q*, Q)

0*yG2)= 0 + 0 -l :_,

Calculating the cross-correlationof two functions.

Autocorrelationfunctions are symmetricalbecausea time shift to the right is the sameas a shift to the left; from eq. (9.43),

0,"(r) : 0.,(-").

0,.(r)

J.

"rr(t)ic(t+r)dt.

(e.47)

The autocorrelationhas its peak valueat the zerotime shift (that is, a data set is most like itself before it is time shifted). If the autocorrelation should have a largevalue at sometime shift Lt + 0, it indicatesthat the settendsto be periodicwith the period At. Hence, the autocorrelationfunction may be thought of as a measureof the repetitiveness of a function. We can expressthe precedingconceptsin integral form applicableto continuous functions. Equations (9.42) and (9.46)now take the forms

d , (" ):J- x(t)/(t+r) dt. ( 9 . 4 81; 5 . 1 4 7 )

( 9 . 4 91 ; 5.161)

9.3.3 Normalized coruelation The autocorrelationvalue at zero shift is called the energyof the Irace:

: O..to) T"i

(e.50; 15.162)

(This terminology is justified on the basis that x, is usually a voltage,current, or velocity,and hencexl is proportional to energy).For the autocorrelationfunction, eq. (9.45)becomes

0..(r)er lX(v)l'.

(9.51;15.161)

CORRELATION

287

T i me ( m s )

(ms)

(al

T i m e( m s )

T i m e( m s )

(cl

(dl

Fig. 9.7 Effect of random noise on an autocorrelation. (From Hatton et al., 1986: 27 8.) (a) Sinusoidal signal r,/: s long, (b) half of autocorrelation of part (a); (c) sinusoid with random n o i s e s u p e r i m p o s e da, n d ( d ) h a l f o f a u t o c o r r e l a t i o n o f p a r t ( c ) .

I n continuous-function notation,

: rr,ll'0, : s,.(o) f J

o, '"r,rr (9.s2;t5.162)

Becausethe zero-shift value of the autocorrelation function is the energyof the trace, lx(l)|, is the energy per unit of time or thepowerof the trace and lX(v)1,is the energyper incrementof frequency,usually called the energydensity or spectraldensity. We often normalizethe autocorrelationfunction by dividing by the energy:

0,,{r)no,.: O*,O)rc

(e.53)

The cross-correlation function is normalizedin a similar manner by dividing by the geometricmean of the energyof the two traces:

0,,,(r).".-:#-**,,,* (e.54)

Normalized correlation valuesmust lie between +1. A value of + I indicatesperfect copy; a value of - I indicatesperfectcopy if one of the tracesis inverted. Purely random noise affects only the zero-shift value of an autocorrelation.A periodic signal of frequencyu buried in noiseincreasesthe autocorrelation value at the frequencyv and hence shows up much more clearly in autocorrelationsthan in a time series

/fie.e.7).

9.3.4 Vibroseisanalysis The signalgj, which our geophonesrecord when we use a Vibroseissource(fig. 9.8d), bears little resemblanceto e,, the impulseresponseof the earth. To obtain a meaningful record, the data are correlated with the Vibroseissweep(control) signal v,. The recorded signalgj is 9,,:v,*e,,, : where we let e', s',* e, * n, * p, in eq. (9.37). Using

DATA PROCESSING

288

-0.21 o.oo

1.06 1.28 1.49 1.70 1.92

0.21 0.43 0.64 0.85

2.13 2.34 2.56 2.77 2.98

3.19 3.41 3.62 3.83

4.05 4.26

4.47

4.69 i

(a)-+q{+r+dn+llxnryt (b) (c)

11-

(d) (e)

tn -{.rln*P#a{f.r*rr.mJ-0.21 o.oo

0.21 0.43 0.64 0.85

1.06 1.28 1.49 1.70 1.92

2j3

2.34 2.56 2.77 2.98

3.19 3.41 3.62 3.83

4.O5 4.26

4.47

4.69

Time (s) Fig. 9.8 Composition of Vibroseis signal and its deconvolution. (From Yilmaz, 1987: 143.) (a) A reflectivity sequence;(b) l0-to-120-Hz sweep;(c) assumedminimum-phase near-surface f i l t e r ; ( d ) a s s u m e de f f e c to f ( a , * r , ,* s l ) i n e q . ( 9 . 3 7 ) ,t h a t i s , t h e

convolution of(a). (b), and (c), what we would expect to observe with Vibroseis (ignoring n, * p,l, (e) cross-correlationof (b) with ( d ) ; a n d ( f ) l 0 - t o - 1 2 0 - H zz e r o - p h a s ef i l t e r e dv e r s i o no f ( a ) .

eq. (9.44), we find for the cross-correlationof the sweepand the recordedsignal

Sometimesthe energyof the stackedtrace is usedinsteadof the amplitude: t+mJ

=e',*(v,trv,)

: ei * Q""(t).

(e.55)

(The next to the last stepis possiblebecauseconvolution is commutative.)Hence, the overall effectis that of convolvingthe earth function with the autocorrelation of the Vibroseissweepsignal.The autocorrelation function, S",(r),is quite sharp and has sizeablevalues only over a very narrow range of time shifts. Therefore, the overlap produced by the passageof a long sweepthrough the earth has been eliminatedalmost entirely(fig. 9.8e).This is shownalso in fig. 9.9, where parts (b) and (c) are the sameVibroseisrecord before and after cross-correlation.

9.3.5 Multichannelcoherence The cross-correlationfunction can only be used as a measureof the coherencebetweentwo traces.As a coherencemeasurefor a large number of traces we could make use of the fact that when we stack several channelstogetheqthe resultingamplitudeis generally largewherethe individual channelsare similar ( coherent) so that they stack in phase,and small wherethey are unlike (incoherent). If we let g,,be the amplitudeof the individual channel i at time l, then the amplitude of the stack at time I will be Ig,, and the squareof this will be the energy. The averageamplitudeof the stackedtraceover a time window from I to t + mL is given by

I llg,,t I

i:L

l+'mL,

/ N

\^

s ls. l' a \?-,""I : c," l-lmA

0,"(/) : 8', * v,, : (v, * e',)* v,,

(e.56)

o.5t)

Both of thesedepend on whether the event within the time gate is strong or weak. Some normalized measureof coherencewould be a better indication of the phaseagreementof tracesthan the foregoing.The ratio of the energyof the stack comparedto the sum of the energiesof the individual componentscould be usedas a coherencemeasure: /

E, :

N

\ r

l1 4s "o ' l,I n

(9.58)

" Itt" " We expect a coherentevent to extend over a time interval; hence,a more meaningful quantity than E, S, (Neidell and Taner, 1971),which is the semblance, ratio of the total energy of the stack the denotes within a gate of length (1 + mL) to the sum of the energyof the componenttraceswithin the sametime gate. Using the same terminology as before,we can write

't,(id' sr.=

(9.5e)

I I k,,)' l

i-l

The semblancewill not only tend to be large when a coherent event is present,but the magnitude of the semblancewill also be sensitiveto the amplitude of the event.Thus, strong eventswill exhibit large semblance and weak eventswill exhibit moderatevalues of semblance,whereasincoherentdata will havevery low semblance.

CORRELATION

289

6

7

8

4

9

5t -_l 031l l0 [ s5 w l o r E e r E

9ry.-"p

(6€tt Hz) (a)

llncorrelatedVlbroselsRecod (b)

Conelated Record (c)

Fig. 9.9 Vibroseis field record and record after crossc o r r e l a t i n gw i t h s w e e p (. F r o m Y i l m a z , 1 9 8 7 :2 1 . ) ( a ) 6 - t o - 6 0 - H z

sweep;(b) uncorrelated record (in three segments);and (c) correlated record.

Semblanceand other coherencemeasuresare used to determinethe valuesof parametersthat will "optimize" a stack.The semblanceis calculatedfor vanous combinationsof time shifts betweenthe component channels,and the optimum time shifts are taken to be thosethat maximizethe semblance.Semblance,therefore, can be used to determine static correctionsor normal-moveout corrections.

on stationaryrandom noise($15.2.12). superimposed The probability that any individual sample will be positive or negativewill dependon the magnitudeof this information bias compared to the noise level. If we record only the sign of eachsample(ignoring completely the magnitudeof each sample)and then sum many samples,we will in effectmeasurethe time variations of this probability and thus the magnitude of the signal bias.This is calledsign-bitrecording. The situation can be thought of as the testing of two mutually exclusivehypotheses:signalpresent(I) or signal absent(,F) (Cochran, 1973).The signal is a quantity r1 for example, a displacementat time t, which can be measuredand comparedwith a preselected threshold value r,,. When r ) ro, we say that hypothesis7 is true. However,this decisionis not nec-

9.3.6 Sign-bit recording In a conventionalVibroseisfield recording, such as shownin fig.9.9, the information about any individual reflectionis distributed over the time duration of the sweep.The information content at any time is small comparedto the noiseand can be thought ofas a bias

l

l

CORRELATION

289

s

2

7

3

8

4

9

5

10

9rypep (660 Hz) (a)

10 llncorrelatedVlbroselsRecord (b)

Conelated Record (c)

Fig. 9.9 Vibroseis lield record and record after crossc o r r e l a t i n gw i t h s w e e p (. F r o m Y i l m a z , 1 9 8 7 :2 1 . ) ( a ) 6 - t o - 6 0 - H z

sweep;(b) uncorrelated record (in three segments);and (c) correlated rccord.

Semblanceand other coherencemeasuresare used to determinethe valuesof parametersthat will "optimize" a stack.The semblanceis calculatedfor varrous combinationsof time shifts betweenthe component channels,and the optimum time shifts are taken to be thosethat maximizethe semblance.Semblance,therefore, can be used to determine static correctionsor normal-moveout corrections.

on stationaryrandom noise($15.2.12). superimposed The probability that any individual sample will be positiveor negativewill depend on the magnitudeof this informationbias comparedto the noiselevel.If we recordonly the sign of eachsample(ignoring completely the magnitudeof each sample)and then sum many samples,we will in effectmeasurethe time variations of this probability and thus the magnitude of the signalbias.This is calledsign-bitrecording. The situation can be thought of as the testing of two mutually exclusivehypotheses:signalpresent(I) or signal absent(F) (Cochran, 1973).The signal is a quantity 4 for example, a displacementat time l, which can be measuredand comparedwith a preselected threshold value r,,. When r ) ro, we say that hvoothesisI is true. However.this decisionis not nec-

9.3.6 Sign-bit recording In a conventionalVibroseisfield recording, such as shownin fig. 9.9,the information about any individual reflectionis distributed over the time duration of the sweep.The information content at any time is small comparedto the noiseand can be thought ofas a bias

290

DATA PROCESSING

essarily correct because there is a finite probability P(rlT) that a signal is present for any value of r and converselya finite probability that the signal is absent for any value of r The probability of a correct decision Q,i"

a': I,u l-'r'I r) dr

(e.60)

and the probability of a "false-alarm" error Q. is given by

o': l-'t'l F)dr I,N

(e.61)

The probability of missingthe signalis

I r) dr. n-: J'_ro

(e.62)

Becausethe signalmust occur for -- ( r ( *- and the probability of a certainty is unity, we must have Q,+ Q_: l,

Q,:

| - Q..

(9.63)

Clearly,we should selectr0 to make Q, a maximum and Q-a minimum. The choiceof r,,dependson what is known a priori about the signaland noise. Sign-bitdetectionconsistsof measuringthe coincidencesin polarity of a group of rutraces.We can specify a sign-bit semblance($9.3.5)as follows. Let x,obe the kth samplein the observationwindow on the ith trace. The product (x*x,o)will be positive when and only when a coincidenceof polarity occurs.The number of coincidencesamong the n tracesat t : k is =' 17: i + l). )l : l sgn(x,,.x,^)u(x,^x,ol. where sgn(y) : yllyl - +-l, the factor u(x,ox,o) ($15.2.5)being includedto eliminatenoncoincidences (that is, when sgn(x*xro): - I ). To normalizethe sum, we divide by the number of coincidenceswhen all traceshavethe samepolarity, that is, by [(r - l) + (rz - 2 ) + . " + 3 + 2 + l l : ! n @- l ) . S u m m i n g o v e r the observationwindow of widttr m, we get S:

, m n ( n - l ) fiflsgn{x,ox,o)r.r(x,or,a)1. i,=, (j:,+l), (e.64)

If we wish to include coincidenceswhen two traces are both zero, we note that sgn(0) and u(0) are both undefined,so we can assignarbitrary valuesto them; when one of the traces is zero, we say that : 0. When both traces are zero at sgn (xoxru)r.r(x,,Jir) t : r, the product is still zero except when the traces coincidein polarity at t : r - l; in this case,the product is * l. Figure 9.10 is a syntheticexampleto show that, in the presenceof noise,the sign-bitmethod givesresults comparable to conventional Vibroseis. Trace I is a synthetic Vibroseis trace resulting from 10 irregularly

spaced reflectors of different reflectivity. Random noise three times the rms amplitude of trace I has been added to it to give trace 2. Trace 3 is a sign-bit representationof trace 2; it flips back and forth between its only two states,positive and negative.Trace 4 is the result of correlating trace 2 with the sweep, that is, it is a conventionalVibroseisoutput, and trace 5 the result of correlatingtrace 2 with a sign-bit version of the sweep.Trace6 is the sum of 20 tracesmanufacturedlike trace 2 exceptwith different noise,and trace 7 the sum of 20 traces, each manufactured like trace 3. Traces8 and 9 representrespectivelya conventional correlatedVibroseistrace and a correlated sign-bit trace. With sign-bitrecording,only one bit is recordedper sampleso that the volume of data is greatly reduced and instrumentationcan be simplified. Instrumentation fidelity requirementsare alsogreatlyrelaxed;geophone nonlinearity, for example, becomesless important becauseonly the sign and not the magnitude of the output is measured.GeocorrMuses sampling boxes in the field with 16 geophonechannelsconnected to each box. The sampling box determines whethereachchanneloutput is positiveor negativeat the samplinginstant and recordsthis information as one of the bits in a l6-bit word that is then relayedto the recording truck. A l6-to-l savingin the number of cablechannelsis thus achieved.Field systemsemploying 1024channelsare now in use.

9.4 Phase considerations The Fourier synthesisof wavetrainsaccordingto eq. (9.2) involvesadding togethercosinewavesof different frequenciesand differentphases.If the samecomponentsare added togetherwith different phaserelations, different waveforms result. Changing the waveformchangesthe location ofa particular peak or trough, and hencemeasurementsof arrival times are affectedby variations in the phase spectra.Because seismicexplorationinvolvesprimarily determiningthe arrival times of events,preservationof proper phase relationshipsduring data processingis essential. Out of all possiblecausal waveletswith the same amplitudespectrum,that waveletwhoseenergybuilds up fastest is called the minimum-delay wavelet; its phaseis alwayslessthan the other waveletswith the sameamplitude spectrumand henceit is also called minimum-phase The simplest wavelet (except for an impulse)is a data setthat containsonly two elements, the set (a, D).The amplitude spectrumof this data set is identical with that of the set (b, a), but no other data set has the same spectrum.lf a > b, energy is concentratedearlierin the waveletin the set(a, b)than in the set (b, a), and hence(a, b) is minimum-phase (or minimum-delay). Larger waveletscan be expressedas the successivetime-domain convolution of twoelement wavelets(see $15.5.6a);a large wavelet is

PHASECONSIDERATIONS

291

' lli)il(f'uri

,l1,.l',' Illlrlr'

'" r,,rfi'f,tJrfi{],'tl't'tf+11lrtf,i,il,l.tttllit,i{rlill,il'dldi4rtiililifliilild*ft[ <.,lfilllllllililililifltll;llliiIlfflilt]tf,lnffilnfii[lll[ilHill;]l'lllltilflrirlilliliitlfiiliqfill .! 4.,'er..:./^q!

...-\*-.

..!:Fr'..f {ry.:{h-r"4fl*

.Ji'-,.i*r

dvrran?.$f

{.,:,:*,*i-,

"t ..1"" "

{ -: * $..tsr - ii ^ J,.:.n

\t

rttlflil,,1rtl[Lil,,ili\riil;ll,^};i,'1,t,',r/r,r,,r'r./r1,,{;]^,.,t!,u+

r-f -,r-i-t" tr r-+' tJ-

1". t1*

Fig. 9.10 Comparison of conventional Vibroseis and sign-bit recordings. (Courtesy of Geophysical Systems.)(Trace l) Uncorrelated record from l0 reflectors; (2) uncorrelated record r l i t h n o i s et h r e e t i m e s t h e r m s a m p l i t u d e o f ( l ) a d d e d ;( 3 ) s i g n brt equivalent of 2: (4) trace 2 correlated with the sweep,that is, a correlated Vibroseis trace; (5) trace 3 correlated with the sisn-

bit version of the sweep,that is, a correlated sign-bit trace; (6) sum of 20 traces manufactured as for trace 2. where each has different random noise; (7) sum of 20 traces manufactured as for 3; (8) correlated trace 6; (9) correlated trace 7. (No1e.'Traces 4, 5, 8, and t have been time shifted.)

minimum-phaseif all of its component waveletsare minimum-phase.Minimum-phasecan also be defined in other ways,for example,by the location of roots in the z-domain(S15.5.6a). Most seismicsourcesgenerate wavesthat are nearly minimum-phaseand the impulse responseof many of the natural filtering processesin the earth are minimum-phase. Minimum-phasedoesnot necessarilymean that the first half-cycleis the largest,however.In the presence of interfering eventsand noise,it is often difficult to tell whethera reflectedwavelethas the sameor oppositesign as the downgoingwavelet,that is, whetherthe reflectioncoefficientis positive or negative.It is also difficult to tell the onset time of a reflection,and it is this time that is neededin determiningreflectordepth. The embedded wavelet can be changed in processingto a zero-phasewavelet(seegl 5.5.6d)to facilitate interpretation.A zero-phase wavelethas its phase spectrumidentically zero, that is, 1(u) : 0 for all v. Such a waveletis symmetricalabout a central peak (or trough), which has higher amplitude than any other peaksor troughs.(An autocorrelationfunction is zero-phase.)We shift the time scale so that the amplitude maximum gives the arrival time. Such a waveletis anticipatorybecausehalf of the waveletprecedesthe arrival time. Some filtering processesrequire that assumption be made about the phase of the signal; generally minimum-phaseis assumed(Sherwood and Trorey, 1965).Deconvolutionbasedupon autocorrelationinlbrmation has to assumethe phasebecausethe phase

information of the waveformwas lost when its autocorrelation was formed. This can be seen from eq. (9.51), where we note that the autocorrelationfunction, $--(l), has the transform, lX(u)1,with zero phase for all valuesof frequency.Thus, all of the phaseinformation presentin X(v) has beenlost in the autocorrelation. However, most signals and natural filtering processesare representedby real, causal functions (see $15.5.6a).The Hilbert transform technique ($15.2.13) can be usedto determinethe phaseinformation if the function is real and causal. Causal waveletsthat are not minimum-phasecan be made minimum-phaseby applying an exponential gain (taper) that cuts down the size of the latter part of the wavelet.Many actual waveletscontain only a few nonminimum-phase roots and can be made minimum-phaseby using an exponential multiplier with a very gentleslope,perhaps0.995',wherer is the time in milliseconds.A criterion for determininshow much taper to apply is discussedin 915.6. CorrelatedVibroseisrecordsare sometimeserroneouslycalled"zero-phase"becausethe effectivewavelet, $""(l) in eq. (9.55), is zero-phase.However, the earth responsee, is usually nearly minimum-phase and thus the result, d""(t),is mixed-phase. It is sometimesbelieved that zero-phasefiltering of minimum-phasesignalsdoes not alter the phase becauseit involvessubtracting zero from the phasespectrum; however,this is only true if the amplitude spectrum is unaltered (see fig. 9.ll and Hatton et al., 1 9 8 6$: 2 . 6 . 5 ) .

. t

292

DATA PROCESSING is nonunique unlessadditional information is available or additional assumptionsare made. Equation (9.36)containsthree unknowns,wr er and r,, but only one known, g,. The most common additional assumptions are lhat w, is minimum-phaseand/or e, has a flat (white) spectrum (at least over some limited bandpass).Additional constraintssometimesusedinclude imposing a maximum length to lfl or use of a multichannelprocedure. Deconvolutionoperationsare sometimescascaded, a deconvolutionto removeone type of distortion being followed by a different kind to remove another type of distortion. Someof the typesof deconvolution are describedin the following sections.Webster( 1978) givesa bibliographyofthe literatureon deconvolution up to that date. 9.5.2 D et erminist ic inver.ge fi lter ing

\ ,

T i me

Where the nature of the distorting filter is known, the inversecan sometimesbe found in direct deterministic manner.In $9.2.4,we gavean exampleof water reverberation whereinwe useda model of a water layer to derive the distorting filter,f,. We also expressedthe inversefilter, i,, by eq. (9.39)but this equationcannot be usedto find the sequencei, (excepton a trial-anderror basis).However,we can transform eq. (9.39) carry out the division, into the ;-domain (see$15.5.3),

I ( z \ : 1 1 P 1 t r f: i ^ ; '

Fig. 9.ll C h a n g e s i n w a v e l e t c h a r a c t e r i s t i c sr e s u l t i n g f r o m zero-phasefiltering. (From Hatton et al., 1986: 24.) (a) Minimum-phase wavelet (10 24 Hz): (b) zero-phase wavelet w i l h s a m ea m p l i t u d e s p e c t r u m ;a n d ( c ) c o n v o l u t i o no f ( a ) w i t h (b).

9.5 Deconvolution and frequency filtering 9.5.I General In $9.2.4,we defineddeconvolutionasconvolvingwith an inversefilter. Equations(9.34)to (9.36) expressed the seismictrace as a convolution of an earth reflectivity function e, and a seriesof distorting filters,whose combinedexpressionis the equivalentwaveletr',. The ultimate objectiveof deconvolutionis to extract the reflectivity function from the seismictrace and thus improve the vertical resolution and recognition of events.Sometimesdeconvolutiononly tries to undo the effectsof someprior filtering. Deconvolutionto extract the reflectivityfunction e,

(e.65)

(providedthe divisiondoesnot "blow up"), and then transform back to get i,. Even wherewe know the nature of the filter, we may haveto determinepart of the solution by trial and error or other techniques;in any case,we still haveto determinethe valueof R in eq. (e.40). solutionsare Deterministic(or semideterministic) also used to removethe filtering effectsof recording and processingsystems.The source waveshapeis sometimesrecorded(though not alwayswith proper ghost effects) and used in a deterministic sourcesignature correction. In marine work, the source waveformis often assumedto be constant;the waveshape may be recorded in deep water with a hydrophone suspendedbelow the source.Another procedure is to monitor each energy release using hydrophonesnear the sourceand calculatechangesin the signaturefrom such monitor records;this procedure corrects for source-to-sourcevariations before stacking.Still anotherprocedureusesthe sea-floorreassuming flection to determinethe sourcewaveshape, that the seafloor is a simple,sharp reflector. 9.5.3Deghostingund recursive filtering With a sourcebelow the baseof the weathering(fig. 9.12), ghost energy is reflectedat the base of the weathering (see $6.3.2b),where the coefficient (approaching from below) is -R (ignoring ghost energy reflectedat the free surface).An impulsefollowed by

I

I

I

DECONVOLUTION AND FREQUENCY FILTERING

293

and write G(z)N(z) : H(z)D(z),

F(z) : 711r11or'r. (e.70) "recursive" part where previous The right side is the output values are used in deriving future output values.

Reflectioncoefficient= R

9.5.4 Deghostingby combininggeophoneand hydrophonerecords

Fig. 9.12

Reflection plus ghost.

its ghost constitutesa filtering action; the transform can be written as F(z) : 1 - Rz',

(e.66)

wherez' representsthe delay associatedwith the twoway traveltime from the source to the reflector producing the ghost.The inversefilter is an infinite series, F t(z): 1l(l - Rz') : | + Rz" + (Rz,), + (R;,)3 + . .. + (9.61\ BecauseR is lessthan unity, this seriesconvergesand so could be usedas a satisfactorydeghostingfilter. An input that includesghost effects,g,, can be convolved with this inversefilter to give a deghostedoutput ft,; in z-transformform, H(z) : G(z)F t(z): C()l(l H(z)(t - Rz'): G(z\, H(z) : G(z) + Rz,H(z).

Rz'), (9.68)

The last term on the right representsa delay of H(z) by r units; this last equation can be written in time notation as h,: 8, + Rh,-,

(e.6e)

Thus, we can determinean output value by adding to the input a proportionateamount of a previousoutput value. If the ghost delay is the sample interval, n: I and hu: 8n' becausetherewas no output prior to zero: hr:gr+RhD, h r : g z+ R h v h . : g t + R h . ," ' Filtering that involvesfeedingback part ofthe output is called feedback or recursivefltering (see $15.7.2).Recursivefiltering allows us to carry out complete deghostingwithout using many terms and thus is economical in computing. In more general terms, we can expressa filter F(4 as a quotient of polynomials(see915.5.3and 15.7.2), H(z) : G@F(z) : G(z)[N()t D(z)|,

If an upgoing compressiongivesa positive kick on a velocity geophone, a downgoing compression will give a negativekick, whereasboth waveswill give a kick of the samekind on a hydrophone(97.5.5).This observationcan be usedto attenuatenear-surfacereverberation when geophonesand hydrophonesare combined in sea-floor cables.The 180' reversalof phase upon reflection at a free surfacecausesa hydrophone to seea rarefactionand thus a polarity reversal of the ghost with respectto the primary impulse,whereasa velocity phone seesalso the reversal of wave direction and hence seesthe ghost and primary with the samepolarity. If R is the sea-floorreflection coefficientand f, and fo are the hydrophone and geophonetransductionconstants(97.5.1d),respectively,the resultingimpulseresponses(fig. 9.13a; seealso(i9.2.Iand 15.4.I ) haveoppositepolaritiesand amplitude ratios of f,,(l + Ryf.;(l - R). This scale factor can be determinedby locatinga sourcedirectly abovethe sea-floorcableand recordingthe respective signalamplitudes.Barr and Sanders( 1989)found that valuesfor R rangedfrom 0.20 to 0.25offshoreLouisiana. Although fig. 9.l3 assumesnormal incidence,R does not changevery much until the critical angle is approached. Figure9.l3b showsthe resultof convolutionof fig. 9.13awith a minimum-phase waveletin both time and frequencydomains.Notchesin the geophoneand hydrophone spectra (caused by interferenceas waves reverberatewithin the water layer) occur at the lobe peaks in each other's spectra,resulting in notchless spectrawhen added together.Over the seismicrange, the hydrophonespectrumrises6 dB/octave,whereas the geophonespectrumis flat, and the phasecharacteristicsdiffer by 90'; theseare the conditions associated with a derivativeoperator (this can be seenby differentiatingthe wave in eq. (2.56) with respectto l). Thus, summinggeophoneand hydrophoneoutputs providesa way of removingreverberation. 9.5.5 Least-squares( Wiener) .filtering Sometimeswe wish to determinethe filter that will do the bestjob ofconverting an input into a desiredoutput. The filter that most nearly accomplishesthis objective in the least-squareserror senseis called the least-squares filter or the Wienerfilter, occasionallythe optimumrther(Robinsonand Treitel, 1967). Let the input data set be g,, the filter that we have

(1 + R)R

dl

o

E

E z o E

(D

9 o

E 2 0 CL E

GEOPHONEOATA

r?(

0 o o

HYDROPHONE DATA E

a o o

Ezo o, E

1.1

100

Time(s)

Frequency(Hz)

(b)

(c)

Fig. 9.13 Reverberation seenby a pressurehydrophoneand a velocitygeophoneon the seafloor. (Courtesyof Haliburton GeophysicalServicesInc.) (a) Impluse responsesof a hy-

200

drophone (above) and a geophone (below). (b) Waveforms and (c) spectra for the source array, the individual elements, and their combination.

DECONVOLUTION AND FREQUENCY FILTERING to determine be f,, and the desired output set be {.,. The actual result of passing g, througli this filter is g, * f, and the "error" or differencebetweenthe actual and the desiredoutputs is Nr,- g, *1. With the least_ squaresmethod ($15.1.6),we add togetherthe squares of the errors, find the partial derivativesof the sum with respectto the variablesl (the elementsof f,), and set thesederivativesequal to zero. This givesihe fol_ lowing simultaneous equations, where g, and A., are Known:

- s,*.r,), : o,

i),t

r:0,1,2,...,n,

(e.7r)

or )

*.f,): o, L, @ - g,*.f,)dJ, 1,"@,

t:0,1,...,n.

\(t

- r;,-r) | !r(+rt -): o

The only terms in the convolution that involve I are thosecontainingg, ,. Hence.

}rn - }sof,-o)s,-,:0,

2&s*,:\lsrs, ,"f,r' The left siae is 6.16';according to eqs. (9.42) and ( 9 . 4 3 )W . e l e t i : , - k and sum over instead of 7 overk.'

d"r(i): \2s, ,s, ,,f,: Y,Lg, ,g, , upon interchangingthe order of summation.The last factor is 0""(i - 7) accordingto eqs.(9.46) and (9.47). Hence, we arrive at the normal eauations;

l-+,,ti-iv: d"o(,),i : 0 , 1 , 2 , . . . , n

(9.73a)

In matrix form, this is

'""(0) 0""(-l) '""(l) I 0*(0) - l) ,r(n) 6o@

- s,* .f,l): 0, (aDtr)(>lh,

(e.7 s)

which is called "minimizing the Z, norm." Much geophysicalprocessingis done as matrix operations(see$15.1.3).Thesemay becomeunstableif some eigenvalueshave the value zero (becausedivision by zerois then involved;seegl5.l.3c).The addition of a small amount of white noise(perhapsas little as 0.0001%but typically 0.5 to 20/o)makesthe eigenvaluesnonzeroand stabilizescalculations. Practical Wiener filtering involves specifying the desiredoutput, the filter length, the filter's zero-time position (or, equivalently,the amount of anticipation component), and the amount of white noise to be addedin the filter design.Truncation and taperingof the input wavelet'sautocorrelationare generallyrequired, recognizingthat estimatesbasedon the data are contaminatedwith noise. Figure 9.14ashowsa minimum-phasewaveletafter being passedby a band-passfilter and figs.9.14b and 9.14c are its frequency-domainrepresentation.The waveletis no longer minimum-phaseafter the bandpass filtering. Figure 9.14d is its autocorrelation, which is the sameas that of the minimum-phasedesired wavelet having the same amplitude spectrum. Figure 9.14eis the minimum-phaseWiener operator to convert fi5. 9.14ainto the desiredminimum-phase wavelet.The amplitude spectrumof fig.9.14e,shown in fig. 9.14f,is simply the inverseof fig. 9.14b,but its phasespectrum(fig. 9.14f) differs from fig. 9.14b becauseit is minimum-phase.The result of convolving f i g . 9 . 1 4 ew i t h f i g . 9 . 1 4 ai s s h o w ni n f i g . 9 . 1 4 h ;t h wavelethas beenshortenedbut not made into a spike; its amplitude spectrumis white (flat) but it is mixed phase.

lI I 10",(o)l "lI l'l: l*::'l'l 0",(o) 4o?n) 0o(l -

I lr.l lo",rnrl (e.73b)

(Seealso problem l5.ll.) The normal equationsfor least-squares filtering also havean integralexpression for continuousfunctions,

t_

These equations can be used to cross-equalizetraces, that is, to make tracesas nearlyalike as possible.Supposewe havea group oftraces to be stacked,such as the componentsof a common-midpoint stack. After the normal-moveoutcorrectionshavebeenmade,the traces may still differ from each other becausethey have passed through different portions of the near surface.The normal equationscan be usedto find the filtersthat will make all the tracesas nearlyas possible like some pilot trace, such as the sum of the traces. This procedure will improve the trace-to-trace coherence before the stack and henceimprove the oualitv ofthe stackedresult. The least-squares method is also called ..minimizing the I, norm." Sometimesother differencesare minimized,for example,the absoluteerror, giving insteadof eq. (9.71)

(e.72)

One equation is obtained for each of the n + I ele_ menlsi1l. Writing the convolutionas a sum usingeq. (9.24)gives

295

d""(" - t) t) dt : o"a(r). (9.74;15.273)

9.5.6 Whitening (a) Spiking deconvolation The normal equations are used to accomplish spiking deconvolution.The desiredoutput is the earth'simpulse response,e,. We assume that e, is random, that is, knowledge of the shallowreflectionsdoesnot help in predictingthe ar-

I

t

t!

I

296 DATA PROCESSING Therefore,if we concern ourselvesonly with positive valuesof t, we have the valuesrequired ; ;1"" .q. fy, the spiking deconvolutionfilter; if,ur, .q. .211) (9.73b)becomes (a)

(e)

0.1 Time(s)

10""(o) 10"(1)

t:

lr-r,

,f.

Jo

(e.78) (0 Frequency(Hz)

Frequency (Hz)

o

E (c)

1250

50 Frequency(Hz)

125

(h)

0

0.1 Time(s)

o.2

0.1 Time(s)

0.2

Fig. 9. l4 Band-pass-filteredminimum_phase wavelet and rts spectra before deconvolution and after friener deconvolution. ( A f t e r Y i l m a z , 1 9 8 7 :l 0 l . ) ( a ) W a v e l e t ; tU unJ ct pf,"r. ""0 ,rnplitudespectra of(a); (d) autocorrelation oftr); i.) inu..r" op"r_ ator calculated from (d) assuming minimum phase: (f and g) phase and amplitude spectra of (ej; (h) result oi lonuoiu,ng ,n_ verse operalor (e) with (a). A spike is not achieved becausethe band-passfiltering destroyed the minimum-pfr"r. "f f "L

rival timesand amplitudesof deeper reflections.Con_ sequently,the autocorrelationof e, is negligiblysmall exceptfor zeroshift. and we can write

: fr E,. Q""(r)

(e.76)

The geophoneinput g, is regardedas the convolution ot e, wlth variousfilters(seeeq. (9.34))(the most im_ portant of which resultsfrom near_surface effects),the effect being representedby the singleequiva_ :l:^rill r e n tn l t e r w , ( s e ee q . ( 9 . 3 5 ) ) : g , : € ,* \ , , . Becausee, (assumedto be minimum_phase)is the de_ sired output h,: h , eq.(9.44)enablesus to write

Q*(t):9,*h_, : { e , * u ' , ) : ( e , *e , ) * n , - "= k 6 , * , r . r ' , : 1,* KWN

(9.77a)

There can be no output from filter u, until after there has.been an input to the filter; this is equivalent to sayingthat g, is causal.Hence,y,, : 0 for i < 0. thur.

0"r(/) : 0

forl<0.

(9.7tb)

The constantkwois usuallysetequal to I when we are only concernedwith the relativevaluesoff : igno.,ng scale factors is common and proper in #uny p.o"_ essingmethods. Note that much of the noiseis random, but presum_ ably the assumptionof minimum_phase dirc.il,nate, againstthe noise. (b) Spiking deconvolutioninfrequency domain (whit_ ening). The ability to transform into the ir"qu.n"y or other domains not only providesalternative computing methods,but also providesinsights as to what deconvolutionmethods imply. Thus, if,intine'of. ."_ sponsee, as random implies equal probabilit]es that the amplitudesat all frequencieiwili be equal and re_ u. of white light, and we call spikingJ".onuo_ ftu,Lton ildr whirening.I-tis equivalentto findingin inverse nlter whosetransficrmis ,f(u),where I(v) : 1151r,

(9.7e)

(see.fig.9.15a),G(u) being the transform of the input, so that the product I(v)G(v) is constant. In applying the inversefilter, the phasehas to be known. Oft.n ,"" start with 0o. assumethat G(v) is minimum_phase so that the waveletcan be determineduniquely irom the o f $ o l s e e$ 1 5 . 5 . 6 cat .n d t h e n ' i n v e rG t ( u )t o :ftltuT oDraln .f(u)(note that (u) is alsominimum_phase; see problem 15.32a). Equatio_n (9.79)appliesto any frequencyvalue, for Ijv,.): llQv,).lf G(v,)shouldbe small,then lluTpli: r.(ur)wril be large.Thus, a whiteningfilter emphasizes the weak frequencycomponents,resulting in improve_ ments to the extent that they are attenuited signals. Above somefrequen.y ,, noir. dominates, so whiten_ p^erjgrmedolly over a limited band-pass. If, as rfn8^rs n n g . 9 . t 5 a . s i g n a lG ( u ) s h o u l db e e s p e c i a l l v weak over a narrow band near frequencyu, (iometimes re_ ferredto as a notch in the signal,p."trl'urr;, it " inu.... filter will magnify noise at thi, fi.qu.n"y,'rorn.ti_.. with disastrousresults.To prevent."..sri* -ugnin_ cation of noise, white nolse, I, is sometimes added when the filter is being designed,the magnitude of , Delngsmail compared to the averageof G(u). This does not substantiallychangethe filier fng. q.iSt.l ut most frequenciesbut makesit smallerat ihl notch tie_ u, (that is, l/[G(u) * e] = l/G(u) exceptwhen 11.1:y i: very small). The white noise is'addedinly for 9J") filter-designpurposesand a ..whitenoir. uaa.Ji "o-_ ment on a seismicsectionthus indicateslessnoise sen_

DECONVOLUTIONAND FREQUENCY FILTERING

29i

autocorrelationvalues,eq. (9.73).The central peak of the autocorrelationfunction representsshifts of less than half the dominant period; it containsmost of the information about averagewaveletshape(fig. 9.16a). On the other hand, peaksand troughsfor greaterlags representrepetition of information, suchas produced by multiples.If our desiredwavelet4, is to b. .rr.n_ tially the sameas the earlypart of the waveletinvolved in g, but is to die out rapidly but smoothly(sincesharp changesinduce ringing - see$15.2.7),we can take for 0*, the early part o[ go multiplied by a suitabletaper that truncatesit after perhapsone cycle (fig. 9.16b), giving us everythingneededto solveeqs.(9.73). vl

Frequency

(c) Delayed spike. Spiking deconvolution often magnifieshigh-frequencynoise, especiallywhen the embeddedwaveletis not minimum-phase.A betterre_ sult can often be obtainedby delayingthe spikeof the desiredwaveletby la samples:

Q)

Q*Fn)

Q,,(r- n) Spectrum of G(u) + addednoise

bo@)

Q,,(m- l) ."

Qo@+t) r:Y,

Qrr(m

r"u :.:

.

Qo(n) 6o(n- rl . '.

0""(o)

^ f,

i Added white noise

vr

- n)

Frequency

X

b) Fig- 9. l5 Spectra of signal before deconvolution (solid curvesl and of inverse filter used to achieve whitening (dashed curves). (a) Without addition of white noise; the bandwidth is usually specified without knowledge of the exact spectrum; (b) wrth white noise added for inverse-filterdesign puiposes.

.f, -f,,,*

I

0

(9.81)

/"

A better result can often be obtained by shifting the shaping-filteroperator, that is, making it two_sided (noncausal),so that it has coefficientsfor both neea_ tive and positive time values.Such a filter has bJth erated by the whitening deconvolution. Equatron anticipotion components(which act on future input (9.78)now becomes values) and memory components(which act on past values).Two-sidedfiltersare sometimescalledshiping Qr,(-n) | flters and they can be used to produce a zero-phasi 0",(l - n) embeddedwavelet.A particular wavelet,suchas a re_ | corded air-gun waveshape, can be usedas the desired - l) (I + €)0".(o)l Q*(n) Qrr(n wavelet in what is sometimes called signature pro_ " l t'essing.Such applicationsof shaping filters are ex_ l r l amplesof waveletprocessing(99.5.9). The effect of a one-sidedfilter using only memory componentsis shown in fig. 9.17. Figure 9.17b is a t.t' smoothed version of the zero-phaseautocorrelation ( e 8 0 1 of fi5.9.17aand fig.9.l7c is a one-sidedversion.FisIt | ure 9.17d is a minimum-phaseinverse operator io Note that adding white noise changesthe minimum_ shortenfi5. 9 .17b,and fig. 9.I 7e is half of iti autocor_ phasecharacteristic ofa waveletlsei gtS.S.Oa;. relation. Figure 9.l7fis the zero-phasewaveletcorre_ The least-squaresinverse filter is designed from spondingwith fig. 9.l7e, but the result of applying it

t';

=lol

DATA PROCESSING

298 to fig. 9.17b gives fig. 9.179; memory components alonecannot achievethe desiredzero-phaseresult (although the multiple has been attenuated).

9.5.7 Predictive ( gapped) deconvolution Predictive deconvolution(Peacock and Treitel, 1969) attempts to removemultiple effects,which can be predicted from knowledgeof the arrival time of the primaries involving the samereflectors.Predictivedeconvolution operators often do not begin to exert an effect until after some time Z (called the prediction /ag), which is usually the two-way traveltime to the first multiple-generatingreflector.We use the portion of $o after the time l, as $"0 in eq. (9.73a)so that the filter predicts the multiples (fig. 9.16c), that is, we write

Io*(; - lf,: 6*(L + i):

First portion Later portionscontain contains repetitive(multiple)information wavelet-shape b) information

(9.82a)

or expressingit in matrix form, eq. (9.82a)becomes

0*(o)

0o(-n)

0""(l - r)

0*(l) dsq(l)

o*"(r+ l)

d*"(L -

l)

..

6,r&)

Portion used for i in eq. (9.82a)

6,,(i

- n) 6rr(L 6r,(l+L-n

+,,(n) 6o{n- l)...

Portion used for Q;zin(8'64) (c)

Fig. 9.16 Determining autocorrelation values to be used in deconvolution. (a) Autocorrelation of a trace; (b) tapering of S** to use for $,,, in eqs. (9.73); and (c) prediction of multiple effects for predictive-deconvolutionfilter design.

0""(o)

+*eL) br,(-L + |

0*(o) 0*(l) 6r,(n

- L)

(9.82b)

This givesa predictionfilter of length n + | and lag L. We can subtractthe predictedtracefrom the observed trace to give the prediction eruor, which is the trace with the nredictedmultiolesremoved:

h,: 8,- 8, ,*.f,'

(e.83)

Where the first multiple-generatingreflector is deep, as with marine data in deep water,the deconvolution operatormay be setto zero over portions of its length (correspondingto the zeros in the inversefilters in eq. (9.40)) to make the computation more economical; this is called gappeddeconvolution(Kunetz and Fourmann.1968). Corrections may be made for variations in the waveletshapefrom location to location. One procedure is to record the initial waveform.Another is to sum the autocorrelationsof all the tracesfrom a single source activation, estimate the waveshapefor this autocorrelationsum assuminsthat the waveform is

minimum-phase,and then apply a Wiener filter to convert this wavelet into a desired constant waveshape.This sort of proceduremay be usedto correct for variationsin waveletshapeproducedby different factors, such as changesin the source,detectors,recording instruments,or near-surfaceconditions.

9.5.8 Other typesof deconvolution Homomorphic or cepstral deconvolution, Kalman filtering, and other techniquesare occasionallyemployed. Homomorphic deconvolutioninvolvesa transformation from a space where functions are convolved (the time domain) to one where they are added (the cepstrumdomain;see$15.6).The transformationof a time-domainfunction g,to a cepstrum-domainfunction g(O is accomplishedin three steps,

g,eG\z). " : Gtzt,I ln{Gtz)} I G(z)<+s(0. J

(9.84;15.235)

The cepstrum-domainequivalentof eq. (9.35)is

c(O: t'(O+ ?(O,

(e.85)

DECONVOLUTION AND FREQUENCY FILTERING

299

a

r-l

Fig. 9.17 Wavelet processing. (From yilmaz, l9g7: 107.) (a) Autocorrelation; (b) same after smoothing; (c) one_sidedversron of (b); (d) spiking-deconvolution operator calculated from 1c.;; (e) minimum-phase inverse of (d), which is often assumed to be the embedded wavelet; (f) zero-phase equivalent of (e) having the.same amplitude spectrum; and (g) shaping filter io convert (e) into (f).

thus the contributionsof the waveleLn(O, and of the reflectivity, a(0, add. The wavelet is uiually slowly varying and its cepstrumlies mainly at low valuei, i whereasthat of the reflection coefficientsis mostly spread out over larger values. Thus, low_passand high-passfiltering in the cepstrumdomain (ialled lift_ ering')achievesa large measureof separation.An in_ verse transformation of the high-pass (reflectivity) portion back to the time domain then completesthe homomorp hic deconvolut ion. The cepstrumof a minimum-phasefunction is one_ sided,that is, g(0 : 0 for ( < 0 if G(z)is minimumphase; this fact is sometimes used in separatins minimum- and maximum-phaseelements.To taki advantageof the one-sided aspect in the cepstral domain, one sometimesforces G(z) to be minimum_ p_laseby applying exponentialgain, that is, by using G'(z): Qz)k",where k < I (Stoffa,Buhl, and Bryan, 1974). The reflectivity tr(z) is then found bv lifterins the high-passportion of g'((). transfiormingback tJ find E'(z), and then applying the inverseexponential

weightingE(z) : 6'1t10-,. Otis and Smith (1977)use spatial averagingin the cepstrum domain as a way of determining source wavelet shape.They assumedthat the source wavelet is stationaryand the earth'sresponseis spatiallynonstationary,so that phasecontributions in the earth's responseat different locations disappear in the aver_ aging. Entropy is a measureof the chaos or lack of order in a system. Primary reflections are nonDredictable from precedingdata and thus lack order. Maximumentropy deconvolutionattempts to extract such reflec_ tions by separatingorderly (for example,equivalent wavelet) from disorderly (for example, signal) ele_ ments.Maximum-entropydeconvolutionis discussed further in 915.7.6d. Most of the foregoingtechniquesassumestationarity, that is, that the statisticsofthe waveshapedo not change with time. However, we know that hlsher frequenciesare attenuated more rapidly than lJwer frequenciesand that peg-legmultiplesand other factors causethe downgoingwavetrainto lengthenwith time. One should deconvolvemore effectivelyif the change in waveshapewith time were accounted for (sie $9.5.11).Kalman filtering (Crump, 1974) and other types of adaptivefiltering attempt to take changeswith time into accountby continuouslyupdating the statis_ tics on which the filters are based.The most common mode of time-variant deconvolution(often abbreviated TV decon; see Clarke, 1968)involvesdesigningone operatorbasedon an autocorrelationof the early por_ tion ofthe data and another basedon an autocorrelation of the late portion of the data, eachdata window being I s or longer so as to give adequatestatistics. The early and late operatorsare then applied at any given time in inverseproportion to the differencein time to the centersof the designwindows,that is, the early operator is graduallyrampedout as the late operator is ramped in. Sometimesthe design windows overlap, and sometimesmore than two desisn windows are used. 9.5.9 Waveletprocessing The convolutional model and Wiener filterins make it possible to replace a known embedded*uu"i-"t *ith u more desirablewavelet,within limitations imposed by the signal-to-noiselevel.The waveletembeddedin the data resultsfrom the convolution of many filters, some of which are space-and time-variant, and the wavelet usually contains some mixed-phase components. Most wavelet-estimating techniques average over a number of traces and sizeabletime windows. The desired wavelet is almost always either minimum- or zero-phase,the output to the final display almost always being zero-phasenowadays. A variety of different processesthat involve determining, assuming,or operating on the effective wavelet shape go under the name waveletprocessing. Someof these(l) attempt to make the waveletshape

rlI

Ia

I

300

DATA PROCESSING

Fig. 9.18 Effectofdeconvolution.(From Yilmaz,1987:85.)(a) Undeconvolved stackedsection,and (b) stackedsectionof deconvolved eathers.

everywherethe same, (2) some change the effective waveletto some"more desirable"shape,and (3) some endeavor to separate the earth's reflectivity from wavelet-shape effects. Wavelet processing,which attempts to make the waveletshapeeverywherethe same,should be done as a prestackprocessso that all the componenttracesto be stacked have the same effective wavelet shape. Low-frequency components are more likely to be stackedin phasethan high-frequencycomponents,so that stacking often acts as a filter attenuatinghigher frequencies;this type of waveletprocessingdecreases this filtering action. Sometimesthe sourcewaveletis actually recordedfor every energyreleasein marine recordingand then usedin deterministicwaveletprocessing.More commonly, the waveletis determined from the autocorrelation function by summing the autocorrelationsof all tracesrecordedfrom the same source,assumingthat the only common elementis the source wavelet, so that the autocorrelation sum ls simply the autocorrelationof the sourcewavelet.Examples of waveletprocessingare shown in figs. 9.l8 a n d9 . 1 9 . The secondtype ofwavelet processing,changingto somemore desirablewaveform,is used to correct for filtering actions (especiallyphase shifts) associated with instrumentation,so as to change hydrophonerecordeddata to look more like geophone-recorded data or to produce a better match betweenlines recorded with different recording instruments.Sometimes the effectivewaveletsassociatedwith certain source types have been measuredand "cataloged,"

and the catalogwaveletis usedin waveletprocessing. Sometimesthe effectivewavelet is determined from the sea-floorreflection. The third type of waveletprocessingattemptsto remove wavelet-shapeeffectsand leave the earth's reflectivity function, that is, to separatew, and e, in eq. (9.35)suchas in the lifteringexamplein 99.5.8.Such processingis usually applied after other processing has removed as much of the noise as possible. Wavelet-processing techniquesusually improve the high-frequencyresponseand, consequently,the resolution. They often precedetrace inversion(95.a.5). An exampleof waveletprocessingof actual seismic data is shownin fig. 9.19.The shorteningof the wavelet reducesringing and permits seeingmore stratigraphicdetail. 9.5.I 0 Frequencyfiltering Reflection signals often dominate over noise only within a limited frequency band. The filter should passfrequencieswhere the signal dominatesand not passthosewherethe noisedominatesin order to optimize the signal-to-noiseratio. Filter panels,such as shown in fig.9.20, which display a portion of record sectionfiltered by a successionof narrow band-pass filters,are often usedto determinethe optimum bandpasslimits. 9.5.1I Time-variantprocessing The frequencyspectrumof seismicreflectionsusually becomeslower with increasine arrival time as the

J 3

i;ff i::i;r;i:iitiiiliiiil,,i-T

b)

(D)

Fig. 9.19 Wavelet processing. (Courtesy of Crant Geophysi_ c a l . ) ( a ) P o r t i o n o f m i g r a t e ds e i s m i cs e c t i o n ,a n d ( b ) t h e s e c t l o n

after processing1o broaden the bandwidth ofembedded wavelet and make it zero-phase.

T I T T DD A I A 'I[TEI oo l||tltllflD

O-a Ns

t-E ltr

n-i

X,

E-|| Hr

PANEI, |t-lllh

l0- Il|r

tl-LNr

ll.ll

lfi

aa-01 llr

'l!+ i

Fig.9.20

F i l t e r p a n e l .( C o u r t e s yo f G r a n t G e o p h y s i c a l . ;

DATA PROCESSING

302

(a)

(b)

(c)

(d)

(e)

(f)

(s)

(h)

(l)

(i)

(k)

(t)

(p)

Fig. 9.21 Tests of deconvolution parameters.A gather and its autocorrelation (below) are shown under various circumstances. (From Yilmaz, 1987:132 4.) (a) The input gather; (b to d) varying autocorrelation windows (between the heavy lines); (e to h)

deconvolution-operator lengths of40, 80, 160, and 240 ms; (i to l) prediction lags of 12, 32, 64, and 128 ms; (m to p) percent prewhitening of I, 4, I 6, and 32n1,.Best choices are (c), (g), 0 or k), and (m).

higher-frequency componentsare attenuatedfasterby absorption,peg-legmultiple, and other natural filter(see$6.5.1and 6.3.2b).Hence,we often ing processes wish to shift the passbandtoward lower frequenctes for later portions of the records,that is, we wish to accomplishtime-variant(TV)filtering. Decisionsas to the time-variant filter parametersare often basedon filter panelssuchas shownin fi1.9.20, the deepestcoherent energy in any passbandbeing taken as the point wherenoisebeginsto dominateover signal. Any discontinuous change, such as an abrupt change in band-passparameters,will produce undesirableeffectson the seismicsection,however,including Gibbs' phenomena ($15.2.7).Changes are therefore usually distributed over a merge zone. For example, filter A might be used down to time l, and filter ,Bbelow time t B GA < /r); the merge zone then is betweentimes l, and t B.A linear ramp may be used in the mergezone; the data in this zone may be filtered with both filters I and B and the data at tA + Lt within the mergezone will be the sum of the resultsof applying these two filters, where the results are weightedaccording to the position within the zone, that is, the weights would be (t" - tn - Lt)l(tB - tA) and L,tl(t" - /r), respectively.More than two fllters and hencemore than one mergezone might be used. In addition to filtering, other processessuch as deconvolution and statics correction are sometimes applied in the time-variant mode following similar procedures.Changesin filter parametersor in the pashouldnot be made in the rametersin other processes region where mapping is to be done lest the effects

of changing parametersbe misinterpretedas having structural or stratigraphicsignificance. 9.5.I 2 Choosingdeconvolutionparameters the selectionof deVlmaz (1987:109 3l) discusses convolutiontime gate,operatorlength,predictionlag, and percent prewhitening for both wavelet shaping and multiple-suppressionpurposes (fig. 9.21); he showsexamplesof the effectsof varying theseparameters.The best time gateshouldexcludethe early part of a record, which contains energycorrespondingto guided waves,and also excludethe deeperpart of the record, whereambient noisedominates(choice(c) in fig. 9.21b-d). Yilmaz observesthat the operator length generally should be chosento include the first energypacketin the autocorrelation;too long an operator may suppressgenuinereflectionswhereasone that is too short may produceovershootand ripples.The 40-msoperator (e) in fig. 9.2le-h leavessome residualenergyattributed to the basic wavelet and reverberatingwavetrain, whereasoperatorslonger than 160ms (h) have little effecton the results.Deconvolutiongenerallyassumesminimum-phaseand zero offset,so it is generally less effective in suppressingmultiples where mixed-phaseor nonzero-offsettracesare involved. As the prediction lag increases,resultsbecomeless spikey,producinga band-limiting effectthat supresses high frequencies,but increasingthe prediction lag is not equivalent to'using spiking deconvolution followed by band limiting. Increasingthe lag tends to produce a waveletwith a duration equal to the predic-

AUTOMATIC STATICSDETERMINATION tion lag and makes deconvolution less effective in broadening the spectrum. Common prediction lags for predictivedeconvolutionare the autocorrelation's first or secondzero crossings(choices(j) or (k) in fig. 9.21i-l), or unity for spiking deconvolution. Prewhiteningpreservesa spiky characterbut adds a low-amplitude, high-frequencytail. The specrrum becomeslessbroadband as the percentprew^hitening increases.Use of spiking deconvolutionwith prewhit_ ening is similar to spiking deconvolutionwithout pre_ whitening followed by band-passfiltering. prewhiien_ ing is similar to adding random noise. Usually, only 0.1 t"olok prewhitening(choice(m) in fig. 9.21m_p)is requiredfor stability. 9.5.I 3 Multichannel deconvolution Most of the foregoingdiscussionsimply that the de_ sign of the deconvolutionoperator is basedon data from the sametraceas that to which it is to be applied. One of the wavelet-processing methodsdescribedwas basedon the sum ofa numberofautocorrelationsand then the application was to all of the componentsof the sum. Occasionally,other multichannel schemes are utilized. The radial multiple-suppressionmethod given by _ Taner(1980)involvesusingdata from one traceas the basisfor designingthe operator to be applied to an_ other trace. For flat reflectors,the angle of incidence is the samefor the first multiple as it is for a primary at halfthe offsetdistance(seefig. 6.35),for the second multiple as for the primary at one-third the offsetdis_ tance,and so on. The reflectivityfor the sea_floorre_ flection changesso much (becauseof the changing angle of incidence)that predictivedeconvolutionap_ plied to the same trace as that used in the operator designdoesnot work well.Tanerachievedbettermultiple attenuationby designingoperatorson traces where the angle of incidencefor primaries was more nearly the sameas for the multiplesto be attenuated. 9.6 Automatic statics determination 9.6.1Interrelationof staticsand normal-moveout ('0rre('Ilons Statics corrections can be determined most easily after normal-moveout corrections have been opti_ mized,but (as will be seen)normal-moveoutdetermi_ nation is best when staticscorrectionsare optimum. Because one of thesedeterminations mustprecedethe otheq the calculationsare often repeatedwith more refinedinputs.Correctionsfor elevationdifferencesor correctionsbasedon uphole or first-break informa_ tion from the field monitor recordsand estimatedve_ locity are usually made before the first automatlc statics determination. This is then followed by normal-moveout determination using these statics ralues.The valuesdeterminedfrom ihe first statrcs and normal-moveoutdeterminationsare applied and then a secondstaticsdeterminationis made.The cycle rrf refining parametervaluesmay be repeatedseveral

303 times to obtain an optimum solution.Marsden 0993) reviewsstaticscorrections. 9.6.2 The surface-consistent model Automatic staticsdeterminationis often basedon a surJace-consistent model that associatesa delay R, with the geophone group at location i and a delay ,! with the sourceat locationT.All data receivedby geophone group i will be delayedby R,,possiblybecausethe geo_ phone group is at a higher elevationor becausethere is a thicker or slowerlow-velocitylayerunderneathit. All data from sourceTwill be delayedby S,, possibly becausethere was a delay betweenthe source firing signal and the actual energyrelease,from the source being in (or on) a medium with lower velocity than other sources,from a higher sourceelevationor shallower shothole,and so on. Following the method of Taneq Koehler, and Alhilali (1974), we refer subscriptsi andj to a common origin and make the station incrementsequal(asin the surfacestackinechart. fig. 8.4b); hence offset distance is proportiJnal to j - i. If there is structure along the line, a delay l,o may be associatedwith the location k (that is, Z, is some sort of averageof time shifts becauseof struc_ ture at different depths below k). For flat reflectors, k : + :(i 7), that is, it is referencedto the midpoint location,and, if the dip is gentle,k is nearlyconstant for common-midpointtraces.If the normal-moveout correctionis only approximate,someresidualnormal moveout Mo wlll remain, and this residual normal moveout will vary as the squareof the offsetdistance. Becausethe residualnormal moveout varieswith arrival time, the delay associatedwith M, will be some sort of average,as was L*. Thus, for the surfaceconsistentmodel, the total time shift for a trace, t,,, will be given by t,i: R, + q + Lk+ Mk(j - i)r.

(9.86)

(The surface-consistent model is not restrictedto determining time shifts for staticscorrection.In surfaceconsistentamplitude adjustment,for example,we assume an attenuationassociatedwith each geophone and an attenuationassociatedwith each source;the subsequentanalysisfollows basicallythe sameprocedure about to be outlined for surface-consistent statics. Likewise,waveletextraction,deconvolution,and other operations are sometimesbased on surfaceconsistent models. Note that "surface-consistent" doesnot necessarilyrequirethat the staticshift, attenuation, and so on are the same when a geophoneis locatedat point P as when the sourceis locatedat p) Although we may not know the amount of time shift to be associatedwith any trace,cross-correlation affordsa meansof determininE t ,i - t^, the time shift ofone trace relativeto another,that producesthe optimum alignmentof the two traces: t,i- t_,- Ri

R , "+ q - s , + L , t , - L _ * , a M,,1(i - i)'- M,*,(n - m)2. (9.87)

DATA PROCESSING 304 ' anY \' It is often (9.90) has a unrque solution for any desired achieve solved in an iterattve manner to values of different assign oi u""ut""v' one can ;;G lnto went that equations I tlo tne different component exfor (9'90))' "1",q qOrror to different terms in eq' values better yield i"f'orieu.ti.u.t someof them ilil, weightedmore heavilvthan others' ffi;;;;iJ;; -'^LiOiiionur someof constraintsare often appliedto the in eq' (9'88)'one can-remgve.9a1of d;;i.; to l'*' limiting by R, or S, "-iGti v u.tween l,*, and R' between relation a small values.One may postulate statsource and receiver the ihat ""T's,, r". example' be similar' especially ics for the samelocatton should using surfacesources' when "'i""..-.i^f' (loc' cit') showedthat.solutiontto.::' rell^e:entm(9.;6t;";. five arbitrarv constantsth,at i'".,o':' L' S" R". for *''.and iolution of which correspondto rhe ;ti6fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff];" trinsic indete.-,nun""t, some sometimes is' arr method' li." 't'iit or the section' that ,nz-,i. it*ffy by the least-squares i;;^;;';;;;;ii tilt overall (b) an deep; in an iterativemanner' i7"n,. u.. too shallow or too the sum structure; minimize to is fictitiousproblem create may The least-squares of ii-t. t..,i"n, which of reai structure'.that is' making of the squaresof the errors: ;il'^i";-;";king or.vlceversa' structureshow up as a staticscorrection ""il;;;it"um a little larger be E = >1: I[,, - t,,,- Ri+ R,', s, + s" should shift allowed combined' - i)' statics geophone - L,*i* L^*,and than possiblesource {,*,(i espeproblems' cycle-jump cause if,is witt sometimes + M^,,(n - *)'] match as-the ringy' .iufiv *t... the data utt 'ornt*liut (e.88) good when the = mlnlmum tJ,J..n iru"es may be almost equally cycle' Usually' addiiional an by OltptaceO R" S'' i.*lt'".. One solve eq' (9'88) for the best set of -M,*,.Th" We wish to preferred the is thift -one' the minimum-,tutt solution is found bv ieast-squares in shifts il small of ;;, might think that cascadinga series one as result setting same give the onit..utiu. solution would : the.case' = o, dElDR, (e.8e) iuig. tnlft, but this is usuallynot _qfla-t, 9') 'h: o ' J implies.that a E l a M ' ' ' : concept o ' dEtdL''': The cross-correlatlon 9t-1-' prlmary ot resultsin the best match as traces of are rnui.tt because.there This resultsin many equations of energy are so as ..n.",iont. Sometimesother types l,o11tt^1ns' many R, as there are geophone-group optlmlze nonrecorrections .it..g ,tt" the calculated and so on' Because .nu.ry S,'ut there are sourcelocations' alignments' reflection than rather with a i;;u;" the best Often, the component tracesare correlated window' rs performedover a mav' for cross-correlation rather than with each other' One obvious that pil;;;;;. so window solutionmay be to narrow the gathersto begin the too made is i*urnpt., selectone of the better window the if ".". i. .^.fitded' However' their rms valueswitlrln a lllelation even if there ;;;iil;""lize determined be will monarrow,an alignment ,u^ them (after removing normal ;il;;uno should-include as trace' pilot are no reflections'The window first a yield to on a first guess) while excluding ".""iU"*a possible reflectionsas ;;.;;.i;".v applying the shifts irt" pl"i ""ce may be refined by nonPrimarY ".-i"q"",i..t energy' and then iterattng determined by cross-correlation' (9'90)andother equatq.aii, ts'asl' and the-n.analyze -uy On. p."".0"... ,tr. 1ij111t^Satners constraints-canbe writtions that expressaiditional pilot trace as one using this pilot trace,^modifyingthe for example'eq' (9'87) appliedbeten in matrix ro'-i$ts't'o); p.o"Z.A.. band-passfiltering is sometimes. window with weightingbecomes fore the correlation pto"""' The correlation (9'91) 7(d,-N=e may changelaterallyto follow the structure' time shift' We do not want to produce an overall : where :-0.and:S, 0,.31:: ttrat IR, On. *uV is to require and weightings'r' extra equatrons adding by this 7f : matrix of coefficients achieved ., "f. tiSZ+l R" S" L'''' and and modifYingeq' (9'88)'wntrng .d : matrix of unknowns'

than k.: Ni + (We usethe subscripti + 7 for ,L rather the sub,t ,o urru.. that the subscriptsare integers: are becau:: icript magnitudesare not important lley the maximizes that tr.tii, "t?* values') ttre strift alignment ".o.t-"o...fution produces the optimum. of the magnitude the and traces, (match) of the two much howindicates quantitatively i.*t-l.t..f"tion data' with.CMP " tttirt ptbd^tt"tt' ffi.*";;;'t""tt some have that traces of we havemany comDrnations Becommon' Mn,in or L,*,, s, R,' ;i,h. ;tk;;ns we have traces' two any "ao,'-"otrelat'e causewe "un unknowns' that ilor. ..iutiu. shift data than we have "overdetermined"set of equattonsto i.,-*. ttu* an uncertaintyin our be satisfied.However,we also have (9'87) that is' opposite sides-of eq."."tti.-.",s,

e ; + \ ( t f t i * ? t *; E ' , , ,* 7,r:.) :minimum,

\>0'

(e.eo)

the relativeem\ being a weightingfactor expressing the equatron of part latter the ;h^;r",; f. iiu.n-,o tt2 t4t' Equation le76: Claerbout' ii;;;;;l;'"see

Mi+.i,

oJ : matrix of time shifts' /'r t"' e : matrix of error terms' solutionis The least-squares lWrW :{ = (7f Z()

Q92; 15'57)

'AUTOMATIC STATICSDETERMINATION Surface-consistent staticssometimesconsiderablvim_ provesmarine as well as land data. Modifications have to be made becausethe characteristicsofhfdrophone groups arc apt to be consistentwith respect to their locationsin the streamerrather than locations along the line. 9.6.3 Maximizing thepower of the stacked trace Another approach assumesthat the optimum static correctionsare those that maximize the power of the stacked trace. A time_shift relation simitar to eq. (9.87)providesthe srarting point, with the i, 5,, Li, and Mr quantities being regarded as independent variables,x,. Appropnate tracesare stacked and the squareof the amplitude(proportional to the powerp) determined.The amount by which tt e po*e.itranges for changesin each variable,that is, (anixStx,, is determinedfor each variable,and Ax. is sellcted so that P increases.This is the method oi rrripir, orrrn, and similar methodsare usedin many data_processing methods.In practice,two problemsare encountered: ( I ) how to find the correct maximum if there are sev_ eral^maxima,and(2) how to get to the maximum with the fewestcalculations To solvethe first problem,one assumesthat the first esttmateis on the slopeof the correct maximum (seis_ mic data are semiperiodicand adjacentmaxima usu_ ally. representcycle jumps). Sometimes a search is made for other maxima so that one can determine. which is the largest.Another techniqueis to make a first solution after filtering out higher fr.lu.n"i., ,o that the maxima are broaderand flwer; tire nrst sofu_ tion is then usedas the starting point for solving the problem with the unfiltereddata. The idealsolutionto the secondproblemis to climb toward the maximum in relativelyfew steps without overshootingthe top by very much. The step size is often.relaredb rtpldx,.Anotirer techniqueisi'o catcu_ late thc curvature(or secondderivative) to estlmate how far awayis the maximum.To minimize.ut.uto_ tions, problems are ofren subdivided, li;iti;g the number of variablesbeing conside."dai one G.. Figures9.22 to 9.24 illustratethe improvement in data quality that can result from appticaiion ot automatic statics. Marked improvemint is otten achieved. 9.6.4 Refraction stat ic.\ Although surface-consistent staticscorrections.which are based on differencesamong traces within the spread length, generallyaccommodatetrace_to_trace variations,they may accumulatesmall errors and do a.poor job at handling long-wavelengthstaticsvaria_ trons (variations of the order of the spread length or larger). Refraction statics correctioni, which are basedon first-break refraction arrival times, provide a meansof dealing with such long_wavelength varia_ tions.

305

I

l

,1

r

fl u

0.5

r

I II

I

I

I

Source Stailca (Fh3t pass)

Ito

I

ms

II

Recelver Steilcs(Ftrstp.ss)

I II

SourcoStailcs(Socond pass)

RocolvorStatlca (Socond Pass)

I

(c) Fig.9.22 Quality improvement resulting from surface_ c o n s r s t e nst t a t i c s .( F r o m y i l m a z , l 9 g 7 : 2 2 2 , 2 2 4 . )( a ) S t a c k w i t h only field staticsapplied; (b) stack after two residual statrss Das_ ses;and (c) diagnosticsafter the first and second oasses.

The refraction first-breaks are picked automati_ cally, usually after an approximaterefractor velocity has been usedto produce reducedrefraction profiles. With CMP data, there is usually appreciablJredun_ dancy, and a logic is used to throw out those rraces that do not appearto involvetravel along the baseof

I

II

I

I l l

l

l

306

DATA PROCESSING

Fig. 9.23 Improvement resulting from applying both refraction and surface-consistentstatics. (From Yilmaz, 1987:229 33.) (a) Stack with only field statics applied; (b) stack after

surface-consistentstatics; (c) stack after refraction statics; and (d) stack after both refraction and surface-consistentstatics.

the LVL. Then the remaining data are averagedfor each location and analyzedby refraction calculation methods,which may be as simple as those discussed in $8.8.2or more elaborate,suchas the plus-or-mrnus method or generalizedreciprocalmethods discussed i n $ l 1 . 5 . 2a n d I 1 . 3 . 3 .

assumedand the calculationrepeated,and so on, until the coherencehas been determinedas a function of both stacking velocity and arrival time. (Sometimes normal moveout is the variable rather than stackine velocity.) Velocity analysis is usually done on commonmidpoint gatherswherethe assumptionof hyperbolic alignmentis often reasonable.Where dips are large,a common reflectingpoint is not achievedand DMO ($9.10.2)or equivalentprocessingmay be required. A velocity-analysisdisplay is shown in fig. 9.25. This is a good analysisbecausethe data involved in frg.9.25aare good. Peakson the peak amplitudetrace (fig. 9.25b)correspondto events.The locationsof the highs yield the velocities(or normal moveouts)that have to be assumedto optimize the stack (hencethe name stackingvelocity),but thesemay not all be primary reflections.Velocity analysesare alsocommonly displayedas contour plots (fig. 9.26) ratherthan as in fis.9.25. Other eventsas well as primariesgive rise to peaks, and hencethe resultshaveto be interpretedto determine the best valuesto be usedto stack the data (see $9.7.3).ln many areaswhere the velocity increases

9.7 Velocity analysis (velocity spectrum) 9.7.I Conventionalvelocity analysis The variation of normal moveout with velocity and arrival time has alreadybeendiscussedin connection with eq. (4.7). Severaltechniquesutilize the variation of normal moveout with record time to find velocity (Garotta and Michon, 1967;Cook and Taner, 1969; Schneider and Backus, 1968; Taner and Koehler, 1969).Most assumea stackingvelocity (4) as discussedin $5.4.4aand apply the normal moveoutsappropriate for the offsetsof the tracesbeing examined as a function of arrival time, and then measurethe coherence(degreeof match) among the tracesavailable to be stacked.Severalmeasuresofcoherencecan be used; some of thesewere discussedin $9.3.5(see eqs.(9.56)to (9.59)).Another stackingvelocityis then

vELOCITY ANALYSIS (VELOCITY SPECTRUM)

307

E

F

(

(km/s)

I/. (km/s)

r";ffi -e;7:8= i;

;"t'$ffi

i2 F

l

0 tr F"

i 2.O

@)

Fig.9.24 Improvement resulting from use of surfaceconsistent statics.(Courtesy of Grant Geophysical.) (a) Section using only the field-determined statics; (b) section using also

statics determined by a surface-consistentprogram; (c) velocity analysis using field statics; and (d) velocity analysis after application of surface-consistentstatics.

more or lessmonotonically with depth, the peaks associatedwith the highestreasonablestackingvelocities are assumedto representprimary reflectionsand peaks associatedwith lower velocitiesare attributed

to multiples of various sorts.In other areas,the relationshipsare not as obvious,and even where the velocity relationshipsare generallyregulaq difficulties may be encountered.

Srackingvelocity(ftls) I

=r €t i t at €l i

t

= l 3l

I

lli ill

i3

flii;:;:

a::::::

l':T".:': iirrirt'r:

_ _ . - - - -

(a)

(b)

Fig: Velocity analysis.(Courresy of petty_Ray Geophysi_ ? ?S^ cal.) (a) Common-midpoint gather showing tfre aata involved in the analysis,(b) amplitude of the stacked i.u". u, a function of stacking velocity at 100-msintervals, and (c) maximurnampli_

(c) tude achievableon stacked traces. The low velocities below 2.7 s are probably multiples and there are few primary reflections below 3.3 s.

VELOCITY ANALYSIS (VELOCITY SPECTRUM)

km

vELocrw.r\,vlrEc

309

on a different stacking velocity. The central two panels, figs. 9.27e and 9.27f, utllize an approximate velocity function; the panelsto the left use velocitiessuccessivelylower by somevelocity incrementand those to the right utilize higher velocities.Such a set of velocity panelsshowswhether increasingor decreasing the velocity will enhanceindividual events.Because stackingvelocity is not necessarilysingle-valued(see fig. 9.28),different eventsmight require different velocities to be optimized.A velocity panel is often run as a check on the interpretation of velocity analyses of the type shown in figs.9.25 and9.26.Velocity panelsare often made of sections(or portions of sections) stackedwith differentvelocitiesas well as of commonmidpoint gathers. 9.7.3 Picking velocity analyses

Fig.9.26 Velocity analysisdisplayed as contours of a measure o f c o h e r e n c e( s e m b l a n c ei n t h i s c a s e ) .( F r o m Y i l m a z , 1 9 8 7 :1 6 8 . )

A compromisehas to be made betweenusing the small amount of data appropriateto a specificspot, in which casethe velocity analysisis apt to be nonde. finitive, and using more data but distributed over a larger area, in which case velocity may be defined better but the velocity measurementsare then averagesover a sizeableregion. The compromiseis often to usedata for three to five adjacentmidpoints.Measurementsare also usuallybasedon all the data within a windoq which is often 50 to 100ms long, in order to increasethe amount of data and henceimprovethe velocitydefinition. 9.7.2 Velocitypanels Velocity panels (fig. 9.27) provide another display tiom which stackingvelocitycan be determined.A set of data is plotted severaltimes,each plot being based

Velocity analysisinvolvesa considerablenumber of calculationsand henceis fairly expensiveto execute; therefore,too few analysesare often run, sometimes only every t/z to 5 km along the line. Where only a limited number of velocity analysesare to be run, their locations should be selectedjudiciously, based on the best availablegeologic information, so that analysesare not wasted in noisy areas and so that changesin geology are adequatelysampled.Where the number of tracesin a CMP gather is large, only every other trace may be usedin order to reducethe cost. Velocity analysesare ordinarily picked by an interpreter. Picking involves selecting the time-velocity values to be used in subsequentprocessing.The velocity-analysisinterpreter often has in mind only achievinga good stack, and stackingcan often tolerate appreciablevelocity errors.Velocity interpretation is time-consumingand henceexpensiveand has significant potential for error, especiallywhen the picker knows little about the local geology and hencedoes not factor this into the interpretation.It is not uncommon for analysesto be picked as stand-aloneoperations and consequentlysuccessiveanalysesmay not evenbe pickedconsistently. Theseerrorsare becoming less frequent today where velocity interpretation is done at a workstationwhereadjacentanalysesalready interpretedcan be displayedalongsidethe new analysisas a guide for picking consistently.A plot (fig. 9.29) of the interval velocities(calculatedby the Dix equation, eq. (5.25)) that a particular interpretation implies is often helpful in interpretingvelocity analyses. The interpreter is ordinarily guided by a set of simplerules(Cochran,1973:1048-9): l. an increasein stacking velocity I{ with increasingdepth is more probablethan a decrease; 2. successivereflections are ordinarily separated by more than 100ms in two-way time; 3. an interval velocity greater than 6700 m/s (22,000ft/s) or lessthan 1430m/s (4700ftls) is unlikelvl

310

DATA PROCESSING +tOO t*FIl5tC *Tit;*

iiixl .r

ti|f u'til r lrtr t.*

(a)

(b)

(c)

(d)

(e)

Fi9.9.21 Velocity panel of a CMP gather. Panels (e) and (f) employ the velocity resulting from a velocity analysis with a mute applied in panel (f). Panels (a) to (d) show results where the stacking velocity is decreased lrom that in (e) by n AV,,

Fig. 9.28 Multivalued stacking velocity values. Reflections B and C arriving at the same time may have different stacking velocities.

4. the differencein interval velocity for successive layers should exceed 2oh (see problem 5.13 for a quick approximatemethod of determining interval velocity); 5. any event at about twice the lo of a previous event and with approximatelythe same ( is probably a multiple and should not be used. Computer picking basedon similar rulesis sometimes used.For example,a possiblepick must satisfyrule 2

(f)

(s)

(h\

(r)

. fh

l:, };

U)

4,3,2,andl, andAV"is sometimes as where n is respectively (g)to (i) showresultswherethestacking muchas200ft/s.Panels isincreased byn AV".(Courtesy of GrantGeophysical.) velocity

and then must passrules 3,4, and 5; if more than one pick passesthesetests,they are testedagainstrule l; if more than one pick is still possible,that with a velocity nearestthe V, of the precedingpick is selected. Multiples are apt to have velocitiesthat are low and diffractions and sideswipe events (for example, diffractions from faults nearly parallel to the hne or reflectionsfor which the line makesa small anglewith the strike direction) are apt to havevelocitiesthat are unreasonablyhigh. Becausethe amount of normal moveout applied varies with arrival time, frequenciesare lowered as offsets increase (fig. 9.30); this is called normalmoveoutstretchand it affectsvelocity-analysispicks. Long-offsettracesare muted ($9.10.3)to avoid excessive stretch effects;clearly the amount of mute applied affects the measured velocities. In the usual case, where velocity varies with depth, the alignment of eventsis actually some other curve rather than a hyperbola.However,the errorsin assuminga hyperbolic alignmentare usually small. The accuracy and resolution of stacking velocity valuesclearly depend on acquisition factors such as the spreadlength,the multiplicity (fold), the recorded bandwidth, the signal-to-noiseratio, and the lack of

VELOCITY ANALYSIS (VELOCITY SPECTRUM)

Q 4oo0

! rcoo 9 zooo rooo o.5

l.o

r'5 2.O 2.5 3.0 3.5 4.O 4.5 5.O T w o - w o lyr m e( s )

Fig. 9.29 Interval-velocity bar graph produced from a veloc: 1 1a n a l y s i s .

311

tervening locations. Values for times between picks are often interpolated linearly, and then the valuesfor traces betweenanalysesare interpolated from these,a process called bilinear interpolation,' this procedure may introduce errors where analysesare inadequately spaced,of poor quality, or picked in a nonsystematic manner.A plot showinginterpolatedvalues(fig. 9.32) providesvaluablecontrol by making the consequences of velocity assumptionsclear. Velocity analysesshould be plotted at the samevertical scaleas the seismicsection so that they can be overlaid on the section to make it easierto identify stacking-velocitypicks with specificevents.The same analyses. eventsshould be picked on successive Analyses should be continuously compared to neighboring analysesto check that variations make geologicsense;comparinganalysesalong a line allows an interpreterto assessthe uncertainty in individual picks and smooth out noise effects.Where data are good, systematicchangesmay indicate stratigraphic changes.Generally,as many eventsshould be picked as possible.While pickingjust a few eventsmay suffice for stackingpurposes,picking many eventsoften disclosesimportant interpretationclues. Whereasvelocity analysesare generallyinterpreted as if reflectorswere horizontal and the seismicline were perpendicular to strike, stacking velocity dependson both quantities.Levin (1971)showedthe dependenceof stacking-velocitymeasurements on dip { and trace E (the anglebetweenthe strikeand the line) for constant-velocityoverburden:

la : V(l - sin2(cos'E)"'

Frg. 9.30 Normal-moveout stretching. (From Yilmaz, 1987: .61.) (a) A signal with a period ?! which after applying NMO : t h a sp e r i o d Z ' > I

near- or far-offsettraces or irregular spacingin the neld. They also depend on processingparameters suchas muting and the weighting of input traces,lo,'ationand length of the time gate,samplingintervals, ,rnd the coherencymeasureused. Gathers are often decimatedfor velocity analysis,perhapsreducing the number of input tracesby l14 andlor subsamplingin rime, to reduceanalysiscosts;in addition, coherence is usually checkedonly for stacking velocitieswithin a window centeredon the expectedvelocities.Figure 9.3 I showsthe effectsof analyzingcombinedadjacent _rathers,subsampling,and insufficient offsets, and \-ilmaz (1987: 173-82)discusses the effectsof other . actorson velocity determinations. '.t 7.4 Usesand limitations of velocity analyses The precisionofreading valuesfrom hard-copyveloc:tl analysesis usually + 10 ms in /oand +50 m/s in (, rut the accuracyis often lessthan this. Velocityvalues rave to be interoolatedfor intermediatetimes and in-

(e.e3)

This relationshipis shown in fig. 9.33. Although the objective of velocity analysesis to achievegood stackeddata, the velocity values also haveinterpretationalimportance($10.5).With seismic data that are not unduly distorted by structuralcomplexities,approximateinterval velocitiescan be obtained from stackingvelocitiesby simplerelationships (seeeq. (5.23)and problem 5.13);however,intervalvelocity valuesdeterminedin this way should be routinely checkedfor reasonableness. 9.7.5 Horizon velocity analysis The determinationof stackingvelocitiescontinuously along a seismicline is calledhorizontalvelocityanalysis. Such analysesare often made for only a singleor a few reflections.Generally,horizonsare pickedby either an autopicker or manually, and analysesare made over a narrow time window about the reflections. The analysisis essentiallythe sameas for a conventional velocity analysis.Coherency is measured within the window as the assumedvelocity is varied and the selectedvelocity is that which maximizesthe coherency.Figure 9.34showshorizontalvelocityanalysesalong five horizons,and fig. 9.35 showsthe improvementin data quality for a reflectionbelowmajor lateral velocity changes,causedby salt diapirism in

t

I

looo 2000 o.o FZ

tooo 2000 3000

o.orr

4000 5000

Scaled Scalod ssnuancoamplitude

tooo

o.o

4000 5000

2000 3000 4000 5000

Scalod

Scal€d

Scalsd amplitude

2.O

Fig. 9.31 Effects of velocity-analysisparameters.(From Hatton et al., 1986:68-9.) (a) Analysis basedon two adjacent CMP'

(b) based on eight adjacent CMP, (c) analysis using only every third trace, and (d) analysisusing only near offsets.

: R . E S E R V A T I O NO F A M P L I T U D E I N F O R M A T I O N

f

v r.?

v v f.L. ? r

f

J I J

T

7000 ft/s

..\

' -

12000 ft/s

Fig. 9.32 Stacking velocity along a seismicline. Values are rnterpolated by the computer from input picks indicated by the

3.0

70'

/ /

2.0 .n

1.0

z

./

2

60' 40'o

6

20" 10' 0'

9 0 6 0 3 0 0 (Strike) (Dip) Angfe bctwecn llnc and dlp Fig. 9.33 Increasein stacking velocity with dip { and the angle b e t w e e nt h e s t r i k e E a n d t h e l i n e d i r e c t i o n .( F r o m L e v i n , 1 9 7 1 . )

this instance.Changesin the velocity in the interval betweenhorizontal velocity analyseson adjacentparallel horizonsare sometimesusedas an interpretation tool to sensestratigraphicchanges. 9.6 Presenation of amplitude information The amplitude of a reflectiondependson the acoustic impedancecontrast at the reflectinginterface.Howeveqother factors,suchas thoselisted in fig. 6.44,often obscure the acoustic-impedance-contrast information. The effects of spherical divergence and

dashes.This line is also shown in fig. 10.33.(Courtesyof Grant Geophysical.)

raypathcurvaturecan be calculatedand correctedfor. The gain of the recording instruments normally is known. Array directivity rarely has a significanteffect on the amplitude of nondipping events and so its effectsare generallyignored. Corrections for offsetdependentamplitude effectsare also usually ignored. Migration can correct for reflectorcurvatureeffects. Remainingeffectsare mostly of two kinds: ( I ) those associatedwith energylossesbecauseof absorption. scattering, transmissivity losses, and peg-leg multiples,and (2) thosethat vary with sourcestrengthand sourcecoupling, geophonesensitivityand geophone coupling, and offset. The effectsin the first group are difficult to determinebut they usuallydo not vary appreciablyalong a line and so may not obscurelateral variations.The high multiplicity of CMP data permits determiningthe secondgroup of effectsin a surfaceconsistentamplitude-correctionprogram(actually,so that the effectsare additiverather than multiplicative, the log of the amplitude rather than the amplitude) similar to automatic statics correction ($9.6.2;see Tanerand Koehler,1981). absorption A correction for frequency-dependent and peg-legmultiples (a Q-correction)is sometimes made: A(t) : l(Q)s"*ro,

(e.e4)

where l(0) refers to some referencetime. BecauseQ is usuallyknown only approximately,it is often taken as 0.01V where Z is the velocity in ftls. One processingroutine adjustsamplitudein several steps.After first correctingfor amplitudeadjustments made in recording, a time-dependent sphericaldivergencecorrection based on assumedvelocity is

280 320 360 400 440 480

-@ 2

-@ -@

lbbrrr€tAdyra.

a

t

r

r

o

tI

t

l -

,I

Fig. 9.34

A stacked section with horizon-velocity analysesoffive horizons. (From Yilmaz, 1987: 184.)

A P P A R E N T - V E L O C I T Y( 2 - D ) F I L T E R I N G

315 9.9 Apparent-velocity (2-D) filtering Apparent-velocity filtering, also called dip,fan, moveout, or pie-slicefiltering (Fail and Grau, 1963;Treitel, Shanks,and Frasier, 1967) for reasonsthat will become obvious, dependsupon the apparent velocity (definedby eq. (4.13a))of a wave as it approachesa recording spread.Equations (2.4) and (4.13) can be combinedto eive V : oolx": 2nvk, (e.es)

Fig. 9.35 Portion of a section across a salt dome orior to migration. (From Yilmaz, 1987: 185..y1a.yConventionally processed,(b) horizon-velocity analysis(HVA) along a base salt reflecIor A (center), and (c) processed utilizing HVA velocities.

applied.Such a correctionmakesthe range of amplitude values smaller and therefore easier to handle. These corrections constitute the "preliminary gain recovery/adjustment" shownin the "editing" phaseof fig. 9.62. Surface-consistent amplitude analysisand/ or correction is then done during one or more of the processingpassesin the "main processing" phase. After velocity has been determined, the spherical divergencecorrection is changedto depend on Vlt, which allows approximately for raypath curvature, using somearbitrary time as a referencevalue.An additional arbitrary exponential gain can be applied to make the range of amplitude values smaller for display purposes.This correctionmay be basedon mean absolute or rms amplitude averagesover time windows a few hundred millisecondsin length and also averagedover many traces.Sometimesthe previous step-by-stepamplitude adjustmentis simply replaced by an arbitrarygain function. Amplitudes are sometimesadjusted so that their rms averagesover a time window (perhaps200 ms rn fength)are equal,this step being called equalization. It should be noted that, becausereflection amplitude varies with incidenceangle (or with offset; see $3.4), CMP stacking does not result in normalincidenceamplitudeseven if the amplitudesof all of the component traces should be correctly preserved (seealso59.10.5).

For a fixed apparentvelocity \,theplot of frequency v versusapparent wavenumberrc, is a straight line. For a seismicspreadalong the x-axis, r, is positiveor negativeaccordingas \is in the positiveor negative directions.For a verticallytravelingsignal,r, : 0 and 4: * and the v-Karepresentationis along the u-axis. For most reflectionsignals,4) V^,",someminimum apparentvelocity,and hencethe reflectionslie within a relatively narrow wedge containing the u-axis, as shown in fig. 9.36a. Coherent noise generallyhas a lower \than reflections(fig. 9.37) and thereforeseparatesfrom them in the v-r, plot, usually calledanf-k plot (frequencyvs. wavenumberplot). We can usetwo-dimensionaltransforms($9.1.4and 15.2.4)to definean apparent-velocityfilter, F ( u , .x , , ): l . l x , ,< l 2rvlV-.1 (9'96) : o, k"")> 2rvtv-. ] that will passthe signalbut rejectthe noise(as shown in fig. 9.36c).Sucha filter that passesa narrow wedge in the u-r, domain is a "pie-slice" filter. Of course, neither signal, noise, nor filter need be symmetric about the y-axis. For example,there are hardly any coherentalignmentsdipping to the left in fig. 8.16a, and so fig. 8.16b if extendedto the left of the u-axis would be essentiallyblank. Apparent-velocityfilters can also be designedto removea noise wedgerather than passa signalwedge;sucha filter is calleda "butterfly" filter. Just as frequenciesabove the Nyquist frequency may alias back into the passbandunlessexcludedby alias filters before the sampling, so spatial sampling involveswrap-aroundaliasing (fig. 9.36b) of data for wavenumbervaluesexceedingthe Nyquist wavenumber (seeeq. (9.33)).The only way to prevent aliasing is to filter beforesampling,which is not possiblewith respectto spatial sampling, or to move the Nyquist points farther out by samplingmore closely. The filter in the space-timedomain (x, t) equivalent to the filter given by eq. (9.96) is obtained by taking the two-dimensionalinverse Fourier transform (see e q .( l 5 . l l 7 ) ) |

.rv

ftx, t) : {112fl | J

l-'ru

| *"

J

''-"' dx, dv F(v. rc,)gr'.,'

,"

or f

f(x, t):

lll2il

|

**!

f

|

*uil

cos (rc,x* 2rvt) drc"dv,

J * r J , n

(e.e7)

DATA PROCESSING

316

because/(x, /) must be real. The convolution of/(x, l) with the input (signal + noise),C@,t), givesthe output h(x, t), h ( x ,t ) : c\x' t) * f(x, t)

j:t

g(o, t) f(x - o, t - r) do dr. (9.98,15.164)

Thisequationcanalsobewrittenin digitalform:

(e.9e)

0

-t(

KN

Wavenumbel

where the space-sampleinterval is usually the trace spacingin the x-direction and the time-sampleinterval in the f-direction. Instead of transforming the 2-D filter to the time domain and calculatingCe, t) * /(x,l) as we did in eq' (9.41),we can transform g(x, t) to the (u, r,) domain, multiply G(v, x") by F(u, r,,) and use the twoto obdimensionalconvolutiontheorem(eq.(15.165)) tain h(x, t\. The use of 2-D filtering to attenuatenoise trains suchas severeground roll on common-sourcegathers is illustrated in fig. 9.38. Using 2-D filtering reduces the amount of muting requiredso that more reflection data can be usedin velocity analysisand in stacking, providing better stacking-velocitydefinition and better attenuationof multiplesin stacking. Figure 9.39showsthat 2-D filtering can be effective in attenuatingsurfacemultipleswherethereis a steady increaseof velocity with depth. 2-D filtering may also be appliedafter stacking(fig. 9.40).

(b)

tr;"T----l

I Passedby anay

l'- ^ A=;.*

I

N

I

ta

)

t

r

---l

I

9.10 Stacking 9.10.1Gathers Common-midpoint stacking is the most important applicationin improving data quality' data-processing The principles involved have already been discussed along with the field proceduresused to acquire the data. The componentdata are sometimesdisplayedas gathers.A common-midpointgather (seefigs. 9.25 to 9.27) has the tracesfor the samemidpoint arranged side by side, and a common-offsetgather has the distance is traces for which the source-to-geophone displayed are Gathers by side. side the samearranged eitherbeforeor after normal-moveoutcorrection.The traceswithin a common-midpointgatherare summed to yield a singlestackedtrace.

Wrycnumbcr(tat rl

9.10.2D M O (dip-moveout ) correction

.a

(m) Wavelength

(c) Fig. 9.36 A seismic gather in the frequency wavenumber domain. (After Sheriff, 1991.)(a) Signal, generally near the u-axis' and noise tend to separate;(b) illustrating wrap-around aliasing where x, is the Nyquist wavenumber;and (c) filtering effectsof frequency, array, and velocity filters.

The result of stacking CMP traces after normalmoveout correction is assumedto be the trace that would be recordedby a coincident source and geophone located at the midpoint' However,the reflecand in fig. 4'9b' tion point is displacedupdip ($4.1.4), the reflectingpoint is R, not P,' this resultsin an offset changeAx given by eq. (4.22a)and a decreasein the

STACKING

317

ll

i i

Fig. 9.37 Three common-source gathers (above) and their / k spectra (below). A, B, and C are high-amplitude, dispersive,coherent noise trains: D is the wrap-around of C and Econsists of

reflection events.As the spatial extent of the noise train becomes wider, its.l:k equivalent becomes narrower; compare F and G. ( F r o m Y i l m a z . 1 9 8 7 :7 0 . )

zero-offsettraveltime Al given by eq. (4.22b). Both effectsare proportional to the squareof the offset,so stackingproducessmearingunlessproper DMO corrections are applied. Also, the velocitiesdetermined in velocity analysesare dip-dependentunlessa DMO correction has been applied. Dip also causespeg-leg multiples to divide into two sets,one with apparent stackingvelocityhigher than the zero-dipstackingvelocity,the other lower(Levinand Shah,1977),so that stackingaltersthe characterofevents that includeappreciablepeg-legenergy. Unlike the classicaldip moveout, which is simply the effect of dip on traveltimefor a common-source record (gather), DMO processingcreatescommonreflection-pointgathers.lt effectivelymoves a reflec-

tion seenon an offsettraceto the location ofthe coincident source-receivertrace that would havethe same reflecting point (fig. 9.41). It thus involves shifting both time and location.The resultis that the reflection moveout no longer dependson dip, reflection-point smearof dipping reflectionsis eliminated,and events with various dips have the same stackingvelocity.It is often carried out as a convolutionin the commonoffsetdomains. Levin ( 1971) showedthat the reflectingpoint moved updip (fig. 9.42a)from that for the coincidentsource geophonetrace by L -- (h' lD) cos { sin {. To avoid reflection-point smearing, offset traces should be gatheredat a point a distancef : (- h' lD) sin { updip. However,such a gather is not hyperbolicbut has the

318

DATA PROCESSING

Offset (m)

Offset (m) 50 500 1(X)015002mo

1s002dr0 r l

(a)

(c)

(d)

Fig. 9.38 Velocity filtering of a gather. (From Yilmaz, 1987: 7 1 . ) ( a ) U n f i l t e r e dg a t h e r ;( b ) / k s p e c t r ao f ( a ) ; ( c ) t h e v e l o c i t y filter eliminating wedge from (b); and (d) the transform of (c) to

the time domain showing how noises A and E have been eliminated but B and D retained.

shapeof the DMO ellipse,

Deregowski, 1985). DMO is usually applied after velocity-dependentNMO, but Gardner'sDMO (Forel and Gardner, 1988)appliesvelocity-dependentDMO prior to velocity-dependentNMO. For further information, the reader is referred to Hale (1991: chaps. 3-4) or Bancroft(1991),who discussseveralmethods and give referencesto original sources. For 3-D surveys,the 2-D ellipsein fig. 4.9b becomes "bowl." Raypaths(assuming an elliptical the velocity is constant) lie in a plane containing the sourceand receiver,and this plane intersectsthe bowl along an ellipsesimilar to that given by eq. (4.19).Thus, 3-D DMO is essentiallythe same as 2-D unlessthe azrmuth changes.Under thesecircumstances, if AS is the azimuthangle,eq. (9.101)becomes

. x'-

Vtn x - h 1: 0 . 2sin{

(9.100)

The DMO correctionmakesthis gatherhyperbolic. Because DMO involves considerable computer time, Hale (1991: 2-9) gives an empirical rule that DMO correctionis requiredwheneverit exceedsonehalf the dominant period. By using eqs. (4.11)and (4.22b),the rule is that DMO processingshould be carried out whenever (4s' 1V' t,) sin' { : (LtJL,x)' (s' vult,)> l, (9.101) where /,, is the zero-offset time, t, is the NMOcorrectedtime, 2s is the offset,u, is the dominant frequency,and { is the dip. Correctionsfor DMO can be made in variousways, including prestack partial migration (Vlmaz and Claerbout, 1980).time-domainfinite-differencemethods or offset continuation (Bolondi, Loinger, and Rocca, 1982),Fourier-domainimplementation(Hale, 1984),and integral(Kirchhoff) methods(Hoskenand

'l l*::l?^*l

(e.102)

A diffraction in location-offset space is called a Cheopspyramid (fig.9.42b); it is not a hyperboloid. Application of NMO changesthe Cheops pyramid into a saddle-shaped'surface (flg.9.42c);DMO makes it into a cylindrical hyperboloid(fig.9.a2il.

STACKING

319

0.0 0 .1 i.2

0.4

o.o

o.l o.2 o.3 o.4 o.5

r i l i l llri :

t;]"ili,,,,:,

ir} rrl:l,l ..;

0,7 0.8 0.9 r.O

r#

v.b

t

o.7 0.8 '":

t . l

t.? t.3 t.4 r.5 1.6

v. :,

l.O t.l 1.2

r.3 .

'

t.4

r.5 t.6

r.8 r.9 2.o 2 .1

.

r.8 " r.9 - 2.O

2.1 2.2 2,3 '-"2.4 2.5 ?.6 2.7 2.8

2.\

2.4

2.8 ?.9

a'Y 1.n

J.U

c

Fig. 9.39 Use of ./ k filtering to attenuate multiples. lFrom H a t t o n e t a l . , 1 9 8 6 :9 8 . ) ( a ) G a t h e r ;( b ) g a t h e rw i t h a p p r o x i m a t e

NMO applied; this gather is then./:.1 filter.ed;and (c) gather after filtering followed by removing the approximate NMO.

"""J (r(?

$

J

Fig. 9.40 Velocity-filtered stacked secrion. (From yilmaz. l98l: 76.) (a) CMP stack contaminated by coherenr noise, and (b) filtered after stacking.

9.10.3Muting First-breaksand the refractionwavetrainsthat follow them are usually so strong that they have to be excluded from the stack to avoid degradingthe quality of shallow reflections(seefig. 9.43). This is done by muting, which involvesarbitrarily assigningvaluesof zero to tracesduring the mute interval. Also, the re-

Fig. 9.41 NMO corrects for the time delay on an offset trace assuming zero dip; DMO moves the data to the correct zerooffset trace for a dipping reflection; migration further moves it to the subsurfacelocation. (After Deregowski, 1986: 13.)

DATA PROCESSING

320

T r a v e li tm e

I

(8)

(b)

=-h;

{c}

(d)

Fig. 9.42 DMO. (From Sheriff, 1991.) (a) Terms involved in reconstruction of the reflecting point assuming constant velocity, (b) a diffraction in location-offsetspaceis not a hyperboloid,

(c) NMO correction makes (b) into a saddle-shapedsurface,and (d) DMO correction along with NMO yields a cylindrical hyperboloid.

flection waveshapeon longer-offsettracesis stretched becauseof rapid changesin the normal moveout (fig. 9.30) and directivity effects of geophone arrays. Stretchingeffectivelychangesthe frequencyspectrum of the wavelet,resulting in attenuationof higher frequencies in subsequentstacking. Therefore, longoffset traces usually are muted before the stretching reaches25o/o.Figure 9.27 also shows the effect of muting. of muting is that the multiplicity of A consequence a stack increasesby steps,the shallowestdata often being a twofold stack, slightly deeper data being a fourfold stack, and so on until the full multiplicity of the stack is achievedafter the muted events have passedbeyond the most distant geophones.To avoid amplitude discontinuitiesassociatedwith changesin the multiplicity, the amplitude is usually divided by the number of nonzerotracesthat havebeenadded. Sometimesan innermute (tail mute) is also applied, setting short-offset traces to zero as air waves or ground roll strikesthe geophones.Tracesnear a shotpoint may becomevery noisy as time after the shot

increases,perhaps becauseof hole nolse (noise produced by oscillation and venting of gasesgenerated by the shot and/or ejectionof material from the borehole). Traces near surface sourcesmay likewise become noisy as time increases. Occasionally,a wedge of data across the gather (suchas a portion dominatedby ground roll) will also be muted (surgicalmute),although it is more common to use apparent-velocityfiltering ($9.9)in such situat10ns.

stctcking 9.I 0.4 Common-midpoint Combining a sequenceof common-midpoint gathers after NMO correction yields a common-midpoint s/ack.Multiples spendmore of their traveltimesin the shallowerpart of the earth than do primarieswith the same traveltimes, and hence usually have smaller stacking velocitiesthan the primaries and so do not align on the NMO-corrected gather. Thus, stacking severely attenuates most multiples. Common-

STACKING

321 Even where DMO has been applied to convert CMp traces to common-reflecting-pointtraces,CMp and zero-offsetsectionsdiffer in important regards.Noises on the two types of sectionsare generallymarkedly different,especiallymultiple noise. Amplitude-variation-with-offset(AVO) differences causereflectioneventsto havedifferentamplitude re_ lations to each other than in the zero-offiet case.a point usually neglectedin inversion,and the assump_ tion of hyperbolicstackingmay havealso changedthe amplitudes of different eventsin different wavs (see Yilmaz, 1987: 244, 251). 9.10.5 Weightedstacking

(b) F i g . 9 . 4 3 D e p e n d e n c eo f r e f l e c t i o nq u a l i t y o n m u t e s e l e c t i o n . ( F r o m Y i l m a z 1 9 8 7 : 1 6 4 . )( a ) A C M p g a r h e r .( b ) The srackcd trace resulting fiom varying the mute; the right trace is the same a s t h e i n s i d e t r a c e o f t h e g a t h e r ;t h e n e x t t h e r e s u l t o f s t a c k i n s the two inside traces; the next stacking the three inside traces't a n d s o o n . T h e b e s t m u t e i n c l u d e sa s m u c h d a t a a s oossible w i t h o u t d e g r a d i n gr e f l e c l i o nq u a l i r y .t c ) M u t e d g a t h e r .

midpoint gathers are sometimes apparent_velocity (.fk) filtered ($9.9)to remove coherent noise trarns beforestacking. Common-midpoint stacking ordinarily assumes , that.all trac,es.inthe gather being stackedhaveequal validity and thus should be given equal weight.The output amplitude is divided by the number of live traces entering the stack, that is, adjustments are made for muted and occasional missing or dead traces. A CMP stackedsectionis often regardedas a zero_ offset section, especially when migrating the data.

In certain situations,unequalweighting(producinga weightedstack) of the tracesin a gathermay yield re_ sults that are better than the CMp stack. Offset_ dependentweighting is sometimesused. The differ_ ence in NMO between primaries and multiples depends,for example, on the square of the offset distancesso that better multiple attenuationmay be achieved by weighting the long-offset traces more heavily than the short-offsettraces (fig. 9.44). Most wejghting is empirical, often varying linearly with offset,the weightsusuallyvarying from 0.5 to 1.5. More complicatedweighting schemesare sometimes used. Where the relations betweenstacking velocitv and time are known accuratelyfor primariJs and for one type of multiples,use of a stackins velocitv diflerent from either can maximize atteriuation of these multiples compared with the primaries even though it doesnot maximizethe primaries;this is the basis of "optimum wide-bandhorizontal stackine" (Schneider,Prince, and Giles, 1965).However,be_ causevanous types of multiples havedifferent stack_ ing velocities, this type ofstackingrarelyproducesop_ timum results. One goal of CMP stackingis to producethe reflec_ tron amplitude appropriate for normal incidence. However,amplitudesvary with incident angle ($3.4), that is, with offset,and especiallyso wherethe intersti_ tial fluid changes.One scheme(Denham, palmeira, and Farrell, 1985) fits amplitude-offset measure_ mentswith a best-fitcurve and then givesthe stacked trace the zero-offsetamplitudevalue.Suchprocessing may be especiallyappropriate as a prelude to one_ dimensional inversion, which assumesnormal inci_ dence. Weighting is also sometimesdone to enhanceco_ herence,weightsbeing basedon a coherencemeasure_ ment ($9.3.5) suchas semblance. Enhancement of certain dips can be achievedin this way. Several iterative or adaptive weighting schemes havebeenused(Naessand Bruland, 1985)for vanous types of noise problems.Estimatesof the sisnal and noise amplitudes are usually required. Weighting (Naess,1979)can be used to suppressabnormal amplitudes.Muting (99.10.3) is a type of weightedstack wherenoisy tracesare givenweightsof zerocompared

DATA PROCESSING

322 s.P o.o

o.o

o.l

o .l

o.2

o.2

o.3

.).i

o.4

o.4

o.5 o.6 o.7 o.8 o.9

S.P.

:

0.5

s*ffi

v'0

o-7

tr*

;1:::-iE s:l

ri

+.

r.o t - l l ' 1

t.3

\

o.8 o'9 r.O

*+*'Er+$---_ry_ff l . l

t.2

r.3

| .4

t.4

t . 5

t.:)

t.6

r.6

t . 7

t.7 (b)

Fig. 9.44 Weighted stacking to attenuate mpltiples. (From H a t t o n e t a l . , 1 9 8 6 :9 7 . ) ( a ) U n w e i g h t e ds t a c k w i t h s t r o n gm u l t i -

ples from the sea floor (e.g.,- o.8s s at left) and interbeds (e.g.. - 1 . 2 s ) ; ( b ) w e i g h t e ds t a c k .

with weightsof one for unmutedtraces.Simply eliminating noisy traces is another form of weighting. Sometimesnoisy traces are replaced with estimates basedon interpolationrather than being simply eliminated;this is equivalentto changingthe weightingof the tracesadjacentto the noisy trace.Diversity stacking ($9.10.6) is anotherform of discriminatingagainst noisethat occasionallyaffectsacquisitionin a nonsystematlcmanner.

recordsare relativelylittle distorted by noise. Under such circumstances,amplitude can be used as a discriminant to determinewhich portions are to be excluded.This often takesthe form of merelyexcluding data where the amplitude exceedssomethreshold,or perhaps some form of inverse weighting might be used.Suchnoiseburstsare often randomly locatedon repeatedrecordingsso that sufficientvertical stacking after the weightingtendsto producerecordsfree from the high-amplitudenoises.

9.I 0.6 Diversity stacking Much data processingis far less exotic than is suggestedby the mathematicalrelationshipsexpressedin the foregoingpages.Some of theseprocessesinvolve merelyexcludingcertain elementsof the data, suchas the muting operationthat has alreadybeendiscussed. It is almost always better to throw away noisy data than to include it (often on the theory that its adverse effects will be averaged out). A very powerful processingtechnique,which is not used as much as it should be,is to simply look at the data and deleteportions that appearto be mainly noise. Diversity stacking is another technique used to achieveimprovementsby excludingnoise.Recordsin high-noiseareas,suchas in cities,often showburstsof large-amplitudenoise,whereasother portions of the

9.I 0.7 Simplanstacking Most sourcesare effectivelypoints and henceseismic waves are spherical or nearly so. An alternative to CMP stacking of component spherical-waverecords is to simulatesectionsthat would havebeengenerated by plane or cylindrical waves;suchsectionsare called "Simplan" sections(Taner,I976). Simplanutilizesreciprocity(S4.3.4)and superpositThe sum ofthe outputsofa geophonefor ion ($2.1.4). a number of in-line point sourcessimulatesthe output from a line source,that is, a cylindrical wave.Figure 9.45 shows a split-spreadrecord and the Simplan trace that resultsfrom simple stackingwithout making any time shifts foi normal-moveoutcorrection. Only thosetracesof the gatherthat lie within the first

STACKING

tzJ

R e c e i v e rc o o r d i n a t e

Traces to be simulated o

Jt

;

J*+ 2 It.+

:al sourcelreccrver Posrtrons ( o m m o n r c c e i v c rd i r e c t l o n

F i g . 9 . 4 5 S y n t h e t i cc o m m o n - s o u r c eg a t h e ra n d S i m p l a n t r a c e . (Courtesy of Grant Geophysical.) (a) Gather showing reflect i o n s s y m m e t r i c a la b o u t t h e t r a c e _ r : - 2 h s i n { . w h e r e ( i s t h e dip, and /t the distance to the reflector, as in fig. 4.2. (b) The Simplan trace that results from summing all the traces;in effect. only the first Fresnelzone contributes.

Fresnelzone make an appreciablecontribution to the Simplan trace. Even moderatedip has little effecton the sizeof the zone,so dip has little effecton the Simplan trace. The first Fresnelzone also includesmore tracesas arrival time increases, so that the rate of amplitude decay on the Simplan trace is less than on the tracesof the gather(the Simplan trace undergoes cylindrical divergencerather than the sphericaldivergenceof the componenttraces).The tracesfrom geophones closely spacedcan be used in the same way as the tracesfrom sourcescloselyspaced.Customary group spacingand range of offset distancesare usually sufficientto avoid undesirableend effects. Split-spreadand Simplan recordscan be simulated from end-on records.Note (fig. 9.46a) that the trace at (yo*,,s*) on the surfacediagram is the sameas the trace at (ro,r**,) by reciprocity.Thus, end-on records can be usedto producea split-spreadrecord for twice

Fig.9.46 Simulating split-spread record from end-on records. (a) Reciprocal relations between traces on a surface stacking chart: traces on one side of the zero-offset line have identical raypaths to traces symmetrically disposed on the opposite side of the line; (b) 96-trace split-spread record simulated from 48trace end-on records (courtesy of Grant Geophysical).

DATA PROCESSING

324

s/km f royporomoter, 2

- _. DIRECT .. . uP otP NODtP DIP

3

4

5

-. {t F

=

3

a @

9 a o .g F

(o)

5

r. Jl /'

Fig.9.47 r p mapping. Reflection hyperbolas in time domain map into ellipses in the r p domain and straight lines (direct w a v e a n d h e a d w a v e s )i n t o p o i n t s . ( F r o m S h e r i f f .1 9 9 1 . )( a ) A n end-on seismicrecord /(-r, /), where .r is the source receiverdistance and l is the arrival time: solid lines indicate no dip, dotted

a n d d a s h e d l i n e s i n d i c a t e u p - d i p a n d d o w n - d i p d i r e c t i o n s ,r e spectivcly.(b) r p domain showing points P, for the direct arrival and P. for the head wave H. The dotted and dashed lines show changesif the profile is in up-dip or down-dip direction.

the number of channels,using the common-source for the two and common-receivertraces,respectively, halves of the split. Figure 9.46b shows a 96-trace split-spreadrecord simulated from 4S-traceend-on records.The stack of these96 tracesyields one Simplan trace. Simplan sectionscontain all primaries, multiples, and diffractionswithout amplitude bias or waveform distortion, whereas CMP stacking emphasizesprimary reflectionscompared to multiples and diffractions.

arelisted the I r domain.Severalof theseapplications in Yilmaz(1981:429).Stoffaet al. ( I 98I ) appliedslant stackingto obtain semblance($9.3.5)and eliminate spatial aliasing ($9.2.2d).Clayton and McMechan (1981)appliedthe techniqueto refractiondatato produce velocity depth models.Gardner and Lu (1991) have collected together papers dealing with slant stackine.

9.11.2 Intelligentinterpolation 9.ll Other prccessing techniques 9.I 1.I r-p transformprocessing(slant stacking) The r-p transform or slant stack is a form of Radon transform (see$9.15and eq. (9.22)).When applied to seismicrecords,the slant stack maps the amplitude g(1,x) from the I x domain to the r p domain (fig. 9.47), the integral in eq. (9.22) becoming a summation. Both reflectionand refraction data can be slant stacked.The inverse transformation can be carried out by fllteredbackprojection,as in 513.5.2(seeeq. ( 13 . 12 ) ) . As in the caseof other transforms,the slant stack is usedbecausecertain operationscan be carried out more easily and efficientlyin the r-p domain than in

Intelligent interpolation is an interpolation process that mimics the interpreter'sability to jump correlate using seismiccharacter.It is often based on crosscorrelation, sometimeson recognition of trace attriIt is usedto interpolatebetweendata butes($9.11.4). spacedfour or five times farther apart than spatial aliasconsiderations($9.2.2d)permit if aliasingduring migration is to be avoided.However,intelligentinterpolation does not alter the resolutionof the resulting data, which is determinedby the original sampling rather than that after interpolation. Intelligent interpolation is also used to permit cheaper3-D acquisition ($12.1.2and 12.1.3)to compensatefor relaxed line-spacingrequirementsand to flll in undersampled grid loops of 2-D coverageto create pseudo-3-D surveys.

O T H E R P R O C E S S I N GT E C H N I Q U E S

32s

I

I

rl T{

r

j I

I

i Fig. 9.48

Automatically picked migration section. (From paturet. 1971.)

9.11.3 Automaticpicking Conceptually,eventscan be picked and graded automatically using coherencemeasuresas criteria (paulson and Merdler,1968;Bois and la Porte,1970;Garotta, l97l). Whenevercoherence exceeds a threshold value,an eventcan be picked,the arrival time, NMO, and dip moveoutbeingdeterminedcorrespondingto the maximum coherence.Grades can be assigned basedon coherencevalues,the distanceover which coherencecan be maintainedbeing includedas a factor. The picks can be automaticallymigratedand plotted, as shown in fig. 9.48. Automatic picking can be expandedto include intersectinglines.The picks can be posted on a map and contoured automatically. Thus, conceptually,the output ofprocessingcould be contoured depth maps of reflecting horizons, and much of the work usuallythought of as interpretation could be automated.However,in the process,many decisionshave to be made. Criteria haveto be specified for determiningwhich eventsare primary reflections and which multiplesare for decidingwhat to do when eventsinterfereor terminate,and so on; the processbreaks down or producesmeaninglessresults if each of thesedecisionshas not been anticipatedand specifiedcorrectly in advance. Although automatic picking was never used very much with 2-D data, its equivalent,horizon tracking (see $12.4),is extensivelyused with 3-D data. Improvements in data quality and the areal density of sampling are largely responsiblefor this success.However,horizon tracking still has to be monitored carefully to producereliableresults. 9.I 1.4 Complex-traceanalysis Let us assumea seismictrace of the form C(A : A(!) cos 2rruf,

(e.103)

wherel(r) variesslowly with respectto cos2rvt,. A(t) is the envelopeof g(t), often called the envelopeamplitude. For A(t) constant, the Hilbert transform ( $ 1 5 . 2 . 1 3o )f g Q )i s g i v e nb y (9.l 04) C(t) <->B,(/) : - A(t) sin 2ivt (seeproblem 15.23a).Thus, we can form a complex signal,l(t), where h(t):

cQ) + jg.(r):

A(t)e-j2rv'

(e.105)

ft(r) being known as the analytical or complex trace (Bracewell, 1965), gL(t)as the quadraturetrace of g(t) (seefig. 9.49). If u is not constant but varies slowly, we define the instantaneous frequency,u,(t),as the time derivativeof the instantaneous phase,y(r); thus,

2nv,(t): : tl' f;en,tt

(e.ro6)

The quantities A(t), lft), v,(t), and other measurements derived from the seismicdata are called attrit butes.

/1 --,

,'ir$e

Comple! seismic t

Fig. 9.49 The complex trace shown as a helix of variable amptitude in the direction ofthe time axis. Projection onto the real plane gives the actual seismic trace and onto the imaginary plane the quadrature trace.

i

DATA PROCESSING

326 To find A(t),1Q), and u,(t),we obtain ftO, either by , atis, e q .( 1 5 . 1 7 6 )t h c,Q) : g(t) * (llltt):

1l/tr) i

t, ,{",*

- t\t,

(e.107) for digital functions(seeproblem 15.23c),or by using eq. (15.177),that is, we calculatethe transform of g(t), set the result equal to zero for negativefrequencies, multiply by 2, and then inverse transform to get h(t). Becausel(t) is real and leiz"'1: l, we seethat

A(t): lh(t)|, l(t):

2rvt : tan I

,,@: L2,;]rrtrlt

t&(r/co)Ll (ero8)

Complex-traceanalysiscan be used in convolution, correlation, semblance,and other types of calculations (Taner. Koehler, and Sheriff, 1979),sometimes facilitating the calculations. Attributes sometimesrevealfeaturesthat are not as obvious otherwise,especiallylateral changesalong the bedding, such as those associatedwith stratigraphicchangesor hydrocarbonaccumulations($10.'/ and 10.8);seeTaner and Sheriff (1977).Phaseplots facilitate picking weak coherent events,and latera'. discontinuitiesin phase facilitate picking reflectiotr terminations as at faults, pinchouts, and so on. Instantaneousfrequencypatterns tend to characterize the interferencepatternsresultingfrom closelyspaced reflectorsand thus aid in correlatinsfrom line to line or acrossfaults.

9.12 Processes to reposltlon data 9.12.1Introduction Seismicdata prior to migration are oriented with respectto the observationpoints. Migration involvesrepositioning data elementsto make their locationsappropriate to the locationsof the associatedreflectors or diffractingpoints.The needto migrateseismicdata to obtain a structural picture was recognizedat the beginning of seismicexploration and the very first seismic reflection data in l92l were migrated (fig. 1.3b). Consider the constant-velocitysituation shown in fig. 9.50 A reflection from a reflector with dip ( at point C underneathE is observedat A and is plotted at C' on an unmigratedsection.Clearly, tan {, : sin {,

(e.10e)

where (, is the apparent dip on the unmigrated section. The reflector lies updip from its apparentlocation, { > €,, and a segmentof reflectionC'D'is shortened to CD by migration. Equation (9.109)is called the migrator'sequation. Migration ordinarily assumesa coincidentsourcereceiver section and is generally carried out after stacking.This usually gives good results where dips

C'D'into principle. of segment Migration Fig.9.50Migration thedip from(.,to (. CDincreases

are small and whereeventswith different dips do not interfereon the unmigratedsection.Migration before stackingalmost alwaysgivesbetter or at leastequivalent results,but is expensivebecausethen many more data haveto be migrated.DMO removesmuch of the need for prestack migration, so that today prestack migration is mainly associatedwith depth migration in areaswherethe velocity distribution is complex. Although the objectiveof migration is to obtain a picture of reflectorsat their correctlocationsin depth, the velocity required for time-to-depthconversionis usuallynot known accuratelyand the result of migration is usuallya migratedtime section,which is a vertically stretchedversionof the depth domain provided velocity variesin the vertical direction only. The pro"depth migration" ($9.12.5)attemptsto account cess for changesin velocity in the horizontal direction as well. Another limitation on migration is the migration aperture,the range of data included in the migration ofeach point; the apertureis often lessthan ideal becauseof the volume of data to be processed. Migration generallyis basedon the premisethat all data elementsrepresenteither primary reflectionsor diffractions.The migration of noise,including energy that does not travel along simple reflection paths, produces meaninglessresults. Migration requires a knowledgeof the velocity distribution; changesin velocity bend raypaths and thus affect migration. Although migration can be extendedto three dimensions with ordinary 2-D seismic lines, we usually assumethat the cross-dipis zero,which resultsin twodimensionalmigration. Ignoring cross-dipsometimes resultsin undermigration,but an undermigratedsection is at least easier to interpret than one not migrated at all. Moreover, cross-dipinformation is often not available,two-dimensionalmigration is appreciably more economical,and the results are often adequate. The simplestapproachto migration is to determine the direction of approach of energy and track the raypath backwards to the reflecting point at half the

P R O C E S S E ST O R E P O S I T I O N D A T A traveltime, or to find the common tangent to wavefronts for half the traveltime;thesemethodswerb extensively used in hand-migrating data. Computer methodsgenerallyinvolvesolutionsof the scalarwave equation,eq. (2.28).We replacethe time with half the traveltime, that is, in effect we start with the energy originating at each reflector, as if each reflector were covered by elementary point sourcesas postulated by Huygens' principle, all actuated at the instant t : 0 (the "exploding-reflector"model). We regard {(x, z, c) as a vertical sectionshowing the wave motion at the point (x, z) at timo / : c, that is, an unmigrated seismic section corresponds to rl(x, 0, l), whereas a migrated seismic section corresponds to tft, z, 0). There are various ways of solving for {(x, z, 0), including (a) integral methods based on Kirchhoff's equation (99.12.2),where the integration is over those elementsin unmigratedspacethat contribute to an elementin migrated space,(b) methods basedon a solution in the frequency-wavenumber domain ($9.12.3),and (c) finite-differencesolutions in the time domain (99.12.4), which accomplishes backward-tracingof seismicwaves in a downwardcontinuationmanner. The methodsdiscussedin the next sectionsaccomplish full-waveformmigration;they involvelargenumbersof calculationsand so are restrictedto computer implementation. 9.I 2.2 Kirchhof (dffiaction-stack) migration Diffraction-stackmigration is basedon a concept of Hagedoorn (1954). We assumeconstant velocity V and convert arrival times to distancesby multiplying by lZ. Figure 9.51a relaresa diffraction pMR and a reflectionMN seenon an unmigrated section.A reflector PQ with dip { passesthrough p at a depth zo, SuPis perpendicularto the reflector.Arcs are swung with centersSo,S,, S", and so on and with radii equal to the distancesto the reflector.Hagedoorncalledthe unmigrated diflraction curve pMR a curve of maximum convexity,becauseno other eventfrom the depth :0 can have greater curvature (see fig. 9.51a). The diffractioncurve is a hyperbolawith apexat p and the unmigratedreflectionis tangent to it at M (seeproblem 9.27). The conceptfor carrying out migration as a manual operationis to plot a diffraction curve for eachdepth and slideit along the unmigratedsection(keepingthe top lined up with zero depth) until a segmentof a reflection is tangentto one of the curves;on the correspondingmigrated section,the reflector is located at the crest of the diffraction curve tangentto the wavefront that passesthrough the point oftangency ofthe reflection to the diffraction curve (fig. 9.51b). The principleis the sameif the velocityis not constantand if the sections,wavefronts,and diffraction curvesare plotted in time rather than in depth. To carry out diffraction-stackmigration,diffraction curves are calculatedfor each point on the section.

321 ,to

sr

s2

s3

.t4

Wavefrontcurvesthrough P Wavefrontcurve through N

Coincident source and receiver

(D)

Fig. 9.51 Wavefront and diffraction curves intersecting at the unmigrated and migrated positions. (a) Unmigrated reflection MN migrates into reflector PQ, (b) relation between wavefront and diffraction curves (from Hagedoorn, 1954).

The data on the unmigratedsectionlying along each diffraction curve are summedto give the amplitude at the respectivepoint on the migrated section.If there is indeedenergyinvolvingthe point at the crestof the diffraction curve, then the addition will produce the value appropriateto the energyinvolving that point; if only noise is present,positive and negativevalues will be equally probablealong the diffraction curve, so the sum will be very small. In effect, diffraction-stack migration treats each element of an unmigrated reflection as a portion of a diffraction, that is, a reflector is thought of as a sequenceof closely spaced diffracting points (fig. 9.52). The relationshipbetweenpoints shown in fig. 9.51b suggeststhat the data at each point could be distributed along the wavefront through that point (wavefront smearing), and when the distributed data for all points are superimposed,they will reinforce wherereflectorsexistbut otherwisepositiveand negative values will be equally probable so the sum will be small. Bursts of noise will not have neighboringelementsto canceltheir effectsand hencewill be smeared out along wavefrontson a migratedsectionto become "smiles"(fig. 9.53). Migration by the method of wavefront smearing producesresultsidentical to diffraction-stackmigra-

328

DATA PROCESSING

:- - l

2 a

2

(a)

"-.'.---:::-:

(b)

Fig. 9.53 A burst of noise on an unmigrated section (a) mi_ grates rnto a wavefront (smile) (b). (From yilmaz, l9g7: 25g.)

1 1 !

1 ! - 1 1 r

1 1 !

k

I

:

Migraringreflectionsas diffracrions. (From yilmaz, 1q._9j?_ 1987:257,258.)(a) A diffraction(below)migratesinro a poinr (above).(b) With a sequence of diffractingloints the diffrac_ tlons tend to mergeto form the reflector.(c) If closelyenough spaced, only thereflectoranddiffractionsat its endsareevrdent.

tion, the only differencebeing in that operationsare performed in a different sequence.The ..common_ tangent" method of migration (Sheriff, l97g) is in effectwavefrontsmearing. A more elegantformulation of diffraction_stackmi_ gration is basedon the Kirchhoffintegral (seeSchnei_ dea 1978). This approach makes it-clear that this techniqueis an integral solution to the wave equa_ tion, as opposed to a finite-differencesolution or a Fourier-transformsolution ofthe waveequation(usu_ ally called "frequency-domainmigration;;. Amplitudes are adjusted for obliquity and diver_ gencebeforesumming along the diffraction curvesin Kirchhoff migration. The former factor givesthe co_ sine of the angle betweenthe direction of travel and the vertical, and the divergencefactor corrects for I,lrzfor 2-D migration or Ilr for 3-D migration. In ad_ dition,.a wavelet-shapingfactor "or..Ct, amplitudes by the i-nverse squareroot ofthe frequencyani phase by 45' for 2-D migration, or by the invers! of the fre_ quency and 90' for 3-D migration. The reasonsfor the wavelet-shaping factor are explainedin Schneider (1978)and Berryhill(1979).Ifwe considercollapsing diffraction hyperbolasas wavepropagationin spheri_ cal coordinates,the near-fieldterms ire generallyne_ glected. How far down a diffraction hyperbola integration (summation)should extend is the aperture_definition problem. In general,the collapseof diffraction hyper_ bolas to points is inverselyrelated to the aperture width. Inadequateaperturewidths effectivelydiscrim_ inate against steep dips and aperture widih can be used-asa dip filter. A generalrule is that the aperture should exceedtwice the horizontal distanceof -rnru_ tion of the steepestdips. Clearly, aperture wi"dth shouldincreasewith depth becausediffractionsflatten

P R O C E S S E ST O R E P O S I T I O N D A T A

329

with depth. Use of a wide aperture also has a detri_ mental effect in tending to organizehorizontal noise wherereflectionsare weak.

ri i rc

9.I 2 .3 Migration in thefrequency-wavenumber domain

H ,

Equation (9.109) provides the basis for freouencv_ domain migration. If velocity is constant,tinesln x, I spacethat havethe sameslope(sameapparentveloc_ ity or same apparent wavenumber) transform by the 2-D Fourier transform into a singleline in r_, to space (fig. 9.54);the separateparallel lines in x, I ipace are distinguishedby different phasesin r,, o space.Fre_ quency-domainmigration changesthe slope of lines in rc..,ro spaceaccording to eq. (9.109); the inverse transform then givesa migratedsectionin x, I space. Thus, conceptually,frequency-domainmigration be_ c9m9l a very simple operation (Robinson, l9g3). (Similar migration, called slant-stack migration, ean done in r, p spaceusing the Radon trinsform; see !e Hubral, 1980.)The problemswith frequency-domain (and slant-stack)migration comeabouf becauseof the assumption that the velocity is constant (see what follows). Stolt (1978) introduced the Fourier-transformmr_ gration method, sometimes called Stolt misratrcn. This methodstartswith eq. (2.30)in two dimensions, the r-axis being along the profile direction and the z_ axis positivevertically downward.Thus,

a#:*KY.3,9

'1

( 9 . ll 0 )

We use eq. (l5.llb) to take the three-dimensronal transform of r!(x, :, /) and obtain \t(x, z, t) r+ V(r,, K., trl) :

K"

[il_*n

z ,t)

exp [-j(r,x * r,z + tol)l d,r d; dl.

x

E q u a t i o n( I 5 .l 4 l ) n o w g i v e s

aru

e+ (jto)'V(r,. r, .o), 3r, D:rlr e+ (jft,)'v(x,. x ,. to). ;x:

a"J, <+ (jk.)rV(x,. Ur,

x,. ro).

z(ort)

Substitutingin eq. (9.1l0), we get t " o ' - V 2 ( r i+ r l ) : 0 .

( e . lrl )

Returningto eq. (9.110),we take the two-dimensional transform of rf(x, ;, /) with respectto r and :, ob_ tainins f- fV , _ - ( r ,K . . r) : | | U ( . y:., r ) J - J -

exp [-j(x,x + rc;)] dr d:.

(9.n2)

(d)

Fig. 9.54 Frequency-domain migration where velocity is constant. (a) Two dipping events on an unmigrated section; (b) these events map into the same line in rc,-
330

DATA PROCESSING

If we restrict the solution to harmonic waves,we can write V,,(*,, r,, t) : V",(^,, K,, 0)e-j.,

(9.113)

(to verify this relation, substituteI : 0); the first factor on the right is the required solution to our problem. To find V,,(rc,, rc,,0), we start by calculating the transformofthe recordeddata with respectto x and 1.. f- tV,,{x..0. or) : | | t|r(x.0. /) J_-J -

exp[-j(rc.x + or)] dr dr;

(9.114)

inverting the transform, we have l - l

rlr(x.0./) : tll2rt, | | V,,(x,.0. or) J _ - Jexp[j(r.x+o/)] dr, do. (9.1l5)

9.I 2.4 Finite-dffirence methodof wave-equation migration

From eqs.(9.I l2) and (9.I I 3), we obtain r | ( x .: , t ) : ( l / 2 r ) '

V.,(r., rc-,0) e i'' | J - J[- * exp [j(r<.x + rc,2)]d r . d r , ;

nence,

r r ( x , 0r.) : ( t l 2 n t )l - f - * , . , , . , K . .,0) J - _ J_ e x p [ j ( r - x + r o r ) ]d r . d r . .

stretching.Gazdagand Squazzero(1984)handle lateral velocity variation by migrating with a number of laterallyconstantvelocitiesand then interpolatingto get the migrated wavefield. Phase-shift ing migr at ion (G azdag, I 978), also called Gazdag migration, carries out migration in the frequencydomain by using a pure phase-shiftingoperator e j*" at each z-step in downward continuation ($9.12.4).The parameterof the maximum dip to be migratedin Gazdagmigrationcan be usedto discriminateagainstdipping coherentnoise,but choosingtoo small a maximum can unintentionally filter out real dips and result in smearingthe data. Use of too-small migration stepstendsto producediscontinuitiesat the step boundaries.Typically,the step size is something lessthan the dominant period, often 20 to 30 ms. Gazdag migration is generallymore sensitiveto velocity error than finite-differencemigration (S9.12.4).

,

(9.116)

Comparingeqs.(9.1l5) and (9.1l6), we seethat V,,(*.,0, o) do : V,,(r,, rc,,0) dr,, or . I . , ( x , . K . , 0 ) : V , , ( x . .g . , l ? t dKr

: W,,(r,,0, to) + (r1rc,)r] '/, [ from eq. (9.111),o being equal to V(r?,+ r])r/r. BecauseV,,(r,, 0,
The concept underlying time-domain migration is downwardcontinuationof the seismicwavefield.Continuation is a familiar processwith gravity and magneticfields(seeTelford,Geldart, and Sheriff, l99O:34, 108).It utilizes the continuity property of fields,one expressionof which is that we can determinethe field over any arbitrary surfaceif we know the field completely over one surface,provided the field satisfies Laplace'sequation,V,S : 0. Evidencesof subsurface featuresspreadout as distancefrom the featuresincreases,or, conversely,the evidencesconvergeon the location of the featuresas they are approached.If we let t' : jVt and assumeharmonic waves,we can wnte the scalarwaveequation(eq. (9.1l0)) as

a',! a',f T

ox'

dz'

I ar{,: ( ^l : ar{, aru arr! + +

l,n dt2

-

Exz dz2 dt'2 : Vrrl(x, z, t') : g,

(e.n7) thus expressingthe waveequation in the form of Laplace'sequation.We know the wavefieldat the surface ofthe earth, z : 0, so our problem is to continuethis field downward to determinewhat geophoneswould see if they were buried at arbitrary depths. We downward-continuein a seriesof steps, effectively loweringthe geophonesgraduallydownward through the earth. At each geophonedepth, we expectto get a clear picture of reflectorsthat lie immediatelybelow the geophones,so we retain this portion of the downward-continuedrecord section from each continuation stepand combinethe upper portion of these to give our completemigratedsection.Figure 9.55 illustratesthe downward continuationof a diffraction. We let I be the one-waytraveltime(half the arrival time for coincident source-detectordata). A plane waveapproachin!the surfaceat the angle0 is givenby {(x, z, t) : A exp {jrlr - (xlV) sinQ (9.118) Q t V ) c o s0 l ) .

P R O C E S S E ST O R E P O S I T I O N D A T A

JJI

7lO 6

tmn - -

;--r-

---. ------

!!9t.

Fig. 9.55 Simulation of downward continuation showing a diffraction hyperbola collapsing to a point. Note that the effec-

(heavy tiveaperture line)shrinks (FromYilmaz, in theprocess. 1987:261.\

If we restrict ourselvesto small anglesof 0, we can approximatesin 0 - 0 and cos 0 - I ]0r, so that e q . ( 9 . 1 l 8 )b e c o m e s $(x, z, t): A exp Ur(l - r\lV - zlV + zTrl2V)).

Lx, Lz, and At*, the plane z : 0 representsthe unmigrated time section(which is our starting point) and the diagonal plane r : f - zlV representsthe migrated time section.(The projection of this diagonal planeonto the l* : 0 planewould alsogivea migrated depth section.)We approximatederivativesby finite differences:

Wenowdefinea newtimescale, /* : , - ,,:i':1, changemeans that our coordinatesystemeffectively rides along on an upcoming wavefront (seeproblem 9.28).We now have {*(x, t, t*) : A exp [jto(t* - x\lV + zTrl2V)]. (9't20\ and

dx'

Er'

dz

dt* 0z

dz

V af

a r u : a r q * -_V2 a & r avt** _ - I dzrp'r dzr- dz, V-atSubstituting into the wave equation (9.110) gives a new waveequation,

*dx2 { * qdz2 g - (\Vldz ? - )dt* 3*l:' eu) For waves that are traveling nearly vertically, the changein rf* with respectto z is small in our moving coordinate system, so we neglect 629*1022. This is called the "15o approximation," and neglectingthis term meansthat we shall not be ableto migratesteep dips well. We thus get

o,22,

(A "45' approximation" givesthe equation

a',lr* dx2 dl*

21,x,2,t*)

_'4L:,t\t*(" (Ax)'

{ t * ( x , z , l * ) - 0 * ( x , z - A z .t * )

rl*(.r. :. /*) :

?^- ^r{'(Ax)2 z - L,z.t*) [V*(x. | 2(Lx), VAz Ar* [ A; Ar* - At*) , t*6, z, t* A,zA,t*

- \ll@

- L^x'z' t*) (Ax)'

- Az, t* - At*) Az At* - 2Lx, z, t*)] V**(x -, _ \t*(x, z

,,o"y

1'

(9.r24)

This is a relation among six elementsof the array, as shownin fig. 9.56b: **(x, t, r*) : a,$*(x, z - L,z, t*) * ar$*(x, z, t* - A,t*) * a.r!*(x,z - Az,r* - At+) + VIao**@- A,x,z, t*) * arrf*(x -2A,x, z, t*)1.

I .. a'{r* arU* : 0 ' - l -l2\ l l/ 2 dx2 dz

-2V'(n

{*(x, z, t* - A,t*) + t*(x, z - L,z,t* -Lt*\ll!,z Af . Equation (9.122)now becomes

3x,

Ty-|)j'ji :'

=V94,/*)

-

r y : q U + ? t 1 a r * : 4 !l q l _* l dz

dx2

a2ilr* dz )t*-

? q : q q . a { : q qq v : g i q dt 0t* dt dt*, dt2 d/*2 a!: qv:. 4u_ - a'{,* dx

a'v*

\Vl dx dt*'

(e.t23) seeClaerbout,1976:198.) If we think of rf* as a three-dimensionalarray (fig. 9.56a),that is, having values at discreteintervals of

This relation can be used to extend the threedimensional array in the *x-, *z-, *l*-directions. Our problem is getting enough of the array to begin with so that we can extendit. For negativez, ** : 0 becausethis refers to wave values in the air above the surfaceof the ground. If our data begin at x : 0, we needvaluesof Q* for x : -Ar and x : -2Ax, which

I i

DATA PROCESSING

J)Z

0 lAz

T

d**""*wah

are not availableto us; we guessthesestartingvalues, and it turns out that the solution is not affected very much if theseare in error, so that a stablesolution can be achieved. Variousalternativesto the foregoingmethod of approximating the derivatives are available, some of which lead to highly stablealgorithmsand permit calculations using a fairly coarsegrid, which of course makes the calculation much more economical.One approachis to take the Fourier transformwith respect to x.' **(x, z, r*)e+Vf (r,, z, t*) so that eq. (9.122)becomes

*lvr(il#:,

\ r ( x ,z - A z , t * l

/''l

I

A$, z - Az,r*

Ar+)

(e.125)

We assumethat we have a table of values of tlri for discretevaluesofz, r*, as illustratedin fig. 9.56c,and we considerthe portion ofthe table centeredbetween the values A, B, C: and D. The approximate value of Vf at this point is 4(A + B + C + D). We can approximated:Vf/dzdl* by the expression

(o;: -

B-A\ ^ ' )

I

ari

A _ B - C + D L,z Ll*

that is, as the differenceof differences.We now write eq. (9.125)in the form

"lll ; I:1lll:' lllllI lll:l|:

(c)

Fig. 9.56 Relationship of elementsin (,r, :, t*)-space. (a) Seismic traces on the top surface. ; : 0, show the unmigrated section; those at successivelayers show what would be recorded by geophones buried at depth :. (b) Elements entering into the time-domain calculation of r!*(x, :, l*). (c) Table of values of r!*(.2,t*).

where e : ivx,?A,zAr*. This box of four compartments is laid over the table of Vf values,and each is multiplied by the overlyingfactor (e * 1),and the sum is set equal to zero. If we know three of the four values,we can calculatethe fourth. Again, some values haveto be guessedto get started. The upper surfacein fig. 9.56a,the unmigratedtime sectionobservedat the surfaceof the earth, thus provides the basis for calculating the time section that would be observedat z : A,z,and so on. In effect,we are calculatingthe output ofgeophonesburied at z : z, as the superpositionof filtered outputs of the geophonesin the layerZ : zr - Az and previouslycalculatedvaluesin the layer2,. The filters are mainly phase shifting to allow for the difference in traveltime from a geophone directly above the buried geophone to thoseat adjacentlocations. In continuing the wave field from z : zt to z : zl * Az, we use the velocity of the layer betweenzr and z, * Az,so that accommodatingvertical variationsof velocity is fairly easy and straightforward in timedomain migration. Velocity Z can also be made a function of x, but often this is not done. A common way of accountingfor lateral velocity variationsis to migrate with different velocitiesand then merge the

i

P R O C E S S E ST O R E P O S I T I O N D A T A

00

JJJ

(b)

0.5

1.5 2.0 2'5

Fig. 9.57 Migration removes the confusion produced by multiple-branch reflections.The deeper data, which are proba_ bly multiples, out-of-plane diffractions, and other tvpes of noise

two sections,that is,,lr: k,lr, + (l - k)rf,, wherek varieslinearly over the mergeregion, the sametype of procedureas is usedto accomplish"time-variant,'fi|tering. Becausedownward continuation proceedsin steps, the size of the step has to be selected.Stepsthat are too large cause undermigration, kinks in reflection continuity, and dispersion, whereas small steps increasecomputation time and costs.The undermigration increaseswith increasinglysteepdips.The dispersive noise is an effect of approximating differential operatorswith differenceoperators.The step size is usuallyselectedsomewhatsmallerthan the dominant period, that is, between20 and 40 ms.

9.12.5Depth migration Migrated time sectionsmay be simply stretchedaccording to a vertical velocity function to give sections

are smeared out becausethey are migrated as if they werc pnmary reflections. (Courtesy oi AGIP) (a) Unmigrated section; (b) migrated.

where the vertical scaleis linear in depth rather than in time (fig. 9.59b).However,wherevelocityvariesappreciablyin the horizontal direction, raypathbending introducesadditionalcomplicationsthat depth migration (Judsonet al., 1980;Schultzand Sherwood,1980; Larner et al., 1981)attemptsto accommodate. Hubral (1977)observedthat the apex of a diffraction curve is where the image ray, a ray that approachesthe surfaceat right angles,emerges.Therefore, if we follow the imageray as it refractsaccording to Snell'slaw down through the earth, it will lead to the correct position ofthe diffracting point evenifvelocity surfacesare not horizontal. This conceptis the heart of depth migration, migration that accommodateshorizontalchangesin velocity.Conventionalmigration collapsesdiffractions to the image-raypositions, so an additional step is needed to move elementsto their correct subsurfacelocations.Larner et al. (198l) develop a two-dimensionalvelocity model, V(r, z), and then ray traceimageraysto locate

aa A JJ+

features.This procedure is illustrated in fig. 9.59c and 9.59d. The velocity model definesthe major velocity surfaces where significant raypath bending occurs; key horizons on a conventionally migrated time section are mappedassumingthat theseare the major velocity interfaces. Clearly, defining the velocity model adequatelyis the key to successfuldepth migration. Specifying velocitiesis a very difficult task becausechoices are not obvious. Robinson (1983) says,"it is at this point that the true inverseproblem is being solved." Detailed knowledgeof the velocity distribution is often not available,especiallyin the structurally complex areas where depth migration is most needed. Howeveq even though velocity errors create depth and location errorsin the final product, the improved structural clarity often makes the procedure worthwhile and an appreciableamount of depth migration is being done today. Subsaltimaging is important in severalareasto locate hydrocarbonstrapped beneathsalt. Appreciable raypath bending occursat the large contrast between the salt and sedimentsand the surfacesof the salt may be quite irregular.Migration is usually done in steps: conventionalmigration first definesthe top of the salt, then the base-of-saltreflectionis definedusingthe salt velocity,and finally migration is completedwith sediment velocities.Subsaltimagingprovidesa severetest of migration accuracyand requiresvery reliabledata, which are usually 3-D data, and processing,often prestackmigration. 9.I 2.6 Hybrid migration The limitations of different migration methods ($9.12.7)can sometimesbe overcomeby flipping back and forth betweenmethods.For example,the Stolt algorithm can be used to improve the performanceof finite-differencemigration, which has difficulty with steepdips.A frequency-domainmigration can be used with a constant low velocity to perform a partial migration, and follow this with a residual finitedifference migration. Residual migration may not work well when too much of the migration is left to the secondmigration step,as is apt to be the casewith steepvelocity gradients. Frequency-spacefinite-differencemigration (Yilmaz, l98l: 309-ll) is sometimescarried out. This method, sometimes called omega-x migration, can handleboth steepdips and lateral-velocityvariations. Kjartansson(1979)implementeda 45oversion,which can be further modified to accommodate even steeperdips. 9.12.7Relativemeritsof dffirent migrationmethods In practical implementation,each migration method involves approximations and limitations that affect data with differentcharacteristicsin differentways,so that one method may migrate better for one data set

DATA PROCESSING but another method might be superior for a different data set, or one implementation of a method may yield better results than another implementation of the same method. Among the characteristicsof the methodsare the following: Diffraction-stackmigration: Migrates steepdips Allows weightingand muting accordingto dip or coherency Aperture can be varied explicitly Usually not adapted to accommodate lateral-velocityvariation Stolt frequency wavenumbermigration: Migratesdips up to spatial-aliasinglimitations Difficult to accommodatelateral velocitv variations Can be appliedto specificlimited areas Is often the most economicalmethod Finite-difference migration: Migratesdips up to 60o 1- 30' with 15' version) Produceslessmigration noise Is effectivein low signal-to-noiseareas Can accommodatelateral velocity variations Frequency-space migration: Migratesdips up to spatial-aliasinglimitations Often the easiestmethod for depth migration (99.13.8) because velocityas a function of spaceis explicit One important point should be made: migration (by almost any method) almost alwaysproducesa result that is closerto the correct picture than failure to migrate,evenwherethe fundamentalassumptionsare grosslyviolated. 9.12.8Resolutionof migratedsections In $6.4.4,it was pointed out that the horizontal resolution of seismicdata is limited by noise and migration considerationsrather than Fresnel-zonesize. as with unmigrateddata. One of the basic assumptions underlying migration is that the data show only the reflectedwavefield;data of any other type (including multiples)will be migratedas if they wereprimary reflected energy and will be rea-rrangedand superimposedon the migrated imagesof reflectors.Noise of limited extent that might be quite recognizableas noise in the unmigrateddata may smearout and degradethe sharpnessof reflectorimages.For example, a noiseburst on a singletracewill becomea complete wavefront(sometimescalled a "smile") on a migrated section(fig. 9.53).Data with a dip componentperpendicular to the line will not be migratedcorrectly,thus degradinghorizontbl resolution.Spatial-aliasingcon-

I

I

D A T A - P R O C E S S I N GP R O C E D U R E S

JJ)

so that the smearproduced by stackingis minor for each of them and then these partial stacks are migratedand the migratedresultsstacked.Thus, the full stackand full noiseattenuationmay be achievedwithout significantreflection-point smear. Turning-wavemigration is anotheraspectof migration that is important today.A turningwaveis a diving wave(94.3.5)that has beenreflectedafter reachingits maximum depth, that is, on its upward travel toward emergenceat the surface.Stackingmay destroysuch arrivals, whose normal moveout may be negative. Turning waves are especiallyused to locate steeply dipping reservoirstruncating againstthe overhanging flanks of salt domes. The migration of unmigrated reflection-timemaps (Kleyn, 1977) often provides an excellent way of achieving correct depth-contour maps. Usually, the unmigratedmap surface(the consequence of picking a reflectionevent, posting, and contouring) and also intermediatevelocity surfacesare approximatedby a grid of small planar elements(fig. 9.60).Raysare then traced,the startingdirectionat the surfacebeinggiven by the attitudeof the elementson the unmigratedmap (which give the angle ofapproach at the surface),the rays being bent in accordancewith Snell'slaw wheneverthey encounteran intermediatevelocityinterface. The consequentdepth map (fig. 9.61)then has data postedin an irregular manner,so theseare usuallyregridded and contoured by the computer.

(b) 9.13 Data-processingprocedures Frg. 9.5U Migration collapsesdiffractions and makes faulting i l e a r e r . ( C o u r t e s yo f G r a n t G e o p h y s i c a l . )( a ) U n m i g r a t e d s e c i r o n : ( b ) m i g r a t e ds e c t i o n .

siderationslimit the amount of dip that can be migrated.The use of a limited migration aperture and the approximationsinvolved in the migration algorithms further restrictresolution.With good data and good migration algorithms, features are sometlmes defined very sharply; for example,faults can sometimesbe locatedto within the trace-spacing distance. Usually,however,the net effectof the variouslimiting lactorslimits the precisionto much lessthan this.Figures9.57and 9.58illustratethe improvedclarity that migrationcan produce. 9 12.9 Other migrationconsiderations Vigrating data beforestackingprovidesan alternative to DMO processing (99.10.2)in avoidingthe smearing of reflecting points for dipping reflections during stacking. Various schemes (Sattlegger and Stiller, 1974;Sattlegger et al., 1980;Schultzand Sherwood. 1980;Jain and Wren, 1980)are sometimesemployed ro obtain the benefitof migration beforestackwithout rncurringexcessive cost. Severalstacksmay be made, eachwith data involvingonly a limited rangeof offsets

9.I 3.I Typicalprocessingsequence (a) Initial steps. Seismicdata processingoften follows a basicsequence(fig.9.62)that is variedto tailor to specificneedsofdata. A small group of people is usually responsiblefor processing.They preparespecificinstructions,choose processingparameters,and monitor the quality of results.This group needsto know the objectivesof the processingin order to make optimal decisions.processingthat is optimal for one set of objectivesmay not be optimal for another set. The first stepafter field tapesare receivedat a processingcenteris to verify the arrangementofdata on the magnetictape. This involvesdumping(displaying the tape'smagneticpattern on a printout) the first few (possibly l0) recordsand comparing with what is expected. With Vibroseis data, format verification includes a check on the Vibroseis sweep length and spectrum. (b) Editing. Editing followsformat verification.The data are rearranged or demultiplexed; field data are usually time-sequential,that is, the first sample for eachchannelis recordedbeforethe secondsamplefor any channel,whereasmost processingrequirestracesequentialdata, that is, all the data for the first channel beforethe data for the secondchannel.

:

Fig. 9.59 Depth migration. (From Hatton, Larner, and Gibe is o n , 1 9 8 1 . ) ( a )U n m i g r a t e dt i m e s e c t i o n ,( b ) I i n i t e - d i f f e r e n cm grated section stretchedvertically to a linear depth scale,(c) ray

tracing through velocity layers, and (d) migration stretched along the raypaths in Part (c).

6

8km

....... .-.."..-.t....".'._-_""t_.__-__r-__

o

338

DATA PROCESSING

sourcethat continuesfor sometime, the equivalentof a source-signature correctionis accomplishedby correlation with the source waveform. Field data encompassa very wide range of amplitude valuesthat havebeenrecordedin encodedform. so the field gain is decodedand a first approximation of a spherical-divergence correctionmay be appliedto give a smallerrangeof valuesfor input to subsequent processes. Sometimesdata are also vertically stacked or resampledto make lessdata for future processing. Resamplingshould be precededby aliasfiltering.

Fig. 9.60 Map migration. The unmigrated map (top) is subdivided into elements and a raypath is traced downward, the direction of the raypath being determined by the valuesat the corners of the elements.Isovelocity surfacesare likewise subdivided into elements,and when a raypath strikes one of the isovelocity elements, it is bent according to Snell's law. The (x, y, z) coordinates when the traveltime is satisfied are contoured to sive a migrated map.

Tracesmay be summedor arraysformed,especially with marine data recordedwith 240 to 250 channel streamers,to reducethe data for further processing. Editing may involvedetectingdead or exceptionally noisy traces. "Bad" data may be zeroed out or replaced with interpolated values. Anomalously high amplitudes,which are probablynoise,may be reduced to zero or to the levelof the surroundingdata. Outputs after editing usually include: (1) a plot of each file, so that one can seewhat data need further editing and what types of noise attenuation are required;(2) a plot ofthe shortest-offset tracefrom each record (near-traceplot), to give a quick look at the geologicstructureand for usein making decisionsas to where velocity analysesare to be made; (3) neartrace autocorrelation plots, which indicate multiple problemsand aid in making deconvolutiondecisrons, (4) a trace-sequential tape that will be usedfor subsequent processing.The field tape is then stored in a tape library. If the input waveform has been recorded,a deterministic source-signaturecorrection ($9.5.2)may be applied to make the effective waveform the same for all records.If the source is Vibroseisor some other

(c) Parameterdetermination. The object of the next sequenceof processesis to determineprocessingparametersthat are data-dependent,such as static time shifts, amplitude adjustments,normal-moveout values,and frequencycontent. Field-geometryinformation is input so the computer can determine which data involvecommon geophonesor haveother factors in common, and so the offset for each trace is known fiornormal-moveoutdeterminations. If the near-traceplot showsstrong coherent noise trains from surfacewaves,shallowrefractionsor other horizontally travelingenergy,apparent-velocityfiltering may be appliedto removethem. Statisticalwavelet contractionmay be applied.The analysisof the statistical characteristics of the data for waveletcontraction and for staticsand velocity analysisshould be based on data that do not include known bad traces and zonesofnonreflection energy,suchas the first-breaks region.A wide window is usuallychosenso as to provide appreciablestatistics,but the window should exclude zones where primary reflections are not expected, such as below the basement. Surfaceconsistentdeconvolutionmay be used. The statics-analysisprogram looks for systematic variations such as would be expectedif time shifts were associatedwith particular source activations, particular geophones,and so on. Preliminary statics, as determinedin the field officefrom first-breakinformation and from the elevationof geophonestations, is usually input beforethe staticsanalysis,so that the staticsanalysisdeterminesresidualstaticserrors.The results of this analysisare output on a control plot. Surface-consistentstatics may be supplementedby refraction-basedstatics(S9.6.4). An analysissimilar to the time-shift analysisof the staticsprogram is carried out for amplitude,to determine systematicamplitudeeffectssuchas might be associatedwith a weak source,poor geophoneplant, and so on, and a control plot is output. Data are then re-sortedto CMP gathers.2-D flltering may be applied to the gathersto improve the subsequentvelocity analysis. Velocity analysesare usuallyrun I to 2 km apart at locations selectedbecausethey are relativelyfree of structural complications.The locations for analyses are based on the near-traceoutput from the editing pass.A first guessas to velocityis input prior to velocity analysisso that the velocityanalysisdeterminesre-

I .t

(b) - t'l

May migration. (Courtesy of Prakla-Seismos.)(a) Unmigrated map; and (b) migrated map.

DATA PROCESSING

340 sidual normal moveout. If the near-traceplots show dip, this information is input becausevelocity determination dependson dip. Compromiseshave to be made as to how much data should be included in an analysis;more data yield better statisticsbut then the resultsdo not apply at specificpoints, so the compromise is betweendetermininga more accurateaverage and a lessaccuratevalue that appliesto a specificlocation. Outputs from velocityanalysismay include(1) a velocity spectralplot, such as in fig. 9.25 or 9.26, which shows the coherenceachievedwhen various stackingvelocitiesare assumed;(2) velocity panels,as shown in fig. 9.27, which show the data stackedaccording to the input-velocityinformation and also according to velocitiesslightly smaller and larger than the input velocity;such panelsallow one to seeifcertain events require different stacking velocity than other events,becausestacking velocity is not necessarily a single-valuedfunction, and also how sensitive the stack is to velocity assumptions;and (3) a graph such as in fig. 9.32 showinghow velocity determrnations at different locations along the lines relate to each other. Velocity analysisis sometimesdone both before and after DMO. Instead of DMO, data are sometimesmigratedbeforestacking.Multiple attenuation by filtering after Radon transforming is sometimesdone after DMO. The data may also be filtered by a sequenceof narrow-band-passfilters yielding a filter panel (fig. 9.20)that is usedto determinesubsequgntfiltering parameters.Autocorrelationsand spectralplots of various sorts may also be output. Preliminary stack sections may be made to show the effectivenessof processingparametersand as an aid in diagnosingadditional problems. (d) Principal processing pass. The main processing pass usually begins with the tape from the editing pass, and the sequenceof operations is almost the same as in the parameter determination pass, the differencesbeing in the valuesthat are input lor statics,normal moveout,and so on. The correct spherical divergencebasedon the actualvelocitiesreplacespreliminary gain assumptions.The main processingpass or portions of it may be repeatedusing more refined values,especiallystaticsand normal-moveoutvalues. The final output from the main processingpasswill be one or more stackedsectionsthat may differ in the processesapplied or in the choicesof parameters,especiallydisplay parameterssuch as amplitude,polarity. and filteringchoices. (e) Other processing. The stackeddata may then be such as miused as input to various other processes, gration (see$9.12)and attribute analysis.Velocity information is requiredfor migration; the velocity may be based on stacking velocity values where dips are not extreme,but in general,the optimum velocity for migration differsfrom the optimum velocitylor stacking. Some empirically determinedpercentageof the

stacking velocity may be used as the migration velocity. Either the stackedor the migrateddata may be further analyzedfor amplitude, frequencycontent, apparent polarity, inverted to acoustic impedance or synthetic-velocitytraces,and so on, and displayedin various ways.The data may also be used in iterative modeling or other types of processing. 9.I 3.2 Interactiveprocessingand workstations Seismicdata processingtoday is a usually a sequence of batch processesthat is often interrupted to insert parameter decisions based on previous results. We would like to make optimum decisionsas to parameter values and processesto be executedand to have processingflow along smoothly and rapidly. Historically, the time required for many processeshas been so great that an operator becomesbored waiting at a terminal for the responseto decisions;the cost of tying up computersor computermemorieswhile waiting for an operator to make decisionshas militated againstinteractiveprocessing.Howeveqworkstations that have enoughmemory and computing power (especiallywhen tied to other computers)to yield results rapidly are currently changingthis situation. Indeed, as computers become faster, memory cheaper,and data storagelarger,seismicprocessingmay becomea geologicinterpretationactivity with real-timevisual feedbackto guide parameterdecisions. One of the most usefulapplications of workstations is simplycheckinginput parametersand instructions for batch processingto make surethat theseare complete and consistent before involving an expensive computerin their execution,and also displayingthe quality-controland parameter-determination outputs (suchas thoselistedin fig. 9.62).The resultsof processingsamplesof the data in different wayscan help visualize the results of complete processing.Workstations can prompt one regarding overlookeddecisions and then communicatethe parametersand instructionsdirectly to a large computer for the actual processlng. Velocity analysis is an especiallycritical process where many decisionshave to be made. Historically, this has been done for occasionalindividual gathers without an easyway to check their consistencyeither internally or with respectto adjacentanalyses.Being able to view at a workstation adjacentvelocity analysesand the interval velocitiesimplied by the operator's picking allows more consistencyand more complete picking. It alsoencouragesintroducinggeologicinterpretation into the picking decisions. 9.14 Generalized inversion In a direct problem, we calculatethe effectsproduced by a model. In an inverseproblem, we try to derive the model from observationof its effects.This is basically the function of interpretation,determining the distribution of the physicalpropertiesof the earth (the

I

I

Processing

Inputs

Hard-copyoutputs

Field Tapes FormatVerification Job request

b0

Demultiplex Editing (kill bad traces) Signature deconvolution Array formin gl tr ace summing Gain recovery/adjust

Vibroseiscorrelation Verticalsumming Resampling

Plot of each file Near-traceplot Autocorrelation plots Field tape to library Job request

o0

(,) I k

c)

l-

Geometry specification Deconvolution Amplitude analysis Statics determination CMP gathering 2-D filtering of gathers Velocity analysis NMO correction DMO correction Mute Trace equalization Stack Coherent noise reiection Predictivedeconvolution Bandpassfiltering

Geometry information Deconvolution pararneters

statics

Mutespecs.

Filter values

plitude equalization

Amplitude plots Statics plots

Velocity plots Velocity panels Velocity along line

Filter panels Spectral plots Autocorrelation plots Stacked section

Jobrequest Migration

Inputvelocity

Wavelet processing o Y

Attribute analysis Inversion

rr

Etc.

Fig. 9.62 Typical processing flow chart. Some of the processing shown is optional, sometimes the sequenceis changed o r o t h e r p r o c e s s i n gi s a d d e d .

constraints

Migrated section High-resolution section Reflectivitv Equivalenf wavelet Color sections Seismic logs

DATA PROCESSING

342 parametersof the problem) from seismicobservations. "inversion" for the algorithmic asWe generallyuse pects of interpretation. We have encounteredonedimensionalinversion(05.4.5)and automatic picking ($9.11.3),and we discuss tomographic methods in $13.5.This discussiondraws heavily from Hatton et al. (1986)and Russell(1988). Inversionis nonunique,that is, a number of different models can lead to the sameset of observations. This is true partly becausemeasurementsare incomplete and also because they involve uncertainties. Often an infinite number of models can lead to the within the uncertaintiesinsameset of measurements volved,and this may lead one to questionthe value of inversion.However,constraintsusually limit physical property values so that the set of possiblesolutions existsonly within narrow bounds. Much of the nonuniquenessoften can be removed by careful model construction. Model construction involves,among other things, determining the model complexity.The selectionof too fine a grid, for example,might permit including an anomalouslayerthat is too thin to be detected(and also lead to excessivecalculation time), whereastoo coarsea grid may involveso much smoothingthat the resultslack utility. Generalizedinversion attempts to determine the spatial distribution of physical propertiesthat could produceobservedseismicdata, usually in an iterative manner.Inversionis usually treatedas a linear problem, that is, measurementsare assumedto bear a linear relation to the parameters.However,many problems are intrinsically nonlinear; such problems are usuallysolvedby a sequenceof linear approximations. A starting model (which may be as simple as a unlform half-space)is given, the errorfield (the difference between the direct-modeling solution and the observeddata) is used to perturb the model in such a way as to make the error field smaller,the new error field is then usedto further perturb the model, and so on. The objectiveis to minimize the error field; iterations are stoppedwhen the error field becomessmaller iterthan somethresholdquantity or when successive ations fail to changethe error field more than some thresholdamount. We must measurethe goodnessof fit in some suitable way. The best estimate to a set of numbers may be the median. The most common error measure is the /easl-squares ft (norm), which measuresthe squareroot of the mean of the squaresof the individual errors; least squaresimplies a Gaussiandistribution of errors. A least absolute deviationfit averages the errorswithout regardfor their sign;it corresponds to the maximum-likelihoodestimatewhen the errors have a Laplace (double exponential) distribution. Theseare specialcasesof I, fits that minimize

v : \tw,e,t,,

(e.126)

where the errors are €,: li - ffip l,being the data points, nr, the values calculated from the model, and

w, are weightingfactors.If p : 1, this yieldsthe leastabsolutedeviation fit and, if p : 2, the least-squares fit. For p : *, we get the minimar or Chebychev fit. The criterion of goodnessof fit is not necessarilytechnical; for example,it can be economic. Models are often formulatedas matricesrepresenting a set of equations to be solved simultaneously. However,the matricesare usually much too large to be solvedby conventionalmatrix methods. Usually, one is searchingfor an extremum (in this case,the minimum error), but the problem is usually phrasedin terms of searchingfor a maximum or hill climbing(rather than valley searching).Hill climbing is sometimes performed by methods such as the ascent.The maximum is discovered methodof steepest by taking an incrementalong the steepestgradientto arrive at the next approximation,the steplength often being proportional to the magnitudeof the gradient. Provision can be made to speedup convergenceand preventoscillationabout the maximum. Convergence (or at leastthe speedofconvergence)is apt to depend on the startingmodel as well as the data and the algorithm used.Convergenceassumesthat the function is continuousand that the initial estimateis alreadyon the lower slopesof the correct maximum (seeLines and Treitel, 1984). "hills" (or "valleys"), However,there may be many and we want to climb the highesthill. Iteration may result in "climbing the wrong hill" and thus converging on a suboptimal solution. We wish to perform global, rather than local, optimization, that is, we want to searchwidely through model spacein order to find the highesthill. We wish for somethingin between a global random search and a local climb. Sometimesfiltering to retain only the low-frequency componentsin early iteration stageshelps get one to the vicinity of the global solution. The simplestmethod of determiningmodel parametersis trial and error, but there is no assurancethat enoughtrials havebeenperformedto arrive at an adequate determinationof the parameters.We might use a random method, such as the trial-and-errorMonte Carlo method,which stepsout in random directions for random distancesin its searchto determine the highesthill. Another method is simulatedannealing(seeVasudevan, Wilson, and Laidlaw, l99l); it usesalgorithms based on an analogy betweenoptimization and the growth of long-range order, such as the growth of largecrystalsin a slowlycooling melt. It is usuallyimplemented by a "drunkard's walk" through model space,where stepsbegin in random fashion, but progressivelybecomebiasedtoward the uphill direction. "enSimulatedannealing has three components;an problem in terms of a ergy function" that definesthe set of parametersand constraints(including interac"order function" that tions betweenparameters),an "temperature" that regumeasurescoherence,and a latesthe system'se4ergyand order; high temperature implies high energyand low order.

PROBLEMS

343

Another class of methods are called geneticalgorithms (see Smith, Scales,and Fischer, 1992; Stoffa and Sen, 1992); they begin with a loose analogy betweenoptimizationand a biologicalsystemcomposed of a relatively few organismsthat react in a relatively complex way. Algorithms try to evolvea population of trial answersin a way mimicking biological evolution (the survival of the fittest). Each model has a "fitness" associatedwith it; the goal is to find the most fit of all possiblemodels.A geneticalgorithm consists of a setof operationsthat we apply to a model population to produce a new population whose averagefitnessexceedsthat ofthe predecessors. The characteristics of models are specifiedby "chromozonestrings." One type of genetic algorithm selectsparents in a somewhatrandom manner but weightedby their fitness("selection");the chromozonesfor a "child" are somewhatrandomly selectedfrom the two parents ("crossover").The child then joins the population, and the least fit member of the population (which might be the child itself) is eliminated. Ar random times,a "mutation," a random changein a member's chromozones,occurs; this permits introducing into the specieschromozoneelementsnot present in the original population. Problems 9.1 Derive the expressionfor the amplitude of frequencycomponents,eq. (9.6), from the equation for Fourier analysis,eq. (9.3). (tlrrt. Multiply both sides of eq. (9.3) by s-j"',' and integrateover one cycle,for example,from - Tl2 to Tl2. Before substituting the valuesat the limits, use Euler'stheorem,e'j' : cos x * j sin x (seeproblem 15.l2a) to expressthe complex exponentialsin terms of sinesand cosines.) 9.2 The techniques and concepts of convolution, aliasing,z-transforms,and so on can be applied to other than the time-frequencydomain. Expressthe sourceand grouppatternsoffig. 8.13aas functionsof .r (horizontalcoordinate)and convolvethe two to verify the effectivepattern shown in fig. 8.13b. 9.3 (a) Becausean impulse6(l) is zero exceptfor I : 0. whereit equals+1 (see$15.2.5), we can apply the Fouriertransformequation(9.12)and find that E(l)e+ - l: showthat 6(/ -

/o) e;' s-j2a(t

pl

and a very steepslopeso that noiseabovethe Nyquist frequency is highly attenuatedrelative to the passband of the system. (a) Assuming an initially flat spectrum,alias filtering filter, and subsequentrewith a 125-H2,72-dBloctave samplingfrom 2 to 4 ms (without additional alias filtering), graph the resulting alias noise versus frequency. (b) Somebelievestandardaliasfilters are unnecessarily severe.Assume a 90-Hz, l2-dBloctave filter and 4-ms sampling and graph the alias noise versusfrequency. 9.6 The sampling theorem applies to uniform sampling. Supposethe spacing between samplesalternatesfrom a to 3a, that is, the impulseresponseof the s a m p l i nigs [ . . . , 6 , 6 , 0 , 0 , 6 , 6 , 0 , 0 , . . . ] . W h a ti s the effecton aliasing? 9.7 (a) Verify the equationfor the inversefilter for water reverberation,eq. (9.a0),by convolvingit with the equationfor water reverberation,eq. (9.38). (b) The spectrumof the water-layerfilter is shown in "singfig. 9.63for n: 1; the largepeaksoccurat the ing frequency."Sketchthe amplitude spectrumof the inversefrlter.(Hint: Time-domainconvolutionsuchas shown in eq. (9.39)correspondsto frequency-domain multiplication, eq. (9.28);the frequencyspectrumof the unit impulseis + l, that is, it is flat.) (c) Verify your sketch of the water-reverberationinversefilter by transficrming |,2R, R' ] el

l)fts-:z"t+

R2e-ranv^,

and calculatingthe value of the spectrumfor u : 0, v, and 2vr. 9.8 Four causal waveletsare given by a, : 12, -ll, b , : 1 4 , l l , c , : [ 6 , - 7 , 2 ] , a n dd , : [ 4 , 9 , 2 ] . C a l c u l a t e the cross-and autocorrelations+"0,6",, $,,, $,,, and $,, in both the time domain and in the frequency domain. 9.9 Fill in the valuesin table9.3.

o

rs)

rb) Show that 6(t) * g(t) : g(t) and 6, * 8, : 8,. rc ) Show that a boxcar of heighti and extendingfrom -u,, to *vo in the frequencydomain hasthe transform h boxr,ne+ ,4 sinc 2rvot : A(sin 2rvot)l2rvot, nhere I : 2hv^,the area ofthe boxcar. 9.4 Show that 4-ms sampling is sufficientto reproduce exactly a signal whose spectrumis confined to :he rangeuoto uo + 125 Hz. (Hint: Modify fig. 9.3 to it this signal.) 9.5 The standardaliasfilter suchas shownin fig.7.46 ras its 3-dB point at about half the Nyquist frequency

o N

z

0

nr

2v, Frequency

3v,

Fig. 9.63 Spectrum of a water-layer filter for a water-bottom reflection coe{icient of 0.5. If z is the water depth and Zis the water velocity, v, : Vl4z.

il

i i , I I I I

t: tt t t

DATA PROCESSING

344 Table9.3Digitalnotationandoperations t:-3

I

-

_ L

I

-

- l

/:0

I

-

T Z

/:+3

t:

i4

/=+5

-2, 1] [2,1, -2a, -3a. -

a, = b,: c. : d,:

4a_,

et =

Ta|_l

,1: t- 1,11. 8,: a,*J, 6,*: 6.. +t (t) $,. (t)

9.10 Show that the semblance,Sr: I, where S, is givenby eq. (9.59),when the N valuesof g,,are identical for all valuesof t in the interval Al. 9 . 1 1 ( a ) C o n v o l v 1e 2 , 5 ,- 2 , l l w i t h [ 6 , - 1 , - l ] . -2, l] with (b) Cross-correlate [2, 5, [6, l, 1]. For what shift are thesefunctionsmost nearly alike? ( c ) C o n v o l v e[ 2 , 5 , - 2 , 1 ] w i t h t - 1 , - 1 , 6 1 .C o m p a r e with the answerin part (b) and explain. ( d ) A u t o c o r r e l a t [e6 " - 1 , - l ] a n d [ 3 , - 5 , - 2 ] . T h e autocorrelationof a function is not unique to that function (see$15.5.6c),for example,other wavelets having the sameautocorrelationsas the precedingare -5,3land - 1 , 6 ] . . . . W h i c ho f t h e f o u r i s l-2, [-1, the minimum-delaywavelet? (e) What is the normalizedautocorrelationof [6, * l, - l]? What is the normalizedcross-correlationin part (b)?What do you concludefrom the magnitudeof the largestvalue of this normalizedcross-correlation? 9.12 (a) Of the four causalwaveletsgiven in problem 9.8, which are minimum-phase? (b) Find 4, * b, and a, * c,by calculatingin the time domain. (c) Repeatpart (b) exceptusing transforms. ( d ) F i n d a , * b , , *7 ' , . (e) Does the maximum value of a minimum-phase function haveto come at I : 0? (f) Can a minimum-phasewaveletbe zero at I : 0? 9.13 Using the waveletsWlz) : (2 - z)'(3 - ;)' and W.() : (4 - z') (9 - z.), calculatethe composite wavelets:W.,(z)+ Wr(z), W,() * zWr(zl, zW,(z) + Wr(z),andz tW,(z)+ Wr(z).Plot the compositewavelets in the time domain. The results illustrate the effectsof phaseshifts (note that all of the composite wavelets have the same frequency spectrum but differentphasesbecausemultiplication by z shifts the phase(see$15.5.6a). 9.14 The following wavelet is approximately m i n i m u m - p h a sI e l:, 1 4 ,5 , - 1 0 ,- 1 2 ,- 6 , 3 , 5 , 2 , 0 , - 1 , - 1 , 0 1 ( f i g . 9 . 6 a a )t,h e s a m p l i n gi n t e r v a lb e i n g2 ms. Use 4u: 2.0 km/s for the velocity in sand, (,, : 1.5 km/s for the velocity in shale,and reflectioncoefficients(scaledup and roundedoff) shale-to-sand= +0. I : sand-to-limestone. (a) Determinethe reflectionwaveshapefor sands0, 2,

4,6,8, and l0 m thick encasedin shale.(A thickness of 6 m is approximatelya quarter wavelength.) (b) Repeatfor the sand overlain by shaleand underlain by limestone. (c) Determinethe waveshapefor two sandseach 6 m thick and separatedby 4.5 m of shale,the sequence "tuned" situabeingencasedin shale;this illustratesa tion ($6.4.3). (d) Repeatparts (a) and (b) with the wavelet[6, I l, 1 4 ,1 4 ,1 0 ,5 , - 2 , - 1 0 ,- I l , - 1 2 , - 1 0 ,- 6 , 0 , 3 , 4 , 5 , 4, 3, l, 0l (fig. 9.64b), a minimum-phase wavelet stretchedout so that it has about half the dominant frequencyof the former wavelet.Comparisonwith the resultsof parts (a) and (b) illustratesthe effectof frequencyon the resolution. (e) Repeatparts (a) and (b) usingthe zero-phasewavel e t [ , l , - 1 , - 4 , - 6 , - 4 , 1 0 , 1 7 , 1 0 ,- 4 , - 6 , - 4 , -1, I, ll (fig. 9.64c),which has the samefrequency spectrumas the waveletusedin part (a). 9 . 1 5 T h e w a v e l e t[ - 0 . 9 5 0 5 , - 0 . 0 1 2 0 ,0 . 9 9 1 5 i]s n o t minimum-phase.How would you make it minimum-

0

l5 0 -15 l5 0 t <

Fig. 9.64 Determining a composite reflection. (a) Minrmumphase wavelet, (b) minimum-phase wavelet of lower frequency, and (c) zero-phasewavelet corresponding to part (a) (but timeshifted and with the polarity reversed).

PROBLEMS

34s

Fig. 9.65 End-on records from a model with lour horizontal larers of velocities: I490, 1895, 2215, and 2440 rnls. (a) Before \ M O c o r r e c t i o n ,a n d ( b ) a f t e r N M O c o r r e c t i o n .

phase,at the sametime changingit as little as pos_ sible?Give two methods. 9.16 Show that the result of passinga minimum_ phase signal through a zero-phasefilter is mixed pnase. 9.17 In $9.5,severaldeconvolutionmethodsweredescribed.List the assumptions of the clifferent methods, such as invariant waveletand randomn.r, oi tt . .._ flectivity or of the noise,that a sourcewaveletrs the sameas a waveletrecordednear the sourceor mea_ sured from a sea-floorreflection by a group offset a l'ewhundred meters,and so on. 9.1.8Assuming the signatureof an air_gunarray is a unit impulse, find the inversefilter loithe recorded rvavelet: [-12, -4, +3, + I j. How many termsshould the filter include to get lo/oaccuracy? 9.19 A sourceis located7 m below the baseof-the LVL..Given that Vr: 2.0 km/s, V, : 0 3 km/s, p" : 2.3 g/cm3,po.= 1.8g/cmr,A : 4 ms and that the're_ flectedsignalis [6, -7, -2.8, 5.6,- 1.6],find the originalwaveletusing:(a) the inversefilter of eq. (g.6T:b) e q .( 9 . 6 9 ) . 9.20 The ghost reflection from the surfaceacts as a notch filter for receiversplanted on the seafloor. {a) Graph the notch frequencyversuswater depth. (b) If air-gunsourcesarefiredat a depthof l0 m, what effectwill the ghost haveon the spectrum? 9.21 Showthat

= 0*"(r- i) (seethe derivationofeq. (9.73a)). 9 . 2 2 C o n s i d e rw a v e l e rA s : [], _2, 3] and B :13,

-2, tl.

(a) Plot cumulativeenergyas function of time. (b) Calculatethe three-elementWiener inverse filters assumingthe desiredoutput is (i) [1,0,0,0]and (ii)

[0,1,0,0];then apply the inverseoperatorsto eachof the wavelets. (c) Add white noisee : 0.01and e : 0.I and repear. 9.23 [nfig.9.32,the hump in the g- to l2 000_ft/s con_ tours might causesome concern. If it is known that this is the samesectionas that shownin fig. 9.29, what would the hump imply? How would you modify the hump to do a betterjob of stacking? 9.24 The horizon velocity analysis for horizon A shown in fig. 9.35 indicateshigher stackingvelociries on opposite sidesof the salt dome and low stackins velocitiesbelow the salt dome. Why? Does this havi geologicalsignificance? The streamerwas about 4 km long. Note that only the portion of the section from 2 to 4 s is shown, the upper 2 s of sectionhaving been cut off. 9.25 On a north-south line, the noise arriving from the south is confinedto the band V,< 6krn/s and the noise arriving from the north is in the band V = 3 km/s. (a) Given that Ax : 50 m, sketchanfk plot such as fig. 9.36b. (b) Repeatfor Ax : 25 m. (c) Calculate/(x, r) for parts (a) and (b) (seeeqs.(9.96) and (9.97)). 9 . 2 6 G i v e nt h e w a v e l e [t 1 0 , - 8 , 0 , 9 , - 1 1 , 6 , 0 , - 7 , 12, -5,0,01, calculatethe following: (a) The quadraturefunction, g,(l). (b) f(t) (add multiplesof n to obtain a monotonically increasingfunction). (c) h(t), A(t), and v(t) at n: 4. Take A : 4 ms. 9.27 Provethe following: (a) The equationof a diffraction curve (curveof maximum convexity;seefig. 9.51)is -- 2 _ - 2- \ _ __ 2 L 0 ,

where O is the origin and 2,,: OP.

346 (b) The unmigrated reflection is tangent to the diffraction curve. (c) The coordinatesof P and the slope of the wavefront at P (hence,the dip also) can be obtained from the recordeddata. 9.28 Show that the coordinatesystem(x, z, t*) in eq. (9.120)in effect "rides along on an upcoming wavefront." 9.29 Derive the finite-differencemigration equation, eq. (9.124). 9.30 Interpret the faulting in figs. 9.58aand 9.58b to seehow much improvement migration makes. "de9.3f (a) The operatorf,: l- l, +ll is calledthe rivative operator"; explain why. (b) What is the integral operator? 9.32 Figure 9.65 shows three reflectionsbefore and after NMO removal. Explain (a) the broadening of the waveletsby the NMO corrections;(b) why the reflectionsdo not have straight alignmentsafter NMO correction.

Referenccs Anstey, N. A. 1964.Correlation techniques A review.Geophys. Prosp.,l2z 355 82. Anstey, N. A. 1970. Seismic Prospecting Instruments, VoL l: Signal Characteristics and Instrument Specificutions. Berlin: Gebriider Borntraeger.

DATA PROCESSING

Crump, M. D. 1974.A Kalman filter approachto the deconvo39: I 13. lution of somesignals.Geophysics, Denham. L. R., R. A. R. Palmeira,and R. C. Farrell. 1985. The zero-offsetstack. Paper read at the 55th SEG Annual Meeting. Deregowski,S. M. 1986.What is DMO (dip moveout)?Flrst Break,4(7):724. Fail, J. P, and G. Grau. 1963.Les filtres en eventail.Geophys. Prosp.,ll:131 63. Finetti.I.. R. Nicolich,and S.Sancin.1971.Reviewof the basic theoretical assumptionsin seismicdigital filtering. Geophys. Prosp., 19:.292-320. Flinn, E. A., E. A. Robinson,and S.Treitel.1967.Specialissue on the MIT GeophysicalAnalysisGroup reports.Geophysics, 32:41I 535. perForel,D., and G. H. F. Gardner.1988.A three-dimensional spective on two-dimensionaldip moveout. Geophysics,33: 604 t0. Gardner, G. H. F., and L. Lu. 1991. Slant'Stack Processing, GeophysicalReprintSeries14.Tulsa:Societyof Exploration Geophysicists. Garotta,R. 1971.Selectionof seismicpickingbasedupon the dip moveout and amplitude of each event. Geophys.Prosp., 19235770. Garotta,R., and D. Michon. 1967.Continuousanalysisof the Geocorrections. velocityfunctionand of thenormal-moveout phys.Prosp.,15: 584 97. Gazdag,J. 1978.Wave-equationmigration by phaseshift. Geophysics,43t 1342-51.

Backus. M. M. 1959. Water reverberations Their nature and 24:233- 61. efimination. Geophy.sics,

1948.Migrationof seismicdata Gazdag,J., and P. Squazzero. s, 49:.124-31. by phaseshift plus interpolation.Geophysic

Bancroft, J. C. I 99 I . A Practical Understanding oJ Migration and Dip Moveout, SEG Course Note Series.Tulsa: Society of Exploration Geophysicists.

Hagedoorn,J. G. 1954.A processof seismicreflectioninterpreProsp,2:85-127. lalion. Geophys.

Barr, F. J., and J. I. Sanders.1989.Attenuation ofwater-column reverberations using pressure and velocity detectors in a water-bottom cable. Paper read at the 59th SEG Annual Meeting. Berryhill, J. R. 1979. Wave-equation datuming. Geophysits,442

132939. Bois, P., and M. la Porte. 1970. Pointe automatiqte. Geophys. Prosp.,18:489-504. Bolondi, F., F. Loinger, and F. Rocca. 1982.OITsetcontinuation of seismic sections. Geophys.Prosp., 302813 28. Bracewell, R. 1965. The Fourier Transform and Its Applications. New York: McGraw-Hill. Chun. J. H.. and C. A. Jacewitz. 1981. Fundamentals of frequency-domain migration. Geophysics,46:'7 17 33. Claerbout, J.F. 1976. Fundamentals of Geophysical Data Pro' cessrrg. New York: McGraw-Hill. Clarke, G. K. C. 1968.Time-varying deconvolution filters. Geophysics, 33:936-44. Clayton, R. W, and G. W McMechan. 1981. Inversion of refraction data by wavefield continuation . Geophysics,'16: 860 8.

Hale,D. 1984.Dip movementby Fouriertransform.Geophy.sic.s, 49:741-5'1. SEGCourseNote Series Hale,D. 1991.Dip MoveoutProcessrrlg 4. Tulsa:Societyof ExplorationGeophysicists. Hatton,L., K. Larner,and B. S. Gibson.1981.Migrationof seismic data from inhomogeneousmedia. Geophysics,46:. 75t 67. Hatton, L., M. H. Worthington,and J. Makin. 1986.Seismic Data Processing;Theoryand Practice.London: BlackwellScientific. Hosken,J. W. J., and S. M. Deregowski.1985.MigrationstratProsp.,33: 1-33. egy.Geophys. Time migration,someray theoreticalaspects. Hubral, P. 19'77. Prosp.,25: 728-45. Geophys. Theodor migration.In Festischrift Hubral, P 1980.Slant-stack Krey,pp.72 8. Hanover,Germany:Prakla-Seismos. Jain,S., and A. E. Wren. 1980.Migrationbeforestack:Proce45:204-12. dure and significance.Geophysics, Judson,D. R., J. Lin, P S.Schultz,and J.W. C. Sherwood.1980' 45l.361-'75. Depth migration after stack.Geophysics,

Cochran, M. D. 1973. Seismic signal detection using sign bits. Geophy sics, 38:. 1042-52

Analysisin Geophysics. Kanasewich,E. R. 1987.TimeSequence Universityof AlbertaPress.

Cook, E. E., and M. T. Taner. 1969, Velocity spectra and their use in stratigraphic and lithologic differentiation. Geophys. Proso..17:433-48.

Kjartansson,E. 1979.Modeling and migrationby the monchromatic 45-degreeequation, Stanford Exploration Project ReportNo. 15,StanfordUniversity,Stanford,Californra.

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Kleyn, A. H. 1977.On the migration of reflection time contour maps. Geophys.Prosp., 25: 12540. Kunetz, G., and J. M. Fourmann. 1968. Efficient deconvolution of marine seismic records. Geophysics,33: 412 23. Larner, K. L., L. Hatton, B. S. Gibson, and I. C. Hsu. 1981. Depth migration of imaged time sections. Geophysics, 46,:, 734 50. Lee, Y W. 1960. Statistical Theorv of Communicatrons New York: John Wiley. Levin, F K. 1971.Apparent velocity from dipping interface reflections. Geophysics,36: 510-16. Levin, F. K., and P M. Shaw. 1977.Peg-leg multiples and dipping reflectors. Geophysics,42:957 81. Lines, L. R., and S. Treitel. 1984. A review of least-squaresinversion and its application to geophysical problems. Geophys. Prosp.,32z159-86. Lines. L. R., and S. Treitel. 1985. Inversion with a srain of salt. Geophysics,50: 99-109. \larsden, D. 1993. Static corrections A review. The Leadins Edse, l2(2):43-9. ll5 20. \{iddleton, D., and J. R. B. Whittlesey. 1968. Seismic models and deterministic operators for marine reverberation.Geophysr c s .3 3 : 5 5 7 8 3 .

Satlegger, J. W., P. K. Stiller, J. A. Echterhoff, and M. K. Hentschke. 1980. Common-offset-plane migration. Geophys. Prosp. , 28: 859-7 1. Schneider,W. A. 1978.Integral formulation for migration in two dimensions and three dimensions. Geophysics,43: 49-76. Schneider,W. A., and M. M. Backus. 1968. Dynamic correlation analysis.Geophysics,33:105 26. Schneider, Vl A., E. R. Prince, and B. F. Giles. 1965. A new data-processing technique for multiple attenuation exploiting differential normal moveout. Geophysics, 30: 348 62. Schultz, P. S., and J. W C. Sherwood. 1980.Depth migration before stack, Geophysics,45:376 93. Sheriff, R. E. 1978. A First Course in Geophysical Erploration and Interpretatlon. Boston: International Human Resources Development Corp. Sheriff, R. E. 1991. Encyclopedit Dit'tionary of Exploration Geophysits, 3d Ed. Tulsa: Society of Exploration Geophysicists. Sherwood, J. W. C., and A. W Trorey. 1965. Minimum-phase and related properties of the response of a horizontally stratified absorptive earth to plane seismic waves.Geophysics,30:

t9t-7. Silverman, D. 1967. The digital processingof seismic data. Geophysics,32:.988 1002.

\lillahn, K. O. 1980. In-seam seismics: Position and develooment. Prakla-SeismosReport,80(2 and 3): l9 30.

S m i t h , M . L . , J . A . S c a l e s ,a n d T . L . F i s c h e r . 1 9 9 2 . G l o b a l search and genetic algorithms. Leading Edge, ll:.22 6.

\aess, O. E. 1979.Attenuation of diffraction noise through very long arrays. Expanded Abstracts, 49th Annual International SEG Meeting, p.32.

Stewart, R. R. 1991. Exploration Seismic Tomography Fundamentals,SEG Course Note Series3. Tulsa: Society of Exploration Geophysicists.

\aess, O. E., and L. Bruland. 1985. Stacking methods other than simple summation. ln Devebpments in GeophysicalExplortttion Methods 6, A. E. Fitch, ed., pp. 189ff Amsterdam: Elsevier.

Stoffa, P. L., P. Buht, and G. M. Bryan. 1974. The application of homomorphic deconvolution to shallow-water marine seismology. Geophysit's,392 401-26.

\eidell, N. S., and M. T. Taner. 1971. Semblanceand other coherency measures for multichannel data. Geophysi<,s,36: :82 9'7.

Stoffa, P L., P. Buhl, J. B. Diebold, and F. Wenzel. 1981.Direct mapping of seismic data to the domain of intercept time and ray parameter - A plane-wave decomposition. Geophysics,46: 255 67.

Otis. R. M., and R. B. Smith. 1977. Homomorphic deconvolution by log spectral averaging. Geophysit's, 42:

n46 57.

Stoffa, P. L., and M. K. Sen. 1992. Nonlinear multiparameter optimization using genetic algorithms: Inversion of plane-wave seismograms.Geophysics,56: 1794 810.

Paturet, D. 1971. Different methods of time depth conversion '.rith and without migration. Geophys.Prosp., 19: 27 4l .

Stolt, R. H. 1978. Migration by Fourier transform. Geophysics,

P.rulson,K. V., and S. C. Merdler. 1968. Automatic seismic relection picking . Geophysic's, 33: 43 l-40.

Taner,M. T. 1976.Simplan,simulatedplane-wave exploration. Paperreadat the 46th Annual SEG Meeting(abstractin Geophysics,42: 186-7).

Peacock,K. L., and S. Treitel. 1969.Predictive deconvolution Theory and practice. Geophysics,34: 155-69. R.trbinson,E. A. 1983. Migration of GeophysicalData. Boston: International Human ResourcesDevelopment Corp. Rrrbinson,E. A., and S. Treitel. 1964.Predictive decomposition ,.i time series with applications to seismic exploration. Geo: h t s i c s , 3 2 : 4 1 88 4 . Robinson, E. A., and S. Treitel. 1967. Principles of digital Wierer filtering. Geophys.Prosp.,15:311-33. L.rbinson, E. A., and S. Treitel. 1980. GeophysicalSignal Analy:-..Englewood Cliffs, N.J.: Prentice Hall. 3.:rssell,B. H. 1988. Introduction to Seismic Inversion Methods. SEG Course Note Series2. Tulsa: Societv of Exoloration Geo:hl sicists. S,tlegger, J. W, and P. K. Stiller. 1974. Section migration :etore stack, after stack, or in between. Geophys.Prosp.,22: :.- 314.

43:2348.

Taner,M. T. 1980.Long-periodsea-floormultiplesand their suppression.GeophysProsp.,28; 30 48. Taner,M. T., and F. Koehler.1969.Velocityspectra:Digital computerderivationand applicationsof velocity functions. Geophysics,34: 859 81. Taner,M. T, and F. Koehler,1981.Surface-consistent correc17 22. ions. Geophysics,461 Taner.M. T.. F. Koehler.and K. A. Alhilali. 1974.Estimation and correctionof near-surfacetime anomalies.Geophysits, 39:441-63. Taner,M. T., F. Koehler,and R. E. Sheriff.1979.Complexseismic traceanalysis.Geophysics,44z l04l 63. Taner,M. T., and R. E. Sheriff.1977.Applicationof amplitude, frequency,and other attributesto stratigraphicand hydrocarbon determinafion. In SeismicStratigraphy- Applications to Hydrocarbon Exploration, C. E. Payton, ed., pp. 301-28.

tl'

L

, I I

t ,

. t

348

DATA PROCESSING

AAPG Memoir 26. Tulsa:American Associationof PetroleumGeologists.

enlatedannealingstaticscomputationusingan order-based 56: 1831-9. ergy function. Geophysics,

Telford,W M., L. P. Geldart,and R. E. Sherifl.1990.Applied Geophysics, 2d ed. Cambridge, U.K.: Cambridge University Press.

Tulsa:Societyof ExploraWebster, G. M. 1978.Deconvolution. tion Geophysicists.

Treitel,S.,J. L. Shanks,and C. W. Frasier.1967.Someaspects 789 900. offan filtering.Geophysics,32: Vasudevan. K., W. G. Wilson,and W. G. Laidlaw.1991.Simu-

Yrlmaz,O. 1987.SeismicData ProcessingTulsa:Societyof ExplorationGeophysicists. Yilmaz, O., and J. F Claerbout.1980.Prestackpartial migras, 45: 1753-79. tion. Geophysic

l0

Geologicinterpretation of reflection data I

I 1

I

Overview [nterpretation.as we use the word in this chapter,involvesdeterminingthe geologicsignificanceof seismic data. This necessarilyinvolvesgeologic terminology and we follow the usagein Batesand Jackson(1987). Interpretation also sometimesincludes data reduction, selectingevents believedto be primary reflections, and locating the reflectorswith which they are associated.Indeed,a number of decisionshave to be made in data processing,acquisition,and evenin the initial planning of a surveythat prejudicethe geologic conclusions and thus could be legitimatelyincludedas part of interpretation. There are few books that concentrateon the geologic interpretation of seismic data; we cite Fitch (1976);Anstey (1977, 1980a,1980b);Sheriff(1980); McQuillin, Bacon, and Barclay(1984);Badley and ; a d l e y( 1 9 8 5 )G ; r i e sa n d D y e r ( 1 9 8 5 ) ; A n s t e y( 1 9 8 4 )B and Brown (1991).Numerousscientificsectionsand their structural/stratigraphic interpretations are shownin Bally (1983-4, 19879). A numberof interpretation papersare publishedin The Leading Edge, and the Societyof Exploration Geophysicistsperiodically publishesreprintsof these(SEG, 1989-92). It is rare that the correctnessor incorrectnessof an interpretationcan be ascertainedbecausethe actual geologyis rarely ever known in adequatedetail. The testofa good interpretationis consistency ratherthan (Anstey,1973).Not only must a good incorrectness terpretationbe consistentwith all the seismicdata, it also must be consistentwith all that is known about the area,including gravity and magneticdata, well information and surfacegeology,as well as geologicand physicalconcepts. One can usuallybe consistentand still havea choice of interpretations,the more so when data are sparse. The interpreter should explore various possibilities, but usually only one interpretation is wanted, that which offers the greatestpossibilitiesfor significant profitable hydrocarbon accumulation(assumingthis is the objective).An interpretermust be optimistic, that is, he must find the good possibilities.The optimistic interpretationis often preferableto the "most probable,"becausethe former will probablycauseadditional work to be done to test (and perhapsmodify) the interpretation,whereasa nonoptimistic interpretation may result in abandoning the area. Management is usually tolerant of optimistic interpretations that are disprovenby subsequentwork, but failing to 349

recognize a possibility is an "unforgivable sin." It should be noted that "success"or "failure," that is, finding or failing to find hydrocarbonsin commercial quantity,is often a poor test of interpretationbecause many factors critical to commercial accumulations cannot be predictedfrom seismicdata. It may also be noted than an optimistic interpretationis not always what is sought.In engineeringand coal seismicwork, for example,one is apt to want to know the worst possible case so that these eventualitiescan be investigatedfurther. Seismic data are usually interpreted by geophysicists or geologists.The ideal interpreter combines training in both fields.He fully understandsthe processesinvolved in the generationand transmissionof seismicwaves,the effectsof the recordingequipment and data processing,and the physicalsignificanceof the seismicdata. His geologicexperiencehelpshim assimilate the massof data, some of it conflicting,and arrive at the most plausiblegeologicpicture.Unfortunately,not all interpretershave the requisite knowledgeand experiencein both geologyand geophysics, and often the next best alternativeis to have a geophysicist geologist team working in close cooperation. Deducing geologicsignificancefrom the aggregate of many minor observationsteststhe ingenuity of an interpreterand his in-depth understandingofphysical principles.For example,downdip thinning of reflection intervalsmight result from a normal increaseof velocity with depth as well as from sedimentthinning, and flow of salt or shalemay causeillusory structure on deeperhorizons.Geometriclocusingproducedby reflectorcurvaturecan produce various effects,especially if the data havenot beenmigratedcorrectly,and energy that comes from a source located off to one sideof the line can interferewith the patternsof other reflectioneventsto produce effectsthat might be interpretederroneously,unlesstheir true nature is recognized.Improper processinglikewisecan createopportunities for misinterpreting data (Tucker and Yorsten,1973). Inasmuchas our interpretationobjectiveis usually locating hydrocarbonaccumulations,this chapterbegins with a summaryof conceptsabout the generation and migration of hydrocarbonsand the typesof situations that trap hydrocarbonaccumulations. A section on interpretation procedures($10.2)includes both discussionsof the philosophy of seismic

GEOLOGIC INTERPRETATION OF REFLECTION DATA

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pectscan also lead to misinterpretation.However,as 3-D work becomesmore common (seefig. 1.21)'the importance of 3-D aspectsis now being recognized more and more (seealso chaP.12). Stratigraphic interpretation involves delineating which representdifferent deposiseismicsequences, tional units, recognizingseismicfaciescharacteristics, which suggestthe depositionalenvironment,and analyzing reflection character variations to locate both stratigraphic changes and hydrocarbon accumulations. Three-D work is especiallyimportant in recognizing stratigraphic features by their distinctive shapes.Hydrocarbon accumulationsare sometimes indicatedby amplitude,velocity,frequency,or waveshapechanges($10.8).The variation of amplitude with angle (or with offset)is one of the newer hydrocarbonindicators(HCl). New roles are being played by seismicmethods in the delineation,description,and surveillanceof reserfurther in $14'4and 14.5. voirs.Theseare discussed Seismicdata are also proving usefulin crustal studies,as aids in indicatingand delineatingdeeperfeawith sedimentary turesthan thoseusuallyassociated exploration. basin

The relation between liquid and gaseoushydrocarFig. l0.l bons generatedby temperature and depth of burial' assuming a geothermal gradient of 22'Clkm. (From Batzie and Wang'

l0.l Basic geologic concepts

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The interpretationof seismicdata in geologicterms ls the objectiveand end productof seismicwork. However,beforediscussingthis most importantand critical phase of interpretation, we shall review briefly somebasicgeologicconceptsthat are fundamentalin petroleumexploration. Petroleumis a resultof the decompositionof plant or animal matter in areasthat are slowly subsiding. Theseareasare usuallyin the seaor alongits margins in lakesor occasionally in coastallagoonsor marshes, inland swamps.Sedimentsare depositedalong with the organic matter, and the rate of deposition of the sedimentsmust be suliciently rapid that at least part of the organicmatter is preservedby burial beforebeing destroyedby decay.Restrictedcirculation and reducing (ratherthan oxidizing)conditionsfavorablefor hydrocarbon preservationare found in the deeper portions of both marine and lacustrine waters. As time goeson and an areacontinuesto sink slowly (becauseof the weight of sedimentsdepositedor because of regional tectonic forces), the organic material is buried deeperand henceis exposedto higher temperEventually,chemicalchangesreaturesand pressures. generation of petroleum,a complex,highly sult in the variablemixture of hydrocarbons,including both liquids and gases(part of the gas being in solution becauseof the high pressure).Temperaturein the earth generallyincreasesat a rate of 20-55'C/km, in some places (for example,Sumatra) by as much as 100'C/km.The habitat of liquid petroleumgeneration (fig. 10.1)is generally65 150'C, which is usuallyin the 1.5 3-km range.At depths of 3 6 km, reservoirs

interpretation,questionsto be answeredby the interpretation, and cautions to help avoid erroneousconclusions.Deducing the geologichistory of the area is one interpretationobjective.Well data haveto be related to seismicdata so that the interpretationis consistentwith both. Structural maps are commonly the foremost interpretation objective.The tectonic setting usually governs which types of structuresare presentand how structuralfeaturesrelateto each other, so a reviewof structural stylesprecedesdiscussionsof the evidences of various geologicalfeatures.Among the featuresexamined are faults, folded and flow structures,reefs, unconformities, channels, and stratigraphic traps. Workstations($10.2.7)are playing increasingroles in interpretation. Modeling providesa major tool in interpretation. Direct modeling, the making of a synthetic seismogram to show what should be expectedfrom a geologic model, helpsin understandingwhat seismicfeatures should be looked for as evidencesof sought-for geologicanomalies.Inversemodeling, the making of or sonic logs from seissyntheticacoustic-impedance mic data, aids in seeingthe geologic significanceof seismicwaveshapevariationsnear well control, especially in locating nearby stratigraphicchangessuggestedby well data. Lateral variationsin velocity can produce the illusion of unreal structural features.Velocity changes affect structural features much more frequently than usually realized.Unrecognizedthree-dimensionalas-

10.l. l Generationand migrationoJ'hydrocarbont

BASIC GEOLOGIC CONCEPTS predominantlycontain gasrather than oil, and at still greaterdepths,the temperatureis apt to be so high as to causegas to decompose. Sedimentaryrocks are porous, generallybeing de_ posited with about 45% porosity. As sedimentsare piled on top, the weight of the overburdencomoacts the rocks and the porosity becomesless (fig. 5.3.y. Someof the water that filled the intersticesin the rock (interstitialwater) escapes during the compactronpro_ cessuntil the pressureof the water equalsthat of the hydrostatichead correspondingto its depth of burial. If the formation watercannot escape,it becomesover_ pressured(see95.3.4). Petroleumcollectsin the pore spacesin the source rock or in a rock adjacent to the sourcerock, inter_ mingled with the remaining water that was buried with the sediments.When a significantfraction of the poresis interconnectedso that fluids can passthrough the rock, the rock is permeable.permeabilityp..-it. the gas,oil, and water to separatepartially becauseof their different densities.The oil and gas tend to rise, and they will eventuallyreachthe surfaceofthe earth and be dissipatedunlessthey encountera barrier (called a trap) that stops the upward fluid migration. Fracturingsometimesplaysan important role in the movementof fluids to the boreholesfrom which they can be produced. 10.1.2TypesoJ traps The essential characteristic of a trap is a porous,permeablebed (B in fig. 10.2a)overlainby an imperme_ ablebed (l), which preventsffuid from escaping.Oil and gascan collectin the reservoirofan antitline un_ til the anticline is filled to the spill pornl. Whereasfig. 10.2a is two-dimensional,similar conditions must hold for the third dimension,the structureformins an invertedbowl. The spill point is the highestpoin-tat which oil or gas can escapelrom the anticline;the contour through the spill point is the closingconrour, and the verticaldistancebetweenthe spill point and the highestpoint on the anticlineis the amountof closure.ln fig. 10.2bthe closingcontour is the -20g5_m contour and the closureis 30 m. The quantitv of oil that can be trappedin the structuredependsupon the amount of closure,the area within the closingcon_ tour, and the thicknessand porosityof the reservoir beds. Figure 10.2amight be the cross-section of an anricline or a dome, and the trap is called an unticlinal trap. OLher structural situations can also provide traps. Figure 10.2cshowslault trapsin which perme_ able beds,overlain by impermeablebeds,are faulted againstimpermeablebeds.A trap existsif thereis also closure in the direction parallel to the fault, for ex_ ample, becauseof folding, as shown by the contours in fig. 10.2d.Figure 10.2eshowspossibletrapsassocr_ ated with thrust faulting. Figure 10.2f shows a .etratigraphic trap in which a permeablebed grades into an impermeablebed. as

351 might result where a sand gradesinto a shale.Sometimes permeablebeds gradually thin and eventually pinch out to form pinchout traps. (Stratigraphic traps of various types are also shownin fig. 10.42.)Closure must also existat right anglesto the diagram,possibly becauseof folding or faulting. Many traps involve both stratigraphicand structuralaspects. Figure 10.2g shows unconformity traps, which may result from permeablebeds onlapping an uncomformity or bedstruncatedby erosionat an unconformity (seealso fig. 10.42).If the permeablebedsare overlain by impermeableones and if there is closure at right anglesto the diagram, hydrocarbonscan be trapped at the unconformity. Figure 10.2hshowsa limestonereef that grew up_ wards on a slowly subsidingplatform. The reef was originally composedof coral or other marine animals with calcareousshellsthat grow prolifically under the proper conditionsof water temperatureand depth. As the reef subsides, sedimentsare depositedaround it. Eventuallythe reefstopsgrowing,perhapsbecauseof a changein the water temperatureor the rate of subsidence, or becausethere is so much sediment suspendedin the waterthat sufficientlight to support reef growth no longer exists,and the reef may be buried. The reef material is often highly porous and covered by impermeablesediments.Reefssometimeproduce archingin overlyingsedimentsbecauseof differentialcompactioneffects,the reef being generallylesscompactiblethan the sedimentson either side of it. The reef may form a trap for hydrocarbonsgeneratedin the reef itself or flowing into it from other beds. Figure 10.2irepresentsa salt dome formed when a mass of salt flows upwards under the pressureresulting from the weightof the overlyingsediments.Below I 1.5km, salt is apt to be buoyantcomparedto densersurroundingsediments,which tend to subside as the basinsubsides,whereasthe salt tendsto remaln at roughly the same depth. Eventually,the salt becomes cut off from the underlying mother salt and may take on a teardrop shape,overhangingdeeper sediments.The salt dome bows up sedimentarybeds, producesfaulting, and affectsthe nature of the beds being deposited. Consequently,traps may be producedover or around the sidesof the dome and sometimeswithin cavitiesin caprockover the salt, the trapping sometimesresulting from dip reversal,faulting, unconformities,or stratigraphicchanges.Becausesalt is impermeable,hydrocarbonsalso may be trapped beneaththe salt. The primary objectiveof a seismicsurveyfor hydrocarbons is usually to locate structuressuch as those shownin fig. 10.2.However,many structuresthat provide excellenttraps do not contain oil or gas in economic quantities.We also try to derivefrom the seismic data as much information as possibleabout the geologichistory of the area and about the nature of the rocks in an effort to form an opinion about the probability of encounteringpetroleum in the structures that we map.

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tion through fault traps; (d) map of the middle permeable bed (f) stratrin (c)l (e) possibletraps associatedwith thrust faulting: (8) graphic tiaps produced by lithologic change and pinchout; the above draping in and a reef (h) in trap a t.aps; in.onfot-ily reef; and (i) possible traps associatedwith a salt dome'

INTERPRETATIONPROCEDURES 10.2Interpretation procedures 10.2.I Fundamentalgeophysicalassumptions Seismicinterpretationgenerallyassumes(l) that the coherent events seen on seismicrecords oi ,rn p.o_ cessedseismicsectionsare reflectionsfrom acoustlc_ impedencecontrastsin the earth, and (2) that these contrastsare associatedwith bedding that represents the geologic structure. Thus, mapping the arrival times of coherent events is relatei to the geologic structure,and by allowing for velocity and migration effects,we obtain a map showing th; geologicstruc_ ture. We also assume(3) that seismicdetail (wave_ shape,amplitude,and so on) is relatedto geologicde_ tail, that is, to the stratigraphyand the niture of the interstitialfluids;we examinethis assumptionfurther in910.7 and10.8. 10.2.2Collet.tionund examinationo/ tlatu (a) Introdaction. The interpreter gathers together all the data relevantto the interpretation,including geologic,well data, and so on. The relevantseismii data usuallyincludeseismicsections, a basemap,and velocity and other data from the field or seneratedin processing. Sometimesthe interpretation is donecon_ currentlywith the 1leldand processingwork, so that the interpreterreceives additionaldata whilecarrylng out the interpretation, and he may be able to feed back conclusionsfrom the preliminaryinterpretation so that field or processing procedures can bechanged or additional work can be carried out, in order to proveor disprovepointsthat are not resolved. Alternativewaysof interpretingdata are almostal_ wayspossible.This "inherentambiguity" existswith almostany data,althoughambiguityin seismicinter_ pretatlonis lessthan with most geophysical and geo_ logic data.Ambiguity arisesbecausedata are incom_ plete and/or inaccurate,and the best way to reduce ambiguityis to add more data.The addeddata misht be more seismicdata, but it also might be information from surface geology, wells, gravity measurements, and so on. The regionalgeologicsettingand concepts about the tectonicstressesto which the resion has beensubjectedshouldalsobe usedas a checkon seis_ mic information. As an example,one sometimesencountersdrsrup_ tions in a seismic reflection. If we explain this as causedby faulting,then we must determinewhat else the fault did. Where did it cut shallowerbeds.or did it die out? Where did it cut deeper beds,or was the lault displacementabsorbedby flowagein mobile salt or shalesediments, or did the fault soleout into the bedding plane?Where is the fault on parallel and in_ tersectinglines,or did it die out laterally?Is the fault a normal fault indicating extensionor a reversefault indicating compression?An interpretationcannot be regardedas completeuntil such questionshave been answeredas completelyas possible.A fault that dies out both shallowerand deeper is difficult to justify

353 (though occasionallythis is the correct inrerprera_ tron). Faultsthat havenot producedeffectson nearby lines may also be difficult to justify. (b) Examining sections. One of the first tasks of an lnterpreterrs to examinethe data for evidencesof mis_ location (do sectionstie properly?)or improper acqui_ sition or processing.Such an examination,'although not conclusive,often uncoversgross errors. Unmi_ grated seismictraces at the intersectionsof seismic linesought to copy.When they do not, mislocationor mislabelingof one or both of the lines is a possible explanation, but differencesin acquisition or pro_ cessingtechniquesalso providepossibleexplanatrons. The title blocks of the sectionsshould be examrned to see what differencesexist. Different_sizearravs. different mixes of offset distances,or different pio_ cessingproceduresmay have resultedin noisecontri_ butionsbeingdifferent.Lack of full multiplicityat the ends of lines or where sourceor geophonespacingis irregular (possiblybecauseof aciesi problems)mav ha.veaffecteddata quality.Featuresthat line up verti_ callyon unmigratedsectionsareespecially suspectbe_ cause geologic features are usually not vertical, whereasthe effectsof staticserrors often are. Occa_ sionally,filesare mixed up and data are assembledin_ correctly.The variousdataelementsshouldbe consis_ tent; if the velocity were assumedto vary. are the assumptionsconsistentwith the structure and the characterof the sections? Are certaindata that show on sectionsmade as intermediatestepsin the pro_ cessingmissingor changedon the finai sections? Un_ explaineddifferencesor departuresfrom what is seo_ logically reasonableshould be investigated, so ihat geologicsignificance is not attributedto errorsin the data. Figure 10.3showsboth ends of a typical seismic section,includingthe data block. The data block is often subdividedinto parts listinginformationabout Iineidentification, dataacquisition,and processine. A g e n e r a l i z el idn ed i r e c t i o na n d h o r i z o n t asl c a l e, o ! U . given.Wherethe horizontalscaleis not indicated,it can be determinedby countingthe numberof traces per centimeter,the trace spacingbeing half the geo_ phone group spacing(assumingno horizontalcom_ positing). The locations at which velocity analyses were run are usually indicated,often with the results of the analysestabulatedas time-velocitypairs.These should be examinedfor consistencyalong the line. The locations of changesin line direction or abrupt surface changes (such as elevation differences or .llng.:r in water depth) should be noted for their pos_ sibleeffectson reflectionquality or attitude.Irregular_ ities in coverageare common in land data becauseof surfaceor accessproblems;theseoften show as irres_ ularitiesin the first-breakpatternsand they may als-o affect reflectionquality and the apparent attitude of reflections.The multiplicity involved in each trace ls sometlmesshown by encoding at the bottom of the section,providing a key to irregularitiesofcoverage.

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"V" at the top of the section' and analysesare indioated by a the tabulated data give stacking (labeled RMS). interval' and a v e r a g ev e l o c i t i e s( s e ee q s . ( 4 . 2 6 ) ,( 5 . 2 5 1 ,a n d ( 5 . 2 0 ) )a s s u m i n g horizontal velocity layering. The solid triangle above SP 27 indicates an intersecting seismic line; TVF stands for time-variant filter, WD fbr water depth, RMS gain means that the RMS amplitu
The vertical scale shown on the section is usually linear in time. Depth equivalentsare sometimesgiven, but theseare only intendedto be approximate;depth determinationscan be made more accuratelyby measuringthe arrival times and convertingtheseto depths using the appropriatevelocity relations.Where processingor displayparametersare changedwith arrival time, the locationsof changesshould be noted so that changesin quality produced by the changesin processingare not interpreted as geologic changes.If changeshavebeenmade in the midst of the objective section, or if the horizon being mapped varies in depth so as to crossthe zone of changedparameters, specialcare has to be employedto avoid possiblemisinterpretations.

(c) Interpretation approaches. Interpretation involves building a model of the prospect area in the interpreter'smind. Some interpretershang all their data on their walls, thus surroundingthemselveswith the data so that they can look from one sectionto another to see interrelationshipsbetter. They develop much of their interpretationsitting back and pondering about which ideas are feasible,rather than being busy all the time timing events,transferring data to basemaps, and drawing contours.lmagination is required in interpretation and it takes time to develop an interpretationthat leadsto new discoveries. Two basicallydifferentapproachesare madeto seismic interpretation:the one focuseson objectives;the other gradually builds up a completeinterpretation.

INTERPRETATIONPROCEDURES

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Often only a few reflectorsare consideredto be of interestbecauseit is alreadyknown that the only prospectlvereservoirs are in one part ofthe section.Such areasusuallycontainwellsin which formationsof interest are identified and then related to seismicsections,eitherby syntheticseismograms or, moresimply, by usingvelocity to relatedepth in the wellsto arrival time on the seismicsections.The associatedeventson the seismicsectionsare followed throughout the region to give structuralrelationships,looking for faults that displacethe reflectionsand watching for reflection character changesthat may indicate thickness changes,sand pinchouts, patch-reef growths, other stratigraphic variations, or hydrocarbon accumulations. The alternativeapproachis to developa more complete interpretation of an entire section. An rnterpretergenerallystartswith the most obviousfeatures, usuallythe strongestreflectionevent or the reflection event that possesses the most distinctive character. and follows this featureas long as it remainsreliable. When the feature being followed deteriorates or changescharacterso that it is not clear what is happening,it is dropped ratherthan "pushed" or extrapolated beyond its region of reliability, and the interpreter returns to it subsequently after he has

developedother featuresof reasonablereliability.By following featuresin the order of their reliability,overall interpretationis developedbeforeattention ls concentratedon the objectivereflections.This type ofapproach usually leadsto a better interpretation,but it is much more time-consuming.An interpretation of the sectionshownin fig. 10.4,for example,might start with mapping of the strong refleclor AA', djstinguishedin part by the following zone of poor reflections .BB'.ReflectorAA' is cut by a normal fault, and following this fault onto intersecting seismic lines might be the next step.Attention might then shift to mapping a shallowerhorizon, perhapsCC,, but here it will be difficult to be sure one is staying with the samereflector.Perhapsone would try to map the base of the B-B'zone next; this will also involveuncertainties, and to resolvesome of theseit may be necessary to go back and revisethe parts interpretedearlier.The petroleumobjectivesare about the middle of the BB, portion, so a map will haveto be made here,but this map can be made more reliably with the aid of more certaln eventsalreadymapped. The interpretationof seismicsectionsis sometimes done from the top downward. Shallow featuressuch as channels,surfacerelief, local pocketsof gas, and irregular coveragebecauseof surfaceobstaclesmay

356

GEOLOGIC INTERPRETATION OF REFLECTION DATA

produce deeperartifacts,so that early recognitionof shallowfeaturesmay preventmisinterpretationof the deeper features.However,sometimesdeep structure such as basementfaulting has prejudicedsubsequent structure,so that it is better to proceedfrom the bot"best way" to tom upward. There is no automatically proceedwith an interpretation. I 0.2.3 Picking reflections Making a seismicstructuremap generallyconsistsof four operations:(l) selectingwhich events,and what point (peak, trough, zero crossing)on eachevent,are to be mapped(picking);(2) measuringthe arrival time of each pick (timing) and converting the values to depthsif a depth map is to be made ratherthan a time map; (3) writing the valueson a base map (posting); and (4) connectingthe postedvaluesto representthe structure (contouring). Other relevant informatlon such as well data, regionaltrends,anticlinal and synclinal axes,the location of gravity highs and lows, inof faultingor indicationsas to the terruptionsbecause relativereliability of posted values(grading)will also be postedon the basemaps. Reflectionsare usuallyidentifiedwith rock units by correlatingwith well-log data, often using a synthetic ($6.2.1)or verticalseismicprofile($13.4) seismogram as the relating device. Clearly, the amplitude and

phasespectraof the waveletsembeddedin both the seismic data and the synthetic seismogram must match. A match is often determinedby varying the spectra(especiallythe phasespectrumof the synthetic seismogram).The match over any appreciableportion of a seismictrace is hardly evergood enoughthat one can be absolutelyconfident that it is correct. A seismologist is apt to attribute matching problemsto errors in the seismicdata, unaccounted-fornoise being the most probablesource.However,the problemsmay also lie in the well-logdata as thesealso involvemeasurementuncertainties. Picking reflectionsis usuallybasedon following the same phasefrom trace to trace, usually a peak or a trough. Wherethe reflectioncharacterchangesbecaue componentbedsthickenor thin or changetheir lithologic makeup, routine following of the same phase may introduceerrors,and the interpretershouldthink about what he is trying to map and take appropriate liberties in interpreting the data. Where peaks (or troughs) split into two events (for example,in figs. 10.5dto 10.5f),whichway to follow the eventdepends on what the interpreter thinks is happening,for example, whetherthere was erosionaltopographyat an uncomformity so that the causeof local thick':ning

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Fig. 10.6 Changing character of unconformity reflections.(a) Polarity reversal as different beds subcrop at unconformity (courtesy of Grant Geophysical). (b) Weak and variable reflection at an angular unconformity (from Fitch, 1976: 68).

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I NTERPRETATION PROCEDURES lies below the unconformity surface or whether the thicknessof the basal membersabove the unconfor_ mity may be changing.The polarity of unconformity reflectionsmay evenreverseas differentbedssubcrop (fig. 10.6)or onlap the unconformity. 10.2.4Mapping reflectinghorizons The horizonsthat we draw on seismicsectionsprovide us with a two-dimensional picture only. A three_ dimensionalpicture is necessaryto determinewhether closureexists,the area within the closingcontour, the location of the highestpoint on the structure.and so on. To obtain three-dimensionalinformation. we usu_ ally run lines in different directions.Most reflection surveysare carried out along a more_or_less rectansu_ lar grid of lines,often with common midpoints at ihe intersectionsof lines to facilitatecorrelatingreflec_ tionson the intersecting profiles. Events picked on one section are compared with those on intersectingsectionsin order to identifv the samehorizons;identificationis made on the basisof character and arrival times. The horizons are now "carried" along the cross-lines,and, ultimately,along all linesin the prospectto the extentthat the qualitv o f t h e d a t ap e r m i t s . When a horizon can be carried all the way around a closedloop, we should end up with the samearnval time with which we started. This closingof loops pro_ vides an important check on reliability-When a loop fails to closewithin a reasonableerror iwhich depends mainly upon the record quality and the accuracyof the weatheringcorrections),the causeof the misclo_ sure (mistie) should be investigatedcarefully. Mi_ gratedsectionshaveto be tied by finding the samere_ flection on the intersectingsections;such tie oornts will be displacedfrom the midpoint by the amount of the migration on each of the lines.Often. misclosure is due to an error in correlatingalongthe line or from lineto line,possiblybecause ofinaccuratecorrections, a changein reflectioncharacter,or error in correlatine acrossfaults. When the dip is different on the twi sidesof a fault or the throw varies along the fault, an incorrectcorrelationacrossthe fault m;y result in misclosures(but not necessarily). After the sourcesof misclosurehavebeencarefullyexaminedand the final misclosure reduced to an acceptablelevel, the re_ maining misclosureis distributedaround the loop. Interpretationis usually done on migratedsections wheneverthese are available.Migrating usually im_ proves the signal/noiseratio, sharpensup fault evi_ dences,and clarifies features whose evidencesare eventshaving different dips, such as pinchouts.How_ ever,becausemigration is basedon only the comoon e n t so f d i p i n t h e i n - l i n ed i r e c t i o n( e i c e p tt o r j _ O data),dipping reflectionswill generallynot time tie at line intersections.It may be necessaryto locate the eventsbeing mapped on the unmigrated sectionsto ascertainthat they tie. Fault planesshould be picked as accuratelyas pos_

357 sibleon seismiclinesas an aid to identifying fault cuts seenon different lines involving the samefault. Fault planesare often locatedby the termination of reflec_ tions that have only small dips, so fault planesoften can be time tied at line intersectionswith sufficient confidenceto determinefault strike (fig. 10.29). After horizons have been carried on the sections, maps are prepared. For example, we might map a shallow horizon, an intermediatehorizonit roughly the depth at which we expectto encounteroil. ifinv is.present,and a deephorizon. We map on a bose^ap, which showsthe locationsof the seismiclines(usualiv be meansof small circlesrepresentingmidpoints) ani other features such as oil wells, rivers, shorelines. roads,land and political boundaries,and so on. Val_ ues representingthe depth of the horizon below the datum plane are posted on the map, usually at the midpoints (although,strictly speaking,they should be postedat the reflectingpoints on the respectivehori_ zo.ns).Whereasinterpretersoften time data at regular midpoint intervals,determiningand postingthe inter_ sectionsof contour values with the picked horizon eliminatesthe needto interpolatealong the lineswhile doing the contouring. Faultsthat havebeenidentified on the record sectionsare drawn on the map and the depth valuesare then contoured. Where unmigrated data are contoured, the unmi_ grated.mapcanbe digitizedand migratedusing map_ migration algorithms. The amount of contour smoothing to be done de_ pends on the assessment of how much noise there is in the data. One does not want to either smooth through real geologic featuresor contour the noise. The amount of noise clearly sets the minimum con_ tour interval that should be used. Depicting small structural featuresbelievedto be real generallysets the contour interval, but larger contour intervalsare often usedbecausesmall featuresmay not be signifi_ cant eventhough real. Dashedcontours interme-diate betweenintegral contours or using different contour intervalsfor differentparts of a map often conveysthe impression of structural relief better than use of a smalleruniform contour interval. After the structuralinformation hasbeenextracted, the next step is to work out as much as possibleof the geologichistory of the area. Fundamentalin this connection is the determination of the ages of the different horizons, preferably according tJ the geo_ logic time scale,but at least relative to one another. Often, seismiclinespasscloseenoughto wells to per_ mit correlatingthe seismichorizonswith geologicho_ rizons in the wells.Refractionvelocities1if aviilabte; may help identify certain horizons. The strongesr, most obvious, and easiestidentifiablereflectionsare often associatedwith unconformities.Occasionally,a particular reflection has a distinctive character that persistsover large areas,permitting not only it to be identified,but also other eventsby their relaiion to it. Notable examplesof persistentidentifiablereflections are the low-frequencyreflectionssometimesassoci_

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and lowermost part of the Jurassicindicates Iittle salt movement had occurred. The left dome began growing in the Jurassic(J), as evidenced by erosion of the top Jurassic;the right dome began growing in the Cretaceous (K). The major dome growth took place during the Lower Tertiary (TL).

INTERPRETATION PROCEDURES ated with massivebasementand the prominent re_ flection from the top of the Ellenburger,a limestone encounteredin Northern Texas. 10.2.5Deducinggeologichistory

359 tions actual flatteningcan be worthwhile. Obviouslv. migration before the flattening is necessaryand the flattenedhorizon should be selectedjudiciously. Inter_ pretation workstationsmake it easyto flatten a hori_ zon by simplesubtraction.Somealso permit removal of fault throws along the fault planesto avoid creating artifacts of faults vertically underneathfault restoral tions. Compaction effects and changes in velocity sincethe depositionofthe flattenedhoiizon shouldbe allowed for, but usually the information required to do this is lacking. The unravelingof the geologichistory of the arears . important in answeringquestionssuch as the follow_ rng: (a) How would the paleotopographyhaveaffected the stratigraphyand the lithology deposited?(b) Was the trap formed prior to, during, o, subsequentto the generationof the oil and gas?(c) Has the trap been tilted sufficientlyto allow any trapped oil to escape? (d) Did displacementof part of a structureby faultine occur beforeor after possibleemplacementof oltt etl though the seismicdata rarely give unambiguousanswersto such questions,often clues can be obtained that, when combined with other information such as surfacegeologyand well data, permit the interpreter mlke intelligentguessesthat improvethe probabil_ !o ity of finding oil. Alertnessto such cluesis ih. ..u.t" of seismicinterpretationand often the distinction be_ tweenan "oil finder" and a routine interpreter.

Seismicsectionsoften subdividenaturally into units. The boundariesbetweenunits are often tire better re_ flectors,and the units often have angular relationsto each other that indicate featuresof geologichistory: periods of tectonism,unconformities,transgressioni, and so on. The boundariesbetweenunits generallyin_ dicate a gap in geologic time and often the unit boundariesseparatesedimentsdepositedin different kinds of environments.Velocity and other seismic measurements, suchas of amplitudeor instantaneous frequency,and their variationsin the direction of the beddingyield additional information. Lithology and/ or stratigrapic situations are usually inferred from many evidencesthat are individually weak but that, taken together,make a coherentpattirn. Isopach maps, which show the thickness of sedi_ ments betweentwo horizons, are uselul in studying paleostructureand structural growth. Ideally, only one rock unit should be encompassedby the interval between horizons, but often the only horizons that can be mapped reliably are separatedby more than one rock unit, so that the resultingisopachmap may show more than one period of movement or more than one depositionaltrend. The interval betweenho_ 10.2.6Integrating well data into an interpretation rizons is often measuredin terms of two_waytrav_ eltime rather than thicknessin metersor feet,li Ueing Wells drilled in the area providegeologicinformatron implied that velocity-variationeffectsare minor com_ that must be consistentwith the interpretation.Bore_ pared to thicknessvariations.Isopachmaps are often hole logs (fig. 10.9)are interpretedto determinefor_ preparedby overlayingmapsof two horizonsand sub_ mation tops, lithology, depositionalenvironment,the tracting the contour valueswhereverthe contours on location of faults (fig. 10.10)and unconformitieswith one map crossthe contours on the other. The differ_ an indication of the amount of secti<--r missing,and encesare recorded on a blank map and then con_ so on. Well logs plotted linearly in time at the seismic toured. If isopachcontours show a irend toward in_ sectionscaleaid in correlating(fig. l0.ll). Although creasedthicknessin a certaindirection,it may suggest a borehole provides an opportunity for actuai mea_ that the region was tilted downward in this direction surements,the resultsavailableto a seismicinterpreter during the period of depositionor that the sourceof are usually interpretationsof measurements. When a the sedimentsis in this direction. Uniform thickness disagreementbetweenwell and seismicdata appears, of a folded competentbed indicatesthat the foldins both well and seismicdata should be reexaminedin came after the deposition,whereasdepositionprobal order to resolvethe problem. bly was contemporaneouswith the growth of an anti_ Synthetic seismogramsmade from well-loe data cline if the thicknessincreasesaway from the crest. providea meansof identifying reflectionswith lorma_ Growth during deposition is usually more favorable tion tops provided the waveletembeddedin the seis_ for petroleum accumulationbecausi it is more likely mic data is close to that used in manufacturing the for reservoir sandsto be depositedon the flanks of synthetic.Vertical seismicprofiles ($13.4)provide an structureswith evenslight relief. even better way of accomplishing this (Hardage, Pale.os^ections (palinspasticsections)can be made by r985). time shifting tracesto flatten somedistinctivehorizon Well information usually has to be projectedinto that can be assumedto havebeendepositedhorizon_ seismicdata, the wells not having sampledthe same tally; the objectiveis to show relationshipsas they ex_ subsurfacelocationsas the seismicdata. Even where yl._dut the lime of depositionof this horizon (seefigs. a wellheadis locatedon a seismicline, dip may result 10.7 and 10.8). In practice, such flattenins is often in the seismicwork seeinga different portion of the done in the interpreter'smind rather than b-yactuallv subsurfacethan the well. The projectionof well infor_ manipulating the data becauseof the cost of re_ mation involvesan interpretationso that the data are processing,but in areasof even moderatecomolica_ projected up- or downdip by the correct amount

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Fig. 10.9 Correlations between well logs within the same field. For each well, the left curve.indicates t[e Sp 1rpon,an".ru, potential) responseand the right curve resistivity. b.uerol cor.elu_ tion lines have been drawn and numbered. Some intervals are rhinner in one well than in another; the interpreter must decide *helher intervals are thinner because section is rnirsing 1as a result of faulting or an unconformity), becauseof stratigraphic r ariations, or becauseof miscorrelation. part of the i_q section r60 rn) is faulted out ofwell Cl 40 m ofthe 6_7 section is laulted irut of well 4 and horizon.5marks an unconformity, explaining rhe thicker 4 5 section in ,4. Obviously, otn., int"rp..iutrons are possibleand a seismicinterpreter should not regard well infor_ mation as infallible.

twhich may vary with depth in both magnitude and direction),making appropriateallowancetr faults or other featuresthat intervene.The seismicsection is us.uallyplotted in time, whereasthe well data are usu_ ally in depth,so that the choiceofan appropriate ve_ iocity for convertingthe one into the oiher iras to be maoe. Even when location problemsare not present, the lell information resultsonly from the rock within a tew centimetersof the borehole(which may havebeen .rlteredby the drilling process),whereasseismic data rncludecontributions from a large Fresnel-zone re_ gion. Well and seismicdata mayte plotted with re_ )pect to different data and time shifts may have been :ntroducedinto the seismicdata in ,."ording or proJesslng.Furthermore, most reflectionsare tle inter_ terencecompositesof severalreflections,and multi_ rles from shallow reflectorsor other noise may also .rffeclthe interference. In consequence. relatin! inter_ :rcesto specificseismiceventsiJnot easyand i"s often Joneincorrectly(seealsog6.2.I and 13.4.2\. .'rt2.7 Workstatiln,t \luch. seismic interpretation today is done inter_ :ctively at workstationsand the usebf workstations is :rpected to.increaserapidly (Brown, 1992). A work_ ,'rllior? consistsof a computer terminal, usually with :so fairly. large high-resolutionscreens,opon *hi.h l.ita can be displayed,often in cotor. ttre display is

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145 t39 t33 ).1

)-,

l2l 54'

lr5 109 54'

0 I l .t

{ 5 6

'7

IJ l0 I I

l : l.l l;l l 5 l(, l 1 lu l9 t0 ll tl l.l :1 t5 l() )1 l8 l9

.r0 .tl .tl -tl 35 -3() 37 ,18 39

Relating well logs to seismic data. The well is loFig. l0.ll "true" cated near the seismicline, the seismicdata are plotted at amplitude and are migrated, and the well logs are plotted at the same time scaleas the seismicsection. Some of the sandsas seen on the SPlog (indicated by excursions to the leit; see Telford, Geldart, and Sheriff, 1990: fig. I l.l2) seem to relate to specific

reflections,assuming the log section is slightly high with respect to the seismicsection. The sandsmarked with a dot are productive in the well. The sand marked.r is nearly the tuning thickness I/4 ($6.4.3),which partially explains why it produces a prominent reflection. (Courtesy of Conoco.)

providesone of the major advantagesof workstations, as it makes it easy to check portions of the data againsteach other to verify their consistency.An interpreter never has time to verify all aspectsof the consistencyof his interpretationand many more can be verified when it is easier and quicker to do so. Workstationsthus providemore consistentinterpretations than are made without them. An interpretationshowing picked horizons,faults, well-log ties, and other featurescan be superimposed on the data. The seismic(and other) data can be manipulated to time shift or color encodethem. Algorithms can be applied to carry out simple dataprocessingoperations ($9.13) or so that attributes

($9.11.4)may be displayed.The facility of being able to seethe effectsofvarying processingparametersencouragestrial-and-error experimentationin order to optimizesought-forfeatures,and the ability to display easily various kinds of attributes permits seeingthe data from various viewpointsto lessenthe likelihood of missingsignificantfeatures.Color displaysconsiderably broaden the dynamic range of data and thus contribute to the ability to seenonobviousfeatures. Workstationspermit the easy flattening of picked horizons to aid in seeingthe attitudes of bedding at the time the picked horizon was deposited,and thus working out the history of structuralchanges.This is apt to be especiallyimportant in areasthat havebeen

INTERPRETATION PROCEDURES involvedin structural inversion,uplifts becomingde_ pressions,normal faults undergoing reverse move_ ment, and so on, as a result of changesin tectonic stresses.

363

The optimum display for one interpretation may not be optimum for another.A regionalinterpretation needsa synopticview and reducedsectionsso that the featuresover a large area can be seenrelativeto each other.The mapping of a prospectrequireslarger sec_ tions 10.2.8Drawing conclusions in order to seedetail to resolvefaults and struc_ reflection clata from tures. Locating stratigraphictraps associatedwith an In mapping seismicdata, one looks for leatls,the pos_ unconformity requiresa full-waveformdisplay of the sibilitiesof hydrocarbontraps that requiremore work unconformity reflection on a fairly large icale. Evi_ to define them completely.Although itructural leads dences of hydrocarbon accumulationsmay require and local amplitude anomaliesmay be fairly evident, displaysat very low amplitude (so that ..brieht spots" the interpreter should also be aleit for subtle clues, becomeevident).displayswith reversedpoiarity.and perhapsto channels,that may indicate stratigraphic displaysof attributessuch as frequencyand veiocity. accumulations.Information from wells in the area A displayof velocity and other attributesmay also as_ may help locate the parts of the sectionwhere stratr_ sist in lithologic identification. Stratigraphic varia_ graphic trapping is most probable. Dip, reflection tions may be more evident if appreciablevertical character,or amplitudevariationsmay iniicate strati_ exaggerationis employed,whereasstructuralinterore_ graphic or porosity changes.Careful study of the tation is usuallyeasierifhorizontal and vertical scales maps,sections,and recordsplus broad experilnceand are nearly the same.The varying requirementsin in_ ample imagination will at times discloseaccumula_ terpretation call for a variety of displaysof seismic tions. data. Sheriff and Farrell (1976) show a section dis_ Whereaswe imply that locating hydrocarbonaccu_ played with various plotting parameters. Feagin mulationsis the goal, this is not il*ays the situation. (1981)discusses plotting parameters. Whateverthe goal, the interpretershould be alert to A good black-and-whitewiggle-tracedisplay has clues.that suggestaspectsother than those being pri_ about 24 dB dynamic range at best, a .ange thut can marily... sou_ght.Interpretation for stratigraphic fea_ be considerablyenlargedby the use of color (Brown. tures ($10.7)should be incorporatedwith structural l99l; Russell, 1992). Color is, however,somewhat rnterpretation(and vice versa).Much seismicwork to_ subjectivebecausedifferent individuals perceiveand day is done to aid in field developmentand produc_ distinguishbetweencolors differently.The ..brighter" tion (White and Sengbush,t987: Sheritr,19921. colors of red and orangeespeciallystand out and are interpretationsare usually cut short befbre -_Seismic most often used to indicate what is perceivedto be all the meaningthat can be found has beenextracted, Clod - high amplitude, amplitude increasing with becausedecisionshaveto be made,or becausethe in_ offset,low velocities,and so on. Color assignmentsto terpreteris neededon anotherproject.The interpreter numericalvaluesare often done in spectralsequence should try to anticipate the questionsthat may be red (largestvalues),orange,yellow,green,and blue. askedof the interpretationandprioritize the work so Someadvocatejuxtaposingcontrasting(almost com_ that at any stageofthe interpretation,the bestanswers plementary)colors, a practice that emphasizessmall can be given,recognizingthat the answersmay change contrasts.Color schemescan emphasizesignificant basedon further study. feature,s, but they can also sometimesemphasizeirrelWhen an interpretationis concluded,a reDort (see evant featuresand obscure significant ones (for ex_ app.G) is usuallyprepared,often borh lor sutmission ample.the relativelycommon useof blue printins on rn writing and for oral presentation.In some ways, a black backgroundmakesit almost invisibleto niany this is the most difficult and most important task of people).Color can be usedto hide as well as to illumi_ the interpreter.He must presenthis findingsin such a nate features. way that the appropriatecourseofaction ii defined as One of the problems with interpretation is that clearlyas possible.The important aspectsshould not thereis often too much data to be examinedand com_ be obscured by presenting u -us of details nor prehended.Superimposeddisplaysmay help in seeing should they be distorted by presentingcarefully se_ the interrelationshipof various data aspetts. Color lectedbut nonrepresentative maps and sections.Evi_ overlaysprovideone way ofadding an additional varidencesto support significant conclusionsshould be able to a display,and color overlaysare employedto given. Alternate interpretationsshould be presented show_amplitude, velocity,frequency,and other aspects and an estimategiven of the reliability of t'heresults (Balch, l97l; Taner and Sheriff, 1977; and Taner, and conclusions.Finally, the interpreier should rec_ Koehler,and Sheriff, 1979). ommendwhat further action should be undertaken. Most often interpretationsare based on misrated sections;featureson theseshould be checkeda-sainst unmigratedsectionsto guard againstpossible,iigru_ 10.2.9Display techniques;color tion errors and to effect ties to intersectingsersmic Display techniquesstronglyaffectthe easewith which lines.Wherean appreciablerangeof depthsis-ofinter_ featurescan be seen.As one truism states,..you can_ est, sectionsplotted so that the vertical scaleis linear not lnterpret what you do not see.,' with depth rather than with time are useful,especially

364

GEOLOGIC INTERPRETATION OF REFLECTION DATA

in working out structural problems. The velocities used in the time-to-depth conversion should be checked,especiallyif the velocitiesvary-horizontally. Where severaloutputs from data processingare available, for example,where there are outputs employing different filtering or specialprocessing,theseshould be examined to see what differencesthey produce. Different displays may prove better for mapping different horizons, but waveshapesand time delays may change with processingparametersand these changeshaveto be taken into accountin the mapping. The various products used to control processing should be examined so that the interpreter understandsmore clearlyexactlywhat was done in the processingand how the decisionsmade there affect the final product. Velocity analysesshould be especially studied for consistencyin picking and to yield clues regarding lithology, high-pressurezones,and so on. Where velocity data exist only in tabular forms, one might wish to redisplaythem graphicallyto facilitate understanding the significanceof variations, especially wherevelocity variesalong the seismicline. 10.3 Evidencesof geologicfeatures I 0.3.I Conceptsfrom structural geology (a) Structural style and plate-tectonicsetting. Many areashavebeensubjectedto fairly simple stressfields that have determined the types and orientations of structuralfeaturespresent,that is, the structuralstyle. The stressfield may have changedwith time, so that the structural style may be different fiordifferent portions of the section,or later patterns may be superimposed on older, different patterns. Because the interpretationof featureson seismicsectionsoften involves some ambiguity, knowledgeof the structural styleappropriateto the regioncan aid in selectingthe most probableinterpretationand in making the interpretationconsistentwith everythingknown about the area,not merelywith the seismicdata alone.It is also important in programming data acquisition in the most economicalmanner. Structural style depends on the tectonic setting (Lowell, 1985),especiallythe location with respectto plate boundaries and the types of boundaries.The typesofplate boundaries(seefig. 10.12)relateto the relative movement of the plates involved:(1) a pullapart zone where plates are separating (divergent boundary),(2) a collision or subduction zone where they are coming together(convergentboundary),and (3) a strike-slipzoneor transformboundarywherethey are sliding past each other. The latter may involve transform faults, a kind of fault that accommodates the changeswhere different portions of pull-apart or subductionzonesare offset, or the changesbetween boundaries of different types. Subsidenceand isostatic adjustment are important factors influencing structural style away from plate boundaries,for ex"passivemargins" of continents. ample,at the First-order or Drimary structural features relieve

I

Fig. 10.12 Plate-tectonicmodel. Magma upwells in rift zones where plates move apart; in convergence (subduction) zones, one plate plunges under another and eventually gets so hot that it melts. Transform faults (7) link offsets in rift and/or convergencezones;transform laults may involve plates sliding by each other over only part of their length, for example, only between the active rifts, other portions having been active before the plates grew at the rift zones. (After Isaacs, Oliver, and Sykes, I968.)

the main stressesproducedby plate movements;they also generatesecondarystressesthat producesecondary structuralfeatures,and thesegeneratetertiary features,and so on. (b) Types of structural style, The major structural stylesas classifiedby Harding and Lowell (1979) are listedin table 10.1.Their first-orderdistinctionis between stylesthat involvethe basementand those that are detachedfrom it. The beginning of sea-floorspreadingseemsto be an uplift resultingfrom heatingby upwellingmagma. This may produce three grabens radiating from a triplejunction (fig. 10.13);twoof thesegenerallytake over and form the rift zone that subsequentlyforms a new ocean.(Local uplifts, suchas salt domes,develop similar faulting patterns.) Sea-floor spreading (fig. 10.14)producesmany more-or-lessparallel normal faults trending perpendicularto the direction of extension, the faults becoming older as distance from the spreadingcentersincreases.The faults are often steepand planar and may dip in eitherdirection.They may be en echelonand throw may shift from one to another.Except near the spreadingcenter,the faults are often inactive.They are sometimesreactivatedby later tectonism,which nr changethe stresspattern and evenreversethe throw direction.The fault blocks are often rotated,occasionallyproducing a high edge that may becomethe venuefor reef development.In intraplateareas,spreadingmay haveoccurredfor only a short time. Compression produces high-angle reverse faults and basementthrusts at convergentboundaries.Secondary compressionis sometimesproduced in other settings. A complex array of featurescan be produced at convergentboundaries.In the subductionofone plate underanothet as in fig. 10.15,portionsofthe oceanic plate may be scrapedoff in a bulldozerlikeaction to produce a melangewedge with thrust faulting. The melanse includes sedimentsderived from the conti-

EVIDENCESOF GEOLOGIC FEATURES nental plate as well as rocks originally remote from the subductionzone. Thrusting and other effectswill also be seenin the forward portions of the continental plateand the affectedzonemay be very wide. The ma_ terialscarried down by the subductingplate may melt and producea volcanicarc parallel to the subducting edge but an appreciabledistance from the trench. Back-arcbasinsand other featuresmay also develop. Occasionally,piecesof the oceanicarc or continentil blocks riding on the subductingplate may adhereto the continental plate and the subducting edge may jump to a new location, so that a variety of complica_ tionsare possible. Wrench-fault assemblagesare commonly assocr_ atedwith transformboundaries(fig. 10.l2). Although the predominantmotion is strike-slip,verticalcomponent of throw is often the most evident aspect.Fea_ turesare usuallyconfinedto a relativelynariow linear zone along the principal strike-slipdirection.Some associatedsecondaryfeatures are illustrated in fie. 10.16. These secondary features are often fairi-y straightand arrangeden echelon.Fault tracesare gen_ erally straight and steepenwith depth. The main stresses may havecomponentsof extensionand com_ pressionperpendicular to the main strike-slipmotion, and irregularities along strike-slipfaultsalso produce extensionalor compressional features.Flower struc_ turesmay occur (fig. 10.l7). Basementwarps are often solitary,very gentlefea_ tures, sometrmeswith associatednormal laultins. They may be of basinsize(Williston Basin)o, o..,i. as regionalarchesor localdomes.They tendto persist over long periodsof time and hencelocalizetrunca_

Fig. 10.13 Rupturing produced by an uplift. (a) A localized uplili tends to produce three sets of faults that produce three g r a b e n s .( b ) A s s t r e s s e sc o n t j n u e . t w o o f t h e s e( , { . B ) t e n c i t o become pull-apart zones, the third (c) may become a /ailetl urm, or uluucopen.

365 Rcd beds and volcantcs

U p w e l l i n gr n a g m a (a)

edbedy O c e a n i cc r u s t

I h i n n e d c o n t i n c n t a lc r u s t

F i g . 1 0 . 1 4 S t r u c t u r e a s s o c i a t e dw i t h r i f t i n g a n d s e a - f l o o r s p r e a d i n g (. a ) E a r l y p h a s eo f r i f l i n g : s o m ec r u s t a lt h i n n i n g , n o r mal faulting mainly down to a central graben or half-graben; infilling sediments are mainly continental. (b) Transition from r i f t i n g t o d r i f t i n g : n e w o c e a n i cc r u s t r e s u l t si n i s o s t a t i cs u b s i dence, possibly restricted circulation and evaporite deposition. (c) Drifting and growth of an ocean,with further subsidence as the oceanic crust gradually cools. Bounding faults are rarely s y m m e t r i c a la n d t h e f a u l t s l i p t e n d s t o s h i f t f r o m o n e f a u l t t o anothcr along strike.

tion, uncomformities,and various types of stratigraphictraps. Thruststend to existas a setof subparallelsalients on the overridingplate at subductionzones.Thrust zonesmay be very broad and of somewhatdifferent forms (fig. 10.18).Thrustsgenerallyparallelthe bedding in incompetentrocksand cut acrossthe bedding in competentrocks; anticlinesoften overlie where thrusts cut acrosscompetentmembers(figs. l0.l8e and 10.l8f). Abnormal pressurezones(g5.3.4)probably provide the d6collement(detachment)zones along which the gliding takesplace,the plate thrust in effect"floating" into place(Gretener,1979).Where massivecarbonatesform a significantpart of the section, thrust sheetsoften involve repeatedslabs(figs. 10.l8a to l0.l8.c); whereductileformationspredominate, hanging-wallfolds are common (fig. 10.l8d). Dips often decrease with depth.The shorteningassociatedwith thrustingand folding may be distributed amongfeaturesin the strikedirectionby perpendicular tear faults (fig. 10.19a)or by transferzones in which parallelfeaturesgrow or decreasein magnitude (fig.10.19b). In folding, the length and volume of beds tend to remaln constant;however,often both cannot be conservedat the sametime, especiallywith intensefolding, and flow and/or faulting occursin somebeds(fig. 10.20).Folding cannot persist to great depths but must give way to flowageor faulting mechanisms. Normal faults detachedfrom basementoccur adia-

GEOLOGIC INTERPRETATION OF REFLECTION DATA

366

Table l0.l Structuralstylesandplate-tectonichabitats Structural style

Characteristics

Dominant

I Plate-tectonic habitat

deformational stress

Pull-apart zones

Fairly high-angle normal laults

Extension

Divergent boundaries ( I ) at spreadlng centers

dipping 60 70'in either

(2) aborted rifts

direction

lntraplate rifts

Rotated fault blocks

Transform boundaries with component of divergence Secondary at convergent boundaries: ( I ) Trench outer slope (2) Arc massif (3) Stable flank of a q.l

foreland and fore-arc basrn

J

(4) Back-arc marginal seas

F q

l! '..l

Compressive faults

z

and basement

High-angle reversefaults, upward imbricating of faults

Compression

thrusts F

Convergent boundaries ( l ) F o r e l a n db a s i n s (mostly) (2) Orogenic belt cores (3) Trench inner slopes

Z

and outer highs

2 q q

Transform boundaries with component of convergence

Wrench-fault assemblages

Strike-slip faulting is primary. secondary featuresat about 30" angle to main trend

Couple

Transform boundaries Convergent boundaries at an angle: ( l ) F o r e l a n db a s i n s (2) Orogenic belts

Fairly narrow trend Faults generally steepenwith

(3) Arc massifs

depth

Divergent boundaries with offset spreading centers

Basement warps

Gentle structure: domes, arches, sags

Isostatic adjustment Heat flow

Plate interlors Passiveboundaries Other areas

Typical profile

EVIDENCESOF GEOLOGIC FEATURES

367

Table 10.1continued Structural style

Characteristics

Dominant

Plate-tectonic habitat

Typical profile

delbrmational stress

Thrust assemblages

Faults sole out at d6collement in incompetent rocks

Compression

Convergent boundaries (l ) Inner slopes of trenches and outer highs (2) Mobile flank of forelands (orogenic belts) Transform boundaries with component of convergence

I -J

:t

!

Growth faults and other normal fault assemblages

j

z

:

Downthrown toward basin or toward center of uplift D i p o f t e n l e s s e n sw i t h d e p r h t f o r growth faults) Often contemporaneous with deposition

Extension

-

i..""a"rr-,"-*iiri.rr.t.,r, re Passiveboundaries

2

Saltstructures

Pillows, domes. salt walls

Plastic flow Solution

Divergent boundaries (rifts provide venue for salt deposition)

Shale structures

Plastic flow (often

Passiveboundaries

involving overpressuring produced by rapid burial)

Drape features

Differential compaction

Volcanic plugs

.9orrce.After Harding and Lowell, 1979.

Igneous intrusions

Subsidingbasins Over reefs

r-re

GEOLOGIC INTERPRETATION OF REFLECTION DATA

368

Back arc basin

Melange wedge

Continental craton

\-__!_--

C o n t i n e n t a lc r u s t

? s n !

Oceanic crust

Mantle

Top of asthenosphere

Zone of transition to eclogite

Upwelling of molten magma

o

o 100

100 Distance(km) an ooeanic plate under a continental plate Fis. 10.15 Tectonic f-eaturesassociatedwith subduction of

*,h"6s \a)

F'ig. 10.16 F'eaturessecondary to a wrench fault (which often m a k e a n g l e so f a b o u t 3 0 ' o r 6 0 " t o t h e p r i m a r y t a u l t ) . ( a ) M a p

Fig. 10.17 Flower structure resulting from strike-slip motton that involves a component of compression. Motion involving a component of expansion produces negative flower structures. Flower structures are usually not symmetrical. (After Lowell' t9'72.\

cent to subsidingbasins,especiallyat the passivemargins of continents.They are generallydownthrown toward the basins and grow contemporaneouslywith deposition as basins subside,as evidencedby thickeninginto faultson the downthrownside(fig. 10.21). The fault planesgenerallydecreasein dip with depth' generally being concave upward. Rotation of the downthrown block in growth faulting producesroli-

view (for symbols, see app. H); (b) isometric diagram showing features on the maP.

ovcr into the fault (sometimescalledreversedrag), the consequentdip reversalpossiblymaking hydrocarbon traps. Growth faults are not only curved in crosssectionbut also in plan view,generallybeing concave toward the depression.Growth faults frequently die out upward. The throw on growth faults often increaseswith depth. They sometimessole-out in the bedding,the movementparallel to the beddingsometimes producing toe .ttru(tures.Toe structuresalso result from downslopesliding (fr5.10.22). Normal faults are often secondary elements of other structuralstyles;for example,normal faults occur on the crestsof folds and abovediapirs.They are often associatedwith shale or salt flowage.Normal faults are probably the most common structural feature becauserocks are especiallyweak under tension. Salt and shalestructurescan be of a variety of types involving flow and withdrawal structures,and sometimes collapsefeaturesresultingfrom salt removal by solution.In fig. 10.23,the systemof growthfaultsdevelopedjust beyond the underlyingshelf edge,where presumablyshale under abnormal pressureacted as a fluid and flowed to the left and up to form shale diapirs. Salt and shaleoften provide the detachment surfacesassociatedwith detached faulting (as with thrusting) and folding. Buoyant salt structuresare es-

ne Mt. anticline P o w e l lV a l l e ya n t i c l i n e

-../

5c *

u.Jor

rNrusr


,

l

(f) Fig. 10.l8 Thrustfaultins.(a, b) Seismicprofile32 km long in Valley-Ridgeprovince of Jast Tenness.;;;l;; i,i.'#.,u,io,l (tiom Harrisand Milici, 1977,): (c) tfr."rii"g i"'d"r"a,i, O..U_ tesproducingthe TurnerValley srrucruref?r"rn6"fi"p,' ,CSf l,

(d) d6collement of theJura Mountains(from Buxtorf, l9l6); (e) "f *uthern Appalachiand6collemen, *iif, un,i"rin., :L:^::i-q:r

orduprex'i'u"u'. ffi:lT:fi.""Hi?,lfi"fJri""eropment

370

GEOLOGIC INTERPRETATION OF REFLECTION DATA factorsthat movemassinto or out of the systembeing considered.lnterpretersoften do not checkto seethat their interpretationsare consistentwith this principle becausecalculations are tedious and often require more extensivedata than are available.Modeling programscan ascertainwhetherdiscrepancies exist.Jones (1988)discusses the balancingof seismicsections.

(al

(b)

Fig. 10.19 Translerofthrow from one thrust fault to another. (After Dahlstrom,1970.)(a) Tearfaults separate thrusts; and (b) faultsgrowor die laterally.

peciallycommon along passivecontinentalmargins, the basinwardprogressionoften being a sequenceof salt-withdrawal structures, pillows, nonpiercement and piercementdomes, and finally, a "wall" of salt (figs.10.24and 10.25);the salt often has movedconsiderabledistancesbasinwardas depositioncontinues and as basin subsidencemovesseaward. Depositionaland structuralfeaturesare often interrelated.Reefsmay grow over shelfedgesor on the upturned edgesof rotatedfault blocks.Differentialcompaction may producedrapeover reefsand shelfedges. Progradation beyond a shelf edge may produce growth faults.The weight of sedimentsdepositedby a river delta systemmay producesubsidencebecauseof isostaticadjustment, affecting faulting patterns and causingsalt and shalemovements. I 0.3.2 Balancingsections Preservationof volume and bed length were mentioned earlier.The total mass cannot changeduring faulting or flowageand the length of competentbeds alsocannot change.Mass can be lost or gainedlocally becauseof erosion or increaseddeposition or other

(c)

Fig. 10.20 Mechanisms that maintain bed length and volume "Concentric" lolding with flow or sein fold interpretation. (a) "simivere distortion as the mechanism (after Goguel, 1962);(b) lar" folding with bedding-plane shear and thickness variations: (c) combination of folding and faulting; the more competent members mainly fault (after Hobbs, Weams, and Williams.

r976).

I I

EVIDENCESOF GEOLOGIC FEATURES

J I I

10.3.3Faulting

Fig. 10.21 Growth faulting. Sediments,especially poorly consolidated ones, slide toward the basin along a concave-upward (/i.r//r(')fault plane. Rotation of the down-dropped block con_ temporaneous with deposition results in thickenine into the t a u l t p r o d u c i n g a r o l l o v e r a n t i c l i n a l a x i s p a r a l l e lw i t h t h e l a u l t plane. The fault is arcuate in plan view with the throw gradually diminishing away from the center of the fault. Seismic data s h o w i n ga f a u l t o f t h i s k i n d a r e s e e ni n f i e . 1 0 . 2 9 .

(a) Introduction. Faults constitute one of the more important hydrocarbontrapping mechanisms(Downey, 1990) and recognizing them and determining their preciselocationsare often the keysto success. Fault nomenclatureis illustratedin fig. 10.26a.The fault quantitiesthat can be measuredon good-quality seismicsectionsare usuallythe vertical throw and the apparent fault-plane dip; horizontal componentsof fault slip can rarely be determined. Ideally reflection events terminate sharply as the point of reflection reachesthe fault plane and then they resumeagain in displacedpositionson the other sideof the fault. In addition, ideally,the reflectionhas a sufficientlydistinctive characterthat the two portions on oppositesidesofthe fault can be recognized and the fault throw determined.In practice,diffractions usually prolong events,so that the locations of fault planes are not clearly evident, although often faults do show clearly as reasonablysharp reflection

(a)

- -

(b) Ftg. 10.22 Seismic section showing downslope sliding and slumping, with toe structures. (Courtesy of Exxon.) (a) CMp section; the marks at the top are 2 km apart so that the vertical

exaggerationis about llX, the sea-floor slope being less than l"; (b) interpretation of lault glide planes and sea-floor multiple (dashed).

l

Fig. 10.23 Shale flowage (Courtesy of Exxon') (a) CMP section with vertical exaggeration of 5x to 8x' decreasing with depth; (b) interpretation indicating the attitude of reflections

(which have not been correlated except for the angular unconformity L/). With depth the faults die out in flowage of the overoressuredshale into shale diapirs D

EVIDENCESOF GEOLOGIC FEATURES

J / J

Fig.10.24 Saltstructures growbasinward assaltincreases rn thickness. eitherbecause moresaltisdeposited or because saltis

pushed basinward assediments aredeposited furtherlandward. (FromTrusheim, 1960.)

terminationson migrated lines that are roughly perpendicular to the faults. Moreover, although sometimes the samereflectioncan be identified unequivocally on the two sidesof a fault, in many cases,we can make only tentativecorrelationsacrossfaults. Campbell (1965)and Sheriff(1982)discusscriteriafor detectingfaultson seismicsections.

10.l3). Normal faulting commonly accommodates stretchingabovea hinge line or at a shelf edgewhere the basin side of the shelf edgesubsidesmore rapidly (fig. 10.23).The location of faulting can be the key to underlyingfeatures,and converselythe underlying featurescan aid in connectingsometimesconfusing faulting evidencesinto a probablepattern.

(b) Characteristicsoffaulting. Faults are produced by unbalanced stressesthat exceedthe strength of rocks,the type of fault dependinglargelyon whether the vertical or horizontal stressesare the larger (fig. 10.26b).Normal faults result when the maximum compressivestressis vertical and the minimum horizontal, often producing fault-plane dip of the order of 50'-60". When the maximum compressivestressis horizontal, thrusts result,often with a fault plane dip of 30'-40'. Where the maximum and minimum stressesare both horizontal, wrench faults result, the faulting often being at about 30. to the maximum stressdirection. Becausevelocity ordinarily increaseswith depth, the samevertical distanceis representedby lesstime as depth increases;in consequence a postdepositional fault with constantattitudeand throw usuallyappears to dip less with depth, that is, has concave-upward curvatureon a time section(fig. 10.27).Furthermore, constantthrow is representedby fewerwavelengthsas depth and velocity increaseand so constant-throw faults appearto die out with depth on a time section. Postfaulting compaction (fig. 10.27)with increased depth of burial also producesconcave-upwardcurvature. Thus, fault tracesare rarely straight on seismic sections. The locationsof faults often are determinedby underlying features. An underlying uplift places the overlying sedimentsunder tension as distancesare stretchedto accommodatethe drape over the structure. Graben faulting forms to relievethe stress,but if the uplift is three-dimensional rather than twodimensional, radial faulting is also required (fig.

(c) Fault example. The two record sections in fig. 10.28join at their north and westendsat right angles. The reflection band consisting of four strong legs markedI can be correlatedreadily acrossthe normal faults. In fig. 10.28a,this eventis down-thrown to the south by about 75 ms (about 2 cycles,ignoring rollover,gl0.3.lb) at its 1.6s arrivaltime;at a velocityof 2500m/s, this representsa vertical throw of about 95 m. The exceptionallystrong event near 2.3 s (marked 1) indicatesa throw of 120 ms (about 3 cycles,the dominant frequencyhavingbecomeslightly lower);at a velocity of 3000m/s, this representsI 80 m of throq so the fault appearsto be growing rapidly with depth. Although the evidencesuggeststhat the fault is a simple break in the shallowsection,at greaterdepths there may be a fault zone of somewidth with subsidiary faults (shown dashed in the figure); perhaps a small piece of the event X can be seen betweenthe faults. Becausedata quality deterioratesin the region under a fault, details often cannot be seenclearly.If the deepercorrelationsare correct, the downthrown eventf,) at 3.5 s is found around 2.9 s on the upthrown side; by assuminga velocity of 3500 m/s, this correspondsto a throw of 1050m. The basisfor correlationacrossthe fault for the ) event is reflectioncharacter;for X, it is strong amplitude; and for other events,the intervals betweenreflections, systematic growth with depth, time ties around loops,and so on. Sometimesthe displacement of an unconformity or other recognizablefeatureindicatesthe amount of throw. Often, however,the throw cannot be determinedclearlv from the seismicdata.

i I

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If the data in figs. 10.28aand 10.28bare transformed into depth sections,we get fig. 10.28c.The components of fault-plane dip are around 55o and 48'. Note that the fault that is nearly straight on a depth section is concaveupwards on a time section becauseof the increasein velocity with depth. If the fault surfaceis actually concaveupwards,the curvature will be accentuatedon a time section.Where the fault was most active (indicated by the most rapid growth in fault throw), the fault surface is most curved. The fault has not completelydied out by the north end ofthe line and hencethe fault traceshouldappear on the intersecting line (fig. 10.28b).As pickedon the E-W section,the fault offsetsthe event I at 1.6 s by only about 25 m, indicatingthat the fault is dying out rapidly toward the east.The fault plane has nearly as much dip in the E-W sectionso that the strike of the fault plane near the intersectionof the two lines is NE-SW and the fault planedips to the southeast.The true dip of the fault plane is about 62o (the apparent dip on sectionsis alwayslessthan the true dip unless the line is perpendicularto the strike of the fault). Fault indicationsare not evident below about 2 s on the E-W sectionso that the fault appearsto havedied out at depth toward the east. In poorly consolidated sediments,such rapid dying out of faults is common. In this instance,we are dealing with a radial fault from a deep salt-coreddiapir locatedjust south and slightly west of theselines;suchradial faults often die out rapidly with distancefrom the uplift. (d) Evidencesfor faulting. A number of the more common faulting evidencescan be seenin the foregoing example. Severaldiffractions can be identified along the fault trace in fig. 10.28abetween 1.9 and 2.5 s. If we had been dealing with migrated sections these diffractions would have been nearly collapsed (but not completelybecausethe fault is not perpen-

dicular to the lines).Terminationsof eventsand offset of reflections (and nonreflection zones) across the fault are other important faulting evidences. Different reflectiondips are often seenon the two sidesof the fault. Someof thesedip changesare real, involving slight rotation of the section as the fault moved along a slightly curved fault plane, drag, and other real phenomena.On the other hand, some (especiallythoseseenthrough the fault plane)are distortions resulting from raypath bending (refraction) in passingthrough the fault planebecauseoflocal velocity changesat the fault. Although the upthrown sediments are most apt to havethe higher velocity at any given level, the polarity and magnitude of velocity contrastsvary down the fault plane as units arejuxtaposedagainstdifferent units, so that the nature of the distortion varies from one place to another. In fact, the distortions may be so great and may change so rapidly as to causemarkeddeteriorationof data quality below the fault, sometimesso great that reflections are almost entirely absent(a "shadow zone") below the fault. This is especiallyapt to be true for CMP sectionsbecauseraypathsfor the componentsstacked togethercrossedthe fault at different places. Occasionally,the fault plane itself generatesa reflection, but generallythe fault plane is a highly variable reflectorbecauseof the rapid changesin velocity contrast along the fault plane. Also, faulting is often distributed over a zone and involves many fracture surfaces.Furthermore,most reflectionrecordingand processingdiscriminateagainstfault-planereflections becauseof the useof arraysand of stackingvelocities that do not optimize such events.In addition, faultplane reflectionson unmigrateddata are usually displacedan appreciabledistancefrom the fault and often the traveltimesto them are so great (becauseof the long slant paths) that they are not recordedand processed.Many of the earlier evidencesfor faulting can be seenfor the growth fault in fig. 10.29(of the type illustratedin fig. 10.21);other faults are also presentin fig. 10.29. The increaseddetail made possibleby 3-D methods often showsnot only more and smallerfaults,but also many that are short along strikeand disconnected(fig. I 2.I 0). The continuity of stratigraphicfeaturesacross faults seenon 3-D horizon slices(fig. 12.14)sometimes providesconvincingevidencethat fault throws havebeenpicked correctly. 10.3.4Foldedandflow structures When subjectedto stress,rocks may fault, fold, or flow, dependingon the magnitudeand duration of the stresses, the strengthof the rocks,the nature of adjacent rocks, and so on. The folding of rocks into anticlinesand domesprovidesmany of the traps in which oil and gas are found. Figure 10.30 shows a migrated seismic section acrossan anticline. Some portions such as l, which are composedof the more competent rocks (for ex-

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ample,limestonesand consolidatedsandstones).tend to maintain their thicknessas they fold. Other portions such as 4 which contain lesscomDetentrocks toften shalesand evaporites),tend to flow and slip along the bedding, resulting in marked variations in thickness within short distances.Geometry places limits on the amount of folding that is possibleand lblded structuresalmost always involve faulting (fig. 10.20c).Nore at C in fig. 10.30how a fault is involved sith the foldingand in fig. 10.31how the forcefold is associatedwith the underlying fault and inverted structure. Arching causes extension; often the sediments break along normal faults and produce graben-type featureson the top. Folding must disappearby faulting or flowage at some depth. Anticlinal curvature tendsto make seismicreflectionsweakeras well as increasethe likelihood of faulting and flowage,so that data quality commonly deterioratesover anticlines. Salt flow often producesanticlinesand domes.In many parts of the world, thick salt depositshavebeen buriedfairly rapidly beneathrelativelyunconsolidated sediment.The sedimentscompact with depth and so rncrease their density,whereasthe salt densityremains

nearly constant.Thus, below some critical depth the salt is lessdensethan the overlyingsediments.Salt behaveslike a very viscousfluid under sufficientpressure,and buoyancymay result in the salt flowing upward to form a salt dome, arching the overlying sedimentsand sometimespiercing through them (fig. 10.25).Piercementdoes not necessarilyimply uplift, however,becausesubsidenceof the sedimentssurrounding a salt plug accomplishesnearly the same structuralresult.Often the velocityin "uplifted" rocks is nearly the sameas that in laterallyadjacentnonuplifted rocks, implying that neither was ever buried deeper;ifthey had been,they would haveirreversibly lost porosity and attaineda higher velocity. Grabensand radial normal faults (whosethrow decreasesawayfrom the dome)often resultfrom arching of the overlying sediments(fig. 10.13), to relievethe stretchingthat accompaniesthe arching. Salt domes tend to form along zones of weaknessin the sediments,suchas a largeregionalfault. The sideof a salt dome may itself be thought of as a fault. Figure 10.32showsa seismicsection acrossa salt dome.Shallowsalt domesare apt to be so evidentthat they can scarcely be misidentified. Becauseof the

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largeimpedancecontrast,the top of the salt dome (or caprockon top of the dome)may be a strongreflector. Steepdips may be seenin the sedimentsadjacentto the salt dome as a result ofthese having beendragged up with the salt as it flowed upward. The sediments often show rapid thinning toward the dome. The salt itself is devoid of primary reflections,although multiples often obscurethis feature,especiallyif AGC is used. Defining the flank of a salt dome preciselyis often economicallyimportant and seismicallydifficult. Frequently,oil accumulationis in a narrow belt adjacent to the dome flank, but becausethe flank is often nearly vertical or even overhung,it usually does not give a recognizablereflectionwith conventionalprocessing.Fortunately,velocitiesare often only slightly affectedby the growth of the dome (exceptfor the velocity in the salt and caprock) so that the steepdips of the sedimentsadjacent to the flanks can be mrgrated fairly accuratelyand the flank outlined by the terminations of often strong, steeplydipping reflections. Evenreflectionsfrom overhungflanksare sometimes used (turning waves,diving wavesthat reflect on their upward leg). Proximity surveys($13.7.1)using wells in or near the salt are also used to define the flanks more accurately.Nevertheless,there remains much art and experiencein definingsalt-domeflanks. Although the weight of sediments prograding toward a salt massmay push the salt seawardfor large distances,the salt in a salt dome generallyhas come from the immediately surrounding region. The removal of the salt from under the sedimentsaround the dome has allowed them to subside,producing a rim syncline.The seismicdata over such synclinesare often very good and aid in mapping the adjacentdome by indicatingthe'volumeof saltinvolved,whenmovement took place (by sedimentthickening),and so on. Such synclines may also help provide closure on neighboringareaswherethe sedimentscontinueto be supportedby residualsalt. Figure 10.25is a portion of a sectionin the North Sea (the horizontal scalehas been compressedso as to display a long line on a short section,producing considerablevertical exaggeration).This line shows deep salt swells that have not pierced through the overlyingsediments(fig. 10.25a),salt that has pierced through someof the sedimentarysection(fig. 10.25b), and also salt that has piercedall the way to the sea floor (fig. 10.25c).The reflectionfrom the baseof the salt is generallycontinuous except for some suggestions of faulting. Distortion is produced by the variable salt thickness.Becausethe salt velocity is greater than that of the adjacent sediments,the base-of-salt eventappearsto be pulled up wherethe salt is thicker. In areaswhere the salt velocity is lower than that of the surroundingsediments,flat reflectorsbeneaththe salt may appear to be depressedwhere the overlying salt is thicker. Figure 10.33showsa salt uplift at a shelfedge.Most of the salt movementoccurredprior to the unconfor-

mity U, although the right side continuedto subside somewhateven after U,, producing the monocline in the shallowersediments.Note the grabenfaulting seen most prominently around 2.0 s. Occasionally,substancesother than salt form flow structures.Poorly consolidatedshale may flow 1fig. 10.23),forming structuresthat strongly resemblesalt domeson reflectionsections;also,at times,shaleflows along with salt, producing a salt dome with a sheath of shale.Magma also sometimesflows into a sedimentary section to produce structural uplifts, including piercementdomes.Shaleoften flows upward on the upthrown side of growth faults. 10.3.5Reefs "reef" as usedby petroleumgeologistscomThe term prisesa wide variety of types,includingboth extensive barrier reefsthat cover large areasand small isolated pinnacle reefs.It includescarbonate structuresbuilt directly by organisms,aggregatescomprising limestone and other related carbonate rocks, as well as banks of interstratified carbonate (and sometimes, also noncarbonate)sediments.Reefdimensionsrange from a few tens of metersto severalkilometers,large reefsbeing tens of kilometersin length,a few kilometers wide and 200 400 m or more in vertical extent. Some reefs grow at the boundary betweendifferent environments,such as the shelf-marginand barrier types shown in fig. 10.34a,whereasothers, such as patch and pinnaclereefs,are surroundedby the same envlronment. We shall describea model reef so that we may develop generalcriteria by which reefscan be recognized in seismicdata, keepingin mind that deviationsfrom the model may result in large variations from these criteria. Our model reef forms in tropical to subtropical waters far enough from river mouths that the water containslittle suspendedsediment.The area ts generallytectonicallyquiet,characterizedby flat-lying beddingthat is more-or-lessuniform overa largearea. The uniformity of the sectionmakesit possibleto attribute significanceto subtlechangesproducedby the reef that might go unnoticed in more tectonicallyactive areas.The reef is the result of the buildup of marine organismsliving in the zoneof waveaction where the water temperatureis suitablefor sustainingactive growth and the water sumcientlyclear to permit significant penetrationof sunlight.The site of the reef is usually a topographic high that provides the proper depth. Although the topographichigh may be due to a structurein the underlyingbeds or basement,such as a tilted fault block, more often it is provided by a previousreef;as a result,reefstend to grow vertically, sometimesachieving thicknessesof 400 m or more and therebyaccentuatingtheir effectson the seismic data. In order for the reef to grow upward, the base must subsideas the reef builds upward, maintaining The its top in the wave zone as the sea transgresses. reef may provide a barrier betweena lagoonal area

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:he hut'krca/) and the ocean basin (Ihe /breree11, so :rat sedimentation(and consequentlytire reflection :.tttern)may be differenton oppositesidesof the reef. 'fhe surrounding basin may be sturved(that is, not -..rresultrcient sedimentsavailableto keep it filled at : : c r a t e a t w h i c h i t i s s u b s i d i n g )a; t t i m e s ,o n l y o n e :ide. more often the ocean side of the reef,may be .:.rrved.Alternatively, the reefmay not be a barrrerto tlrr\'€rlert of the sedimentsand in this caseit will be ..lrroundedby the samesediments. Erosionof the reef :jr.n providesdetritus for depositionadiacentto the ::el-.resultingin /breset bedswith dips up to 20., but -.ually with smallerdips,often of only l_2.. The pn_ :-..rrvreef may possessconsiderableporosity, which ::.rkesit a good potential reservoirrock. but other or_ i:nlSlnS s.uchas spongespenetrateinto and replace :-.uchof the reef rock, alteringthe porosity in the pro_ -:>s,Usually,the actual biohermalportion of the reef :re portion produced by reef-building organisms) --1nnotbe distinguishedfrom other portions exceptby ':.: examination of samples,and the entire comolex :::rrciat€d with the reef is called the ..reef,"only per_ -.ps l5% of which is biohermal.Eventuallv.the envi_ - rment for the reel organismschangesso that they -:n oo longer continueto live and build the reef; thii --rsht come about becauseof changesin the water ':nperature, a changeto more turbid water,possibly

An lncreasein the rate of subsidence so that the or_ ganic buildup cannot keep pace (called clrowningof the reef), or various combinationsof circumstances. Subsequently,the reef may becomeburied by deepwater shalesrhat may provideboth an impeimeable cap to the porous reef and sufficient hydrocarbons that the reef becomesa petroleum reservoir.Additional sedimentsmay continue to be deposited,their weight compactingthe sedimentsthat surroundthe reef more than they compact the relativelyrigid reef; thus, the overlyingsedimentsthat were depositedflat may developa drapeover the reef.The interior of the reef may be more porous and lessrigid than the edges so that somedifferentialcompactionmay occur over the reef itself. Basedon the foregoingmodel, we developthe reef criteria illustratedin fig. 10.35.The top of the reefmay be outlined by reflections(fig. 10.35a),perhapswith someonlappingreflections,but the reef interior is apt to be a reflection void (fig. 10.35b). Occasionally, diffractions from beds terminating at the reef (fig. 10.35c)and the abrupt terminationof reflectionsfrom the surroundingsediments(fig. 10.35d)can be usedto Iocate reef flanks, but the preciseoutline of reefs is usually difficult to determine.If the reef provided a barrier to sedimentation,the pattern of reflections may differ on oppositesidesof the reef (fig. 10.35e).

GEOLOGIC INTERPRETATION OF REFLECTION DATA

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Overlyingreflectionsmay show small relief (usually only a few milliseconds in magnitude) becauseof differentialcompaction,the effectdecreasingwith distanceabovethe reef(fig. 10.35f).A velocitydifference betweenthe reef materialsand the surroundingsedimentsmay causethe traveltimeto flat-lyingreflections below the reef to vary (Davis, 1972),and this velocity differencemay producepseudo-structure on reflecting horizons below the reef. Usually, the velocity in the reef limestone is greater than that in surrounding shales,so that the reef may be indicatedby time thinning betweenreflectionsaboveand below the reefand by a pseudo-highunder the reef (fig. 10.35g);the mag-

nitude of such an anomaly is small, usually lessthan 20 ms. Sometimes,however,the reef may be surrounded by evaporitesor other rocks with higher velocity than that of the porous reef limestoneso that the time anomaly is reversed(fig. 10.35h).The hingeline or high that localizedthe reef developmentmay (figs.10.35iand 10.35j). alsobe detectable Barrier reefs, which separate different environments, are often fairly evident, but small patch reefs may be so subtlethat seismicmapping is feasibleonly in good record areas.Of importanceis geologicinformation about the nature of the sedimentsand the environment of deposition, so that one knows before-

EVIDENCESOF GEOLOGIC FEATURES

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F'ig. 10.35 Criteria fbr reef identification. (After Bubb and Hatlelid. 1977.) (al Reef outlined by reflections; (b) reflection void; (c) diflractions from reef edges; (d) abrupt termrnatron ( ) l r e f l e c t i o n s :( e ) c h a n g e i n r e f f e c t i o n p a i l e r ; o n opposrle sides of reef; (f) differential compaction over the reef (isopach thinning); (g) velocity anomaly underneath the reef where r,"", >. tr,"..",",,,,,*, or (h) where 4.*, < 4--,"u,",,. (i) Reef location on a hingeline or shelfedge, and f), on a struitural uplift.

385 hand in what portion of the section reefs are more likely to occur.Subtlefeaturesin the part of a seismic sectionwherereefsare expectedmay be interpretedas reefs whereassimilar featureselsewhereare ignored. K u h m e( 1 9 8 7 d ) i s c u s s er se e fi n t e r p r e t a t i o n . Similarities between reefs and salt featurescause problems at times. The lagoonal areas behind and around reefs often provide conditions for evaporite depositionand salt is frequently presentin the same portion where reefsare expected.The amount of salt may not be thick enoughto produceflow featuresbut differential solution of salt beds followed by the collapseof the overlyingsediments into the void-thuscreated may produce seismicfeaturesthat are similar in many waysto thosethat indicatereefs. A seismicline acrossa barrier reef is shown in fig. 10.36;note the changein reflectionpattern acrossthe reef, the differential compaction and velocity uplift evidences,and the changein regional attitude of reflectionsbeneaththe reefthat indicatesa weak hineeline. A line acrosspatch reefsis shown in fie. 10.j7. Patchreefsare usuallymuch smallerin verticil extent than theseand, consequently,often difficult to locate. Typical seismic facies characteristicsof different carbonatedepositionalsettingsare illustratedin fie. r0.34b.

10.3.6Uncon/brmities Unconformitiesrepresenta missingsequence of rock, a time period during which rocks were being eroded

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reflections (,4') is to the right; the forereef showing an entirely different reflection patten (A) is to the left. (Courtesy of Conoco.)

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F i g . 1 0 . 3 7 T w o p a t c h r e e f si n t h c E t o s h a B a s i n o l ' N a m i b i a . C d c n o t s s t h c c a r b o n a t e p o r t i o n o f t h e s e c t i o n ;B i s t h e b a s e o f the reefi. The region of reefs is indicated approximately by the

a r r o w s b e l o w t h c s c c t i o n .T h c r e c l ' t o t h c l c f i h a s a b o u t 2 1 0 m ( 8 5 m s ) t h i c k n c s s t. h e o n e t o t h e r i g h t 3 0 0 m ( 1 2 0m s ) . ( C l o u r t e s y

away,or at least not deposited.Conditions probably changedduring the hiatus,so that the natureof the sedimentsabovethe unconformity are often different contrast from thosebelowand an acoustic-impedance is likely to existat the unconformity.Hence,unconformities are usuallygood reflectors.They frequentlyinvolvesomeangularity betweenthe beddingbelow and above.and this also tends to make them stand out as reflectors.The result is that unconformitiesare often among the easiestand most distinctive reflectorsto map. On the other hand, the rocks that an unconformity separatesoften vary from one location to another so that the contrastat the interfacechangesand hencethe unconformity reflectionvariesin amplitude (seefig. 10.38)and sometimesevenin polarity (seefig. 10.6a).Theremay be largeregionsoverwhich the beds aboveand belowthe unconformity parallelthe unconformity so that there is no angularity to distinguish the unconformity reflectionfrom other reflections.In such regions,the unconformity has to be mapped by correlatingit along the beddingwith placeswherethe unconformity can be identifiedby angularitieson the seismicsectionor whereit can be identifiedfrom well or other typesofdata. Rather prominent unconformities can be seen in figs.9.59 and 10.38,mainly evidencedby angularities and fairly strong reflectionsfrom the unconformities themselves. Various types of hydrocarbontraps are associated with unconformities both (l) pinchoutsand truncations of reservoirbedsbelow the unconformity where

the unconformity constitutesa seal, and also (2) laid down on variationsin the sediments stratigraphic the unconformity(seealsofig. 10.42).Unconformities are involved in most stratigraphic traps. Streams flowing across the unconformity surface may have erodedvalleysinto the surface,and the streamdeposits may constitutethe reservoiror sometimesthe seal.

o f E t o s h a P et r o l c u m .)

10.3.7Channels Ancient stream channelsare involvedwith a number (fig. 10.39a).The relief of oil and gas accumulations associatedwith large river valleysmay be sufficientto give structural evidences,but most often the seismic effectsare slight. The velocity of the sedimentsinfilling a channel may differ from that of the sediments into which the channelis cut so as to distort underlying reflections,especiallywherethe channelsare deep (fig. 10.39b).Figure10.40represents a shallowseismic time slicefrom a 3-D data set(seealso$12.3);it shows the pattern of a meanderingstream.This suggeststhat evidencesof very minor featuresare containedin seismic data if we can but find an economical.feasible way of extractingthem. Profiler surveysin deepwater often revealchannel cuts and fill (fig. 10.41),indicating that channelsare important in the deep marine environmentas well as on land. Turbidity currents at places clearly have erodeddeepchannelsand havesometimesbuilt up extensiveleveesystemsunder deep marine conditions. (A turbidity current is a density current in water

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F i g . 1 0 . 3 9 C h a n n e l s o n s e i s m i cs e c t i o n s .( F r o m M c Q u i l l i n . B a c o n ,a n d B a r c l a y ,1 9 8 4 :I 8 3 . 2 4 5 . )( a ) G a s - f i l l e dC o l o n y c h a n nel szrnds:their amplitudes correlate with the net pay thicknesses:(b) seismic line across the East Kingfish field' Bass

Straits. Australia; Miocene channels lilled with higher veloctty sediments produce significant velocity anomalies at the deeper Latrobe producing horizon.

causedby dilTerentamountsof solidsin suspension; they are important in submarineerosionand deposition and in hydrocarbonaccumulations.)Somechannels undoubtedly result from the sea level lowering and some from marine erosional processes.Brown "destructive shelf" and Fisher (1980) developed a concept relating the erosion of channelsinto shelves to the lack of new material availablefor deposition; this concept has helped discover accumulationsin fans on the slopesoffshoreBrazil and South Africa.

I 0.3.8 StratigruPhictraPs Rittenhouse(1972) gives a classificationschemefor stratigraphictraps,summarizedin table l0'2. His firstorder division dependson whetherthe traps are or are not adjacentto unconformities.Illustrationsof some of theseare given in fig. 10.42. A large portion of the hydrocarbonaccumulations remaining to be found probably involve stratigraphic traps. Marr (1971), Lyons and Dobrin (1972), and Dobrin (1977) discuss seismic evidencesof strati-

EVIDENCESOF GEOLOGIC FEATURES

389

Table 10.2 Classificationof stratigraphic traps I. Not adjacentto unconformities A. Facies-change trapsinvolvingcurrent-transportedreservoir rock (l) Eolian(dunesor sheets) (2) Alluvial fan (3) Alluvial valley(braidedstream,channelfill, point bar) (4) Deltaic (distributarymouth or finger bars,sheet, channelfill) (5) Nondeltaiccoastal(beach,barrier bar. spit. tidal delta or flat.y (6) Shallowmarine (tidal bar, sand belt, washover, shelfedge,shallowturbidite or winnowing) (7) Deep marine (marine fan, deepturbidite or winnowing) B. Noncurrent-transportedreservoirrock (l) Gravity (slump) (2) Biogeniccarbonate(shelf-marginreef,patch reef. algal buildup or blanket) C. Diagenetictraps ( I ) Changefrom nonreservoirto reservoir (a) Replacementand leached(dolomitized) (b) Leached (c) Brecciated (d) Fractured (2) Changefrom reservorr[o nonreservolr (a) Compaction (physicalor chemical) (b) Cementation ll. Adjacentto unconformilies A. Trapsbelow unconformities ( I ) Sealsaboveunconformity (a) Subcropat unconformity (b) Topography(valleyflank or shoulder,dipslope,escarpment,valley,beveled) (2) Sealbelow unconformity (a) Mineralcement (b) Tar seal (c) Weatheringproduct B. Trapsaboveunconformities (l) Reservoirlocation controlledby unconformiryropography (a) On two sides(valley,canyon,fill) (b) On one side(lake or coastalcliff, valleyside, flank of hill or structure) (2) Transgressive Source:After Rittenhouse.I972. graphic traps drawn from published case histones. They paint a discouraging picture; most stratigraphic accumulations have been found by searching for something else, that is, by serendipity. Sheriff (1980) includes the following quote: Stratigraphiccasehistorieshad one important moral. While the discoveryof stratigraphicaccumulationswas not generally attributed to a sound explorationprogram, the genrus lay in being alert when a surpriseoccurred.Often the surpnse occurs in the record from a borehole; some portion differs from what we expectedin such a way as to suggestthe possibilityof a stratigraphictrap nearby.But where?This is where reflection-character analysiscomesinto its power; it can help us locate the nearby accumulation that the unexpectedin a well suggests.It can help us searchlor strati-

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Fig. 10.40 Time slice(912.3) througha 3-D datasetshowing reflectivityvariationsalong(mostly)the samereflector.The pattern showsa meandering streamchannel.(From Brown,Dahm, andGraebner. 1981.)

Fig. 10.41 Tracing of a profiler section showing channel cut and fill. A present-daychannel can be seen in the sea-floor reflection at about 750 m depth, with indications ofa natural levee to the left of it. Earlier channels and leveesare also indicated.

graphic traps directly rather than relying on luck and statistics.

In the last few years,the searchfor stratigraphictraps has changedsignificantly.Patternsrecognizablein the seismicdata (910.7)sometimesindicate the environment of deposition and thus narrow the choicesof rock types,and this hasled to a number of discoveries where the stratigraphyhas been accuratelypredicted from seismicpatterns. Usually, seismicstratigraphic techniquesare combined with well data and geologic insight. Further improvementsin techniqueswill expand considerablythe circumstancesunder which this can be done successfully. 10.3.9Integration with other geophysicaldata Although it should go without sayingthat an interpreter should utilize boreholeand other geologicdata in his interpretation,he should also utilize other types of geophysicaldata, especiallygravity and magnetic data. Where available,thesedata should be examined to seeif they suggestanything not otherwiseevident and whether the gravity and magneticfields are consistent with the mapped features.The nature of a diapir is not alwaysevidentfrom examinationof seismic data alone,and other data may reducethe ambi-

GEOLOGIC INTERPRETATION OF REFLECTION DATA

390

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guities. A gravity model can be constructedfrom a seismicstructural interpretationby assigningdensity valuesto various portions of the section;the gravity field calculatedfrom this model then can be compared with the measuredgravity field. Sheriff (1989: 102, 106-9) showsan exampleof comparing seismicand gravity expressions of a salt dome.Sucha comparison may revealareasof disagreementthat call for a reexamination of the seismicinterpretation.The depth of basement,which may not be evident from seismic data, may be indicatedby magneticdata. Refraction velocitiesmay help in the identificationof the nature of certain reflectors.Especiallywhere seismicrecord quality is poor, such as in areasof karst or volcanics on the surface,magnetotelluricsoundingsmay be useful in reducinginterpretationalambiguities. 10.4 Modeling 10.4.1Intoduction Interpretation of seismic data invariably involves "model" of the portion of the earth a conceptual involved in seismicmeasurements.The model is a simplification of the actual earth in which the only elementsincluded are those expectedto be most important in affectingthe measurements.For example, the identificationof stackingvelocity with rms velocity is basedon a model in which velocitydoesnot vary in the horizontal direction,and staticscorrectionsare basedon a model in which travelthrough the weathering is vertical regardlessof raypath direction below the weathering.A model may be an actual physical model, mathematicalexpressions,or merely a rather vaguemental picture. Modeling is often subdividedinto two types, forward and inverse.Forward or direct modeling involves computing the effectsof a model and inversemodeling

involvescalculating a possiblemodel from observation of effects.Inversemodelingin a senseincludesthe entire interpretation processand invariably involves "modeling" without uncertaintyand ambiguity.Often a precedingadjectiveimpliesforward modeling.Modeling is important as an aid to understandhow various types of possiblefeaturesmight appearin seismic data (Edwards,1988;Fagin, l99l; Noah, Hofland, and Lemke. 1992). In forward modeling,expectedvaluesare calculated from the model and comparedwith actual measurements,differences("errors") being attributedto either inaccuraciesin the model or factors not accounted for. Modeling is usually iterative;the model is altered in an effort to account for errors,errors from the altered model are calculated,and so on, until the errors have been reducedto what is consideredacceptable. "prove" that Adequateagreement,however,does not the model correspondsto the actual earth; a different model might also provideadequateagreement. I 0.4.2Physicalmodeling Many geologicphenomenaare too complicatedto be amenableto theoretical treatment; hence, modeling sometimesinvolvesexperimentswith miniaturephysical models(fig. 10.a3).However,modelsmust be geometrically,kinematically,and dynamically similar to the systemsbeing modeled(Hubbert, 193'7)if the resultsare to be useful.Geometricsimilarity is achieved by making anglesin the model equal to those in the systemand correspondinglengthsproportional. If L is the ratio of lengths,the ratios of areasand volumes Kinematic are proportional to \2 and }'3,respectively. similarity concernsthe ratio of times, r, required to effectsimilar changesin position or shape.The ratios will be trit and tr/r2,and ofvelocitiesand accelerations angular velocity and angular accelerationratios will

MODELING

391

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be l/r and l/tr. Dynamic similarity concerns the ratio of massdistributions,p; this fixesdensityratios,pi\r. rorces actlng on correspondingmass elements must be such that the motions and ihanges in shape pro_ duced are geometricallyand kineiratically similar; lbrce ratio is p),/f. Dimensionlessquantities (like Poisson'sratio) must have the samenumerical value. Thus, there are only three independentvalues, 1,. r. anop. might, for example,rvishto representI km by .^We l0 cm in a model; hence,the ratio of model to actual distanceis L: l0 a. In practice,seismicvel,ocrty is restricted by available materials, and the ratio of model to actual velocity can range only by a very small amount, that is, ),/r = l. Becausef has already beenselected,this restrictst. If the model material has the samevelocity as the earth, t : l0 a and we must usefrequenciesl0a timeswhat is usedin the earth (be_ cau.se_frequency ratios dependon l/r). The densityof availablemodel materialsis probably about the same as that of earth materials;bicauseit " O.nritv ,utio p/\3 : l, this determinesthe massratio p -li-,r. 11 rve wish to model severaltypes of things simultane_ ously,suchas variousmodesof wav. prJpugution, u,_

wave simulation, and sourcesand receiversare moved over them to obtain seismicdata, motions being controlled by a computer to simulate various field-recording u..ung.rn.ntr. Directronal transducersare also held directly against solid models to study S-wavesand wave conversion.

tenuation,and so on, we must be surethat the relevant physicalpropertiesare consistentwith our model ra_ tios ),, r, and p. Examplesof physicalmodeling are shownin figs.6.2I , 6.28, and6.43. I 0.4.3 Computermodeling More commonly,modeling is done by computer and severalexampleshave been given (for example,figs. 2 . 3 2 ,3 . 3 ,6 .18 , 6 . 2 3t o 6 . 2 5 ,6 . 3 7 ,6 . 3 8 ,6 . 4 0 ,a n d6 . 4 1 ) . Many types of algorithmsare usedin computermod_ eling, ranging from simply convolvinga waveletwith a sequenceof reflection coefficients,tracins ravs through models where the raypathsbend in u..or_ dance with Snell's laq to full-waveform methods basedon relationssuchas Kirchhoff's equation(2.42\, or wave-equationmethods such as used in misration ($9 13.4)(99.13.a;Gazdag,t98t). Synthetic seismogramsmay help in determining . how stratigraphic changes might affect ,"ir-i" .."_ ords,.and raypath modeling ($10.4.5)in determining the distortionsthat complicatedvelocity distributions produce.Where the algorithmsand moiels are good, the resemblanceto actual seismograms is good. Mod_

I

392

GEOLOGIC INTERPRETATION OF REFLECTION DATA

eling is an invaluablepedagogicaltool (seeHilterman, 1970),but it invariablyinvolvesassumptionsand approximations that should not be forgotten when drawing conclusions. I 0.4.4 Syntheticseismograms The one-dimensionalsyntheticseismogramwas introduced in $6.2.1;it is simply a waveletconvolvedwith the reflectivity assumingzero offset and horizontal layering, the velocity and density values most often being thosemeasuredfrom boreholelogs in a well. Its most common use.isin identifyingreflectionson CMP sectionswith specificinterfacesin the earth, but a perfect match shouldnot be expected.CMP sectionstend to averageamplitudesover a range of offsetsand involve dip effects that are not generally allowed for with synthetic seismograms,as well as including can be noisesof variouskinds. Syntheticseismograms generalizedto allow for multiples and other types of eventsthat may be superimposedon the primariesonly synthetic.Horizontal changesmay be made in the reflectivityso that a sequenceof one-dimensional synthetictracessimulatesa common-midpointline recorded over the changingreflectivity. Two-dimensional synthetic seismogramsare not limited to vertical travel nor to zero offset.They permit modelingdiffractionsand the dependenceon offand amplitude.Mode set of arrival times,waveshape, conversionmay be allowed for. A variety of methods are used, some p ritting dipping interfaces and some employing srmptifications(such as the scalar form of the waveequationallowing for P-wavesonly). Trorey (1977) approximatedreflectors'bya seriesof semiinfiniteplane strips and basedhis method on the Kirchhoff equation (2.42). Other methods utilize wave-equationmethodsof the type usedin migration processing($9.12.3and9.l2.4). A common model is the exploding-reflectormodel; each reflecting interface is assumedto be a distributed source detonated at time l : 0, the sourcedensity being proportional to the reflectivityat the interface;seismicwavesare radiated upward at half the actual velocity (to give the traveltime for two-way travel). The record receivedat the surfacesimulatesa common-midpoint sectionin many (but not all) regards.Wavesmay be tracked by someof the methods.More elaboratemethods allow for mode conversion,the variation of reflectivitywith incident angle,surfacewaves,head waves,and so on. I 0.4.5 Ray-tracemodeling Wherevelocity variesin other than a very simpleway, the tracing of raypaths through a model obeying Snell'slaw at eachvelocitychangeis one way of developing an understandingof how a seismicsectionrelatesto a portion of the earth wherevelocity complications exist. Horizontal changes in velocity especiallycan distort structural pictures ($10.5)and make it difficult to appreciatethe significanceof structural evidences.

Taner, Cook, and Neidell (1970) carried out ray tracing for severalmodels, one of which is shown in fig. 10.44. An illustration of ray-trace modeling for a thrust-fault situation is shown in fig. 10.45. Downward ray tracing for offset traces is not feasible excepton a trial-and-errorbasis becauseinitially we do not know the startingdirection.Where sourceand detector are coincident, as is assumedon commonmidpoint sections,the raypath to and from a reflector must be coincidentand so must strike the reflectorat right angles.This makesit easyto trace rays upward. If incrementsalong the reflectorsare constant,then the density of emergentraypathsat the surfacemay give a qualitative indication of amplitude variations. Note the buried-focuseffectsin fig. 10.44d.Ray-trace modeling is also used in depth migration (see fig. 9.59c). Ray-trace modeling is useful in seeing how stacking-velocitymeasurementsget distorted by velocity complications(May and Covey,l98l). Taner et al. (loc. cit.) also tracedraysto obtain simulatedgathers on which to make velocity analyses(fig. 10.a6). They did this iterativelyon a trial-and-errorbasisbecausethe reflection-pointshifts updip with offsetin a manner not easily predicted.The stacking velocities determinedfor fig. 10.46do not bear any simplerelation to rms velocities,as was pointed out in $5.4.4a, and. in fact. the curve of arrival time versus offset is not even a hyperbola, so that the valuesof stacking velocity obtained from best-fithyperbolasdependon the mix of offset data used in the calculation. Most ray tracing doesnot allow for mode conversion. 10.5 Laterat variations in velocity 10.5.1Gradualchanges Often, velocity variations in the horizontal direction are sufficientlygradualthat their effectscan be treated as a second-ordercorrection. This situation is especially common in Tertiary basins filled mainly with clasticsedimentsthat havenot been subjectto uplift. The horizontal variations often result from gradual changesin the lithology,for example,as distancefrom the sourceof the sedimentsincreases.Sometimes,the vertical velocity function is changedslightly from location to location and the horizontal gradient is otherwiseignored in plotting data. A common variation of this techniqueis to map reflectorsusing a single function for the area and then to add locationdependentdepth correctionsto the mappedvalues. Lateral effectsalso resultfrom changesin the thickness of a water layer on top of the sediments.The changein velocity with depth effectivelybeginsat the seafloor, variationswithin the water layer usually being insignificant.The velocity of the sedimentsis not affected greatly by the amount of water overburden; the difference between overburden and interstitial pressuresis usually the factor determining velocity (see $5.1.2) and, becausethe water layer increases both pressuresby the same amount, it does not

L \TERAL VARIATIONS IN VELOCITY

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quent time section; (e) gathers for modeling offset traces; note the triplication of reflecting points for the left gathers and the reflection-point smear for the steeplydipping one.

:hangethe differentialpressureor velocity.Ofcourse, :he averagevelocity down to a reflectoris affectedby :nclusionof more travel path at water velocity.Figure 10.47showsa seismicline that goes from shallow to deep water; much of the apparent dip is a velocity elTectrather than real dip (comparefigs. 10.46aand 10.46b).The apparentdip can be correctedby chang:ng the velocity function with water depth when making depth calculations.

Lateral velocity changesalso affect the horizontal positions of features(fig. 9.59).This is illustratedfor a diffractingpoint and a simpletwoJayermodel in fig. 10.48(seealso problem 10.l4). The crestof a diffraction usually locatesthe diffracting point, but lateral changesofvelocity shift the crestofthe diffraction. If we consider more complicatedmodels, for example, two dipping layerswith different strikes,seriousdistortions exist that rwou\d be lery dimcult to unravel

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10.5.2Suddenchanges Where lateral changesin velocity are more sudden, correction may not be simple.Considerthe effectsof the sea-floorrelief in fig. 10.49.The velocitiesof the sedimentsimmediatelybelowthe canyonprobablyare markedly different from those of their lateral equivalents becauseof the differencesin overburden,but at large depths,the effectsof the canyon probably vanish. Furthermore,the sedimentsbelow the bottom of the canyon may be in fluid-pressureequilibrium with their lateralequivalents,thus havingfluid pressureappropriate to the uneroded thickness whereas their overburdenpressureis lessbecauseof the erosion,so that they are overpressured.A "correct" method of removingthe velocityeffectis not evident,and usually

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the method adopted is the empirical one that producesthe most sensibleresults.In areasof purely erosional relief,suchas that in fig. 10.49,this may not be too difficult, but where structural complicationsaccompany,and perhapscause,the sea-floorrelief, objective criteria may be lacking. The flow of salt into lensesand domesmay produce velocity anomaliesin the sectionbelow them. Salt velocity, about 4.5 km/s, may be either higher or lower than that of the laterally adjacent sediments,and so the velocity anomaliesresulting from a salt lens may be either a pull-up or a push-down.A pull-up is most common becauselower-velocityclastic rocks are the most common lateral equivalent,but lime-rich sediments, anhydrite, or other high-velocity rocks may producepush-down,and in someareas(suchas Mississippi)both can occur.Figure 10.50showsa salt pillow with a consequentpull-up. Similar velocityeffects

can result from other situations,such as reefing(see figs.10.359and 10.35h),whereeitherpull-upor pushdown can occur,dependingon how the velocity in the reef (which dependsin part on the reef's porosity) compareswith the velocity of the lateralequivalents. Velocity complicationsmay be very drastic in regions of compressionalor thrust tectonics. Figure 10.51showsvelocity effectsresulting from the overthrusting of high-velocity sedimentsin the Rocky Mountain thrust belt. The complicationsinvolve not only velocity pull-up, but also fictitious faulting evidences,phantom diffractions,and other effects.The usual solutionsto such extremecomplicationsare to trace rays through a model of the sectionin an effort to achieve reasonableagreementwith what is observed(see$10.4.5),but this approachis subjectto the uncertaintiesof modeling,mainly lack of information as to how to construct the model and determineap-

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GEOLOGIC INTERPRETATION OF REFLECTION DATA

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propriate velocities; such information is apt to be lacking whereneededmost. 10.6 Three-dimensional interpretation of 2-D data Reflectingpoints lie updip from the points where the reflections are observed. Migration accommodates the components of dip in the in-line direction (although not alwayscorrectly),so that most of the problemsin three-dimensionalmapping result becausethe component of dip perpendicular to the line is unknown or not taken into account properly.The subsurfacetrace(line of reflectingpoints) liesin the direction updip from the seismicline and this should be allowed for in mapping data oriented along seismic lines (seeSheriff, 1978,chap. 2l), that is, data ought to be posted on maps at the reflectingpoints on the reflector being mapped rather than at the source points on the surface.Wheredata are limited to a grid of a few lines,the first stepin mappingis to determine reflectingpoints wherecross-informationpermits this to be done, as at line intersections(fig. 10.52).Reflectingpoints in betweenthe points wheresuchdeterminationscan be madethen often can be inferredwith adequateaccuracy.An alternativethat is sometimes

feasibleis to map unmigrateddata and then migrate the maps to produce a correct structural map (see f i e .9 . 6 1 ) . The techniquesof acquiring data specificallyfor three-dimensional(3-D) analysis are discussedin in $12.2,of 3-D display $12.1,of 3-D data processing in $12.3,and of 3-D interpretationin $12.4and 12.5. As of 1994, 3-D is one of the fastestgrowing areas of geophysics.Most 3-D work has been (and is still) devoted to detailing fields after hydrocarbonshave been discoveredin order to optimize field development and exploitation (Brown, 1991; Sheritr, 1992), and 3-D techniques have been very cost-effective. Thereseemsto be unanimousagreementthat 3-D surveys result in clearer and more accuratepictures of geologicaldetail and that their costs are more than repaid by the eliminationof unnecessarywellsand by enablingthe recoveryof more hydrocarbonsthrough the discoveryof isolatedpools that otherwisemight be missed.3-D techniquesare now also being usedfor explorationin a number of areas. In areaswhere2-D lines are particularly denseand of good quality, interpolation is sometimesused to yield a uniform data grid that is then processedas 3-D data. Clearly,the 2-D data must be compatible, that is, must involve the samefrequencies,sameembeddedwavelet,and so on. The result,called a 2%-D survey, has some of the benefits of, but is usually markedly inferior to, a true 3-D survey. 10.7Stratigraphic interpretation 10.7.1Introduction Extracting nonstructural information from seismic data is called seismic'stratigraphy or seismic-Jacies analysis.Faclesrefersto the sum total of featuresthat characterizethe environmentin which a sedimentwas deposited.Faciesinvolves,among other things, sedimentary structure,the form of bedding,original attitude, and the shape,thickness,thicknessvariations, and continuity of sedimentaryunits. Our interesthere is in inferring stratigraphy rather than in locating in $10.3.8),though the stratigraphictraps (discussed implications for stratigraphic traps are obvtous. Books on seismicstratigraphyinclude Sheritr (1980) and Hardage(1987);many casehistoriesshowingapplicationsin various circumstancesare given by Halbouty ( 1982),Berg and Woolverton (1985),Van Wagoner et al. (1990), and the three volumes edited by "classic" reference is Payton Bally (1987 9). The (1977) and one of the best referencesis Wilgus et al. (1e88). Depositionalpatternssuchas progradation,pinchouts, and channelscan sometimesbe seenin seismic data (for example,in figs. 10.40and 10.53)although many stratigraphicfeaturesare too small to be resolvable(Sheriff,1977)or too gradualto see.Depositional patterns are sometimesassociatedwith depositional energy(which determinesthe degreeof separationof fine particles from coarse),lithology, porosity, and

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F i g . 1 0 . 5 2 3 - D e f l e c r si n i n r e r p r e r i n g2 _ D < l a t a .( a ) 2_D mi_ g r a t e d .i n t e r s e c t i n gs e i s m i cl i n e s w i t h o n e h o r i z o n timed at the s o u r c e - p o i n tl o c a l i o n s : ( b ) c o n t o u r s o f d a t a p l o t t e d at source

other physicalpropertiesthat are important in hydro_ carbon reservoirs. Seismic stratigraphy is often divided into several suDareas: l. Seismic-sequence analysis, separatingout time-depositionalunits based on deiectrns unconformities or changes in seismic patl terns; 2. Seismic-facies analysis,determiningdeposr_ tional environment from seismic_ieflection characteristics; 3. Reflection-character analysis,examiningthe lateral variation of individual reflection events,or seriesof events,to locate where stratigraphic changes occur and identifv their nature;the primary tool for this is mod_ eling by both synthetic seismogramsand seismiclogs, as alreadydiscussedin $6.2.1, 5.4.5,and 10.4.4.

p o i n t s s h o w f i c t i o u ss t r u c t u r e ta n d ( c ) w h e n d a t a a r e p l o t t e d al r e f f e t ' t i n gp o i n t s .c o n t o u r s i n d i c a t es i r n p l es t r u c t u r e .

4. Detection of hydrocarbonindicators,which will be discussed in $10.8. I 0.7.2 Sequence stratigraphy Seismicstratigraphyhasevolvedinto sequencestrarlg_ raphy as depositionalfeaturesthat were first seenin seismicdata have been recognizedin outcropsand in well-logand paleontologicaldata. The conceptunderlyingsequencestratigraphyis that sedimentdepositionis controlled by four factors: l. Subsidence of the crust becauseof tectonic and/or isostatic reasons; this creates the space(accommodation)availableto receive sediments.Thermal expansion and/or tectonic forcescauserocks to rise so that their higher portions are exposed and eroded. Upon cooling,rocksbecomedenserand sink to restoreisostaticequilibrium. The cooling

402

GEOLOGIC INTERPRETATIONOF REFLECTION DATA

t.5 D'

F i g . 1 0 . 5 3 S e c t i o ns h o w i n g p r o g r a d i n g u n t t A A ' w i t h t o p l a p w h e r e r e l l e c t i o n sp i n c h o u t a t t h e t o p o f t h e u n i t a n d < l o w n l a p w h e r e r e f l e c t i o n sc o n v e r g ea t t h e b a s e .N o t e a l s o t h e c h a n n e la t

'

is a very slow process becauseof the extremely low thermal conductivityof rocks (Jurassicoceanicbasementis still losing its heat of formation). Isostaticadjustmentis also a very slow process,although not as slow as conductiveheat loss (isostaticrebound from Pleistocene continentalglaciation is still going on). The filling up of the accommodationspaceadds to the load, producingfurther subsidence and additionalaccommodation;sedimentvolume ultimately may be up to three timesthe original accommodation. 2. Sediment inflow, which provides the sediments for infilling the accommodation. 3. Eustasy,risesand falls of absolutesea level. The combination of subsidenceand eustasy determinesrelative sea level. which in turn fixes the amount of accommodation. Eustatic changesare generallyrapid compared with subsidenceand they predominatein determiningthe localeof sedimentdeposition. The depositionalpatternsleft in sedimentary rocks thus mainly document the eustatrc variations. Sea-levelchangesoccur on several time scales,and the more rapid sealevel rises and falls (paracyc'le.s) produce changes within the sequenceof rock being deposited; these changesmay produce seismic reflections. Seismic reflectionsthus indicate the

surfaceof the solidearth at the time of sedim e n td e p o s i t i o n( { 1 0 . 7 . 3 ) . 4. Climate,which mainlydeterminesthe nature ol the sedimentsbeing deposited.For example, carbonatesare apt to dominate in w a r mc l i m a t e s The interplayof thesefactors,especiallythose of factors2 and 3, determines depositionalpatterns.The effectscan be seenin rock outcrop,well-log,paleontologic,and seismicdata. The heat injectedby thermal/tectoniceventsis conducted away very slowly, producing gradual subsidence.Theselong-term subsidenceeffectsoccur at a rate that is relatively constant compared with the more rapidchangesresultingfrom eustasy(fig. 10.54). The net effecton sealevel is the superpositionof the tectonic(thermal)and eustaticeffects.Whereabsolute sealevelis rising, it will increasethe rate of growth of accommodation,but whereabsolutesealevelis falling the net effect will dependon the relativeratesof the tectonic subsidenceand the eustatic fall. Where net sea-level fall results,we havea "type-1" situationand a significant type-l unconformity involving subareal erosion; where the effectsof the two causesnearly canceleach other,a "Iype-2" situation results. Clearly, relative (rather than absolute) sea-level changesare what affects deposition, that is, subsidence will produce the sameeffectsas sea-levelrise. We expecteffectsto be local if causedby local subsidencebut widespreadif causedby eustaticchanges.

>I R \ T I G R A P H I C I N T E R P R E T A T I O N

I

403

Onesequence

u..f9

Hsr | rcr I rsr I xsr lsusrlrsr : : . t ) . 5 ; l C h a n g e si n r e l a l i v es e a l c v e l .( a ) A s s u m e de u s t a t i c r - : r : l o l l . w h i c h i s r a p i d c o m p a r e dw i t h t e c t o n i cs u b s i d e n c e( b ) . --,...nic s u b s i d e n c ei s n e a r l y l i n e a r o v e r t h i s s h o r t t i m e s p a n . , - . u q h l o g a r i t h m i co v e r a l o n g t i m e . T h c s u m o f t h e s et w o ( c ) : . l . t h e r e l a t i v ec h a n g co l s e a l e v e lt h a t p r o v i d e st h e a c c o m m o : . : . r n s p a c ef o r s e d i m e n t sT . h e w e i g h t o f d e p o s i t e ds e d i m e n t s l r o d u c c f u r t h e r s u b s i d e n c eb y i s o s t a t r ca d j u s t m e n t .a l s o a - : : - t c n n p r o c e s s .S y s t c m t r a c t s ( . s 1 0 . 7 . 5a) r e i n d i c a t e d a t t h e -

::!)nt.

\1rn) of the effectsof sea-level changesappearto be - :rtenlporaneous in widelyseparated basins. -

-1Time .signifit'unceo./refec'tions

-:-.pliedin seismic-sequence analysisis the concept :-.1rthe attitudeof seismicreflections is that of depo..ilt'rrol time surfacesrather than facies surfaces.A ' '1i! sur/aceindicatesa surfacethat at one time was :: surlaceof the solid earth. The passageof major :i-rrfirs,floods,and other short-term eventsredistrib-:e sediments within very short periodsof time along .:ne surfaces,whereasthe long periodsbetweensuch i..ents often do not leave a record becausethe new .edimentsbrought in are rearrangedby the next ma-.r event. Stratal surfacesthus follow time surfaces. Because the thicknesses of stratalunits are generally :nall. much smallerthan the seismicresolvingpower, ::e1 only producevery minor reflectioncontributions, rut thesetend to interferein essentiallythe sameway rrer a widespreadarea becausethe stratal surfaces are generally parallel over a wide area and change r ery slowly laterally.The interferenceproducesthe coherent lineups of reflectionevents.The fact that seis-

Fig. 10.55 The nature of'facies surfaces. (Data for a and b from Vail, Todd, and Sangree, 1977b.)(a) Facies surface based on data from two wells 17 km apart: the SP-log curves distinguish the sand from surrounding shale. (b) Redrawing of the facies surface based on intervening well-control points; the major portions parallel stratal or time surfaces.Seismicdata show reflections parallel to the time surfaces onlapping the unconformity. (c) Classical picture of sand-rich sediments tn a prograding/aggradingsystem suggestsa reflection along the facies boundary ,1,4', which does not show (d) Occasional major storms and other catastrophic events rework the sand-rich sedi ments and spread them along time surfaces, which is the attitude of reflections.

404

GEOLOGIC INTERPRETATION OF REFLECTION DATA

mic reflectionsparallel time surfacesis well established by many observations but it is somewhat contradictory to intuitive feelings that reflections should be due to changesin rock nature,suchas from sandto shalealong faciessurfaces.Faciessurfacesare often basedon fairly widespreadcontrol (for example, on well control) so that the detailedinformation as to how to draw the facies surfaceis not available.The major portions of correctlydrawn faciessurfacesparallel time surfaces(figs.10.55and 10.56). I 0. 7.4 Depositionalmodels The angularities between reflections where one of them terminates(fig. 10.58)are the principal seismic evidencesof seismic stratigraphy.Where the data quality and resolution are sufficient,we observethat reflectionsgroup themselvesnaturally into packages bounded by systematicreflection angularities;these packagescorrelatewith sedimentpackages,groupsof sedimentsdepositedcontemporaneouslybetweeneustaticevents. Two types of angularitiesoccur at the bottom of reflectionpackages:onlap and downlap.Onlapis indtcated by reflectionsthat are horizontal or dip away from their terminations and dow'nlapby reflections that dip toward their terminations.Except for dip direction,onlap and downlap often look very similar in seismicdata. Subsequentrotation, becauseof differential subsidenceor other reasons,may havechanged the dips so that presentattitudesno longer havethe original implications, that onlap indicates locations that are proximal (depositionclose to the source of sediments,that is, on the landwardside of a sediment package)and downlap locationsthat are distal (deposition distant from the sedimentsource). Three types of angularitiesare seenat the top of reflectionpackages:toplap, erosionaltruncation, and apparent truncation. Erosional truncation indicates that the sedimentpackageformerly extendedhigher than it doestoday but that portions were removedby erosion,whereastoplap indicatesdepositionnear sea level and that the sedimentpackagenever extended significantlyhigher in the section.With good seismic data quality, toplap sometimescan be distinguished from erosionaltruncation becausetherewerechanges in the depositionalenvironmentnear toplap and consequentlyreflectionsare changeablein attitude and character,whereasno such changesoccurred at erosional truncation terminations. Apparent truncation (like downlap) is a consequenceof sedimentstarvation, that is, insufficient sedimentswere availableto continuea resolvablereflection.Sometimesreflections terminate because of starvation other than at boundaries,as in divergentpatterns sediment-package associatedwith differential subsidence.Over large portions of sediment-package boundaries,reflections parallel the boundaries,a situation called concordance. Diagrams from Vail, Mitchum, and Thompson

(1977) indicate the depositional patterns expected from sea-levelrisesand falls (fig. 10.57).A relativerise of sealevelcan be producedby either an absoluteseaThe primary evidence levelrise or by land subsidence. in seismicdata for a sea-levelrise is a coastalonlap pattern, the progressivetermination of reflectionsln the landward direction. Whereas a sea-levelrise is (coastlinemovusuallyassociatedwith a transgression ing landward) over an unconformity (fig. 10.57a),it can also be associatedwith coastalregression(coastline moving seaward;seefig. 10.57b)provided the influx of sedimentsis large enough. Coastal onlap is seen in both situations.The primary evidencefor a stillstandof sea level (fig. 10.57c)is toplap.A fall of sealevel(fig. 10.57d)would exposepreviouslydeposited sedimentsto erosion, so erosionaltruncation is the primary evidencefor a sea-levelfall. Where sediment packagesare thick enoughand noisesufficiently low reflectionsshowingthesefeaturescan be seenin seismic data and used to determine the sea-level changes. The techniqueusedby Vail et al. (loc. cit.) is to first mark reflectionangularitieson a seismicsectionand then draw the unconformity boundariesthat join onlaps (heavy solid lines in fig. 10.58)and downlaps (dov,nlapsurfaces,indicated by dashed lines in fig. 10.58).A line connectingonlapsmarksa relativesealevel fall and indicatesan unconformity (and a sequenceboundary);this boundary is then continued through regions where angularities are not evident (the correlativeconformity).The procedurecontinues with mapping the boundariesover a grid of lines,constructing maps of structural relief and isopachthicknessfor the intervals betweenthe boundaries.subdividing the sequencesaccording to seismic-facies evidences($10.7.6),relating them to adjacent sequences,and finally attributing stratigraphicsignificanceto them. This procedureis illustratedby the examples shown in fig. 10.59.(Some people use the downlap surfaceof maximum flooding, which generally separates transgressiveand highstand tracts (S10.7.5), boundary.) as the sequence The package of sedimentsbetween the sequence that is, a three-dimensional boundariesis a sequence, set of faciesdepositedcontemporaneouslyand linked by depositionalprocessesand environments.For example, fluvial deposition on land, deltaic deposition near the coastline,a strand plain off to the side,reefs near the shelf edge,and turbidite fans at the baseof the slope might all be contemporaneousand parts of the samesequence.Downlap surfacesoccur within sequences. Vail et al. (loc. cit.) believe that many sea-level changeswere contemporaneousworldwide and they developeda eustatic-levelchart showing the worldwide pattern (Haq, Hardenbol, and Vail, 1988).By correlatinglocal coastalonlap chartswith the master chart, one can sometimesdate reflection eustatic-level eventswith ratherhigh precision. In the early daysof seismicstratigraphy,the magni-

STRATIGRAPHIC INTERPRETATION

405

Pebble Shale Kingak

I-ig. 10.56 Line on the North Slopeof Alaska. The top of the Torok lbrmation in contact with transgressivesediments (and t h e n n o n m a r i n eN a n u s h u k s e d i m e n t s i)s a i a c i e ss u r l a c e .A s t h e sea level has varied, the deltas have stepped successivelyhigher

in the section, thus jumping fiom one time surface to another. ( U S G S d a t a . ) ( a ) S e i s m i cl i n e a n d ( b ) i n t e r p r e t a t i o no f t h e t o p Torok formation.

tudes of sea-levelfalls and rises were determined simply by measuringthe vertical distancesbetween onlap points; this led to estimatesthat wereunreasonably large.Howevegonlap can occur both well below sealevel (marine onlap) and abovesealevel,and recognition of this removedthe early objections.Periodic sea-levelchangesare now widely accepted.They appear to occur with severalperiodicities(table 10.3). Wilgus et al. (1988) (and especiallyGreenleeand \{oore, 1988)discusscalculatingsea-levelchanges.

ability of sediments.Accommodationis dominatedby the rate of eustaticchange.Let us conceptuallymodel one simple eustatic cycle of sea-levelfall and rise, starting with a sufficiently rapid eustatic fall that, produces when superimposedon tectonicsubsidence, a net fall of relativesealevel(as in fig. 10.54).The sea will regressand exposemore area to aerial erosion.If the sealevel falls below the shelf edge(fig. 10.60a),a type-l uncon/brmitywill result and sedimentswill be depositeddirectly in deep water rather than on the the result is depositionof bypa.ssing); shelf(,sediment a lov,standor basin-fioor.fan (or a slope fan). Coarse sedimentswill probably be included in the lowstand fan. The steepslopeswill result in instability of sedimentsrestingat the angleof repose,producingslumps

10.7.5Systemtac'ts Depositionalpatternsdependon whether accommodation is growingor decreasingas well as on the avail-

Final

r,,r_-*#*

Relativerise

Q)

(b)

(c''

Terrigenousinflux *

Erosional truncation

I n it i a l Relative fall of sealevel

(d)

E Nonmarine "o..t.l o"norit/?O B

t$

-"u'?a/,surface

Littoral deposits

El Marine deposits (e,|

Fig. 10.57 Patterns associatedwith relative sea-levelchanges. (After Vail, Todd, and Sangree,1977b.)(a) Relative sea-levelrise produces a transgressionif terrigenous influx is low and (b) a regressionif terrigenousinflux overwhelmsthe effectsof the rise. (c) Progradation associatedwith stillstand ofsea level. (d) Grad-

ual sea-levelfall produces downward shift in pattern but tops of patterns are eroded. (e) Rapid sealevel fall produces major seawardshift in the locale ofonlap; the pattern indicates a gradual rise, then a sudden lall between units 5 and 6, followed by another gradual rise.

Table 10.3Eustaticcycles Order of cyclicity

Period

Magnitude

Cause

Terminology

=100m -50 m

Plate tectonicsand continentalbreakup Tectonicsand eustasy Glacial eustasy Orbital (Milankovitch) cycles Climatic changes

Supercycles Cycles

-50 Ma

2 3 4 5 o

3-50 Ma t/z 3 Ma 80-500 ka 30 80 ka l0-30 ka

I

J

Paracycles

SUnFACE

rnutCairijr

.,, 5E Terminology for reflection terminations and therr r: to sequenceboundaries and downlap surfaces.(After .rm. Vail. and Thompson, 1977.)

-

1.O

3{.IGOCENE : O N O E N S E D. SECTION

6 o z O - 2.O U I 3 0 . OM a U

= tr

OLIGOCENE

W E D G E ?- 3 . O

(a)

2.2

:.;'::-i;:'*:

*-i'{h

qSsR

iS

t.li -\'\:nl -r"'L.

(b) Frg. 10.59 Picking reflectionterminationsto indicate se(a) Sectionon eastcoastof New Zealand(from Loutit quences. et al.. 1988):and (b) sectionfrom Midland Basin,Texas,show-

I

ing reflectionterminations(arrows)used to locate sequence boundaries(from Sarg.1988).

Ty?e 1 SequenceBarndary

Mlxlmum Floodlng gurlecc

Iyp€ 1 SequenoBoundary TopSlopeFan Downlap Surlace TopBasinFloorFanDownlap Surface

TypeI SequenceBoundary

Fig. 10.60 Schematic of system tracts on a passive margin. (After Posamentier.Jervey, and Vail. 1988.) (a) Sea-levellowstand producing basin-floor and slope mounds, level-channel complexes, and (not shown) lowstand prograding wedges; (b)

TopSlopeFan

transgressivetract topped with a maximum flooding surface; and (c) highstand tract of aggrading followed by prograding deposition.

! . R .{ T I G R A P H I C I N T E R P R E T A T I O N

409

,_: slides.a condition that will encourageturbidity Roksandic(1978),Sangreeand Widmier (1979), and _... -11this time, riverswill cut (incise)"valleys into Brown and Fisher (1980); table 10.4 concernsfacies --; erposed shelf. classifications. Classificationsare sometimesbasedon \s the rate of eustaticfall becomesabout the same reflectionterminations(fig. 10.5g),reflectioncharac_ :: i:3 r&t€of subsidence,relativesealevelwill remain teristics(abundance,continuity,amplitude,amplitude -. -_:hl1constant and a lowstanl will result.Incoming consistency, and so on), stratal patterns(fig. l0.6ta), -:,:::'tent volume will decreasebecausegradients of and the-external ':: .:nd shapesofsequences (ng. t-0.OtU). subjectto erosionwill no longeibe increas_ Parallelreflectionssuggestuniform depositionon a . :urbidityflowswill continueand tend to build up stableor uniformly subsidingsurface,whereasdiver_ . ;.-.pof the lowstandfan or on the slope;the result gent reflectionsindicatevariation in the rate of depo: :3fosition of a,slopefanAs sealevelbeginsto rise, sition from one area to anotheror elsegradual tilting. :.;ri:reflts will begin to uggrade(build up--ward) and Chaotic reflectionssuggesteither relativelyhigh depo_ ' : .,ltde (build outward) to producea lowitandwedge. sitional energy,variability of conditionsduring depo_ > :reralley filling may also occur.The overallpa&_ sition, or disruptionafter deposition,suchas can be .:: of sedimentsinvolvinglowstandfan, slope fan, produced_byslumping or sliding or turbidity_current .:; iosstand wedgeis called a lov)standsystem tract. flow.A reflection-freeinterval suggestsunifoim lithol_ I: the eustaticfall had been smallerit might have ogy such as a relativelyhomogeneousmarine shale, ::::ll balancedout the effectsof subsidenc"e on sea salt, or massivecarbonates;however,distinguishing :'.el and a sea-levelstillstandand would have re_ reflection-freepatternsfrom multiples and nJise that .-.::d. The sedimentswould first progradeand then obscuresreflectionsmay be difficult. ::iidd€ as relativesealevelbeginsto riseslowly. This Reflectionterminationssuch as onlap (fig. 10.5g) .. -.uld produce a shelf-marginsystem trac.t.A type-2 and downlap (sometimescalled offiap), alriady ae_ ,':-,tnfbrmityseparatesa shelf_margintract from the scribedin $10.7.4,give a geneticcontext:onlao is the -:derlying highstandsystemtract (seewhat follows). the landwardedgeof a unit. whereasdownlapresults .\s a eustaticrise becomesgreater,it will have the from inadequate sediment supply (starvatibn) and ,.me effecton relativesea level as the tectonic subsi_ thus is the seawardedgeof a unit. ::nce: thus,relativesealevelwill riserapidlyand ac_ Oblique progradational patterns (fig. 10.62a)are -,.mmodationwill increaserapidly.The ioastline will characterizedby toplap angularity (aiso sometimes ::rnsgressover the shelf (fig. 10.60b)producing za_ called offiap) and reflection-character ":,;cflooding.Because variability.The the shelfcan now accommodate tops of obliquepatternsindicateperiodsduring which :!rre sediments,few sedimentswill be transported far sealevel was not changingmarkedly (stillstands)and ::!rm the coast,resultingin relativelythin deeo_water depositionnear the wave base,with consequenthigh :eposrts called a conden.sed .re(.tiun.The condensed depositionalenergy.Thus, the tops ofoblique patterns :3ctionis often rich in both numbersof fossilsoecr_ often contain relativelyclean sands.Sigmbid'progra_ :rensand species, and usuallyprovidesthe bestpale_ dational patterns,on the other hand, are character_ ..nrological agedating.The packageof sediments de_ ized by gentle S-shapedreflectionsof rather uniform :rrsited during the rapid sea-levelrise is called a character,the tops of the reflectionsexhibitins con_ -t'Lt nsgres.ltvesls tem t ra( t. cordancewith the top of the sequence unit. Thise in_ .\s the rate of sea-levelrise slowsdown, becomes dicaterelativesea-levelriseand usuallyconsistoffine_ :tatic,and beginsto fall, sedimentsfirst aggradeand grainedsediments,sometimescalcareous. :hen.prograde (fig. 10.60c). The packageoiiediments The three-dimensionalshapeof units providesthe .s calleda highstandsystemtact. A eustaticfall at the principalbasisfor classification in basiniettings(fig. :nd of the highstandsystemtractmarksthe top of the 10.62b).Units that drape over preexistingtopography S€Qtrsng. that beganwith the precedingsea_level fall. ar3 generally low-energy fine-grained pelagic untts. Of course,eustaticvariationswill not usually be a Those with mounded tops or chaotic refleclionsare srmplecycle,as assumedin the foregoing.Small, more generallyvariable-to-high-energy deposits. rapid oscillationssuperimposedon la.g-e.os.illtions High-reflection continuity suggests continuous result in parasequences. The sequenceiwill also not strata depositedin an environmentthat was relatively alwaysoccur in the foregoingorder and the local set_ quiet and uniform over a widespreadarea, such as ting and tectonicsituationswill affectthe patterns that marine shalesinterbeddedwith iilts and calcareous develop.Nevertheless,the system_tractconcepts are shales.Fluvial sedimentswith interbeddedclavs and central to sequencestratigraphy. coalssometimesproducestrongreflections. The lateral equivalents of units sometimespro_ 10.7.6 Seismic'-facies analysis vide the key to identification.Thus, a low_reflectron_ amplitude Seismicfucies ($10.7.1)concernsthe distinctive .facies representingprodelta shales may char_ grade landward into a facies of high continuiri acteristicsthat make one group of reflections look and amplitude resulting from interbeddedsilts and/ differentfrom adjacentreflections;inferences u, to tt . or sands, whereas a low-reflection_amplitudesand depositionalenvironmentare drawn from seismic fa_ facies may grade landward into a nonmarine. low_ cies.Analysis and classificationschemesare given by continuity, variable-amplitudefacies. The prodelta

G E O L O G I C I N T E R P R E T A T I O N O F R E F T , E C T I O ND A T A

4IO Table 10.4Seismic-faciesclassification Regionalsetting

Basisof distinction

Subdivisions

Interpretation

Shelf

Reflectioncharacter Unit shape:widespread sheetor gentlewedge Reflectionsgenerally parallelor divergent

High continuity, high amplitude

Generallymarine Possiblycut by alternatingneritic submarinecanyons shale/limestone, Distinguishon basisof interbedded high/low location comparedto energydeposits,or other lacies shallowmarine clasticstransported mainly by waveaction

Mounded shape

Self margin prograded slope

Other characteristics

Variablecontinuity,low Fluvial or nearshore amplitude,occasional clastics,fl uvial/wavehigh amplitude tlansport processes (delta platform), or low-energyturbidity current or wave transport

Distinguishon basisof location comparedto other facies Shale-proneif seaward of unit above Sand-proneif seaward of unit below

Low continuity,variable amplitude

Nonmarine clastics, fluvial or marginalmanne

Occasionalhigh amplitudeand high continuity lrom coal members

Variablecontinuity and amplitude

Delta complex

Internal reflections gently sigmoid to divergent Occasionalhigh amplitudes

Local reflectionvoid

Reef

Seefig. 10.35

Adequatesediment supply Shelf margin deltaic High energydepositsin updip portions Occasionallydue to strong currentsln deepwater

Moderatecontinuity and amplitude, reflectionsvariable Foreset(clinoform) dips to 10'(averaging 4 5"), steeperdips are calcareous Often fan-shaped (includingmultiple fans)

Internal reflectionpattern Oblique,fan-shapedor overlappingfans

Sigmoid,elongatelens/ fan

High continuity,high to Low sedimentsupply moderateamplitude, Low depositionalenergy uniform reflections

STRATIGRAPHIC INTERPRETATION

411

Table 10.4 Seismic-faciesclassifcation Regionalsetting

Basisof distinction

Basinslope,basin Overall unit shape iloor

Subdivisions

Interpretation

Drape

Sheetdrape Deep marine hemipelagic;mainly clay Low energy

High continuity,low amplitude Drapesover preexisting topography

Mounded

Contourite

Variablecontinuity and amplitude

Deep Low energy

Fan-shaped Variableenergy,slump/ turbidity currents

Fill

Slopefront fill

Low energy Deep marine clay and silt

On^lapping Low-velocityturbidity fill currents

Mounded onlap fill or chaotic fill

Canyon fill

Source:After Sangreeand Widmier, 1979.

Other characteristics

Discontinuous,variable amplitude At mouth of submarine canyons Compositiondepends on what was eroded up above

Variablecontinuity and amplitude Fan-shapedto extensive along slope

High continuity, variableamplitude

High or variable-energy Overall mound in a turbidites topographiclow, gougecommon at base Discontinuous,variable amplitude

Variablesuperimposed strata Coarseturbiditesto hemipelagic

Variablecontinuity and amplitude

GEOLOGIC

+tz

INTERPRETATION

OF REFLECTION

DATA

AGGRAOATIONAT OFFTAP SHETF E D G E oEuouE

SIGMOID OFFIAP

OFFTAP CHANNEL/OVERBANK COMPTEX

MASS FLOW

APPARENT TRUNCATION

S L O P EF R O N T FI L L

MOUND

(a)

Sheetdrape Onlap-fill

(low energy) Moundedonlap-fill

(usuallylow energy)

(high energy)

Chaotic-fill (hi8h energy)

Fan-comPlex (high energy)

(b) Fig. 10.61 Seismic-faciespatterns. (a) Patterns on sersmlcsections. and (b) three-dimensional shapes of basinal sequences (from Sangreeand Widmier. 1979).

shalemay gradebasinwardinto a prograded-slopefacies,whereasthe sandmay gradebasinwardinto highcontinuity,high-amplitudemarine facies. are shown classifications Examplesof seismic-facies i n f i g s .1 0 . 6 2a n d 1 0 . 6 3 . I 0. 7.7 ReJlect ion-character analysis Reflection-characteranalysis involves study of the trace-to-tracechangesin the waveshapeof one or more reflectionswith the objectiveof locatingand determining the nature of changesin the stratigraphyor fluid in the pore spaces.Specialdisplaysmay be used

to make it easierto seethe changes,such as enlarged displaysof the portion of the section being studied, (Tanerand Sheriff, displaysof attributemeasurements lgll:Taner, Koehler,and Sheriff,1979)suchas envelope amplitude, instantaneousfrequency,and so on displays(45'4.5),which often (S9.11.4), or seismic-log color. involve ($6.2.1)are often usedto deSyntheticseismograms termine the nature of the stratigraphicchangethat a change of waveshapeindicates.The various stratigraphicchangesthat are regardedas reasonablepossiUitlti.,sare modeled (Harms and Tackenberg,1972; Neidell and Poggiagliolmi,1977)and matched with

STRATIGR APHIC INTERPRETATION

4i3

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ll

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tldt

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f

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^i\

Ic*r...-?l

\t'i+*

i

/

+

I

/ \

'n",,,^,L Deposiulnat

\

D

+ ^

\"tPotitionarrimit

tinlt

sequ.nce not present

opdwn

r-

Degosrirona\ t$$l Delta plain

/ o \ ;""t""^ \t{$$

Delta sloPe

(c)

Fig. 10.62 Stratigraphic interpretation in east Texas. (After Ramsayer,1979.\ (a) Portion ol a seismicsectionl unconformities bounding the unit are mapped over a grid of lines; portrons of synthetic seismogramsfrom two wells are superimposed;the top and base ofthe sequencemapped in (b) are indicated at the Ieft. (b) Characteristics of reflections within the unrr are

mapped; Top = toplap at top of unit, C - concordance at top of unit. on - onlap at base of unit. dwn - downlap at base of' u n i t . T h i n - u n i t n o t t h i c k e n o u g ht o s e ei n t e r n a lp a t t e r n .O b : o b l i q u e p a t t e r n i n t h e b o d y o f t h e u n i t . p - p a r a l l e lr e f l e c t i o n s i n i n t e r i o r o f u n i t ; s o l i d a r r o w si n d i c a t ed i r e c t i o no f o n l a p . o p e n a r r o w s t h e d i r e c t i o n o f d o w n l a p . ( c ) F a c i e si n t e r p r e t a t i o n .

the observedwaveforms.Clement(1977)describesthe useof reflection-character analysisin mapping a sand associatedwith channelson an unconformity surface in Oklahoma.A distinctivereflectionwas present(fig. 10.64)where the sandswere more than 6 metersin thickness,in which situationthey usuallywere also porous. Severalsuccessfulwells were drilled on the basisof the predictionsfrom reflectioncharacter,but

one well encountered tight indurated interbedded sandstoneand shale that gavevery similar reflection character.This study thus illustratesboth successful applicationof thesetechniquesand alsothe ambiguity ofconclusionsbasedon reflectioncharacter. Maureau and van Wrlhe (1979) successfullyused seismiclogs to predict high-porosityzonesin Permian carbonatesin the Netherlands.Lindseth (1979) re-

E

t;

P - ll t o o e

l

Yg

;Y 300

paraJ.lel high

and dipping,

oaraIIel.

-E ll z o o

$Y

discontinuous,

rcderate

-

amplitude hi.oh amoiitude-

prograding,

ob).ique,

oblique

redium

amplri.ude

lov

amplltude,

t.o subparallel,

100

conLinuous

.

E

t I t

.-'Y E

dl scont inuous

300 paral).e1.,

moderate

amplitude,

conLinuous

300 disconlinuous.

very

loy

amplitude

400 (b)

350

depositional

estimated

environrent

titholoqy

gravel/

fluviatile

-

Tegelen

small aqui fers

Eaassruls

clay

marlne

hydrogeological scherotr i z a L l

sand/claY

100

eE

Formation

aqurLarq

? ll

o lzoo

:Y

shallow marine j.c

sand

shallow

sand

oosterbout

highly

Breda

Permeabl'e

Boom

aquj.c).ude

E !

U/o.

perreable

della

marine

I I l

V €

deltai,c

300 marine

clay / fj.ne

sand

400 (c) Fig. 10.63 Section showing seismic-faciescharacterizations. (From Meekes and van Will, l99l: private communication.) (a)

CJ'aY

300 350

Seismicsecrion; (b) seismicfacies;and (c) depositional envrronment, lithology. and hydrogeologiccharacter.

HYDROCARBON INDICATORS

)_>) !

)l)-l_r>>i_l !

415

>*t

rll.i D >-

) rl i..I-

i

Fig. 10.64 Evidencesof a channel sand. (From Clement. 1 9 7 7 . )( a ) P o r t i o n o f s e i s m i cs e c t i o na c r o s sa c h a n n e l s h o w i n g developntent of an event where channel sand is more than 6 m r h i c k : t h e w e l l a t I j u s t m i s s e dt h e c h a n n c l .( b ) M o d e l s o f l o r s

portsmappingporosityin Devoniancarbonates in Al_ berta and other reflection-character analysisstudies usingseismiclogs.Seismiclog examplesare shownin platesI and 4. l0.E Hydrocarbon indicators The velocityand densityof sedimentary rocksdepend on porosityand on the propertiesof the fluids filling the pore space,as was discussedin $5.2.4and 5.2.1. The dependence ofporosity on densityis the straight_ lbrward relation given by eq. (5.6), but the depen_ denceof porosityon velocitywherethereis a mixture of fluidsin the pore spaceis not as simple.The change in velocityresultingfrom a changein the interstitial fl uid is often marked,as shownin fig. 5.27, resultingin amplitudeanomaliesassociated with accumulations. The extensiveuseof automaticgain control obscured theseamplitudeeffectsuntil about 1970whenrecosnl_ tion of their usefulness in locatingaccumulationiof hydrocarbonsbecamewidespread.Becausethe anom_ aly is most often one of locally increasedamplitude (asin fig. 10.65),they werecalled,,brightspots."Soon other types of anomalieswere found to be associated rvith hydrocarbonaccumulationsunder some condi_ tions(seetable10.5and Blackburn,1986a).The rela_ tion between hydrocarbon indicators and hydrocarbon accumulationsis not simple and universal, however,and many bright spots turned out to result lrom changesother than commercialhydrocarbonac_ cumulations. The effect of interstitial fluids on velocity depends on the structure of the rock and is generallygreater and simplerfor relativelyunconsolidatedclasticrocks. Thus, the effectsare generallygreaterfor young rocks than for older, and hydrocarbonindicator technology is especiallyapplicableto Tertiary clasticbasins;these are mostly offshorearound the peripheryof the conti_ nents,but being offshoreis irrelevantexceptthat marine data are often ofbetter quality than land data and henceanomalieseasierto see.The effecton velocityis generally greater (and more complex) for gaseous

for various sand thicknessesand seismic traces at locations l, I , a n d C , t h e f i r s t o f e a c h p a i r i s a s y n t h e t r ct r a c ea n d t h e s e c o n d an observed seismictrace.

Table 10.5Hydrocarbonindicators Structuralcrestor againsta lault? Local increasein amplitude? Local decrease in amplitude? Discordantflat reflector? Local waveshapechange?

Trappinglocation

Bright spot Dim spot Flat spot Polarity reversalor local phasing Reservoirlimits consistent'l (lf reservoirbaseand top separately visible) Polaritiesconsistent? (lf zero-phase data) Low frequenciesunderneath? Low-frequencyshadow Time sagunderneath? Velocitysag Lower amplitudesunderAmplitudeshadow neath(and sometimes above) Increasein amplitudewith AVO anomaly offset'l P-wavebut no S-wave S-wavesupport anomaly? Data deteriorationabove Gas chimney (and perhapsminor bright spots)

Sourt'e: Alter Brown.1991.

than for liquid hydrocarbons,as indicatedin fig. 5.27. Replacementof brine by hydrocarbonsas the interstitial fluid almost alwaysresultsin a lowering of velocity, but the effecton reflectionsdependsalso on the properties of the rock overlying (and underlying) the reservoirrock. Ifthe overlyingrock has higher velocity than brine-filled reservoir rock, lowering the reservoirrock velocity by filling it with hydrocarbon increasesthe contrast and henceincreasesthe amplitude of the reflection from the top of the reservoiq giving a bright spot; this is common in many Tertiary clasticbasins.If, on the other hand, the overlyingrock hasa velocityappreciablylower than the reservoir,the effectofhydrocarbonsis to decreasethe contrast,producing a "dim spot"; this indicator is shown in fig. 10.66,wherecarbonatereservoirrocks are cappedby

4t6

GEOLOGIC INTERPRETATION OF REFLECTION DATA

I

Fig. 10.65 Seismicsection showing amplitude and flat-spot hydrocarbon indicators,Gulf of Mexico. (From Brown. l99l: 2 8 5 ).

shales.Wherethe overlyingrock has a velocityslightly smaller than that of the reservoirrock, lowering the reservoir rock velocity by hydrocarbonsmay invert the sign ofthe reflection,producing a "polarity reversal" over the reservoir:this can be seen in olates 2 a n d3 . Where a well-definedfluid contact is present,especially a gas-oil or gas water contact,the contrastmay be great enough to give a fairly strong reffectionthat may stand out on seismicrecordsbecauseof its flat attitudein contrastto the dipping attitudesof other reflections.This is a "flat spot" and, whereseen,it is usually the most definitiveand informativeof the hydrocarbon indicators.Flat spots can be seenin figs. 6 . 1 0 ,1 0 . 6 5 1, 0 . 6 7 1, 0 . 6 8a, n d p l a t e s2 , 3 , a n d5 . H o w ever,reservoirthicknessesare usuallysmall compared to the resolvablelimit, so that the reflectionsfrom the reservoircap, fluid contact, and the baseof the reservoir generallyinterferewith one another to provide a compositereflectionthat may show variousphaseand amplitudechangesas the componentreflectionsinterfere in differentways.Thus, "phasing" can also be regardedas a hydrocarbonindicator. The loweringof velocity in a hydrocarbonaccumulation will also affectreflectionsfrom deeperreflectors by increasingarrival times to causea "sag" in reflections seenthrough the reservoiqbut the magnitudeof the sag is usually small becausemost reservoirsare not very thick. Sagsare especiallyprominent in plate 2. The lowering of velocity can also bend raypaths passing through the reservoir (as indicated in fig. 6.19a), resulting in distortion of deeper reflection events;sometimesthe effectis simply a degradationof reflection quality under the reservoir, a "shadow zone." The high amplitude associatedwith a bright spot often resultsin a lowering of the amplitudesof entire tracesby processingproceduresthat make the mean energy of all traces the same (amplitude nor-

I

malizing to removenear-surfaceor recording-system amplitude effects).A consequenceis that a loweramplitude shadowzone may exist abovea bright spot as well as below it. The high amplitude associatedwith a bright spot also affectsmultiplesinvolvingthe reservoirreflectors. Occasionally,sectionsare made to emphasizemultiples by using lower stackingvelocitiesthat attenuate primary reflections;an increasein amplitude of multiples seen on such a section also can be used as a hydrocarbonindicator. A loweringof instantaneous frequency($9.1l.a) is often observedimmediatelyunder hydrocarbonaccumulations.Such"low-frequencyshadows"seemto be confinedto a couple ofcycles below (not a1)accumulations. No adequateexplanationis available(see Taner,Koehler,and Sheriff, 1979);proposedexplanations generallyinvolve either the removal of higher frequenciesbecauseof absorption or other mechanisms,or improper stackingbecauseof erroneousvelocity assumptions or raypathdistortion. Specialdisplaysare often used to enhancehydrocarbon indicators and help in their detection and analysis.The most common is a low-amplitude display in which specialeffortsare made to preserveamplitude relations; reflection events are subdued on such displaysso that the amplitude buildups associated with bright spots stand out more clearly (fig. 10.11 is a low-amplitudedisplayof part of line ^Bof fig. 10.29). Sections are also often displayed in variable-areamode with both normal and inverted polarity, becausehydrocarbon-indicatoreffects are sometimesmore evidenton one than the other. Measurementsof amplitude, envelopeamplitude, amplitude ratios,phase,frequency,velocity,and so on, may be displayed so as to make lateral changes along reflectors more evident. Complex-trace analysis ($9.11.4)may be usedto makesuchmeasurements, the

HYDROCARBON INDICATORS

Frg l0 66

417

Dim spot associatedwith gas accumulation in porous carbonatcs overlain by shales.(courtesy ofreledyne.)

t,

I' i' 1 . 8 .' ' .

-l

v aJ r-' ,-," <)+,

'"i4M.

l*t

=*

StsS:'ffi

F i - q .I 0 . 6 7 D u a l - p o l a r i t y l i n e s a c r o s sa g a s c o n d c n s a t e rcser_ r o i r i n t h e N o r w e g i a n N o r t h S e a .T h e f l a t s p o t d i s a p p e a r so v e r l h e c r e s to f t h e s t r u c t u r eo n t h e r i g h t s e c t i o n .i n d i c a t i n s rhat the

rcservoir is completely filled with hydrooarbons over thrs por_ t i o n . ( C o u r t e s yo f E l f A q u i t a i n e N o r g e a / s . )

resultsoften beingdisplayedas color overlayson seis_ mic sectionsto facilitate correlation with structural evidences. Hydrocarbon indicators are also used cuantita_ tively to predict reservoir thicknessesand hydrocarbon volumes.Amplitude and dominant-periodmea_ surementsbased on model studies of a wedse pinchout (such as that in figs. 6.40b and 6.40c.1 aie shown in fig. 6.42. Dominant-periodor peak-to_ trough time measurementscan be used to indicate

thicknesseswhen greater than a quarter wavelength and amplitude measurementswhen smaller than a quarterwavelength.Note that in the thin-bedcase,the amplitude is a measure of the net thicknessrather than the grossthickness,as was illustratedin fig. 6.43. Hun (1978)mappedhydrocarbonproductivitybased on amplitude measurements,and many examplesof their quantitativeuse can be found in Brown (1991) and Sheriff (1992). The use of amplitude measurements, however, requires an amplitude calibration

GEOLOGIC INTERPRETATION OF REFLECTION DATA

418

l

I a o z

1 0 ()

U U-)

z

= F

2 0

= F

o

+

I

2 I

F i g . 1 0 . 6 8 M i n i m u m - p h a s e s e i s m i cs e c t i o n a c r o s s t h e T r o l l Field. oflshore Norway. The reservoir is Iilled with water at ,4, but contains gas at B ('indicates thc gas water contact and

(that is, what would the amplitude be if the reservorr werevery thick?),which often is not possible,or well control. Quantitativeanalysisalso requiresextremely careful processing,good data, and assumptionsabout the natureof the rock sequence, which may not always be true.Modelingis almostalwaysinvolvedin quantitative analysis. Another hydrocarbonindicator is the variation of amplitude with angle of incidence (or with offset, AVO); this was discussed in $3.4(seealso Hilterman, 1990).The usualexpectation is that amplitudewill decreasewith offset unlessgas is present,in which case amplitudewill increase.Whereasthis is often the case, the situation is not this simple,as sometimesgas will result in an amplitude decreaseand sometimesother situations will result in an amplitude increase with offset.Modeling using realisticvaluesof P- and S-wavevelocitiesand densitiesis usually required to determine the significanceof AVO measurements. AVO has to be measuredon gathersbeforestacking, are wherenoiseis usuallylarge,so that measurements not very accurate.Nonuniform acquisition (inadequate minimum or maximum offset distances,line bends, or skips), array directivity effects,processing that doesnot preserveamplitude,residualstatics,failure to migrate data, out-of-the-planedata, and other factorscan introducesizeableerrors. Comparisonof S- and P-wavesections($13.1.4 and

I

3XM 2 MILES

"flat D is a s p o t " d i p p r n g s l i g h t l yb e c a u s eo l - o v e r l y i n gv e l o c i t y c h a n g e s (. C o u r t e s yo l ' N o r s k H y d r o . )

l3.1.5)is anotherindicator.Examplesof other hydrocarbonindicatorsare shownin $12.5. Plate 4 showsseismicdata that have been inverted to seismic-logform and color coded to illustraterelative acousticimpedance(or relative velocity), low acousticimpedancevaluesthat are coloredblue being associatedwith hydrocarbons.The resulting horizon slice($12.3)through3-D data nicelymapsthe hydrocarbonaccumulation. Plate 5 shows a slice through a 3-D data volume parallel to a fault ($12.5).Hydrocarbonaccumulationsagainstthe fault can be seenboth by their amplitude and flat-spoteffects. 10.9 Crustal studies The field and processingtechniquesusedin petroleum seismic exploration - common-midpoint recording, vertical stacking, statics corrections,deconvolution, velocity analysis,the use of long streamers,precise navigation at sea, large Vibroseis sourcesand long sweepson land, and so on - are beginningto be applied to studiesof the Earth'scrust. Marine seismiclines reveal details of the crustal structure,such as subductionzones(fig. 10.69)and the base of the oceanic crustal plate. In the United States, the Consortium for Continental Reflection Profiling (COCORP) has been running a seriesof

PROBLEMS 419

,_- 7.O

-

8.0

-

9.0

_ r0.0

-

Fig. I0.69

CMp secrior neit ion,rt t ;;',";";;l# -ii. ff ,::i:i.:.; :i'l1#llJr]i,..],* is.riaing unacr 1,,n"".r., ,1,:::1,,,T fr:"ic^ptate .n" orscontrnuous reftecrion

BB, ii interpre,".j ;;;;j;;

M o h o r o v i c i cd i s c o n t i n u i t yT . h e i n t e r v a lv e l o c i t y b e t w e e nI a n d B calculated liom stacking velocities L O O i,-tn-,ir, so rhe 2_s t t m e i n t e r v a l b e l w e e nt h e m r e p r e s e n t s a t h i c k n e s so l . a b o u t 6 k m l b r t h e o c e a n i cc r u s r .( F r o m M a t s u z a w a" ; ; 1 . , ' i ; ; ; ;

n,",..

f rom rhe

linesto study the structure of the earth,scrust, and s i m i l a rs t u d i e sa r e b e i n g ."_.J'.ri'in'o*.. .oun_ tries. The preliminary results have U.* .^.,ting. Many complexitieshave.b.* founJi" ii. ..urt, ,n_ regions of layered ..n."tionJ tf,"i rrrr.r, ..0_ :19:t lmentary rather than

plane,find the surfacelra_ce and strikeofthe fault [see t h ed e r i v a r i o n o f e q . ( 4 .l 7 ) 1 . O) At whatdepthwouldyou look for this fault in well D located500 m N30.W-from well (? (c) Another fault cuts wells ,4 and C at depthsof 1300 an_d1000 m, respectlvely,and is known to strike N20'Vl Wheredoesit cut well B? 10.3 Try to matchthe section shownin fig. 9.24bwith the structuralstylesshownrn table 10.t. With which might it be compatible? (Note tfratng. l.Z+Cl, unrn,_ grated but assumethat it is n.orlv the strike.)Do the velocitydata i.rp.niicutu. to from problem 5.lg

aouot.arv,.;il;;i1",r.,T"fi ;."r.l,Jlix",:r,lr:;

-c""tl Appalachians (Cooi<et al., 1979; ri."r", ""0 oliver, 1980)in<Jicares that the .;yr;"il;.";;d other rocksseenon thesurfacefrur. U".n-i[.uri" i"", oi._ tancefrom the eastandrhat ..d_.ntJ.y loiu, _u, underlieportionsof the tow_angl; ;;;:.'ii. ,0.u. of "thin-skintectonics,, ur. U.g'i'*lngi"?n"'ir. ."" ceptsof howcontinents werelormed. Problems I0.l Thelinein fis. 10.I I shows partof lineB of fig. 10.29(theweilis licatea ut w iiir.',"O.ZSj,;r, ,i,ll relativeamplitudeinformatronpreserved and plotted j:i.:,,h_.,"rgesramplit ud.. ;* ;;; .;ioo.oi *" :: conclusrons can bedrawnfrom lig. 10.t t itiai a.. t.s, evidentfromfig. 10.29? 1:..2 Well8 is 500m due eastof well andwell lal C is600 m duenorthor_ae rauri'rii"tA i,"^a c at depthsof 800,1000and600m, ..rp."ii".inlir"rn_ ing att wels are venicaluna rn.-ilui;';;;i;. ,. "

l0.i How_doyou reconcile_ the contradictorydips betweenrhe 5- and 6_kmmarksar rhe rop oiifi. ,..rion in fig. 8.5?Whar strucruralstyle i. ..;;;;,;;? How would you draw faults? 10.5-Four linesforming a grid are shownin fig. 10.29. (a) Map the three horizons encounteredut t.:SS, I .830,and 2.660s ar the inrersecti.r, oiiri. "".f".i,V i "rA C A velocityanalysisat this location gi".riirn. (stackingvelocirv)oairs as rofforr,-o.ioo,, -ztiibl"jrju "Vr, 0600,

83o;osbo,rqoo;

^r ,roo, _r.zoo, i.q00, t;60_0,2|40; 2.000,2280: zJ00,zaqi;ana't-.iio, zcto. thecurvedfaultplane. !l) Y"p (c) E s t i m a t ea n d m a p r h e t h r o wo n thelauh. 10.6 (a) In fig. 9.44,rhe reflection uiuUoul'O.O, up_

r l.u

420

GEOLOGIC INTERPRETATIONOF REFLECTION DATA

pearsto be faulted at S.P.5; draw in the fault and describeits probabletype and characteristics. (b) Note the changeswith location in the interval times betweendifferent reflectionsin fig. 9.44. How can thesebe explained? 10.7 Figure 10.33showsa salt uplift at a shelfedge. (a) How could one tell that this featureis not caused by reef growth instead? (b) Could it havebeencausedby shaleflowage? (c) Does the relief abovethe unconformity U, indicate postunconformitysalt movement,renewedactivity at the shelf edge (downdrop along faulting at the shelf edge), or differential compaction because of the weight of the postunconformitysection? 10.8 lf the nature of a flow structure(such as shown in figs.10.32or 10.33)shouldnot be clear,how might be used gravity,magnetic,or refractionmeasurements to distinguish betweensalt, shale.or igneousflows? Betweentheseand a reef? 10.9 Figure 9.63bmapstwo separatehigh closureson anticlinalnose. a northeast-plunging (a) Assumingthat the only existingcontrol is that shown by the lines marked by diagonal slashesin fig. 9.63a (which is on a differenthorizon), what additional program would you recommendto check out in the interpretation before recommendweaknesses ing a well to test for hydrocarbonaccumulation'J (b) Can you find a fault for which the indicateddirection of throw is clearlywrong? 10.10 How would evidencesof thickening/thinning around a salt dome or in a folded structurebe distorted on an unmigratedtime section?On a migrated time section? l0.ll What kind of featureshowsin fig. 9.19 about 75(f,of the way acrossfrom the left end of the section at about 2.5 s?What characteristicshelp to identify it? 10.12 Interpretthe sectionsshown in figs.(a) 9.59b, ( b ) 9 . 6 1 d ,( c ) 1 0 . 3 1a, n d ( d ) 1 0 . 3 8A . s s u m et h a t o u t data are not important.(Pickeventsthat of-the-plane involveangularitiesbetweenprimary reflectionsrn order to identify unconformities and or seismic sequenceboundaries.)Deducethe geologichistory. 10.13 An obviousunconformityis evidentat approximately 1.5 s in fig. 9.18; preciselywherewould you positionit? Is it associated with the sameeventat oppositesidesof this section? 10.14 In fig. 10.48,V, : 2.00km/s. Z. : 4.00 km/s, the horizon dips l0' and the vertical depth of the diffractingpoint P is | 000 m, the interfacebetweenZ, and V. being 350 m vertically above P Compare the diffraction curvewith that which would havebeenobserved if V, : V. : 3.00 kmls. (Hint: Ray trace enoughrays to roughly definethe diffraction curves.) 10.15 Attempt a stratigraphic interpretation of fig. 10.53.CC' dividesnonmarinefrom marinesediments. Doesthe surfacechannelcreatefictitious deepeffects? 10.16 What stratigraphicleaturescan be seenin fig. 9.57?What can be said about the geologichistory? 10.17 For the bright spot shown in fig. 10.67,what do you think is the polarity of the embeddedwavelet?

L

Whereis the top and bottom of the gasaccumulation? What is the maximum thicknessof the gas column, assuminga velocity of 1800m/s?Why do the reflections from the reservoirtop and bottom not converge at the pinchout edgeof the gas reservoir? 10.18 Using the minimum-phasewaveletof problem for a sand enclosed 9.14a,determinethe waveshapes in shale where the two-way traveltime through the sand is 12 ms when the upper part containsgas,the two-way traveltimethrough the gas-sandportion be0, 2, 4, 6,8, 10, and 12 ms. Plot the ing successively traces side by side shifted successivelyby 2 ms as would be the case where the gas-watercontact was horizontal. This illustratesa bright-spot,flat-spot situation. Take the reflection coemcientsfor shale to gas-sandas -0.1, gas-sandto water*sandas +0.15, and water-sandto shaleas -0.05. (b) Repeat using the zero-phasewaveletof problem 9.14. References Anstey, N. A. 1973. How do we ktrow we are right? Geophvs. Prosp.,2l:407'.11. Anstey, N. A. 1974. The Nev'Seismk IHRDC Press.

lnterpteler Boston:

Anstey, N. A,.1977. Sei.smitlnterpretution: The Physitul Aspects. B o s t o n :I H R D C P r e s s . . oston:IHRDC Press. A n s t e y ,N . A . 1 9 8 0 a .S i m p l e S e i s m i t sB Anstey. N. A. 1980b.Seismic Erplorution.lbr Sandstone Reser' loir.r. Boston: IHRDC Press. Badley. M. E. 1985. Prattical Sei:smit'Interprel4li.)n Boston: IHRDC Press. Badlcy, M. 8., and N. A. Anstey. 1984. Basic Interprctution, Video Library for Exploration and Production SpecialistsGP501.Boston:IHRDC Press. : n e w d i m e n s i o ni n s e i s B a l c h , A . H . 1 9 7 1 .C o l o r s o n a g r a m sA mic data interpretation. Geophysit's,36:1074 98. Bally, A. W, ed. 1983 4. Seismit ExpressionoJ Strutturul Style,s' V o l s . 1 , 2 . a n d 3 . A A P G S t u d i e si n G e o l o g y 1 5 .T u l s a : A m e r i can Association of Petroleum Geologists. Bally, A. W, ed. 1987 9. Atlas oJ Sei,smitStratigraphy, Yols. l, 2, and 3, AAPG Studiesin Geology 27. Tulsa: American Assoc i a t i o n o f P e t r o l e u mG e o l o g i s t s . Bates, R. L., and J. A. Jackson. 1987.A GlossaryoJ Geologt',3d ed. Falls Church. Va: American Geological Institute. Batzie,M., and Z. Wang. 1992.Seismicproperties of pore fluids. Gt'tryhysit.t.57: I 396 1408. Berg, O. R., and D. G. Wolverton, eds. 1985. Seismic Stratigrapht' II, AAPG Memoir 39. Tulsa: American Association of Petroleum Geologists. Blackburn, C. J. 1986a.Direct hydrocarbon detection: Some examples. Er2 Geophys.,l7:59 66. Blackburn, G. J. 1986b. Depth conversion: A comparison of m e t h o d s .E x p . G e o p h v s .1, 1 : 6 7 - 7 3 . Bouvier, J. D., C. H. Kaars-sijpesteijn, D. F. Kluesner, C. C' Onyejekwe, and R. C. van der Pal. 1989. Three-dimensional seismic interpretation and fault-sealing investigations, Nun River Field, Nigeria. Bull. AAPG,13:1397 414. Boyer, S. 8., and D. Elliott. 1982. Thrust systems Bull. AAPG' 6 6 : 1 1 9 61 2 3 0 .

I

REFERENCES

Brown, A. R. 1991.Interpretation of Three_DimensionalSeismic Data, 3d ed., AAPG Memoir 42. Tulsa: American Association of Petroleum Geologists. Brown, A. R. 1992.Seismic interpretation today and tomorrow The Leading Edge, tl(tt):10 15. Brown, A. R., C. G. Dahm, and R.J. Graebner.l9gl. Strati_ uraphic case history using three-dimensional seismic data in rhe Gulf of Thailand: A case history. Geophys. prosp., 29: t27 49. Brown, L. F., and W L. Fisher. 1980. Seismic Stratigraphic Interpretation and petroleum Exploration, AAPG Contin;ing Education Course Notes 16. Tulsa: American Association oi Petroleum Geologists. Bubb. J. N., and W G. Hatlelid. 1977. Seismic recognition of '-arbonate buildups. In ,serimic Srratigraphl. _ Appllcution.s to Hrdrr\arbon Erploration, C. E. payton, .a., p-p. lg5-204, AAPG Memoir 26. Tulsa: American Associati& of petro_ JeumGeologists. Buxtori A. 1916. Prognosen und Befunden beim Hauenstein_ basrsund GrenchenbergTunnel und die Bedeutuns der letzern tIr die Geologie des Juragebirges.Verh. Nuturf de:ell. Ba.se!, 27:185 254. Campbell, n F 1965. Fault criteria. Geophysics,30:976 97. C i e m e n t , W . A . 1 9 7 7 .C a s e h i s t o r y o f g e o s e i s m i cm o d e l i n g o f basal -\4orrow-Springer sandstones, Watonga_Chickasha trend, Geary, Oklahoma. In SersrzicStrurigrapil, Apptica_ ru,)n.t.toHydrocarhon Explorutitn, C.E. payton, id.. pp.'451 -6. A A P G M e m o i r 2 6 . T u l s a :A m e r i c a n A i s o c i a t i o n o f p e t r o _ ieum Geologists. Cook. F. A., D. S.Albaugh, L. D. Brown, S. Kaufman. J. E. Olir c r . a n d R . D . H a t c h e r . 1 9 7 9 . T h i n - s k i n n e d t e c t o n i c si n the crvstalline southern Appalachians. Geology,7: 563.7. C'rrok.F. A., L. D. Brown, and J. E. Oliver. 19g0. The sourhern \ppalachians and the growth of continents. Sci. Amer., 243 ( . 1 ) :1 5 6 6 8 . Dahlstrom, C. D. A. 1970. Structural geology in the eastern rrrargin of the Canadian Rocky Mountains.-Fetro. Geol. Bull., l8:132 406. D a h m , C . G . , a n d R . J . G r a e b n e r .1 9 g 2 .F i e l d d e v e l o p m e n tw i t h t h r e e - d i m e n s i o n asle i s m i cm e t h o d s i n t h e G u l f o f T h a i l a n d : A case history. Geophl'.sics, 47: 149 76. Dalis. T. L. 1972. Velocity variations around Leduc reefs. Ceo_ nlt.t.sits,37: 584 604. D i m i t r o p o u l o s ,K . , a n d J . D o n a t o . 1 9 g 3 .T h e g r a v i t ya n o m a l y of. the St. George'sChannel Basin, southern IriJh Se; A possrble erplanation in terms ol salt migration. J. Geol. Sot.'Lonrktn. ll0:239 44. Dobrin, M. B. 1977.Seismicexploration lor stratigraphic traps. ln Seismu.S_tratigraphy Appticutions to Hvdrocaihon Erptora_ r t t t n ,C . E . P a y t o n ,e d . , p p . 3 2 9 5 2 . A A p G M e m o i r 2 6 . T u l s a : \merican Association of petroleum Geologists. Downey. M. W. 1990. Faulting and hydrocarbon entrapmenr. T h e L e a d i n gE d S e . 9 ( l ) : 2 0 3 . E d w a r d s ,S . 1 9 8 8 . U s e s a n d a b u s e so f s e i s m i cm o d e l i n s . 7 l e Lt,tding Edge. 8(4): 42 6 Fagin, S. W. 1991. Seisnic Modeling oJ' Geologit. Srructures. Tulsa: Society of Exploration Geophysicists. Feagin, F. J. 1981. Seismic data display and reflection per_ ceptability. Geoph.ysit's, 46: 106-20. Fitch. A. A. 1916. Seismic ReJlectionInterpretatiorn.Berlin: Gebnider Borntriiger. Fontaine, J. M., R. Cussey,J. Lacaze, R. Lanaud, and L. yapaudjian. 1987. Seismic interpretation of carbonate deoosr_ tronal environments.Bull. AApG. 7l: 2gl_97.

421 Gallup, V{ B. 1951. Geology of Turner Valley oil and gas field, Alberta. Bull. AAPG, 35: 797_g21. Gazdag, J. 1981. Modeling of the acoustic wave equation with translorm methods. Geophysics.z16:854 9. Goguel, J. 1962. Tectonics.San Francisco: W H. Freeman. Greenlee, S. M., and T. C. Moore. 1988. Recoenition and inter_ pretation of depositional sequencesand calculation of sea_level changes from stratigraphic data Offshore New Jersey and Al_ abama Tertiary. ln Sea-Leyel Changes; An Integrated Approach, C. K. Wilgus et al., eds., pp.329-56, Society oaEconomic pale_ ontologists and Mineralogists Special publication 42. Gretener, P. E. 1979. Pore Pressure;Fundamentals, General Ram_ rJtcations,and Implications./br Structural Geology (revised), Ed_ ucation Course Note Series4. Tulsa: American Association of Petroleum Geologists. Gries, R. R., and R. C. Dyer. eds. 1985. Seismic Exploration of the Rockl' Mountuin Region. Denver: Rocky Mouniain Association of Geologists and Denver Geophysical Society. Halbouty, M. T., ed. 1982. The Deliberate Search for the Subtle Trap. AAPG Memoir 32. Tulsa: American Assoiiation of pe_ troleum Geologists. H a q . B . U . , J . H a r d e n b o l , a n d p R . V a i l . l 9 g g . M e s o z o i ca n d Cenozoic.chronostratigraphyand cycles of sea-levelchange. In Sea-l,evel Changes: An Integrated Approach, C. K. Wilgus et al., eds.. pp. 7l 108, Society of Economic paleontologists and Mineralogists Special Publication 42. Hardage, B. A., ed. 1985. Vertical SeismicproJiling,part A; prin_ crplr,^r: London: Geophysical press. Hardage, B. A.. ed. 1987. Seismit'Statigraphl, London: Geo_ physical Press. Harding, T. P., and J. D. Lowell. 1979. Structural styles, their plate-tectonic habitats, and hydrocarbon traps in petrol.u. p r o v i n c e s .B u l l . A A P G . 6 3 : l 0 l 6 5 8 . Harms, J. C., and P. Tackenberg. 1972. Seismic signatures of s e d i m e n t a t i o nm o d e l s .G c r T l i 1 . r r r .3. 7 r .: 4 5 5 8 . Harris, L. D., and R. C. Milici. 1977. Characteristicsof thin_ skinned styles of deformation in the Southern Aooalachians and potential hydrocarbon traps. U.S. Geological Survey prof. P a p e r I 0 1 8 , W a s h i n g t o n ,D . C . : U . S .G e o l o g i i a l S u r v e y Hilterman, F. 1970. Three-dimensional seismicmodelins. Gcc,fl1.rit'.r, 35: 1020 17. Hilterman. F. 1990. Is AVO the seismic signature of lithology? A case history of Ship Shoal South Addition. The Leatlins Edgt, lO(61: 15 22. Hobbs, B. 8., W. D. Weams, and p Fr.Williams. 1976.Outtine ol Struttural Geology. New York: John Wiley. Hubbert, M. K. 1937. Scale models and geologic structure. Geol. Sot. Amer. Bull., 48: 1459,520. H u n , F 1 9 7 8 .C o r r e l a t i o nb e t w e e ns e i s m i cr e f l e c t i o na m p l i t u d e and well productivity A case study. Geophl,s prc,ip.. 26: t57 62. I s a a c s , 8 . ,J . O l i v e r ,a n d L . R . S y k e s .1 9 6 8 .S e i s m o l o g ya n d t h e new global tectonics.J Geophys.Res. 73: 5855 99. Jones,P. B. 1988. Balanced cross-sectons An aid to structural interpretation. The Leuding Edge, 7(B): 29ff. Kuhme, A. K. 1987.Seismicinterpretation of reefs.The Leadinp 6rige, 6(8): 601T. Larner, K. L., L. Hatton. B. S. Gibson, and L. C. Hsu. l9gl. Depth migration of imaged time sections. Geophtsits, 46: 734 50. Lindseth, R. O. 1979.Synthetic sonic logs A processfor stratrgraphic interprelalion. Geophysics,44:3 26. Loutit. T. S., J. Hardenbol, P. R. Vail, and G. R. Baum. l9gg.

Aaa

GEOLOGIC INTERPRETATION OF REFLECTION DATA

Condensedsections:The key to age determination and correlation of continental margin sequences. ln Sea-Level Changes: An Integrated Approach, C. K. Wilgus et al., eds.,pp. 183 213, Society of Economic Paleontologists and Mineralogists Special Publication 42. Lowell, J. D. 1972. Spitzbergen Tertiary orogenic belt and the Spitzbergenfracture zone. Geol. Soc.Amer. Bull.,83: 3091-102. Lowell, J D. 1985. Structural Stttles in Petoleum Explorqtion. Tulsa: OGCI Publications. Lyons, P L., and M. B. Dobrin. 1972. Seismic exploration for stratigraphic traps. In Stratigraphic Oil and Gas Fields Classifcation, Exploration Methods, and Case Histories, R. E. King, ed., pp.225 43, AAPG Memoir 16. Tulsa: American Association of Petroleum Geologists. Marr, J. D. I 971. Seismicstratigraphicexploration Part l. Geophysics,36: 533 53; Part ll, Geophysics,36:533-53; Part IIl, Geophysics,36:676 89. Matsuzawa, 4., T. Tamano, Y. Aoki, and T. Ikawa. 1979. Structure of the Japan Trench subduction zone from multi-channel seismicreflection records. In Marine Geology,pp. l7l 82. Amsterdam: Elsevier. Maureau, G. T., and D. H. van Wijhe. 1979. Prediction of porosity in the Permian carbonate of eastern Netherlands using seismic data. Geophysics,442 1502,17. May, B. T., and J. D. Covey. 1981. An inverse ray method for computing geologic structures from seismic reflections: Zero offset case. Geophysics,46:268 87. McQuillin, R., M. Bacon, and W Barclay. 1984.An Introduction to SeismicInterpretation,2d ed. London: Graham & Trotman. Meekes,A.. and R. van Will. 1991. Private communication. Mitchum, R. M., P R. Vail, and S. Thompson. 1977.The depositional sequenceas a basic unit for stratigraphic analysis. In Seismic Stratigraphy Applications in Hydrocarbon Analysis, C. E. Payton, ed., pp. 53-62, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Morgan, L., and W Dowdall. 1983. The Atlantic continental margin. In Seismic Expression of Structural Styles,A. W. Bally, ed., AAPG Studies in Geology 15. Tulsa: American Association of Petroleum Geologists. Neidell, N. S., and E. Poggiagliolmi. 1977. Stratigraphic modeling and interpretation. ln Seismic Stratigraphy Applications in Hydrocarbon Analysis, C. E. Payton, ed., pp. 389-416, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Noah, J. T., G. S. Hofland, and K. Lemke. 1992. Seismic interpretation of meander channel point-bar depositsusing realistic seismicmodeling techniques.The Leading Edge I l(8): 13,18. Payton, C. E., ed. 19'77. Seismic Stratigraphy Applications to Hydrocarbon Exploration, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Plawman, T. L. 1983. Fault with reversal of displacement, central Montana. ln SeismicExpressionof Structural Styles,A.W. Bally, ed., AAPC Studies in Geology 15. Tulsa: American Association of Petroleum Geologists. Posamentier,H. V(, M. T. Jervey,and P R. Vail. 1988. Eustatic controls on clastic deposition. ln Sea-LevelChanges:An Integrated Approach, C. K. Wilgus et al., eds., pp. 109-54, Society of Economic Paleontologistsand Mineralogists Special Publicalion 42. Ramsayer, G. R. 1979. Seismic stratigraphy, a fundamental exploration tool, OTC Paper 3568. Dallas: Offshore Technology Conference. Rittenhouse, G. 1972.Stratigraphic trap classification.ln Stratt graphic Oil and Gas Fields Classifcation, Exploration Methods, and Case Histories, R. E. King, ed., pp. 14-28, AAPG

Memoir 16. Tulsa: American Association of Petroleum Geologlsts. Roksandic, M. M. 1978. Seismic facies analysis concepts. Geophy. Prosp.,26: 383-98. Russell,B. H. 1992.Using color in seismicdisplays.The Leading Edge, ll(9)z 13-18. Sangree,J. B., and J. M. Widmier. 1979. Interpretation of depositional facies from seismic daIa. Geophysic,s,44: 131-60. Sarg, J. F. 1988. Carbonate sequencestratigraphy.In Sea-Level Changes: An Integrated Approach, C. K. Wilgus et al., eds., pp. 155-81, Society of Economic Paleontologists and Mineralogists Special Publication 42. SEG (Society of Exploration Geophysicists). 1989-92. Seismic Interpretation Senzs, Vol. I, 1989; Vol. 2, 1990; Vol. 3, 1992. Tulsa: Society of Exploration Geophysicists. Sheriff. R. E. 1977. Limitations on resolution of seismic data and geologic detail derivable from them. ln Sei.smicStratigraph!' Applications to Hydrocarbon Exploration, C. E. Payton, ed., pp. 3-14, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Sheriff, R. E. 1978. A First Course in GeophysicalErploration and Interpretation. Boston: International Human Resources Development Corp. Sheriff, R. E. 1980. Seismic Stratigraplry.Boston: International Human ResourcesDevelopment Corp. Sheriff, R. E. 1982. Structural Interpretation of Seismic Data, AAPG Continuing Education Course Note Series 23. Tulsa: American Association of Petroleum Geologists. Sheriff, R. E. 1989. Geophysical Metfutds. Englewood Cliffs, N.J.: Prentice Hall. Sheriff, R. E. 1991. Encyclopedic Dictionary of Exploration Geophysit's,3d ed. Tulsa: Society of Exploration Geophysicists. Sheriff, R. E., ed. 1992. ReservoirGeophysics.Tulsa: Society of Exploration Geophysicists. Sheriff, R. E, and J. Farrell. 1976.Display parametersof marrne geophysicaldata, OTC Paper 2567. Tulsa: Society of Exploration Geophysicists. Stone, D. S. 1985. Geologic interpretation of seismic profiles, Big Horn Basin, Wyoming.ln SeismicExploration of the Rocky Mountain Region, R. R. Gries and R. C. Dyer, eds., pp. 16586. Denver: Rocky Mountain Association of Geologists and Denver Geophysical Society. Taner. M. T.. E. E. Cook. and N. S. Neidell. 1970. Limitations of the reflection seismicmethod. Lessonsfrom computer srmularions. Geophysics,35: 551 73. Taner, M. T., F. Koehler, and R. E. Sheriff. 1979.Complex seismic trace analysis. Geophysics,44:1041 66. Taner, M. T., and R. E. Sheriff. 1977.Application of amplitude, frequency,and other attributes to stratigraphic and hydrocarApplications to bon determination. In Selsrrlc Stratigraphy Hydrocarbon Exploration, C. E. Payton, ed., pp. 30i-28, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. Taylor, J. C. M. 1981. Late Permian Zechstein: Course Notes No. l,Introduction to the Petroleum Geology oJ the North Sea. London: Joint Association for Petroleum Exploration Courses. Telford, W. M, L. P Geldart, and R. E. Sheriff. 1990. Applied Geophysics.2d ed. Cambridge, U.K.: Cambridge University Press. Trorey, A. W. 1977. Diffractions for arbitrary source receiver localjons. Geophyics, 42: 1177 82. Trusheim, F. 1960. Mechariism of salt migration in northern Germany. Bull. AAPG,441 1519-41.

i

REF E R E N C E S

Tucker, P M., and H. J. Yorsten. 1973. Pitfalts in Seismic Interr)retation.Tulsa: Society of Exploration Geophysicists. Vail. P R., R. M. Mitchum, and S. Thompson. 1977a.Relative changesofsea level from coastal onlap. In SeismicStratigraphy .lpplications to Hydrotarbon Exploration, C. E. payton, ed., :p. 63 8l, AAPG Memoir 26. Tulsa: American Association of Petroleum Geologists. \ail. P R., R. G. Todd, and J. B. Sangree,1977b.Chronostratruraphic significanceof seismicreflections.ln SeismicStratwraApplications to Hydrocarbon Exploration, C. E. payion. !h.t ed.. pp. 99-l16, AAPG Memoir 26. Tulsa: American Assocration of Petroleum Geologists. \an Wagoner,J. C., R. M. Mitchum, K. M. Campion. and V D. Rahmanian. 1990. Siticlastic SequenceStrutigruphl. in Well Lttgs, Cores,and Outcrops; Conceptsfor High-Resolution Corre.,ttion of Time and Facies,AAPG Methods in Exoloration Se: r e s7 . T u l s a :A m e r i c a n A s s o c i a t i o no f p e l r o l e u mG e o l o g i s t s .

+zJ

Wanslow, J. B. 1983. Piercement salt dome, Upper Continental Slope. In Seismic Expression of Structural Styles, A. W Bally, ed., AAPG Studies in Geology 15. Tulsa: American Assocration of Petroleum Geologists. White, J. E., and R. L. Sengbush. 1987. Production Seismology. L o n d o n : G e o p h y s i c a lP r e s s . Wilgus, C. K., B. S. Hastings, H. Possamentier,J. Van Wagoner, C. A. Ross, and C. G. S. C. Kendall. eds. 1988. Sea-Level Changes: An Integrated Approach. Society of Economic paleontologists and Mineralogists Special Publication 42. Williams, W D., and J. S. Dixon. 1983.Seismicinteroretation of the Wyoming overthrust belt. In Serimlc Erpre,s,siinof Srructural Styles,A. W Bally, ed., pp. l3 22. AAPG Srudies in Geology 15. Tulsa: American Association of Petroleum Geologists.

tl

Refraction Methods

Overview Refraction work generally can be divided between :hat designedto define deep structural featuresand :hat aimed at defining the near surface,the latter in,-ludingmost engineeringstudies and determining ::atics correctionsfor reflectiondata processing.The jata to definedeepstructuralfeatureshaveoften been rsufficient(simplyto keepcostsdown) and the inter:retation oversimplified(for example,assumingcon::.int overburdenvelocity and nearly planar refrac: ris ): consequently, preciseresultswere not expected. l:aditional interpretation was a hand-craft art em:...r ing many ingeneous methods(we describea few), : -r little work of this type is donetoday.On the other -:nd. considerablenear-surfacerefraction is being : :.e. The need for simplification is much less and - ::hods suchas the GRM (911.3.3) havebeendevel::d Io accomplishmore preciseresults.Differences -"::.ieen achieving different refraction objectives . =':-'tmainly the scaleof acquisitionoperations. R.:tractionand reflectionwork are similar in many ,'r3s'IS.much differentin others.The similaritiesare --:,:rent that reflectionfield crewssometimesdo re-:-irtrn profiling,thoughoften not with the efficiency : cr€w specificallydesigned for refraction. The - i::;nces between reflection and refraction field :r. tor definingdeepfeaturesmostly result from the -,: source-to-geophone distancesemployedin re-:,ri,-rfl.The energyinput to the ground must be - -.:.: tor refraction work and explosivescontinue to ': -.. Jominantenergysourcewherelongerdistances - -: :-.Lrl\edalthough other seismicsourcesare also - .;: The longer travel paths result in the higher fre- -:-,i3S being mostly absorbedso that refraction -.-': :13generallyof low frequencycomparedwith rer:-' r. data. Consequently,refraction geophones "- : ,-.'*er natural frequenciesthan reflectiongeo-- -::. although the responseof the latter is often - -: - -:i3 tor satisfactoryrefractionrecording.Most - :':. :3rSrnic equipmentcan be usedlor refraction, " - :. rr,i trlder analog equipmentdoesnot haveade. - : -.*-liequencyresponse. ' ,: :eliaction techniquesinvolvehead waves and ' : -:tilrg of memberswhosevelocitiesare signifi-..r:er than those of any overlyingrocks.Such . ::; nrrt alwayspresentand maps of the high: - -. -:tits that are presentmay not be relatedto :i -: i.. .-rbjectives, so refraction methods are not .-- -:r 3 rn many situations.Evenwhereapplicable,

425

refraction surveyingis usually slower than reflection surveying becausethe large offset distancesinvolve more moving time and createproblemsof communications and logistics.However,refraction profilesare often not as closely spaced as reflection lines and hencethe cost of mapping an arca is not necessarily greater. Most refraction work involves in-line profiling ($11.1.1),especiallythe use of reversedprofiles. Broadsideand fan-shootingand placingthe geophone in a deepboreholeare methodsusedfor certainobjectives. Small-scalerefraction is used in studiesfor the foundationsof structuresand other engineeringproblems. Marine refraction involvesspecial operational problems. The computation ol refraction data is discussedin {11.2; data have to be correctedfor elevationand near-surfacevariationsas with reflectiondata. The essential in refraction interpretation is correlating eventsthat involvethe samerefractinginterfaces;ambiguitiescan often be removedif more data are available.Once refractioneventshavebeencorrelated.refractor depths and dips can be found using the formulas given in $4.3, but in addition a variety of methodsare availablefor more complicatedsituations and for routine interpretation of large amounts of data. Thesemethodscan be divided into threegroups: those using relatively complex formulas based on , e l a y - t i mm , nd t h o s ei n $ 4 . 3( $ l 1 . 3 ) d e e t h o d s( $ l 1 . 4 ) a wavefrontmethods($l 1.5). Interpretation of engineering refraction data ($14.1.2) is generallymuch simplerand more straightforward than that of large-scalesurveysbecausethe near-surfacelayersare relativelyfew and little detail is usuallyrequired. 1l.l Field techniques I 1.1.1In-line refractionpro;t'tling The basic refraction field method involvesshooting reversedrefraction profiles, a long linear spread of many geophonegroupswith a sourceat eachend, the distancebeing great enough that the dominant portion of the travel path is as a head wavein the refractor or refractors being mapped. Usually, it is not practical to record simultaneouslyso many geophone groups spread over such a long distance,and hence refraction profiles are usually recorded in segments. By referringto fig. I I . I a, which showsa singlerefrac-

426

REFRACTION METHODS

Fig. I l.l Reversedrefraction profiles. (a) Time distance plot for continuous reversedprofiling, and (b) section showing single refractor. Reciprocal times are indicated by t"

tor, the spreadof geophonegroups might be laid out betweenC and D and shots at C and G fired to give two records;the spreadwould then be movedbetween D and E and shotsflred at C and G as before,and so on to devefop the complete reversedprofile CDEFG. The charge size is often varied for the different segments becauselarger chargesare required when the offsetbecomesgreater.Usually,one or two groupswill be repeatedfor successive segmentsto increasethe reliability of the time tie betweensegments. The sourceat C can also be usedto record a profile to the left of C and the sourceat G a profile to the right of G Note that the reciprocaltime t,is the same for the reversedprofiles and that the intercept times for profilesshot in different directionsfrom the same source point are equal. These equalitiesare exceedingly valuable in identifying segmentsof complex time distance curves where several refractors are present.In simple situations,the reversedprofile can be constructedwithout having to actually shoot it by using the reciprocaltime and intercepttime information. However,usually situationsof interestare sufficiently complicatedthat this procedurecannot be carried out reliably. The reversedprofles shot from C and G in fig. 1l.l aflow the mapping of the refractor from L to M. The reversedprofile to the left of C permits mapping as far as K but no coverageis obtained for the portion Kl. Hence, continuouscoverageon the refractor requires an overlap of the reversedprofiles;a reversed profile between A and E (shown dashedin fig. 1I . la) would providecoveragebetweenU and V thus including gap Kl as well as duplicatingthe coverageUK and LV Wtth perfectdata, the duplicatecoveragedoesnot yield new information, but in actual profiling, it provides valuable checksthat increasethe reliability of the interpretation. In caseswherereversedprofilesare not essential,a

split refractionspread(ACE in fig. I l.l with sourceat C) can be usedwith a savingin the numberof source locations required. Howeveq becausethe updip and downdip apparent velocitiesare not obtained from the samepart of the refractor,faulting, curvature of the refractor,lateralvelocityvariations,and so on can render the method useless. If we havethe two-refractorsituationin fig. I1.2, first-break coverageon the shallow refractor is obtained from L to K and from M to ,Vwhen the sources are at C and G the correspondingcoverageon the deeperrefractor is from p to S and from R to P lf we are able to resolvethe refraction eventsthat arrive later than the first-breaks,calledsecondarrivals refractions,we can increasethe coverage or sec'ondury obtainedwith a singleprofile.With analogrecording, it is difficult, and sometimesimpossible,to adjust the gain to optimize both the first-breaksand the second arrivals at the same time; if the gain is too low, the first-breaksmay be weak and ambiguitiesin timing may result,whereasif the gain is too high, the secondary refractions may be unpickable.Becauseof this difficulty,prior to magnetic-taperecording,refraction mapping was generally based on first-breaks only. With magnetic-tape recording, playbacks can be made at severalgains so that each event can be displayedunder optimum conditions. In order to economizeon field work, the portions of the time distancecurvesthat do not add information necessaryto map the refractor of interest often are not shot wherethey can be predictedreasonablyaccuratefy.Thus, the portions CP and GQ of the reversed profilein fig. I l.la are often omitted. Where a singlerefractor is being followed, a series of short refraction profiles are often shot rather than a long profile. In fig. I 1.3,geophonesfrom C to E are usedwith sourcepointC, from D to F with sourceD and so on. The oortions of the time-distancecurves

FIELD TECHNIQUES

427

Fig. I L2 Reversed refraction profiles for two_refractor sase. (a) Time distance plot, and (b) section showing the two re_ fractors.

been obtained for a long proflle from sourcepointC becauseofrefraction eventsfrom deeperhorizons. An efficientmethod commonly usedin engineering surveys ($$14. 1.2) is the four-shot method (Milsom, 1989).A spreadofgeophonesis laid out with sources at each end (for "short shots',)and also with sources offset in-line from each end (for .,long shots"). The offsetpoints are locatedfarther from the spreadthan the criticaldistance(x' of eq. (4.39;1;this aisurescov_ erageon the refractor over the entire spread leneth and often givesbettermeasuresof the apparentvelic_ ities and intercept times than can be obtained by fit_ ting a straight line to the small segmentof arrival trmesrecordedwith the short shots.parallelismof the time-distancecurves for long and short shots indi_ cates travel in the same refractor. The nearest seo_ phone is usuallymoved half a geophoneinterval aivay for the short shotsto give a better measureof V,. An exampleof this method is shown in fis. 14.2. I l. I .2 Broadsiderefraction andfan-shooting

Fig. | 1.3 Unreversed refraction profiles lor a single refraclor

or 02 ____T________-_

03 ___I-

o1

o5

, \ A--+ , /

L : I

: ( rt

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\ L-\

_

1\ \ t

t

I

r

t

t

t t

_B r

r

t u l -

_D

Fig. I 1.4 Broadside refraction profiling with geophonesalong the central line and sources along the outside lines. Refractor depths are sometimes attributed as applying along lines lB and CD

attributableto the refractor being mapped are then translatedparallelto themselvesuntil they connectto_ getherto make a compositetime-distanie curve such as that shownby the dashedline. The compositecurve may differ from the curve that would actually have

ln broadsiderefractionshooting,sourcesand spreads are locatedalong parallel lines (seefig. I 1.4) selected so that the desired refraction event can be mapped with a minimum of interferencefrom other events. Where the refraction event can be clearly distin_ guished from other arrivals, it provides a very eco_ nomical method of profiling becauseall the data yield information about the refractor.However,usually,the criteria for identifying the refraction event are based on in-line measurements(suchas the apparentveloc_ ity or the relationshipto other events)and thesecrite_ ria are not availableon broadsiderecordswhere the offset distanceis essentiallyconstant.Thus. if the refractor should unexpectedlychangeits depth or ifanother refractionarrival should appear,one might end up mapping the wrong horizon. Consequently,broad_ side refraction shootingshould be combinedwith oc_ casionalin-line profilesin order to checkon the identity of the horizon being mapped. The first extensiveuseof refractionwasin searchins for salt domes by the fan-shooting technique lsei A salt dome insertsa high-velocitymassinto {i1.2.3). an otherwiselow-velocitysectionso that horizontally traveling energyarrives earlier than if the salt dome were not present; the differencein arrival time be_ tween that actually observedand that expectedwith no salt dome presentis called a lead. In fan-shootmz (fig. ll.5). geophonesare locatedin differentdirections from the sourceat roughly the sameoffset distances,the desireto maintain constantoffsetdistance usually being sacrificedin favor of locationsthat are more readily accessible. The leadsshown by overlapping fans then roughlylocaterhe high-velocitymais. This method is not usedfor preciseshapedefinition. I l.1.3 Gardner',s methodof definingsalt domes Much of the petroleum associatedwith production from the flanks of salt domes lies close io the salt

REFRACTION METHODS

428 Geophonelocations

\ \ - 7/

Craph of time leads frorn shotpoint.4

_ A p p r o x i m a t eo u t l i n e of salt dotne responsrblefor leads

2/

;' Normal pr(llile (.4 ( ) shcre no dotDe l\ pte\enl establirhcr nonnrl tilne di\ttncc r c l a tr o n s l r i p

(;riLph ol trnrc lcad\ l r o r r r\ h o t p o r n l B

F i g . l l . 5 F - a n - s h o o t i n gL. e a d t i m e s o b t a i n e d b y s u b t r a c t i n g observed times from expected times are graphed concentrically a b o u t t h e s o u r c e s (. F r o m N e t t l e t o n , 1 9 4 0 . )

flanks,so that the accuratemapping of the flanks is of considerableeconomicinterest.The method of Gardner ( I 949)involveslocatinggeophonesin a deepboreholedrilledinto the salt for this purposeand shooting from various locationson the surface.The travelpath for each shot is partially through relatively lowvelocity sedimentsand partially through high-velocity salt. For given traveltimeand geophonelocation, the locus of possiblepoints of entry of the travel paths into the salt is a surface (aplanatic surJace)that is roughly a paraboloid(fig. 11.6a).A surfacetangent to the paraboloidsfor all measurementsfor various combinationsof source-detectorlocationsdefinesthe salt dome (fig. I l.6b). Variations of Gardner's method employ shooting from surface locations into geophoneslocated in a deepwell near to but not within the salt. Thesemethods are also used occasionallyto definebodies other than salt.

I 1.1.4Marine refraction Becauserefraction recording requires that there be appreciabledistancebetweenthe source and the recording locations, two ships have usually been required for marine refraction recording.To shoot a reversedrefraction profile in one traverse requires

L

three ships - a shooting ship at each end while the recordingship travelsbetweenthem. For the shooting ships to travel the considerabledistancesbetween source points takes appreciabletime becauseof the relativelylow maximum speedof shipsand hencethe high production rates that make marine reflection work economicalare not realizedin refractionshooting. Consequently,marine refraction work is relatively expensive. The sonobuoy(fig. I 1.7)permits recordinga refraction profile with only one ship' The sonobuoy ls an expendablelistening station that radios the information it receivesback to the shooting ship. The sonobuoy is merely thrown overboard;the salt water activatesbatteriesin the sonobuoyas well as other devices that causea radio antennato be extendedupward and one or two hydrophonesto be suspendedbeneaththe buoy. As the ship travels away from the buoy' shots are fired and the signalsreceivedby the hydrophones are radioedback to the ship,wherethey are recorded. The arrival time of the wave that travels directly through the water from the sourceto the hydrophone is usedto give the offsetdistance.After a givenlength of time, the buoy sinks itself and is not recovered' Sonobuoysmake it practical to record unreversedrefraction profiles while carrying out reflection profiling, the only additional cost being that of the sonobuoys.

REFRACTION DATA REDUCTION AND PROCESSING

*

429

;

,1{fl J,, sa";-=j$

., JA '+' t l l

lLl

i

. .. \ l , \.

gfl-

r l.?-,. r"-^-",.. |-. ,> -

-\

'tiTt:::i:1:,:J"'pr ^ -_Q".,::t,g,,,..

--ra

.\ -h---

- ,^.'" -illll

\

-.-*-

\

:-J.

^

!-'r uil(,ph,i.s 3rr Jrrnd I BJn.n.\rrc(uv{dl

---'\

R...

t-t,i\t,t,

1",) ,/ii/,/'/h

l

,a;f

',7'7'

lfuth*

r::'r;'

h>-

Shotoornl il

I

H,J,,nh,tu. >! ./4

t' -"-t>"4 F i g . I 1 . 7 S o n o b u o y operation. (Courtesy of Select Internat i o n a l I n c .)

'-i-.-----_

I - i g . l l . 6 O u t l i n i n g a s a l t d o m e u s i n g a g e o p h o n ew i t h i n the s a l t . l A f t e r G a r d n e r , 1 9 4 9 . )( a ) p l a n v i e w o f a n a r r l a n a t i c sur_ l a c e ,a n d ( b ) i s o m e t r i cv i e w o f a r a y p a r ha n d p a r a b o l o i d of the ; r p l i r n a t iscu r l a c e .

ll.2 Refraction data reduction and processing Refractiondata haveto be correctedfor elevationand weathering variations, as with reflection data. The correction methods are essentiallythe same except that often geophonesare too far from the source to record the refraction at the baseof the LVL and thus there may be no weatheringdata along much of the line. Additional sourcesmay be used for special re_ l i a c t i o nw e a t h e r i n ign f o r m a t i o n . Whereasthe effect of corrections on the effective source-to-geophone distanceis usually small for re_ flection data, this is often not so for refraction travel pathsabovethe refractor,becausethesemay haveap_ preciablehorizontal components.Hence, the refer_ encedatum should be near the surfaceto minimize sucherrors. The identification of refraction events is usually simplerthan reflectionevents.Traveltimesare usually availablefor a relatively long range of offsets,and henceit is easyto separatereflectionsand diffracrrons uith their curved alignments from the direct wave. surl-acewaves, and refractions with their straight alignments.The direct wave and surf'acewaves are

easily distinguishedfrom refractionsbecauseof the lower velocitiesof the former. Usually,the only prob_ lem is in identifying the different refraction events when severalrefractorsare present. Wherecompleterefractionprofilesfrom zero offset to largeoffsetsare available,playbackof the data with judiciousselections of filtersand automaticsain con_ trol may allow one to correlatereflectionevlnts with refraction events,thus adding useful informauon ro eachtypeof interpretation(fig. I 1.8).The most prom_ inent reflectionsmay not correspond to the most promlnent refractions. Recordsectionsare very useful,especiallyin studying secondarrivals.The refraction profile in fig. I 1.9 shows the direct wave as the first arrival near the source and refractions from successivelydeeper re_ fractorsbecomethe first arrivalsas the offsetdistance increases.Following the lirst arrivals, the continua_ tions of various eventsare seenafter each has been overtakenby a deeperevent. Numerous other events are also seenin the zone of secondarrivals; most of theseare refractionsthat never becomefirst arrivals. or multiply-reflectedrefractions(seefig. 6.39). Another useful refraction playbacktechniqueis to display the data as a redut'edrefractionsection(fig. I l.l0), wherearrival times have been shiftedby the amount xlVo, where.r is the offsetdistance,and Zois a valuenear the refractorvelocity.lf Vowere.*u"tly equal to the refractor velocity,the residual times would be the delay times (which wiil be discussedin $11.4)and relief on the reducedrefractionsection would correlate with refractor relief (althoush dis_ placed from the subsurfacelocation of the.-relief). However,evenif Zois only approximatelycorrect,the use of reduced sections improves considerablythe pickability of refraction events,especiallysecondary refractions. Often, the chief problemin refractioninterpretation techniquesis the assumptionof constani-velocitv

SP336 53,464'

SPlO n t

Fig. I 1.8 Section showing the continuity of some reflectionswith their respectiverefraction events.Recorded by sonobuoy

,l

M a r i n e r e f r a c t i o np r o l i l e .( F r o r n I n g h a r n . 1 9 7 5 :1 3 0 . )

Distance r J J

3 o

0, - N N

o o

I

o ot : o o

sj

3

(b)

o

J J

o 0r -N

N

o o

9

-Ot

U,

o 5 o o

@

(a) R e d u c e dr e f r a c t i o ns e c t i o n .( C o u r t e s yo f P e t t y - R a y Fig. ll.l0 (b) reduced GJophysical.; (a) Conventional refraction sectionl ut SiOti -lt to align highest velocity events; arld (c) reduced at

(c) 2 7 3 5 n V s . S u b t r a c t r n g,v/l/o makes it easier to separate events and simplifies picking.

BASIC-FORMULA INTERPRETATION METHODS layers,hence raypaths that are made up of straight_ line segments;this is usuallynot true, esiecially inlhe shallowestlayers.When using equationsin $4.3to cal_ culate refractor depths, the biggest improvement in the resultsis often due to a more realisticassumption for Z, basedon information other than that obtain_ able from the refraction data themselves(Laski, t973). Problemssometimesresult from a hidden zone, a layerwhosevelocityis lower than that of the overlying bed so that it never carriesa head wave.Energy ihai would approach it at the critical angle cannot get through the shallowerrefractors,and hence there is no indication ofits presencein the refractionarrivals. The low_velocity of the hidden layer, however, rn_ creasesthe arrival times of deeperrefractorsrelative to what would be observedif tiri niaden zone had the same velocity as the overlying bed, hence results in exaggerationof their depths.Another situation,which is also referredto at times as a ..hiddenzone,,,is that of a layer whose velocity is higher than those of the overlying beds but that never producesfirst arrivals despitethis, becausethe layer is too thin and/or its velocityis not sufficientlygreaterthan those of.the overlying beds. Such a bed createsa secondarrival, but the secondarrival may not be recognizedas a dis_ tlnct event. Refraction interpretation often is based solely on ^ first arrivals,primarily becausethis permits accurate determinationof the traveltimes.When we usesecond arrivals,.we usually have to pick a later cycle in the wavetrainand estimatetraveltimefrom the measured trme.However,velocitiesbasedon secondarrivals will be accurateand much useful information is available through their study. Refraction interpretation often involves ..strip_ ping," which is in effectthe removal of one layer at a time (Slotnick,1950).In this method,the problemis solvedfor the first refractor,after which the portions of the timedistance curve for the deeper refractors are adjustedto give the result that wo;ld have been obtained if the source and geophoneshad been lo_ cated on the first refractinghorizon. The adiustment consistsof subtractingthe traveltimesalongihe slant pathsfrom sourcedown to the refracto, und up from the refractor to the geophones,also of decreasing the offsetsby the componentsof the slant paths parallel to the refractor.The new time_distancecurve ls now solvedfor the secondrefractinglayer,after which this layercan be strippedoff and the piocesscontinued for deeperrefractors.

433

two refractors,especiallywhen theseare not Darallel. The basicformulasare commonly usedin the inter_ pretation of engineeringsurveys ($14.1.2)and de_ termining static corrections for reflection seismic work (98.8.2). One of the simplestrefraction interpretationmeth_ ods is the ABC method.With the arrangementshown in fig. I l.l 1, sourcesare locatedat thelnd points of the spread,A and B. Lel tABbe the surface_to_surt-ace traveltimefrom A to .B,and so on; then (seeproblem 11.4) hr. : (ll2)(tr.n * tr." - tn") [V,V2l(Vr2_ V,z!tD1,

( rl . r )

where V, is the overburdenvelocity, and V.the refrac_ tor velocity.(The depth-conversion factor. F :

V,V2l(VrI *

V,lcos0, (ll.2a) often occurs in refraction time-to-depthconversions, for example,in eq. (4.38): h:

Ft,l2,

where /, is the intercept time.) Frequently,V2>> and we can replacethe lactor F by V,; tien hr.:

( V , 1 2 ) ( t . n* t r . " - t u " ) ,

the error in ft.. being lessthan 60l,if V, > 32,. This method assumesthat the overburdenis essentiailyho_ mogeneous, the depth variationsare smooth,the velocity contrastis large,and the dip small.Depth calculationsusingeq. (l 1.3)are generallygood because they dependon the measurementof only one velocity, V,, and three traveltimes.Whereasrefractor dip can be determinedfrom differencesin apparent veiocity as seen on reversedprofiles, it is more often deter_ mined from a seriesof measurementsof depth at different locationsof Cl Better accuracyis given by the four-shot method (gll.l.l), which is efficient for many applications where only a local profile is needed(a .,iounding") rather than a profile line. An application to an engi_ neeringproblemis givenin gl4.1.2. 11.3.2Adachi'smethod Adachi (1954)derivedequationssimilarto eq.(4.42) for the caseof severalbeds with the same st;ike but differentdips. His method departsfrom the usual pa_ rametersand usesvertical thicknessesand anelesof incidenceand refractionmeasuredwith respecito the vertical(seefig. I l.l2). The derivationof Adachi'sfor_ mula is straightforwardbut involveslengthy trigono_

ll3 Basic-formula interpretation methods I 1.3.I Usingba.sic. formulas The basicformulas of $4.3are usedto interpret small amounts of data where the refractorsare aisumed to be planar. Even wheretheseconditionsare met. rnter_ pretationis usuallydifllcult when there are more than

Fig. I L I I

ABC refraction method for determining deoth

REFRACTION METHODS

434

Notation used in Adachi's lbrmula.

metric manipulation (see Johnson, 1976), and we merelyquote the result:

,"- =

" t l ng ' V,

* i

1 l ( . oo, , + c o sB , ) .( 1 1 . 4 )

-V,

where /- is the traveltimeof the refraction at the nth interface(separatinglayersof velocitiesV, and V, ,), ct,and B, are the anglesbetweenthe vertical and the downgoing and upgoing rays in the ith layer,respectively, ft, the vertical thicknessof the ith layer under the source.The anglesa,, b, (seefig. I I . 12) are angles of incidence,a',, b', anglesof refraction, all measured relativeto the normal, and (,*, : dip of the ith interface.Then ai : sin-tl(V,,,|V,) sin a), bj = sin- '[( 4 . ,l v,) sin b), ct,:a,*{,*,, 9 : a,- t,*, a r , : a ' ,* ( , * . , , B,*,: b,| €,*,. For the refractionalong the rth interface,a,: h,,: 0,,, the critical angle. Assumingreversedprofiles,we measureV,, the apparent velocities,Vr, and V"r, and the intercepts,l',, and t,, as usual. For the first interface, : sin t(Vjlv2,i), 9, : sin-'(VJV,,) 0ctr 0 , t : a r - b , : j ( o ,+ 9 , ) ,

valuesof ct,, B,, and then find the other angles(note that (, is now known): t(Vtl V.,), ct, : sin t(Vtl V3,), B, : sin b,:9,+t, ar:a,-t, d! = sin tl(V.lV,)sina,l, bl : sin tf(VrlV,)sinb,l, a,:a'.+ly P.:bi-ty a z : b . : 0 . . : l ( c t .+ 9 , ) : ) A ' + t : 1 , v . = v , l s i n0 , . . {, : l(a, B.). t.,:

( h , l V r ) ( c ocst r + c o s P , ) + (h.lV.)(coso, + cos Br),

/r, being found from the last relation. In principle,this iterativeprocedurecan be continued indefinitely,but in practice,as with all refraction schemes,the errors and difficultiesmount rapidly as the number of layers lncreases. Adachi's formula is best suited to simple cases where the refractorsare plane, no velocity or structural problems exist, and the refractors are shallow. When theseconditions are not met, the formula, in common with other similar ones, may be of limited value.Often, one is not surethat formulasare applicable to a specificreal situation.Where there are more than two refracting horizons, it is often difficult to identify equivalentupdip and downdip segments,especially if the refractors are not plane or if the dip and/or strike change.

€,:l(..,-8,)' reciprocalmethod(GRM ) I1.3.3 Generalized

(seeeq. (4.50)) Vr:

V , l s i n0 , r ,

h , : V r t r " l ( c ocst , * c o s B , ) .

To solve for the secondinterface,we calculatenew

The GRM method (Palmer,1980)is capableof mapping highly irregular refractorsusing reversedprofiles and is relativelyinsensitiveto dip up to about 20'. It

BASIC-FORMULA INTERPRETATIONMETHODS is alsoableto resolvelateralvariationsin the refractor r,elocity(Palmel 1986, l99l); this is especiallyimportant in engineering(wherelow velocity may indicate low rock strength) and groundwater studies (where it may indicate high porosity). The GRM is well suitedfor computer implementation. The GRM involvesselectingseveralpairs of points (X, Y) and making a seriesof calculationsresulting in determining an optimum distancebetweenthem, )f {o,, which approximatesthe critical distancex' in eq. (4.39). Methods for determining XY",, are given toward the end of this section,but approximatevalues often suflice. Our discussionfollows Palmer's 1980 book exceptfor the notation and order oftopics. Figure ll.l3 showsfour bedswith the samestrike but different dips {,. Depths zn, and zBi aremeasured normal to each interface;o, and B, are downdip and updip anglesofincidence,respectively, the anglesat S and ?"also being critical angles.To get the traveltime ti', we consider a plane wavefront PQ passing through A at time t : 0; the wave arrives at (, alter traveling a distance zAt cos o, at time / : (:{r cos ur)lV, The wavefrontreachesR at the time

+5)

Let X and )/ be two points separated by 2a (fi5. l1.l4b). The GRM is basedon the use of a velocityanalysisfunction T, and a time-depthfunction T. referred to G, the midpoint between X and Y. They are defined by ( t u , - t x B+ t A ) 1 2 , T" : (!n, * t r, - t AB- xYlv,')|2, Tr:

27, : .1'\t . i . l, o,cos 0r + z r, cos$,ll V, + AY(S,I V,ll , - .'-' Q r ,c o s 0 i + z " , c o s B ) l V , + X B ( S , | V , ) J LL

. cosc, + :r,cos + .I La 'En, B)IV,+ AB(S,IV,)l : [>

Qz^i

- z.\.r)cos oi I z cos V) B,l ,j

+ (AY - XB + AB)(S,tV,).

: ). r--.coso )/2. t

t

.{ similar expressionholds for /u". Becausethe wave is critically refractedat R and 14the time from R to Z rs RVlVo.Generalizing,we get for n layers

L o c a t i n gP s o t h a t A P : a , e q . ( 1 1 . 7 )i s n o w u s e dt o expressthis result in terms of z. and AG. We have Z , t :i Z r i + a S f ,

The distanceRV : YJ : .6'lcos(€, - t.). Continuing in this manner,we get RV: AB cos l, cos (t. - €,) x cos ({. - t.). In general,we have : )

,t,cos c{i + z,cos B,)lV,+ AB(S,IV,,).

S,-cos{,,

(l 1.6)

( s e ep r o b l e mI 1 . 7 ) . Let X be any point updip from A in fig. I l.l4a. We can expressdepths at A in terms of depths at X as follows: :z^IAH:zr,+AXsinl, : zn I A'H' : zn 4 A'X' sin ((, 9,1 : zn I AX cos€, sin ({, 6,;. For theTth layer,

+ AX. Si,

(rr.1)

: . o r . ( , c o s ( { ,- € , ) . . . c o s ( { r ,€;_J xsin({,-t,,), j>1, :sin{,, i:1.

Z.ri:zpi+(2a-

AqSf,

z r j : z r j - A C , S f, so that n - l

f r:

cr,* cos 9,)l2V >l zpr(cos n l ("or cr,- cos p)Sf DV + S,lV,l

+ ACII

(lr.s)

w h e r eS , : c o s { , c o s( { , - € , ) . . . c o s ( { , , - € , . ) . We assumehenceforththat the dip increasesslowlyso that {, - €, , : 0. In this case,

{t

(l 1.9)

where V,' is an apparent velocity (definedbelow by eq. (ll.ll)). The refractorvelocity V,can be found from T, and the depth from Io. B y u s i n ge q .( l 1 . 5 )i n e q .( l 1 . 8 ) ,

l

7

(11.8)

(I 1 . 1 0 ) This equationshowsthat, for a fixed a, AG and Trare relatedlinearly; hence,the slope dl,./dx is the coefficient of AG in eq. (l Ll0); we definean apparentvelocity V,' such that dT,ldx: llV:.

(ll.ll)

When the dip variesslowly,^f is small (becauseof the sinefactor in eq. (l 1.7))and S, reducesto cos {,_, (see e q . ( l 1 . 6 ) ) ;t h u s ,t h e c o e f f i c i e not f A G t n e q . ( 1 1 . 1 0 ) b e c o m e cs o s l , , 1 V ,a, n d f r o m e q . ( 1 1 . 1 1 ) ,

4-

v',cos{, ,.

(n.t2)

Thus, if the dip {, , is known, V, can be found from the apparentvelocity V,';if not, \' canbe usedas the refractor velocity. I f w e s u b s t i t u t e q . ( 1 1 . 5 )i n e q . ( 1 1 . 9 ) , I

n

F

T.: rIL

l

k * c o s 0 r + z r y c o s Bv,,) l

+ ir1s,tv,- uv,,)1.

REFRACTION METHODS

436

Fis. I l.l3

T h e l a s t t e r m v a n i s h e sb y e q s .( 1 1 . 6 )a n d ( l l . l 2 ) ; moreover,if the dip is small, z.r.i- zr;i- ;r,. Thus, I., reducesto n

I

Palmer (1980: 13, 14) statesthat T. is similar to Hagedoorn'splus value ($11.5.2)when X)' : 0, to Hales' "critical reflectiontime" (r' in 911.5.3)when XY : XY"o,,and to the mean of the geophonedelay t i m e sa t X a n d Y ( $ l 1 . 4 . 1 ) . We definea depth-conversionfactor, V,,: 2V,l(cos ct,* cosB,).(Comparethis with eq.(l 1.2),notingthat V,^: 2F becauseof the factor ll2 in eq. ( I I .9).) We c a n n o w w r i t ee q .( 1 1 . 1 3i)n t h e f o r m n

:

l

T

,

L't;t

n

l

-- \

L

I

- - c ti ' ' r vn ' t

tll'14)

For zero dip,V,0".i..,

on usingSnell'slaw;theprimessignifythatthevelocitiesareobtained fromeq.(l l.l l). ) ,eg e tf o r t h e S e t t i nxg: A G : 0 i n e q .( 1 1 . 1 0w interceptof thevelocity-analysis function n

l

\ i -"fcos 0i + cos 9,ll2v, ,-

(l l . l 6 ) Comparing with eq. ( I I . I 3), we seethat the intercept of In is approximatelyequal to the time-depthat P The GRM can use averagevelocitiesto determine the depth to a refractor without referenceto the actual layering. Assuming horizontal plane layers,eq. (ll.l3) becomes n

l F

To= L(",coso',)l V,, '

s i nc r : 7 t v ; , , t t t . t Z 1

7 , , = ( z , c o so ) / Z

the lorm of eq. ( I L I 7) is preserved. When R and S in fig. 1l.l4b coincide,the distance XI is the criticaldistance,X1"", (seeeq. (4.39)): ^ X Yo p r : I

E r

> - t-' t. . t a n 0I .

H

I

We define the XI,,n, for the single constant-velocity case: XY : 2:L.tans.

( l 1.20)

Eliminating:, and o betweeneqs.( l 1.l8) and ( I 1.20), we find that

v : v " l w l t x y + 2 7 , ; v : ) l t , ) (. l t . 2 l )

V,,: V,lcoso,: ViV;ll(V,'),- (V;),1',,, ( tl . l s )

T t u: - r'r.r

If we replacethe actual section by a single layer of thickness:t.: L zGiandconstantvelocity Z with an angleof incidenceo such that

l

r^ G - l,S u - r , f c o s o . , - l c o s B ) l V(, 1 1 . 1 3 )

'r ( '

R e f r a c t o r sw i t h t h e s a m es t r i k e b u t d i l l e r e n t d i o s

sincr,:VilV,.

(tt.tz)

lf x)' is assumedto be X{.,, which can be found by methodsdescribedin the next paragraph,7,, and V', a r e n o w g i v e nb y e q s . ( 1 1 . 9a) n d ( l l . l l ) . E q u a t i o n (ll.2l) then gives V, after which a and then:r are g i v e nb y e q . ( l 1 . 1 8 ) . To achievemaximum accuracy,the GRM requires a knowledgeof the critical distance,X{,n,, that is, the value when the forward and reverserays leavethe refractor at the same point. Determining this value is "potentially the most confusingaspectof the GRM" (Palmer, 1980: 34). Approximate values sulfice for most purposes(X{0, is relativelyinsensitiveto dip); however,accuratevaluesare requiredifthere are hidden zones ($4.3.1).Palmer gives three methods of finding XY.,,: (a) from knowledgeof the thicknesses and velocitiesof all layers,for example,from borehole information; (b) from the separationof distinctivefeatures on forward and reverseprofiles,such as sharp changesofslope; (c) from trial calculationsof 7",.and

B.{SIC-FORMULA INTERPRETATION METHODS

437

---
(a)

b+ +Q--------4

A

+cl+e+

P

X

G

Y

a = A P = X G = G Y = g q .b = A G (b) Fig. I L14

Relations for points on the surface. (a) "Depths" at A and,x (b) Raypaths to xand

f" curvesfor a seriesof X)r-values;X{0, corresponds to the simplest ?n"curve and the moit detailed ?"., crlrve. Palmer (loc. cit.: 55-7) lists the following stepsfor computation of GRM data: (a) near-surfaceirregularitiesare studiedand their effectsremoved,(b) each point on the traveltimecurvesis assignedto a specific refractot (c) the optimum XI-values are determined, (d) refractor velocities are calculated using eqs. ( l l . l l ) , ( l l . l 2 ) , o r ( l l . 2 l ) ; ( e )d e p t h sa r e f o u n df r o m the time-depthfunction and depth-conversion factors. Hidden zones"are the most common sourceof errors in the majority of refractioninterpretationmethods" (Palmer,loc. cit: 39). When data are good, hidden zones may be defined by comparison of the optimum X)z-valuesobtained from I, and To curves rvith thoseobtainedfrom eq. (l l.l9); differences be-

)4

tween the two sets of values presumablyare due to hidden layers. In a recentpaper,Palmer (1991)illustratedthe application of the GRM to studies of shallow layers where the velocity and/or depth change laterally within a narrow zone, for example,becauseof faulting, a low-velocityshearzone,or a sink hole. Because the anomalousfeaturesare narrow and shallow,geophone and sourcespacingsare small (3 and 36 m in one examplewheredepthswere l0 to 20 m). We illustrate with one of Palmer's models; fig. I l.l5a showsa low-velocityshearzone (or weathered dike) sandwichedbetween two high-velocity layers. F i g u r e sl l . l 5 b a n d l 1 . l 5 c s h o wc u r v e so f T r a n d T . (eqs.(11.8)and (11.9))plotted for XY valuesfrom 0 to 150 m, a range that includesX{",; the curves are displayedvertically at lO-msintervals.

T

1 o a o

t

!

I a !

i

!

!

!

!

,

!

a

a

u

-

-

a

slAllox

E

U

!

rulra€R

J

.

o

Q

a

.

-

-

t

(a) i

i

x

r

t'

st^rt(N xtLo€n x'.':.':.1",]',;,-j-':,1';,;t?

ra

ta

(b) Statbn lqttta. a1

aa

Fig. 11.15 Model resultfor changingrefractorvelocityfrom 6000 to 2100 to 4000 m/s. Station spacing = l0 m. (From Palmer,1991.)(a) I" curvesfor XY :0 (bottomcurve)to 150

a9

m, (b) Io curves for same Xf-values, and (c) depth sectionbased on results obtained in parts (a) and (b).

DELAY.TIME INTERPRETATION METHODS The left and right portions of the curves in fig. I 1.15bare parallel straight lines with slopesof 6000 m/s and 4000m/s, respectively. Startingat the left side, the 6000 m/s straight line extending farthest to the right is the line for XY = 70 m; rhe data begin to depart from this line at about station 44.5. All the 6000 m/s lines are now extendedto station 44.5.On the right side,the 4000 m/s line extendingfarthest to the left is that for XY : 90 m, which reachesstation 5l.5, so all 4000-m/slinesareextendedto station51.5. Stations 44.5 and 51.5 are now taken as the limiting points for the 6000-and 4000-m/szones.The ends of the straight lines at stations 44.5 and 51.5 for each XY-valuearejoined by straightlines;theseshort lines are assumedto be correctedZ" lines for the intervening low-velocity zone. Their slopes give a 2100-m/s value in this case.The 70- and 90-m X)z-valuesare XYo*for the 6000-and 4000-m/szones,respectively; note that the deviationsbetweenthe observedvalues and the straightlines(includingthe short low-velocity linesjoining them) are minimal for theseX)z-values. The time-depthfunction Io is shownin fig. I l.l5c; in calculating Zo values close to the lateral velocity changes,the XYIVi term in eq. (11.9)is a distanceweightedaveragebasedon the adjacentvelocity values.The "pull-up" in the time-depth valuesis due to the low-velocitylayer.The depth-conversionfactor rs calculatedfrom eq.(l l.l5) using Z,: 1.60km/s, Z, : 6 . 0 0 ,2 . 1 0 ,a n d 4 . 0 0 k m / s ;t h e v a i u e sa r e 1 . 6 6 . 2 . 4 7 . and 1.75,respectively. Equation( I l. l4) now givesthe depths shown in fig. ll.l5d; the "pull-up" in fig. ll.l5c has been wiped out by the higherdepthconversionvalue over the shearzone. ll.4 Delay-time interaretation methods l 1.4.1Delay time The concept of delay time, introduced by Gardner (1939),is widely usedin routine refractioninterpretation, mainly becausethe various schemesbasedupon the use of delay times are less susceptibleto the difficultiesencounteredwhen we attempt to use eqs. (4.36) to (4.50) and (11.4) with refractorsthat are curved or irregular. Assuming that the refraction times have been correctedfor elevationand weathering, the delaytime associatedwith the path SMNG in fig. I 1.16is the observedrefractiontime at G /., minus the time required for the wave to travel from p to e (the projectionof the path on the refractor)at the velocity Vr. Writing E for the delay time, we have

the geophonedelay time becausethey are associated with the portions of the path down from the source and up to the geophone. An approximatevalue of 6 can be found by assuming that the dip is small enough that PQ is approximately equal to the geophoneoffsetn. In this case,

E:E"+6":t"

Providedthe dip is lessthan about l0', this relationis sufficientlyaccuratefor most purposes.If we substitute the value of /. obtained from eqs.(4.37),(4.47), and (4.48),we seethat E is equal to the intercepttime for a horizontal refractor but not for a dipping refractor. Many interpretationschemesusing delaytime have been given in the literature, for example, Gardner (1939,1967),Barthelmes(1946),Tarrant (1956),Wyrobek (1956),and Barry (1967).We shalldescribeonly the latter three. The methods describedby Wyrobek and Tarrant are suitable for unreversed profiles, whereasthat of Barry works best with reversedprofiles. I 1.4.2Baruy'smethod The schemedescribedby Barry, like many basedon delay times, requiresthat we resolvethe total delay time 6 into its component parts, 6" and 6.. In fig. 11.17,we show a geophoneR for which data are recorded from sourcesat A and I The ray BN is reflectedat the critical angle; hence,Q is the first geophone to record the head wave from ,B. Let En, be the sourcedelay time for sourceA, 6ro and Eo* the geophonedelay times for geophonesat Q and R, 6r, and 6r^ the total delaytimesfor the pathsAMNQ and AMPR. Then, E/o:E/M+6Np E1R:6/M+6pR

AE:6rn-6r^:E"n-E"^. For the sourceat fl the sourcedelay time 6r, is approximatelyequal to 6"o providedthe dip is small. In this case, Er^:6"r*E"^-b\p+EpR.

P Q _ ( s M+ N G + r \ _ r 0 v, \ v, v.l v. (SM + NG\ IPM + NO\

6 : r '. _

\ v : l ISM \V,

t

-

t

, / \ v PIA ING l + l V.I \V.

-

l

2 l //O\ V.I

:6" * 6,, (r1.22) where E"and 6* are known as Ihe sourcedelay time and

(r r .23)

Fig. I I . 16

Illustrating delay time.

REFRACTION METHODS

440

reflector.Denoting the delay time associatedwith the path QR by E,, we have 6 r : p l V t - ( p c o s$ ) / Z r ; hence, p: Fig. I L 17

Determining source and geophone delay times.

The geophonedelaytimes are now given by

6"0:I(SBR+AE)l 6r^:l(6rR-Ab)J

V,6rl(l - sin 0. cos $).

This is the polar equation of an ellipse.An ellipseis the locus of a point Q $g I I . l9b) that movesso that the ratio (QRlQI[) is constant(equalto the eccentricity e, which is < I for an ellipse), pl(h+pcos$):e,

(1t.24) hence,

p:Ehl(l-ecos0). Thus, it is possibleto find the geophonedelaytime at R provided we have data from two sourceson the sameside and we can find point Q. If we assumethat the bed is horizontal at N and is at a depth ft* we have /r- :

Z,Er^,/cos 0,,

BQ = 2h* tan 0, : 2 V r 6 u r ( t a n 0 , / c o s0 . ) : 2 V r 6 " , t a n 20 , . .

(11,27)

(il.25)

(tl.26)

The sourcedelay time E* is assumedto be equal to half the intercepttime at ^8,this allowsus to calculate an approximatevalue of BQ and thus determinethe delay times for all geophonesto the right of Q for which data from A and ^Bwere recorded. The interpretation involves the following steps, which are illustratedin fig. I l.l8: (a) the correctedtraveltimesare plotted, (b) the total delaytimes are calculatedand plotted at the geophonepositions, (c) the "geophoneoffset distances"(PP' in fig. I l.l7) are calculatedusing the relationPP' : V r 6 o ^t a n 2 0 (, s e ep r o b l e m1 1 . 8 )a n d t h e delaytimes in (b) are then shiftedtoward the sourceby theseamounts, (d ) the shiftedcurvesin (c) for the reversedprofiles should be parallel;any divergenceis due to an incorrect value of Z, hence,the value of V, is adjusted and steps (b) and (c) repeated until the curves are parallel (with practiceonly one adjustmentis usually necessaryJ, (e) the total delay times are separated into sourceand geophonedelay times, the latter beingplotted at the points of entry and emergence from the refractor (S and 7 in fig. I 1.17);the delay-timescalecan be converted into depthifrequired usingeq. (11.25).

I1.4.3 Taruant's method Tarrant (1956)usesdelay times to locatethe point Q (fig. I l.l9a) at which the energyarriving at R left the

(l 1.28)

The major axis,2a: po=o* po=.: 2ehl(l - e2).The semiminoraxis,b, can be found by writing y = p sin - €2) tt2.The $ and finding-y-"*;this givesb : eh(l distancefrom the focus,R, to the centerofthe ellipse, @ is equalto pla-o- a: ehl(l - e) - ehl(l - e' 1: ec. If we take e : sin 0, and h : Vr6,,eq. (11.28) becomeseq.(11.27). For a horizontal refractor,we havethe ellipsein fig. l l . l 9 c , w i t h a : V r 6 ,t a n 0 , s e c0 , , b : I z 1 6t*a n 0 , , and OR : Vr6, tan20,.Also RQ : blcos 0, : a and IOQR : tan-t(ONb) : 9,, OQ : OR cot g, : Vr6rtan 0,. We can approximatethe ellipsein the vicinity of Q with a circle of the same radius of curvature. If we write the equationof the ellipsein the Cartesianform, (xla)' +(Ylb1z:1' the radiusofcurvature,4 becomes r:[l+(y,)r],,.|y,,, -(bla)' (xly), y" : -(bla)' (y - xy')ly', : where y' : 0 and !" : bla2. Hence' r : a2lb = at Q, Y' : Vz6,tan 0, sec20,and the center C is at 2,,6*/cos30, the point (0, r - b), that is, (0, 4 6* tan30.).Also, ICRO : tan \(COIRO) : 0,; hence,ICRQ is a right angle. To apply the method,we must determinethe velocities, i/, and V, and the delay time at the source 6,. Then we compute 6" from the formula, E,:/^-(xlVr)-6". We are now ableto compute OR, OQ, and then locate C by drawing RC perpendicularto RQ. From Q, we draw an arc of the circle to representthe refracting surfacein the vicinity of Q, lf the dip is not zero, the point of emergenceis Q' , the arc QQ' increasingwith the dip. Even for moderatedip the elliptical arc QQ' will be close to the circular arc through Q and thus the envelopeof circular arcs will outline the refractor closely. Tarrant'smethod is usefulwhen the dip is moderate or large and the refractor is curved or irregular.The principal limitation is in the determinationof L'

DELAY-TIME INTERPRETATION METHODS

441 Distance(km)

,*f

F

F

g

, ra[

i

Travcltinlcs

1206 1 2 . r 0g 12.t4 I2 18

I

9

E q

F

Average offsct delay time

g 9

l r

Fig. ll.l8

2.8 3.0

I l l u s t r a t i n g t h e d e l a y - t i m e m e t h o d o f i n t e r p r e t i n g r e v e r s e d p r o f i l e s . ( A f t e r B a r r y1, 9 6 7 . )

I 1.4.4 Wyrobek'smethod To illustrateWyrobek'smethod, we assumea seriesof unreversedprofiles,as shown in the upper part offig. I1.20. The various stepsin the interpretationare as follows: (a) the correctedtraveltimesare plotted and the intercepttimes measured, (b) the total delay time E is calculatedfor each geophoneposition for each source and the values plotted at the geophoneposition (if necessary, a valueof 4 is assumed);by moving the various segmentsup or down a composite curve similar to a phantom horizon is obtained, (c) the intercept times divided by 2 are plotted and comparedwith the compositedelay-time curve;divergencebetweenthe two curvesindicatesan incorrectvalueof Z, (seewhat follows),hence,the valueusedin step(b) is varied until the two curves are "parallel," after which the half-intercepttime curve is completed by interpolation and extrapolationto coverthe samerangeas the compositedelaytime curve, (d) the half-intercepttime curve is changedto a depth curve by using eq. (a.38),namely, s, h : , r V , t , l c oo

tcl

Fig. I l.l9 Illustrating Tarrant's method. (a) Relation between the receiver R and the emergent point Q, (b) showing that the locus of Q is an ellipse with R at one focus, and (c) geometry of the ellipse through Q

(note that we are ignoring the differencebetween the vertical depth /r and the slant depthsh, and huin eqs.(4.45) and (4.46). Wyrobek's method depends on the fact that the curve of E is approximately parallel to the halfintercept time curve. For proof of this result, the reader is referred to problem 11.10. Wyrobek's

REFRACTION METHODS

442

StcP (a) (travcltimes)

---v----------

'/

StcP (6) (total delay times)

compositcllay{ime

SteP (c) . (half-intcrccPtttmes)

lt,

I

Stcp (d) (dcpth)

I unreversed Fig. I 1.20 [llustrattng Wyrobek's method using profi les. (After WYrobek, r9 5 6 . )

the method doesnot requirereversedprofilesbecause the upon interceptat a sourcepoint doesnot depend out' laid is direction in which the cable Delay-time methods are subject to certain errors that must be guarded against' As the source-tog.opno.r. distaice increases,the refraction-wavetrain to later 6."orn.t longer and the energy peak shifts cycles different that danger the ;yJ.- There'is thus error the that and profiles different on picked *itt U. time' will be inte.p..ted as an increasein sourcedelay obviis usually error the lf sufficientdata are available themous. Variations in refractor velocity manifest total-delayoffset the of divergences local selvesin if time curvesfor pairs of reversedprofiles'.However' in travel refraction some data that do not represent inthe refractor under considlration are accidentally the if as same the be to is apt cluded,the appearance *l:t' refractor velocity were varying' In situationtvelocltles same severalrefractorsthat havenearly the not be u.. pi.t.nt, unambiguousinterpretationmay possible. 11.5Wavefront interpretation

methods

method 11.5.I Thornburgh's graphical Wavefront reconstruction, usually by interprerefraction several of basis the means,forms is one by paper classic The techniques. i;il"' are those itto.nUutgtt (t^q:O);other important papers

with 1 . 6 0 0+ A r r . 1 . 6 0 0+ A r c . . " B y d r a w i n ga r c s can we ' ' ' VrLtr, ' VtL'tu, centersB, C, . .. and radii = (AZ) accu' as s 1'600 I for establishthe wavefront rately as we wish. Similarly, other refraction wave: l '400 s, can be confronis, such as that shown for / direct r1.""t.a at any desired traveltime interval' The circles the course of are S source wavefronts from the shown in the diagram. In fig. I |.22, ie show a seriesof wavefrontschosen shown so thaionly wavesthat will be first arrivals are interests the in eliminated being (all secondaryarrivals of simplicityj. Betweenthe sourceS and the crossover polnt b lseeeq. (4.40))the direct wavearrivesfirst' To horizon itre righi of C, the wave refracted at the first from refraction the of right G the "i.i"""t first until, to refraction' shallower the overtakes horizon the deeper the diThe iwo systemsof wavefrontsrepresenting. hoshallow the from rect wave und th. refractedwave line' this ABC; line dashed the rizon intersectalong curveby Thornburgh' passes calledthe coincident-time through the points where the intersectingwavefronts is a truu. itt. same traveltimes. The curve DEFG The horizon' deeper the for coincident-time curve at refractors the to tangent are curves coincident-time critical the reaches ray incident the where i;"i D which ungf. tt.. problem I I . l2), whereasthe points at are surface the meet the- coincident-time curves timethe of slopes in the changes abrupt marked by distancePlot. Becausethe coincident-timecurve is tangentto the one refractor,the latter can be found when we have critical depth' dip, the as such data' other ptonf. unA ;;;[ - or a secondprofile (not necessarilyreversed) and beJausewe now havetwo coincident-timecurves curves' the to tangent the refractor is the common When reversedprofiles are available,the construcof lotion of wavefrontsprovidesan elegantmethod illustrated is principle cuting ttte refractor.The basic and i" ng] f f .Z:, which showstwo wavefronts,MCD interan at intersecting B and A at PCE, fromsources travmediate point C. Obviously, the sum of the two reciprocal the to eltimesfiom ,4 and .B to C is equal the time betweenA and B, l,' If we had reconstructed without curve time-distance the from two wavefronts we knowing where the refractor RS was located' not PCQ' would draw the wavefronts as MCN and of waveMCD and PCE Therefore,if we draw pairs the travof sum the that B such fronts from A and points the pass through must refractor the 1., .tti-.t is of interseciionof the appropriatepairs of wavefronts'

(t?l-l' Hales(1958)' iv C"ta".. irS+Si,Baumgarte and Schenck (1967)' Rockwell ilug"ooo.nirqssi, (1967). ' reconfigu.. 11.21illustratesthe basic method of that wavefront refraction The struciing wavefronts' at times " ' : C' B, reached s 1.600 / at reachei,4

plus-minusmethod I 1.5.2Hagedoorn's utilizesa conHagedoorn'sp lus-minusmethod (1959) When the rejust described' that to similar strriction wavefronts fractor is hortzontal, ihe intersecting diamondform milliseconds Jrawn at intervals of A vertiwhose i'24; (fig. I ;h;|ed fls;*s lo-rlz^oltal.and 0" respecZ,A/cos and VrL' to equal are cal diagonals at each ti".fv. if we add together the two traveltimes

WAVEFRONT INTERPRETATION METHODS

443

2 04

'=N

06

Fig. I l.2l

s

R e c o n s t r u c t i o no f w a v e f r o n t s .

a

Fig. 11.22 First-arrival wavefronts for three layerswith velocitres in the ratios 2:3:4. The dashed curves,4BC and DEFG are c a l l e dc o i n c i d e n t - t i m ec u r v e s .( A f t e r T h o r n b u r g h , 1 9 3 0 . )

the "plus" lines to plot the refractor shape. The differencebetweentwo traveltimesat an intersection is called the "minus" value;it is constantalong vertical lines passingthrough the intersectionsof wave"minus" lines fronts. The distancebetweensuccessive as shown in fig. 11.24 is VrL,; hence, a continuous checkon Z, is possible.Although dip altersthe forgoing relations,the changesare small for moderatedip, and the assumptionis made that the "plus" lines are still parallelto the refractorand that the "minus" lines do not convergeor diverge. Fig. I1.23 Determining refractor position from wavefront intersectlons.

intersectionand subtract 1,, the resulting "plus" values equal 0 on the refractoq +2A on the horizontal line through the first set of intersectionsvertically abovethose defining the refractoq +4A on the next line up, and so on. Becausethe distancebetweeneach pair of adjacentlines is Z,A/cos0., we can use any of

IL5.3 Hales'graphicalmethod Graphical methodsare well suitedto many refraction interpretation problems.When carried out carefully, they often give the requisiteaccuracyrapidly and they are satisfyingto make becausethe picture unfolds as one carriesout the interpretation. Hales' method (1958)is useful where the depth to the refractorvariesappreciably,a situationoften associated with variation of overburdenand refractor ve-

REFRACTION METHODS

444

R e f r a c t o;r' p l u s ' v a l u e= 0 'plus'value=2A

' t 6 ^ "

. m , n u su' a l u e = f ,-

'{ l0A

F i g . I 1 . 2 4 l l l u s t r a t i n gt h e p l u s m i n u sm e t h o d

locities.The method requiresreversedprofiles.The essenceof the method is the schemefor locating pairs of points such as A and B (fig. 11.25a),which havea common point of emergenceQ, when the dip and depth of the refractor are not initially known. The interpretationprocedurewill be describedfirst and then the propositionswill be proven. Given reversedrefraction profiles,as shown in fig. ll.25b, we selectan arbitrary point B at which the arrival time is t^r. The point K is located such that KB : (t, - t*r).A line through K at the angle o : tan-t(Vt sin 0,) intersectsthe reversedprofile at time /.r, at location ,4, which is the point on the reversedprofile associatedwith the same point on the refractor (Q on fig. ll.25a) as B The time r' (fig. I 1.25b)and the distancex'can now be readfrom the reversedprofile plot. A line is drawn through I at the critical angle 0, (fig. 11.25c),which intersectsthe perpendicularbisectorof AB at C. An arc is then drawn of radius p : VJ'l(2 cos 0,). The refractor is the common tangent to arcs drawn in this way. The angle a given aboveis not preciselythe correct angle o'', but the error is negligible,as will be shown. To establishthe soundnessof this method,consider the geometryof the triangle AQB (fig. I 1.25d),where Q is the refractingpoint. The refractedwavesfrom R to .Band from S to I (fig. ll.25a) leavethe refractor at Q. The circle that passesthrough A, Q, and ,B is drawn and the valuesof the severalanglescan be determined in terms of critical angle 0. and dip (. The distanceCQ : p can be found by noting that

o' Pcos =3[,=i3;i[ However,AN : CN tan q : CGtan € : BG hence, adding the two expressionsfor p cos 0. gives p:

( A Q + B Q ) I Q c o s0 , ) .

From fig. ll.2la, we seethat t * , * t r n : t , + ( A Q + B Q ) lV , : hence, AQ+ BQ: VJ',

p:

V,t'l\2cos0,)

(note that t' is the traveltimefor a reflectionwith path AQB). Further, AB:x':AH+HB _ AQ sin 0, sin(l,n+ {) = tAQ * t0) :

*

80 sin 0, sin(jn - {)

crn

fl

.;r; VJ' sin O,/cos(.

T h e a n g l e so : tan'(2, sin 0,)and cr' : tanr ( x ' l t ' ) a r e e q u ailf { : 0 . I f € * 0 , a ' > o s o I w i l l b e locatedtoo closeto B by the amount Lx', t"nand t' will be slightlytoo smallby the amountAl', and p will be too small by Ap. Referringto fig. ll.25e, Lt'lL,x' : slope of traveltimecurve : sin(0, + {)/2, (for the downdip traveltimecurve) a^ _

J O ' :

V , L , t ' _ A x ' s i n ( 0 ,+ t ) 2cos0, 2cos0,

The point C from which p is measuredalso movesto C'(fig.ll.25e): CC' : Ax'l(2 cos 0,) CQ - C'Q' : CC' cos(jrr - e, - €) : Ax' sin(O,+ 0l(2 cos 0,), which is exactlyequal to Ap. Hence,the only effectof neglectingdip is to displacethe refractingpoint updip by the amountjAx'. Hales' method requiresknowledgeof V.,and Vrin

(d)

F i g . 1 1 . 2 5 H a l e s ' g r a p h i c a lm e t h o d . ( a ) R e l a t i o n b e t w e e nr w o : ! ' c c i v e r sA a n d B h a v i n g a c o m m o n e m e r g e n tp o i n t p , . ( b ) g e o _ n e t r i c a l p r o p e r t i e so f p o i n t s o n t h e t r a v e l t i m ec u r v e s c o r r e _

sponding to ,4 and B (construction lines are dashed); (c) construction for locating p,. (d) properties of circumscribed circle through A, B. an
REFRACTION METHODS

446 order to calculateo. Variation of Vrcan be accommodatedby calculatingVrfrom the slopesof the respective traveltime curves at B and aLA (an approximation of the location of A will usually suffice).Variations of Z, with depth (usuallyan increasewith increasing depth) cen b€ accommodatedby iterating the calculation. 11.6Geologic inte4lretation

of refraction data

Much of what is termed refractioninterpretation,especially the application of equations such as eqs. "computa(4.36)to (4.52),should properlybe termed tion." The geologicalinterpretationof refractiondata, to distinguish it from computation, is much cruder than that of reflectiondata and usually much more restrictedin rangeof depthsinvolved,detail and prerefractiondata cision.Under favorablecircumstances, can yield both structural and stratigraphicdata but usually only structuralinformation is obtained. ln virgin areas,refraction is often done with the twin objectivesof determiningroughly (l) the shape of the basin, including depth to basement,(2) the nature or rock type of the major lithological units based on their velocities.Velocitiesin the range 2-3 km/s generallydenote sandsand shales,whereasvelocities of 5-6 km/s usuallydenotelimestone,dolomite,or anhydrite.Crystallinebasementrefractionsoften havea characteristicenvelopeand are very strong.Velocities of the variousrock typesoverlap(asshownin fig. 5.5); hence,refraction velocitiesgenerallydo not identify rock typesuniquely.When outcropsandior well information are available,the interpretationmay be more reliable. Structural interpretationis usually simpleprovided the data permit accuratematching of up-and-down dip velocitiesand velocity complicationsare not present. Faulting is sometimesindicatedmore clearly on refraction recordsand the displacementfound more accuratelythan is the casewith reflectiondata. However,the usual paucity of refractorshardly ever permits us to find the variation of displacementwith depth, curvatureof the fault plane, and so on, information that under favorable circumstancescan be found from reflectionsections. If enoughdata are available,interpretationalambiguitiesoften can be resolved.However,in an effort to keepsurveycostsdown, only the minimum amount of data may be obtained(or lessthan the minimum) and someof the checksthat increasecertainty and remove ambiguitiesmay not be possible.

11.2 The velocity of salt is nearly constant at 4.5'7 km/s.Calculatethe amount of lead time per kilometer of salt diameter as a function of depth assumingthe sedimentshavethe LouisianaGulf Coastvelocitydistribution shown in fig. I 1.28. 11.3 In refraction mapping of the 5.75-km/slayer at about 0.6 km depth in the Illinois Basin,the overlying "hidden layer."Using the velocity-depth shaleforms a data from fig. 11.28, determine approximatelyhow much error neglectof the hidden layer will involve. I1.4 Proveeq. (1l.l ) assumingthat the surfaceis horizontal and the refractor is a plane. 11.5 Given the data in table I Ll for a reversedrefraction profile with source points l, B, use Adachi's method to find velocities,depths,and dips. 11.6 (a) Solveproblem ll.5 by strippingotr ($11.2) the shallow layer (usethe samevelocitiesas in problem I 1.5for the purposeof comparison). (b) Compareyour resultswith thosein problem I 1.5. (c) What are some of the advantagesand disadvantagesof stripping? ll.7 Proveeq. (11.6).(Hint:Wtite cos ((,, ' €,,,): is integer; (€, lcl an where cos[((, , €, .) + €,")], , expandand drop the product ofsines.Treat the factor cos (€, . - t,,) in the sameway and continuethe processuntil you arrive at the factor cos ((' {,")' then setnr:0:€".) 11.8 Showthat PP' in fig. I l.17 is givenby PP' : VzE"^ tan20,. 11.9 SourceB is 2 km eastolsource l. The data in table I 1.2 were obtainedwith cablesextendingeastward from A and B with geophonesat 200-m intervals. Interpret the data using Barry's method. Note that -x is the distancemeasuredftom A. Take V,: 2.5 km/s and assumethat the delay-timecurve for the

time (t)) Total delaY

(,-;\ 1 tr ,

l,

-1,

Problems 11.1 Early refraction work searchingfor salt domes "lead" to in the Gulf Coast considereda significant be of the order of 0.250 s. Assuming a range of 3t/z miles (5.63 km), a normal sedimentvelocity at saltdome depth of 2.74 km/s, and salt velocity of 4.57 km/s, how much salt travel would this indicate?

Fig. I I .26 Demonstrating the parallelism of the curves of total delaytrmeand the half-interceptrme.

PROBLEMS

447

Table I 1.1Reversed refractiondata

0.00 3.00

05

1 0

1.5

2.0

2.5

3.0

3.5

40

0.25 2.90

0.50 2.80

0.74 2.68

0.98 2.52

1.24 / - . +|

1.50 /.)|

1.70 2.20

I .81 2.07

5.5

l8

2.16 1.65

2.28 1.50

2.38 1.40

50km l.9l 1.91

202s 1.80 s

7.0

7.5

8.0

8.5

9.0

9.5

10.0km

2..+.1

2.56 t.t2

2.64 1.00

2.72 0.75

2.80 0.49

2.89 0.23

3.00s 0.00s

t.2s

Table I 1.2 Time-distancedata

1 . 6k m t8 -r.0 1.6 -1.8 -1.0 t 1

1.4 ;1.6 .1.8 5.0 5.2 5.4 5.6 5.8 6.0 o.l

6.4 6.6 6.8 1.0 7.2 1.4 7.6 7.8 8.0

TA

tR

1.02 s t.05 t.l0 t.l4 l.l8 t.20 t.26 t.32 1.35 1.39 1.45 1.50 1.56 l .59 1.62 t.66 t.'12 1.73 1.80 1.85 l .91 t.97 2.00 2.02 2.05 2.10 2.13 2.16

0 . 2 5s 0.34 0.43 0.52 0.61 0.70 0.78 0.87 0.96 1.05 l.l0 1.14 |.20 |.22 1.28 l.31 1.36 t.42 |.47 r.53 1.56 1.59 1.63 1.67 l.'70 |.'73 1.78 l.8l

reversedprofiles is sufficientlyparallel to yours that s t e p( d ) i n g l 1 . 4 . 2c a n b e o m i t t e d . ll.l0 Prove that the half-interceptcurve referred to in the discussionof Wyrobek'smethod in gll.4.4 is parallel to the curve of the total delay time 6 (seefig. 11.26).Note that the reciprocaltime can be written (seeeqs.(4.47)and (4.48)

,,:;l(+ +r,,) +(i..,,,)l 11.11 SourcesC, D, E, F, and Gin fig. ll.3 are 5 km apart. The data in table I 1.3are lor three profilesCE, DF, and EG with sourcesat C, D, and E, no data being recordedfor offsetslessthan 3 km. For profiles from

,indc=#

^.oco

\"g"'y't'E ,4--

\

B)

/

:f D--

/---

/ X""o

Fig. I 1.27 Deriving the properties of the coincident-time c u rve.

F and G the interceptswere 1.52and |.60 s, respectively. Use Wyrobek'smethod to interpret the data. 11.12 Using fig.11.27,showthe following: (a) DE, the "wavefront for I : 0," is at a depth SD : 2h : 2z cos0,. (b) After DE' reachesl, wavefrontssuch as BFcoincide with the headwavewavefronts. (c) The coincident-timecurve AH is a parabola (d) Taking DE and DS as the )r:-and /-axes, the equation of AH is 4hy : x2 + 4h2. (e) The coincident-timecurve is tangentto the refractor at A. ll.l3 Interpretthe data in table I 1.4usingthe plusminus method. 11.14 The data in table ll.5 show refractiontraveltimes for geophonesspaced400 m apart between sourcesA and B, which are separatedby 12 km. The columnsin the table headedtj and t| give secondarrivals.Interpret the data using;(a) eqs.(4.45)to (4.50); (b) Tarrant'smethod; (c) the wavefrontmethod illustrated in fig. I 1.23;(d) Hales'method. On the basisof your results,comparethe methodsin terms of (l) time involved;(2) effectofrefractor curvature;(3) effectof random errors; and (4) suitability for (i) routine production or (ii) specialeffort wherehigh accuracyis essential. 11.15 Constructthe expectedtime-distancecurve for

448

REFRACTION METHODS

TableI 1.3Refraction data

3.00km 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.60 6.80 7.00 7.20 7.40 '7.60 7.80 8.00 8.20 8.40 8.60 8.80 9.00 9.20 9.40 9.60 9.80 10.00

Table I 1.4 Time-distancedata

I ru

to,

Isc

1 . 1 s8 1.22 1.24 1.28 1.35 1.38 t.4l 1.47 I. 5 1 1.53 1.58 1.63 l.65 1.69 t.74 1.78 1.82 1.87 1.90 t.94 1.97 2.01 2.06 2.10 2.t4 2.17 2.20

1 . 2 0s |.29 1.38 t.4s 1.54 l.60 1.70 1.74 1.77 1.80 t.82 I.85 l.91 1.95 t.97 1.99 2.03 2.08 2.12 2.16 2.20 2.25 2.30

1 . 1 s9 1.28 1.35 t,43 50 58 68 t.'76 1.82 1.89 2.00 2.06 2.15 2.21 2.29 2.38

2.30 2.32 2.35 2.38 2.44 2.47 2.50 2.54

L--)-)

2.37 2.41 2.45 2.4'l 2.52 2.55 2.61 2.64 2.68 z-tJ

2.78 2.82

1.+-) 1.40

2.49 2.54 2.57 2.60 2.65 2.68 2 . 7| 2.74 2.7'7 2.82 2.85 2.89 2.93 2.9'7 3.00 3.04 3.07 3 .l 0

the Java Sea velocity-depth relation shown in fig. 11.28.Is it feasibleto map the top of the 4.25-km/s limestoneat a depth of about 0.9 km by the use of head waves?What problemsare likely to be encountered? I l.16 In early refractionexplorationfor salt domes,a "blind spot" (the regionB C in fig. I1.29)was found when the dome lay directly on the line betweenthe sourceand the geophone,that is, the arrivalswereoften too weak to the detected.This wascalled"absorption of the wave" by the salt dome. What is the true explanationfor this "absorption"? 11.17 How many distinctly separatehead wavesare indicatedin fig. I 1.9?What are their apparentvelocities?Calculatethe depthsand velocitiesof the respective refractorsassuming(a) no dip, (b) 5' dip to the right, and (c) 5'dip to the left. References Adachi, R. 1954. On a proof of fundamental formula concern-

0.0km 0.4 0.8 t.2 1.6 2.0 2.4 2.8 3.2 J.O

4.0 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2 '7.6 8.0 8.4 8.8 9.2 9.6 10.0 10.4 10.8 tt.2 l1.6 12.0

lA

tB

0.00s 0 .1 5 0.28 0.44 0.52 0.63 0.70 0.'76 0.84 0.91 0.9s 1.04 1.12 1.16 1.25 l.30

2 . 3 0s 2.23

I.JJ

1.40 1. 5 1 1.5'7 1.60 1.72 t.78 l.80 I.91 1.93 2.04 2.0'7 2.1'7 2.20 2.30

z .t J

2.09 2.04 1.98 1.92 1.85 1.80 1.'72 1.64 t.60 L55 t.4'7 1.40 t -Jz

1.28 1.24 I.18 t.l0 L04 0.96 0.90 0.83 0.'t6 0.66 0.52 0.39 0.25 0.t2 0.00

ing refraction method ofgeophysical prospecting and some remarks. Kumamoto J. St'i., Ser. A, 2: l8 23. Barry, K. M. 1967. Delay time and its application to refractron profile interpretation. ln Seismic Refraction Prospetting,A.W. Musgrave, ed., pp. 348 61. Tulsa: Society of Exploration Geophysicists. Barthelmes, A. J. 1946. Application of continuous profiling to refraction shooting. Geophysic,s, ll:24 42. Barton, D. C. 1929. The seismic method of mapping geologic structure. ln Geophysical Prospecting, pp. 572 674. New York: American Institute of Mining and Metallurgical Engineers. Baumgarte, J. von. 1955. Konstruktive Darstellung von seismischenHorizonten unter Beriicksichtigung der Strahlenbrechung im Raum. Geophys.Prosp.,3z 126-62. Gardner, L. W. 1939. An areal plan of mapping subsurface structure by refraction shooting. Geophy.sits, 4:247 59. Gardner, L. W. 1949. Seismographdetermination of salt-dome boundary using well detector deep on dome flank. Geophl,sits, 14:29 38. Gardner, L. W. 1967.Refraction seismographprofile interpretation. In Seismic Refraction Prospecting,A. W Musgrave, ed., pp. 338 47. Tulsa: Society of Exploration Geophysicists.

i

.l

n

I

i Velocity

I

ti

5 \ \

l

i

'-'--""""'' \ i \'-"-.""'i

F i g . I l . 2 t l V e l o c i r y d e p t h r e l a t i o n s h i pl b r w e l l s i n t h e I l l i n o i s B a s i n ( s o l i d c u r v e ) ,J a v aS e a ( d o t t e d c u r v e ) .a n d L o u i s i a n aG u _ i l f C o a s t ( d a s h e dc u r v e ) .

Table I 1.5 Time-distance data TA

0.00km 0.40 0.80 t.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 4.80 5.20 5.60 6.00 6.40 6.80 7.20 7.60 8.00 8.40 8.80 9.20 9.60 10.00 10.40 10.80 I 1.20 l 1.60 12.00

0.000s 0 . 18 2 0.320 0.504 0.680 0.862 0.997 1.170 |.342 1.495 1.6'7'7 1.82t 1.942 2.103 2 . 15 0 2.208 2.330 1 tr11

2.504 2.602 2.658 2.720 2.744 2.'/60 2.855 2.920 2.980 3.065 3.168 3.230 3 . 3l 0

I8

3 . 3 1s0 3 . 18 2 3.t40 3.063 2.917 2.839 2 . 7t 4 2.681 2.s70 2.505 2.442 2.380 2 . 3l 8 2.220 2.t25 2.030 2.003 1.862 1.743 t.622 1.610 1.482 1.329 1.r40 t.018 0.863 0.660 0.503 0.340 0 . 19 8 0.000

t:

1 . 6 8s2 1.760 1.858 1.881 t.962 2.053

vl

ti 1.561 1.440 1.288 1.202 | .t7'7 1.082 Fig. 11.29 Travelpaths through a salt dome according to Snell's law. (From Barron, 1929.) (a) Verrical section; (b) plan r iew.

i

450 Hagedoorn, J. G. 1959.The plus minus method of interpreting seismicrefraction sections.Geophys.Prosp.,7:158 82. Hales, F. V{ 1958. An accurate graphical method for interpreting seismic refraction lines. Geophys.Prosp., 6z 285-94. Ingham, A. 1975. Sea Surveying.New York: John Wiley. Johnson, S. H. 1976. Interpretation of split-dip refraction data in terms of plane dipping layers. Geophysics,4l: 418 24 Laski, J. D. 1973. Computation of the time-distance curve for a dipping refractor and velocity increasing with depth in the overburden. Geophys Prosp., 2l: 366-78. Milsom, J. 1989. Field Geophysics.New York: John Wiley. Nettleton, L.L. 1940. Geophysical Prospecting for Oil. New York: McGraw-Hill. Palmer, D. 1980. The Generalized Reciprocal Method of Seismic Refraction Interpretation. Tulsa: Society of Exploration Geophysicists. Palmer, D. 1986. Handbook of Geophysical Exploration, Vol. l3: Refraction Sersmics. London: Geophysical Press.

REFRACTION METHODS Palmer. D. 1991. The resolution of narrow low-velocity zones with the generalized reciprocal method. Geophys. Prosp, 39:. 1 0 3 16 0 . Rockwell, D. W. 1967. A general wavefront method. In Seismit Refraction Prospecting, A. W. Musgrave, ed., pp. 363-415. Tulsa: Society of Exploration Geophysicists. Schenck, F. L. 1967. Refraction solutions and wavefront targeting. In SeismicRefraction Prospecting, A' W Musgrave, ed , pp. 416 25. Tulsa: Society of Exploration Geophysicists. Slotnick, M. M. 1950.A graphical method for the interpretation of refraction profile data. Geophysics,15: 163 80. Tarrant, L. H. 1956. A rapid method of determining the form of a seismic refractor from line profile results. Geophys Prosp.' 4 : 1 3 19 . Thornburgh, H. R. 1930. Wavefront diagrams in seismic interpretation. Bull. AAPG, 14: 185 200. Wyrobek, S. M. 1956. Application of delay and intercept times in the interpretation of multilayer refraction time distance curves.Geophys.Prosp.,4: ll2 30.

t2

3-D methods

Overview

I

portant new display techniques.In order to disptay Seismicdata are usuallycollectedalong lines and manipulatethe hugevolume of data involvedin a of tra_ verse that form some sort of grid and 3-D survey,most 3-D interpretationis done at work_ the three_ dimensional(3-D) pictureof structureis deduced s t a t r o n (sS 1 2 . 4 ) . by rnterpolating between the lines. However, The interpretationof 3-D data to extractstructural features seenon such seismiclinesmay be located and stratigraphicinformation is discussedin $12.5. off to the sideof the linesratherthan underneaththe One of the best ways to improve interpretationis to hnesand small but important features(like faults) can reviewcasehistories,a number of which have been occur betweenthe Iines.Theseproduceerrorsin interpreta_ published.Brown (1991)is the best singlereference tion.f -D surveysendeavorto obtain data on 3-D techniques. Applicationsof 3-D to reservorr unifbrmly distributedover an area rather than merely delineationare givenin Sherifffl992). along lines,in order to correctlylocatethe geologic features that produce the seismicevidences.3_D techniques alsosignificantlyreducespatialnoise. 12.13-D acquisition A l t h o u g h3 - D d a t aa r e n o t i n e x p e n s i vteo a c q u t r e . process,or lnterpret,their value is becoming | 2.L l A cqui.titi,tnrsrlylrr'*r,n,., widely -growrng recognrzed,and 3-D We would like to attributevariationsseenin our seis_ _ls9ne of the lastest areasof geophysics. Much 3_D work involve,tn. a._ mic data to geologiclactors rather than acouisition tailingof fieldsonceoil has beendiscovered in order f a c t o r sT. h u s .i d e a l l yw e w i s ht o h a v eu n i l o r ma c q u i s i _ to optimize field development and exploitation tion conditionsand a uniform surfacedistributionof (Dahm and Graebner,t9g2; Galbraith uni Bro*n, CMPs, that is, (l) data distributedon a uniform erid 1982),and 3-D has provenvery cost effective in this with (2) the sameCMp multiplicity(3) utilizing the regard.Thereis nearlyunanimousagreement that 3_ samemix of offsetdistancesand (4) the samemix of D surveysresultin clearerand more iccurateplctures azimuths. of geologicdetail.The costsare more than repard by 3-D seismicdata can be acquire<J in a number of theeliminationof unnecessary development holesand ways.The usual method in marine work is to run a rmprudentinvestments, by the increaiein recoverable seriesof closelyspacedIines;in transitionand in some reserves throughbetterlocationofproductionand in_ land work, to havegeophones laid out in two or more lectionwells,and by the discoveryof isolatedpoolsin parallellineswhile the sourcemovesabout between A reservolrthat otherwisemight be missed.-l-D is also the lines;in other land work geophonesare laid out beingusedincreasinglyfor explorationbefore the drit_ on lines at right anglesto sourcelines.Whateverthe ling of an explorationwell, for the samereasons. The method of acquiringthe data, the goal is to achieve cost-effectiveness of 3-D seismicmethodsfor ShellIn_ uniform samplingover the area. A unilorm grid of ternationaland Exxonis discussed by Nestvold( 1992) rectangularbln.sis set up and data are includedwithin and Greenlee,Gaskins,and Johnson .|gg4l. a particularbin if the midpoint falls within that bin. The techniques for acquiring3_Ddata,discussed in The data within eachbin wiil be subsequently stacked $12.1,are usuallyquite differentin the marine envr_ and migrated.To avoidbias,eachbin shouldcontain ronmentfrom thoseon land. Most large3_D surveys the samenumberof tracesand with the sameset of have been marine becausegenerallyinvestments are offsets.Marine work usually satisfiesthis uniformity larger and risks greaterand hencet-heneedis greater requirement reasonablywell, but land work often for improvedfield definition. Howevequre of j_D on doesnot achieveit. The direction in which the survey land is growingas the major improvements that it pro_ ls run may introducea slightbias.The bins shouldbe videsbecomemore widely recognized. smallto avoidlosingresolution,but not so smallthat Except in a few very lmportant regards,mainly many bins turn up empty.The output tracesfor empty dealingwith migration,processing 3_o lata iSfZ.Zlii bins are usually synthesizedby averagingthe stacked not very different from that of 2_D processing. How_ tracesfrom adjacentbins. eveqdisplaying3-D data(912.3)so tirat an intlrpreter The samplingshould be denseenough (bin size can understandand extract the significantelements smallenough)to avoidaliasingduring processing and provideschallenges that haveled to a numberof.im_ interpretation(98.3.10). To avoid spatialaliasing,at 451

3-D METHODS

A<1

leasttwo surfacesamplesshould be obtainedfor each apparent wavelength present. The apparent waveand length is smallestfor the highest frequencyumax the steepestdip €-"-; thus the maximum spatial sampling Ar is given by

Ax<

v 4u-._sin (..^

(t2.1)

The surfacesamplingintervalsshould generallybe 20 to 100m, the subsurfacesamplingbeing l0 to 50 m, and the samplingin the dip direction is often greater than in the strike direction. Vertical samplingis usuvelocity),so the dimensions ally 2 ms (3 m at 3000-m/s of a potentially resolvablevolumecell (rrlxel) is of the order of 3 x l5 x 25 mr assuminggroup intervaland line spacingof 30 and 50 m. Becausethis volume is so small,the subsurface detail presentin 3-D data is very great. Determiningthe small bin sizerequiredto prevent aliasingrequiressomeknowledgeof the geologyand In practice. achievingsmallbin sizemay be expensive. compromisesare made and design parametersare basedpartly on the availablehardwareand the nature of the objective.Geologicalconstraints(for example, the preknowledgethat dips are small, faults are not to imagefaultcloselyspaced,and it is not necessary plane reflections)may permit relaxing the spatial aliasingconstraints.Becausemost data are migrated and spatialaliasingcreatesnoisein migration.this often becomesthe limiting constraint.Intelligentinterp o l a t i o n( $ 9 . 1 1 . 2c)a n b e u s e dt o r e l a xt h e s a m p l i n g constraintsimposedby migration,but it doesuot improvethe intrinsicresolution.Resolutionof smallleaturessuchas riverchannelsgenerallyrequiresthreeor more sampleswithin the smallfeature. Eventsmigrateupdip.so if the dip nearthe edgeof the prospectis away from the prospect,data must be collectedover a fringe area surroundingthe desired subsurface areathat is to be imaged.The width of this fringe area M, assumingstraightraypaths,is (ll2)Vt sin (; however,raypathcurvatureenlargesthis and we approximatethe migration distanceas M :

t/c Zl sin (,

(12.2)

where Z is the averagevelocity,I the two-way record time,and { the dip. In order to properlydetermineseismicamplitudes (neededfor hydrocarbon indicator and stratigraphic studies)by the migration process(which effectively sums amplitudesover the Fresnelzone,$6.2.3),the fringe area should be further enlargedby the radius of the first Fresnelzone(eq.(6.6a)),

^ : Vr lxr ,

( r2 . 3 )

To achievefull multiplicity, we must add the line-end taper L (half the streamer length in marine work). Thus, in order to migrateproperly and achieveproper

amplitude and multiplicity, the fringe width xr."r. shouldbe

= ,rn * L, * L -r,.,,n*. 2 f,r, ,^^. *

(n.4)

Note that 3-D multiplicity usuallydoesnot haveto be as largeas requiredfor 2-D work, halfthe required2D multiplicityoften beingquite enough. The sparse3-D or exploration 3-D technique is sometimesusedto decrease3-D acquisitioncosts,but data quality is also decreased.Lines are spacedfour or five times more widely apart than the sampling theorem requires and the omitted lines are interpolatedduring processingby intelligentinterpolation ($9.| 1.2),a processbasedon seismiccharacter.Intelligentinterpolationpreventsaliasingduring 3-D migration; however,the resultingdata have only the lower resolution determinedby the spacingof the lines as collectedrather than that after interpolation. Sometimesirregulargrids of 2-D lines are transformed into a regular 3-D grid by filling in the undersampled grid loops using intelligentinterpolation. This approach yields data of even poorer resolution than the exploration3-D technique;again,the resolution is limitedby the originaldatadensity.Interpreters of 3-D data should take into accounthow the data so that unrealisticexpecwerecollectedand processed tationsofsubsurfacedetailcan be avoided.In general, widely spaced3-D data are inappropriate for reservoir studies. I2.1.2 Marine 3-D ut'tlui.sition Marine 3-D dataaregenerallyacquiredby a boat towing a hydrophonestreamerand an array of air guns while it traversesback and forth acrossthe area being surveyed(fig. l2.la). ln conventionalmarineacquisition, one line of data is acquiredon each traverse. Lines are normally orientedin the dip direction so that the spacingin this direction (the hydrophone is smallerthan that in the strikedigroup separation) rection(the line spacing). The demand for lessexpensive3-D surveysled to boatstowing two or more streamers(fig. 7.24),which to collecttwo or are pulledto the sidesby paravanes, Sometimestwo airmore linesof data simultaneously. gun arraysseparatedby paravanesare fired alternately to double againthe numberof linescollectedon a traverse(figs.12.lb and 12.1c).As many as six parallel streamershave been towed by a single ship. Sometimes two acquisition ships. each towing two (or three)streamersand one or two air-gunarrays,with the sourcearraysfiring one at a time, are usedto collect up to 12(or l8) linesof datawith onepass.(However,becauseonly one sourcecan be usedat any time and the shipsare travelingcontinuously,this increases the minimum bin size and changesthe mix of traces betweendifferentbins.)

ACQUISITION 453 When data are acquiredon parallel lines, apprecia_ ble time is lost while the boat movesfrom-one line to rhenext(asin fig. l2.la, usuallyat least I hour).Boam sometimescollect 3_D data in circles(fig. 12.2),either successively_overlapping circles of tLJ same radius rlike a stretchedand flattenedcoil spring) or ln a spi_ ral. to minimizelost trme. Navigation accuracy must be sufficient to place data in. the proper bins. This .equi..ment apptiesto

thepositions of alleGment, orit . u"qui_ 1:l:1115 sltlon system,including

thoseof the sourcesand eich hydrophonegroup. positioning of the ship is usually accomplishedby radiotrilateration from fixed base

1.3)supplemenred by Gps ($7.t.5); rhe l:l::l: !t? rsatways ,:^.i,]"1 derermined redundintty, hope_ ::i: to an accuracy rulty.

of l0 m. Positioningthe hydrophonegroups relative to the , boat is.generallydetermined fiom a seriesof com_ passes in the streamer(whichoften drifts offline by a lew degrees)and lrom measurements of the location of the tail buoy. These are often supplemented by measurements with pingerst, transducersthat sendout acousticpulseswhose traveltimesto tuned receivers on the boat or in the streamerare measured and used to locatethe boat, sources,and streamers with respect ro eac.hother ($7.1.7).Anchoredpingers($i.f .O)are sometimesused to provide locations-for survey ol a restrictedarea. A huge, highly oue.determined vol_ umeof navigationdata resuitslnconsiderable redun_ dancy.The volumeof navigation/positioning daramay exceedthe volume of seismicdata. The "navrgation data reduction to determine which traces-fail into whichbinsis doneby computersboth in real time and atsoIn postprocessing. 3-D data can be obtainedin a one_boat operatron if the streameris pulledat an angleto ttreslisilic fine. Wherenatural oceancurrentsare present, the sersmic c3n be orientedperpendiculaito the current so ljne rlar rle streameris pulled off_line.Figure l2.ld is a pror ot streamerlocationfor a line in such an area. However, one has relatively little control ou.. tt. amount of cross-coverage obtained and the number of tracesin different bins is apt to Uei..eguta.. AI.o, the tracesthat fall into bins larther frorn ih.lin. ur. those with longer offsets,producing a bias that may make residual normal moveout appear as fictitious cross-dip. Collecting3-D data in an areaof platforms . or other invotvesspeciatproblemsio uuoiJingoU_ :.?r1ll.,t-o"r separate source and recording :,:::rl :.-erimes Doarsare usedso that the area adjacentto a platform can be undershot. In shallowwater and whereobstaclesinterfere with towing streamers,a sh)athrechniqueu.ing g.ofhon., or hydrophoneslaid on rhe ,.i floo, iru'y 6. .__ ployed.Two or more bottom cablescontaining receiv_ ers are laid in parallel lines and connected tJ an an_ chored recording barge. A source boat then moves

over the area betweenthem (fig. 12.3)to acquire the 3-D data. All or parts of the liyout are then moved and the operation is repeatedto extend the survev area. 12.1.3Land 3-D acquisition The cost of a 3-D land survey is generallypropor_ tional to the number of sourcepoints involved,and hence surveysusually employ many more geophone groupsthan sourcepoints.Cost-effective desienis dis_ cussedby Rosencrans(1992)and Beeet at. (i-gq+). Data are often acquiredusingperpendicularsource and geophonelines (fig. 12.4a),which is called, patch shooting.This arrangementgivessinglefoldcoverage over a rectanglehalf the length of the geophoneliie by half the length of the sourceline. HJwever.the off_ set distancesand azimuth directions vary systemati_ cally over the midpoint coverage,which may introduce bias in processingand interpretation.Difierencesln normal moveout may be interpretedas dip and vice versa,and the lack of multiplicityat midpoint locations may causeproblemsin determiningstatic cor_ rections.The sourcepoint and geophoneJtationloca_ tions do not need to be the .ami, .o the midpoint locationsare not necessarily equallyspaced. In order to achievemultiple coverage,severalgeo_ phone and sourcelinesmay be used.Commonlv. four or six parallel geophonelines are used with pirhaps srx or eight source lines perpendicular to t'he eeo_ phone lines, especiallywhen the Vibroseismeth; is used. The use of parallel geophone lines builds up multiplicity in one directionand the use of parallel sourcelines builds multiplicity in the perpendicular direction, so that the total multiplicity achievedis the product of geophone-line and .ou.ciline multiplici_ ties.Two setsof vibrators employingdifferent sweeps ($7.3.1e) can be usedsimultaneously to nearlydouble the speedof data acquisition.The minimum bin size is halflhe geophonegroupintervalby halfthe source_ point interval.Multiplicity can also be increased by increasingthe bin size; for example,doubling linear dimensionsquadruplesthe bin areaand the miltiplic_ ity (assumingthe samedistributionof CMps). After the data for one patch are acquired,tire patch is moved to the next location (for ixample, to the north) and the operation is repeated. Successive patchesoften overlapto further increasethe multiplic_ rty over a strip (a block)of the subsurface.Successive blocksmay also overlap(for example,to the east),still increasingthe multipticity. Multipte fold in lr.!h9l both line and cross-linedirectionsaids in determinins staticcorrections. Land data are sometimesalso acquiredby a swath techniquewhereinseveralparallel geophonelines are usedwith one or two sourceunits in a ioll_alongtech_ nique similar to that used in conventionalZ_dCUp acquisition(fig. n.aU. This resultsin a swathof 3_D

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(b) Arrangements for 3-D marine acquisition. (a) After Fig. l2.l recording a line, ships generally turn with a fairly large radius becauseof their long tows in order to record the next line, and then double back to achievethe close line spacingrequired; solid lines show where recording was done and dashed lines indicate moves between lines. (b) Use of paravanes to tow units to the side of the ship. Arrangement shows two source arrays and a

single streamer to give two lines of midpoints. (c) Typical spacing when towing two sourcesand two streamersto acquire four -+ '4 lines of midpoints; the two source arrays fire alternately; I l' streamer I into source from indicates locations of midpoints and so on. (d) Feathering of streamer due to cross-current Location of boat and 48 streamer groups (plotted for every 25th station).

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3-D METHODS

456

Fig. 12.2 Marine acquisitionusingcircleshooting.(a) Spiral for shootingaround a salt dome and (b) overlappingcircles.

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the peripheryand the bisectorsthat divide the rectangle into quadrants.However,it giveslow multiplicity over most of the area and poorly determinedstatic corrections,and so generallyprovides poorer data. The loop method can be used around irregular loops to acquiredata in areasof difficult access;of course, somedegradationresultsfrom the consequentirregular distributionof midpoints. The land 3-D acquisitionmethods generallysacrifice someuniformity of CMP coverageand uniformity of offset distancesand azimuths representedwithin the bins. Someloss of CMP multiplicity can be tolerated becausethe data in nearby bins in many directions can be consultedin decidingwhich eventsare to be honored. As a rule of thumb, 3-D work requires only half the multiplicity of 2-D work to achievecomparable data quality. In many land areas, accessrestrictionsinterferewith any of the foregoingacquisition methods.

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Fig. 12.3 Swath technique for obtaining lines of midpoints spaced25 m apart with two lines of geophonesin bottom cables (A and B) 150 m apart, using three lines ofsources (1,2,3) spaced 50 m apart. Lines ofmidpoints are dashed; 1 ,4 indicates locations of midpoints from source I into bottom cable l, and so on. Following this acquisition, cable ,4 may be moved 150 m beyond B to acquire the next swath.

coveragewhosewidth is half that of the geophoneline spacingtimes the number of parallel geophonelines. With l2-fold in-line coverageand two parallellinesof sourcepositions,24-foldmultiplicity can be achieved. As with the use of patches,successiveswathsoften overlapto increasethe multiplicity. There are a number of variationsof the patch and swath techniques.Sources sometimes move along short segmentsbetween geophonelines (fig. 12.4c). Then, after one block ofdata has beenrecorded,some of the geophonelines are moved to cover the next block, and the source-linesegmentsfor it are staggered with respectto those for the precedingblock, "brick-wall" pattern. Sourcessometimes creating a "zig-zag" betweengeophone lines. These variations are used to improve the mixes of offsetsor azimuths within bins while improvingfield efficiency.Geophone groups are sometimesarranged in a checkerboard "button patch" arrangementas sourcestraverseparallel lines. In areasof limited access.data are sometimescollectedwith sourcesand receiversplaced around rectangles(fig. 12.4c).This loop methodpermitsacquiring data over the interior of the rectangle,which may not and it providesmultiple coveragealong be accessible,

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Fig. 12.4 Arrangements for 3-D land acquisition (a) Perpendicular lines of sourcesand geophonesacquire data over rectangle with half the dimensions of the source and geophone lines; often, several lines of geophones and sources are used to give multiple coverage. (b) Arrangement for acquiring a block of data using four parallel geophone lines with two vibrator sourcesoperating simultaneouslyto produce swath of eight midpoint lines; for the next swath, geophone lines I and 2 will be moved to 5 and 6 and vibrators will move down line 4 and midway between 4 and 6. (c) Loop layout with geophones and source locations around perimeter of an area; midpoints are shown as dots; 40 geophone groups and 40 source locations might be located at 160-m intervals around the four sides of a souare 1.6 km on a side.

3-D PROCESSING Land 3-D acquisition methods also may lead to poor static corrections,so acquisition arrangem"nt, usually include some2-D CMp portions so that con_ ventional statics analysiscan be used to provide a framework for determiningstaticsfor the iemaining bins. Becausethe sourcesare at different azimuthsto the geophone arrays, array-directivity patterns will differ; windmill arrays (fig. 8.lag) are occasionally usedto minimize array-directivityeffects. Field designoften has to copewith accessproblems that prevent any land-acquisitionsystemfrom being completely uniform. Field computers do the bookkeeping for the sometimesvery complicatedadjustments that are required. They usually include pro_ gramsto displaythe arealdistribution of multipliiity, offsets,and azimuths within bins. Theseare used to analyzethe effectsof arbitrary sourceand geophone geom€trychangesto assistin designingcompromises to achievebetter distributions while the crew is still acquiringthe data. l2.t 3-D prooesslng The processingof 3-D data is similar to that of 2_D data in most regards.CMp stackingattenuatesmulti_ ples and improves the signal-to-noiseratio. Diomoveout (DMO) correction (99.10.2)in 3_D is re_ quired (in addition to the normal-moveout(NMO) correction) to remove reflection-pointsmear.DMO shifts data into different bins prior to migration. DMO and migration require that the midpoints have beendeterminedand the data binned. Thire is often a delay betweenthe end of a surveyand mapping the binned CMPs, but this information is now 6iing oU_ tainedin real time on most marinesurveysand many land surveys. A numberof processing stepsdo not dependcriti_ cally on the CMP locations.Theseincludetestingthe data integrity, gain recovery,deconvolution,mutlng, scafing,and velocityanalysis.2-D brutesta(*s (stacks made without correctionsor refined processing),al_ though not of the quality ultimatelyexpectedfrom the survey,can give an indication of data qualitv.prelimi_ nary time slicesmade of the near-tracedata help lo_ catetaresthat suggestpositioningproblemsthat need to be examinedmore carefully.Theseoperationsare sometimesdone in the field. Velocityanalysesare usuallyrun in conventional 2_ D fashionon a rather coarsegrid over the prospectin order to make 3-D velocity maps.Theseincorporate also preliminary structural information and velocity data that are availablefrom wells.Data should be reexaminedwherethe velocitymapsare not consistent with the structure. Marine data may need a static correctionsfor tidal variations. Static corrections are almost always re_ quired with land data; both conventional2-D refrac_ tion and surface-consistentstatics are often used. Where there is sufficientredundancyin both in_line and cross-linedirections,3-D staticsanalysismay be used.

457 After being binned, DMO corrected,and stacked, data are migrated (fig. 12.5)to repositiondipping reflections,collapsediffractionsfrom discontinuities,focus energy spread over Fresnel zones ($6.2.3),and nearly eliminateout-of-the-planenoise.One to three additional tracesare often interpolated(g9.11.2)to avoid aliasing in the migration process.Migration greatlyreducesspatialnoiseand helpsclarify and correctly locate structural anomalies;it also greatly improves stratigraphicresolution by imaging the data dispersedover the Fresnelzone. Migration is the major factor in improving the signal-to-noiseratio and interpretabilityof 3-D over 2-D data. Sometimesmigration is performedbeforestacking. Usually, in-lines in the dip direction are migrated first and then traces are sorted and migrated in the cross direction. This procedurecloselyapproximates true 3-D migration where the velocity field is simple. Sometimesa more expensiveand more accurate3-D migralion is done in one operation.Figure 12.6shows severalexamplesof the improvementof 3-D over 2-D migration. Eliminating the energycoming from out of the planeof the sectionby 3-D migration often results in major data-qualityimprovement. The accuracyofmigration dependson the accuracy of the velocityfield used.Some2-D linesselectedfrom the 3-D volumemay be migratedwith differentvelocities to help selectmigration velocities.Where structure is complex,depth migration may be required,but inadequateknowledgeof the velocity field may limit the benefits of depth migration. Occasionally, prestackdepth migration is performed. Waveletprocessingto reshapethe embeddedwavelet is an important aspect of 3-D processing.Interpretersgenerallyprefer zero phasefor maximum interpretability(Brown, 1992)and henceconversionto a zero-phaseembeddedwaveletis often done late in

i /

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Q) Fig. 12.5 Illustrating 3-D migration. The data to be migrated lie on a hyperboloid appropriate to the stacking velocity I(; the aperture ($9.1.31)should be centered near /0. 3-D migration is often done by first doing 2-D migration in the y-direction, sorting, and then doing a second 2-D migration in the x-direction; this economizes on computing time. The effect is to move data from I to B in the first migration and then from ,B to ln in the second migration. If stacking velocity trj is used for the first migration rather than the correct (, the error created is usually small. The aperture in the first migration would also be different than an aperture symmetrical about tu. (a) Isometric sketch of (.t, ,ll t) space and (b) stacking velocity versus time.

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ratio achievableby 3-D migration (right) oi CMP stacked sections (left).

f ISPLAY OF 3.D DATA

Fault outcrop at surlace East Set of crosetyspaced N'S lines Top of section shown in (b)

l'5rl rary re (d)

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Fault s l r c e( e )

Frg. 12.7 Three-dimensional data obtained from a set of - r \ ) s e l ys p a c e dN S l i n e s .( F r o m S h e r i f f ,l 9 9 l : 3 0 0 . )( a ) I s o m e t r i c :r.rgram of the volume thesedata occupy.The easternmostN S €ction is shown (a "line") along with an E W section (a "cross.:ne") made from the southernmost traces on each N S line. (b)

The data set with the top portion removed; the top now constitutes a time slice. (c) The data sliced along one reflection (h) constitutes a horizon slice. (d) An arbitrary line cuts through the data volume, perhaps to connect well locations. (e) A fault slice runs parallel to a lault displaced a small distance from it.

the processing sequence. Brown (loc. cit.) emphasizes the importanceof zero phasein 3-D interpretation. \ zero-phasewaveletis symmetricalwith most of its .nergy concentratedin the central lobe, so reflectrons .ireeasierto recognizein the presence of other events and noise.Comparedto a minimum-phase(or 90.) rlavelet,zero phasegivesonly one major trace excursion ratherthan two, so that thereis lessambiguityin determiningthe polarity and identity of a reflection. Howeveq ascertaining the phase of an embedded saveletis extremelydifficultin most instances.

horizon; such horizon slices are used for studying stratigraphyand reservoirproperties.Horizon slices are alsomadealong surfacesparallelto nearbypicked horizons; that may be picked more reliably than the horizon of interest(seeplate l4). This method helps show polarity and phase changesthat may be produced by stratigraphicfeaturesand that might cause difficultieswith tracking the particular horizon. The volume can be sliced parallel to a fault surfaceQfault s/ices,'seefig. 12.20and plate 5) to study fault-related structure,secondaryfaulting (fig. 12.19),and the sealing ability of faults. Combinations of vertical, hori-

123 Display of 3-D data .{ 3-D seismicsurvey providesa volume of data for rnterpretation.If the sampling has been adequate, eachdata point in the volumeis lessthan half of the apparentwavelengthfrom adjacentdata points in every directionand thereare no areaswherealiasingoccurs, a more accurateand consistentinterpretation should thereforeresult. Sampling is, however,often not closeenough,and an interpretermay haveto facI o r t h i sf a c t i n t o h i s i n t e r p r e t a t i o n . The 3-D volume can be slicedin various ways (fig. 12.7),for example,along threeorthogonalplanesproducing linesand cross-lines (verticalsections)and titne s/ices(also called hori:ontal sectionsand SeiscroprM sections)(fig. 12.8).The 3-D volume may also be slicedverticallyalong diagonallinesconnectingwells, or in the plane of a deviatedwell, or in zig-zaglines to tie wells (arbitrary lines). A volume can also be sliced along a reflection (an interpretedhorizon) to show the spatial distribution of amplitude over this

Data volume

I

I I

Horizontalseclions

Tracked

Horzon s rce

sections

Faull slce

Composde displays

Altrrbute displays: Envelope amplitude Inslanlaneous phase Inslanlaneous fequency Sersmr log (Velocily)

Smoothed maps

I

Dip/azrmuth dtsplays llluminalon lsun-shade)

Fig. 12.8 Nomenclature for 3-D products. The data volume can be sliced vertically, horizontally, or along (or parallel to) tracked horizons. Composite displaysare combinations of vertical and horizontal slicessuch as shown in plate 7. Transformed data produce attribute displays,including dip/azimuth (seeplate 6) and illumination displays.

3-D METHODS

460 zontal, and other sections (plate 7) permit an interpreter to appreciate the three-dimensionality of the subsurfaceand interpret the data more meaningfully. Combination displays (plate 7) can be made easily with an interactive system. Interactive systems also permit isometricdisplaysand the rotation of isometric displaysso that they can be viewed from different vantagepoints. A time sliceshowsall the seismiceventsfor a particular seismictime (or depth if time-depth conversion has been performed).The attitude of a feature on a time sliceindicatesits local strike.Picking a reflection on a time slice is equivalentto mapping a time contour, and successive contours can be mappeddirectly by following the sameevent on successive time slices (fig. 12.9).This procedurethus yieldsa structuralcontour map without the intermediatestagesof timing and posting. Likewise picking a fault on a sequence of time slicesyieldsa fault-surfacemap directly.Typically the number of interpretablefaults increasessignificantly (fig. 12.10),perhaps l0-fold. The ability to seethe strike and continuity of faults often resultsin markedly changedinterpretations.The shapeof the structural contours generallyrevealsfine detail that cannot be extractedwith conventionalmapping techniques; therefore,an interpreter of 3-D data should not expectcontoursto be as smooth as thoseon maps made from 2-D data. Attributes of various kinds can be calculatedfor 3D data to createattribute volumesthat can then be slicedthroughjust like the 3-D data volumeitself.The common attributes such as envelopeamplitude, instantaneousphaseor frequency($9.11.4),or acoustic impedance (or velocity) generated by onedimensionalinversion(seismiclogs;see$5.4.5)can be displayed.Automatically tracked structural contour maps may be manipulatedto revealsubtle lineations indicating faults (fig. l2.l I ). Thesemanipulationsinclude (a) calculatingvertical derivativesof a horizon surface to display the attributes of dip magnitude and dip azimuth(plate6), (b) smoothingand subtractinga smoothedversionfrom the data to yield a residual,(c) subtractingthe arrival times or amplitudesof successive horizon slices to yield difference displays, (d) effectivelyilluminating a horizon slice with light in sucha direction that shadowsemphasizefeaturessuch as faults (sunshadeor artificial illumination displays), and (e) multiplying a time map by a velocity map to give a depth map. Color is an important factor in 3-D displays.Color increasesthe visual dynamic range and permits us to see and interpret more detail. A gradational color scheme is generally used in which two distinctly different colors or ranges of colors representpeaks and troughs,with the intensityproportional to amplitude values.The most commonly usedcolors are red for positivereflections(peakson SEG-standardzerophasesections)and blue for negativeones,with the color intensity proportional to the amplitudes.Contrastingcolor schemescan be usedto emphasizesmall

changesin the valuesdisplayed;howeveqthesecan be misleadingif the valuesselectedfor the color contrasts are not well-chosen.Introducing a bias in the color assignment,as in platesl4 and 15,can alsoemphasize features.The channelin the horizon slice of plate 15 is more clearly defined by limiting the range of red values. 12.4 Interactive 3-D interpretation Most 3-D interpretationis done at interactiveworkstations.The data are held on magneticor compact (optical) disk and are viewed and interpretedon the screenof a color monitor (TV tube). The benefitsof working interactivelyare many.Much time is savedby eliminating the paper handling associatedwith conventional methods.The interpretercan composethe data display or combination of displays(plate 7) that he thinks best suits study of the problem at hand. Color is availableas a standarddisplayand colorscan be chosento suit individual preferencesand specific problems.An interpretercan move through his data quickly, with fewer distractions,and maintain a productive idea flow. With the incorporation of well and other data into data bases.more information (for example, well logs) can be reviewedeasilyto seehow it contributesto understandingthe seismicdata, so that the final interpretationshould be more consistentand hencebetter. Becauseinteractivesystemsmake it so easy to perform checks,more checkingis performed and interpretationalinconsistencies can be seenmuch easier.Interactive3-D interpretationdoes not necessarily accomplishquickerinterpretation,but it is more completeand accurate. An important function of an interactivesystemis horizon tacking, mapping the same reflection throughout the 3-D volume. Usually, severalmodes are available.Manual tracking,also known aspoint or streamtracking,dependsonly on the judgment of the interpreteras he movesa cursor over the data. Automotic tracking of a chosen reflection is efficient in most cases,but it has to be monitored to make certain it doesnot jump to the wrong event at faults, unconformities, or where data quality deteriorates.Automatic tracking is highly desirablefor amplitude studies becausethe computer can track the maximum amplitude (which is very difficult for a person to do) or fit a smooth curve to amplitude valuesin order to interpolatethe maximum amplitudewhereit doesnot fall on a samplevalue.The computer can then store the time and amplitude values for later use. These times and amplitudes can relate directly to the reflectivity contrastifthe data are zero-phase,the event has a good signal-to-noiseratio, and the sameevent has been tracked everywhere.Automatic horizon tracking ofzero-phasedata over a reservoirreflection yields a horizon slice of the reservoir interface.Becauseit may be difficult to track a horizon that involveschangingcontrasts,especiallyifthe changesinvolve reversingpolarities, tracking is often done on somenearbyhorizon believedto be conformablewith

I

]-D INTERPRETATION

461

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Frg. 12.9 Successive time slicesthrougha 3-D data volume. l-ineupson a time-slicemap constitutea time contour.Tracins

the sameeventon successive timeslicesdevelops a structure contour map.(AfterBrown,1983: I183.)

the desired horizon and then the horizon slice is made a fixed time below the tracked horizon (as in plate l4).

Faultsare recognizedon time slicesin the sameway as on vertical sections,mainly by reflectiondiscontinuitiesand offsets(f,g. 12.11,and plates8 and 9). Just as picking a fault on a seismicsectionis difficult where the fault nearly parallelsthe seismicline (so that the lault surfacenearly parallelsreflectionevents),picking faults on time slices is difficult where the fault nearly parallelsreflectionevents.However,the useof both vertical sectionsand time slicesremovesmost of the problems in fault picking. Dip and azimuth dis-

Il.5 3-D interaretation The most completeinterpretationrequiresthe use of all types of sectionsto help an interpreterseeand ex_ tract maximum subsurfacedetail. 3-D displaysgener_ ally reveal enormously greater detail than *ui for_ merly believedpresentin seismicdata.

Fig. 12.10 Structuremapsresultingfrom 2-D (left) and 3-D (right)surveysshowingthe additionaldetailderivedfrom the 3-

D s u r v e y s .( F r o m B r o w n , l 9 9 l : 5 7 , 5 8 . ) ( a ) G u l f o f T h a i l a n d survey,and (b) Chile survey.

:.D INTERPRETATION

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plays(plate6) oftenrevealsmallfaultsbecause the dio along horizonsoften changesin the immediatevicin_ ity of faults.3-D data haverevealedthat manv faults previouslymapped as continuousare actuilly en_ echelonsystems(seefig. l2.l0b). Horizon slices are made routinely by interactrve systems. Guidedor automatictrackingusuallyfollows the maximumamplitudeof a reflection;the timesgen_ erated by the tracking provide a time-structuremap and the amplitudesprovidea horizonslice.A honzon slice shows how the reflectivity varies along the re_ flector. It yields an uninterestingpicture lor ionstant reflectivity,ofcourse, but wherethe reflectivitvvanes. the pattern may suggestthe reasonfor the reflectivity variation(Enachescu, 1993).Obviously.the valueof a horizon slicedependscriticallyon th; correctness of the structuralinterpretationon which it was based. The horizonfor the sliceshownin plate8 was cut bv

a numberof faults,and the continuityof the stream pattern on the horizon sliceconfirms that the correlation acrossthe faults is correct and that the faults postdatethe unconformity. The most striking features revealed by horizon slicesare stratigraphicbecausethe map-styleview often showsdistinctiveshapesrelatableto depositional systems.Viewsof ancientdepositionon horizon slices sometimesresemblethe views of modern deposition seenfrom an airplane:channels,offshorebars, point bars (sanddepositsin river meanders),crevassesplays (wherea river broke through its leveeduring a flood), sand bars, karst topography, depositional edges, pinchouts,and so on. The picturesrevealedon horizon slicesoften are almost impossibleto seeby interpreting vertical sectionsalone. The patternsin fig. 10.40and plates8, 10, 15,and l6 are easilyidentifiedas streamchannels.A horizon

464 slice along an erosionalunconformity may show onlap patterns above the unconformity or the subcrop outt.rn below the unconformity (plate l1)' The horizon slice in plate 12 shows a turbidite fan (containing gas). Color display is valuable for showing trend patterns.In plate 12,the dark reds indicating the highest amplitudeslocatethe zonesof better reservoirquality (producibility).The low-amplitudelineationsrunning gNg to E are faults. The high-amplitude lineation running approximatelyWNW is caused in part by tuning (46.4.3)betweenthe reflectionsfrom the top of the reservoirand the fluid contact,as the reservotr thins to the north. Similarity of the amplitude patterns of the reflectionsassociatedwith the top and baseof the reservoirtendsto confirm that the patterns are causedprincipally by lateral variationswithin the reservoirrather than in the host rock' Amplitude patternsoften can be interpretedin terms of the reservotr extent or lithology changes,variations in reservoir quality (especiallyporosity) or thickness (if below tuning thickness).An amplitudelineationthat follows structuralstrike and/or is parallel to the downdip reservoir limit may be a tuning phenomenon($6'4'3and 14.5),indicatingwherethe reservoirthicknessequals the tuning thickness.With knowledgeof the reservoir thickness,tuning effectscan be subduedto produce "detuned" horizon slices. Fault slicesare usuallymade by slicingthrough a 3D volume parallel to a fault surfacebut displaceda small distance(perhaps25 to 50 m) into the downthrown and upthrownfault blocks(fig' 12'l2a)' This is done to avoidthe distortionsoften presentvery near the fault itself. The shifts betweencorrelativeevents on thesetwo slicesgivesa map of how the throw on the fault variesalong the fault surface(fig' l2'l2b; see also plate 5). If the lithologiesassociated-withthe different eventsseenon the fault slicescan be identified, perhapsbasedon nearby well control, then perhaps it can be determinedwhich potential reservoirs are juxtaposedagainstimpermeablerocks acrossthe fauli and thus where the fault will seal and provide trapping conditions.The additivecolorsachievableby suierimposing displaysin complementarycolors are usefulin somesituations,for example,in supertmposing up- and down-thrown fault slicesto seewhere lithologiesare juxtaposed againsteach other acrossa fault.J-D seismicdata are also usedto predict sealing at faults becauseof the smearingof ductile clay (Bouvier et al., 1989;Jev et al., 1993)'Fault slicessometimes show up splinter faults that were generatedby the stresseswithin a fault block as it moved along a curved fault surface(seeplate l3). Workstationsmay permit restoringthe 3-D volume to its former situation when a particular horizon was beingdeposited,basedon the assumptionthat the ho.irori *us horizontal at the time of deposition' This involvesnot only picking the horizon and identifying it in different fault blocks, as is done in constructing a horizon slice,but also moving the deeperreflections

3-D METHODS ("unfaulting the faults") to remove the fault heave and throw (iee fig. 10.26a).The resultingrestoredvolume can then be examinedin order to work out the history of deformationsand the effectsdeformations may have had on Paleofluid flow. Amplitude anomaliesattributable to hydrocarbon accumulationsshow in much the same fashion as in vertical sections.The patternsin plates4, 10, and 12 show the amplitude variation of a hydrocarbonindicator and thus outline the productiveareas'Horizon slices through a reservoir zone sometimesdelineate areasof maximum porosity.This is the casewith the lighter colorsextendingSSEfrom well L-7 in plate I I ' wiere the reflectivityof one subcroppingmember of the Lisburn formation producesa dim spot' The limiting factorson the utility of horizon slicesare the resolut-ion, signal-to-noiseratio, and amplitude and phase Preservatlon' ^ Clearly, reflectioneventsmust be correctly identified in oider to make usefulhorizon maps' Data from boreholesare usuallyusedto identify correctly reflecseismiclogs, and tion events.Syntheticseismograms. Figure 12'13 used' all profiles are seismic vertical showsthe correlationbetweena VSP and 3-D seismic data. A multitude of productivereservoirsfrom a verofchanneldepositsexistin this area' tical sequence two horizon slicesseparatedby only shows l7 Plate 4 ms (about l6 ft), which is about \/10 of the dominant frequencyand well below the usuallystatedtheoretical risolvable limits of \/4. The differencesbetween these slices relate to the separate channel and systems.Pressurehistory,bottom-holepre.ssures' interferencetests (where something,usually induced changesin pressure,is done to one well to see if it affecisanotherwell) indicatethat a number of the reservoirs that produce from roughly the sameintervals are not connected(seefig. 12.14)'The pressuredata help in interpreting the seismicdata and vice versa (Hardage, 1993). ' Once reservoir reflections of sufficient signal-tonoise ratio have been identified, an attempt can be madeto interprettheir amplitudechangesin termsof reservoirproperties.Without well control, the effects of properiiesaffectingamplitude (lithology,nature of fluid, pressure,hydrocarbonsaturation,porosity'ratto of net-to-grossreservoirthicknesses,and tuning) are usually inseparable,but often some of them are known from wellsalreadydrilled so that lateralamplitude variations can be reasonablyascribedto variations in porosity,the ratio of net-to-grosssand thickn.rr, uni/o. the product ofporosity and net reservoir thickness(see Sireriff, 1992).Determining porositythicknessirom 3-D data in the Prudhoe Bay oil field is discussedby Stanulonisand Tran (1992)' For the common type of clastic reservoirwhere the acoustic impedancebf th. ..t"tuoir sand is lower than that of the surroundingshales,higher amplitudegenerallyindicatesbetter reservoirquality regardlessof whether this is due to higher porosity,higher ratio of net-tosrosssand thicknesses,increasedreservoirthickness'

3D selsmrc data sel

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able formations by ties to well control, reflections on opposite sidesof the fault can be superimposedto indicate sealingquality ol fault.

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l l s e p a r a t er e s e r v o i rl e v e l sb e t w e e n l . l a n d 1 . 5 s . P l a t e s 1 4 t o l6 show data from this survey. (After Hardage, 1993; courtesy of the Texas Bureau of Economic Geology.)

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f,l eerrorations [-_l sptuy l-l':Jlchannel F i g . 1 2 . 1 4 T h e r e s e r v o i ri n t e r v a l f o r t h e f o u r w e l l s s h o w n t n plate 17. Each of the four wells has different pressurehistories and bottom-hole pressures(BHP). The separations shown are w e l l - h e a dd i s t a n c e sa; l t h o u g h t h e w e l l sa r e a l l p r e s u m a b l yv e r t l -

or higher hydrocarbonsaturation (tuning effectsexcepted). Problems l2.l Reconcilethe spatialsamplingequation(12.1) w i t h e q .( 8 . 2 a ) . 12.2 (a) For an operation involving towing two sourcesand three streamers,what spacingsare requiredto achievea minimum bin sizeof 25 x 25 m? (b) To achieveuniform midpoint line spacing,what ship tracks? distancewill separatesuccessive 12.3 Assumingthat hydrophonegroupcentersin fig. l2.ld are 50 m apart and that the ship speedis 6 knots, calculatethe cross-currentat two locations. 12.4 (a) Whereasseismicships sometimestow three or more streamers,they only rarely usemore than two (array) sources.Why? (b) Marine shallow-waterswathand patch techniques often use more source than geophone locations, whereasthe practice is usually the opposite on land.

whv? 12.5 Conventionalmarine operationsinvolvea taper in the CMP multiplicityfor half the streamerlength at eachend of a line. How much taper is involvedwith the circlemethodsillustratedin figs.12.2aand 12.2b? 12.6 A survey over a marine prospect used a single patch with six parallellinesof 96 receivergroupseach in sea-floorcables with a 50-m group interval, the

c a l . t h e y p r o b a b l ym a y d e v i a t eb y 1 " . T h e v e r t i c a lt i c k s a r e l 0 f t ( 3 m ) a p a r t . C h a n g i n gt h e p r e s s u r ei n w e l l I 7 5 i n a n i n t e r f e r e n c e test did not aflect the pressurein well 202. (Al1er Hardage, 199-l: courtesy of the Texas Bureau of Economic Geology.)

lines being 400 m apart' The geophone/hydrophone sourceboat towing an air-gun sourcetraversed20 linesperpendicularto the receiverlinesspaced250m apart with air-gun pops every 50 m, thus covering nearlythe sameareaas that occupiedby the receiver lines. (a) What is the minimum bin sizethat can be used'l (b) How much multiplicity will be achievedover differentparts of the surveYarea? (c) Assumethat the 3000-mdeepobjectivehorizonts a nearlyflat erosionalsurfaceand that the trapping is stratigraphic,so that amplitudesmust be mapped accurately.If the averagevelocity is 2500 m/s, how largean area can be mappedwith confidencel(d) Assumethat the objectiveformationsdip 20" away from one edgeof the area. How large an area can be mappedconfidentlY? 12.i-Assumea land surveyemployingsix E W lines of | 12 geophonegroupseach with geophone-group spacingof 35 m, the geophonelines being 400 m apart. Fifteen N-S vibrator lines are spaced300 m apart with the sourcepoints betweenthe four center linesspacedat 70-m intervals. (a) What is the minimum bin size?What pattern ol multiplicity is achieved?How much variation of offset and azimuth mix is involved? (b) If three geophonelines are moved for successtve parallel acquisition blocks, what is the effect on the multiplicity, offset,ar.rdazimuth mixes'J

R E F E RE N C E S

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I r g . 1 2 . 1 5 A l o o p l a y o u t f b r 3 - D s u r v e y i n g .S o u r c e p o i n t s are : n t l i c a t c db y X . g e o p h o n cg r o u p c e n t e r sb y O .

1 2 . 8 I n o n e 3 - D t e c h n i q u es, o u r c ep o i n t s ( x ) a n d r e o p h o n e(sC ) a r e l a i d o u t a s s h o w ni n f i g . 1 2 . 1 5a; l l thc geophonesare recordedlor each sourceDolnt. This arrangementemploys 4g geophone stations, :pilced50 m apart,and 48 sourcepoints,spaced100 nr apart. r a ) L o c a t ea l l t h e " m i d p o i n t s 'a, n d d e t e r m i n e t h e i rr e _ .pectrvemultiplicity.(Hint; lJtilizesymmetry to save * o r k .) i b ) N o t e t h a t s o m eo f t h e m i d p o i n t sf a l l o u t s i d e the \quare.lf this layout is repeatedwith common geo_ phonelines,thesepoints will fit in adjacentsquares. \\'hat effectwill this haveon multiplicitv,l 1 2 . 9A n E W f a u l t c u t s t h e , t . r i . t r r . s h o w nr n f i s . 11.9.How largeis its throw assumingthe velocitya't :hc mappedhorizonis 3000m/s?Drew a depthcross .s'ction l b r a n a r b i t r a r yI i n ep e r p e n d i c u l at or i h e t a u l t . \\'hat sort of fault do you think is involved? 12.10Locatepossiblefaultson fig. 12.I I and indicate :heirpossiblethrows.Sketchdips on oppositesidesof .rnelault lor an arbitraryline alongthe iault. Assume :hat Iines/cross-lines are25 m apartand that the domi_ :rantfrequencyis 40 Hz. l2.ll Locateseveralplaceswherelaultsshowon one ..f the displaysin plate6 but not on the other. 12.12Interpretthe faultsin plate9. 1 2 . 1 3W h a t a r e t h e a d v a n t a g easn d d i s a d v a n t a g o e fs. .r -l-D marine survey recordedin the dip directron Jtrmpor€d to one in the strikedirectionin iermsof (a) -ross-linesmear,(b) DMO, (c) spatialaliasing,and 1d.y , elocityestimation? 12.14Will a saltdomeappearlargeror smallerbased rn 2-D migrationof a coarsegrid of linesthan on a nigrated3-D survey? A swathsurveyis to be recordedusingl0 paral_ lJ.l5 ei geophonelinesspaced50 m apart,eachiontaining . I stationsspaced50 m apart.Two sourcelinesoer_

467 pendicular to the geophonelines are located at the endsofthe geophonelines;eachcontainseight source locationsspaced125 m apart symmetricaliylocated with respectto the geophonelines. What will be the minimum bin dimensionsand what multiplicity will be achieved? 12.16 A land surveylayout is shown in fis. 12.16. (a) In the southern 213of the area whereipacing was regular,what is the smallestbin size that should be used?What is the best multiplicity achieved?How wide is the multiplicitytaperarea?What is the small_ est bin sizeif squarebins are desired?For the best_ multiplicitysquarebins,what are the offsetand azi_ muth ranges? (b) Answer the questionsin part (a) if four of the smallestsquarebins are compositedto give a larger squarebin? (c) How much degradationis causedby the irregular spacingin the northwestern part of the area,assumlng the largersquarebins are used? 12.17 Copy fig. 12.13and cut along the junction be_ tweenthe VSPand the 3-D data.Slidethe two up and down to ascertainthe confldencein the match. As_ suminga velocityof 6000ftls, how much shift is in_ volved?What would be the effecton the match if the VSP and 3-D embeddedwaveletswere lg0o out of phase'? If theywere90oout of phase?How mucheffect might a changein waveletshapehave. 12.18 Assumethat the wellsindicatedin fig. 12.14deviateby l"; how much might this changethe bottomhole separations? 1 2 . 1 9T h e w e l l ss h o w ni n p l a t el 5 w e r ea l l d r i l l e db e fore the 3-D survey.What changesin well locations would you expectil the 3-D surveyhad beenavailable?Wherewould you recommenddrilling new wells now assumlngthe horizonis flat and that hishercolor i n t e n s i t yi n d i c a t ebs e t t e rp r o d u c i b i l i t y ? 12.20 Interpretchannelsystemsin plate l7 as best you can.Considerthe locationuncertainties indicated by problem 12.18.What amplitudelactorsaffectthe interpretation?This is the land surveydiscussedin p r o b l e mI 2 . 1 6 . References A b r i e l , W . L . . P S . M e a l e .J . S . T i s s u e ,a n d R . M . W r i e h t . 1 9 9 1 . M o d e r n t e c h n o l o g yi n a n o l d u r e a : B a y M a r c h a n d r e v i s i t e < J . Tltc Lcading Edge. 10(6):21 35. B e e ,M . F l . J . M . B e a r d e n .E . F l H e r k e n h o f f ,H . S u p i y a n t o ,a n d B. Kocstoer. 1994.Eliicient 3-D seismicsurveys in a jungle en_ vrronment. Fir.ytBreuk, 12:253 9. Bouvier, J. D.. C. H. Kaars-Srjpesteijn, D. F. Kluesner, C. C. Onyejekwe, and R. C. van der Pal. 1989. Three-<jimensional s e i s m i c i n t e r p r e t a t i o n a n d f a u l t s e a l i n g i n v e s t i g a t i o n sN , un River Field,Nigeria.Bull. AApG, i3:1397 1414. B r o w n , A . R . 1 9 8 3 . S t r u c t u r a l i n t e r p r e t a t i o nf r o m h o r i z o n t a l seismicsections.Geopltl,sits,48: ll79 94. B r o w n , A . R . 1 9 8 5 .T h e r o l e o f h o r i z o n t a l s e i s m i cs e c t i o n si n stratigraphic interpretation. Sei.vni c S ta r igrapi_r,.Lf,O. R. Berg and D. G. Woolverton, eds., pp. 37 48, AAPG Memoir 39. Tulsa: American Association of Petroleum Geoloeists.

I

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in four northward for the next swath. The area was mapped access Vibrator shown)' are swaths fourth (the and first swaths (After in the northern part was restricted to the trails shown' Economtc Hardage, 1993; courtesy of the Texas Bureau of GeologY.)

R EFERENCES

469 Brown Seismic ^A. ry 1991 Interpretarionof Three-dimensional Duta.Jd ed.AApC Memoir42.Tulsa:American Assoctatlon of PetroleumGeologists. Brown,A. R. 1992.Definingreservoirproperties.ln Reservor fulia: Societyof feophysics,R. E. Sheriff,ed., pp. lSS_ZO^0. ExplorationGeophysicists Brown,A. R., G. S. Edwards,and R. E. Howard. 19g7.Fault slicing- A new approachto the interpretationoi iault de_ tarl. Geophysics. 3): l3l9 27. Dahm, C. G., and R. J. Graebner.19g2.Field developments with three-dimensional seismicmethodsin the Gulf of Thai_ land A casehistory.Geophysics,4T:149 76. Enachescu, M. E. 1993.Amplitudeinterpretation of 3_Dreflec_ tion data. TheLeadingEdge,12:673 95. Galbraith,R: M., and A. R. Brown. 19g2.Fieldappraisal with three-dimensional seismicsurveys, offshoreTrinidad.Ceophys_ ics,47: 177-95. Greenlee,S. M., G. M. Gaskins,and M. G. Johnson. 1994.3_ D seismicbenefitsfrom explorationthrougha.u.loprn.nt, nn txxon perspectle.TheLeadingEdee,13:7304.

Hardage, R. A. 1993. Notes for Re.seryoirGeophysics Short Course. Tulsa: Society of Exploration Geophysicists. Jeq B. I., C. H. Kaars-Sijpesteijn, M. p A. M. peters, N. L. Watts, and J. Y Wilkie. 1993.Akaso Field, Nigeria: Use of integrated 3-D seismic, lault slicing, clay smeiring, and RFT pressure data on fault trapping and dynamic leakage. Butt. AAPG,77: I 389 1404. Nestvold, E. O. 1992.3-D seismic:Is the promise fulfilled? Iie Leuding Edge, ll(6): 12 19. Rosencrans,R. D. 1992. Cost-effective3-D seismic survev de_ sign. The Leading Edge, tl(3);17,24. Stanulonis, S. F., and H. V Tran. 1992. Method to determrne porosity thickness directly from 3-D seismicamplitude within carbonare pool. prudhoe Bay. Ttv'LtttJing Etlge, !!g_Li$yrl,. ll(t): l4 20 Sheriff, R. E. 1991.EncyclopedicDit.tionarl,tl Exploration Geo_ phy.sics,3d ed. Tulsa: Society of Exploration Geophysicists. Sheriff, R. 8., ed. 1992. ReservoirGeophvsics.Tulsa: Society of Exploration Geophysicists.

13

Specialized techniques

Overview Lesser-usedtechniquesare discussedin this chapter. Seismologistsshould be familiar with ttrem because they provide the most efficient means of galning neededinformation under specialcircumstances. S-wavesdependon differentelasticproperties _ than P-wavesand henceyield additional iniormation when combinedwith p-wave studies(gl3.l). Especrally in anisotropicsituations,,as is likely wherefracturrngis present,they may yield more definitive information than obtainable from p-waves.Additional infbrma_ tlon can also be obtainedby treatingwavemotion as a vectorwith three-component recording($13.2)rather than dealingwith only the componeniof motlon in one direction. waves trapped in low_velocity channels ....S^.ll-i. ($13.3) can be usedto obtain informationabout the propertiesof the channels.However. their analysisis difficult becausethey are highly dispersrve. seismicprofiling (VSp) ($l3 4) providesone -Vertrcal of the bestmeansof relating,eflection'events to the specificinterfacesinvolved in their generation. VSp alsoprovidesthe meansto see,with liigherresolution than availablewith surfacedata, whaimay Lie ahead of the drill bit or what changesmay lie to'the sideof a borehole. Tomographicmethods(913.5)provide a different kind of approachto invertingt.uulltir. (ana, in ttreory at least,amplitude)measurements to determrne distributionsof velocity(and absorptiveproperties). Although their use is relativelyn.* unO the best meansof applicationare still being developed, they are especiallyapplicableto resolving boiehole_to_ boreholemeasurements. The time-lapsetechnique(g13.6)consists of. re_ peatingmeasurements in order to determinechanges that may have occurred over time. This technrque is usedmainly in reservoirstudies. In additionto velocitysurveys,VSp,and cross_hole surveys,measurements within boreholes($13.7)include surveysto determinehow closea boreiole rs to the flank of a salt dome. Waveform logging allows analysisof the velocitiesof different*auJiooi.s, ano the boreholeteleviewerprovidesin effecta picture of the wall ofa boreholeand showsfeaturessuchas frac_ turing. Passive seismic measurementsrely on natural sourcesto generateseismicwaves.Joint inversion uses a different kind of measurement(such as of gravrty) 471

as an aid in seismic interpretation. Geostatistical methods interpret rock properties from geophysical measurementson a statistical basis, allowing for changesbecauseofvarious unknown factors. 13.1Exploration with Swaves 13.L I Why explore with S-waves Nearly all seismicexploration is carried out with p_ waves,the assumptionbeing that p-wavesalone are involved and that any S-waveenergy presentmerely contributes to the noise. However,conversionof p_ wavesat interfacesmeansthat S-wavesare involvedin seismicobservations evenif we wish to avoidthem. P-waveshave advantagesover S-waves:they are easierto generate,only a single mode exists,they travel faster and so arrive first, and they are easierto interpret.However,S-wavesalso haveadvantages:(l) S-wavevelocity dependson different propertiesthan P-wavevelocity(p versus(I + 2p)-see eqs.(2.5g) and (2.59));(2) S-waveshave two modes (Strzand SH-see 92.4.l), which is both a complicationand a potentialadvantage. Thus.S-wave..oiry dilTerentinformation from P-waves.If both p- and S-waveveloc_ itiescan be measured,then we havea sourceof addi_ tional informationabout the subsurface. Figure5.12 suggeststhat such information should indicatelithol_ ogy, and fig. 5.26 suggeststhat it should also indicate the fluid contained within a rock's pore space.The shear modulus p is most important in engineering studiesbecauseit relatesto the ability of the earth to support structures.The shear modulus along fault planes seemsto change in anticipation of earth_ quakes.S-waveexplorationis the subiectof a book e d i t e db y D a n b o ma n d D o m e n i c o( 1 9 g 7 ) . The SZ-mode involveswave motion within the nearly vertical plane that contains the raypath, whereas the S.I/-mode involves horizontal motion. The SZ-mode is involved in conversion at near_ horizontal interfaces,but the S.F1-mode is not. The potential advantagesof S-wave exploration have resulted in appreciableeffort being devoted to developingS-wavetechniques.However,a methodology has not yet evolved. 13.1.2S-w,ave re<:ording on land Becausewe are usually primarily interestedin waves travelingmore nearly vertically than horizontally,we

SPECIALIZEDTECHNIQUES

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',e;;1.'lt: Jrr F i g . l 3 . l H o r i z o n t a l v i b r a t o r f o r g e n e r a t i n gS - w a v e s .( C o u r tesy of Conoco.) (a) Truck-mounted vibrator: the weight of the truck is used to keep the vibrator in firm contact with the

ground. (b) Detail of vibrator pad showing teeth that are triangular in cross section to maintain ground coupling as the horizontal movement of the pad compacts the soil during a sweep.

must induce approximatelyhorizontal motion in order to generateS-waves,it is much more difficult to couple horizontal motion to the earth than vertical motion. Virtually all S-wavesourcesalso generatePwaves.To minimizeconfusionas to waveidentity,one usually attempts to generateSl1-waves(so that convertedwavesare not involved)and then look only for the S/1-componentwith horizontal geophones. There are basicallyfour waysof generatingS-waves

(in addition to relyingon mode conversion):(l) use of a horizontal vibrator, (2) a horizontal impact, (3) impact at an angle,and (4) using horizontal asymmetry. S-wavevibrators usuallyhavetriangular teeththat dig into the underlyingmaterialto maintain coupling (fig. l3.l); as earth compactiondue to the vibratory motion tends to creategaps,the triangular teeth dig in farther and maintain coupling. Of course,this also leavesgapingholesso that the useofhorizontal vibra-

EXPLORATION WITH .S-WAVES

EARTH UOIlol{

413 tors often involveslargedamageclaims. A horizontal blow (fig. 13.2)againsta block held firmly againstthe earth, usually by the weight of the vehicle (Hasbrouck, 1987; Layotte, 1987),generates S-waves.Striking a steelblock with a sledgehammer is usedto generateS-wavesin engineeringstudiesand a large 1700-kghammer (MarthorrM) has been used wheremore energyis required (Layotte, 1987).Vertical stackingis usuallyrequiredto build up the energy sufficiently;blows are struck alternatelyagainstopposite sidesof the block and then the polarity of the records is alternated before stacking to minimize Pwavesgeneratedincidentally.

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Fig. 13.3 Omnipulseseismicsource.(Courtesyof Bolt Tech(a) Truck-mounted nologies.) unit that can be tiltedup to 45oto eithersideto produceS-waves with oppositepolarities.(b) Re-

lease of air from an air gun pushes the reaction mass upward as horizontal and vertical comDonents of thrust are exerted on the sround.

474 GeneratingS-wavesby an impact at an anglegenerates appreciableP-wavesas well. Repeatingthe impact at 180' azimuth reversesthe polarity of the Swaves;when the recordsare subtracted,the S-waves impactswill add whereasthe resultingfrom successive P-waveswill nearly cancel.An impact at an anglecan be producedby dropping a weight constrainedto fall "horizontal weight drop"); the weight at an angle (a can alsobe propelledby compressedair or someother force ("enhancedweight drop"), as with the Omnip u l s e( f i g . 1 3 . 3 ) . Utilizing earth asymmetryis the basis of the SyslaprM technique.It uses three holes drilled close to each other (fig. l3.a). An explosivedetonatedin the center hole gives a conventionalP-wave record and produceslocal changesabout the central hole, which result in horizontal asymmetry in the two nearby explosionsin the outsideholesthus holes.Successive generateS-wavesout of phasewith eachother in addition to P-waves,so that subtractingthe two records, as is done with the angularimpact sources,addsthe S-waves while nearly canceling the P-waves.Two closelyspacedP-wavevibratorsoperating180' out of phase have also been used as an ,S-wavesource (ShoverlM). Recordingcan be done with horizontalgeophones and perorientedin-linewith the sourcefor SZ-waves pendicular to the line lor SH-waves. Threeare also used(see{i13.2). componentgeophones

S P E C IA . L I Z E D T E C H N I Q U E S

lP

I 3.1.3 S-tvaverecordingat sea Shear waves are not transmitted through water, so marine sources cannot generate S-waves nor hydrophones detect them. However, where the water bottom is hard, there is significantmode converslon (fig. 13.5)of downgoingP-wavesto,S-waves at appreciable anglesof incidenceat the water/rockinterface (Tathan,and Stoffa, 1976).The SZ-wavesreconvertto P-waveson return to the solid liquid interfaceln an equally efficient conversionand so can be detected with conventional pressure detectors. Very long streamers(-6 km) are usedto obtain largeincidence angles.The long group length used in conventional marine streamersdiscriminatesagainstwavesthat approach at large emergentangles,and a streamerfor optimizing SZ-wave detection must employ short groups. Time shifts can be introduced and the data from several short groups combined to simulate a longer group, as with beam steering.P-wavesand Swaves can be further separated in processing by fi ltering. apparent-velocity and displayingS-wavedatu I 3.L4 Processing S-wavesprocessing,although similar to P-waveprocessingin many regards, is different in several respects. Near-surfacevariationsfor S-wavesare often quite large and static correctionsare essentialin order to

F i g . 1 3 . 4 T h e S y s l a pm e t h o d . ( C o u r t e s yo f C G G . ) ( a ) E x p l o sion in the center hole generatesmainly P-waves;(b) becauseof the asymmetry produced by the explosion in the center hole' the explosion in the right hole generatesP-wavesand SH-waves;(c) the Sl1-waves generated by an explosion in the left hole have opposite polarity to those in (b).

stackS-waveland data and produceusablerecordsections (Anno, 1987).The large S-wavedelaysresult from the sensitivityof S-wavevelocity to rock-matrix variationsin the near surfaceand they are not necessarily proportional to P-wave static delays, which mainly result from the fluids in the near-surfacesediments. Automatic staticsprogramsmust searchover time intervals larger than the staticsdifferencesand this is apt to involve cycle skipping becauseof the

EXPLORATION WITH S-WAVES

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e

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RECEIVERS

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ITDPOINT (a) Fig. 13.6 Raypaths of common-midpoint gather for modeconverted data.

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E >

,"

/

ta

40-

o

t--gog

; 0 I. 5

3.0 4.5 Seafloorvelocity(km/s)

6.0

(c) Fig. 13.5 PSSP reflectionsgeneratedby conversion at the sea f l o o r . ( A i t e r T a t h a m a n d S t o f f a , 1 9 7 6 . )( a ) G e o m e t r y ; ( b ) c o n _ version coefljcient versus angle of incidence in the water: the curves are labeled with sea-floor velocity in km/s, (c) maxinrum a m p l i t u d e a n d w i d t h o f m a i n l o b e v e r s u ss e a - f l o o rv e l o c i t v .

largemagnitudes of S-wavestatics.Hand editinsis often required. Becauseof the asymmetryof raypathsinvolvedwith convertedwaves(fig. 13.6),their stackingis not as simple as that of P-waves,even with horizontal reflectorsand simplevelocitydistributions.Gathersof traces having a common conversionpoint are discussedby Tarhamand McCormack( | 99| t. BecauseS-waveresultsare usually interpretedin conjunction with P-waveresults,S-wavesectionsare normally displayedwith a verticaltime scaledouble that usedfor the P-wavesectionsto compensate approximatelyfor the roughly l:2 ratio of S-waveto pwavevelocities,B/ct(seefig. 13.7). 13.1.5Interpretationand useof'S-waveclata The objectiveof S-wavemeasurements is usuallythe ratio of S-waveto P-wave velocities(p/ct). Thrs re_ quires identifying S- and p-wave reflectionsfrom the

samereflectingrnterface.Becausea P-wavereflector is not necessarilyan S-wavereflector,this identification can be very difficult and often this is the most critical aspect of S-waveinterpretation. Plotting Swave sectionsat double the vertical scaleusedfor Pwavesections,as indicatedin $13.1.4,helpsconsiderably by making the two sectionsappearmore nearly the same.IndependentidentificationusingP- and Swavesyntheticseismograms or P- and S-wavevertical seismicprolilesis highlydesirable. Usually, S- and P-wavereflection identification is basedon somegeometricfeaturesuchas terminations at faults, overall structuralcharacter,truncationsat a shelf edge or unconformity,or prograding stratigraphiccharacter(fig. I 3.8).Other reflections arethen identifiedby referenceto theseevents,assumingthat S- and P-reflectionamplitudesvary in similarfashion ( f i g .1 3 . 9 ) . Figure 5.12showedthat the ratio of S- to P-wave velocities.B/ct,is an indicatorof lithology.Howeveq whereasthe domainsof sandstones, limestones, and dolomitesgenerallydo not overlap,the domain of shalescan overlapthoseof the other lithologies,limiting the usefulness of B/o as a lithologyindicator.Tat h a m a n d K r u g ( 1 9 8 5 )i n t e r p r e tl i t h o l o g y u s i n g a time-average equationassumingB/ctvaluesof 0.45for shale,0.54 for carbonates,0.59 for sandstone,and 0.67 lor gas; they show a tetrahedrongraph 1fig. 13.10)to illustratevariousproportionsof thesecomponents. Generally, B/ct decreasesas porosity and carbonate/sandratio increaseand as the sand/shale ratio decreases. The velocity ratio is also affectedby pore shape.If porosity is known from the wellsand is cross-plotted againstthe velocityratio, the resultant distributionsometimes can be interpretedin termsof pore aspectratio, that is, the relative abundancesof sphericalporesand thin cracks.Examplesof interpretationsusingS-waveare givenby Fix, Robertson,and P r i t c h e t(t 1 9 8 7 )a n d E n s l e y( 1 9 8 9 ) . S-wavesare not affectedby fluids, so hydrocarbon indicators such as bright spots,flat spots,dim spots, and polarity reversalsdo not occur on S-wavesections (exceptin subtleways,mainly through their effecton density). Consequently,if such indicators are observedon P-wavessectionsbut not on S-wavesections (fig. 13.9),this is a gas-reservoir indicator.

SPECIALIZEDTECHNIQUES

416

\al

Fig. 13.7 Comparison of P- and S-wave records. (Courtesy of CGG.) (a) P-wave record and (b) S-wave record displayed at double the timing speed to make comparison of eventseasier.

In concept the variation of seismicamplitude with angle of incidenceor offset (AVO; see$3.4)contains the sameB/ctinformation as the joint study of P- and S-wavesections.AVO information is alreadyavailable in CMP data, so there has beena decreasein interest in S-wavedata. Tatham and Krug (1985)show field examplesin which S-waveswere used to define carbonate-clastic facieschanges,locatedolomitization(fig. l3.l l), and distinguishbetweensandand shale(fig. 13.12). I 3.1.6 S-wavebireJringence A major useof S-wavesis in fracturedetection.Fractures produce anisotropy that can lead to birefringence ($2.6.2).Most fracture studies involve threecomponentrecording($13.2).An S-wavevibrating in a preferreddirection travelingthrough an anisotropic medium may be split into two mutually perpendicular S-wavevibrations, parallel to and perpendicularto the anisotropicaxis, that travel with different velocities. With orientedfractures,an ,S-wavevibrating parallel to the fracture direction seesunfracturedrock and so travelsat a higher velocity than the S-wavevibrating perpendicularto the fracturedirection (fig. 13.13). The differencein velocitiesoften is a measureof the fracture density.Oriented S-wavesourcesand detectors can be usedto determinethe azimuth of the fractures(Crampin,1987).

13.2Three-component recording 13.2.IAcquisition The motion associatedwith a wave arrival is a vector quantity, and measuringits three orthogonal componentsrequiresthe useofthree orthogonalgeophones. The projectionof the vectormotion onto a plane produceshodograms(Sheriff,l99l) such as those in figs. 2.15band 2.15d.The coordinatesystemfor measuring the threecomponentsofa vector can be rotatedabout any axis;in particular,they can be rotatedinto a natural coordinatesystem,suchas one dictatedby the orientationof an arriving wave(fig. I 3.I 4) or the orientation of a fracture system.Such a coordinaterotation is sometimescalledAlford rotation or polarizationfil' tering. Orienting three separategeophonesin orthogonal directionsin the field is tedious,so usually the three geophones are housed together. Most threecomponent phones use one vertical phone and two horizontal phoneswhosecharacteristicsnearly match thoseof the verticalphone (seealso Garotta, 1987). Becauseit is difficult to designhorizontal and vertical geophoneswith the same characteristics,sometimes three identical orthogonal phones, each making a 54.7oangleto the vertical(Gal'perinarrangement),are used(seeproblem 13.2).Three-componentgeophones are used in boreholesfor VSP surveys($13.4);they must be gimbel-mountedto maintain their attitude to vertical in directionalholes.

P

S

STRUCTURAF L E A T U R E S( F A U l t i N g ) 1. 5 2.O2.5-

4 J

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1.6S T R A T I G R A P H IC CH A R A C T E R

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Fig. 13.8 ldentifying featureson P-wave sections(left column) and on S-wave sections(right column) to correlate events from

the same reflecting interfaces (From Tatham and McCormack' 1991:178.)

IV A

4 I

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(a) F i g . 1 3 . 9 C o m p a r i s o no f P - a n d S - w a v er e c o r d sa c r o s s t h e P u tah Sink sas field. California. The P-wave section (a) shows a

Slarge amplitude associatedwith the gas accumulation; the waie section (b) does not. (After Tatham and Krug' 1985 )

w A

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SPECIALIZEDTECHNIQUES

480

GAS

3 j

t F

Fig. 13.10 VrlV,valuesas a function ofproportions ofsandstone, shale, carbonate, and gas-saturatedpore space. Shown are the four faces of a tetrahedron representing possible combinations of these lithologies. (From Tatham and Krug, 1985:149.)

A field-recordingtechniquesometimesadopted in fracture studies is to successivelyrecord from P-, SH-, and SZ-sources with three-component geophones; this is called nine-componentrecordingbecauseseparatesectionscan be generatedfor each of the three geophone outputs for each of the three sources(fig. 13.15).If the sourceswerepure and the earth isotropic, three of the nine sectionswould be conventionalP-, SH-, and SZ-sections,two would show convertedwaves(P to SV and SZ to P), and the other four would be blank. Howeveqif azimuthal anisotropy (such as may be causedby vertical fractures) is presentand if the StI- and SZ-sourcesand geophonesare not oriented parallel and perpendicular to the natural orientation, then S-wavesplitting (S2.6.2)will result in energyshowingon all panels.Polarization filtering then can be used to rotate the sourcesand geophonesinto the natural coordinate system,thus determiningthe orientation of the natural system and of the fractures.It should be noted that, if fracture systems at different depths have different orientations,the detectionsystemmay show only the orientationof the shallowestone.Winterstein and Meadows (1992a, 1992b) use a layer-stripping techniqueto resolvethe natural orientationsof successivelayers. Although three-componentrecording must be regarded as a researchrather than operational tool as of 1994,it showspromise in severaltypes of studies, especiallyin determiningthe orientation and density of fractureporosity (Ebrom and Sheriff,1992).Lewis, Davis,and Vuillermoz(1992)usethree-componentrecording to resolvefracture problemsin a 3-D survey.

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Fig. l3.l 1 Effect of carbonate-sand transition on I/r/tr/"values, Midland Basin, Texas. V"lVrvahtes for the Glorieta and Wolfcamp intervals indicate the carbonate-to-clastic lacies change associatedwith the carbonate shelf edge. (From Tatham and Krug, 1985: 166 8.) (a) P-wave data, (b) S-wave data, and (c) V"lV.values for the intervals shown in parts (a) and (b).

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(c) Fig. 13.12 Dependenceofslowness ratios on thickness ofsand beds surrounded by shale, Horse Butte, Wyoming. (From Ta-

tham and Krug, 1985: 174-7.) (a) P-wave data, (b) S-wavedata, and (c) Al,/At (or VrlV.) values and interpreted sand thickness.

AUSTIN CHALK 4.7

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5.4 F i g . 1 3 . 1 3 F a s t ( S l ) a n d s l o w ( S 2 ) S - w a v es e c t i o n sA t t h e l e f t cnd. the respectivearrival times of the Austin chalk reflection

SH-RECEIVER

SV.RECEIVER

Fig. 13.14 Rotation of SH- and SZ-geophone traces into the pline of the S-wave motion. (From Tatham and McCormack' 1 9 9 1 :1 6 0 l . )

are 4.85 and 4.90 s. The 52 reflection dims where the liacturtng i n t h e A u s t i n c h a l k i s c s p e c i a l l yi n t c n s e ( l ' r o m M u e l l e r ' 1 9 9 2)

ROTATED 52.RECEIVER

C H A N N E L W A V E S( N O R M A L - M O D E P R O P A G A T I O N )

483

SOURCE MOTION

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F i g . l l l 5 N i n e - c o m p o n c n ts e c t i o n sr c c o r d e db y p - . . S l l - , a n d . S / - g e o p h o n e lsi r r P - . S H - . a n d S Z - s o u r c e sW . ithout anisotronv

( o r c r o s s - d i p )t h e s e c t i o n sm a r k c d ' l w o u l d b c b l a n k . ( F r o m Tat h a m a n d M c C o r r n a c kl,9 9 l : l l l . )

:111ililililil||il1il[l

l -1. 2.2 Rtlari:ut ion./iltaring Polarizationlilteringsometimes involvesmerelyrotating the coordinatesysteminto another orienrarron. but it may also involveshifting the phase.adjusting t h e a m p l i t u d e sa,n d t h e ns t a c k i n go r t h o g o n acl o m p o nents.Differentpropagationmodesand directionsof wavetravelinvolvesystematic relationships amongthe components.so polarizationfilteringcan be usedto prelerentiallyselector rejectparticularmodes,such a s g r o u n dr o l l ( l i g . 1 3 . 1 6 )O . t h e re x a m p l e os l t n e a p plicationof polarizationfilteringfor separating p- ancl S - c h a n n ew l a r e s i n c o a l - s e a ms t u d i e s( $ 1 4 . 2 . 5a) r e s h o w ni n f i g s .1 3 . 1 7a n d 1 3 . 2 2 .

. FEFLECTOFS WITH GROUTD ROLL . POLANIZATION FIITER 9 fILTEN POINTS

:-o!

133 Channel waves (normal-mode propagation) Under certain circumstances. wave energy may be trappedwithin a layer that then channelsthe waves. Such waves are called chunnelwavas,guitletl n,uvc,t, truppetll'.ry(,.r,or .\eon1\'uves (where they propagate rn a coal seam).This phenomenonis also known as tluveguideor normul-motlepropugation. Two kindsof boundaryconditionscan producethis situation:(a) the impedancecontrastis so great that the reflectioncoefficientis very large (nearly unity); (b) waveswithin the waveguideare incident on the boundaryat an angle greaterthan the critical angle so that total reflectionoccursand little energyleaks through the boundary (exceptlor evanescentwaves.

F i g . 1 3 . 1 6 U s e o 1 ' p o l a r i z a t i o nf i l t e r i n g t o a t t e n u a t eg r o u n d roll. (From Tatham and McCormack. l99l: 133 4.)

and convertedwaves). $2.3.I, which can be neglected. Polarity reversesat an interfaceof type (a) but not at type (b). A common exampleol the first type is the surfaceof a water layer and of the secondtype the seafloor. This propagationis somewhatanalogousto organ-pipereverberation(Lapedes,1978),a water

SPECIALIZEDTECHNIQUES

484

(a)

F ' i g . 1 3 . 1 7 U s c o f p o l a r i z a t i o nf i l t e r i n g t o s c p a r a t eP - a n d S w a v e so n t r a n s m i s s i o no f i n - s c a m d a t a . ( F r o m M i l l a h n ' 1 9 8 0) ( a ) R e c o r d s o f c o m p o n e n t s p e r p e n d i c u l a ra n d p a r a l l e l t o t h e

(Dt gallery in which measurementswere made and (b) atler polarization filtering and rotation to emphasizecomponents perpendicular and parallel to the source receiver direction.

layer correspondingto an open organ pipe with a node at the sea-floorinterface,an antinodeat the free surface,and a coal seamto a closedorgan pipe with nodesat both boundaries,the acousticimpedanceof the coal being much lower than the overlying(roof) and underlying(floor) lormations. Figure l3.l8a shows waves bouncing back and forth at differentangleswithin a waveguide.Fbr most ofthe angles,thereis destructiveinterferencebetween the differentwaves,but for certainangles,thereis constructive interference and consequently a strong buildup of energy reflected at these angles. In fig. | 3.I 8b, wavefrontAC hasbeenreflectedupwardat the lowerboundaryat the angle0, where0 > 0,, the critical angle.A parallelwavefrontthat occupiedthe same position AC earlier,then was later reflectedat the upper and lower boundaries,following raypathssuch as EFGH and BDAI, now coincideswith the later waveEF + FG + GH : BD -t DA,we front at,4C Because phase differencebetweenthe two wavesis seethat the r(BD + DA\ + mr I e, wherelz is 0 or l, nrn is the sum ofphase reversalson reflectionat the two boundaries,and e is a phaseshift that occurs when 0 > 0, (Officer. 1958: 200 l). For a water layer, m : l, whereasm : 0 for a coal seam. For constructiveinterference,we must have -r.,(BD + DA) + mt I e 2nn. Because DA + BD : /r/cos0 + (ft/cos0)cos20 : 2ft cos 0.

F'ig. 13.18 Waveguidephenomenon. (a) Wavesbouncing back and forth in a layer oi velocity ( becauseof nearly perfect reflectivity at the boundaries, (b) construction to show reinforcement conditions, and (c) phase and group velocity relationship'

we have 2r,ft cos g:

(4nhv,lV,)cos0 : (2n - m)r - e,

or v, : l(2n - m) - (elr)lV,l(4h cos 0)' (13.1)

Neglectinge lor the moment.we get constructiveinterferencewhen (13.2) v , . : ( 2 n - m ) V , l ( 4 hc o s0 ) .

CHANNEL WAVES (NORMAL-MODE PROPAGATION) For a water layer,m -_ 1; hence, .l v, : V,l(4hcos 0 ). v r : 3 V , l ( 4 hc o s0 ) : 3 u , .I

:

rt:':l

I

v,: (2n _ I )v,.

)

which correspondsto an open organ pipe (exceptfor the factor cos 0). For a coal seam,m : 0, and

v, : Vrl(2hcos0), v, : 2Vtl(2hcos 0) : 2r,,1 v n :n v t ,

(13.4)

I

which is analogousto a closedorgan pipe. Thus, provided the original wave generatedby the sourcecontains the appropriatefrequencies,normal-modepropagationconsistsof a seriesof wavesof frequenciesv, and its odd or evenharmonicspropagatingalong the waveguideby reflection at angles 0 that satisfy eq. ( 1 3 . 3 o) r ( 1 3 . 4 ) . In addition to the upward propagatingset of wavefronts parallel to AC, there is a symmetrical downward-propagatingset parallel to PQ in fig.

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Because0 is a functionoffrequency(seefig. (13'2)to (13.4)), Z is also frequency-dependent'so that the wavemotion is disPersive. The minimum value of 0 is the critical angle 0,; hence,there is a minimum cutoff frequencyv0,where (for a water layer)

(1 3 . 6 )

v , , : V , l ( 4 hc o s0 , ) ,

the correspondingphasevelocity V being V : V,lsin but Z decreases. v increases 0, : V..As 0 increases, In the limit, 0 -+ jn (the grazing angle),v --1 -, afld

tan fe: (p,/p,)[tan' 0- (VtlV2cos 0)2]'/2

U//

l

0.4 0.6

13.18cand the interferencebetweenthe two setscreatesa standingwavepattern along the perpendicular to the waveguide.As a result, the wave motion is propagatedparallel to the boundariesof the waveguide.The velocity Z, is the phasevelocity normal to the wavefronts,but there is a different phasevelocity V in the direction of the effectivewave propagation. By referringto fig. 13.18c,wavefrontsAC and PQintersectat R, and therewill be a local buildup of energy maximum propagatesin the here.This energy-density direction RR'',if AC and A' C', alsoPQ and P' Q', are the wavefrontpositionsone time unit apart, then the phaseat R movesto R' in one time unit so that V : RR', that is,

V -+ V,. If we do not neglecte, the formulasare more complicated,but the resultsare basicallythe same.Officer ( 1 9 5 8 )s h o w st h a t f o r e > 0 , ,

\ N

1.0 0.8

I

lh*.J-

ER_I {Tt

v

485

2

4

6

810

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0=

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e,, l.rt

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]

t r :z r

(a)

Fig. 13.19 Channel-wave propagation for a liquid layer on an elastic substratum. (a) Phaseand group velocity versus normalized frequency where ctrlct, : 2r/3, o, : 0.5, o" : 0.25. and p"lp, : 2.5 (from Ewing, Jardetsky.and Press, 1957). (b) First-

mode wavetrain from a source 4 km away. (c) The highfrequency portion ofpart (b), which is called Ihe water wave; rls onset is sometimesused in marine refraction work to determine the range. (Parts (b) and (c) from Clay and Medwin' 1977.)

SPECIALIZEDTECHNIQUES

486 Typical curves of Z versus u for a water layer are s h o w ni n f i g . 1 3 . 1 9 fao r n : 1 , 2 . The group velocity U is given by IJ:v*rdv=v-vdv. dto du For a water layer, fig. l3.l9a shows that the term v(dVldv) is never positive; hence, U < Z Moreover, although v(dVldv) increasesin magnitudeat first as u increasesfrom the value u,,,eventuallythe term approacheszero as u approachesinfinity (becausethe derivativegoesto zero faster than u goesto infinity). As a result of thesefactors, U has the value 4 at the to a minimum U,,, cutoff frequencyu0,then decreases at somefrequencyv,,,afrer which it increasesasymptotically to the value V, at v = *. A normal-mode wavetrain for a water layer is shown in fig. 13.19b.The first arrival is a waveof frequencyv,,that has traveledwith the maximum group velocity Z,; this is followed by wavesof increasingv and decreasingU until U reachesthe value Z, at which time a very high-frequencywave, which also on the has traveledwith velocity 2,, is superimposed first wave. Following this, the frequenciesand group velocitiesof the two wavesapproachu,,,and U,,,,respectively.The burst of energy beginning with that travelingat velocity Z, to that of the energytraveling at L/,,,,theoften-abruptend of the normal-modewavetrain, is calledthe Airy phase(asin fig. l9c).

Clearly,a channelwavein water must be a P-wave, but in solids,severalother typesofchannel wavescan exist. Love wavesand SZ-wavesin the surfacelayer can be explainedas normal-modepropagation(Grant and West, 1965:8l-5). Figure 13.20showsrecordsfrom five geophonesat differentpositionswithin a coal seam.The amplitudes in the roof and floor are relativelyweak. The fundamental modes have their largestamplitudesnear the centerof the coal seam,whereassecondharmonics (with double the fundamentalfrequencies)havetheir largestamplitudesat nearly ll4 and 3/4 points with a node near the center.Different modes can be generated preferentiallyby varying the sourcelocation up or down within the coal seam.Variationsin elastic constantsof the coal and bounding lithologiesproduce asymmetriesand complexwaveforms. Mason,Buchanan,and Booer(1980)studiedchannel wavespropagatingin coal seamsand found the wave motion to be very complex. They found both and PSH-modes,calledfrrson or pscudo-Lov€w*ar€s, (fig. v':aves pseudtt-Rayleigh or Krey SZ-modes, called 13.22b).These modes are all highly dispersive(fig. 13.21).Wavesobservedwith orthogonalgeophones usuallyhaveto be rotated($13.2.2)to separatewave modes,as in fig. 13.22(which also showshodograms for two portionsof the wavetrain).The eventsshown in figs. 14.6and 14.7are channelwaves.

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o

-

Olrirncat

In

mat.tr

o o-

a

t

t

: ; : i ! l l : i i F I i i : i I lf E lxs)

(a)

Fig. 13.20 Channel waves observed at different elevations within a coal seam. (From Reguiero, 1990.)(a) Positions of geophones and (b) records of the respectivegeophones.

(b)

VERTICAL SEISMICPROFILING (VSP)

48'7 13.4 Yertical seismic profiting (VSP) I3.4.I General

5

20

Fig. 13.21 Dispersion of the fundamental mode of channel wavesin a coal seam.(From Mason. Buchanan, and Booer, I 980.)

Usually,the seismicsourceand geophonesare located at or very near the surface.Most borehole surveys, such as conventional well-velocity surveys ($5.4.2), measureonly the traveltimeof the first energy.In contrast, verticalseismicprofling (Kennett, Ireson, and Conn, 1980;Cassell,1984;Fitch, 1984)involvesrecording the complete waveform at regularly and closelyspaceddepth stations.Extracting velocity information is only one of the objectivesof VSPs.They becamemore common in the late 1970sand 1980s. but the cost of occupyinga boreholefor the time required continuesto deter their greateruse. A VSP generallygivesbetterdata than surfaceseismic methods becausethe energy does not have to travel as far and thereforeundergoeslessattenuation. the resolutionof a VSP is usuallyapConsequently, preciablybetter than that of surfaceseismicdata. 13.4.2VSP typesund their uses

UNROINTEO

l l R I B. , , . , . ^, ,^. .. . .

...,1.11'

I nl8 ^rAr,rr,ur lr r"vf ^ilill^^1,^^l,,,llllr, .r-"r.rrvy' vffyilv,||eiluilll\, \ t r \

',\*"1\\

The most common VSP in vertical (or near-vertical) holes,a zero-offsetZSP (figs. 13.23aand 13.23b),uses a singlesourcelocatednear the wellhead.ln an ofset ZSP (fig. 13.23c), the sourceis locatedsomedistance from the wellhead,often 900 2100 m, to give data away from the borehole. An alternativeway of obtaining such data is a walkaway VSP (fi9. 13.23d),in which the sourcelocationsare moved to successively farger distancesfrom the wellhead. Azimuthal VSP surveyslocatesourcesin different directionsfrom the wellheadto investigate changeswith azimuth.Combinationsof the foregoingare usedin deviatedholes in directionalZSP surveys($13.6.5);in marine directional VSP surveys,a sourceboat often travelsso that it is verticallyabovethe geophonewith anothersource locatednear the wellhead.Balch and Lee (1984)and Hardage(1985)describevariousVSP techniques.

8O]FIEO

T

2()() (c)

Fig.13.22 Records from orthogonal geophoneswithin a channel. (From Reguiero, 1990.) (a) Unrotated records, (b) hodograms of portions of the wavetrains involving Krey waves (left) and Evison waves (right), and (c) records after rotation.

Fig. 13.23 VSP raypaths. (a and b) Zero-offset VSPs (WS is small); (c) offset VSP (l/,S large); and (d) walkaway VSP (tr/S, >> WS,). Downgoing arrivals at a well geophone G are shown in part (a) and upgoing waves in (b, c, and d). A, a downgoing direct wave (first arrival); B and C, downgoing multiples involving reflectors above or below the geophone; O an upgoing primary reflection; and E, an upgoing multiple.

S P E C I A L IZ E D T E C H N I Q U E S

488 The basictask of a zero-offsetVSP is to match seismic eventsto specificinterfaces.Becausethe depth of the well geophoneis accuratelyknown, time-depth relationshipsare establishedprecisely,and thus reliable reflectionidentificationand subsurfaceseismicvelocities are obtained(seefig. 12.13).Zero-offsetVSPsare also used to identify multiples (see fig. 13.29) and other events,and to study reflectionsfrom below the bottom ofthe borehole(becauseofthe increasedresolution) to aid in decidingif a well should be deepened ("looking aheadof the bit"). Payne(1994)claimsl525-m resolutionwhen looking ahead600-1200m. "look to the Offsetand walkawayVSPsare usedto side" of the boreholeto seeif a major change,suchas a fault or reef. occurs near the borehole and hence that the findings from the borehole may not apply to the nearby region. The greater resolution of a VSP may help delineate small faults, stratigraphic changes,and thin reservoirsands.An objectiveformation may havebeenfaulted out or a well may be on a nonproductiveside of a fault so that sidetracking may encounterproduction (Puckett, | 99| ). Detection of reservoirsjust missedis especiallyusefulwherehydrocarbon indicators($10.8)are observed.Surveys usinggeophonesin deepholesare also usedin searching for and defining nearby features such as salt d o m e s( s e e$ l l . l . 3 a n d 1 3 . 7I.) . Use of a three-componentborehole geophone yieldsadditional information as to the direction from which energyapproachesthe geophoneand also helps distinguish converted-waveenergy (Noble et al., 1988).S-waveVSPscan be acquiredon land using Swave sources and a three-component geophone ( $ l- ? . 2 . 1 ) . Much work is now underway(Hardage,1992;Massell. 1992)to find a source that can be used in the borehole.This would allow a multitude of geophones to be located on the surface. Such a reverseVSP

o.o

nilE (d

o.4

13.24 ReverberatoryVSP tracesin a casedhole wherethe cement bond betweenthe casingand the boreholewall is absent. (Balchand Lee,1984:158.)

would markedlyreducesurveytime and cost.By using multiple-sourcelocationson the surface,one can generate a 3-D VSP; however,such surveystoday would requireso much time in the boreholethat they are run only rarely, but they should be feasible once more effectivedownholesourcesare developed.A borehole energysourcemust be nondestructiveto the well. One schemeusesthe vibrationsof the activedrill bit as the energysource;this involvescontinuousrecording for several minutes and cross-correlationof geophone outputs with the signal that travelsup the drill stem, somewhatsimilar to Vibroseisprocessing. VSPs can also be used(at least in theory) to study absorption(from amplitudeand waveformchangesin the downgoing wave; see $6.5), crack orientation (from S-wavebirefringence;see$2'6.2),permeability (from tube waves),dip, variationsof reflectivitywith incident angle,convertedwaves,and so on' 13.4.3Recordinga VSP Ordinarily, the well geophoneis lowered to the botwell tom ofthe boreholeon standardseven-conductor cablesand stoppedfor eachrecordingon the way out of the hole. Both open and casedholes are used for recording VSPs.The geophonemust be firmly coupled to the formation for recording(unlessthe phone is locked to the boreholewall upgoing reflectionsare usually too weak to be recordedbecauseof the noise often present).It is important that the casingbe well cementedto the formation(fig. l3.24),althoughsatisfactory resultsare sometimesobtainedin older poorly cementedholes where the formations have collapsed onto the casingover time. Recordingoften cannot be done wheremultiple-casingstringshavebeen set (see fig.13.27b). The depth sampling should conform to sampling theoremconstraints;when V: 2000m/s and u : 100 Hz, \ : 20 m, so the samplinginterval should be less than 10 m ($9.2.2c).However, sampling is usually coarserthan this in order to cut costs,and 20 to 50 m depth samplingis common. Someslackis givento the cable at each recording to prevent disturbancefrom energytravelingdown the cable.Thus, only one (or at most a very few) geophonelocationscan be occupied at a time. Consequently,a VSP survey is timeconsumingand expensive,the cost of occupying the boreholebeingthe major exPense. To achievethe samewaveform for each of the many sourceimpulsesrequired,the sourcewaveformshould be very repeatable.Sourcesare usually kept small becausetheseare richer in high frequencies.The most common marine sourceis a small air-gun array,but a single,largeair gun is preferredby somebecauseit is more repeatable,although its spectrumis poorer. On land a water-filledpit 5 m deep and 6 to 7 m across can be usedwith an air gun 2 to 4 m deep; the mud pit is sometimesused for this. The waveformsfrom explosivesin boreholesare generallynot sufficiently reproducible unless special precautions are taken,

VERTICAL SEISMICPROFILING (VSP) such as centeringsmall chargesinside a heavycasing or a borehole of large (-l-m) diameter. Explosive chargesare usuallysmall.often0.I to 1.5kg: the vertical stackingofrecords ofseveralsmall chargesis preferable to using a larger charge. Impulsive surface sourcessuch as weight droppersare sometimesused. The most common land sourceis Vibroseis,which has the added advantageof distributing the energy over time so that backgroundnoise,often largenear a well, is averagedout. Usually, severalsource impulsesor Vibroseissweepsare stackedto build up t-hesignal strength. Much of the surfacenoise is attenuatedby stacking the recordsfrom severalsourceimpulsesat eachdepth. A monitor geophone(or hydrophone)located near the sourceis usedto verify the constancy of the sourcewaveform. Well geophonesare necessarilyrather massive(fig. 13.25)in order to withstandwell pressures and temperatures.An ideal VSP phone (Hardage,lgg5: 47 52) would have (l) three orthogonal sensorswith identicalresponsecharacteristics, (2) a meansof determiningsondeorientation(tilt and azimuth),(3) a retractabledevicefor locking to the boreholewall, (4) a mechanismto determinethe coupling,(5) downhole digitizing(now ofren l2 bits),(6) smalldiameterand weight, and (7) meansfor measuringat severaldownhole locationssimultaneously. A boreholegeophone with all thesefeaturesdoesnot yet exist. A boreholeis an efficienttransmitterof tube waves becausethese attenuate very slowly with distance. Tube waves(92.5.5)can be generatedwheneverthe boreholefluid is disturbed,as by the interactionof groundroll (the most likelycause)or of a p-waveat a contrast in the borehole (e.g., air water surface, changein boreholediameteras at the baseof surface casing,boreholesonde,bottom of borehole,or at a particularly permeableformation). Tube waves are also reflectedat suchcontrasts.The most troublesome tube waves, which sometimesobscure reflections. are Stoneleywaves($2.5.3)that travelat about the pwave velocity in the borehole ffuid. Generally,tube wavescannot be eliminatedby frequencyfiltering, but often their generationby ground roll can be weakened considerablyby lowering the fluid level in the borehole. Tube-waveinterferenceis substantiallyreduced by firmly clampingthe well geophoneto the borehole wall becauseaxial tube-wavemotion is very much smaller in the wall than in the boreholefluid. Although in theory tube wavesgive information about formation permeability,this is usually not a VSp objective. The needfor detailedinformation about the region immediatelysurroundinga boreholeis often greatest in the marine environmentwhere well costs are very high and many wells are drilled from a platform. Thus, deviatedwells are common and many VSps are run in deviatedholes.Well geophonesmust be gimbalmounted to maintain correct vertical attitude. Schimschal(1986)makesthe following points regardingVSP acquisition:(l) Make suresondedepth

489

D O W N H O LVES P S P I K EP L A N T E D .AN'GEOPHONE GEOPHONEPACKAGE PACKAGE T Y P I C A LP A R A M E T E R S : T Y P I C A LP A R A M E T E R S : L € N G T H- 3 m L E N G T H- l O c m D I A M E T E R l- O c m OIAMETER-3cm MASS - | O0 kg g m s MASS 2OO

F'ig. 13.25 Comparison to scale of conventional land geophone and well geophone. The well geophone has a pivot arm for locking the geophone against the borehole wall. (From Har-

dage, 1985: 36.) is zeroedat the well head; (2) record five to six levels as the tool is being lowered;(3) checkthe depth reading, and determinethe gain and numberof recordsto be stackedto achievethe requiredsignal-to-noiseratio at total depth; (4) take at least five records and monitor at everylevel;(5) slackthe cableafteranchoring; (6) reoccupythe down levelsand checkboth times and waveforms:(7) avoid washed-outzoneseven if it causesunevenspacing;and (8) recheckdepth at well head. 13.4.4VSPproces.sing A portion of a zero-offset VSP is shownin fig. 13.26a. The slopeof the first breaks(direct-wavetraveltimes) givesthe velocity.Reflectionshavea slopeoppositeto the first breaks.By using this difference,it is possible to separatedowngoingwaves(which consistof direct wavesand multiples involving an evennumber of reflections,as in fig. 13.23a)from upgoing waves(reflections and multiples involving an odd number of reflections;seefig. 13.23b).The upgoing wavesmay be 30 dB belowthe downgoingwaves.One way of separating downgoingfrom upgoing wavesis to subtract or add the direct arrival times.Subtractingaligns the traveltimes of the downgoing waves horizontally, making theseeventsmuch more readily seen.Adding the traveltimesof the direct arrival emphasizesthe upward-travelingwaves(fig. 13.26b),but the downgoing wavesare usually still evident becausethey are stronger.The separationcan be done more completely by apparentvelocity filtering (fig. 13.27);it is usually

SP E C I A L I Z E D T E C H N I Q U E S

490

Velocity (m/s)

888

oooil 1; i t : 7 '

ti tl

^ :

o : 2.0

t+ . t , t 1800 i

) I.

l i i{--

i1 l]

t l

1 l 3000I { t . !

1 I

(h\

and Fig. 13.26 Vertical seismicprofile. (After Kennett, lreson' geoC o n n . 1 9 8 0 : 6 8 0 , 6 8 2 , 6 8 3 . )( a ) E a c h t r a c e i s r e c o r d e db y a (b) The ohone in the borehole using an air gun at the surface travdirect-wave by the shifted has been iame except each trace

more effectiveto attenuatethe downgoingwaveswith a narrow band-rejectfk filter than to simply passthe upward-travelingwavesthrough a narrow band-pass however,in.fk filter. Filtering in the fk domain, 'volves smearingand wraparoundaliasing($9'9)'Median filtering(Hardage,1985:189-94)often produces better separation.Downgoing and upgoing S-waves can be separatedin the sameway as P-waves' All the downgoingenergy(exceptfor the P- and Swavefirst arrivals and tube waves)must be multiples' Becausewe know both the input and the desiredoutput (a singlespike),a Wienerfilter ($9'5'5)can be designedto ..-ou. surfacemultiplesalmost completely and lisP deconrolution).Moreover, the downgoing upgoing multiples differ mainly by an additional reRection at ot n.u. the surface(which is apt to be a simpleinterface);therefore,the upgoingmultiple pattern will be nearly the sameas that of the downgoing multiples, so that the deconvolution filter for the downgoingmultipleswill effectivelyremovethe upgoing multiples.The samedeconvolutionoperator may be appliedto nearby surfaceseismicdata (fig' l3'28)' Th. t.uc.. of the upgoing VSP are often stacked togetherto yield the pattern of primary reflectionsfor coirelatingto conventionalsurfaceseismicdata' Only the portionsjust below the well geophone(CC' in fig'

(c)

\d)

reeltime, thus horizontally aligning upcoming eventsincluding the flections. (c) Portion of the surface reflection record across well. (d) Sonic log in the well.

13.29)are stackedin a coruidor'slack,theseportlons are most apt to be relativelyfree of peg-legmultiples' Corridor itacks are usually better than synthetic (fig' ,.ir-ogrurn, made from well-log measurements the because i::Ol iot relatingreflectionsto interfaces and frequencies measurementsare made at seismic ofthe are not sensitiveto logginguncertainties'Stacks points reflection involving VSPs offset fo.tions of nearestthe boreholeare also usedfor this purpose' With offset and walkaway VSPs, the reflection points move awayfrom the boreholeas the geophoneio-reflectordistanceincreases'For a vertical borehole points and horizontal reflectors,this gives reflection VSPA l3'3lb' iocatedas shownin figs. 13'31aand (and (fig' 13'32) data the rc-inp tans/brm relocates place the.reflections grid) to regular a resamplesto at theieflecting points assuminghorizontal reflectors' also' This transform incorporatesa NMO correction combined be can VSPs Vuttioffs.t (but copianar) be exwith this transform, and conceptuallyit can tended to 3-D for azimuthal VSPs' VSP-Io-CRP or untransforms can accommodatedeviated holes can geophysicist experienced u.uul g.o.n.tries, and an seelng lrom geometry recording the often figure out the transformPattern. reMigration of offset VSP data to move dipptng

-6?

o

W A V E L E N G T (HM E T E R S ) -lOO -250 (I) 250 IOO

6?

.,,.' .;,',;,,, ;.:';,;',;;'.t.:;/l I

#',,t,;l$ t',.'t..tll

AMPLITUDE

N F t!

z

t! D g trJ

4o

L

5o

=-a

I ffl

oro-2odB -zo ro -40dB

E|

-40 ro -60dB

l-]

rESSrHAN--60dB

6

-16

-10 0 10 (per km) WAVENUMBER (a)

16

T | M E( S ) l.o

2.o

,H-

q9

3.O

4.O

+ CEMENT

* rTrOusPtC, 'EcMAESNrTN c

T F

+ CEMENT

oLJ

o

2300

2900

+ O P E NH O L E

3500 |

(b) F i g . 1 3 . 2 7 f k f i l t e r i n g o f V S P d a t a . ( A f t e r H a r d a g e , 1 9 8 5 :l l l , l 1 3 . ) ( a ) f k p l o t o f V S P d a t a ; a n a p p a r e n tv e l o c i t yf i l t e r p a s s i n g only data between the straight lines (lz: 1800 and 4000 m/s)

rejects most of the nonupgoing energy. (b) VSP data after the indicated /:k filtering and AGC; the shallow data are poor because of a double-casing string.

SPECIALIZED TECHNIQUES

492

F U

DEPTH

T I

3 0 -

c

U

t

I

T c Fig. 13.28 Surface seismicdata (a) after conventional deconvolution and (b) after deconvolution with VSP-deriveddeconvolution operator. (From Kennett, Ireson, and Conn, 1980:

fie.10.) flectionsto the reflectorlocationsis somewhatdifferent from that of CMP data. After a VSP-Io-CRP transform, reflectionsare horizontal if the reflectors are horizontal but curve if the reflectors dip (fig. 13.33).VSP migrationcorrectsthis. Alignments on a VSP are not straight becausevelocity varieswith depth.They can be made straightby stretchingthe time scale(fig. 13.34).Such stretching beforeseparatingdowngoingfrom upgoingwavefields improvesthe separation.VSPsoften contain gapsbecauseof acquisitionproblems(suchas causedby poor contact with the formations);interpolation along the straight alignmentsof upgoing or downgoing waves can fill in for the missingdata. Much VSP processingis the sameas usedwith conventional seismic data. This includes wavelet processingto shortenthe embeddedwaveletand shapeit to the surfacedata to which it is to be matched.Data are often displayedusing AGC with a fairly long time constant,or, whererelativeamplitude is required(for example,beforeinversion),a correctionassumingthat amplitude variesproportionally to llV2t ($6.5'l) may be applied,followed by a small exponentialramp (or very slow AGC) to allow for absorption,transmission, and other losses. 13.4.5VSPplanning Ray-tracemodeling is used in planning VSP surveys to determinesourceoffsetsfor the requiredsubsurface coverage,frequencycontent required to resolvefeatures,Fresnel-zoneeffects,eventsto be expected,and so on, to make sure that objectivesare achieved.It is also used as an aid in understandingthe VSP evidencesofgeologic featuressuchas dip, faults,angular

Fig. 13.29 Model of a VSP showing all multiples for four layers-.Data from the region CC' just following the first arrivals is relatively free from intrabed multiples.

unconformities,diffractions,and so on. The objectivesof VSP surveys(paraphrasedfrom Gilpatrick and Fouquet,1989)are listed in table l3' l ' Examplesof VSP applicationsto thesevarious objectives can be found in Balch and Lee (1984)and Hardage(1985).Cramer(1988)appliesVSP to the mapping of a point-bar sand and Noble et al. (1988)to resolvingstructuralproblemsin the Vulcan Gas Field'

13.5 Setsmic tomograPhY 13.5.1General "tomos," or "section") Tomography(from the Greek rn.uni t picture of a cross-sectionof an object' In practice, the term denotes determining the internal propertiesof an object from external measurements on rays that passedthrough the object' X-ray tomography has beenusedfor sometime in medicalexamination and in nondestructivetesting.The computerassistedtomography (CAT-scan) technique uses Xrays that have penetrateda body along many raypaths in many directionsand tomographyis usedto,explain the losi in intensity of the X-rays becauseof the absorptivepropertiesof differentparts of the body' Tomographicanalysisusually assumeseither that the propirty being determinedis a continuousfunction of position (transform methods) or that a medium is composed'ofa finite numberof elements,each of which hai a discretevalue of the property' The first method implies a continuous distribution of rays

S E I S M I CT O M O G R A P H Y

493

WELL P 0.5

VSP

SYN

WELL z VSP

ONE MI L E

SYN

ffi

*it,;.,li,,iiijFI? t'ry-qnirrt .tiq._rFf

rh,r ry>k>l+

t t ||

I#:Si']

?.

2. Fig. 13.30 Surlace seismic data with inserted VSP corridor stacks and synthetic seismograms.The match with the corridor

(From at thearrows. stacksis betterthanwith the synthetics H a r d a g1e9, 8 5 : 2 8 3 . ) (hence,an infinite number of rays).Seismictomography,which usesa quite limited numberof rays,clearly is bettersuitedto the secondclassof techniques.However,to introduce some important concepts,we shall first discussintegralmethods.Our discussionfollows closelythat of Stewart( l99l ) exceptfor notation. I 3.5.2 Tomographicconcepts In fig. 13.35a,we show several rays passing from sourcesS, to receiversR, that registervaluesthat depend on someproperty C(x,y) of the medium M. The recordedvaluesare those of the Radon transform of g(x, y) along raypathssuch as R,S, in fig. 13.35b(see also eq. (9.21)). In the following, we use G(d, 0) for the Radon transform (projection) of C(x, y), Gr(u, v) : Gr(p, 0) for the 2-D Fourier transform of g(x, y), G,(p,O) for the l-D Fourier transform of qe, q We wish to find the valuesof g(x,y) usingonly the projections.We start with the Fourier projection theorem,which statesthat the 2-D Fourier transform of the object C@,y) is equal to the l-D transform of the projectionsG(1, 0). The proof is as follows. The 2-D Fourier transform of g(x, y) is (see eq. (9.19))

Fig. 13.31 Loci ofreflection points for VSPs for flat reflectors. (From Balch and Lee, 1984: 82-5.) (a) For an offset VSP, (b) a walkaway VSP, (c) a VSP in a directional well, and (d) a VSP in a directional well where the source moves to stay above the well geophone.

G,(u,v):

fJ

j2'("*'1')dxdJ, (13.8) tO,.,,)e

(note that g(r, l) : 0 outsidethe object,so the infinite

S P E CI A L I Z E D T E C H N I Q U E S

494

OFFSET (Kft) 0.00

0 . 50

r.00

|.60

2.00

2.t0

t . 00

g,

o UJ

;

tr

tr

=

5.0 (b)

(a)

Fig. 13.32 The VSP-to-CRP transtbrm. (Frorn Dillon and T h o m s o n , 1 9 8 4 :t i g s . I I a n d 1 4 . )( a ) I l l u s t r a t i n gt h e c o n c e p to f d i s p l a y i n go f T s e V t S P t r a c e sa l o n g m i d p o i n t l o c i c u r v e ss u c h a s

s h o w n i n l i g . l 3 . 3 l a . ( b ) V S P - t o - C R Pt r a n s l t r r m e dd a t a l i o m a n ofl.setVSP

f- rl- l-

0 + l sin0 c,tr 0) : I ll I r(,t,r) E(-r'cos J _ L J - J -

{ ) d Y d' ll l e ' " ' ' d { . lnterchangingthe order of integration(see5s15.2.31 gives

c,,(p,e) : F i g . 1 3 . 3 3 D i p p i n g r e f l e c t i o n sa l i e r V S P - t o - C R P t r a n s l b r m h a v ec u r v e d a l i g n m e n t s R . e f l e c t i o ni n t e r s e c t i o n w s ith thc downg o i n g d i r e c t w a v c ( 1 . & a n d ( ' ) d o n o t r e q u i r em i g r a t i o n .( F r o m H a r d a g e .1 9 8 5 : 2 1 0 . )

r'pD(r o cos Jli" ,,,[J_. + y sino - e)dl] dx o;, c" o+'sin0)dx dL]' ..v)sr2'otr [f_t-

limits are merelya convenience and the actuallimits are the boundsof the medium M).Replacing (u, r,) with polar coordinates(p, 0), wherer.r: p cos 0, and , : p sin 0, Gr@,r,)becomesG-(p,0): G-t O 0) :

i- f|

|

ft r, f')e

jhp(\ cos0+' sino) d,r d.t'

Thel-D,runr*rl Jt,n. prol...ionin eq.(9.21) is

(becausethe inner integrandis zeroexceptwhen ( : ,r cos 0 * ,r'sin 0). Thus. G , , ( pe.) : G - ( p ,0 ) ,

(13.9)

that is. the 2-D Fburiertransformof g(.t,t') is equal to the l-D Fouriertransformofthe projection. is the mapping from the ((, 0) doBat'kpro.jet'tion main back to the (.r, r) domain; it is approximately equivalentto inversetransformation.Backprojection

495

S E I S M I CT O M O G R A P H Y Table 13.1Objectiveso.f VSP surveys

I

Objective

How achieved

Refleclorrdentification \ Surface-to-borehole correlationI Increased resolutionat depth t

Upgoing wavestudieson zero-offsetVSP

Time depthconversion ) Enhanced v e l o c i t ya n a l y s i s) Log calibrarion t

First-breakstudieson zero-offsetVSP

Multipleidentification \ Deconvolutionoperator t

Downgoing wavestudieson zero-offsetVSP

Improve poor data area

All types,especiallyoffsetVSP

Predict aheadof bit

Upgoing wavestudieson zero-offsetVSP

Structuralimaging

Walkawayor offsetVSP with presurveymodeling

Delineatesalt dome

Proximity surveywith sourceover dome

Seeingabove/belowbit on deviatedwells

Zero-offset.offset,or walkawayVSP Stratigraphic imaging (channels. faults,reeli, pinchouts) Multiple-source locationswith offsetVSP AVO studies Researchstudy on offsetVSP with prcsurveymodeling P/S-wave analvsis \ Polarization studies t Fractureorientation t

Researchstudy on offsetVSP,three-componentphonc

Attenuationanalysi(

Rescarchstudy on zero-offsctVSP

Secondaryrecovery Tomographicstudies Permeability studies

Researchstudy on offsetVSP M u l t i p l cw c l l s m . u l t i p l eo f f s e r s Tube-waveanalysisresearchstudy

Alier Gilpatrickand Fouquet,1989

consistsof summingall projectionsto which a certain property value g(.r,,,r',)has contributed,that rs, we sum the measuredvalues lor all rays that passed through the point (.r,,-l',).This operationcan be expressedby an integral, the backprojectionl(.r, 1') being

: ir(.r, 1'y

J"

: J,

",r, o,oo "r coso r -r'sino,g)do,

(r3.r0)

w h e r eG ( ( , 0 ) i s g i v e nb y e q . ( 9 . 2 1 ) . Althougheq.( 13.l0) doesnot reproducethe object, that is, i(,r, ,r') is not equal to g(.r.,r'),it does give an approximatepictureof g(r, 1').By usinga ratherloose analogy.just as a seismicreflectionappearsas a ringy embeddedwaveletratherthan as a spike,the backprojection is a blurredimage,and we can sharpenthe imageby deconvolution(filtering).We obtain g(,x,1,)exactly by transformingthe projections,filtering,and then takingthe backprojection. To showthis.we start with the inverse2-D Fourier transform of G..(u,r') (see e q .( 1 3 . 8 ) ) :

ATr

TIM[+ ATt

x !J

F (L U

l

v

I

F i g . 1 3 . 3 4 S t r e t c h i n gt h e V S P t i m e s c a l et o g e t c o n s i s t e n t i m e i n c r e m c n t sb e t w e e nt r a c e ss t r a i g h t e n sa l i g n m e n t s .

i.t

. s ( . \ r- . ): |

l.''

l 1 . 1 s , , ' ',,",d' r d r : | G..{r,

J - J .

(p. 0). we have Changingto polar coordinates g(-r, .lr,l :

fi

G_(p,e)e,r""pdp d0, ( l3.l I )

S P E C I A L IZ E D T E C H N I Q U E S

496

R3

R4

R1

R1

R2

R2

(a)

(b)

F i g . 1 3 . 3 5 T o m o g r a p h i cc o n c e p t s .( a ) R a y p a t h sf r o m v a r l o u s s o u r c e st o v a r i o u sr e c e i v e r ssl o l u t i o n c a n b c o b t a i n e d i f e n o u g h

raypaths penetrate the medium in all directions. (b) 1n practice a grid of uniform cells are usually assumed.

wherewe have usedthe relations{ : "xcos 0 * .1'sin "area" : dp(pd0). We wish to 0, and that dy d,y : changethe limits of the integrals;recallthat replacing 0 with 0 * n changesthe signsof cos 0, sin 0, and (, g) - l( p, 0 + t). and that in polar coordinates,.f(p, W e n o w s e p a r a t et h e 0 i n t e g r a li n e q . ( l 3 . l l ) i n t o parts;thus,

| -D Fourier transform of the projections.Moreover,

t r i.-

dp de s(.Y. l') : I I G"(e.0)er:",',p J,,J,, fr" l.

-

ndpo . | | c _ ( e . g 1 s , :d" p J" J., Thelowerintegralis equalto

-J,lo'-

0+T)e+r' "p,pdp d0

:II

dp d0, G,(-p, 0+rr)e i:'r',lpl

because-p : lpl when p is negative.Therefore,eq ( 1 3 . 1 l )c a nb e w r i t t e n tf

f -

l

l

st.r,ut : | | G"(p.0)o' *'lpldPd0. ,ltt ,l

s(x,-y):

ulot, J"",tr,

(13.12)

so that g(.r,l) is obtainedby the backprojectionof the Fourier-transformedfi ltered projections. 13.5.3Solution.fbru limitetl nunrbero.fdixrate cells We divide the medium M into m cellsor pixelsand assume17rayspassingthrough M (fi9. 13.35b).Each pixel has a property value g(.t, 1') and the recorded value at R, is a projectionequal to the sum of the productsr{g,, wherer{ is the path length of the ith ra1 in thejth cell.Assumingthe usualcasewhere8, is the we write for the traveltime slowness. t ' : t l i g , + r t ' t g t + ' ' ' + r t , , s , :, 2 d ' ,g , , (13.13) i:1.2.....n. Of course.any givenray will not passthroughall cells so that many of the r{ are zero. If we measureamplitude insteadof traveltime,we can take the logarithm o f e q . ( 2 . 1 1 0 )a,n d e q .( 1 3 . 1 3b) e c o m e s

\n(A;tA'):Lair,

(13.14;

-

o r b y e q .( 1 3 . 9 ) ,

: o)d"o;o1ar]ae. s(r,.v) [[l_o,The inner integral,which we shall denoteby Gr,(f, 0), is a filtered version of C,(p, 0), which in turn is the

Equation(13.14)can be written in matrix form as T : %9,

(13.15)

whereI is a n X ,fl matrix relating column matrices T and 9. Given 9, we wish to find L This is onll possibleif we know 9, at least approximately.When 3 is known exactlyand n : ,?1,an exact solution can

SEISMICTOMOGRAPHY be obtained.However,the important caseis when the number of raypathsexceedsthe number of cells (r > m) and I is known only approximately;in this situation, a numberof solutionmethodsare available(Herman, 1980).A least-squares solutionis possiblebut is time-consuming.Stewart(1991: 2-21, 2-28) describes a solution basedon backprojectionthat givesan approximateresult.More suitablesolutionsare siven bv the algebraicreconstructiontechnique(ARTJand simultaneousreconstructiontechnique(SIRT) or one of the methodsdescribedby Justiceet al. (1992). The basicproblem is to solvea setof n linear equations with n unknowns(n > m), where the m \ n coefficientsare known only approximately.ART starts by guessingvaluesof the parametersgj and usesthese valuesto calculatet1. The differences"between calculated and observedvaluesof t2 are usedto vary the g, so as to minimize the error (difference).The adjusted valuesof q are used in the secondequation and another adjustmentof g, is made.The processis continued until all ofthe equationshavebeenusedand then the entire processis iterated.Stewart ,1991:2- 2g to 2 34)givesfurtherdetails;seealsoGordon (1983)and Herman, Lent, and Rowland (1913\. In using ART, the adjusted parametersobtained from one equation are used in the next eouation. In S I R T ( S t e w a r t 1, 9 9 1 . . 2 - 3t4o 2 3 6 ; D i n e sa n d L y t l e , 1979),the errors are calCulatedfor all equationsusins the first guess,and the average..ro., ui. usedto ad] just the parametersfor a secondpassthrough the set of equations. I 3.5.4 Cross-hole measurements Unlike well-loggingand VSP measurementsthat involve a single borehole, cross-holestudies involve sources and detectors in different boreholes (fig. 13.36).Usually,severalreceiverlocationsin one borehole recorddata from a number of sourcelocationsin another borehole,and sometimesrecording is done on the surfaceat the sametime. The recordingprocedure is time-consumingbecauseusually only one or a very few receiverand source locations can be occupied at a time, but developmentsare underwayto permit the simultaneoususe of severalreceiversand to increasethe sourceenergy.The objectiveofcross-hole studiesis to learn about the region betweenthe boreholes(Wonget al., 1987;Rutledge,1989;Lines,l99l). Nearby boreholesare most often availablein developing and producing oil fields, so cross-holestudies usually havereservoirgeophysicsobjectives. Tomography is usually employed in interpreting cross-holedata. Both slowness(l/velocity) and absorption satisfythe requirementsfor tomographicimage reconstruction(Deans, 1983).In theory. we can measureboth traveltimeand attenuationand solvefor the unknown velocity field and absorption distribu_ tion (Q-map)in the interveningregion. However,usually, we try only to determinethe velocity distribution in the interveningregion from measurementsof seis-

497 mic traveltimesbecauseamplitudescannot be measured with sufficientaccuracyto determine absorption. Most applications to date use only the firstarrival information in their solutions. The often-madeanalogybetweenseismictomography and CAT-scanningshould take into account that in a CAT-scan,the X-rays travel in straight lines in many directions,whereasin seismictomography,the different directionsraypathscan take are limited and the raypathsare not straight (fig. 13.36b). Although generallythe number of observationsis much largerthan the number of cellsand the problem is overdetermined,somecellsmay not have been traversed.so that their slownesses cannot be determined and many of the travel paths may have traversedthe samesubsetof cells so that their individual contributions cannot be separated.All measurementsinvolve uncertainty,and the reliability of a determinationdepends on the number and range of directions of the raypathstraversinga cell. In CAT-scanning,we can haveraypathstraversingthe body in many directions, but seismicapplicationsare constrainedby the boreholes availablefor measurement.Also, the zone of major interest is often near the bottom of the boreholes,so that only nearly horizontal travel is possible for raypathsin this zone. The seismicproblem is unlike the CAT-scanproblem in another very important regard: Seismicraypaths bend (fig. 13.36b)appreciablyas the velocity changes.This makes the seismicproblem nonlinear becausea changein the slownessof any cell changes not only the traveltime,but also the raypath.Forward modeling to recompute the traveltimesthrough the modified model has to be carried out anew for each ofthe variousraypathsbeforeeachiteration ofthe tomographicalgorithm. Many ray-tracingmethodsare not suitablefor forward modeling becauseSnell'slaw does not apply at the cell boundaries,which exist only as a mathematical device and do not correspondwith actual interfaces(that is, they have arbitrary rather than natural orientations).Figure 13.37aillustratesthis problem; rays only incrementallydifferent can strike near the corner of a cell where the velocity contrast is large. The ray that strikes the upper boundary first will be bent in an entirely differentdirection from that of the ray that strikesthe side boundary first, and hencethe corner will producea major discontinuitywith no raypaths at all entering a fairly large region unless diffraction is taken into account. To accuratelyaccount for the actual observedarrivals, the forwardmodeling algorithm must include arrivals that result from diffraction and critical refraction.The procedure of Justice et al. (1992) involves computing the full acoustic wavefield at closely spaced increments in time, thus allowing for diffraction, and then tracing rays along orthogonaltrajectoriesbackward(in time) through the wavefronts to find the minimum-time ray connectinga sourceto a receiver. The tomographic solution is almost always itera-

lV!ll

-1i-

ro. 2

Tfansilla? Ldtbna

(d)

(a)

L CO18

LAco 26

L A C Q1 8

LAco 26

zs?

377 a3? a17 521 55? 58? 6l? 647 671 1a1

?95

(b)

F i g . 1 3 . 3 6 C r o s s - b o r e h o l em e t h o d s ( a ) S o u r c e s a n d g e o phones in nearby boreholes provide criss-crossingraypaths (b) in an actual situation, raypaths bend and concentrate in high-

(c)

(c) Velocity vclocity cells. leaving rnany cells poorly sampled(b) (from Bois
T I M E - L A P S EM E A S U R E M E N T S

499 and detectorlocationsare limited to existinsavailable boreholes(or the surface).travelpathshive much larger horizontal than vertical components.Figure 13.36bshowsraypathsin a real situationinvolving high-and low-velocitylayering(fig. 13.36c); the shortest traveltimesmaximize travel in the high-velocity layersso that many low-velocitycellsare not traversed and thus cannot be investigated.(However,channel waves(913.3)travelpreferentiallyin low-velocitylayersand thus in conceptprovidea methodfor their investigation.) Also, becausemost raypathstraversethe samecentral cells of the higher-velocityregions,an increasein the slownessof one of thesecellscan be compensated by a decreasein a neighboringcell so that their individualvaluescannotbe determinedvery accurately.The lact that many boreholesstop near the zoneof interestpreventsadequatesamplingof cellsat that depth. Theoretically,many of the foregoingproblemscan be eliminatedby consideringentire recordedtraces rather than only the first arrivals, but theseapplications are still in the researchstage.Techniquesemploying reflected(fig. 13.36d)as well as direct travelpathsare beingdeveloped. The foregoingassumes all travelis within the plane connectingsourcesto receivers; clearly,many realistic casesrequireallowancefor travel outsidethis plane. Tomographicmethodsare also sometimesusedwith (reflecborehole-to-surface and surface-to-surface tion) measurements.

(a)

Tlm rdudoh:0.{

rc

llm rooluion :0.5m

Tlm rdu{on:1.0

ro

E F

t*o t00 0 CROSSHOLE

1OO

DISTANCE (FT) (b)

F i g . 1 3 . 3 7 P r o b l e m sw i t h m o d e l i n g c r o s s - h o l et r a v e l t i m e s(. a ) S u b d i v i s i o no f s p a c ei n t o c e l l sw i t h a r b i t r a r y b o u n d a r i e sm e a n s that Snell's law cannot be applied at the cell boundaries: two nearly parallel raypaths may take off in quite different directrons becausethey strike different boundaries first. (b) F-orsiven un_ c e r t a i n t i e sr, a y p a t h sc a n l i e a n y w h e r ew i t h i n t h e s h a d e i r e g r o n s (which are called /czllc.r):uncertainties are respectively0.4, 0.6, a n d 1 . 0 m s ( J u s t i c ee t a l . , 1 9 9 2 : 3 2 4 ) .

trve.As statedearlier,a startingslowness distribution is assumed,raypathsare trackedthrough it, an<Jthe problemis solvedfor the changesin slowness that produce a closersolution.The model is then moclified. new raypathstraced,and the processrepeateduntil the slowness distributionmatchesobservations within acceptable tolerances. Constantslownessis often assumedfor the startingmodel. Still another problem is that the solution is very sensitiveto small measurementerrors.Justiceet al. ( 1 9 9 2 ) i l l u s t r a th e i s( f i g . 1 3 . 3 7 b ) b ys h o w i n gt h e l a r g e volumesthroughwhich raypathscould passallowing for small traveltimeuncertainties.Relativelyminor deviationsof a well bore,for example.can introduce s i g n i f i c a ntti m i n ge r r o r s Considerationof a few simplemodels brings out other problemsinvolvedin a solution.Because source

13.6 Time-lapse measurements In principle.any measurementcan be repeatedat different times in order to determinechangesthat haveoccurredduring the interveningtime.In particular, measurements on a reservoirat different times during its productionhistory sometimescan be used to monitor changesbecauseof movementof fluids within the reservoir.Repetitive3-D surveys,sometimescalled4-D surveys(the fourth dimensionbeing time), are usedfor reservoirsurveillance (914.4.5). It is important that the data collectionand processing be identicalfrom one surveyto the next;hence,geophones are commonly cementedat the bottom of shallowboreholesto equalizegeophoneplanting.Vertical and horizontal sectionsfrom the first survey (bu.sesurvev)can be subtractedfrom those at later times to give difJerent'ese('tions,which display the changesin a very sensitiveway.Time-lapsemethods are alsoappliedto cross-hole and other typesof measurements. (EOR) operations using Enhanced-oil-recovery thermalmethods(fire and steamfloods)in heavy-oil reservoirshavebeenmonitored experimentally.Heating reducesthe viscosityof the oil and allows it to flow moreeasily.An increase in temperature markedly decreases the seismicvelocityof heavy-oilreservoirs (see$5.2.6),and sometimeswe can observechanges associatedwith the loweredvelocityand/or depressing

I

500 of underlying reflections.In a fire flood' a portion of the oil is burned in place,the burn being controlledby the injection of oxygenor air. The wastegasesfrom the combustionaid in pushingthe oil toward production wells. The asymmetricexpansionof the bright spot in fig. 13.38aindicatesthat the fire flood did not proceed uniformly away from the injection well where the in-situ combustionwas initiated. In a steamflood, injectedsteamheatsthe reservoir.Differencesections or differencetime slices,such as fig. 13.38b,show areas interpreted as heated zones' The expanding heatedzonesresultingfrom steaminjection have also beenmonitored using cross-welltomogramsin a timelapseway. The movementof reflectionsassociatedwith an oil/ water contact can sometimesbe monitored in water flooding. A water flood may also produce visible changesbecauseofthe temperaturechangesresulting from the invading cold water. Changesin a gas cap may producevisible changesin a bright spot or gasoil or gas-waterreflections.The fluid fronts in carbon dioxide floods sometimescan be monitored. 13.7Borehole studies I 3.7.I Salt-proximitysurveYs Precisedefinition of the flanks of a salt dome are important becausehydrocarbonaccumulationsare often Bedstruncatingat the salt trappedthere(see$10.3.4). dome often have considerabledip, so that an appreciable amount of attic oil will be left behind if a well is not drilled close to the salt seal. Salt proximity surveysare sometimesrun in wells drilled into the salt (see$l1.1.3and fig. 13.39),but more often they are run in wells drilled close to the salt; they can be thought of as a form of VSP survey.The energy sourceis placed on the surfaceabovethe salt and a geophonein the well recordsarrivals at severalpositions. For each source-geophonecombination, an aplanaticcurve(the locus of all points wherethe sum of salt and sedimenttraveltimessatisfiesthe measured arrival time) is constructed;seefig. I 1.6.The envelope ofthe aplanaticcurvesis then the interpretedposition of the salt face.Where the sourcemovesto different azimuths,a 3-D solution can be obtained. Steeplydipping reflectionsfrom the sediment salt interfacecan also be migratedto locatethe salt flank (Ratcliff, Gray, and Whitmore, 1992).Diving waves that reflect on their upward leg from overhanging flanks (turning waves)can also be migrated. 13.7.2Sonicwaveformlogging Severaltypes of logs record the entire seismictrace rather than just the first arrival (Jamesand Nutt, 1985).The sonic waveformlog is an extensionof the regularsoniclog in the samesenseas a verticalseismic profile is an extensionof the well-velocitysurvey.A iecordedwaveformlog (as in fig' 5.39,an array-sonic log) permits some control over measuring transit

SP E C I A L I Z E - D T E C H N I Q U E S time by tracking the first arrival. In addition to measuring the P-wave arrival, an ,S-wavearrival can be observedand tracked where the S-wave velocity is sreaterthan the P-wavevelocity in the boreholefluid' the dipole sonic log also yields this information' Determining S-wavevelocityis generallypossiblein wellconsolidatedbut not in poorly consolidatedsections' The S-wavearrivals can be used to generatea shear transit-time log. This is valuableadditional information that permits one to calculatethe velocity ratio VrlV, and Poisson'sratio, and thus get some indication of lithology.An S-wavesyntheticseismogramcan be generatedfor event identification on S-wavesections. A sonicwaveformlog in a deviatedwell can record reflectionsfrom nearbyacousticinterfaces'Sonic imagesare also recordedwith specialtools' I 3.7.3Boreholeteleviewer The boreholetelevieweris an ultrasonic acousticde"pictures" of the borehole wall with vice that takes little penetrationinto the rock formations.The tool is held centrallyin the well bore and an acoustictransducer operating at megahertzfrequenciesspins on a vertical axis;a compassrecordingis madeeachtime it passesthrough north. The traveltimeof the recorded iignal measuresthe borehole radius and the amplitude measuresthe wall's reflectivity. Figure l3'40 showsteleviewersectionsdisplayingboth of theseparametersagainstgeographicaldirection and depth in the borehole. Anomalously long traveltimesor low amplitudes indicate fractures, and the televiewerts used to identify fracturesand measuretheir orientations. Fracturesstriking east-westand dipping steeply to the north are clearlyvisiblein fig. l3'40' The borehole televiewercompeteswith the formation mlcroscanner (which uses dipmeter and microresistivity seeTelford,Geldart' and Sheriff,1990: measurements; seeingfrac$11.4and 11.2.5)as a boreholetool for tures.

13.EPassive seismic methods Seismicapplications that do not involve controlled sourcesare said to bepassive.Activitieswithin a reservoir. such as fluid flow within a fracture system,periodic strain accumulation and release due to gas buildup and flow, thermally inducedfailurecausedby a fireflood,rock breakageduring hydraulicfracturing' subsidencedue to fluid withdrawal, or gas flow in a blown-out well, may createmicroseismicevents'Plotting the locations of observeddiscreteeventsmight givi usefulinformation (Dobecki, 1992)'Suchseismic Jventsare apt to be extremelyweak (perhapsof Richter magnitude-3). Dobecki refersto passivemapping of conibustionin a Canadianheavy-oilEOR project' combustion-gasificationstudies in a Wyoming coal seam, and hydraulic fracturing in coal-bed methane production. Downhole seismicarrayshavebeenusedto monltor

M

M€IERS

!5J

rmr'

03

o z O

c !

o.

(Mrd.burn -

(Posl burn -

( D r eb u r n )

(Pre burn)

90-

c

a , r ' l

D P

lt

,r

a t I

r0 L I NE AMPLITUOE 5

----

0

'

.

I OBSERVATIONW3O6 A I N J E C T I O NW E L L NE L L a P R O O U C T T OW

N

O -I-

8 METERS (2ooFr)

(b) Fig. 13.38 Differences induced by a fire flood. (From Greaves and Fulp, 1992: 313, 314.) (a) Vertical sections before, durins,

and after a fire flood (4 and l0 months having lapsed). (b) Difference time slices through the zone of the fire flood.

SPECIALIZEDTECHNIQUES

502

that are not usedin determiningthe parameters(hidclentlata\ are usedto checkthe reliability of the analysis. A measureof the confidencethat can be placed in the calculationsis usualiy generated.Sheriff (1992) showsa number of geostatisticalmethods'

s

"sttzstocx

Problems

(a)

(b)

Fig. 13.39 Salt-proximitysurveys.(From Steinmannand Riihmkorf, 1968.)(a) Geophonein a well in a salt dome,a n d tbl rn a wellneara saltflank

the growth of hydraulicfracturesand determinetheir orientation in petroleum reservoirsand in a hot dryrock geothermalproject (Dobecki,loc. cit.). Passive may help to describenatural stresspatmeasurements terns and thus give information about fractures. 13.9Joint inversion Joint inversioncombinesdifferentkinds of data sets to producea betterinterpretationofthe subsurfaceby Interreducingthe number of possiblealternatives. pretationshoulddealwith all data,andjoint inversion providesa formal way of achievingthis. Gravity and seismicdata providethe most common combination (Ramo and Bradley,1982;Lineset al.' 1993)as both explicitly <Jependon density.Sheriff (1987: 106-8) showshow a gravitymap over a salt dome can be used to check on the plausibility of the interpretation of seismicdata over the salt dome' 13.10Geostatistical methods Geophysicalmeasurementsvary becauseof changes in rock propertiesof severaltypes,as was seenin chap' 5, and usuallyone cannot determinethe distribution of any singleproperty without uncertaintiesbeing introducedbecauseof possiblechangesin other properties.Geostatisticsattemptsto cope in a statisticalway with changesintroduced by unknown parametersso as to yield a probable distribution of a sought-for property basedon somecombinationof the available measurements.Geostatistics deals with relations among seismicamplitude, seismicvelocity, well-log measurements,density,lithology, bed thickness,porosity, interstitial fluid, and possiblyother factors' It generallyassumesthat the property being sought is given by a linear relation among the availablemeaiurements and that the relation changesslowly becauseof factorsthat are not measured.Valuesare often made exact at control points, and availabledata

13.1 In a marine survey,the water depth is I km and a reflector is 3 km below the sea floor' Use flg' 13'5 to determinethe optimum range of offset for S-wave seneration.The P-wave velocity just below the sea ioor is 2.8 km/s and the water velocity is l '5 km/s' 13.2 Show that the angle between the vertical and each of three orthogonal geophones'equally inclined to the vertical,is 54.74".(Hint: Find the direction cosinesof a line passingthrough the origin of a set of orthogonalaxesand equallyinclinedto the -t-, y-, and :-axes.) t3.3 (a) In fig. l3.l9b, vn: 40 Hz; find the water depth using data from the figure' (Uj fina the frequenciesthat are reinforced when the rays are reflectedat anglesof 30' and 40' to the vertical. modeswith frequencies (c) Calculate0 for successive o f 1 8 0a n d 3 0 0H z . (d) Find Z for cases(a), (b)' and (c)' 13.4 A sourceis offset 1000m from a vertical well in which a geophoneis suspended;a horizontal reflector is presentat 2000m; V : 3 km/s. (aj f ina the traveltimeboth graphicallyand by calculaiion for a geophoneat depths; : 800, 1200,1600' and 2400m; find the valuesAl/A:. (b) Repeat for reflector dips of E : 'l-1" where the reflectorsintercept the well at the samedepths as in part (a). (a) changeif ic) By how much do the valuesin part source' the toward the well deviatesby 3' 13.5 A sourceis offset 2400m west of a vertical well in which a geophoneis suspended'There is a vertical = N-S fault 600 m west o[ the well with velocitiesZ respecfault, the of east 2.50and 3.00km/s westand tively,and a horizontal reflectoris presentat 2000m' (a) Find the reflectiontraveltimefor geophonedepths of 250 and 1000m. (b) What is the deepestgeophonelocation for which the reflectioncan be seen. 13.6 An outcropping salt dome has roughly vertical flanks.A sourceis locatedon the salt and a geophone Table 13.2Surveyto defne salt-domeflan ; (ml

/ (s)

; (m)

/ (s)

500 750 1000 t250 1500

0.325 0.337 0.366 0.397 0.422

1750 2000 2250 2500

0.45'7 0.490 0.530 0.580

503

R E FE RE N C E S

Co!1. lor rurpanalon o n d a l a c l r i c o l c o n n o c f o i-r

Dolo rlripr from ruccarrira tavolulionr orc dlrplotad lo lorn unrropDrd picluia of borahola foll

Iofor for lurnlne lmar orrtnblt I o e n a f o m a f a rt o t raoting oorlh dl aoci

?avolulion

0lpplne South

Piaroalactfic f r o n t d u c a t b o lh randr oul sonic baon oad rrala ctr r . l l . c l . d !aom

S o n i c b a o n Ir oc,a r halir on bor.holl roll cr f00l i r r o i r a d

{al

(b)

(c)

F'ig. 13.40 Borehole televiewer.(From Zemanek et al., 1970.) (a) Schematic of sonde in the borehole, (b) appearance of two plane fractures. and (c) the fractures shown in part (b).

is suspended in a verticalwell in the sediments1400 m from the sourcepoint. Determinethe outlineof the salt dome from the t z data in table 13.2.Take the velocitiesin the salt and adjacentformation as 5.00 and 3.00 knls. (Hint: Draw circleson transparencies as in fig. I 1.6and overlaytheseto find points on the aplanaticsurfacesfor eachgeophoneposition.) 13.7 Graph Poisson'sratio vs. P-wavetraveltime for the five eventsin fig. 13.7. References Anno, P. D. 1987. Two critical aspects of shear-waveanalysis: Staticssolutions and reflectioncorrelations. ln Sheur-WuveExp l o r a t i o n ,S . H . D a n b o m a n d S . N . D o m e n i c o ,e d s . .p p . 4 8 6 1 , Vol. I in Geophysical Development Series.Tulsa: Society of Exploration Geophysicists. Balch, A. H., and M. W Lee. 1984. Vertical Sei.smitProfiling: Tethnique,Applicutions,and Case Hi,storie.s.Boston: International Human ResourcesDevelopment Corp. Bois, P, M. La Porte, M. Lavergne,and G. Thomas. 1972.Wellto-well seismicmeasurements.Geophl'sits,37: 471 80. Cassell,B. 1984.Vertical seismicprofiles: An introduction. Fir,rl Breuk, 2(lll:9 19. Clay, C. S., and H. Medwin. 1977. Atoustitul Oceanogrupht. New York: John Wiley. Cramer, P W 1988. Reservoir development using offset VSP techniquesin the Denver-JulesburgBasin. I Petrol. Tech.,February: 197 205. Crampin, S. 1987. Crack porosity and alignment from shearwave VSPs. In Shear-Il/aveErploration, S. H. Danbom and S. N. Domenico. eds.,pp. 227 5I.Yol. I in Geophysical Development Series.Tulsa: Society of Exploration Geophysicists. D a n b o m , S . H . , a n d S . N . D o m e n i c o , e d s . , 1 9 8 7 .S h e a r - W a v e Exploration, Vol. I in Geophysical Development Series.Tulsa: Society of Exploration Geophysicists.

Deans, S. R. 1983. The Radon Tran.sfbrmund Some of lts Appliratlon.r. New York: John Wiley. D i l l o n , P B . , a n d R . C . T h o m s o n . 1 9 8 4 .O f f s e t V S P s u r v e y sa n d their image reconstruction. Geophys.Prosp.,32:790 9ll. D i n e s , K . A . , a n d R . J . L y t l e . 1 9 7 9 .C o m p u t e r i z e dg e o p h y s i c a l tomography. Proc. IEEE,87: 1065 73. Dobecki, T. L. 1992.Alternative technologies.ln ReservoirGeoplr1,sfu.r, R. E. Sheriff, ed.. pp. 335 43. Vol. 7 in Investigations in Geophysics Series.Tulsa: Society of Exploration Geophysicrsts. E b r o m , D . , a n d R . E . S h e r i f f . 1 9 9 2 .A n i s o t r o p y a n d r e s e r v o i r development. ln Reserwtir Geophysics,R. E. Sheriff, ed., pp. 355 6 I . Tulsa: Society of Exploration Geophysicists. E n s l e y ,R . A . 1 9 8 9 .A n a l y s i s o f c o m p r e s s i o n a a l nd shear-wave seismic data from the Prudhoe Bay field. The Leuding Edge, 8(ll): l0 31. Ewing. W M., W. S. Jardetsky.and F Press.1957.Elastic Wuve.t in Lut'ered Mediu. New York: McGraw-Hill. F i t c h . A . A . 1 9 8 4 . I n t e r p r e t a t i o n o f v e r t i c a l s e i s m i cp r o f i l e s . F i r s tB r e a k , 2 ( 6 ) : 1 9 2 3 . F i x . J . E . . J . D . R o b e r t s o n .a n d W C . P r i t c h e t t . 1 9 8 7 . S h e a r wave reflections in three West Texas basins with high-velocity surface rocks. ln Sheur-WaveErplorution, S. H. Danbom and S. N. Domenico, eds.,pp. 180 96, Vol. I in Geophysical Development Series.Tulsa: Society of Exploration Geophysicists. G a r o t t a , R . 1 9 8 7 .T w o - c o m p o n e n ta c q u i s i t i o na s a r o u t i n e p r o cedure. In Shear-WaveExplorution, S. H. Danbom and S. N. Domenico, eds., pp. 122 36, Vol. I in Geophysical Development Series.Tulsa: Society of Exploration Geophysicists. Gilpatrick, R., and D. Fouquet. 1989.A user'sguide to conventional VSP acquisition. Geophl'sics,The Leading Edge o.fErpkr r a t i o n ,S ( 3 ) : 3 4 9 . Grant. F S., and G. F. West. Interpretation Theory in Applied Geopht'sit's.New York: McGraw-Hill. Greaves, R. J., and T. J. Fulp. 1992. Three-dimensional seismic Get monitoring of an enhanced recovery process.In Re,servctir

504 physics,R. E. Sheriff, ed., pp.309-20. Tulsa: Society of Exploration Geophysicists.

Officer, C. B. 1974. Introduction to Theotetical Geoplr1'sics.New York: Springer-Verlag.

Hardage, B. A. 1985. Vertical Seismic Profling: Part A: Principles,2d ed. London: Geophysical Press.

Payne,M. A.1994. Looking ahead with vertical seismicprofiles' Geophy'sics,59: I 182-91.

Hardage, B. A. 1992. Crosswell surteying and reverse VSP' London: Geophysical Press.

Puckett. M. 1991. Offset VSP: A tool for development drilling. The Leading Edge, l0(8): 18-24.

Hasbrouck, W. P 1987. Hammer-impact shear-wavestudies ln Shear-WaveExploration, S' H. Danbom and S. N. Domentco' eds., pp. 97-12i, Vol. I in Geophysical Development Series' Tulsa: Society of Exploration Geophysicists

Ramo, A. O., and J. W Bradley. 1982. Bright spots, milligals, and gammas. Geophysics,47: 1693-1705.

Herman, G. T. 1980. Image Reconstruction from Projections' New York: Academic Press. Herman. G. T.. A. Lent, and S. W Rowland. 1973.ART: Mathematics and applications. J. Theor. Biol.,42:1 32. James, A., and W. L. Nutt. 1985. New techniques in borehole eophYs.,16:349 56. seismic. Justice, J. H., A. A. Vassiliou, M. E. Mathisen, S. Singh' P' S' Cunningham, and P R. Hutt. 1992. Acoustic tomography in ,.r.ruoii surveillance. ln Reservoir Geophysics, R E' Sheriff' ed. Tulsa: Society of Exploration Geophysicists. Kennett, P, R. L. Ireson, and P J. Conn. 1980. Vertical seismic profiles: Their application in exploration geophysics. Geophys' Prosp.,28:676 99. Lapedes, D. N., ed. 19'18.McGraw-Hill Dictionary of Physics and Mathematics. New York: McGraw-Hill. Layotte, P. C. 1987. Marthor, an S-wave impulse source' In Shear-WaveExploration, S. H. Danbom and S. N. Domenico' eds., pp. 79 96. Vol. I in Geophysical Development Series' Tulsa: Society of Exploration Geophysicists. Lewis, C., T. L. Davis, and C. Vuillermoz. 1992' Threedimensional multicomponent imaging of reServoirheterogeneity, Silo Field, Wyoming. Geophysits,56:2048-56. Lines, L. 1991.Applications oftomography to borehole and reflection seismology. The Leading EdSe, l0(7')l ll 17. Lines. L. R.. M. Miller, H. Tan, R. Chambers, and S' Treitel' 1993. Integrated interpretation of borehole and crosswelldata irom a West Texas field. The Leading Edge, l2(l)z 13-16' Mason, I. M., D. J. Buchanan, and A. K. Booer. 1980. Fault location by underground seismic survey. ln Coal Geophysics' pp' 341 55, SEG Reprint Series 6. Tulsa: Society of Exploration Geophysicists. Massell, W. 1992. Emerging geophysicaltechnologies'In Reservoir Geophysics,R. E. Sheriff, ed., pp. 344'54. Tulsa: Society of Exploration GeoPhYsicists Millahn, K. O. 1980. In-seam seismics:Position and developmenr. Prakla-SeismosReport,80(2 & 3): 19 30. Mueller, M. C. 1992. Using shear waves to predict lateral variability in vertical fracture intensity. The Leading Edge, ll(2):

29-35. Noble. M. D., R. A. Lambert, H. Ahmed, and J. Lyons' 1988' Applications of three-component VSP data on the-interpretation of the Vulcan Gas Field and its impact on field development. Firsr Break.6: l3l-49. Oficer, C. B. 1958. Intodu('tion to the Theoty of Sound Transmission. New York: McGraw-Hill.

I

I i I

I

l l

\

S P E C I A L IZ E D T E C H N I Q U E S

Ratcliff, D. W., S. H. Gray, and N. D Whitmore. 1992. Seismic imaging of salt structures in the Gulf of Mexico' The Leading Edge, ll(4): 15-31. Reguiero, J. 1990. Seam waves: What are they used for'! The Leading Edge, 9(4)t 19 23; 9(8): 32-4' Rutledge, J. T. 1989. Interwell seismic surveying workshop: An oueruie*. The Leading Eclge,S(6\38 40' Schimschal, U. 1986. VSP Interpretation and Applications' Houston: SchlumbergerEducational Services' Sheriff, R. E. 1989. Geophysical Methods' Englewood Cliffs' N.J.: Prentice Hall. Sheriff, R. E. 1991. Encyclopedic Dictionary of Exploration Geophysics,3ded. Tulsa: Society of Exploration Geophysicists' Sheriff, R. E., ed., 1992. ReservoirGeophysics Tulsa: Society of Exploration GeoPhYsicists. Steinmann, V. W, and H. A. Riihmkorf. lg68 SeismischeMessunsen zur Salzstockflankenbestinmung:Eine Case History' Zeitschrift Jilr Geophysik, A; 45'l 68. Stewart, R. R. 1991. Exploration Seismic Tomogruphy Funda^"nroir, Course Notes Series,Vol. 3. Tulsa: Society of Exploration GeoPhYsicists. Tatham. R. H., and E' H. Krug 1985. VplV, interpretation ln Developmentsin Geophysics,6, A A Fitch, ed, pp 139-88' New York: Elsevier. Tatham, R. H., and M. D. McCormack. l99l Multicomponent Seismology in Petroleum Exploration Tulsa: Society of Exploration GeoPhYsicists. Tatham. R. H., and P L. Stoffa. 19'76.VplV" a potential hydrocarbon indicator. Geophysics,4l:. 837-49' Telford. w. M., L. P. Geldart, and R. E. Sheriff. 1990' Applied Geophysics.Cambridge, UK: Cambridge University Press' poWinterstein, D. F., and M. A. Meadows. l992a Shear-wave larizations and subsurface stressdirections at Lost Hills field' Geophysics,56: I 33 l-48 Winterstein, D. F., and M. A. Meadows' l992b Changes tn shear-wave polarization azimuth with depth in Cymric and Railroad Gap oil flelds. Geophysits,56z1349 64' Wong, J., N. Bregman, G. West, and P Hurley l98T Crosshole seisiiic scanning and tomography. The Leading Edse' 6(l): 36-41. Zemanek, J., E. E. Glenn, L. J. Norton, and R L' Caldwell' 1970. Formation evaluation by inspection with the borehole televiewer. Geophysics, 35: 254-69.

14

Specializedapplications

Overview Whereasthe most important application of seismic methodshas alwaysbeento hydrocarbonexploration, the areasoffastest growth are applicationsto ground_ water,environmental,and engineeringgeophyiics and to improving the economicsand effici-eicyof coal and hydrocarbon extraction. The applicationsof seismic methodsto nonexplorationactivitiesis the subject of this chapter. l4.l Engineering applications 14.L I Objectivesof engineeringwork Seismicrefraction and reflectionmethods give infor_ mation of value to civil engineers.It is neiessary to map in detail the geology,especiallyfaulting, for the engineeringof large structuressuch as tunnJis or nu_ clear power plants. Mapping the depth to bedrock is the most common engineeringapplication. Seismic studiesare often used in conlunciion with borehole results to interpolate betweenholes and reduce the number of boreholesrequiredfor an evaluation. Seismrc methodsare also usedto map voids such as cav_ ernsand abandonedcoal mines,buriedchannels, and shallow faults. Groundwater studies are often suffi_ ciently similar to engineeringwork that the discus_ s t o n si n $ 1 41. . 2a n d 1 4 .1 . - ta p p l yr o r h e ma l s o . Most engineeringinterest is only in very shallow . data; targetsare generallyshalloweithan 30 m, often only l0 to 15 m and sometimesonly 3 m deep. Occa_ sionally, engineeringinterest extends to deep data, perhapsfor tunnel constructionor nuclearuu.t. dir_ posalsites. Both P- and S-waves are used in borehole_to_ boreholestudiesand p- and S-waveboreholeloggrng is used in engineeringstudies.Measuring the S_wave velocityis especially importantbecause iiis a measure oI the shearstrengthand henceinvolvedin the ability of the earth to support structures.passivesersmic measurementsare sometimesused to detect micro_ seismiceventsthat might precedelandslidesor move_ ment along faults. The equipment and methods for engineeringgeo_ . physicalstudiesare usually simple,oftJn b..uur" .n_ grneeringpro3ectsdo not allow much money for them. Large energyis generallynot required,,o th. rou... might be a sledgehammer striking a steelplate on the ground or another small seismi. .oui". (57.2.4) (Miller et al., 1986).The energyis usuallydetecied by

505

moving-coil geophonessimilar to those describedin $7.5.1.The numberof channelsis often l2 or less.and the amplifiersand cameraare usually combined in a small metal suitcase.Marine applicationsusually in_ volve"proflling" (98.6.3). 14.1.2Refractionsurveyson land Refraction methods (chap. ll) are generally better than reflectionfor locating bedrock and the warer ta_ ble, determiningrock rippability (from seismicveloc_ ity), and identifying buried fracture or shearzonesin lgneous or metamorphic terrain in groundwaterex_ ploration. Early engineeringrefractionoften employeda timer triggered when the amplitude exceeded a preset threshold to measurethe first-break time, or a timer plus oscilloscopefor viewing the time pick. present_ day refraction is usually multichannel employing a signal-enhancement recorder ($7.6.5) that vertically stacksthe data to build up the signalstrength.The usefulrangeof a hammer blow canbe increaiedfrom 30 to 100 m with a signal-enhancement recorder. Sourcesused for shallowrefraction(\7.2.4\ include small explosivecharges,a hammer bloq weight drop, or "enhanced"weightdrop. The velocityin shallowlayersoften variesconsrder_ ably with location. The observed head wave olten comes from the base of the weatheredlayer (which may not be the same as the top of bedrock) or the water table. Reversedprofilesare generallyneededto distinguish whether data indicatq r.u..ui refractors, refractor relief,or refractor velocity changes. Fracture zonesmay lower the velocity appreciably . (sometimesfrom 5000 to 2500 n/s), bui their detec_ tion often requirescloselyspacedgeophonesbecause they are apt to be narrow.Refractorv-elocityis an in_ dicator of the difficulty to be expectedin cutting rnto bedrock(fig. 14.l); velocitieslower than about 1200 m/s usuallyindicatethat bedrockis ..trenchable"(that a ditch-digging machine can cut through it easily), lower than 2100 to 2400 mls that it is ;rippable"-(a bulldozer can cut through it without requiring blast_ ing). The absenceofa high-velocityrefractionirsually indicatesthat bedrock will not be encounteredshal_ lower than about one-third the source-geophonedis_ tance. Scott et al. (1963)describethe useof seismicmea_ surementsin the Straight Creek Highway tunnel bore

SPECIALIZEDAPPLICATIONS

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to locatefracturezones,the height ofthe tensionarch abovethe tunnel. the stablerock load, and potential weaknessfactors as an aid in designingtunnel lining and supports.They found a nearly linear correlation betweenseismicvelocity and the rate and cost of tunnel constructton. Refraction is also used with Sf/-waves. A horizontal hammer blow (fig. 13.2)or other types of Svelocityis can be used',S-wave wavesources($13.1.2) strength. material of indicator an important Although variousrefractioninterpretationmethods applications(Mooney,1977)' are usedin engineering t e t h o d s( $ 1 1 ' l . l a n d 1 1 . 3 . 1 ) t h e A B C a n d f o u r - s h om method, the simplest.gives ABC are commonest.The good depth calculationswheredepth variessmoothly and wherethe velocitycontrastis large.The four-shot method, which is only slightly more complicatedand can be carried out quite efficiently,givesbetter coverageof the refractorand is more effectivein many situations.An exampleis shown in fig. 14.2. on land 14.1.3ReflectionsurveYs are the centralissues Resolutionand cost effectiveness in engineeringreflectionsurveys.Reflectionmethods are often better than refraction methods at mapplng very shallow faults, cavitiessuch as limestonecaves or abandonedmines, and stratigraphicfeatures.It is

also sometimesusedfor bedrockand water-tablemapping, including water-table pulldown produced by pumping. Sourcesgivingdata to depthsof45 m includesledge hammers, blasting caps, buffalo guns, and rifles; sourcesfor data to 900 m include Betsy,50-caliperrifle, miniSosie,weight drops,and small explosives(see t a b l e7 . 1 ) . With 24-channel equipment and source spacing equal to the geophonegroup interval, CMP stacking yields l2-fold data. Steeplesand Miller (1990) acquired l2-fold CMP data usinga rifle fired every35 s, using a 1-m sourceinterval with a four-mancrew at a cost of $5-25lsourcepoint.Their crew consistedof an observer,shooter,jug hustler,and linesman(who lays out cablesahead of the jug hustler and picks up the cables and phones after use). Using one to three geophones/channeland l/4-ms sampling, they recorded frequenciesup to I kHz, achievingresolution o f 0 . 5t o 2 m . Shallow reflectionwork involvesa number of pitfalls. Steeplesand Miller (loc. cit.) use 200-300-Hz low-cut filters to attenuateair wave,ground roll, and shallow refractions.Thesemay spatially alias' and a walkawaynoise test with l/4-m geophonespacingis to checkthat aliasingis not occurrtng' recommended Silencersare usedwith gun sourcesto suppressthe air wave.Shallowhigh-velocitystringers(such as nearsurfacelimestone)causedifficulties.Near-surfaceelevation and velocity problems can be severefor the high frequenciesusually required to achievethe requiredresolution.

srrv()'s enginccring 14.1.4A4urint' Much of the cost of a marinesurveyinvolvesthe ship used.The small(relativeto petroleumsurvey)budgets for marine engineeringsurveys frequently preclude optimum outfitting so that they are often run with shipsof opportunity.Once the ship costsare covered' add-on surveysare relativelyinexpensiveand hencea numberof differentsensorsare often employed'These may include (fig. la.3) profilersoperatingat different energylevelsand frequenciesto give different resolutions and penetrations,side-scansonars,magnetometers,and so on. Objectivesmight be to determinesea-flooror bedrock relief. to locate sand lensesthat might support platforms, hazardsto projectedconstructionsuch as faults, mud-flow gullies,collapsestructures,gas seepage, shallow gas accumulations,sea-floor resources sich as sand or gravelthat can be dredged,areasalready dredged, shipwrecks,downed aircraft, refuse dumped at sea(for its possibleeffecton pollution and fisheiies),to inspectin-placestructuressuch as pipelines, or to get geologicdata for subseatunnel construction(Tychsenand Nielsen,1990)' Fathometersdesigned to map the sea floor operating at around 100 kHz provide virtually no penetration into the sedimentswhereas subbottom pro-

ENGINEERING APPLICATIONS

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filers record reflectionsto about 30 m with resolution of about 50 cm. Thesegenerallyoperatein the l- to l0-kHz range at speedsup to about 12 knots. The sourceis often a piezoelectrictransducer.Gas bubbles in the water from sea-floorseepsare also sometimes visibleon profiler records.Motion sensorsare usedto compensatefor the distortion becauseof the heaveof

andsolidtriangles. glesandlong-shot datawith solidsquares ( F r o mM i l s o m1. 9 8 9 : 1 7 3 . )

the ship or sourcebird; this distortion can also be compensatedfor in processingby smoothingthe seafloor reflection. Analog recordersfor single-channel systemsusually have a dynamic range of only about 20 dB, but digital recordingwithout this limitation is becomingmore common. Higher-power systems employ sparkeq air-gun, water-gun, and sleeve gun sources (see $7.4.4). Sourcesand/or streamersare often towed at depths around3.75m, so that the surfaceghostwill enhance 100Hz. Their penetrationbelowthe seafloor is about I s (1000m). The higher-powersystemsoften tow a 300- to 500-m 12- to 24-channelstreamer.They are usually digitally recordedand processedin much the sameway as conventionalseismicdata. Side-scansonargenerallyusesa towfish(fig. 8.25), or on-board equipment to project high-intensity, high-frequencyacousticbursts in fan-shapedbeams; the frequenciesare in the range 50 to 200 kHz (usually, 100kHz) and the beamsare narrow horizontally (:1") but broad vertically (:40"). Beams are projected alternatelyto the left and right of the towfish or ship. Sea-floortopographyand objectsto one side of the ship backscatterthe energy.The arrival time of the backscatteredenergy(echo)recordedat the towfish yields the distance to the backscatterer.Echos

508 scansare displayedside by side after from successive distance;this givesthe sonar imslant for correcting aseof the seafloor in real time. Horizontal rangesare uiually 75 to 300 m, dependingon the height of the towfisir abovethe sea floor. Pulse length is generally 0.1 to 2 ms and resolution is of the order of 15 cm' Tow speedsare 3 to 8 knots.

SPECIALIZEDAPPLICATIONS nating impedancecontrastsof largemagnitude'Interferenie betweencomponent reflectionsproducesthe observedseismicrefleitions.Short-pathpeg-legmultiplesare generallystrong,with consequent^lengthentng of tn. piopugatingwavelet,weakeningof the.wavelet onset,andsevereattenuationofhigh frequencies'Pegleg multiplesmay mask deeperreflections,so that only can be mapped seisth"euppei part of coal sequences 1980)' Voogd, de and (Koefoed mically

l4.2Coal geoPhYsics 14.2.1Obiectivesof coal geophysics

14.2.3Longwallmining

Before the 1970s,the coal industry made little use of seismicmethods.Relativelylargereservesof coal were known. most coalfieldshavingbeendiscoveredin outcrops with little geology employed.Geophysicalexploiation for new coalfieldsis rare. Historically'it was ielatively easy to transfer from one working face to anotherand insuranceagainstcoal depletionwas provided by opening spare working faces.Howeveq as mining became more mechanizedin the 1960s,the cost of moving the mining equipmentto a new face increasedso much that it becamemore important to anticipategeologicfaults or other featuresthat might interruptproduction.The main mining problemis to find out about disturbances(faults, washouts,abandoned workings,and so on) in time for continuity of productionto be maintained.Maintainingcontinuity of production is the main objectiveof most coal geophysics.A secondaryproblem involvesensurlngthat itt. -uin accessdrifts in a new mine are driven in directionsthat support the easiestmining' In the past,most coal explorationwasdoneby drilling boriholes (Bond, Algeq and Schmidt, 1969)' Hivever, during the last 20 yearsor so' the seismic method, although more expensive,has been used increasingly to supplement the data obtained from borehoies.The seismicmethod is able to furnish continuous profiles,which, in conjunction with borehole information, enables us to answer vital questions about the deposit,whether it is flat-lying or dipping, disturbedor not, the quantity ofcoal present,the best extractionmethod, disturbanceswe may needto provide for, and other questionscritical to the evaluation of a deposit(Gochioco,1990).

Virtually all of the coal production in the United Kingdom (and much elsewhere)is obtained from long:wall coalfaces(Mason, Buchanan, and Booer' To preparea longwall face,two paralleltunnels tSS-0). about 200 m apart are driven into the seamoff a main drift. The faceis establishedby driving a third smaller tunnel through the coal to connectthe two tunnels'A track is laid to support the face conveyor and coalshearingmachineand coal is won by horizontal millins as ihe cutter travelsbetweenthe tunnel ends' As .ulting proceeds,hydraulic props that support the tunneiroof and the track are snakedforward and the unsupportedroof collapsesbehind the machineto relieve'siressat the face. A longwall face may take 6 monthsand U.S.$10million to setup, and the investment can be recoveredover l2 to l8 months of operations provided no serious problems appeal' A fault with a throw no greaterthan the seamthicknesscan be crossed(the averagethicknessof seamsmined in the United Kingdom is a little over I m), although at the cost of output. However,a larger fault probably (1976) will causeabandonmentof the face' Clarke at production outs the economiclimit for British coal In one m' 3 than greater i faults/km' if their throw is recent year, approxrmatelyhalf of the longwall faces in Britain wereabandonedshort of their plannedlifetimes.To sustainproduction,collieriesmust maintain expensivestandbyfacesas insuranceagainstencounjumps signifiteiing large faults; mining profitability in advance' mapped are faults cantly where

I4.2.2 PropertiesoJ coal Coal generallyhas markedlylow velocity and density compared with surrounding rocks (Greenhalghand Emeison, 1986). In European Carboniferouscoals, velocity is often 1200m/s and density I . I g/cm3,compared with valuesas high as 3600m/s and2;6 glcm3 ior surroundingrocks.Thus, reflectioncoefficientsare very high. The interfacesbetweencoal and adjacent .o"kt tnuk. excellentreflectorsbut causemultiple and transmissionproblems. Coal also has the required propertiesfor a waveguide.Profitablec.oalseamsmay te lessthan 2 m thick so that individual coal members are usually not resolved.Coal is often laid down by cyclic deposition,producing a seriesof rapidly alter-

14.2.4Surfaceseismicmethods Most of the surfaceseismicsurveysfor coal havebeen in Great Britain and Germany,whereUpper Carboniferouscoals are found at 200 to 1000m (0'150 to 0.800 s) (Ziolkowski and Lerwill, 1978; Fairbairn' Holt, and Padget,1986),and in Australia' Explosives in boreholesaie the most common source'but Sosie and Vibroseisare sometimesused(Gochioco'l99l)' Chargesare usually 114to ll2 kg or 800 to 1500 whacier blows.Typically,48 groups are usedwith l0 group to 12 geophonesper group spacedover a l0-m 500 to around limited are offsets intervil. i4a*imum this disbeyond severe becoming conversion m, mod. tance.Dominant frequenciesin the coal measuresare : 100Hz; with coal velocities- 3 km/s, resolution: 7.5 m. Samplingis usually I ms (aliasfilter cuts at 250

COAL GEOPHYSICS

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to 375 Hz) and frequenciesbelow 40 Hz arenormally filteredout to reduceground-rolleffects.Staticcorrections are very important; LVL velocitiesare often as low as 300 m/s so that the wavelengthin the weathering: 1-. Improvedfault definition by surfaceseismicis illustrated by fig. 14.4.Triassicvelocitiesare around 2000 m/s, but the Permiansectionabovethe Carboniferous coal measuresconsists of very high velocity (4000 m/s) marls and limestonesthat producemultiplesthat

angular unconformity at the base of the Permian. The apparent dip in the Carbonil'erous coal measures between A and B is largely a velocity anomaly produoed by Triassic thickness changes.Faults at 0.4 s occur at the arrows marked fl

interferewith the coal reflections.Figure 14.5showsa channel(washout)cutting into a coal seam. 14.2.5In-seammethods Although surface-reflection methods can detect throws of severalmetersif the data are of high quality, locating faults by projecting surface and borehole data onto the coal seam is often unreliable.In-seam methodsallow investigationof faults within the seam

SPECIALIZEDAPPLICATIONS I

I

- Sand Channel

0.1

. "1t:f: 0t

Fig. 14.5 High-frequency Vibroseis data recorded across a s h a l l o w c o a l s e a m s h o w i n g a n e r o s i o n a ls a n d c h a n n e l .( F r o n r Chapman.Brown, and Fair, l98l: 1665.)

itsell for example,faults with throwsslightlysmaller than the seamthicknesscan be locatedusinechannel waves($I 3.3). Hasbrouch, Hadsell, and Major ( 1978) used sourcesand receiversin boreholeswithin coal seams to investigate seamcontinuityby measuringthe sersmic transmissionproperties.Geophonesare sometimes cementedin coal seamsencounteredin boreholesfor later usewith in-seammethods. Borges (1969) describesmeasurements to locate fault zonesin coal minesfrom the amplitudeof waves travelingin the coal seams.Faultswith throwsof 1.5 to 4 m producedamplitudereductionsof 50 to 70'2, for wavespassingthrough the faults, whereasthrows greaterthan 4 m resultedin 70 to 90'%reductionin amplitude.Sauland Higson(1971)carriedout an extensiveevaluationof the method.Holesweredrilled2 to 3 m into the rock aboveor below the coal for both the shots and geophorles.The shots were located along one roadwayand the geophonesalong an adjacent roadway.The shots were coupled to the rock by careful stemming(tamping) and the geophoneswere attachedto rock bolts.Their conclusionswerethat the method was soundlybasedand that faults produceattenuation;howeveqtheir resultsvaried too widely for them to establisha relation betweenamplitude and the degreeof disturbance. Mine drifts (roadways)and galleriesprovide more extensiveaccessto a coal seam than do boreholes

(Mason,Greenhalgh,and Hatherly,1985).An applic a t i o no f t h i s t e c h n i q u ei s g i v e nb y M i l l a h n ( 1 9 8 0 ) . Assumegalleriesin a coal seam,as illustratedin the map of fig. 14.6a,with geophonesplantedalong gallery A. The transmission record obtained from a sourceat B is shownin fig. 14.6b;it showsP- and Swaveheadwavesthat travelin the higher-velocityrock bounding the coal seam and channelwaveswith a trailingamplitudebuildup,the Airy phase($13.3and fig. 13.19).If a fault with throw largerthan the coalseamthicknesshad cut the seambetweenthe source and the receivers(as would be the casefrom source C). the channelwaveswould not be seen.Severalreflection recordsobtainedwith sourcesalong gallery,1 are shownin fig. 14.6c;thesecontain reflectedchannel waves,but their phasesare not sufficientlycoherentto allow them to be picked easily.Complex-tracetechare usedto obtain the amplitudeenniques(99.11.4) velope (fig. 14.6d)and stackingto make the highamplitudeAiry phasestand out as a distinct arrival (hg. 14.6e).The amplitudeenvelopeof the Airy phase is similarlydeterminedon the transmissionrecords. which yieldsthe group velocityto be usedin stacking the reflectedrecords.Figure 14.7 shows a reflection from a lault cut by another lault. Mason, Buchanan,and Booer (1980)usedsource and receiverarraysin coal seamsexposedin drifts to study channel waves propagating in the waveguide formed by the coal seamand the host rock. The wave

Gallery ,.l

S-wove

Arry Chose

+ wova

t

s

gur0e0 by 5eofr

(b)

I

ll

t

I

[,( r] I

[.^ffi il

(d)

F i g . 1 4 . 6 I n - s e a mm e r h o d s .( F r o m M i l l a h n , 1 9 g 0 . )( a ) Diagra_ I n a t r c n l a p ( n o t t o s c a l e )s h o w i n g l o c a t i o n o f g e o p h o n e s and s o u r c e s . Ba n d C . ( b ) t r a n s m i s s i o n r e c o r c lf r . o ms o u r c e . B , . ( c ) reflection records of' common-midpoint type obtainetj liom

(.) sources in gallery 1,. (d) display of envelopes of records in p a r t ( c ) ; a n d ( e ) 6 - l b l d s t a c k o f r e c o r d ss u c h a s t h o s e s h o w n i n part (c).

S P E C I A L IZ E D A P P L I C A T I O N S

512

solve mining problems (Buchanan and Jackson, 1986).Gravity has been used to detectmining subsidence. Dikes intruding into coal beds have been mappedby magneticand resistivitymethods.The low conductivity of coal makes in-seamradar a possible tool in probing for faults. 143 Groundwater, environmental' archaeologicalo and geothermal applications

I

1 0 0m

Fig.14.7 In-seam reflection survey in roadway 1064 showing reflection from a fault interrupted by a cross-fault.The surfacewave event along the roadway is called a roadv,uy mode. (From B u c h a n a n ,1 9 8 5 :1 8 - 1 9 . )

motion is very complex; an Sl1-mode (Evison or pseudo-Lovewave) as well as various P-SZ-modes (Krey or pseudo-Rayleighwaves)may be generated. Thesemodesare all highly dispersive(fig. 13.21).A surfacewave along the wall may also be generated. in a hole I m into the seam) Sources(1|4-kgexplosive and geophonesare usually placed in the center of a coal seam. Interpretation, basedmainly on the S.f1mode, permits mapping faults in a 3-m seam with throws as small as I m by observationsof wavesreflected by, as well as transmitted through, the fault planes. 14.2.6 Miscellaneousaspectso/ coal geophysics Passivegeophonearrays are sometimesused in regions of high mechanicalstressto monitor the acoustic emissionsthat precederockbursts.Seismicmethods have been used to search for lost abandoned mining drifts that might resultin subsidenceproblems or flooding of current workings if accidentallyencountered. Nonseismicgeophysicalmethods are also used to

Groundwater, environmental, and geothermal concerns have in common the fact that they involve mainly the movement of fluids (Ward, 1990). The presenceor absenceof fluids in pore spacesaffects densityand seismicvelocity,and thus the strengthsof reflectionsand refractions.Seismicmethodsoften can map a surfaceseparatingregionswherethe interstitial fluid is liquid from oneswhereit is gaseous.The techniques employed are the engineeringseismic techniquesdescribedin $14.1(Dobeckiand Romig, 1985; Lankston,1990). The water table often can be mapped by both refraction and reflection.The nature of a liquid, however,usuallydoes not markedly alter velocity or density, and so seismic methods are not good at the natureof the liquid. distinguishing Environmentalgeophysicsusually involvesthe flow (1991)citesa number of ahazardousliquid. Steeples havebeen used methods in which seismic of incidents to solveproblemswheresuchliquids are leaking' The instanceshe cites relate to mapping geologicalfeatures that control fluid flow, that is, mapping buried topography,detectingundergroundcavities,and especially locating faults. One of Steeples'examplesresulted in changingthe mapped direction of flow of a hazardousfluid by 90'; prior mapping basedon well data had given the wrong answerbecauseof aliasing with the limited number of wells.Seismicmethodsare usuallyin competitionwith drilling, but drilling may endanger the integrity of an impermeablemember, whereasseismicmethodsdo not involvethis risk. Passive seismicmethods ({i13.8)have been used for geothermal studies. Reflection work has been used in searchesfor chambers(tombs)that may contain archaeologicalartifacts and for other featureswhoseelasticproperties may differ from thoseof the surroundings.Soonawala, Holloway, and Tomsons (1990) describethe use of seismicmethodsin nuclearwastedisposal. 14.4 Hydrocarbon-reserroir applications 14.4.I Introduction Petroleumengineersneed realistic models of oil and gas fieldsin order to make development-production decisions,but historicallygeophysicistshave contributed little to this decision making except indirectly through maps and reports generallypreparedfor exploration purposes.Theseupstream data are usually not reviewedin the light of information obtainedsub-

H Y D R O C A R B O N - R E S E R V O I RA P P L I C A T I O N S sequently.However,this situation is changingrapidly, and more geophysical methods, ,u.h i, hieh_ resolution, 3-D, VSp, cross-borehole,and time_laise seismology,are being appliedto reservoirproblems.

513

picture of the geometryand stratigraphyand changes within the reservoir,and productionand pressuredata (and occasionallytracer data) give information about the reservoirconnectivity.Seismicdata usually provide the only sourceof detailedinformation about areal distributions even though seismicdata lack the Reservoirgeophysicsis sometimessubdividedinto vertical resolutionofborehole logs and cores. three areas: Porosityis determinedfrom measurements on cores 1. Reservoir delineation; the use of seismic and well logs using relationshipsthat are somewhat empirical. Becauseporosity is one of the major facmethodsto definereservoirlimits and locate tors determiningseismicvelocity and reflectivity,vefaults and other barriers to fluid flow; such locity and amplitude measurementscan sometimes information providesconstraintson the enei_ yield porosity information. Permeability generally neer'smodels of fields and, even in ..well_ correlates with porosity. Permeability may differ understood" fields,sometimesresultsin the acrossa reservoirand from zone to zone within a resdiscoveryof new prospectswithin old fields. 2. Reservoirdescription;the useof seismicmea_ ervoiq and often it variesdirectionally.A reservoirbed usuallyincludesshaleor other laminationsthat sepasurementsto help defineinternal featuresof rate it into zones that behave somewhat indepena field,suchas net and grossthicknesses (res_ dently.Flow may occur within one part of a bed while ervoirformationsoften includesomenonporouszones),porosityand porosity_thickness, bypassingother parts. Both horizontaland vertical variationsof porosity and permeabilityaffect fluid lithology,and poisson,sratio. flow. 3. Reservoir .surveillance; monitoring the A geologicalpictureis developedl-romthe core,log, changes within a reservoiras productionpro_ and seismicdata. An assessment ceeds,especially of the environment the progressofenhancedre_ of depositionis basedon thesedata,supplemented covery. by paleontologicindicatorsof depth and temperatureat SherilT(1992) describesthe application of seismic the time of depositiongivenby isotoperatio measuremethods to the solution of reservoirmanagement ments. problemsand to increasingthe profitabilityoihydro_ Initially, the recoverablereservesof a field are esticarbonfields. matedbasedon the reservoirvolume,porosity,saturation values,and recovery tactor. The hydrocarbons I 4.4.2 The nature of' hydrocarbonre.servoir.g originallyin placeare determinedfiom structureand isopach maps, porosity and hydrocarbon saturation This sectionsummarizes someof the principlesof pe_ maps, and cross-sections basedon log correlations troleumengineering. Petroleumengineersare mainly and seismicdata. Seismicmeasurements (including interested in producing hydrocarbons at a profii. hydrocarbonindicators;see910.8)play a major role More specifically,they want information about the in defining reservoirvolume and how reservoirsare pore spacesin a rock (porosity),how they are inter_ broken up into separatepools.Reserveestimatesare connected(permeability), the natureof the fluids fill_ usually modified considerablyafter the production ing the pore spaces(fluid saturations),the energy/ historyprovidespressure-decline values. pressurethat may causethe fluids to flow (r/riyc),the Hydrocarbonsare pushednaturally from reservorr vertical and areal distribution of pore-connected pore spacesby water or gas.The gasfor gastlrive may spaces,and barriersto fluid flow (sealingfaults,strati_ comefrom the expansionofa gascap as the pressure graphic barriers,and so on). Thesehave to be deter_ declinesor from gasthat comesout of solutionas the mined lrom the data available,which probablv con_ pressuredeclines.Natural waterttriverequiresconnecs i s to f tion with an aquifersurroundinga hydrocarbonaccu(a) surfaceseismic,gravity,and other geophysi_ mulationthat is at least 100times the volumeof the reservoir.The amount of recoverablehydrocarbons cal and geologicaldata, (b) VSP and other boreholeseismicmeasuredependson a numberof factors,includingthe interstiments, occasionally borehole_to_borehole tial porosity,pore sizes,shapes,and interconnectivity, saturationsof oil, gas, and water, interfacial surf-ace measurements, ( c ) boreholelogsofvarious types, tensionsand wetability,and fracture spacing,width, (d) corestaken in boreholes. and orientation. (e) analyses of fluids recovered in drill-stem Primary re('overyusesthe natural energypresentin a reservoir to drive the fluids to the producing boretests, holes.After primary production declines,the natural (f) production and pressuredata, l o \ drilling-rate logs, mud logs, energy is often supplementedby gas or water injecand other simition; this is called secondaryrecovery.More exotic lar data. means of stimulatingproduction involve injection of Well logs,well-to-welllog correlations,seismicdata. misciblefluids, surfactants,or other chemicals,injecand geologicalbackgroundknowledgegive an overall tion of steam,and settingfire to the hydrocarbonsin

514 a reservoir.Thesemethods are called enhancedoil recover,v(EOR). EOR floods tend to flow through the more permeable beds or channelsand thus may bypassportlons of a reservoir.Seismicmethodsmay help monitor the changesin a reservoir that result from productton wheregasis involved(for example,as a result of a fire flood) or where the seismicvelocity is changed sigas a result of the nificantly(for example,decreased temperatureincreaseresultingfrom steaminjection). Reflectionsfrom gas-oil, gas water,and/or oil water contacts can sometimesbe mapped. Obviously' we needto know how fluids actuallyflow through a reservoir in order to optimize (and, often, to make economical)EOR projects. Reservoircontinuity is mainly establishedfrom pressurehistoriesand from observingthe pressure changesin one well when conditionsin anotherwell change.When productionin a well is stopped.the early buildup of pressuremainly dependson the immediatevicinity of the well, the influenceof more recurve motepartsof the reservoiraffectingthe pressure (chap. tomocross-hole l2) and later. 3-D seismic what help in understanding graphicmethods($13.5.4) goeson betweenwells. and permeabilityanisotropymay Inhomogeneities of causechanneling water or gas,bypassingpockets (and leavingthem as unproducible). of hydrocarbons Bypassedregionssometimesmay be invadedlater if sufficientfluid is injected.or bypassedoil can be recoveredby drilling additionalintermediatewells. Multiphasesituationsresultin complexsequences May (1984) of conditionsas productionprogresses. givesa scenariofor the very simplecaseof a uniform poroussandcontainingonly oil with dissolvedgas: smallamountsof gasevolved is reduced. As the pressure thcrestsin theporesandincrcase accumulate fromsolution mobilc. tanceto flowof oil withoutthegasitselfbccoming eventuincreases. thegasbubbles but,astheconcentration phase, andfiom thcnon allyjoin up to lbrm a continuous both fluidsmovethroughthe fbrmation.. . . [T]hepermeandtheperabilityof theporousformationto gasincreases. producHence.with continued to oil decreases. meability gas/oilratiotendsto rise,thistendency tion.theproduced volof thcliquid-phasc bothby shrinkagc beingaccentuated with lossof disin its viscosity umeand by the increase gas. solved The distribution of porosity. permeability,and otherpropertiesoverthe areaof a reservoirare incorporated in a reservoirsimulationmodel.The models are updatedas additionalinformationbecomesavailable.The validity of a model is judged by how closely actualresults(fluid-flowrates,bottom-holepressures, and so on) match predictionsbasedon the model. A generalrule of thumb is that a model can be used to predict reasonablywell for about the samelength of time as it has matchedpast history.Scenariosof production programsand strategies(for example'varylng the number of wellsand their locations,preferentially producingdifferentwells,injectingwater or gasto en-

S P E CI A L I Z E D A P P L I C A T I O N S hance the drive, and so on) are run on the model to determinehow to optimize the economicreturn' Obviously,very realisticmaps and data are required to achievethis end result. delineation I 4.1.3 Reservoir Brown and Gilbert ( 1992)statethat accuratedetermination of the physicalboundariesand internal segmentation of a reservoir have significant impact on and productionoperationsthroughouta development field'slile. Historically,delineationand infill drilling and reservoirmodeling have been driven mainly by well data and a geologicmodel with relativelylittle geophysical input except for prediscovery seismic
R E F E R E NC E S

515

and changes in these other lactors often are not known.Geostatistical methods({13.10)areoften used to changethe proportionality factorsto yield porosity or porosity-thickness maps. Sheriff (1992)contains eight casehistoriesthat employ differentgeostatistical methodsto determinenet pay and lithology. Intrinsicambiguityis one of the major problemsin reservoir description (de Buyl and Hardage, 1992): any measurementcan be aflectedby severalunknown factors.Hence,well data or someother indeoendent informationsourcemust be calledou to narrow the field of possibleexplanations(de Buyl, 1989).Values of physicalpropertiesare determinedat wellsand often are assumedto vary slowlyand smoothlybetween wells,that is, valuesare determinedby interpolation and extrapolation. Bettervaluescan often be obtained using seismicdata, ordinarily the only information availableexceptat the well locations. High-qualityseismicdata are neededfor reservoir description.Most often 3-D data that havebeenacquiredand processed to preserveamplitudeinlormat i o n a r e u s e d .O c c a s i o n a l l yc.r o s s - h o l (ei s 1 3 . 5 . 4o)r other data are used.

0 .{

Fig. l4 8

l0

60

Engineering

90 Dl.trnc.(n)

120

t50

lto C

refraction profile.

mentsinvolvingCO. flooding,steaminjection(Matthews, 1992),and in-situ combustion(Greavesand Fulp, 1987)havebeensuccessfully monitored(seefig. 13.38).lt appearsthat other typesof EOR could also be monitored.The greatestpotential,however,is ficr monitoringwaterfloods,the mostcommonsecondary recoverymechanism,and this may becomea major seismicapplication. Problems

I 4.4.5 Rescrtoirsurveillunce The objectiveof reservoirsurveillanceis monitoring the movementof fluidswithin a reservoir. This usually meansrepeatingmeasurements at differenttimes to seehow the measurements havechanged;this is the time-lapseprocedure(\13.6).AlthoLrghany type of measurement can be usedin time-lapse rnode.usually 3-D or cross-hole data are used(Justice.1992).lnterpretationoflen involvessubtractingone surveyfrom a n o t h e rt o b r i n g o u t t h e c h a n g e sA. s s t a t e di n \ 1 3 . 6 , the two surveysmust be conductedunderas nearlyas possibleidenticalconditionsso that observedchanges can be attributedto the fluid movement.Moreover. the changesbeingmeasuredmust be largeenoughto standout from the backgroundnoise. The quantitiesmostoftenmeasuredareseismrc amplitudeor velocity.Gas.oil, and waterfilling a rock's pore spaceschangethe velocityand density(\5.2.7) and thereforethe contrastwith adjacentformations and reflectivity.Sometimes,changes in temperature are involved(because injectedwateris cold or becausea reservoiris heatedto lower the oil viscosity) and this alsomay changethe seismicvelocity(95.2.6). Few time-lapsesurveyshave been carriedout becauseof the largecostsinvolved.SomeEOR experi-

X

19ms 29 39 50

75m 90 los t20

59ms 62 65 68

1 3 5m 150 165 180

References A b r i e l , W L . , P S . N e a l e .J . S . T i s s u e .a n d R . M . W r i g h t . 1 9 9 1 . M o d e r n t e c h n o l o g yi n a n o l d a r e a : B a y M a r c h a n d r e v i s i t e d . Lcuding Edga, 10(6):21 35, B o n d . L . O . . R , P A l g e r . a n d A . W S c h m i d t . 1 9 6 9 .W e t l l o g a p plications in coal mining and rock mechanics.ln Coul Geophl'si r ' , i ,D . J . B u c h a n a n a n d L . J . J a c k s o n .e d s . , p p . 2 8 3 5 , S E G Geophysics Reprint Series 6. Tulsa: Society of Exploration G e o p h y s i c i s t s1. 9 8 6 .

Table 14.I DataJrom a dam-site.surve1,

15m 30 45 60

l4.l To find the depthto bedrockin a damsitesurvey, l 2 g e o p h o n ews e r el a i d o u t a t i n t e r v a l o s f l5 m along a straightline away from the source,offsetsranging from l5 to 180m. Determinethe depthof overburden from the data in table l4.l assuminga singlelayer above the refractor.By how much does the depth differ if we assumetwo layersabovethe refractor? 14.2 The time distanceobservations in fig. 14.9constitutean engineering refractionproblem. (a) Solve for the first layer for both pairs of reversed profilesand show that the layer has a thicknessof about 2.9 and 3.8 m at sourcesA and C (dip about 0.3"). (b) Apply eq.(4.41)to get approximatethicknesses of the secondlayer. (c) What is the dip of the deeperinterface? (d) Why are the answersin parts (b) and (c) approximate? 14.3 Fine the depthsand velocitiesof refractorsin fig.14.2.

7 2m s 16 78 83

B o r g e s .E . 1 9 6 9 .E i n n e u e ss e i s m i s c h eVse r f a h r e ns a n o r t e n v o n Verwurfen und Auswaschungenin Floz. Gluckuu/ Forstltli.,4: 201 8. Bouvier. J. D.. C. H. Kaars-Sijpesteijn, D. Fl Kluesner. C. C. O n v e i e k w e .a n d R . C . v a n d e r P a l . 1 9 8 9 .T h r e e - d i m e n s i o n a l

516 seismic interpretation and fault sealing investigations, Nun River Field. Nigeria. Bull. AAPG,73:1397 414. Brown, A. R., and O. E. Gilbert. 1992. Reservoir delineation: Characterizing the trap. In Reserwtir Geophysits, R. E. Sheriff, ed., pp. 7l 2. Tulsa: Society of Exploration Geophysicists. Buchanan, D. J. 1985. In-seam seismology: A method for detecting faults in coal seams. ln Developmentsin Geophy'.sical E.tploration Metfutds, 5, A. A. Fitch, ed. London: Applied Science Publishers. Buchanan, D. J., and L. J. Jackson. 1986. Coal Geopi_yslc.r, SEG Geophysics Reprint Series 6. Tulsa: Society of Exploration Geophysicists. C h a p m a n , W . L . , G . L . B r o w n , a n d D . W . F a i r . 1 9 8 1 .T h e V i broseis system: A high-frequency tool. Geophvsit.s, 46: 1657 66. C l a r k e , A . M . 1 9 7 6 . W h y m o d e r n e x p l o r a r i o nh a s l i t t l e t o d o with geology and much more to do with mining. Collierl, Guurdian Annu. Rev.,224: 323 36. Also in Cou! Geophysits, D. J. Buchanan and L. J. Jackson, eds., pp. l0 24, SEG Geophysics Reprint Series 6. Tulsa: Society of Rxploration Geophysicists.

SPECIALIZED APPLICATIONS

36. Also in Coal Geophysic.r,D. J. Buchanan and L. J. Jackson, eds.,pp. 341-55, SEG GeophysicsReprint Series6. Tulsa: Socie t y o f E x p l o r a t i o nG e o p h y s i c i s t s . Mason, J. M., S. A. Greenhalgh, and P Hatherly. 1985. Underground seismicmapping of coal seam discontinuities at West Wallsend No. 2 Colliery. Explor. Geophy.,16:357 64. Matthews, L. 1992. 3-D seismicmonitoring of an in-situ thermal process: Athabasca, Canada. In Reservoir Geophysics, R. E. Sheriff, ed., pp. 301 8. Tulsa: Society of Exploration Geophysicists. May, C. J. 1984. Reservoir engineering. ln Modern Petroleum Tethnokryy,5thed., G. D. Hobson, ed., New York: John Wiley. Millahn, K. O. 1980. In-seam seismics:Position and development. Prakla-Seismo,tReport,80(2): l9 30. Miller, R. D., S. E. Pullen, J. S. Waldner, and F. P. Haeni. 1986. Field comparison of shallow seismic sources. Geoph,-sit..s, 5l: 2067 92. Milsom, J. 1989. Field Geophysits.New York: John Wiley. Mooney, H. M. 1977. Hundbook oJ' Engineering Geophy5i:'s. M i n n e a p o l i s :B i s o n I n s t r u m e n t s .

de Buyl, M. 1989.Optimum field development with seismrcreflection data. The Leading Edge,8(4\: l4 20.

Nolen-Hoeksema, R. C. 1990. The future role of geophysicsin reservoir enginecring. Thc Leuding Edge,9(12)t 89 97.

de Buyl, M., and B. A. Hardage. 1992.Defining reservoir properties. In ReservoirGeophl,sits,R. E. SherilT,ed., pp. 185 8. Tulsa: Society of Exploration Geophysicists.

S a u l .T . , a n d G . R . H i g s o n . l 9 7 l . T h e d e t e c t i o no f f a u l t s i n c o a l panefs by a seismic transmission method. Internat. J. Rotk Mcclt.Min. Str., 8:483 99.

D o b e c k i , T . L . , a n d P . R . R o m i g . I 9 8 5 . G e o t e c h n r c a la n d groundwater geophysics.Gatphysics,50: 2621 38.

S c o t t . J . H . , F . T . L e e . R . D . C a r r o l l . a n d C . S . R o b i n s o n .1 9 6 8 . l e a s u r e m e n t st o e n g i n e e r i n g T h c r c l a t i o n s h i po f ' g e o p h y s i c am and construction parametersin the Straight Creek Tunnel pilot boring, Colorado. lntcrnut. J. Rock Meclt. Min. Sti.,5: I 10.

F a i r b a i r n ,C . M . , J . M . H o l t , a n d N . J . P a d g c t . 1 9 8 6 .C a s c h i s t o r i e so f t h e u s c o f t h e s u r f a c es e i s m i cm e t h o d i n t h c U . K . c o a l mining industry. ln Coul Geophr'.slr'.r, D. J. Buchanan and L. J. J a c k s o n ,e d s . ,p p . 1 8 8 2 0 3 . S E G G e o p h y s i c sR e p r i n t S e r i e s6 . Tulsa: Society of Exploration Geophysicists. G o c h i o c o , L . M . 1 9 9 0 .S e i s m i cs u r v e y sf o r c o a l e x p l o r a t i o na n d mine planning. T'heLeuding Edge,9(4):25 8. G o c h i o c o , L . M . 1 9 9 1 .A d v a n c e si n s e i s m i cr e f l c c t i o np r o f i l i n g lor US coal exploration. The Lauding Edgc, 10(12\:24 9 G r e a v e s ,R . J . , a n d T . J . F ' u l p . 1 9 8 7 .T h r e e - d i m e n s i o n asl e i s m i c monitoring of an enhanced oil recovery process. Gcoph.y,sit.t, 52: ll75 87. G r e e n h a l g h ,S . A . , a n d D . W E m e r s o n .1 9 8 6 .E l a s t i cp r o p e r t i e s of'coal measurerocks lrom the Sydney Basin, New South Wales. 6,rplor Geophys., 17:. l5l 63. H a s b r o u c h ,W . P , F . A . H a d s e l l ,a n d M . W M a j o r . 1 9 7 8 .I n s t r u mentation lor a coal seismic systenr.Paper read at 48th SEG A n n u a l M e e t i n g .S a n F r a n c i s c o . Justice. J. H. 1992. Geophysical methods lbr reservoir surveillance. In ReservoirGcophysits,R. E. Sherifl, ed., pp. 281 4. Tulsa: Society of Exploration Geophysicists. Koefoed, O., and N. de Voogd. 1980. The linear properties ol thin layers, with an application to synthetic seismogramsover coal seams.Geophy,;k's,45: 1254 68. Also in Coul Geoph;'sits, D . J . B u c h a n a na n d L . J . J a c k s o n ,e d s . ,p p . I l 0 2 4 , S E G G e o physics Reprint Series 6. Tulsa: Society of Exploration Geophysicists. Lankston, R. W 1990. High-resolution refraction seismicdata acquisition and interpretation. ln Geotethnitul und Ent,ironnu'ntul Geophy-slr'.r, S. H. Ward, ed., Vol. l, pp. 45 73. M a s o n , I . M . , D . J . B u c h a n a n ,a n d A . K . B o o e r . 1 9 8 0 .F a u l t 1 o cation by underground seismicsurvey.Proc. IEEE, Fl27:322

Sherifl. R. 8.. ed. 1992. Rcscrwir Geoph.v.tit.s. Tulsa: Society of E x p l o r a t i o nG e o p h y s i c i s t s . S o o n a w a l a ,N . M . . A . L . H o l l o w a y ,a n d D . K . T o m s o n s . 1 9 9 0 . Geophysical mcthodology fbr the Canadian nuclear flel waste management program. ln Geotuhnitul und EnvirorunentalGeoS. H. Ward, ed., Vol. l, pp. 309 31. Tulsa: Society of 7l/i,1,.irr'.r, E x p l o r a t i o nG e o p h y s i c i s t s . S t e e p l e sD , . 1 9 9 1 . U s e s a n d t e c h n i q u e so f e n v i r o n m e n t a lg e o physics. Gcopfiysits, tha Lading Edge ol E-rplorurion, lO(9):

3 03 1 . S t c e p l c sD , . W , a n d R . D . M i l l e r . 1 9 9 0 .S e i s m i cr e f l e c t i o nm e t h ods applied to engineering, environmental, and groundwater problems. ln Geotechnirul untl Envintnmental Geophl'sics,S. H. W a r d , e d . , V o l . I , p p . I 3 0 . T u l s a :S o c i e t yo f E x p l o r a t i o nG e o physicists. Trabant, P K. 1984.Applied high-resolution geophysicalmethods. Boston: International Human Resources Develonment Corp. Tychsen,J., and T. Nielsen. 1990.Seismic reflection used in the sea environment. ln Geotethnital und EnvironnrentalGeophysir'.r,S. H. Ward, ed., Vol. l, pp. 3l 44. Tulsa: Society of Exploration Geophysicists. Ward, S. H., ed. 1990. Geotechniculand EnvironmentalGeophl,sir'.rl 3 vols.. Investigations in Geophysics 5. Tulsa: Society of Exploration Geophysicists. Z i o l k o w s k i , A . , a n d W . E . L e r w i l l . 1 9 7 8 .A s i m p l e a p p r o a c ht o high resolution seismic profiling for coal. Geoph. Prosp., 27: 360 93. Also in Coal Gcophysics,D. J. Buchanan and L. J. Jackson, eds., pp. 154 87, SEG Geophysics Reprint Series 6. Tulsa: Society of Exploration Geophysicists,1986.

l5

Background mathematics

Overview This chapterservesas an appendixto this book rather than a portion of the main text. It can be omitted by those already familiar with mathematicsor by those who wish to take the mathematicson faith. More extensivetreatmentscan be found in Wylie ( I 966),Pipes and Harvill (1970),Robinson(1967a,1967b,1967c), Cassandet al. (1971),Robinsonand Treitel (1973, 1980),Kanasewich(1973), Bath (1974), Kulhrlnek (1976),Claerbout(1976),Silviaand Robinson(1979), Potterand Goldberg(1987). We begin with short summariesof determinants, vector analysis,matrix analysis,infinite series,complex numbers, the methods of least squares,finite differences,numerical solution of differential equations,and partial fractions(gl5.l). Most of the chapter involves the mathematics of data processing, especially Fourier (S15.2), Laplace (415.3), and :-transforms(415.5),and almost all deal with linear systems($l5.4).The cepstrumis discussed (Q15.6) and a final sectiondealswith filtering.ln contrastto chap. 9, which dealtalmostentirelywith digitaldata,much of this chapterdealswith continuousfunctions,although someportionselaborateon digital considerations,especially 8l 5.5. l5.l Summaries of basic concepts I5.l.l Determinants A determinant,det (a),is a squarearrayof n x n numbers, au, called elemenls;I and 7 designatingthe row and column,respectively: .., a,

. . u '. : ... u,,,,

det(a) =

(r5.r)

The minor M,, of elementa,,is the determinantof order(n - l) foimed by deletingthe ith row and theTth column of det (a).The product (- l)'ti l,t is Ihe cot''actor of a,,.The value of a determinantis definedas det(a):

\{- |),*io,,M,

Tt-

l)i*ia,,Mf ]

det(a) :

t 2 3 4 5 6

-l)r't t

8 9 0 * (-l;t*:2

4 6

5 6 9 0 4 5

+ ( - l ) ' * r3

8 0 8 9 : l(5 x 0 - 6 x 9) - 2(4x 0 - 6 x 8) +3(4x9-5x8):30. A determinantis thus a singlenumber. E q u a t i o n s( 1 5 . 1 )a n d ( 1 5 . 2 )c a n b e u s e dt o d e r i v e the following rules (seePotter and Goldberg, 1987: 2 2 6 7 : W y l i e ,1 9 6 6 : 4 0 3l 0 f o r p r o o f s ) : L If all the elementsof one row are zero,or if the elementsof one row are proportionalto the correspondingelementsof anotherrow, then the determinantequalszero; 2. Multiplying all the elementsof a row by a constantmultipliesthe determinantby the sameconstant: 3. Interchanging any two rowschangesthe sign of the determinant; 4. Interchangingrows with columnsdoes not changethe value; 5. Equation(15.2)givesthe samevalueregardlessof the row selected; 6. Any row can be multipliedby a constantand addedto anotherrow without changingthe v a l u eo l t h e d e t e r m i n a n t : 7. "Row" can be replacedwith "column" in any of the precedingrules. Determinantscan be usedto solvea systemof linear equations: a t r x r * d r ) . Y .+ " ' - d t , . \ , = b , . I

1.'1'. . .'.'ll. . .].":".': : 3: l(r5.3a)

ontxt *

en:xt *

"'

*

u,,.r,:

,,,

)

,,,,, =

where the summation is taken along one row (row expansion;hence,i : constant)or along one column (column expansion,/ : constant); the result is the samein both cases(seerule 5 below).As an example, we expand by the first row:

Cramer'srule statesthat .x, : [det(a),)]/[det(a)],

(15.3b)

518

BACKGROUND MATHEMATICS

where w1]l q12

'..

q1n

421

421

...

Q2n

a,t

a,z

.,,

A,,

det(a) :

and det(a,)is det(a)with the rth column replacedby b , , b r , . . . , b " ( P i p e sa n d H a r v i l l , 1 9 7 0 :1 0 1 ,P o t t e r and Goldberg, 1987:232-3). W h e na l l t h e c o n s t a n tbs , , b r , . . . , b , i n e q . ( 1 5 . 3 a ) are zero, the set of equationsis said Lo be homogeneous.Obviously,one solution (the trivial solution)is when all the x, are zero. Nontrivial solutions exist when det (a) : 0 (seeproblem 15.1a). 15.1.2Vectoranalysis (a) Basic definitions. A scalar quantity, such as temperature,has magnitudeonly, whereasa vectorquantity, such as force, has both magnitudeand direction. Vectors(representedby boldface type) can be added to givea resultant,as shownin fig. 15.1a.Subtraction is equivalentto reversingone of the vectorsand then adding.Multiplication of a vector by a scalarchanges the magnitudeof the vector but not its direction (except that multiplication by negativenumbersreverses the direction). A vectorcan be resolvedinto componentsalongcoordinateaxes(fig. 15.lb) and expressed as a vector sum of its components: A : a,i * a,] * a-k,

(15.4)

wherei, j, and k are unit vectorsalong the x-, y-, and (Vectorscan also be expressedin z-axes,respectively. other coordinate systems such as cylindrical or spherical.) Vectors can be added by adding corresponding components.Thus, if

(1 5 . 5 )

If A has direction cosines((, m, n), then

(15.6)

(b) Vectorproducts. The scalaror dot producl oftwo vectors,A and B, is written A . B; it is a scalar:

A.B:lAllBlcos0,

A .B : B .A , :0, : lAl lBl,

whenAIB, when A ll B.

Also.

i . i : i . i : k . k : 1 ,

i . j : j . k = k . i : 0 ;

hence, A.B:

B.A:

a , b , - ta , . b , .a*- b _ ( 1 5 . 8 )

and

The vector,or crosr-,productof A and B, A X B, is a vector definedby the relation

The magnitudeof a vector,written lAl, is given by

A:lAl({i+mi+nk).

massthat suffersdisplacementB, A ' B givesthe work done on the mass.Obviously,

A : : A . A : a 2 , * a 2 ,l a | .

A : a,i + a,j + a-k, etc., _ A + 28 3C : (a,+ 2b,_ 3c,)i | (a, * 2b, - 3c,.)i + (a, + 2b. - 3c")k.

lAl : (al * al,-t al)1t2.

F i g . l 5 . l O p e r a t i o n s o n v e c t o r s .( a ) A d d i t i o n a n d s u b t r a c t i o n , (b) resolution into components, (c) cross-product, (d) scalar function of position, and (e) curvilenear coordinates.

(1 5 . 7 )

where 0 is the smallerangle betweenthe vectors(fig. 15.1c).Thus,the dot productequalsthe magnitudeof one of the vectorstimes the projection of the second vector onto the first. If A is a force actine on a ooint

A x B : ( l A l l B ls i n 0 ) q ,

(15.9)

where 1 is a unit vector perpendicularto the plane containingA and B and in the direction ofadvanceof a right-handedscrewrotated lrom A to B through the angle 0 (fig. l5.lc). Becausethe direction of q dependson the sequence, A x B :-B x A . : 0, : lAl lBh,

when A ll B, when A A B.

The magnitudeof A X B equalsthe area of the parallelogramdefinedby A and B. The torque about an arrs of rotation is given by a vector product (seeproblern

SUMMARIES OF BASIC CONCEPTS 15.3).Applying eq. (15.9)to the unit vectorsi, j, k gives ixj:k, ixt<:i, i x i : i x j : k x k : 0 .

rs.4).

A X B :

a,

i

k (ls.l0)

at. a-

b. b, h, Productsof more than two vectorscan be formed in variousways(seeproblems15.5). (c) Vectoroperators. Let rf(,r, y, z) be a scalar function of position (for example,temperature).If the valueis r! at P(-t,y, z)in fig. 15.Id,then the valueat a nearby point Q is rf + drf, where d{i =

tUd.r*oUd, *oUddy'

d-x

V . l r ,V . V x A , V x V x A , V ( V ' A ) , a n d s o o n ( s e e p r o b l e m1 5 . 7 ) .

kri:i;

A x B can be expressedas a determinant (problem

i

519

0:

( i * , - 1 *- i * 1 * x ) r i a .+ j d l * k d : ) ay rt: I \d.r

(d) Orthogonal curvilinear coordinqtes. Although Cartesian coordinates are usually the most convenient, at timescylindrical,spherical,or other orthogonal curvilinearcoordinateslead to simplerresults.We the surfacesu, : write u,, u, u.for suchcoordinates, : : : (c, constant)being orthogonal. cr, u, c2,u1 cr When a, changesby du,, the elementof length, ds',is given by ds' : (ft, du,)2I (h. durf + (ft. du.)r, where/r, : h,@,,u.,u.) is a variablescalefactor. To express rf in curvilinear coordinates,we note hence. to drfi(ft,da,); that drf/dxcorresponds

: v+.l, ,t ;:y,",

whereu, are unit vectorsalong the curvilinearaxes.To obtain V . A, we useGauss'theorem(Pipesand Harv i l l , 1 9 7 0 :9 0 9 ; P o t t e ra n d G o l d b e r g ,1 9 8 7 : 3 5 57 ) , which statesthat

=J) . J J J^"d v A d . / , tff

Vrl .dr; thus.

(1s.il)

drf/dr:V,1,'r,,

where r, is a unit vector along dr. Vector Vrf (pronounced "del {") is the gradientof rf (grad rf); it is a vector in the direction of the maximum rate of increaseof rf and its magnitudeequalsthis maximum rate of increase(seeproblem 15.6b).The rate of increaseof rf in an arbitrary direction p, where p is a unit vector,is givenby VQ ' p (seeproblem15.6c). The vector operator,del, Y : i(A/A,r)+ j(a/Ay) + k(alaz),is often usedas if it werea vector.Thus, we can take the dot productof V and a vectorA, called the divergenc'e of A, or div A:

divA:v.A:

^!,*o:, .i-t

dy

*o.u (15.12) d:

We can also takethe vectorproduct of V and A. called the curl:

i a a i

curlA:VxA:

6x d v 6z a,

a,

(15.13)

: Laplacian of \t.

[r:r,

where surface .9/ encloses volume V', and the outward-drawnnormal is positive.Applying this to an elementof volumedV"(fi9. l5.le), we get ': V ' A dV surfaceintegralover the six faces, -- -AJ4du, h.du.) + Alh2 du, h. dur)

*

a''] du'dr'" loou"o'0"'t

* similarexpressions for the otherpairsof faces. Thus, ( A , h . h .+t v . A ( r r d u , h . d u , h . d u: , , oau_tA.h,h,1 [ri, + ^d (A,h,h.lfdu,durdu,. dut

v.A : , ], f "' v,hhk), (ls.l6) hrh.h,-Au, i, j, k being in cyclic order. The Laplacian,V'U, is equal to div grad rf, that is,

az

on usingeq. (15.10).OperatorV can be appliedmore than once.For example, div grad {, : V' V{, : O'rll :

ff

J

k d

rl5 I5\

. #. #]r (ls.l4)

In the sameway, we can form products such as V x

v'q,: ' ,+>+(Yri+\ (r517) h,h-h,'Au,\h, au,l 15.1.3Matrix analysis (a) Definitions. A matrix is a rectangular array of numbersa,, arrangedin r rows and s columns;an entire matrix is here indicatedby script type:

520

BACKGROUND MATHEMATICS

,:11,,' :;,,,ll llo"o" o'' ll

(ls.l8)

The orderof a matrix is r X s. If r : I, we havea row matrix; if s : I , a columnmatrix; theseare also called vectorsof thefirst and secondkinds, respectively.A null matrix t has zerosfor all elements.The transposeof a matrix has rows and columns interchanged;thus,

dtt d:t '.' t r

-

i l r- l o4o- 2, tI l l

ll ll ..r o l l :ll" ./ ' : l l t 2| -' oll ll' ll,

ll

at: a:: ar: ...Il

at,d:,r,,

.4 ism x n and thatof ,t9 isnx p, theorder of 6 is m x p.When more than two matricesare multiplied, products can be formed in pairs; thus, . I '/9't : (, A.8)6 -- .'Z (.-196). The transposeof a product is the product of the transposesof the individual ma: ./tr,'l'r tricesin inverseorder,that is, (,'l',8)' (seeproblem 15.10b). It is sometimesconvenientto partition a matrix, that is, to representit as a matrix whoseelementsare submatricesof the original matrix. For example,

(l5l9t

: : , lll l

A matrix of order r X r is a squarematrix. The printipal diagonalof a squarematrix has the elementsa,,.A diagonalmatrix is a squarematrix with zeros for all elementsthat are not on the principaldiagonal,that is, a,, -- 0 if i + j, and at least one of the a,, t 0. An identitymatrix .V is a diagonalmatrix wherea,,: I for all i. A matrix with zerosbelow (above)the principal diagonal is called an upper (lower) triangularmatrix. A ,symmetric matix equalsits transpose,that is, , I : , lr, and a skeu,-symmetit'matrix equalsthe negative of its transpose,. I : -. I r. A symmetric matrix where all the elementsalong any diagonal parallel to the principal diagonal are the same is a Toeplitz matrix. The colactorof an elementa,. of a squarematrix is (- lft' timesthe determinantformed by deletingthe rth row and the ,rth column. The adjoint of a square matrix, adj(. I ), is the transposeof the matrix . I with eachelementreplacedby its cofactor.The determinant of a ,square matrix, det(. I ), is a singlenumber given by det(. I ) = L, a*A r : 2r a,rA, wherelu is the cofactor of a,*.The inverseof a squarematrix can be found by dividing the adjoint by the determinant [if de(. I ) + 0], that is, . 'l | : [/det(, I )ladj( /),. /-t. I : ,V (15.20) (seeproblem 15.2b). (b) Matrix operations. Operations performed on matrices change the values of the matrix elements. Corresponding elementscan be added,that is, if { : , 1, + ,.8 , c,, : a,, * b,.;matricesmust be of the same order to be added.Matrices can be multiplied by scalars,that is,if U : k, 1',d,,: ka,,.In matrixmultiplication, the ith row of the first matrix is multiplied element by element by the jth column of the second matrix and the products are summedto give the yth elementof the product matrix, that is, if 5 : . l./9 , e,,:Lre,rbo,. The first matrix must havethe samenumber of columnsas the secondmatrix has rows for matricesto be multiplied. ln 5 : . l.B , if the order of

ll'ololl

where

;iil ": l i-2' o' l,lll o : -'*ll' 0Oil

il,

.n : ll0 o r l l , . / : l l I 0 l l To add , 'l to a similar 4 x 5 matrix .'4, 'A must be partitioned in the sameway,that is,

ull /) : l l . r v/ llv ll' where.Vis3 x 3. lt is 3 x 2, and so on. Then

l l. r * . 7 Q - / / l lr l ./ +., =ll rn -V .'t+7t'll When partitioning matricesthat are to be multiplied, the submatricesmust be comformable.Thus, if '6 :. /.'8. w h e r e . I i s m x n , , . t i i s n x p , w e c a np a r t i t i o n . I and.tl as follows:

, : ,l lt l " = l l ; 1 , 1

' f b e i n g a x bU; , a \ c ' ; 5 , d x b ; . 7 , d x t : ;! ( , 6 x i ; , i l . b x k ; 7 , r ' x j , . V / , c x k ; a + d : m ,b *c:n,j+k:p,then

' ' : l ll lrr ': tt ++.vu 77

't.7i + q.vlll ;t .vl + .v.v/ ll

Matrices can be used to solve simultaneousequations. lf we write a set of linear equationsas atit

+ er.x, I aBx3+ "' :

a"rx, * errx, I ar.x. I

-t "'

-

at:xz t

41.x3

antXr t

anzxz t

Q,txz t

attxt

"' : :

(15.211

SUMMARIES OF BASIC CONCEPTS

521

and let , 4 be the elementsa,,,JL- be a column matrix with elementsx,r : x,, '6 be a column matrix with elementsc,.then we can write

0",(0) 0",(+l) +"0G,!) 0

.4,X:' 6 and solve for ,I':

0 . r ' t .l . ! , ' : , y ' - t { , .r' : .4-t6 b e c a u s e, l , t , i

I (15.22) : .v.l ) This solution requiresthat, 4 be a squarematrix and that the equationsare independent,that is, det(.'t) * 0 (seeproblem 15.1b). Convolution, e, * b, : c,, can be performedby the : '6 if ,4 is a matrix of the form operation,4,8 indicated in eq. (15.23) and ./9 and 5 are column matrices.For simplicity,we assumethat a, and 6, both haven * I data points, zerosbeing added to achieve this. Then, 0 "'

ao 0

0

:

:

:

'

.

.

Q , a , t 0

a

,

a

l

ll*ll_ ll,.ll ,:,

Lr+

".an

0

ll'i' " tl

"' ll"' €r €, € o - .€. .t llo

#*(-n) ;

l|;lt 0",(- l) 0"r(0) 0"r(+l)

( 1s.25)

Q"uftn)

Autocorrelationis given by

I

(ts.26)

6uu: UrU.

, ( l 5.23)

:

0

"' o ll". o o,

cl

0 Q o

This gives the same values as eq. (9.42). The crosscorrelation matrix is a Toeplitz matrix. Another schemefor cross-correlating e, with d, is given by

llo o '.'"u

co

a t a o 0 " ' 0

(ts.24)

The autocorrelationis also a Toeplitzmatrix. The Wienerfilter normal equations,eq. (9.73),can be expressedin matrix form as

vl"

(ts.27l w h e r er ' ,: L r a , o b r : Z o a r b - ^ .T h u s ,e q .( 1 5 . 2 3g) i v e s the sameresultas eq. (9.23).Note that matrix. I is o f o r d e rp x ( n + l ) , w h e r ep : 2 n + l . Cross-correlationcan be written as i4 rU : $",,, that is.

eo et

0

0 e"

e,,

€o €t

0 0

where

0""=

0 u

eo 0 r u

v:

en €n-r

ll:: -":lll:l?11; il1il ,

(ts.29)

llo-,t,1 ll

The filter .7 is givenby

o " o

.v: do 0 dt do

0

0

0

' d l

0 do o

d, 0

d

,

d

d,,

o ,

d,,l

J

(1 5 . 2 8 )

0

6;"'610.

( 15.30)

Sometimes,the solution of a matrix equation rnvolvesthe inversionof a matrix that is not square,as in.'l.B:6,where. / isofsizemXn,,B of size n x p, and t of size m x p. To solve for .B , we multiply by . .l', . 1,r,.1 .4 :.

/,r{

(note that , l'r. .l is alwayssquare),then multiply by (. l'. l) | to get..4: ../):(. lr.'l)-t. lr' {.

(15.31)

BACKGROUND MATHEMATICS

s22 (c) Characteristic equation of a matrix; eigenvalues. On replacingc, with /i, we can write eq. (15.21) in matrix form as

, ' a . r:' a

(1 5 . 3 2 )

Column matrices .,5- and '/ are vectorsof the second a linear transformakind, and eq. (15.32)represents tion of veclor .-l' into vectolq .If U : \,Jf,', \ beinga constant,1/ is saidto be in the samedirection as .,{'. The condition for this is that

.1,-{.:a:},.t. or

( . . r- | \ . v ) . t ' : ( . arrxr* "' I

-t (azz - ^]]' *

"*"

a,,x. + "' + (a,,-\)x,

anit +

100,. :

0.

( r5 . 3 3 ) Theseequationshave one or more solutionsif and only if the determinant of the coefficientsvanishes ( $ 1 5 . 1 . 1E ) .x p a n d i n gt h e d e t e r m i n a n tw, e g e t a n n t h order equation in \ of the form

x "- 9 , \ ,' * g . t r ,-:. . . + ( - l r p " : 0 .

Maclaurin'sseriescan be used to derive many useful infinite series.For example,rf "f(x) : e', then/'(0) : | : f"(0) /"(0), so e ' : I + x l l t + x 2 l 2 !+ x 3 l 3 t+ " ' . ( 1 5 . 3 7 ) cos .rr, (d' ldx' ) Similarly, since (d/dx)(sin x) ( s i nx ) : - s i n x , a n d s o o n , / ( 0 ) : 0 : .f"(0): "', f' (0) : +l, f"'(10): -l, . . . ; therefore, f"(Ol : (15.38) t x 5 l 5 !- x 1 l 7 t+ . . . . sinx : x - x3l3+ (15.39)

(c) Binomial series. The binomial seriesis obtained by the expansionof the function (a + b)'. We can write this in the form a'(l * x)", wherex: bla. Let lal > 16lso that lxl < l, then we can expand(l + .r)'in a seriesthat is finite ifn is a positiveintegerbut is otherwise infinite. Writing/(x) : (1 + "r)', we have ' 1 . = :u n , f'tO1: n(l + x)'' , f ( 0 ): I , -- n(n - l), and so on, .f"(x) so that

(15.34)

This is the characteristicequationof matrix . 1, and its roots are called the characteristicroots (values)or eigenvaluesof , 'l . Becausethe determinant of the coeflicientsin eq. ( 1 5 . 3 3v) a n i s h ew s h e n e v e\r i s a r o o t o f e q . ( 1 5 . 3 4 )i,f a root happensto be zero,the determinantreducesto det(.I ) so that det(.I ) : 0. In this case,eq. (15.20) showsthat . I ' is infinite(doesnot exist).

l . x 1= t + n x + +

n(n ' l'l , ., .r-+"'

n ( n - l ) \ n - 2 ) " ' ( n -r + l ) x'+ "'. (15.40) yl.

This seriesis valid for all finite values of n (Wylie. 1966: 695). The binomialseriesis usedfrequentlyto obtain approximations,especiallyof the following functions: l l l ( l + . Y ) r ' r :| + - r - . . r t+ - - . Y-l " ' . ( 1 5 . 4 1 ) 2 8 16

I 5.1.4 Seriesexpansion,s (a) Taylor's series. Taylor's series is discussedin most advancedmathematicstexts,for example,Potter and Goldberg(1987:84)and Pipesand Harvill (1970: 34l 3). The seriesenablesus to find the changein/(x) when x changesby ft in terms of powersof ft and the derivativesoflr). The seriescan be written .f(r + h) :/(r)

(15.36)

ar,x,: 0,

. ]

. . .:':'

"f(x): l0) + x/'(0) + (x' l2l)f'(O)+ "' + (x'lnl)f(0) + "'.

In the sameway, c o sn : | - x 2 l 2 t+ x a l 4 l- x 6 l 6 !+ " ' .

equations: This is equivalentto n homogeneous (a,,-\)x, *

placeh with x in eq. (15.35),we get

+ hf'(x) + (h' l2r.)f'(x)

+ . . . + f h ' - t l ( n- r ) l l J ' ' ( x ) + R ( € ) ,

(1 5 . 3 s )

wheref'(x), f'(x), . . . ,-f' '(x) are derivativesof orders 1 , 2 , .. . , n - l , t : k h ,w h e r e 0< k < l , a n dR ( { ): (h'lnl)fG) : remainderafter n terms.Obviously,R({) is the error when we truncate the seriesafter r terms; hence,the error is of the order of ft'. The larger /r, the more terms we require to achievea given accuracy.In practice,two or three terms are usually sufficient. (b) Maclaurin'sseries. If we set,t : 0 and then re-

1 3 5 (l + v) rir: | - ^t - lx'- '"r'- "" (15'42) 2 8 16 (15.43) (l + x) t: I -,t *.x: - tr * "', ( l + . r ) : : I - 2 - r* 3 x 2- 4 , r r+ " ' . ( 1 5 . 4 4 ) 15.1.5Complernumbers The squareroots of negativenumbers are imaginar.t numbers,and numbersthat are partly real and partl] imaginary arecomplexnumbers.If we write j : { Ithat is, j' : - I (somewriters usei insteadof i), rie can write, for example, the imaginary number

rl-s:{t{-r:3j. A complexnumber,z : (1+ jb, can be represented by plotting in the complexplane where the direction of imaginarynumbersis at right anglesto the real direction,as in fig. 15.2.We can also expresscompler n u m b e r si n p o l a rf o r m : ::

a+j6:

r ( c o s 0+ j s i n0 ) : 1 s t o

(l5.4it

SUMMARIES OF BASIC CONCEPTS - modulus (seeproblem 15.l2a),wherer : (a2* b2.)t/2 of z : lzl and 0 : tan t(bla) : arg(z).The conjugate complexof ;, :-, is definedas Z : a - jb = rlcos e-- j sin 0; : re jo(seefig. 15.2). The sum (or difference)of complexnumbersis ob_ tained by adding (or subtracting)the real and imagr_ nary parts.lf z,: a -t jb, z.: c + jd, then(:, t ,.) : (a t 4 + j(b -+ d). A complex number is zero oniy if both its real and imaginaryparts are zero;hence,rwo complexnumbersare equal only if both their real and lmaglnaryparts are equal.Multiplication and division obey the usual algebraicrules. For example(seealso p r o b l e mI 5 . l 2 b ) ,

:i. = (a+ jhtrc+ jd) :(uc-hdlLj(atl-bcy : r , r , [ c o s ( g+, 0 , )+ j s i n ( o + , 0.)] :1",1,9j{ot*o:).

_t_ :u

I I

(15'46)

f )

+ jh _ {u+ jbtg.- jd) (c+ jd(c- jd1 c.Jd

:(qc+bd)+j(bc-ad) c 1+ d : (r,/r,)[cos (0, - 0.) + j sin(0,- 0,)] o:r. : (r,/r,)eit{}r

BecauseE is a function of the parametersa* only, the minimum is given by dE 6ar

2\(/-,

- cto-

alxi

... -o,,*,)(-rf)

: o,

- T ri + a , \ x f * t + . . . + o , , ) * f - , ": I r f r , , k:0,1,2,...,m.

(15.50)

There are m * | such normal equations,so we can solvefor the m * I unknowns,a,. Sometimeswe wish to find a least-squares solution subjectto a certainconditionon the unknownDarameters(constraint ), for example,we could require that at: eqand/or a, I a, + at 0. We can write each constraintin the form C(a,,ar, . . . , a,,) : 0. Because the a,'sare chosenso that |Eli)a : 0 (and OCl\a : 0), we can write the least-squares conditionwith con_ straintsin the form A

+ \C;:9, al,o,(E

( ts . 4 7 )

The rth root of :, ;0, can be found by writing : : rero- .i: (r,,e,nu;, [r,,(cos0,, + j sin 0,,)],, : rfi(cosn8,,I j sin r0,,) by de Moivre'stheorem(seeproblem l5.l2a). Hence. :"" : zt : r,,(cos0,,* j sin 0,,), r,,: r"n, 0,,: (0 + 2nk)ln, k : 0 , 1 , 2 , 3 ., . . n - t .

s23

i:0,

l, ...,m,

(15.51)

\ havingthe samesignificancehereas in the Lagrange methodof undeterminedmultipliers(pipesand Hu.vill, 1970:968).Thesem * l equationsand the equa_ tron C(a,,a., , a,,,): 0 sufficeto solve for L and

€ 'a

(1 s . 4 8 )

Figure 15.3showsroots plottedin polar form for the casewhere: : r : rei2.: real.n : 5 and6.

I 5.I .6 MethodoJ'leastsquares ( a) Basicmethod. Let us assumethat we wish to ob_ tain the "best-fit" curve of orderm, l i = a o* a l , i

a . x i + . . . t a , , , r , (, ,1' 5 . 4 9 )

to representa setof r pairs of measuredvalues(-ru -y,). If n : m + l, the curvewill passthroughall r points, (x,,,t,).If n > m + l, the curvewill not passthrough all n points and we seekthe ',best-fit"curve such that the sum of the squaresof the ,,errors" betweenthe curveand eachpoint (.r,,1,) is a minimum,the errors e, being the differencesbetweenthe measuredvalues 1, and thosegivenby the curve.Thus,

Fig. 15.2 Geometrical representationof complex numbers.

€ i : ! , - ( a u* a r r * . . . I a , , , x ' , , , ) , i:1,2,3,...,n, and we wish to minimize E. where .i : ) , | y , - ( a , , t a ) x i+ . . . + a , , , x ' i , ) l r . E: 1,ei i:i i

Fig. 15.3 Roots of a complex number.

524

BACKGROUND MATHEMATI CS

the m * I valuesof a,. The extensionto severalconstraintsinvolvessolving

This can be partitioned thus:

,-u,illl ;ll=,:"ll;ll

! ( t * } , c \'l : o d./,\ 7'

Equation(15.50)can be written in matrix form as

'f

: .)t'*' l*'

(1 s . 5 2 )

where

rl"l l'il ,, ll e,:il,,"illl,, :il-,y,",, "-,'||ll

Tr,', !.''*r .,{'+

il ll il",-llll,-ll

:

-''il il; ll ll -, = .. "; llI d,--.o)lll llll : il. ,""ll" ", ,,,ll","llll","l and

! "r"'

,|[|I ' r:

Then,

, 1 )

s-,,

L'' ,.' '

_, /il ll

!r',,1,

Because..{,'* is square,

. -l* : (.I*) ,a*

where the first column of ..8 is - :/, and the rest is .I) We can accommodatethis by taking the first column as the Othcolumn, that is, b,, : - ! , and bu : xu, i>0. Individual errors are given by

(1 s . 5 3 )

:llI tt , t(il E:fe; ;,,ll

x'-v,Y")ll

problem usingmatrices If we solvethe least-squares from the beginning,we can obtain a more generalresult that is alsowell adaptedto computercalculations. We write eq. (15.49)in the form

/ll,,,,ll

.rt;.: a,

:ilr /,|.nll ;ll

where

Yr: '''-ll lllx,, l .Y..... x,, ll lllo, l ' lll l lllll,' ' l l / ' : l l : : , l l . , r =, ll ll . a =, llll il ll ll*",*.,...x,,,, ll

i ,,,, lil ll ll

il ll llr, ll

xi,, /, being known, rz,unknown, and n ) m. (In general, x, can be any m x n known quantities,including powersof x, as in eq. (15.49). Becausewe have more equationsthan unknowns, the equationscannot hold exactly; writing I as the c o l u m nm a t r i xo f t h e e r r o r se , ,i : 1 , 2 , . . . , n , w e obtain

, t ;t - u : r .

,r?,:.,ll 1tl

t,,,,ll: ./?r'8.

(15.54)

Setting derivativesof C with respectto a, equal to zero gives

fo:o:llo..oro oll./?. 0

This can be simplified by writing (Claerbout, 1976:

r07)

lll:,,.,,,llllj lflI

+l1 l i

t' ll ' ' ' 2

0 I 0 0

SUMMARIES OF BASIC CONCEPTS

il ll/ 1 tl ll

: -rll r r 0... " o | 0..'oll.,? 1t 2a,

525 do this by multiplying the error e, and the ith row of ..r')by a weightingfactor {u',. Then,

because.4 : ..4r.8 is symmetrical.This resultcan be written

ll't

l l r l l ',,,, llll -- ll = o.

r = r. 2 , 3 , . . . m .

"ll.

\

.x, , y ,., \ - ,, l,ll l l l l l : t I . / ' l l . / ?-ll ll -l -l ,l ll-,r, l l l /. l l l l / llll . lll. ll ll I

ll'lll

If we combine the m equations,we get

:/ lll: , "ll ll l

(15.58)

except that we lack the Oth row of 'y'|. We define a quantity u by the relation

= r,,,' ro,,llll 11r,,, ; il v ;

(I 5.55)

we now have

llJll o'",, ,l 4lt'l:l[[[[[[ ll: l; ll ll'll

:tl;.,,llll u t't tt11 :tl- ) , / / , . )l l l l " ) - /y''ll

E : \ n ' , c i : lt l . / ' l l >

ll I

where.l?,* is the weightedform of .4 : ..r']t.4, that is, the product of ..r']r and./]when the ith row of ./] and the ith column of .4 | are multiplied by {u',. Equation(15.57)is still valid exceptthat the ith row of .l'and '/y'andtheith columnof .11 aremultiplied by {n',. Constraintscan be consideredas additionalequaexactly, tionsin the unknownsa, that must be satisfied that is, they have infinite weightsin comparisonwith the error equations.We can write k linear constraint equations,k < m, in the form

: (' .

(15.59)

(l 5 . 5 6 ) where

/y',//-r/, / llll I

r :

that is,

llvll

l | l lrl' :'u | /+- .1,'r.1," 'U/' ' t "Il llll

F b l l o w i n gC l a e r b o u t( 1 9 1 6 :I l 2 l 3 ) . w e a s s i g nt h e weight {l' to each constraintequation,insert the weightedleft-hand side of eq. (15.59)in the error equations.derivethe result,and then let l approach infinity. Without constraints,we had the result (seeeqs. ( 1 5 . 5 4a) n d ( 1 5 . 5 6 ) ) .

ll'll Thus. we have

v : '/y'' '// - /y''.2'. t ' and

ll:ll ll,ll hence, . I : ('l'r'1,) '.)r'''2/.

il'll1ll ,rll,illl:ll

' ll lll ; (15.57)

At timeswe wish to giveextra weightto one or (-r',,-x,,,-r,:,. . , -r,,,,). moresetsof observations Wecan

ll -r', -',,

t , , , )

B A CK G R O U N D M A T H E M A T I C S

s26

Equations(15.59)and (15.62),which give the solutions for the nr unknowns a, and the k unknowns \,, can be combinedin the form

with constraints,this becomes

['il:

ll -r, -t,,

\

t,,,"1 I

.r'([1

l lo ' n/ I * / ' / l l )]

llr

iltlll

"ll-

t15.60t

llt that is.

(../)r., +

i l ,=ll, l t l=7* l;.ll 1s/'/ ll,ll ll

We write

, l l - r :l . tlt + e ' / ' f lr':', ll

"

ll

+x:'1'!+"'

lil l ' /tll l

where . lif is the matrix that gives the desiredsolution, . 1,f is a similar matrix with different unknowns' and so on. Substituting,we get

"'):

V*

e t'.'{r'6,l'tr : a lt + 6r{.'/rl: xo'.,/9r./9, Because

r**

Y^*l*)

: : r - L x , x r , _ Z Y , Y r, , the sumsbeingoverall appropriatevaluesof i' Squaring E, gives

-E^):

: ] - ( t . " . r ,) ( I r ' , \7''' 1\7

V*'

]

, , ,u , ,

,)

+2(Ix,r- ,)(l r'-.,) Summtng over the full range of k gives the total squarederror:

' 1i:"ll ;ll:'

from eq. (15.59),the first equationis satisfiedautomatically; hence,it provides no new information' In 't : ''/,whete the secondequation,we substitute /f / is a k x I matrix whose elementsare the equivalents of Lagrangianundeterminedmultipliers'Then

lt + '{1'/' -- Vx. ..r']r..4,

,t)

+(I r,r-,)(;,r'- ,) - z r r ( \ x , r ^ ,* I t r -

Equatingpowersof e gives

( I 5.63)

(b) Multitraceleastsquates. ln $9'5'5and 15'l'6a' filtering of a single trace' we discussedleast-squares Extensionto multitracesituationsinvolvesconsiderably more complexmathematics,so we shall consider the caseof two tracesfirst and then discussgeneralization to ,?traces.Our treatmentis basedlargely on that of Schneideret al. ( l964)' but we havesimplified their notation; to do so, we have departedliom our usualnotatlon. We considertwo tracesx, and y,, which are inputs * to two filters X, and I,. The filter outputs are X, x, close be as output each that and Y, * y,, and we require as possibleto a singledesiredoutput' We write z, for the sum of the desired outputs; hence, the error at t i m e I : k i s ( s e ee q . ( 9 . 2 3 ) ) Er:ir-(X^*

(.r,,,+1'r''t)<.t,t+ e . / . f +

lF"ll

ll;; ; ; ll=

llc,oc,, .

(15.62)

-;[(;t'"- 'X]1'-)]

.'I[(Io"-,)(]r,-)]

SUMMARIESOF BASICCONCEPTS Interchangingthe order of summation,we get

r': ).--;"' ))xx,/I,, ' \ 7 " ' ., ' ' l) 7" a7

527 (note that the dex, has been The solutio written as

+ I I y y! 'ftL\ L .l ' i ' t r,k. ) "i'

being a dummy innd ( 1 5 . 6 5 bc) a n b e

,41 tl il_

ll',,

I

v,ll

li ll :'

- -zlxll-? ' ' \ ? - ^ *, ' ^ ' )l -zt + ' ,y/t\ / i - A .,, ' t , \I

i l - , llrts.oea; u"

ltll ,

*'\+rr(fn''^')

tl

ll

where Because all sumsare overall appropriatevaluesof the indices,we can replacethe index k in the second, third, and sixth summationswith a new index r : fr - i, moreover, I^:*.l-* , : 0..(-i) from eq. (9.42), and so on. Thus,

* : l.i .Z\rx x(1,,.,,.u ,,) - :Ix,o.,r-i)- 2t {0. l-il + z\)x,,(|r,,,,,, ,,) _.i) FF: 0..(0) * \Ix,x,o,,(i-i) + II y,yid.,,(i - ,r LF n. r, ,Q , ( - r. ), - t" L\ -r ', ,o, . , ( - i l t

- ,. + zl\x r,o,,(i

(r5.64)

N e x t , w e v a r y X , , ,a n d { , , , n t : 0 , l , 2 , . . find the minimum F. This requires that

A E ^ A E : n" : dx,,, ;)y,,,,

,lVto

I ?

T o v e r i f yt h a t t h i s g i v e st h e s o l u t i o no f e q . ( 1 5 . 6 5 ) , we can selectany row of ]]r ]],lor example,the second, multiplythe row bV I U ]1,equatethe resultto tr2,, and comparewith eq. ( 15.65).The productol the second row and ll U ll gives r , U , I r , , U ,* . . . * r , , U r : V , . It is sufllcientto show that the first row of the produ c t s g i v e s e q . ( 1 5 . 6 5 a )( t h e s e c o n dr o w g i v e s e q . ( 1 5 . 6 5 b )T ) .h u s ,

+ 0,,( +l) y,,1 + [d,,( 0) + ts,,( - l) x,, x 0,,( 0) r ,] - l) x" + 0.,( l- r t/) r "l: 9,.( l) , + "' + td,,( N that is. I

nr:0.1.2,...,N

Somecare must be taken in differentiatingthe sums. BecauseX,,,{and )r,,,)occursonce in a strmmation, we have

: ,, ,t),,(z*)

lltll .:llt.8ll r:()

. I; r,r,(Tr,,,,,,,,,,)

i

l l o , , t 'or , , r - s r l l o,,r'rll' ll0,,r-,'r

d [ -

I

. , . l L X 6 . t i ) l: 6 . '( - r r ) ; rtA,,,L7 '' J ll

= L X , 6 , , t n- i t

,', {>rl4*d,,tii l l l J)

|

, \-.,, + LX'Q"Q - ml

: zlx,6,.{n- i) :

-\'., , t).A.Q..Un -

tl.

i

Obviously,similar resultshold for differentiationwith respectto {,,. Carrying out the differentiationsof eq. (15.64),we get the followingnormal equations.

- * - r): S,.(n), (15.65a) Tx,O,.t" il Iy,g,,(nt il + I yb,,(m i) :g,.(rn) (15.65b) \xA,,f*

N

for , O , , t r{ + i ) + I f) O , , r r i ) : 0 , . ( l ) , w h i c h i s e q . ( 1 5 . 6 5 a()b e c a u s 0 e, , ( - l

0 , , ( l- r ) ) .

+ l):

Derivationof the normal equationslor the general caseis givenby Simpsonet al. (1963)and Wigginsand Robinson( 1965)discussthe solution.The matrix ll r I in both the two-tracecaseand the generalcaseis a Toeplitz matrix; hence, the Levinson aleorithm (g15.5.6c) can be used.Wigginsand Robins-onalso g i v eo t h e rr e c u r s i vseo l u t i o n s .

15.1.7 Finire difJbrent.es The calculusof finite differenceshas wide application in many practicalsituations,for example,calculations with data in digital form, interpolation, numerical differentiationand integration,numericalsolution of differentialequations.We shall discussthe basicrelations assuminga function ./(.r) that is given at equal intervalsA, that is. we discussthe discretefunction/(x + nL,),n : 0, + l, -r2, .. . . We shallusethe notation J,, : ./(x,,,+ rA) (note that/u : ./(x,,)).

BACKGROUND M ATHEMATICS

528 We definethe following operators:

E{f,} : J,,.' LVJ : f,*,- -f,

(1 5 . 6 7 )

,{t} : olo'.-'l ..

(ls'6e)

(1 5 . 6 8 )

= l \ \o+ra

+ rA) : E'U,I: (l + A)Y;' /("xn

l'n+l

D-,t{

,}:l

(Wylie, 1966),which enablesus to interpolatebetween is known as the For"f^ ind f,: in this case,the result (a Backward Gregorymul a w o n y r go N e r i ard Gr:e for Newton iormula existsfor use near the upper end of the tabulatedvalues;seeWylie, loc' cit')' An expressionfor the derivative, D{fi}, can be found as follows:

" tt * 'L*'(', ^'- )'t

:

(1s.70)

f(xtdr.

J,o

Theseare respectivelythe delay,difference'derivative' and integrationoperators. The fiist and seconddifferencesare given by

Differentiatingwith respectto r (that is, treatingr as if it werea continuousvariable)gives

d f . r " + r A ) -l ( L * r , - t A ,

L V J : f.*,- f,

^\; - ^,'l ,\ 'L, + ...1"f,,. + '' ..' ' 'j:

Adr

- att'} Ev.t: ata{I,}}= LLf..'} applicationsof the preceding,we find By successive that

A ' U ; ':I1 , , , - r f , , . , + + "'+

( - l t) t'

We let r go to zero and obtain

rt-l\^

2t J,"

d/(x" + rAl

D r r , )df= |; ; l = .., , , 1 a a ,

't(t l) '.'.'t''f (r-

ru l

l)l

\='h

(15.71)

+ (-t)'f,. Also,

tl5.75l

:

Repeatingthe Process,we find that

Att,) : f,., - .f,= EU,,\ l,:

: j{: l",t - r' D.y;t -

hence, (Sometimes the difference is referencedto 't * (n '+ ll2)Linsteadof .r * nA, in which casel{f -','} is calledthe centraldilJ'erencc.) Combinationsof operatorsare often used, for examole.

r/d /\l I =/d/\l :'l.t;l)l ElD{t,}' ] l;,ll ".t^

* ,lA, ..')f,.

(1s.72)

a:r-1.

'\'

=t;: \g+l

/,:. a{D{r,}}:l,:.,The operatorsobey the basiclawsof algebrasuchas associition,distribution, and commutation' We can apply Taylor'sseries(eq' (15'35))to EU;] to obtain

:1.r,, + A) : fl.r,,) - A/'(.r,,) EL|,,I I "'' + + (Nl2r.)f"(ru) !rtS.Z:l = [ + ( A D t+ ( l / 2 ! X A D1) r" ' 1l ( r ,) . 1 ' = ernUu).

)

To obtain an expressionfor interpolation between (15'12)to write ./(,tn)and/(rn + A), we useeq' E,:tlrlr,=!+",1\

r ( ' - 1 ) A ,* . . . *

"')l,' ^(A lA'* iA'- iA'+ ( 1 5 . 7 6 ) I

(15.74)

Although r is normally u po,itiut integer in this ex( * l ( pressioi, the equationis still valid when 0 r

(1s.17)

We encounterthe operator(-1, l) in a numberof resituations,for example,to describethe successive (fig' 6'42) bed flectionsfrom the top and baseof a thin or to describea reflectionfollowed by its ghost from a perfectreflector(fig.9-12).This is the differenceop= D' erato.of eq. ( 15.68);eq. ( 15'76)showsthat [lA' operathe of the derivativeoperator.Thus, the effect tor (-1, l) on the waveshapeis approximatelythe sameas that of the derivative. The integral D '{f,},can be found as follows: f',, r

D 'tDtt )] = |

-,f("n) /'(x) dx : /(rn + A)

tto

= a{d,}' so that

otD: L. wehave and(15.73), Fromeqs.(15.72) E:1+A:exP{AD}' or , =

I L

I l n( l + A ): . ( A - l A ' + l A . A

Thus, pr:|.tD:A(l-

(1 s . 7 8 )

s29

SUMMARIES OF BASIC CONCEPTS Therefore.

Twice the sum of the first and third equationsminus the secondgives

D-,Uuj= A(t + jA -,r.A. * ;A, - ...)f,. ( l s.79)

lq:

if we multiply togetherthe operatorsin eqs.(15.76) and (15.79), we find that the product is I in a g r e e m e nwt i t h e q . ( 1 5 . 7 8 ) .

because(xn,yn)can be any one ofthe setofvalues,this relation appliesto any four successive values,that is, !,*r : l, . + (4L,13)(2y,z !,_t + 2y',). (1 5 . 8 2 )

I5. 1.8 Numericalsolutionof differentialequations Often, one must solvedifferentialequationsby numerical methods, generallybecausethe equation is too complex to be solved analytically or the data are in digital form. The basicproblem is that of finding y(.r), where dyldx : ,"' : JU, y), ), : lo when x : xn, .flr, "v) being given either in functional form or as a table of valuesof fx, y) at equal intervals A of x (if the valuesare at unequalintervals,it is usuallynecessary to interpolateto get evenlyspacedvalues). We useTaylor'sseries(seeeq. (15.73)to write

!c, + @L,13)(2yi y', + 2y'.):

lf we know y, for four consecutivevalues, r : n, n - l, n - 2, n - 3, we can find yj for thesevalues; hence,obtain1,,*,from eq. (15.82).To get started,we need the values of !0, !t, !2, ly the derivatives, yi, y'r, y!, can then be found by substitution in the differentialequation after which we can calculateyo. Becauseyu is known, we useTaylor'sseriesto find y,, l:, .t,i to do this, we must know 1", -y",/"', and so on, at r : x...We have

.rr : j/(xo * A) : /(xo) + A+ = y,, + LAx, yn),

y' : .l(x,y),

: :f * lt ,'.

|

d,Y

dr;

and so on.

OX

lz :. l(x, + 2L) * y, * L,f(x,,+ A, l,),

( 15.80)

ln*t : y, + L,f(x,,+ nL, y,). This method, called lhe Euler Cauchy method, involvescalculatingsuccessive valuesof the derivative, /(.r,, * rA, ,1,,),and using Taylor'sseriesto find the approximatevalueof y,*,. The accuracyis low unlessA is small,but in this casethe computingtime can become excessive. Many methods have been devised, most of which involve higher-orderterms in the Taylor series,to increasethe accuracyand decreasethe computingtime. We shalldescribebriefly two of these. Milne'smethodstartsfrom eq. ( 15.75)with r : 1 . 2 . 3 4. . Writing y in placeof /,

O'l ,' : d''l \:

we get

\O+/_l

"rl: ltr * lA. *:A,+ ,,'A.)-i'., rj : i(A + 1A'+iA, - ,'"A*).r',, r: : lrr +:A'+ fA'* jA')'r',,, .'; : ff + lA. + ?A,* jlA.r.i,.. U s i n ge q . ( 1 5 . 7 1 )t,h e s eb e c o m e -li : (-3.rrn l0-y, + 181,, !'::( y\:

lo( -yu+

6.1',*

+ 8],8-v, 6 y , 1 8 y , + 1 0 y .*

r'./t2A.l .ua)/12A. I .

3

tr^,'rrtrll)

)': : ( 310 l6y, t 36y.- 48y.+ 25 (1 5 . 8 1 )

The derivativescan be evaluatedin the neighborhood yu) and so I t, !2, !-t can be found to any desired of (,vn, accuracy after which the differential equation gives y i , y ! , y ! , a n d e q . ( 1 5 . 8 2g) i v e sy o . Like mostmethods,Milne'smethodis subjectto errors that at times are cumulative.A checkis furnished by the equationobtainedby adding the secondand fourth equationsin eq. (15.81)to four timesthe third, the generalresultbeing t , * r : 1 , , + ] A ( r j , + 4 y l ,+ t ' , r J . ( 1 5 . 8 3 ) , e f i n d y i , - , ,t h e n O n c ey , * , i s f o u n d b y e q . ( 1 5 . 8 2 )w s u b s t i t u t ien e q . ( 1 5 . 8 3t)o g e t a c h e c kv a l u eo f j / , * , . Up to this point, we haveusedderivativesevaluated at r0 + rA, r integral. Higher accuracywill result in generalif we usederivativesevaluatedat intermediate points suchas r + (r + l)A. This conceptis basicin the Runge Kutta method.The following accountsummarizes a more detaileddiscussionby Potter and Goldberg( 1987:498 502). Considerthe caseof the first-orderequation, dyldt : ./(y, t). Starting from the point (y,, t"), we wish to find the valueof y,*, corresponding to l,*r : /, + A. We write !,*r:

!, + A(dylAl) : y. + Lf(y, t).

(1 5 . 8 4 )

In the Euler-Cauchymethod,./(.itl) is the valueof the derivativeat the initial point (y,, l,), but to get higher accuracy,we selecta value of .f(y, t) at a point in betweenpoints(v,, t,) and(_y,*,,l.*,). Let this intermediate point be (y, + q Lf,, t, + pL), wheref, is the slope at ( t",, t,) andp, q are constantsto be determined.Takingl as the slope at the intermediatepoint, we assign to / the value

f:

af, + bf,,

(I 5 . 8 5 )

530 BACKGROUND MATHEMATICS

whereagainconstants a andb are to be determined. is equivalentto Theslopel canbefoundusingT"yl"ra;;j;, Y' : z(x,!), I : loatx : xu, "f,:I+ q L,feftayl+ pAafta\ (15.86) z' : f(x, y), Ly : q : p L), whererhe partial z : y[ at xn. lTl:^j,hl !f,, tt oenvarrves are evaluated (1,, t,). By using eqs. We solvethe secondequationfor;(x, _ay 1) and then solve (15.85) and(15.86), eq.(15.84) nor'f."*..'-' the first equationto g.t y : _y(n). !,*t:!,+A(af,+bf,) : y, + * LbV, I5. 1.9 PartialJraction,s lo/, ! t L,f,@flay+ p L,(dflafil : y, + L,(a + b)f, + Nlhqf,taftayt + Lprcita\l.' It is often convenientto expressa function ofthe fbrm ( 1 5 . 8 7 ) N(x)lD(x) in partial fraction.g,that is. To get valuesfor the c_onstants a, b, p, andq, we N ( x )A, A, A expand "' + Taylort ,..i., una ,!,*, in a second_order o t * l x e , x equate a . + thecoefficients * ''u,,, with thosein.q. f f S.SZi. fnur, 8,, !, B + ! !,*t : !, + L(dytdt)+ (L2t2t)(drytdt) 1 -" ' + ( x ) - b,Y' ( . v - 6r)' G b,y' : y, + Af,+ (15.88) :L,@fldt). f wherethea,'saresinglerootsof D(r). and 6, is a multi_ But ple root of order r. 6bviously. ryfrl.'Oi*, "* df_af

ay,61

d' t

At

6 ' )v

At

!,*,=!, + ^r,+ A,

-af ',A, uy

iotyno_ mials in x, and we take the o.a". of lfi")'i.r, ,fi"" ,f,", o f D ( . r )( i f t h i s i s n o t s o . w e c a r r y out longdivision and the remainderwill be a fraction "f rfririyp"f . fo find the valuesof l, g,, we note that we can write

df cll

,{,) (r5 8e) brl.

fl(D _ D(.r)

,\r(r)

k $ - u ) ( . r - a . ) . . . ( x_ u , , , | ( _ a h,y,

Comparingeqs.(15.g7)and (15.g9), we find that a+b:1,

bq:!2:bp.

Becausewe havefour unknowns and three equations, we are at libertyto assignan arbitrary valueto one of the

constants; usually, i isset.qr;i;; i/il. t. wr,." b : ll2,weEeta : t/2,p : I : u, *f,"""t': r, o : 0,p : ttz : q. rn theflrit "ur., .q. f ii.iii'u".o_., r , . r = r , + a l ,+ t ( t ^ t " a l * t a l l \2" dy 2 arl

:t,+

= y, +

I

f

T,

f

r

"r

A,, :

)t v * ./(y, + Lf,,t, + L)1. )t1.,,r, + ]A).

(15.90)

and

y,=ylatx:rn,

(15.911

ro geta..we-rr,ipryl"ir,l,o.,uy j: 1.^'lT:,*.t r . r - D r , ' ,a n d t h e n s e t . Y = D , ; thus.

r(.I) r-: " ,**(r)

where the double asteriskmeans that the factor 1r _ 6,)'has beendeleted.To get B, we differentiui. on.. ,, beforesettingx:b,:

).

Higher-order differential equations . can be solved by reducing them to simultaneors n.*o.j., .quu_ trons.Thus,the equation, f :!o,

N(*) | o*(x) |

wherethe asteriskmeansthat tt.'fu.to. ,r - a, is de_ leted from D(x). We can ger all of the .o.m#nts x,

Thus,given(l,, r,l,we can get],*r with the sameaccu_ racy as that ofa second_order Tiyior,s ,".i.. (not. tf,ut eq..(15.88) is equivalentto eq. (15.90)) The Runge_Kutta techniqu. .un 6. developedto provide even higher accuracy; of course,the equiva_ lent of eq. (15.90)rhencontains ,; ;.;il;; provide the increasedaccuracy(porter ""d C;i;;;;g,, 1987: s0l

y":-f(x,y),

. . .

wherek is thecoefficient of thehighestpowerin D(r). To find A,, we multrplyUotf,r ,lies oitf,;;;;ro,ng by r ani then setl: ,,,^t.""r..,n" ;111ssion !, tactor cancelson the left and in the firsi termLn tf,e righ^t, andappears in all other,".-, on tfr. ,igfr,,,0., wefind

2 ^L "J' r [*' l t ,t*(Y1' d,y:. !rd!t1] l 1

When b : l, we get r,*t : r, + Af(y, *

8 " + lx - b,),

In general,

4 : i t#il,]I 'r:61

I d, ri(\) B .. : .'t I ll

ar. [o**trljl

( 1 5 . 9)2

531

F O U R I E R S E R I E SA N D F O U R I E R T R A N S F O R M S As an example of the above,let it be required to find the inverseLaplacetransform (see$15.3)of s2-2 s(s' - 5s + 6)(s- l)'

n integral,

t:-l s(s- 2)(s- 3)(s- 1,):'

f-'"

A, A. T T .r .s-2

a

s--/-

s(.s- 2)(s - 3)(s- 1)'

,

8

2

,

m*n (15.95)

rinmocosnodo: o,

( l 5.96)

(^r- l)'

.,n,rodo: n,

(15.91)

nodg: n. f'.'".o.,

(15.98)

f.'"

A. .g-3 B

l

f.'-

s- I

and n

s2-2 | , t (r'- 2)(s 3Xs- l)' 1.=n

'

^

"'r A.: B,: -t

If we multiply both sidesof eq. (15.93)by cos nornl and integrateover the period I we get

.sr-2 i'' = 7 I .(.r- 2)(.r- tt.l 12' .s:-2 | | sft 2)(s 3)l

s . = d l" - ' '

f.'"

r 3'

1 2 - 7| A' , = -l :-r, - _ s(s 3)(.r l)rl

I 2'

:

'lls(s - 2)(s- 3)2s ls(r- 2)(s- 3) - 2)(s- 3) + s(^s - 3) (^rr- 2){(^s , : 1 + .(s- 2)Ill,=, ;

ln?

(2lT) |

a,:

g ( t )c o sr o r n ld / ;

(15.99)

J ,,.

likewiseif we multiply both sidesof eq. (15.93)bysin r?to.land integrateover the period ?l we get ptl)

ll 3)ll

- 2Xrds [s(.s

(r5.e4)

"o, mo cosro do: o,

Then, -i

,ln,,o sinnodo: o,

I

h,,: Qln I

g(t)sinnor,,r dl.

(15.100)

J ,,.

In particular,for n : 0, tu,,: (llT)

-

J

,

l1L2

|

n

g(t) d/ : averagevalueof g(/); ( l 5 . l 0 )l

hence,a,,: 0 wheneverg(l) is an odd function. The sineand cosineseriescan be combinedinto one seriesby introducingphaseangles,1,,: 15.2Fourierseriesand Fouriertransforms

r \ g(t) : ',(u+ L r',,cos(reo,/ 1,,), (15.102) n=l

I 5.2.I Fourierserie,s Let g(t) be a periodic function with period I that is, g ( t ' rn T ) : C ( / ) , n : 0 , 1 , 2 , . . . . Providedthat g(/) is reasonablywell-behaved,that is, providedgO obeysthe Dirichletconditions:( 1) it has at most a finite numberof maxima.minima.and discontinuitiesin an interval I and (2) f+ tl)

ttt')tat

I

'l

rtz

is finite, then gO can be expandedin a Fourier series: . : (15.93) g(tl : ',ao+ L\u, cos r?o0/* b, sin rornf), n = l

whereor,,: 2nv,,: 2trlT. We havewritten ]a,,instead of ao so that all values of a, are given by the same formula, eq. (15.99).To obtain coefficientsa,, and b,, we use the fact that, for any value d and for m and

where

cj,: ai,+ bi,: (',- ut,i 1,,: 0; ^y,,: tan t1h,,lu"l.

r > 0.

i

,,r. rO1

)

as Equation(15.102)showsthat g(l) can be expressed an infinite seriesof harmonicsof the fundamentalfrequency tou.Constants c, and ^y,give the amplitudes and phaseanglesof the harmonicsand are referredto as the amplitude spectrum and phasespectrumof g(l). (Frequencyspectrumis used for both the amplitude spectrum and the combined amplitude and phase spectra.)For very large n, the amplitudes must get smaller,that is, lim,-- c, : 0, becauseotherwise I Tlz

I urtllat

J r n

would not be finite.

BACKGROUND MATHEMATICS

532 Equation(15.93) canbewritten g(t) : (ttT)

lr/2

As Z approachesinfinity, 1/T becomesinfinitesimal; hence,

s(.y)dy

|

llT -t dvu: dal2r'

.-Tt2

+ Qtni

|fl1

) ,,.r,

,, cos[rtoo(r t')] dY.( I 5. I 04)

wherevariableI in eqs.(15.99)and (15.100)has been replacedby the dummy variabley. Often, in practical work. the function g(l) is given only at equal intervals,A, for exampleSft) : S@L), ) (15.103) n : 0 , 1 , . . . , m . E q u a t i o n( s1 5 . 9 3()1, 5 . 9 9t o still hold exceptthat the sums in eqs. (15.93)and (15.102)are finite sumsand the integralsare replaced by sums. For further details the reader can consult Wylie (1966).An interestingaspectof Fourier series is that the finiteseriesobtainedby discardingall terms in eq. (15.93)abovea certainvalueofn is the bestfit for g(r) in the least-squaressense (see problem ( 15 . 15 a ) . The two infinite seriesin eq. (15.93)can be combined using Euler'sformula (seeproblem l5.l2a) to givean exponentialform of the series: 9\t) :

€ 2

0ner"or;

fl5.r05)

*": io,l a-,,: or

(15.107)

J

The integral with respectto I is calculatedfirst, the result being a function of or; then in the secondintegration, t'-rdisappearsand we haveagaina function of t. The factor ll2tr can be combined with dto to give du, but then or in the exponentialtermsshouldbe replacedby 2ru. I 5.2.3 Fourier trttnsforms If we write

then

lrtz

g(t)e=i'-o' dt, tLt

n:0,1,2,...,*. At times we wish to representa function g(t) by a +'.n Fourier seriesin an interval such as t-tf regardless of the valuesoutsidethis interval (for example,see$15.2.l2 and 15.5.I ); the Fourierseriesthen repeatsthe sameportion of g(l) eachtime I increases or decreases by ?l

I 5.2.2 Fourier integral When g(t) is periodic, the Fourier coeflrcientsconstitute a discretefrequencyspectrumwith components g(t) repeatsat longer at intervalsof oo. If 7 increases, intervals while the frequency components occur at smaller and smaller intervals.When 7"becomesinfinite, the frequencyspectrumbecomescontinuousand the sum in eq. (15.105)becomesthe Fourierintegral. We shall give a heuristicdemonstrationof the transition from the Fourier seriesto the Fourier integral as Z approachesinfinity; a rigorousderivationcan be found in Churchill (1963). We can substituteeq. ( 1 5 . 1 0 6i n) e q .( 1 5 . 1 0 5o) ,b t a i n i n g

c(t):,i

I

, { ( / ) 'e- 'd r l e ' 'd' t / 2 n .

J_LJ_

J

(15.106)

J

f- rf-

g ( r r: I l l

G(o): I g(/;e,-'ar

)(a"+ib,)

o." : (l/fl

The differencebetweenadjacentharmonics,nto,,and (n + 1)ou,becomesinfinitesimal,that is, nco,, becomes a continuousvariable
dr]e^.( uf . -lf",.g(/)e-r"o'

_

: Fouriertransform of g(l),

(15.108)

| ^. tUl : (ll2rll G(o)ei-'dor J -

: inverseFourier trans/brmof G(
s(r)<-)G(o).

(r5.ll0)

of the Fourier A sulhcientconditionfor the existence transform of a function g(r) is that Jl-tgtrttdr be finite. However,this condition is not necessary,and a second, somewhatmore complicated,condition can be stated(seePapoulis,1962:9). The Fourier integralcan be written in severalways. Assuming,q(l) is real,as is usuallythe case,and noting t h a t I i n s i d et h e s q u a r eb r a c k e t si n e q . ( 1 5 . 1 0 7i)s a d u m m yv a r i a b l ew. e c a n w r i t e

: (tt2n)l_r-[l*-,r,;. ,',ay]o, e(r) ')dydto = (12")f g(r)e*,'" J = (lzr)J- g(-y){cos to(r- -r,) J+ j sin o(r - 1')) d_y,dor

F O U R I E R S E R I E SA N D F O U R I E R T R A N S F O R M S

f- ftft) : (ll2t1l I g(/) costo(r- y) dy d, J-_J__

(because g(l) is real) : (ll2rll f * t - S(yl (cost-rrcos.,l I J -J__

* sin c,rrsin ory) dy do ft(r) : (ll2trll R(or)cos tol dor tI

X(r,r)sin r,rrdo,

Although the independent variables are usually time and frequencyin eqs.(15.108)and (15.109y, thii is not necessarilyso. We can, for example,calculate the Fourier transform with respectto x so that cohas inverse-lengthdimensionsinstead of inverse-time.in which caseo in the precedingequationsbecomesr, the spatialfrequencyor angular wavenumber.

15.2.4Multidimensional Fourier seriesand transforms

J -

-lll2nll

s33

( 1 5 .l 1l )

where t

R(or) : I S(yl cos r,rydy J _

The Fourierseriesin eq. (15.93)was definedfor g(l), a function of one variableonly. We can expandg(x, l) in a Fourier seriesif we assumethat g(x, l) is also periodic in r with "period" equal to r, : 2rl\ (equivalent to oo,,: 2nl7).Then : : g(r, l) : L L tr,,,,cos /??Kox cos tlool

: c'osinetransJbrmof g(t)

f o,,, cos tnKoxsln ,?(oo/ *<.,,,,sin mKoxcosnort,l - t d , , , s i n nr . x s i nn o r , , l ) ,

(rs.l12)

and - X(a) : f I r{.r.t sin oy dy J _ .

: sine transform of g(t). If we expressthe exponentialterm in eq. (l5.l0g) in terms of cos to/ and sin o/ and comparewith the foregoing.we find that ,

: G(<'r) i[;]"1;:t'' I

(15rr3a)

h,,, = (41|\n

and

i

1(or;: tan I[X(o)/R(or)] : phasespe(,trumof g(r).

/

( ' 5r r 3 b )

As an example, the minimum-phase wavelet ($15.5.6a) shown in fig. 15.4acan be represented bv the amplitudeand phasespectrashownin figs.15.4b and 15.4c.Figure 15.4dis a variablearea disnlavof individualfrequencycomponents(at intervalsoi0.5 Hz) and their sum (leftmosttrace);the componenrs cancelat times earlier than 0, lorm the waveietfrom 0 to about 0.125s, and then cancelat latertimes.The loweramplitudesin fig. 15.4daround32 to 35 Hz cor_ respondto the notch in fig. 15.4b. I f t h e p h a s e^ y- 0 i n e q . ( l 5 . l l 3 ) , w e h a v ea z e r o _ phasewavelet.Then eachcomponentaddsin phaseat zero time to yield a symmetricalzero-phase wavelet, half of which occursbeforetime zero,as shown in fis. 15.5afor equal-amplitude componentsup to 32 H;. If the phasewere a linear function of lrequencv^/ : k u . e a c h c o m p o n e n tw o u l d b e s h i l t e db v t h e - s a m e time and the result would be the linear-phasewavelet shownin fig. 15.5b.Ifthe phaseT = 90o,the resultis a 90owaveletshownin fie. 15.5c.

I

| J

t\t2

1fl1

|

^,a.

s(x. /) cos mKox

,ra

X s i nr c o odl r d t

( 1 5 . 11 5 )

Similarly,the Fourier transform equations(l5.l0g) a n d ( 1 5 . 1 0 9b) e c o m e Ctx,o) :

where A(a) : [Rr(ro)+ K(.)l',. : umplitude spet'trumof g(t)

(15.1l4)

the coefficientsbeing given by equationssimilar to e q s .( 1 5 . 9 9a) n d ( 1 5 . 1 0 0 )f ,o r e x a m p l e ,

f-

f-

(l5.l16) t , " , * 1 . ' + . , r6 l , yd / , J| _ |J _ \_ f- tF ( r , r ) = 1 l ( 2 n ) , 1| | G ( x ,o ; E ' " ' - - , ' d xd o r . J-J-

The extensionto any numberof dimensionsr is obvious (the factor ll(2n): becomesll(2n),). For further details,seeFail and Grau (1963)and Treitel,Shanks, and Frasier(1961\. I 5.2.5 SpecialJunctions We shall have many occasionsto use certain special functions,especiallystep(/),the unit slep,definedby

"'0"'] 1,. :r: l

(r5 rr8)

Obviously,step(/) hasa discontinuityat r : 0. Multiplying a function C(t) by stepO "wipes out" the function for negativevaluesof I and leavesit unchanged for positivevaluesof l. A unit step shifted t., unlts to the right can be written step(/- ru):multiplicationby step(l - t,,)wipes out a function for all valuesof / less than t,,.Moreover,k step(l)is a stepof strengthk. To get the transform of step(l),we define

100.00 l0,00 60.00 r0.00 ?0.00 0,00 -?0.00

)

-{0,00

U

-60.00

c 3

00.00

/ \ i v

l

l 1/t /

- rm,00

(a)

l \ // V

tt

t

t

t

l

t

: ) = L

c

| 00. o0

I 00.oo

90.00

ltt. il

40.00

lo.0o

t0.00

t?.00 16.00

U cf

50,00 r0,00

0.e -!6.00 -12.o0

t0,00

U)

?0.00

I L

|0.00 o.@

3

(b)

l

(M5EC)

TI H t

40.00 U O :l

t

i

?

1

e

1

?

. 1

3 6

"

3

;

8

-ls.00 - r{ r . 0 0 -ls.00

.

;

(H FBEOUENC YZ ]

?

n

;

;

d

,

i

;

;

3 t

3 t l n

F R E O U E N (CHI Z ]

(c)

-0.2

0 o o

E F

64 (d)

96

112

(Hz) Frequency

Fig. 15.4 Synthesisof a wavelet by summing sinusoids.(From Yilmaz, 1987: 12 13.) (a) A minimum-phase time-domain wavelet, (b) amplitude spectrum, (c) phase spectrum, and (d) display

t.

80

of the frequency compon€nts and their sum (left trace). Note how the phase (P) (follow the same peak) has the same shape as the phase spectrum in part (c).

F O U R I E R S E R I E SA N D F O U R I E R T R A N S F O R M S

535

-0,1 9 0 o *

5

0.1

1 ! F.oqulney(Xt)

t8 Frequ€ncy(Hz)

t8 Froqwncy (H:l

(a)

(b)

(c)

Fig. 15.5 Synthesisof equal-amplitude sinusoidswith different phase shifts; right-hand traces show the sums. (From yilmaz, 1 9 8 7 : 1 5 , 1 6 . ) ( a ) E a c h c o m p o n e n t w i t h z e r o p h a s e ,( b ) p h a s e

shift linear with frequency, and (c) each component with 90' phase shift.

The simplestway of proving the relation, sgn(l) <_>2/jo,

(r5.r20)

is to calculatethe inversetransform:

ror+j 'iner)a, 0, : ,t^,L,1".,-' l,* .,il_(tes : rl t'nrrdr. ?fiJ,.

0)

because(costo/)/o is odd and the portions for negative and positiveor will cancel,whereas(sin tot)/o is even. The definite integral has the values0, jn, or -jrr ac cordingas I : 0, / > 0, or r < 0 (Weast,"l975, integral 621);hence,the right sideequalssgn(l).Thus, s t e p f t ) : [ + s g n ( r ) ] / 2 e n 6 ( r , r ) +l / j c o ( l 5 . l 2 l ) (using the transform pair I <+ 2n6(to); see eqs. ( t s . t 2 4 )a n d ( I 5 .I 2 8 ) ) . The unit step,step(r),is useful in dealing with discontinuous functions such as g(l) in fig. 15.6,which has valuesg(a+) and g(a-) at the discontinuity r : d as we approach from the right and from the left, respectively. We write S,(r)for the continuousfunction obtained by using g(r) to the left of I : a and the dashedcurve to the right of t : a, which is merelyg(r) displacedparallel to the vertical axis by the amount of the discontinuity.Then, g(l) : C,.O + [g(a+) g(a-)lstep(/ - a). The derivariveof g(r) is the sameas that of g.(t) exceptfor an impulse(seeeqs.(15.124) and (15.129))of strengthC@+) - g(a-) at t : a. The gate or boxcar,box,(r), is defined by

box"(r):0, t--:a; I - , r o- , = : l. l' o) , f ( 1 5 . 1 2 2 ) :0 .

,=)o

a

t

Fig. 15.6 Application of step(r)to describedisconrinuous functions.

The boxcar thus has a width a and unit height, with area a. It can be expressedas the differenceof two steps(seeproblem 15.18a).The transformofbox,(l) is box,(l) <-;

f"

g j-r d1 : I J .r,

e+i6a/2 -

e-j*/2

Jor

- 2 sin(aal2) : a sinc\aar2), o)

(rs.t23)

where sinc 6 : (sin 0)/0. The transform of box,(r) is shownin fig. 15.7a. The unit impulse or Dirac delta, 6(t), is defined by the relation

l-ut"u,dr

: g(o)'

(15.124a)

536

BACKGROUND MATHEMATICS

that is, 6(l) is an operator that setsthe argumentequal to zero. [The unit impulse,6(l), is an exampleof a distribution (a readily understoodaccount of distributionsis given in Papoulis,1962:app. l).) A distribution,e(t), is an operator that causesg(t) to be replacedby some function of g(r), dtg(r)1, rhat is, f-

d/ : Otc(/)1. I o(r)g(r) J

Thus, the spectrumof the unit impulseis flat, that is, all frequenciesare presentand have the sameamplitude and zero phase (seefig. 15.7c).Conversely,the transform of a constantin the time domain is an impulsein the frequencydomatn: t-

I*

r.-

f-

| e ' - ' d r = | c o s r o r d r - Ij s i n < , - r r d r J * J J _ : 2r'6(
(1s.128)

*

The integral sign does not have its usual significance here; it is used only becauseof certain analogiesbetweenthe abovedefinition and real integrals;thus eq. (15.124a\can be taken to mean

: g(0) E(r)g(r)

(15.124b)

The derivativeof a distributionis defined by the relation

using eq. (15.126)and noting that the sine integral vanishesbecausethe sine is odd. We now consider the relation betweenstep(/)and 6O. The derivativeof step(/)is everywherezero except at the origin, where we may think of it as a distribution: ds f- d(step(r)t f- 1 steP(I)-" dl | " "r(/) dr : d r d r J _ J _ '

ff(do/dl)c(r)dr = c(txdg/dt) 61 : -g(dg/dr); | | i" ",fr.. words, O.rO,l, " distribution that attributes to dg/d1 the same functional value that cr(l) does to g(r), exceptfor the minus sign.Thus, f-

dr : -dsldrl,=o.l I taalarls(r)

: - l-i: dr: -g(r)li: s(o)' becauseg(+-) : 0. Thus, (didl) step(r): 6(/).

A comb is a seriesof equally spacedunit impulses, the seriesusuallybeingconsideredinfinite:

' J -

To give a physical concept of the impulse (rather than a rigorousmathematicalanalysis),we start with a unit boxcaqbox,,(t)la,and let a approachzero while holding the area at unity, so that the height approachesinfinity as the width approacheszero: lim[box,,(l)/a]: 6(1;. In the limit, EO is an impulseof infiniteheightbut of zero duration, the "strength" of the impulse being unity. Furthermore, E(/ - /u) is a unit impulse occurring at t : to.If we multiply C(r) by 6(t - t0), the result is the value of g(l) at the time the impulse occurs,that is, g(ln).Papoulis(1962:280)showsthat d(I) :

.. sln (r)I llm

(rs.l25)

We can derive another expressionfor 6(l) from this result:

l:

cos<'lrdto :

:rl J ,

cosor/do

t 1,,

.,-- Jo

: 2n timll

et

r'l : 2n D(/).

( 15 . 12 6 )

u U_ @ n A ) ,( l s . l 3 o )

wheren is integral,""d A';;fixed time interval.Multiplication of gO by comb(l) replacesthe continuous function with a digitizedfunction sampledat intervals of A. The transform of the comb is derived in the next sectlon. A linear ramp is ramp(/) : /, = 0

0=t
t' er I t e i . , d t ramp(r)

: 1l/tor)[e 1 . r ( l + j r o b )- l ] .

(r5.r31)

rln

An exponentialdecay,e-k', for t > 0 and k positive, has the transform e r ' s t e p ( l )e

f-

e i'step(I)er-'dl

I I

(15.132)

(seefig. 15.7e). A double-.sided exponentialdecay, e rt'tfor k positive, has the transform f(,

e r,t <-) |

The Fourier transform of 6(l) is

r'' ar = . -'l : + l . atrr.- [ 6(r)e J _ I

combo: I

. r l e , k + t a ) , d=t l l ( k + j , u ) J,

: 2 lim Il" cost'-rldor: 2 lim sin'r l'

(15.129)

sr,s i.,dl -

J-

(l5.l2l)

< + 2 k l ( k ' +G ) (seefig.15.7f).

r/e x,/dr | e

J,

(15.133)

F i g . 1 5 . 7 E x a m p l e so f f u n c t i o n sa n d t h e i r F o u r i e r t r a n s f o r m s . ( a ) B o x , , ( t )(: b ) s i n c ( a t l 2 )(; c ) b ( / ) l ( d ) 6 ( 1- 1 , , )(; e ) e { ' s t e p ( l ) :( l )

( o o ) s t e p ( l- / 0 ) ;a n d ( h ) , e r , s t e p ( I ) (. N o k . a a n d k e r r , r(; g ) e are positive constants.)

538

BACKGROUND MATHEMATICS

15.2.6Theoremson Fourier transforms

the applicationofeqs. (15.136)and (15.137),we start w i t h e q s .( l 5 . l 2 l ) , ( 1 5 . 1 2 3 )a,n d ( 1 5 . 1 3 1a) n d o b t a i n :

Many theoremsexistregardingthe propertiesof Fourier transforms;the most important are listed below: g(l) <-+(co), kg(t) e kG(a);

(a) shifted step: step(/ - d e s r.'o[T[6(to) + l/jo];

(15.134)

krgr(t)+ krgr(t)<+ k,G,(w)+ krGr(a);(15.135)

e-je'box,(t)e a sinc[ja(to+ k)]; (c) shifted ramp: i.6(l + jtob) - l], g(t - t) er (l/to2)e-,.,0[e

Shift theorems: CQ a) <+ e-,*6(o); e-i",g(!)er G(to+ a).

(15.136) (15.137)

g(at) <->(l4al)G\oola); (lllal)sQh) <+ G(aa).

g ( t- t ) : ( t

(15.138) (15.139)

qt) e 2rg(-oo):

(15.150)

fortu< t
Equation (15.136)showsthat the effectof a time shift is to leavethe magnitudeof the transform, A(a), unchangedand to add a linear shift to the phase,"y(or). This is illustrated in fig. 15.7 by comparing (c) with (d) or (e) with (g). The proof of eq. (15.138)is straightforwardif we take c positiveat first and then considerthe necessary changeswhen a is negative.As an example,if

(15.140)

Derivativetheorems:

af;,i'*o'<,,),G(,); (l5.l4l) (1s.142\

0
CQ):!2t, : 0,

Integral theorems: cql dt <->(llj'.d)G(a);

for all other l,

t h e nf r o m e q s (. l 5 . l 3 l ) a n d ( 1 5 . 1 3 8 ) ,

(15.143)

'l _-

cQ)e('):)at1o7

-(rtjt)g(t)- o1,;or. f-

* tjab) - ll. : (l/2or21[e-rn,r(I

(1s.144)

An important corollaryof eq. (15.138)is obtained b y s e t t i n ga : - 1 , r e s u l t i n gi n

Convolution theorems:

- r) d1(15.t45) * s,(t):f - s,{r)r,{r s,tt) J_ e+ G,(co)Gr(orl:

J

2rg,(t)gr(t) <-l G,(co)* G(to). Cross-correlationtheorem:

(15.146)

f l d , , ( I ) : l t J r ) t r k+ r ) d r l,rr.ror1 J -

<+ C1u1C,1r),

)

wherethe superscribedbar meansthe complexconjugate.The convolutionand cross-correlationtheorems are discussed in $15.2.8and 915.2.9,respectively. T o p r o v e( 1 5 . l 3 6 )w . ewrite

gQ-a)el

- t),

= o

Symmetrytheorems:

I

(15.149)

where

Scalingtheorems:

ll I

(l5.148)

(b) windowed exponential;

C(-4 e->G(-<,r).

Equation(15.138)can be usedto illustratean important relationbetweena time function and its transform, namely,that the more the time function is concentrated,the more spread out is the transform and vice versa.Thus, in fig. 15.7eto h, the largerk is, the more closelythe time function approachesa spikebut at the same time the magnitude of the transform, l(o), broadensout. In the limiting caseof E(r), the time function is concentratedentirely at one instant, whereasthe transform is spreaduniformly from -tO to.

Equation(15.140)can be provedby substituting-r for r in eq. (15.109),giving 2ngl_ il :

t-

g(t-u1st-'61 .,: f - g(y)e j-o+")dy, (y: I J f: e j."l g(y)e-t-rdy: J -

t - a),

r'' I G(to;e dto.

and then interchangingthe symbols/ and o (because this doesnot affectthe equation);thus, 2rg(-to) :

fI

G ( r ) e ' - ,d t .

6-r*(o).

The proof of eq. (15.137) is similar. As examples of

( 1 5 . l 5)l

A s a n e x a m p l ew, e h a v e ; ; ; . ,

(ts.t32)

e-k'step(t)er 1/(k + jt'r),

F O U R I E R S E R I E SA N D F O U R I E R T R A N S F O R M S

539

and hence Il(k + jt) +s 2tek- step(-
Fig. 15.8 Fourier transforms of (a) cos o't and (b) sin o,,1.

(15.152) becausebox,(o) is even(seefig. 15.7b). The derivativetheoremsare of fundamentalimportance. The proof follows directly from eqs. (15.109) and (15.108)if we differentiatewith respectto / and or,respectively. Thus, A dg_ llt G(t) l ler-'; dto or zlrJ * dl

E(r)<-++l;

= I l _ lioGor)lej'' dt'-t' 2-l

E(/ * /o) <+ e=j.,0;

(l/2)tD(t+ rJ + 6(r - ro)l<+ cosorlo; (l/2j)16(t* to)- 6(r - rJl <-+sin oto;

nence

(ls.l54)

dgldt <+joG(o).

l t+ 2n6(o);

Differentiatingn times,we get eq. (l5.l4l):

eajoot<) 2tr6(o T o); cos t,ro/ e; n[E(to * ar) + 6(o - or)]; sin oul e+ jrr[E(o * ,J - 6(to - <,r)].

d's(t) <->(to),G(ro). dr The most important applicationof eq. (15.141)is in the solution of differentialequationswhereit permits us to replacederivativeswith a function of or.It also enablesus to obtain new transform pairs by differentiation of a known pair. An interestingapplicationof this theoremis 6(r) : (d/dr) stepO <+ jto[rr6(<,r) + l/jor] : I (usrngeqs. (15.129)and (l5.l2l) and the facr rhat orE(or): 6;. Equation (15.143) is easily proved using eq. ( l 5 . l 4 l ) .W ew r i t e f' f' : g,(il = | Sftlar I g ( x )d x e + G , ( t o ) . J J dg,(t)ldt : g(t) e+ joG,(to) : G(to), wherewe have used Leibnitz' rule to differentiatethe integral on the right. Leibnitz'rule statesthat, given F(0 : I2',',', O(r, \) d\, then the derivative of r'(l) with respectto t is given by dF

fn' aO

dr

J,,,at

OA + O(I. D)

db -

dr

da

6 ( t . 0 l' d r

(seeKaplan, 1952:220).Therefore, o, <-+( I /jto)G(
Equation (15.144)can be proved in the same way, starting from eq. (15.142).Equations (15.145)to (15.147)will be discussedlater. , 5 . 1 3 6 )( ,1 5 . 1 3 7a) n d ( 1 5 . 1 4 0 ) , U s i n ge q s .( 1 5 . 1 2 7 () 1 we can write the following transform pairs:

The transforms of sin to.t and cos tootare shown in fig.15.8. To obtain the transform of the comb, we find its Fourier seriesexpansionusingeq. (15.104)and then find the transform of the series;replacingg,(y) in eq. (15.104)with E(./)and writing I: A, we get comb(t):

I

afr - nL):

,,:T,io.

ll! + (2tL) ) cos noor, n='

U s i n ge q . ( 1 5 . 1 5 4 )w, e o b t a i n € € L d(l ,?A)(+ (t)u ,f

d(trr- maol.

that ls. comb(r)e+ <,-ro comb(to).

(15.155)

15.2.7 Gibbs'Phenomerutn At a discontinuity,the Fourier integral (and the Fourier series)givesthe averagevalue,][g(a-) + g(a+)] in fig. 15.6.However,S@t e), e infiiitesimal, does not approachthe averagevaluesmoothlyas e goesto zero. For simplicity,we take a : 0 and write c@ : c,.@+ [g(0+) - c(0-)] step(r).

BACKGROUND MATHEMATICS

540 Equation(15.109)gives f-^ g(r) : lim (ll2trl I G(or)ei''d0) \rJr

: lim( tt2n\l ^ '@

^{[ -r,r,.

,*' 6y]si-6,.

J^tJ-

Interchangingthe order of integration(see$15.2.8), we have * /f r \ l.

" ar,r)d"r g(r): limttlUtJ c(/) (J _ n.''" ^ 0, " .1 : li m( ttz.,rtl g (" f) .1 " " ^+d

J(r_I)l

J*

^

F i g . 1 5 . 9 G i b b s ' p h e n o m e n o na t a d i s c o n t i n u i t y .

sin\(r - ]r) : r i n . ,[ -. s" ( y ) o, n ( t- y l ^ _ _ J_ l-* : lim {e,(y) + [g(0+) | Ie-J -

sinr(t - /) or' g(o- )l step(/)) = [

J_

T\r

y)

sin'r{r - rr) , ,t) lim l r | t - y l o,

- -r) sin'\'(/ + [g(o+)- c(o-l]rimI o, ltg n-Ju

Y)

f-: + ( I / T ) [ C ( O +) I 9 , ( / ) 6 ( I , r / )d . 1 ' J -

| ^r

rt0-)l lim I sinc.vd.r. ^r_J _ and taking\(/ - "v): r so that dl = usingeq.( 15.125) -dx/\ (we do not go to the limit in the right-hand integralbecausewe wish to study the behaviorof this term as \ goesto infinity). The right-hand term can be written

- c(0-)1(J'_ sincx dx c'(t): (l/rr)[g(0+)

-g

l,'

.inc., o").

BecauseJ1- sinc x dx : l'n (seethe derivation of eq. ( 1 5 . l 2 l . ) )w, e h a v ef i n a l l y . c'(t) : [g(0+) g(0-)][l/2 + (l/n) lim I sinc x dxl. The graph of the secondfactor is shownin fig. 15.9. As \ -+ -, the peak values do not change,but the ripple moves toward the discontinuity from both sides.At the discontinuityI : 0, we have for g(t), g ( 0 ): g , ( 0 )+ l [ 8 ( 0 + ) - 8 ( 0 - ) ] :

however,an infinitesimaldistanceawayon either side, "overshoot" of about l8%' (9" at the top we havean and9'k at the bottom). Gibbs' phenomenonis important wheneverdiscontinuitiesare present,for example,in the applicationof filters and windows.If we multiply by a boxcar in the at the edgesof the time domain, the discontinuities "ringing." The objectiveof "winboxcar will produce dow carpentry," that is, of shapingwindows,is to remove discontinuities(and discontinuitiesin derivatives)so as to minimizeringing.Seealso$15.7.5. I 5.2.8 Convolutiontheorem The t'onvolutionof two functions,g,(l) and gr(/), usua l l y w r i f t e n i n t h e f o r m g , ( / ) * g . ( r ) ,e q . ( 1 5 . 1 4 5 )i ,s definedas -| -

g,(r)* g.(/):

J_

s,(t)r,(r t)dr.

We seethat g,(r - r) oertJt.. g,(t) displacedI units to the right, whereas8:(-r + l) is g.(t - r) reflectedin the vertical axis.Thus, convolutioninvolvesreflecting one of the curves in the vertical axis (often called "folding"), shifting it by t units, multiplying correof the two curves,and summing spondingcoordinates from -- to *- (seefig. l5.lOa).The valuedepends only on the time shift t and is independentof which curve is shiftedand reflected,that is, g , ( t )* g . ( t ) : 8 . ( t )* g ' ( r ) '

(15.156)

This is illustratedin fig. l5.l0b. The convolutiontheorem.eq. (15.145),can be proved as follows:

.I f- r ' fs , ( I ) * 8 . ( I )| el+l g , ( r ) 9-. (rr) d r l e ' - ' d r ' .l _LJ

*

We assumethat g,O and g,(l) havetransforms,that is,

I B , ( , ) l d
-

i: 1,2.

F O U R I E R S E R I E SA N D F O U R I E R T R A N S F O R M S

G,(co) *

541

G,(')erlrlz';f c,1yl[j-4ft) x ej(.,+),), o"] o,

I I

<+tt tnll

I I

Y

x : a - / ,

dy _c,(.r,)ej,,

I G,(x)sr.,6*.

<+ ( | I 2T)[2Tg (t)][2T s,(t)] <+ 2ng,(t)gr(t). I 5.2.9 Cross-correlat ion theorem T,he cros.c-correlation Junction, S,.(t), defined by eq.

(ts.t47), reflected and shifted

g, reflected and

shifted

\b) F i g . 1 5 . 1 0 C o n v o l u t i o n i n t h e t i m e d o m a i n . ( a ) R e f l e c t i n ga n d s h i f l i n g g , ( r ) a n d ( b ) g e o m e t r i c a l d e m o n s t r a t i o n t h a t g , 1 r . ;* .C,(I)= g,(1) * g,(r.).The overlap is the same regardlessof which is reflectedand shilied.

We shall assume that this relation means that J--k,(r)l'dl is alsofinite.In this case,the orderof integratron can be interchanged (papoulis, 1962: 27), giving

-

f

- r)e,.,ar]a", r,f"f s2(t [-

j.r,.',or]o" J-r ,",[f e:(/)e

(wherel:t-r), ffsr(/) * gr(1)e+ | g,(r)e r-,dr I gr(/)e ,.' dli J _ J _

e G,(o)G,(ar). The inverserelation,eq. (15.146),can be provedas follows; G,(to) *

is closely related to the convolution function. Obviously,$,r(/) is the resultof displacingg,(r) r units to the left and summingthe productsof the ordinates.It is evidentthat

S,r(/): Sr,(-r).

o

s,(t)* s.(t)*

I

o , . ( / )= | g l l l g : ( rr r ) d r . .,_

G.(or) <-+rtt2nyl f i- o,,r,

J _ t J_ x G.(trr- i,f dr,]er'' dor.

(15.157)

The cross-correlationfunction can be regardedas a convolutionof two functions,the secondof which has beenreversedin time:

0 , , , ( r ) :g l t ) * g , ( - t ) : 0r,(/):g,(-r)*g.(r); g,(l) * g.(t) : cross-correlation of g,(r)with Cret), : cross-correlation of g:(r)with sl-t).

( r 5 . 15 8 )

Theserelationscan be verifiedby drawingcurvesof glt), gr(t),gl-t), and gr(-t), and checkinggeometrically the previousrelations.They can also be verified by substitutingin the integrals;for example, f: g,r(/) g,(t) * s.Gt) : I g,(r)gr(t+ r) dt J

*

(note that for zero shift, the argumentof the functions in the integral is r, not l, hence replacingg,(l) with g:(-r) on the left-handsidechangesthe sign of t in gr(r)). The cross-correlation theorem,eq. (15.147),states that d , r { l )< - +C t r u r t C , t ( ,:) l O , . t t - r ) . wtrere-,1rj is the conjugarecomplex of G,(to).The proof is straightforwardif we note the first relation of eq. (15.158)and changethe signofr in g,(r - r) in the first integralin the derivationof eq. ( I 5.145)(Sl5.2.8). This changesthe first exponentialin the next to the

BACKGROUND MATHEMATICS

542 last line of the derivationto si", and eq. (15.147)follows immediately.Writing @''(
The autocorrelationfunction assumesits greatest valuefor zeroshift,that is. 0,,(0)>0,,(r), (seeproblem 15.22a). I 5.2.1I Multidimensionalconvolution Equation(15.145)can be extendedto more than one dimension.In two dimensions,we definethe convolution by the expression

t0 , , ( 0 ) : d , , t 0 :; I g , ( " ) 9 , ( " ) d "

f- fg , ( x ./ ) * g r t x .r ) : | | g , ( o ." t

f- -

x Cr(x - o, I - t) do dt'

J _ J -

J -

do lll2rl I G,(to)G,(or) J_

I I |\

t

(15.163)

t+0

(15.164)

( 1 5 . r 6 0 ) We can derivethe two-dimensionalconvolutiontheorem as follows.Using eq. (15.116),we have

( l / 2 t ) | G , ( o ) G , ( t ' r ) d o .I J _ I

g,(x, /) '* gr(x, t)

.l f-f-rf-le , l I l l l t , l " . r )x g , ( x - a - t -r ) d o d t l

This is referred to as Parseval'stheorem.

X e

J -.f -LJ -J j(K-t+@/) djr d/.

I 5.2.I0 Autocorrelation

Interchangingthe order of integrationgives

When gr(/) : g,(t), the cross-correlationfunction becomesthe autocorrelationof g'(r)' that is,

l- fg r ( x ,r ) , *g , { x ,r ) e ; | | g , ( o ." )

f-

0 , , ( I ) : I B , { " ) B ,+t rr ) d t < +l G , ( t o ) l ' .

J - J -

- f x ll 'lf l g , ( * - o 1. - r ) L J_ J _ _ I r t & -' " d x

(15.161)

J -

Clearly, 0,,(/) : 0,,(-/), so that Q,, ts an even function. theorem,eq. (15.160)'beWhen I : 0, Parseval's comes

: ,r,t"lt' do. dr : (1/2r)flG,(o)l' S,,(0) f (15.162) ln most cases,g,(t) is a voltage,current, velocity,displacement,and so on, so that lg,(r)ltis proportional to ih. .n..gy; adopting this point of view and choosing the units properly,we can say that $,,(0) is the total energyof g,(l). The quantity (l/2n)lG,(o)1' is thus the energyin the spectrumbetweenthe frequencieso and or * dor; it is called the energydensity or spectralden' -o [e *o; sit)r.We note that the integrationis from by the cosines and sines replaced we if we recall that exponentialterms e-j'', it is clear that we must recombine terms for +o to get the result, that is, energydensity of frequency, : lG'(-to)|' + lG,(+or)12. W h e n g , ( l ) i s a r e a l f u n c t i o n .G , { - < , r ): G r ( t , r()s e e the total enproblem 15.17b)so that 2lG,(+or)1' gives ergy density for the frequencyor (if we are interested only in relativeenergies,as is usually the case,we can forgetdoubling).

dll do dr'

X s f-

*

fr)s irr---'r |I | t,to' , ._ J _

'[[ Ll -l X e

[ - r , , ' - o .I - r ) -

j ( k ( r - o ) + o o - " ) ) d ( x-

o)

x d(l - "tl ooa'. I

rf- l-

I

dtl ,- ll t,,o.r)e r{r'"-''do L J- JI X Gr(rc,ar), + + G ' ( r co, r ) G , ( r r, o ) ,

(15'165)

which is the equivalentof eq. (15.145). I 5.2.12 Randomfunctions The periodic and aperiodic functions that we have been consideringhitherto have one property in common: If the processthat generatesone of thesefunctions is repeatedexactly,the same function is generated. However, in many instancesrepetition of the processgivesa differentresult eachtime, for example, of microseismsgive a function g(t) that measurements neverrepeatsno matter how often we repeatthe mea-

F O U R I E R S E R I E SA N D F O U R I E R T R A N S F O R M S surement.Functions of this kind that cannot be predictedexactlyno matter how often we repeatthe measurementsare calledrandomfunctions. Becauserandom functionscannot be predicted,we useprobability theory to deducetheir properties.The set of functions obtained if an experiment were repeatedan infinite number of times is called an ensemble. lf we arrive at the samevalue for some property of the ensemble(for example,the averagepower density for the frequencyto),whetherwe averagethe values for each of the tracesat a certain instant in time or averageall valuesfor one trace,then the ensemble is said to be ergodicand we can determinethe statistical propertiesusing a sufficientlylong portion of one function ratherthan havingto makemeasurements on many functions. A .stationarytime series is one whose statistical propertiesare independentof the locationof the ori_ gin t : 0. It can be shownthat ergodicensembles must also be stationary(seeLee, 1960:208 9; Bendatand Piersol, 1966: ll l2). We assumethat the random trme serieswith which we deal are ergodic and sta_ tionary. The autocorrelatktnof'a randomfunt.tionis defined by the equation d ' , ( / ):

l i m{ l l 2 \ l 7f-

E t 6 t s J t* r ) d r . J

Note that this definition differs in form from that in e q .( l 5 . l 6 l ) b y t h ef a c t o r l l 2 T a s w e l la s t h e l i m i t i n e process.Becausea random function does not aol proach zero as / approaches-r -, the integral for the Fourier transform,eq. (15.108),does not converge and G(or)doesnot exist.Thus, thereis no equivalent o f e q .( l 5 . l 6 l ) f o r r a n d o mf u n c t i o n sW . h e nr : 0 . lt'

0 , , ( 0:) tin ttt2Tll gi(r) dr T

: mean squarevalueof g,(t) The portion of one measurementof a random functron, g,(l), includedin the interval(-7, +T) will be writtengi(r).lf we express gi(r) asa Fourierserieswith period 2T the seriesrepresentsg,(l) exactly in the interval (- T +T) but not elsewherebecauseit repeats gi(t) in each interval of length 27. ln the limit as ?"--r -, ci(1)becomesg,(t).We write gi,O for the autocorrelationfunctionof giO:

0;,(/):

l' ,siilsi|

+ t) dr

Wecanexpanddi,(l) in a Fourierseries; di,trl : L a,e"-u' ":-: T . , , - ^ , fl l ' d' ,, ., ( l ) e i " .- "l , d l l Lc"" lrrl ',,,'

usingeq.,'r;;;

-r;.

di,(t) by 2T andlet Z -+ -, $i,(t) approaches S,,(r)as defined in eq.(15.166). Thus, -

lT

Oll(/)= lim I e,x'(ll2\l Si,(r)e,^0,61, ^-': ' ; , ' = l l l 2 n ) l e ; - , d o l d , , { r ) e, - , d r , J _

J-..

: { l l 2 r - y l O , , { o ; e i -d, t o . I

where fO , , { t o :) | 0 , , ( r ) . - , - , d 1 .

( r 5 . 16 7 )

J -

Thus, although a random function doesnot havea Fourier transform, its autocorrelationfunction has a Fouriertransform:0,,(l) e O,,(or).Also,

0 , ' ( 0 :)

^ nnTf

d, ,si(r)

: ( l l 2 n ) l O , , ( c od) t - r .

(15.168)

(15.t66)

1'

J

543

: nrr rr *l aiuia.

Becauseof the factor ll2T, Qrr(a) gives the power density, not energy density as with aperiodic funct i o n s . E q u a t i o n s( 1 5 . 1 6 7 )a n d ( 1 5 . 1 6 8 )e x p r e s st h e Wiener autocorrelation theorem, that the Fourier transform of the autocorrelationof a random function existsand givesthe power-densityspectrum.The autocorrelationfunction and the power-densityspectrum are real, evenfunctionsand $,,(l) has its greatest value at the origin (seeproblem 15.22b). The cros,s-correlation of randomfunctions is defined in a mannersimilarto eq. (15.166).Let g,(r) andgr(t) be random functionsfrom differentensembles, for example, the input and output noise of an amplifier; then t

l I

g r ( r ) g . (+I r ) d r . ( 1 5 . 1 6 9 t

0 ' . ( r ) :I i m . - l T--LtJ

T

E q u a t i o n( 1 5 . 1 5 7 )0, . , ( r ) : 0 , , ( - r ) , h o l d s f o r b o t h randomand aperiodicfunctions. I 5.2.I 3 Hilbert transfbrms The Hilbert transform is a special form of Fourier transform. Let SQ)be any real function and c(/) (-) G(') : R(co)+ jX(o). Now g(r) can always be divided into even and odd parts,&(/) and g"(r) (seefig. 15.l I ), where

s,(tl= ]lslrl* s(-14. I

: ]tsur- c(-r)1.)I g"ft)

rts.lTo)

BACKGROUND MATHEMATI CS

544 Becauseg"(/) is even, fg.(/)e+ | g(r) "ot ot dr : R(<'r) (15'171) Jn

( s e ee q . ( l 5 . l l 2 ) ) . L i k e w i s e . s.(/) e jx(<,l).

0s'172)

: > If g(r) is also causal(see$15.5'6),C"Q) g.(t) for I -C.(l) for I < 0, that is' 0 and g.(l) : c"(l) : g.(r) sgn(t)' g"(r) : s"(t) sgn(/) t h a t s g n ( r<) + 2 l j a ( s e ee q ' . ecalling ( s e ee q .( 1 5 . 1 l 9 ) )R ( 1 5 . 1 2 0 )w ) ,e o b t a i n jx(.'r) : (l/2t)R(o) * (2lja) o n u s i n ge q . ( 1 5 . 1 4 6 )T.h u s ' I _ fX l . l T: - - V | [ R ( v ] / ( < ' r l ' ) l d r ( 1 5 ' 1 7 3 ) f I J _

: _ R ( t o )* ( l / t r o ) , where.% indicatesthat we take the Cauchyprinciple value at r,r: y (seePapoulis,1962:.910 and footnote on p. l0). Likewise, R\a):

I'

f. 7 | l x ( v t l ( a - " r ' ) l d " t( 1 5 ' 1 7 4 1

gO and the Fig. l5.l I Relation between a real causal function g,,(t)' respectively' g,.(1) and functions' o<Jd and deiived even

:1,,'-i',,,,r. ) e f i n et h e H i l b e r t E q u a t i o n s( 1 5 . 1 7 3 )a n d ( 1 5 . 1 7 4 d transform. Given either R(or)or X(o), the other can be calculated. writing G(o) : l(to)girt't and taking logarithms gives l n [ G ( o ) ]: l n [ , 4 ( t o )+l j 1 ( t o ) : : in this case,R(o) : ln[l(or)] and X(or) 1(o)' Beto' l(to) frequency the of amplitude the is causel(o) is the squareroot of the Fourier transformol the autoknowingl(to), correlationfunction(seeeq. (15.161)); (15'173)' The Hilbert we can calculatery(to)by eq. the phase of calculation the allows transform thus from the autocorrelationfunction: l f f'( t o ): X ( a ) = - " 7' f r | { t n [ . a t l t ] l t ( ' ) - I ) ] d - v J_ (l5.l7s) Given the real part of the Fourier transform of a real, causaltime function, the Hilbert transform enables us to find the correspondingtime function' A similar problem is: Given.the real part of a complex time function whose imaginary part has a transform +90o out of phasewith that of the real part, find the complex time function. Let f(t) be a complex time function. where f (t) = x(t) + jY(t); F(<,,): X(o) + jY(r) : X(or)[l + jQ(t)],

phaseby +90'' O(to) being a filter that changesthe but has no effect on the amplitude spectrum;0(o) ts referred to as a quadruture.filter,and y(t) as the quudraturetrace. Becauseerint2: * j, we can selectQ(to) equal to +j, -j, or *j sgn(r,r).Selectionof the first two choices givesq(r) : -+jE(r)from eq. (15.127\,which is not a -j sgn(ro)'Then, iseful result.We thus selectQ(o) : -jx(o) sgn(o) l(/) e X(a)Q@): : jx(r) sgn(-t'r). ) ,e h a v e U s i n ge q s (. 1 5 . 1 2 0()1, 5 . 1 4 0a) ,n d ( 1 5 ' 1 4 5w (15'176) r(r) : x(/) * (llrt), that is, .t(l) and -l(l) form a Hilbert transform pair ( s e ee q .( 1 5 . 1 7 3 ) ) . Although/(t) can be found by usingeq' (15'176)'it is easierto calculateF(or) and transform to get ll)'

Thus, F(t):

X(o) + jf(t'r) : X(otXl : X(o){l + sgn(to)}' :0,to<0, : 2X(a), to ) 0, : 2X(o)step (or).

+ i0(,)l'

(15.177)

) n d ( 1 5 . 1 7 7 f)i n d applicationin E q u a t i o n s( 1 5 . 1 7 6 a analysis($9.I 1.4). complex-trace

LAPLACE TRANSFORM

545

r-f,. "or

l5J Laplace transform

( -^rI -

The Laplacetransformis closelyrelatedto the Fourier transform. If we do not usedistributions,many functions such as sin at and cos a/ do not have Fourier transformsbecausethe integral giving the transform does not converge.However,if we multiply the function by a convergencefactor a rrrtr, o being real and positive and large enough that lim,_*_[s "tlg(1)]: 0, ,n.n . "tttg(/)has a Fourier transform that is calledthe Laplace transform of g(t). lf gQ1: 0 for I ( 0, we get the one-sidedLaplace trans/brm:

c(t)el,. ,"rtrl.r.,dt-

where,r: o * jt'-r,the real part of s beinglargeenough that lim,,_[e "g(l)] : 0. (The Laplacetransform, G(^r'), is distinguishedfrom the Fourier transform, G(or),by the variables insteadof t'r.) The inverseLaplace transformationbecomes

: ttrzntl d<,; _c(,')ei""

(notethattherealpartofs is largerthantherealpart of k, so that e fi+')1: 0 for t :

fdtrl | G(s)e','.r",r/ Gg)e",d^1 (15.178b)

"l: /e :', s t e p (
J

,'d/: . '1,:.: *" atr; <+ J-atr;e

*_);

cos .// e+.s/(s:* crr):I sin zrl <+ ul1.s)+ u:11J

-':]' coshz'rr - ''11'.

] sinh rrr e ul(,\)- u.). J

'

(r5.r84)

The last four resultscan be obtaineddirectlyby integration or by substitutingk : -+ja or k : +a in eq. ( 1 5 . 1 8 2a) n d c o m b i n i n ge x p o n e n t i at e l r m st o g e t c o s a t , a n d s o o n . E q u a t i o n s( 1 5 . 1 7 9 )(,1 5 . 1 8 0 )a, n d ( 1 5 . 1 8 2 )s h o u l d b e c o m p a r e dw i t h e q s . ( l 5 . l 2 l ) , ( l 5 . 1 2 )7. a n d ( I 5 .I 3 2 ) .

Most of the theoremson Fourier transformshave counterpartslbr Laplacetransforms.The most useful theoremsare listedbelow.Note that a ) 0 in all cases.

wherethe path of integrationis a line to the right of the origin parallelto the imaginaryaxis suchthat the integral converges.The calculationof Laplacetransforms is usuallyrelativelysimplein comparisonwith Fouriertransforms.but the inversetransformationis generallydifficult. The Fouriertransformis more convenientwhenwe wish to discuss properties that depend upon frequencyand/orphase.lt is very usefulin certainareas of probabilitytheoryand in solvinglineardifferential equationswith boundaryconditionsthat can be expressedin Fourier or Fourier Besselseries.The Laplacetransform is very usefulwhen we investigatethe analyticalpropertiesof the transform(as in circuit analysis)and in solvinglinear differentialequations with constantcoefficients when initial conditionsare glven. The Laplacetransform of somecommon functions can be easilyderived.Thus,

Jn

(ls.l82)

15.3.2Theorcmson Laplut'etran.sJbrms

or

: (|t2nll

: 1 er,<-) *,."0,: _ti*-.';ils*fr 1,,-.

"dr: G(s), J-rtrf. (15.178a)

g(t) : (ll2T)

(1 5 . 1 8 1 )

lu

15.3.1Introduction

e -g(t)

r)l: i, t'

( r 5 . 17 9 )

g(r) <-+G(s); (15.185) k s ( t )e k c ( t ) : k , g , ( t )+ k . g . ( t )< + k , G , ( r )+ k , 6 . ( s )(. 1 5 . 1 8 6 ) Shift theorems: t:(

r.r)step(r- u) e e ,.'G(s); (15.187) e " , 9 ( t )< + G ( . r+ a ) . ( 1 5 . 1 8 8 )

Scalingtheorems: g(ut)<+ (lla)G(slu): (llu)g(tla)<-+G(a,s).

( 15.189) (15.190)

Derivativetheorems: d,,e(/ ) - .s" rg(0 *).-' <-).r"G(s) dt"

.r" rgr(0-r)

_..._9,,(0+); ( t's?) e d"G(s)/ds'.

( l 5 . l 9 )l

(15.192)

Integraltheorems:

ar<+(l/s)G(s); J'sfrl

(15.1e3)

( t t t ) s ( t ) *| . ' - o ( ro) r .

(t5.le4)

J.

Convolution theorem:

( I 5 . 18 0 )

I' - r)dr e+G,(s)G"(s). g,(r)*,9.(1):I s,(r)s.(r .,0

(15.195)

BACKGROUND MATHEMATICS

546 In eq. (l5.l9l), the symbolsg(0+) and g'(0+) denote the values of g(l) and its rth derivative at / : 0 when / approacheszero from the positiveside. The proofs of most of the precedingtheoremsare simple.For example,for eq. (15.187)we have

- a) step(r- a)e "'dt

-l sl(t)* s2(/)

[rU

:

- a)e" dt,

ftu

(r+d)d/ r:

: fdrl.

g,(l) : 0, t < 0. As shownin fig. 15.10a,g'(t) : 0, t < 0, whereascr(! - r): 0, t > I' Thus,changingthe limits to +- would not changethe value of the integral. The convolution theorem, eq. (15.195),can be proved as follows: I "dr' sJlls2l- r)dtle 1 | Jo "lo f-rl'

f-rf. I "dr' - r)drle - r)step(r .tl 1l g,(r)s2(I tJ" J"

t - a,

: .-"'Jsfr)" "'d.v: e "'qs).

wherethe changeof the upper limit and the insertion of step(t - t) have not changedthe value of the integral becausestep(f- r) is unity for r ( / and zero for t > L Changingthe order of integrationgives

r-

b y p a r t s .g e t t i n g F o r e q .( l 5 . l 9 l ) . w e i n t e g r a t e

o.t "l- - ,-r,|.-r(I)e.,d/. = g(r)e * l?.t. ,161 " dr J,,dr J,, I -c(0+) : + sG(s). By successiveapplicationsof this result, the general formulais obtained.For eq. (15.192),we write G(s): Jlg(r)e "dl, and then differentiatewith respectto s', giving

t't"'' "o' s{r)j,te";dt: J,,tI

d :G i - a

::

and hence -tg(t) <->dG(s)/ds.Successivedifferentia)' ewrite ) .o p r o v ee q .( 1 5 . 1 9 3 w t i o n sg i v ee q .( 1 5 . 1 9 2 T 1,,g(t) dt e->G,(s),then differentiate,obtaining s(t) <-> sG,(s):G(s);hence,G,(s): (lis)G(r). The conversets proved by writing f-

f-rf" d rr f IJ . c o t o . ' J= |. lLlJrNr ( r ) e J d s . l-

rf*

I

: I g t r rI le " a s l o r ' ) tJ' Jn

on changingthe order of integration'Then i'-

f-

J.

J,,-

^r,,-

f-/r\

"dr: I c 6 ) 0 . ' =| g ( r-)1"rI,a t : I l l l s ( r ) e J,,\1/

hence,(l/r)g(r) <+ Jlc0) as. When Laplacetransformsare being used,the convolution of two functions,g'(l) and gr(l), is usuallydefined by the expression

g,(r)* g,(0: Is,(")rdr- r) dt. (15'196) This definition appearsto differ from that givenin eq' (15.145)becauseof the different limits. Howeveqthe differenceis only apparentand is due to the fact that

rf-

g , ( I )* ,g , ( r )e + | s f r ) l |

*,tt

I - r ) s t e p ( l- r ) e " d r l d t .

t'Ju

J" t@

usingeq'(15'187), e+l s,(r)e''G.(s)dt, J0

<-+G,(s)G.(s).

15.4Linear systems I5.4.I Introduttion We use the term systemto denote a group of objects so relatedthat, when an input is appliedat one polnt, an output is generatedat anotherpoint. We may know little or nothing of the detailed workings of the system. A linear systemis one in which output li(r) is proportional to input g(l), that is, if gO - > h(t'1,then kgo ) kh(t), where k is a constant and the arrow representsthe effect of the system.(Cheng, 1959,has a good accountof linear systems.)By consideringg(l) io be the sum of two signals,we seethat theprinciple of'superpositiorholds, namelY, k,g,(t) + k.g,(t)-+ kthlt) + k.h,(t)' (15'197) A systemis time-invariantif Ihe sameinput produces the sameoutput regardlessof the time when the input is applied,that is, kg(t - t,,) -+ kh(t - to)

(15.198)

for all valuesof /n. In principle at least, systemscan be describedby meani of differentialequationsrelating output to input. The correspondingdifferentialequationsfor linear systemsare lineat and the coefficientsof the equation are constantswhen the systemis time-invariant' We shall study the piopertiesof differentialequations describinglineaq time-invariantsystems.Our drscussion will not dependon the order of the equationand

541

D I G I T A L S Y S T E M SA N D z - T R A N S F O R M S becausemany systemsare representedby secondorder equations,we considerthe equation d'hQ) -d;;t

dhtt\ * o,* a ) h ( t l: s ( t l . a)r'

(15.199)

dl)

- l?(o+). sH(s)

where &(0+) and ft'(O+) are the valuesof h(t) and dh(t)ldt, respectively, at / : 0. S u b s t i t u t i nign t h e e q u a t i o ng i v e s (s2* a,s + a.)H(s)- [(s + a,)h(0+) + ft'(0+)] = G(s). Solving for t1(s),we have

H(r) :

( I 5.203)

But

f r o m e q .( l 5 . l 9 l ) , h e n c e , - h'(o+), <-+s2H(s)- s/r(o+.1

o1'1'dr)

F(s) : s{(s)

e sr,6) - f"(0+) : rG) - f,(0+) dl1f,Q)ltdt

U s i n ge q . ( 1 5 . 1 9 1 )w, e g e t o:;(l)

tweenJ(l)andf (t) asfollows.Fromthepreceding,

G(t)+ (s + a,)ft(0+)+ ft'(0+)l l2-larsl-u"

( I 5.200)

The numeratorin eq. (15.200)is calledthe total excitation transform;it dependsin part upon the excitation (input),g(l), and in part upon the initial conditions of the system.When the initial output and its first derivativeare zero,that is, lr(0+; : 0 : ft'(O+), the systemis said to be initially relaxedand the total excitation translorm reducesto the input transform, Gk). For convenience.we shall assume henceforth that the systemis initially relaxed. The quantity, l/(sr * a,s + a,) is known asthe tran,sJbrJunction,F(.r).It is a function only of the properties of the system.Thus, the transformof the output correspondingto any input g(l) is given by H(s) : f(s)G(.s)

( l 5 . 2r0)

We apply the convolution theorem to this equation and obtain

t' h(t) : .f (t) * g(r) : | ilr)f(t - r) dr,

f (t) : dv,Olldr+ r(0+) 6(r). (15.204) Becausef(t) : 0, t < 0, the step responsehas a discontinuity at the origin of magnitude//0+) Thus, the impulseresponse,/(r),is equal to the derivativeof the step response./"(l) plus an impulse at the origin of magnitudeJ(0+). 15.4.2Linear systemsin seriesand parallel Assume that two linear systemswith transfer functions Fr(s)and F,(s)are connectedin seriesso that the output of the first systemis the input of the second system;then FI,(s): F,(s)G(s); hlt):.f,(t)*g(t); H.(t) : F,(s)FI,(s) : F,(r)F,(s)G(.s);

h.(t): .f.(t)* Ult) * s(t)1, : lf.Q)*.I(r)l* g(r).

( r5.205)

Thus, two systemsin seriesare equivalentto a single systemwhose transfer function is the product of the two individual transfer functions. Obviously,this result holds for any number of systemsin series. When an input g(r) is appliedto two systemsin parallel,the outputswill be superimposed so that H(s) :11,(.s) + H,(s) : G(s)f.,(s)+ c(s)f,f), : G(^r)tr,(s) + f,(s)l (15.206) Thus, the equivalenttransfer function for systemsin parallelis the sum of the individual transferfunctions.

(1s.202)

JO

w h e r e F ( se) J Q ) . If we apply a unit impulse 6(l) to a linear system, t h e n 6 ( r ) < + G ( r ) : * 1 f r o m e q . ( 1 5 . 1 8 0 )s, o t h a t l{s) : f(s) and h(t): f(t). The response/(r)to a unit impulseis the impulseresponse (often calledthe "unit impulse response");it is the inversetransform of the transferfunction. In principle,we can predict the behavior of a linear system for an arbitrary input by applying a unit impulse and measuringthe impulse response, then applyingeq. (15.202)to get ft(/). Let us apply a unit step function to the input of a l i n e a rs y s t e mT. h e n f r o m e q s .( 1 5 . 1 7 9a) n d ( 1 5 . 2 0 1 ) , we obtain 1{s) :4,G) : (l/s)F(s), where {,(s) e -f"(t) : step response(also called the "unit step response").We can find the relation be-

15.5Digital systems and z-transforms I 5.5.I Sampling theorem When a continuousfunction, g(t), is sampledat regular intervals,A, we obtain a seriesof values,g(r?A).The sampledfunction, which we write g,, can be regarded as the product of g(t) and a comb (seeeq. (15.130)) with spacingA: c,:

$ g0rA) 6(r - nL). L

(15.201)

(Although the comb extendsfrom -- to *-, we need to sum only over the rangefor which S(t) + 0) When we sample,we discard values of g(t) in between the samplingpoints and henceapparentlylose information. However, the sampling theorem states that g(/) can be recovered exactly from the sampled

548

BACKGROUND MATHEMATICS

valuesprovidedg(t) has no frequenciesabovethe Nyquist frequency,y,r : j(samRlinefrequency): ll2L^. The Fourier transformof g(r), G(r,r),thus must be zero for ltol> 0r, 0r : 2nv,: rrlA, so that

g(t): (tl2n)f- c1'1.-'a', J : (1l2rll

T h e i n t e g r a l si n e q s .( 1 5 . 1 4 5 )(,1 5 . 1 4 1 )a, n d ( 1 5 . 1 6 1 ) involvecontinuousfunctions.When the functions are sampledfunctions,the integralsbecomesummations. To show this, we start with continuousfunctionsfil) and g(r) so thatfit) * gO is given by

G(to)e,-'dto. ( 15.208)

If we expandG(r,r)in a Fourier seriesin the interval -t'r. to to., the Fourier serieswill repeatG(or)in each intervalof width 2to-, although G(o) is zero outside the interval (-o", tor). The fundamentalperiod for : A. the seriesis l/2t'r";hence,to,,:2rlT and n'lo-r.u ) i t h o rr e T h e r e f o r eu,s i n ge q s .( 1 5 . 1 0 5a) n d ( 1 5 . 1 0 6w placingt, we get G(a,t: )

I5.5.2 Convolutionand coruelationof samnled functions

o,e,,-r.

-, : f,rr.,['n' o1ry.'^".0r. C o m p a r i s o nw i t h e q . ( 1 5 . 2 0 8s) h o w st h a t i f w e t a k e t : rA. then cr,:(rr/
flt)* g(t):

: )

Gtorl:

- jrn(t'.r/to,)]. {rrlt'-r,)r(rAlexp[

This equation'r,*, ,n. correctvaluesof G(or)in the interval -t^l- to *or-, but is not zero outsidethis interval as it should be to representg(l) exactly.To correct for this, we can multiply by a boxcar,box,,(to), extending from -<,r- to t.ru, that is, G(o) is given exactlyby C(o) : box,,(to) ) (n/to,)g(rA)exp[-jrn(orlr,i.)]. (15.209) E q u a t i o n( l 5 .I 5 2 )g i v e s t'l.u sinc tll_l e> n. box.,,,.(ttr).

.f,: LlftAtbtr - kAl (seeeq.(ts.207)),;.;".

6(r- *or]*r,- r) dr J_[},t,-^) - t)lD(r- k^)dr} : f{f 'r,o^)s(l

J,* s(t):

Applying eq.(15.124)to eachintegral,we getfor each term in the sum the value /(kA)g(l - kA). If we now sampleg(t), I becomesa multiple of A; hence,g(t kL) : C, ^, so that we have(notethat dr + A)

Using this result,we can write the inversetransform o f e q .( 1 5 . 2 0 9i)n t h e f o r m L g(rA) sinc (
(r5.21r)

d,l"):

L,\.frsr,,.

(ts.2t2)

Usually,we setA : I in theseequations(for example, eqs.(9.23),(9.24),(9.41),(9.42),and (9.46)),but care must be takenin somecases. 15.5.3z-tunsfbrms :-transformsare a specialform of transformusefulfor calculationsinvolvingdigital (sampled)functions.We take the Fourier transform of both sides of eq. ( 1 5 . 2 0 7 )u,s i n ge q . ( 1 5 . 1 5 4a) n d o b t a i n i

g{,?A)e,-^.

If we write - : e r-4,we get

or- sinc [to"(t - k)] <+ n box..^(o)e rk"'.

g(r\ :

Ll./rt, *.

In the sameway,we lind thal

g, e+ G(to):

U s i n ge q . ( 1 5 .1 3 6 ) w , ef i n d t h a t

- r) dr.

Replacing./(t) by the sampledfunction

./.,* g,:

hence,

lfr,r^,

(15.210)

This result,no*, ,nla the functionsinc(o-r - rn): sin n'(t/A - r)ln(tlL - r) providesperfectinterpolation to give g(/) for all valuesof /, not merely lor the sampling instantsrA. When G(to)* 0 for valuesof ltol) o., the loregoing proofbreaks down and we are not ableto recoverg(/) from the sampledvalues(seediscussionin Q9.2.2).

G ( t o :)

Le@L):':G(:),

(15.213)

where G(:) ,r rn" t-rr"r*rm of g,,that is, g, <->G(z). T h u s ,i f g , : 1 , 2 , - 5 , 4 , - 6 1 , G ( : ) : 1 t 2 : - 5 : 2 - t 423- 6:a. Negativepowersof : denotevaluesof time past; thus tf g, : t2, 6, -i, O, S1(the superscribed arrowdenotesI : 0), then G(;) : 2z 2 * 6z | - 1 * 5:2.It is evidentthat multiplicationby : is equivalent to delayingthe time function by one sampleinterval and division by : to advancingit one sampleinterval. (We could havetaken the Laplacetransform to get G(:) in terms of : : e 'A.The Fourier form is more the convenientfor studyingfrequencycharacteristics,

D I G I T A L S Y S T E M SA N D e - T R A N S F o R M S Laplace form for examiningstability,as when study_ ing filters.A differenceis tliat s.ung., over that part of the complex plane to the right oi the vertical line through o, rhat is, Re[s]= o 1gtS.3.ty, whereasthe ter_ minusof z : s-iorrlieson the unit circlewith the center at the origin.) (In.signal analysis,z is often defined as z = d.A, resultingin a polynomial in which the z's have nega_ tive exponentsas compared with the preceding.The convention used here is more common in seismic data processing.) Clearly,z, hencealso G(z),is a periodic function of o with period 2nlA. As ro increasesfrom c ro c + 2nlL,, c being any real number,the terminusof z goes around the unit circle (;l : I ) with centerat the origin once in the clockwisedirection; this follows from the relation 3 : s-jor : cos t_rA_ j sin
: c@ : tsrep(/) ifnotuf, - nL), : r step(r) <+ ltn!):,: r In_n-O

n-l

: <-s:A /nz, t : zA,l(l- z)r.(15.214) Again,let

549 Table l5.l Finding amplitudeanclphasespectra

0o

' ){ero:),

* i1.oo, * 1j'*"ry, * .... - eoo:).

<+ l/(l

(r5 . 2 1 5 )

Settingk:0gives

step(/) e t/(t - z). Settingk:

(15.216)

+j0gives

erd'step(/) e+ '

I I -

-

:ej"a

I ff -rcosOAl-1:iinel; (ts.2t7)

rationalizingthe denominator,we get

.' .l=;.TJ #,+ i,i:,t'".u u^i

Equatingreal and imaginarypartsgives l-_-cos0A l-2:cos0Ar:r' : sinoA s i n o / s t e' p ( / )e ) l-2zcos0A*z:'

Phase

5

5.00 s.67 5.39 6 .1 5 I 1.00 6 .l 5 5.39 5.67

0.0. 85.9. 203.6 293.0 360.0" 67.0" 156.4

90" I35" 180" 225" 270" 315.

5 + ) i

t+"'lz-i+",8

-11

t+al2+ i+",12 s-2j l-{2+j4\D

I

J

( l 5 . 2 81 )

774 )o

As an example of the use of z_transformsin de_ termining the frequency and phase characteristics of digital functions, we shall obtain the spectra of f u n c t i o ng , : 1 . . . , 0 , 0 , _ 2 , 0 , l , Z , _ 2 , the^dieital ) , U , 0 , . . . l . T h e t r a n s f o r m iqsr ) : - 2 2 - . 2+ | + 3z - 222+ 5^r : -2e:j.r + I + 3e .t.,r_ 2e ziorr * Se__r,r-r. Substitutingvaluesof to givesvaluesof Gk) that arc co.mplexin generaland from which we "un g.t the am_ plitude and phasespectraas functionsof o."Tabte I 5.I givestypical valuesof G(o.r)for a few values of or. 15.5.4 Calculat ion oJ'z-t ransforms; Fast Fourier Transform Generally,we must calculatez-transformsfor many more valuesof the argument arA than we did in thl lastsectionto obtainthe requiredprecision.Consider the function 8': G(z) :

8 o '8 t , 9 y . . . ' 8 , t , ,. g,z ! g2z2 + ... * g,, ,2,

gn t

It is convenientto taken : 2k,wherek is integral(this can alwaysbe achievedby adding zerosto and to .sj 'equal calculatethe transformfor incrementsof orf to 2rln, that is, we take aA : r(2n/il, r : 0, l, ;" . . ., n -,1 (note that G(z)repeatsfor r > n). If we let q : e r2'h, then 2 = s irtzrtnt : q,. The various values of G(), G,, can be written in matrix form

c. llG,

ll

ll I

l

-- illlr ;

q'

ll":

ll'

q

l q

2 qo

il'

llo l l '

ntn 1

(cos0l + j sin 0/) step(r)

cos ur steD(I)e

Amplitude

t-..12-j4..la

Aaa

: - nL); g(t) : e('step(,r) ie*"^a (t ' J ' ; : et'step(/) e+ )er'r:'

Qz)

t)

n2\n q

tt

TIilI il (ts.2t9)

This method requires12 multiplicationsand n2 addi_ tions. Becausea seismictrace often has a few thou_ sand values, millions of calculationsare necessary. The fast Fourier transform (FFT) is an ingeniousal_ gorithm for calculatingG(z)with only n logrn calcula_ tions. For n - 2to : 1024,the difference'i, b",*."n lOaand2x106. The fast Fourier transform (Cooley and Tukey, 1965)dependsupon doublingprocesses ty which a se_

BACKGROUND MATHEMATICS

550 ries is built up from (or decomposedinto) shorter series. Let us take the time series C1:Co,CpC2,..

tCzn,t'

C(z): co I crz * crz2+ "' * ct, ,z2n-t,

X, p l,

X o ,1 6 , X 1 ,l v . . . ,

r,

We write x6; x1'

x,:

. . 1 Jfn 1i

X(z) : xo I x'z + "' + xn 'zn-t' rr f O ' t l "

l t

l) ' ) n ' l '

Thus.

Y(z): yo * y,z * "' * ln,,z'-', where the valuesx,, Jrioccur at intervalsof 2A, not A as in c,. r : 0, 1,2, We calculateC(z) fot z : Q',Q : s-izrt2n, . . . (2n - l), whereasX(z), Y(z)are calculatedfor the : q 2 ,r : 0 ' l , 2 , . . . , ( n - l ) . v a l u e s( q ' ) 'q, ' : s i z t t n Writing x, for the value of X(z) for z : q'',

x,: > r,q"'1

,ls'2201 I ' : 0 . 1 . . . . .n I '

;i

Y,: > y,s"'I i:n

)

(15.223)

The utility of z-transformsfor digital processing arisesbecausethey can be written by inspectionand manipulatedas simple polynomials.For example,in g 9 . 2L. .w e c o n v o l v e d f: ( l . - t . j t w i t t rg , = t l . ] . - j l . 'r:2and G(z) : I The z-transformsare F(:) : I - z I - jz2;we have * ),

.f,* &,a F(z)G(z):,r._:i):t..,r! _ ::;,, ll _ As anotherexample,in $9.2.4,eq. (9'38) gavethe water reverberationfilter for n : I as - 2 R , 3 R ' , - 4 R 3 , 5 R 4 ,. . . ) .f,: (1,

2 n l

^ \ C , : L c , Q , ' .=r 0 . 1 . . .. . ( 2 n- l ) t=0 Forr:0,1,...,(n-l),

so that

't) C,: (xn* *,qt'* ,.qo'+ "' + x, fl'\' * q'(y,,* y,q" * Yrqo'+ "' + !,-rq''" "),

:

+ q'\Y'q"" \''0"' : X,+ q,Y,.

(ts.22t)

Whenr : n,(n 1' l), . . ., (2n- l),wemustmanipulate the exponentsto expressC, in terms of X,, Y,. Thus, 2 n l

: ( 2 n- l ) . (n I/ : U c i q ' 'r. n , * l ) . . . . .

: We write r : n + lr? so that qri : q\n+n)I: q"'q''' . ence, l ) b e c a u s eq n t : ( e - t n ) tH (-l),q-',m:0,1, fr n-l

H(z) : F(z)G(z)<+ h, : f, * g,.

t h u s , il .g ,: ( 1 ,- : , - i , i , - l

But,

C,:

15.5.5Application of z-transformsto digital systems By digital systems,we refer to linear systemsin which the input and output are sampledfunctions.If the system is analog,eachelementof the digital input g, gives rise to a continuousoutput h(t - nL).In this case,we sample the output in synchronismwith the input to get ft,. Then, F(
it into two series: and decompose Ct:

of singleelementsof c, with a tremendoussavingin work when n is large.

n

"', | - 2 R z r 3 R 2 2-2 4 R 3 2*3 5 R a z+a and the inverse filteq i,, is such thar f, 'r l, : 6, <+ I (eq. (9.39)).We can solvefor I(z)by divisionbecause divisionby polynomialsis a properoperation: f,-

Kz\ : llF(z) : | + 2Rz * R2z2' Thus, (z)e (1, 2R, R' ), which is eq. (9.40). As a third example,in $9.3'1,we cross-correlated x , : ( 1 , - t , l ) w l t t ry , : ( 1 . j , - j t F r o m e q . ( 9 . 4 4 ) , we haveS,'(") : x, * Y,. Hence, 2 -

I

'': Z*'n'-'- q'"LY'q''""

o-t

I

I

m:0,1,...,(n-l),

c,: x^ - q-Y-, : X,_,- 4'-"Y,, r:

2

(rs.222)

n , ( n+ 1 ) , . .. , ( 2 n- l ) .

To calculateC, usingeq. (15'219)requires2(2n)' : 8r2 arithmetical operations. To find X, from eq. (15.220) requires 2n2; hence, to get C, from eqs' (15.221)and (15.222)requiresslightly more than 4n2 operations,a 50% saving.Becausen is a multiple of 2, doubline can be continueduntil the subseriesconsist

(seefig. 9.6f).

I 5.5.6 Phaseconsiderations (a) Minimum-phase wavelets. We define a causal function as a real function that is zero for negative time, that is,l0 : 0, r < 0' A physicallyrealizable function is a causal function that has finite energy,

551

D I G I T A L S Y S T E M SA N D z - T R A N S F O R M S that is Ji W)1,dt is finite. A minimum-delayfunction is a physicallyrealizablefunction whosetransformhas an inverse,that is,l(t) <+ F(z), and llF(z) is finite. F(z) is said to be minimum-phase.The reasonsfor the terms "minimum-delay" and "minimum-phase" will be apparent later. (The literature is replete with different definitions of "minimum-phase,"the most common of which will be derivedfrom the foregoing definition.) Considera simplewavelet,w,: (a, -b) with transform W(z) : (a - bz). Then,

r _

r

I ml t / ( z ) f

nelw(z

_ r / ,_ b _ \ '

W\:) a-b-

aU

a'l

lf lblal< l, we can expand(notethat lzl : 1) and get

I W(:\

rl.,b

( t + _ - +/ tb_ -\ t,+ . . .I ) ul cr \a I ) : convergentseries.

lf lblal> l, the seriesis divergent.Thus, for minimumphase,lal > lbl.The wavelet(b, a) is maximum-phase. When a: tb, the transformis a(l + z) and the inversebecomesinfinite when I and the denominator never becomes zero, hence l"yl curve of lm{W(z)} versusRe{ W(:)} doei not enclose the origin (seefig. 15.l2a).Also, 1is periodicwith frequencyvaluesrepeatingeachtime roAincreasesby 2n (seefig. 15.12c).If lcl < l, the denominatorbecomes zero twice as toA increasesby 2n, hence1 assumesall values between 0 and 2r. that is. the curve of lm{W(Z)} versus Re{ W(z)} enclosesthe origin (fig. l5.l2b). Each time oA increasesby 2n, y also increasesby 2r (fig. 15.12c).The phase of the maximum-phasewaveletincreaseswithout limit while the phaseof the minimum-phasewaveletis alwaysbetween -lr and +]rr, henceis alwayslessthan that of the maxiinum-pha"se wavelet. l f ( a ,- b , z ) i, : 1 , 2 , . . . , n , a r e a l lm i n i m u m - p h a s e , the sum 2,(a, - b) may or may not be minimumphase becauselIlJ is not necessarilygreater than ll,b,l. On the other hand, the product [1,(a,- b7) @orresponding to time-domain convolution) is always minimum-phasebecauseeach term, hencethe product also, is cyclic with frequency,hencethe curve of lm{W(z)} versusRe{ W(z)} cannot enclosethe origin (if the curve enclosesthe origin, the phaseincreases without limit). If all of the terms (a,- b,z)in the product are maximum-phase,the product is maximumphase.If someof the factors are maximum-phaseand someminimum-phase,the product is mixed-phase. lf @, - b;) is minimum phase, (a,lb,)is a root.

F i g . 1 5 . 1 2 V a r i a t i o no l p h a s e 1 f o r ( a ) m i n i m u m - p h a s e w a v e l e t and (b) maximum-phase wavelet as ol increasesfrom 0 to 2n; (c) graphs of 1 versus oA for parts (a) and (b).

Hence

- z): o fi@,,u

(ts.224)

j:o

This equation has the constant term (a, a. "' a,lb, b, ...b"); if we add a constantto the left-handsideof eq. (15.224),the roots are no longer the same,and so the expression on the left-hand side is no longer minimum-phasein general. Consider the function fl,(a, - b, z) lfl,(a, - b,z) where all the factors are minimum-phase.The phaseof the function is the sum of the phasesof the numerator minus the sum of the phasesof the denominator; becauseall of the phasesare cyclic, the phase of the function is also cyclic, hence the function is minimum-phase. Becausea, - b,zis minimum-phaseif lc,l > lb,l,the root of the equation a, - b,z : 0, namely,a,lb,, lies outside the unit circle. Similarly, the roots of a maximum-phasefunction lie inside the unit circle. When fl,(a,,- b,z)lfl,(a,- b,z) is minimum-phase,all of the roots of the numerator and denominator lie outsidethe unit circle;becauseroots of the denominator are calledpoles,we can say that all roots and poles of a minimum-phasefunction lie outsidethe unit circle. (When z is defined as e*j'A, these rules are reversed.) The roots z : 0, + 1 follow the precedingrule. If W(z) is multiplied by z'^, m tntegral, the wavelet is shifted without changein shape,but the phaseis increasedby -jltloA; the roots are those of Z(z) plus

BACKGROUND MATHEMATICS

552 the lactor --* and root ? : 0. However,the graph of z=*W(z) now enclosesthe origin as in fig. l5'12b, hencez'"'W(z) is not minimum-phase.The convention when we are determining the phase of an isolated waveletis to have I : 0 for the first nonzeroelement so that roots: : 0 do not occur. Becausethe expansionof 1/(l + z) is divergent,the expansionof (l -+ z)W(z) is also divergent,hencethe product is not minimum-phase.When a root is only slightly larger than unity, problemsare often encountered becausethe expansionconvergesslowly. "minimum-phase" and "maximumThe terms phase" imply a comparison,and in fact they refer to the set of waveletsthat have a given frequencysp_ectrum. The four wavelets(a - bz), (b - az), @ bz), (6 az) all have the same frequencyspectrumas ls easily verified by multiplying by the conjugatecomplexes;two of the four are minimum-phase,two maxlmum. (Other waveletswith the samespectrumcan be obtained by multiplying by any complex constant c', where lcl : l, so that there is an infinite number of waveletswith the samespectrum.)When a and b are real, as is most often the case,the four waveletsreduce to two.

circle,it is clearthat an autocorrelationfunction is not unique to one function. The precedingmethod requiresthat we find the factors of the spectrumand this is time-consumingwhen r is large, hence other methods are used (seeClaerbout, 1976:chap. 3). One method utilizes the special propertiesof the Toeplitz matrix to reducelabor- Let R(z) be a given spectrum and W(z) the minimumphase wavelet to be determined; W(z) must have an inversethat we denote V(z).Then, R(:) : W(z)W(z),

R(z)V(z): W(,).

hence

BecauseW(z)is minimum-phase,we can write V(z) : b , , * b , zl b . z 2+ " ' . Moreover,if a r z * a ' 2 2I " ' I a ' 2 " Il(z):tnl then W e ) : a o + a F + a . z 2+ " ' +

a,z'

and R ( z ) : ( r , z ' + " ' * r , z t * r , ,* r F + " ' * r"z,). Then,

( b) Energy relations. The intensity of a wave is proportional to the amplitude squared at any instant, whereasthe total energyis proportional to the sum of the amplitudessquaredfrom I : 0 up to a given instant. For the wavelet(a, b), the total energyrs a2,a2 r b2aL I : 0 and / : A, respectively.For the wavelet (b, a), the values are b2, b2 + a2. If (a, b) is the minimum-phasewavelet,the energy builds up faster than for the maximum-phasewavelet(b, a). Because the two waveletshave the same frequencyspectrum, the only differenceis in the rate of buildup of energy' Theseresultscan be extendedto more complexwavelets;for example,Robinson( 1962)showedthat the total energyat a time I for a minimum-phasewaveletis greater than or equal to that for any other wavelet with the same spectrum.Claerbout (1963) showed "center of gravity" of the minimum-phase that the waveletis closer to t : 0 than for any other wavelet with the samespectrum.Theseprinciplesexplain the "minimum-delay." origin of the term (c) Determining the minimum-phase wavelet for a given spectrum. The spectrum of the wavelet (a bz),a and b being complexin general,is (a bz)(ii t ). a b z ' z + ( a 2+ b ' ) d b z( n o t et h a t z : 6z \: spectrum the alb and toot Thus. the wavelethas the has this root plus the root bla (when a and b are real, the two roots are reciprocals).When a spectrumis of order 2n, it must haven pairs of roots of the form (2,, l/2,). Half the roots will be outsidethe unit circle,half inside. The minimum-phasewavelet is obtained by multiplying togethern factors of the form (2, z), fot the z, outside the unit circle. Becauseother wavelets can be obtainedby multiplying togethern factorscorrespondingto roots both inside and outside the unit

r,z' + "' + r rz I I rn]-r,zI "' * r,z') X ( b o+ b r z I b , z 2* " ' ) : C a o *- a r zI I d , z 2 + " ' + - a , z ' ) . Equatingcoefficientsfor positivepowersof ; gives: r-,b,:-an, z o ' . r n b nrl , r b ,+ " ' + -- 0, "' * + r-,*rb, + rob, zt'.rrb,., z 2 ' . r . b , r +r r b ,+ " ' * r , * , b , : . 0 , : z ' ' .r , b oI r , . , b , + " ' +

rub,: 0.

(We assumethat (n + l) valuesof b, are sufficientto give W(z) : llV(z) with the requiredprecision;if not, more terms can be obtained by equatingco-efficients of :"' to zero,m > n.) We note that au : l/bu, hence we have(n + l) equationsto solvefor the (n + l) unknowns,b,. The solution can be written in terms of a Toeplitzmatrix,t2 as,'t?'.ti : V/' :

:llllj;lll[l ,

(ts.22s)

-aulbo : lllbol'. Solving wherer, : r -,, b', : b,lbo,w : theseequationsfor the (n + l) unknownsb, in terms of the known r,, we find V(z), then W(z)by inverting Z(z). The straightforwardsolution of eq' (15.225)requirescomputer time proportional to le3and memory proportional to n2.The Levinsonrecursionalgorithm'

D I G I T A L S Y S T E M SA N D z - T R A N S F O R M S which we describenow,reducesthesequantitiesby the factor n. The method is basedon forming new equations by selectinga k x k matrix from the upper lefthand corner of .ti and the top k rows of ,.8 and V,/.., for example,

553 by a constantc* and subtractit from eq.(15.227): fo

ft

ft

fo

rk

ili,lll;lll ll|llll llllli;:illll

rk "'

lll:r'

(t5.226\

ro

.

wherethe asteriskssignify that b! and u'* are different from the corresponding quantitiesin eq. (15.225)because they satisfy a different set of equations.The Levinson recursion algorithm shows how to obtain the solution for the (k + I )th casewhen we know the solution for the kth case.Thus, we can start with k : l, that is, rn X I : ni then usethe algorithm to get the solution for /c : 2. that is. lor

(ts.229)

We wish this equation to reduce to the equation for the (fr + I )th case,namely,

ll"''llll' ll- ll,.ll ll'," llllur ll lloll

(15.230)

then continue the processuntil we get to k : n * l; the final stepgivesthe solutionof eq.( 15.225) whereas intermediate solutionsare discarded. We startwith eq. (15.226)in which allof the b! and n,* are known becausewe havesolvedthe equation, and we wish to solvethe equationfor k + I in terms of these known quantities. We write the following e q u a t i o nt h a t d e f i n e st h e q u a n t i t ye .

Comparingeqs.(15.229)and (15.230),we seethat, to get 0 for the bottom elementof V/', we must have e - (.||* : 0, that is, c* : e/w*. Also, w** must equal (w* - cre): w*[ - (elw*),].Finally, on the left, we haveb,t* : bi - cobf-,(note that bf* = -c* because bf : 0, b,T: l) Finally,allof the quantities,6f* and u,**, in eq. (15.230)can be found in terms of the k n o w ns o l u t i o no f e q . ( 1 5 . 2 2 6 ) . (d) Zero-phase and linear-phasewavelets. We note that (z' + :-'l. : 2 cos nurA,n integral; becausethe imaginary part is zero, the function has zero phase. Zero-phasewaveletscan be obtained by multiplying pairs of elementarywaveletssuchas (l - az)(s2-tl ) : a - - t - ( l + a 2 )* a z : W , ( z )B. e c a u steh e p h a s e is zero, ry() : W"(7),hence the spectrumis

lll"lllll:lllllll0

Obviously,

k

e = r a+ r , , b f + . . . + r , b f, = L r l r f , henceis a knownquantity.We now ",;".r," ,.8 and V/' to geI

W i Q ) : { a z ' - ( l + a r )* a z } 2 : e2z 2 - 2a(1 I u2)z I t (aat 4a2t l) - 2a(l + a2)z* a222. If lal < 1, the minimum-phasewavelet with the sameamplitude spectrumis (l - az)2 = 1 - 2a: * a222.Therefore,if a minimum-phasewaveletcan be written fIIl - a,z),,,,where n is a multipleof 2, then an equivalent zero-phasewavelet can be found by replacingeach pair of factors,(l - a,z)2by (l - a,z) x (a,2- l). A zero-phasewaveletis symmetricalabout the origin, henceis neither causalnor physically realizable. The maximum amplitude is at I : 0. Roots of a zerophasewaveletoccurin pairs,(a,,lla,),oneof eachpair being insidethe unit circle,one outside.

lll''llllll:lllll ( 15.228)

(one can verify that the last two equations are the sameby direct expansion).We multiply eq. (15.228)

554

BACKGROUND MATHEMATICS

If we multiply a zero-phasewaveletby 2", that is, delay it by r sample intervals, we get W'(1:

z'W"(z).

Becausez' : e i'-, where c is constant, Wrhaslinear phase, that is, ^yr : cto. Moreoveg LI/r(z)is merely \Q) displaced n time units and has the same roots (plus z : 0) and is symmetricalabout t : nL. Linearphasewaveletsare physically realizableif they start after t -- 0. Zero-phasewaveletscan be obtained from linear-phasewaveletsby time shifting. Linear-phase waveletsare often called "zero-ohase." 15.5.7Integralrelationsfor inversez-transforms Although z-transformsand inversez-transformscan be written down by inspectionwhen g, is a time series, there are occasions when the equivalent of eq. (15.109)is more convenient.As <,rincreasesfrom 0 to 2rlL, (or from -nlA to +niA), z goes once around the unit circlein the clockwisedirection(see915.53). Therefore.we have C, = (ll2nl

l:r/a

J,, Y

: (tt2rjL) orr' h+,,dz, f

( r 5 . 2 3t r

where the integration is in the counterclockwisedirection. 15.6Cepstrum analysis Transformation from the time domain into the frequencydomain permits the equivalentof convolution to be carried out by the simpleroperationof multiplication. Transformation from the frequency domain into the cepstrumdomain permits such operationsto be carriedout by the evensimplerprocessof addition. Moreover, in some cases,frequenciesthat overlap in the frequencydomain are separatedsufficientlyin the cepstrumdomain so that filtering can be carried out more efficiently(Ulrych, 1971.) The cepstrum,3(0, is given by an inverse transform of the log of the frequency spectrum:

do. (r5.232) E(O: lrlzn;f rnlqo)lei.< The transformationfrom the time domain is usually carried out in three steos:

c(r): (r)

G(to) : l(o)lst,t't,

ae, f s{e)e-''

G(ro) :

)t,

exp [G((l)

(rs.234)

fG(or)e cQ) : (U2T)) G(o)ej-'dto. (Other definitions of cepstrum are also used (for example, Ulrich, 1971; Bith, 1974),but the previous definition is the most common in seismic data analysis.) For a discretefunction 9,, we use the z-transform and the earlier stepsbecome

qz):

) r,'',

G{.) : ln(lg,z'), l r ^ ; : (l l2njL)fhQ2)2- tL+|' 17. 6t

(15.235)

usingeq. (15.231).To return to the time domain,we have

bCl:L3,r" -'tnlez)1, G ( z ) : e x p l O ( z: .l l* o ( | E r , ) . g, e Gk).

,,, ],,,

The variable ( is called the quefrency,a permutation "cepstrum" is perof the lettersin frequency,just as a "spectrum." mutation of the letters in The cepstrum can be expressedin terms of its lampitudeh(0 and its saphe ilLl.

g(o : a(g;e:"'tet.

(1s.237)

The equivalentof filtering in the time or frequencydomains is called liftering when performed in the cepstrum domain. An essentialstep in going to the cepstrumdomain is finding the phase1(o), usually by means of eq. (15.113).Becausetan (0 + rn) : tan 0, each value calculatedfor 1(or)is ambiguousby nn. This ambiguity must be removedbefore transforming to the cep"uncracking" strum domain, an operation called the phaseambiguity.One method is to utilize the fact that 1(to) is continuousand add a to 1(o) in a trial-anderror approach,the values adopted being those that make the slope of 1(to) as smooth as possible.An alternativeis to take the derivativeof the expressionfor 1(to)in eq. (15.1l3b), a procedurethat doesnot introduce the ambiguities: d-;;r y ( t o: ) d l ' -l I {X(tot/R(o)}J O- ltan

: lnlc(o)l+ jr(r), lnlG(co)l

e g(0:

3(0 e G(,) :

T

ffo)er-^ dt'r

: 1tt2nt d c(--rr"[e'-,dz(-jA)1.

C(t) e

Thus, the essentialfeature that characterizescepstrum analysisis taking the logarithm beforemaking the inverse transformation. To return to the time domain. the precedingthree stepsare reversed:

+ jr(rlt",',Or] rrlzn.lJtrnlAo,)l (15.233)

R(to)[dX(o)/do]_49194(,I9el tR(t'r)]t + {X(c,r)}'

(15.238)

FILTERING

555

Stoffa,Buhl, and Bryan(1974)discussthe deconvolution of marine reverberatory noise, starting by weighting an observedtime seriesg, to give another seriesg,, whosez-transformis

Ge) : \r,a,2,, where a is a constant slightly smaller than one that makesg, minimum-phase(seealso $9.4).We may then associatethe slowly varying components with the sourceand the reverberation,and the rapidly varying componentswith the reflector series.The nonlinear operation of taking the logarithm results in undersampling of G(z), but the weighting lessensalias effects.(Stoffa et al. suggeststarting with a : 0.94, presumablyfor A : 0.004s, and then increasinga until aliasingbeginsto createproblems,to ascertainthe largestvalue ofa that can be used.)

15.7Filtering I5.7.I Introduction Filters aredevicesthat passor lail to passinformation basedon somemeasurablediscriminant. Usually,the discriminant is the frequencyand the filter alters the amplitude and/or phase spectraof signalsthat pass through it. Analog filtering was discussedbriefly in $7.6.3and digital filteringin 99.5.Although someof the following discussionis applicableto analog filtering, emphasiswill be on digital filtering.The literature dealing with filtering is vast; many referencesare givenin Blackmanand Tukey(1958);Lee (1960);Finetti, Nicolich,and Sancin(1971);Bdth (1974);Kulhrinek(1976).The followingdiscussionis basedto a considerableextenton Kulhrlnek. Most of the filters with which we deal are assumed to be linear to facilitate calculation,and so we here assumelinearity.Digital filters are more versatilethan analog filters,partly becausewe are not restrictedto physicallyrealizablecomponentssuch as capacitors, inductances,and resistances, partly becausewith digital filters we know future valuesof the signal as well as the presentand past valuesupon which analogfilters must act. The output ofa physicalsystemcannot precedethe input; thus, when g(l) and h(t) are respectivelythe input and output signals,if g(l) : 0 for r < 0, then l(r) : 0, t < 0 for analog filters. This is not necessarilythe casefor digital filters. A filter is stableif the output is flnite for any finite input. The output in the time domain for a linear system is given by eq. ( 15.202).A filterflt) will be stable provided that

. *o. f 'ru,'0,

(15.239)

This requires that f(t) be finite everywhere and approach zero as t approaches-t-- (Treitel and Robinson.1964).

We may write eq. (15.201)in the form

rG) : H(s)/c(s). The right-hand sideis almost invariablyexpressibleas the ratio of two polynomialsin s with the numerator of lower order than the denominator(if this is not the case,long divisionleadsto terms in r,, wheren is positive, and these usually give rise to instability). Applying the method of partial fractions,we can write

Fg):f1,tu-s,), wheres, is one of the n roots of the equation G(s): 0. Taking the inversetransform gives J-

:

ftl

(s,/): I't,exp

Lt,exp

[(a,+ jbt)tl

becausein g.n..ut ,, i. "ornoiJ*. Forl(l) ro remain finite as / approachesinfinity, all of the 4, must be negative; thus, the roots s,, usually called thepoles of ,F(s), must lie on the left-hand side of the complexplane. The preceding discussionapplies equally well to digital filters,in which case eq.(15.239)becomes

I m .-

(ts.240)

I 5.7.2 Filter synthesisand.analysis Filterscan be designedeither by requiringthat a given input producea desiredoutput (flter syntheni) or by investigatingthe effectsofa given filter on various input signals Qftlteranalysls).As an example of filter synthesis,we design a filter to transform a sampled input g, into a desiredoutput Nt,,where t

n

l

g , : L c o 6 ( r- k A ) , k:0

n

l

h,:Z&*s(r - frA). t:0

For simplicity,we assumem : n, zerosbeing added to g, or ff.,to achievethis.From eq. (15.201)we obtain F(z): .%(z)lG(z)

(15.241)

Because.%(z) and G(z) arepolynomialsin z, long division givesa polynomial that may be of infinite order. In practice,infinite polynomialsmust be truncatedto a reasonablenumber of terms.Once we haveF(;), we can getJ,; then the output y, for any input x, can be found by convolution: y,: f,* x,. We could alsogety, by usingeq. (15.241)to write Y() : F(z)X(z) : {.%(z)l G(z)}x('), or

Y(z)G(): x(z).%(z); hence Y,*g,:xt*fLt,

o r b y e q .( 1 5 . 2 1 1 ) , \.

L!,

\.

*8t= Lx,'^ho'

r:0,1...,fl -1,

BACKGROUND MATHEMATICS

556 which is a set of r equationsin the r unknowns,y*. Settingr : 0, l, . . ., we obtain the solutions lo : xo&J\o, lt : (xlto'r xoft.r)lgolgJCo, :

where2,, z, are roots of the denominator. Referenceto eq. (15.246)shows that the secondorder recursivefilter is equivalentto two first-order filters in series,the transferfunctionsbeing

e^)ll-:)

hence

t,:

l J . \- / : \ \Lx, rf,rtso) \Lt,_sJs,)

wherethe prime meansthat g, is delayedone unit. Equations(15.242)and (15.243)give a solutionin terms of present(r,) and past inputs (xnto x,-,) and past outputs ("yuto -y,-,). Filters of this type arepredictive,recursive,or feedback.Such equationscan be solvediteratively,yn being found first, then !p 1., and so on, a type of calculation convenientwith digital computers. As an exampleof filter analysis,we take a specific caseof eq. (15.242): a x , - b ) ' , , r, : 0 ,

1,2,...,n- 1

' ) L,

be stableprovided thar lzlz,l < l, that is, lb -+ (bz 4c)'''l > 2lcl. In general,recursivefilters of any order can be replacedby first-orderfilters in series. Equations(15.241)and (15.248)can be generalized as follows:

J,ro, wherewehaveseta : 1 : fi (thiscanalwaysbedone by introducinga scalefactor).Then, x , : Y ,* . f r l , , + " ' + - f , Y o ,

we have and on taking;-transforms, X(z): Y(z)(l+ .f,z+ "' * J, ,z'-') or Y(z):.X()l(l * f,z r "' * J, ,z' '). (15.249) (15.244) filter of order Thus,the generalform of the feedback

( c o r r e s p o n d i tnogg , : [ , b , 0 , 0 , . . . ] , & : [ a , 0 , 0 , . . . l), a and b beingreal,and we determinethe filter properties.Taking :-transforms of the sequencesobt a i n e db y g i v i n g r t h ev a l u e s 0l , 2 , . . . , n - I i n e q . ( 1 5 . 2 4 4 )w, e g e t Y(:):oY1='-b:Y(:)' where the factor : takes into account the delay of g, by one unit. Solving for X(:) gives Y(:):oY1t'1(l +b;)' F() : Y()lX(:): al(t + bz). (ls.24s) Providedlb:l < l, we can expandthe right-hand side and obtain F(;z): o11 I bz)-t: oit-trrl', 05.246) hence -ab, ab2,-ab3, . . .). f, : (o, Equation(15.240)showsthat lbl < I for the filter to very be stable.The seriesin eq. (15.246)may converge slowly and a recursivesolution may be better than finding/, as before,then calculating/, * .ls,. A second-orderrecursivefilter can be definedbv -v,: 4x, bY,-, cl, z.

1 t.

tts.242) Comparisonwith eq. (15.240)showsthat the filter will

Becausethe systemis linear,there is no lossofgenerality by setting&, : * l. We also note that the initial value of k in the secondsummationmeansthat g, has beendelayedone time unit (see$ 15.5.3,alsoproblem 15.38).Therefore.we can write - (Y, * g,)', (15'243) -f, : rr * h,

,,:

, | -

(ts.247)

Then,

,,t ls

pF(:):t//f ,:.

( l s.2s0)

,=(l

I 5. 7.3 Frequency.filtering Frequencyfilters are classifiedas low-pass,high-pass, or band-passaccording as they discriminateagainst frequenciesabove or below a certain limiting frequency or outside of a given band of frequencies. "Ideal" filters of thesetypes are the following: Low-pass:

F,(to) : +1, lol < lto,,l,

,,, ,, ', : 0, lorl> ltonl; ] High-pass: Jr@) :0, lol < lto,,l,

,',,,,, : +1,lol> lonl; ] Band-pass: flro;:

+l' lo,l < lol
Y(z) : oY1t1- bzY() - c:2Y(z),

,,,, : 0, lor,l> lol or lorl> lto,l.] , ' , ( l 5.248)

Thesefilters are discontinuousat o)6,o)1,to,. The discontinuities will cause some ringing. Obviously, a band-passfilter is equivalentto a low-passfilter with

FILTERING lool : ltorlin serieswith a high-passfilter with ltonl: lo,l. The low-pass filter can be obtained from eq. ( 15 . 15 2 ) Fr(a) : box,-,,(or)e+ (ou/n) sinc (tool): .f,0. il5.254) For digital functions,provided lr,,l ( r, : n/A, this becomes f' : (1h) |
f-

./,,Q): (l2r) | ej-'dor+ (ll2rr)l ei',,do J_ J_,,,,, - (l/rr)J.,cos (15.256) becauseei'' : cos arl + j sin rrrland sin tol is odd. Then

J,,():

(15.251)

Clrangingto digital functions,becausea,lrnL.: nr, :. .fi' : (llrl L [t-r,sinc (ror"A) or,,sinc (nro,,A)]

s i n c( t . r , , r A ) ., * n .

: ( l / n " ) ( t - r-" t l n ) ,

n:

.5

1.0

1.5

T i m e( s )

2.0 0 20 40 60 80 100

(a)

(b)

F i g . 1 5 . 1 3 C h i r p f i l t e r .( A f t e r K u l h r i n e k , 1 9 7 6 . )( a ) l m p u l s e r e s p o n s ea n d ( b ) a m p l i t u d es p e c t r u m .T h e f r e q u e n c yi n c r e a s e sl i n early from l0 to 30 Hz in 2 s.

c o n s t a n t s( c o m p a r ee q .

where <1,,, t,.lr are

(7.5)):

G,(or)- constant, t,ro{ or ( orr, : Q, ot ( to,,,ro ) ro,. Such an operation, called chirp .filtering, is roughly equivalentto band-passfiltering, as shown in fig. 15.13. If g(t)is the input to a filterwhoseimpulseresponse, ./'-(t), is given by

then clearly4,,(co): Glurland the output of the filter is given by H(a) : I'n,(t'r)G(o): 1G(or)1., hencelr(t) : autocorrelationof .q(t).Filters of this type are calledmutt'hador conjuguteftlters.When the input consistsof g(l) plus randomnoise,the output is mainly the autocorrelatioq. of .g(t)becausethe crosscorrelationof g(l) and the noisewill be approximately zerofor all shifts.

I 5.7.1 Buttcruorthfilters

The Buttervorthfilter is a common form of low-pass I filter; it can be definedby ( 1 5 . 2 5 e ) I

0. J

As for /,,(1),/i' : -"fi (exceptat I : 0) when both filters have the same cutoff frequency,t,ro.Thus, the designofhigh-passand band-pass filtersis essentially the sameas that of low-passfilters. To achievean ideal low-passfilter requiresan infinite serieslor /,t, which is impossible.The result of usinga finiteseriesfor//, is to introduceripples,both within and without the passband,the effectbeingespeciallynoticeablenear the cutoff frequency(Gibbs' phenomenon;see$15.2.7).Theseeffectsresult from the discontinuityand can be partially overcomeby rnultiplying// by a smoothing window fur.rction(see $15.7.5). A noisy signal may be cross-correlatedwith a Vibroseis-type signal, g , ( t ) : s i n { t o , *, ( o r - a ) t l L } t , :0, l<0andl>L,

0

f,Q): ser),

( l / r r ) s i na r r l l , , , , , (o-r,,/rrr) sinc (to,,l)

becauselim. - _ (sin o/) : I (seePapoulis , 1962:278). Thus, /,,(r) : -J',G) provided both lilters have the samecutoff frequencyt ,,. For a digital filter, the responseshould be zero abovethe Nyquistfrequencyto avoidaliasing.Changi n g t h e l i m i t i n e q . ( 1 5 . 2 5 7l)r o m * - t o * c o , ug i v e s . / , , Q ) : ( l / n l ) ( s i nr , l - t - s i n o o 1 ) . ( 1 5 . 2 5 8 )

- -,r;;,i

Swept-frequency waveform

0< t < L,

lF(o)l' : l/[ + (o-rlto,,)r"], (15.260) where or,,is the "cutoff" frequency,and n determrnes the sharpness of the cutoff Curvesof lF(o)l for vario u sv a l u e so f n a r es h o w ni n f i g . 1 5 . 1 4 . To investigatethe stability of the filteq we use a L a p l a c e - t r a n s f o rdm e f ni it i o n : ll1s)l'= l/il + (-l)".tr"1,

(15.261)

where.s: o * j(arlo,,).This functionhasno zeros,but has 2n polesgivenby the roots(seefig. 15.3)of

'.,': ;i lllT

Theseroots are of the form 1*, * jb). a and b being real and positive.Thus, the roots are symmetrical aboutthe realand imaginaryaxes.Whenn is odd, two roots reduceto +1. Becauself(s)l' : fl(r)F(r),if the roots a + jb (and - I whenn is odd) are assignedto

558

BACKGROUND MATHEMATICS

r ,'Iz' ,.(r,r):

I - l* j(t'r/to"). I +j(t'rltoo) I +(orlrrro):

The first filter has a linear phaseand the secondhas the phase "Y(<'r) : -tan '(o/ton)'

15.7.5Windows

0 o

l

2

3

u

s

l

'

'

:

o

Fig. 15.14 Amplitude responseof a Butterworth {ilter.

F(s),and +a+jb (and +l) to F1.r), F1,r)is stable.For the nth order digital filter, 4 , G ) : [ ( s+ l ) ( s* u , * j b , ) ( s+ a , - j b , ) X (s * c, + jb.)(st ct. - jb.) ...1 ' (the factor (.s+ l) is omitted when n is even). The expressionfor {,(s')can be expressedas partial fractions($15.1.9),transformedto the time domain and then to the :-domain to get the digital filter.Thus, for n:2, the rootsof ,f : - I are(+ | -rj)/(2)'/r;hence

-r\"' A(s):

[s+ (t + j/r/2]t.'+ (r - j/{21

t+ '-,vJz] : jr[, + (r j)tl2 , * ,, U s i n ge q . ( 1 5 . 1 8 2a) n d w r i t i n g i l t l Z : r * , w e g e t .lW : 0llDle-o+j)r* " r,-irr-lstep(f/Zl*) : (r/Ze-'.sinr*) step({2r*). Applying eq. (15.218)and the results of problem 15.2'7. we obtain

: {2[r_ rr,, #;S;1e;,,; *] Equation (15.260)gives no information about the phasecharacteristicsof the filter. We can consrrucr Butterworth filters with different Dhasecharacreristics, for example,

We often wish to selecta portion of a signalfor study, or we may with to "smooth" a function such as a transform. We can achievetheseobjectivesby multiplying the signal or transform by a windowor gate,a function that variesin a more-or-lessconvenientmanner within an interval and is zerooutsidethis interval. We representa window by w(t) <->W(a), the symbolism emphasizingthat we can think of the window as being in either the time or frequencydomain ("window" is also usedin a third senseto denotemerelyan interval of time, especiallyan interval in which data can be recordedfree from interferenceby noise such as ground roll). The result of applying a window to a signal,g(l), in the time domain is to give h".(t)-- s!)w(t) <->(ll2n)G(a) * W(a). Using a time-domainwindow is a form of frequency filteringso that the transformofthe part ofthe signal selectedby the window is distorted in both amplitude and phasein comparisonwith G(o;. The lollowinglist includesthe morecommonlyused time-dornainwindows (seeBath, 1914l.157-64).The formulasgivew(l)in the interval0 < ltl < T n(l) being zero for Vl > T (exceptfor the Gaussianwindow, (ft), wherethe rangeis +-). Obviously,similar windows can be'appliedin the frequencydomain, the inverse transformsbeingobtainedusingeq. (15.140). (a) Boxcarwindow: w(l) : boxrlt) <->W(a) : 2Z sinc t_r?l (b) Sinc window: w(r) : sinc (rtlT) <->W(a),

wtrt=Ti

(-lr

(15.263)

rfil2n+lll2n+l)t X [(tt - toT)t'+r+ (rr + loT)2"+r1.

(c) Fej6r kernel window: w(t) : tin.z,trtlT) <->W(a),

ri w(.j,t=

( l)'

21tr:f,1' (2n + l)(2n + l)l

for n : ll2,

(15.262)

x [(2n I i.l7)z'+t -2(iuol1t"*z + (2n -

uoT)z'+21.

(ls.264)

FILTERING

559

(d) Cosinewindow: n(/):+1, 0
I

l

(ts.26s) (e) Hanning window: ,(t):)+jcos(nttT!, w ( t )e

(1s.266)

w ( a ) : z { s i n co I + (2aT sin <,rT)/[n2- (,o]")tl].

(f) Hamming window:

If

w(t : 0.54 + 0.46cos (nllT),

w(/) <-) W(a) : ?n{L08sinc o?" +(0.92aT sin oZ)/[.n2- ( o r 7 ' ) ' l ) . J

(ts.267) (g) Triangular window: tt(t) : (l - ltllT) <+ W(a) : T sinc,(aTl2).

( l s.268) (h) Gaussian window: w(l):exp(-alr), a>0, <+ W(a) : tlr,la exp(-a2l4a)

|

called the (. norm (seeClaerbout, l9j6: l2l, 123);it provides the maximum-likelihoodestimateif the errors have a Gaussianor normal distribution, where P(e), the probability of an error e,, equals P(ej) : ll I o(2r)t /21exp[- t(e,I o),], (t 5.2j 0) where o is the standarddeviation (that is, the sctuare root of the variance) Another criterion sometimesusedto findl is the f , norm that minimizesllerl (Claerboutand Muir, 1973; Claerbout, 1976: 123;Tayloq 1981).It providesthe maximum-likelihood estimate if the errors have a Laplacian or one-.sided exponentialdistribution, where the probabilityol an error e, is

P ( e :, ) ( l / p )e x p [ - e , / p ] . , , - 2 :o'

etlo'

| ,,r.rr,,

where p is the mean absoluteerror and the variance is pr. Useof the (, norm is lesssensitiveto errorsthan the Wieneror f, norm. The fo norm that minimizes Ie1 is used in minjmum-entropyfiltering (615.7.6e).pursimttnious decdnvolution(Postic, Fourmann, and Claerbout. 1980) minimizes (ler,')t/t,l(\e11,,r, where p is very slightly larger than 4. Occasionally.the minimux, Chebychev,or (.- norm is used,for example,in array design(Rietsch,1979).

,',,un,

Combinationsof windows are also used; for ex_ ample, the discontinuoussidesof a boxcar can be modified with a cosinetaper.A common techniqueis to apply a window for example,a boxcar,in the time domain, then smooth the resulting transform by applyinga secondwindow in the frequencvdomain. T h e e f f e c ot n t h es p e c t r u mo f g ( l ) o i a p p l y i n ga w i n dow in the time domainis determinedby lZ(to).Thus, curves of W(a) give some idea of the effect of using the window. In general,the wider the window and the gentler the fall-off, the less effect on the soectrum. Bdth (1974: 164-71),Blackman and Tukey (1958), and Kurita (1969)discusspracticalaspectsof ..win_ dow carpentry."

15.7.6Optimum.filters (a) Introduction. The filters discussedup to this point havebeenbasedon periodicor aperiodicinputs. If the input noiseis a stationaryrandom function, we may designa filter that will givean "optimum', output according to some criterion. Among the most important and most widely used of such criteria is the Wieneror least-squares criterion. To apply this criterion, we compare the output of the filter with some "desired" output, the differencebeing the ..error', in the output; we then designthe filter to minimize the power (or energy)of the error by applying the prrnciple of least squares.The Wiener criterion is also

(h) Wiener (least-squares)filtering. he input

Let us write for

s(t) : s,Q) + c,,Q), 05.212) wherethe subscriptsreferto "signal" and ..noise."If h(t) and rt(r) denote the "actual" and ,.desired"outputs, their difference,the error, is e(l). Then, the energy of the error, 4 is the sum of the squaresof e(t). For continuousfunctions.this sives lr' E : l i m( t t 2 n I I h ( )- n u f d r I-.

J

.

: trm aDTf, {[J-n",s(t- r)a"]- ar,r]'a, : tm (v27-) {l',[_r"*(, r) dr x

- ol a'] ar f-_tt"txt

-2f f'

- aa,fa6at ,[-r"rru 1

+ | n,u)il|. ) J -

where the squareof the integral giving l(l) has been written as the product of two integralsthat are identical exceptfor the dummy variablesof integration.Interchangingthe order ofintegration gives

BACKGROUND MATHEMATICS

560

n:

a' trrzn J-rt"r [rif _tota" - ') d4 t *, - r)c(t f, - ") dr] - z yoto" orzn [g I ,h(t)s(t f *

dt , v2n 1,T ] ,h(l)

: rr"ro" f(o)Q,,ft J- f

o) do

-'J"

./(r)9"*(r)dr + $**(0),

u s i n g e q s( .1 5 . 1 6 6a)n d ( 1 5 . 1 6 9 ) . E is a funcBecausewe havespecifiedg(l) and NL(t), tion ofl(t) only, and so the problem reducesto that of finding the function /(/) that will minimize E Determination of the form ofll) in generalinvolvesthe calculusof variationsand we onlv statethe result:

J

Jlo)O-,(t

- o) do : 0**(t),

r > 0.

(15.213\

This integral equation, known as the Wiener HopJ equdtion,holdsfor a causallinearsystem,the function /(l) satisfyingthis equationbeingthe impulseresponse of the desiredoptimum filter. The solution of eq. (15.213)is lengthyand complicated,mainly because of the requirementthat r > 0. For details,the reader is referredto Lee (1960:360 7, 389 92). (c) Prediction-errorfiltering. Considera filter/, : 1,. l'., . . . , ./',,that is designedto predict the causaltime s e r i e s g , : . 9 gr ,r ,. . . , g , , , . m > n , o n et i m eu n i ta h e a d basedon currentand past valuesof 9,. For example, at I : 3A, the filter predictsgo basedon the current valueg. and pastvalues9,,,9,,9.. In general,the predictedvalueof g' is o

6t

:

:

l o

., lAl

+

|

+

l'o

/:^/

I

+

',. +

+ .

Treitel, 1980).Moreover,the previousdiscussiondealt with predictionbasedon current and past values,that is,forward prediction.Wheneverthe entire set of valuesofg, has beenrecorded,backwardpredictionbased on future valuesis also possible,as is predictionbased on combinations of the two. The data for adjacent tracesmay also be availableand thesecan be usedin multichannelprediction methods (Claerbout, 1976: 13940). The prediction concept implies that all of the data involved are parts of the same ensemble,hence the statisticsof the ensembleare central to the prediction concept.Whether or not the ensembleis ergodicand l2) and the natureofthe distribution stationary($15.2. are clearly relevantto the prediction method. criterion to deIf we usethe Wiener (least-squares) termine the filter f, we obtain the following normal equations(seeproblem15.43a,alsocomparewith eqs. ( 9 . 7 3 )a n d ( t 5 . 2 1 )t o ( 1 5 . 2 9 ) ) : .a :0**(r). r:1,2,...,n. ) . . 1 - ^ 6 r , 1k,)

A simple expressioncan be obtained for the error p o w e rE b y u s i n ge q s .( 1 5 . 2 7 5a) n d ( 1 5 . 2 7 6 )N. o t i n g g,,,we have that e,,:

' : i ; , i : i ( 2 r ' r -r , ) '

=I tEt-t,^tr,r,,) - 2c,(2/rc,-) . r;], where we usedifferent summation indicesk, ( to get the squareof the first term. Interchangingthe order of summationgives

r q

J,Alr

^ = L J ^ 8 ,* . rL=Jr ^ F , r:r

i:

l'2""'t't' .5'214\

wherewe can usethe upper limit n because8, o : 0, k >.7. Note that the loregoingimpliesthat the predicted valueof gu is zero.The error in the predictionof g, is e,,where

,,= + L . J^ 9 ,

s , : > J ^ c ,^ . ( 1 5 . 2 1 5 1

t h e f i l t e r - 1 , . / , , . 1 , ,... , . / , t h e w h e r e . d:, - t . ; : c a l l predit'tion-error of .filter length(n + l).for unit prediction distance. In the foregoing section, the prediction distance was one unit, but it is possibleto designfilters for prediction distance (span)p, p integral (Robinson and

(15.216)

E:y^l\l;(},"", ,)] - rLrr(Lrs r) * 0,.(o), r _ l _ = ) n l f t , a k - ( t l - 2 > f , 6" , , (k-)+ d , " ( o ) ?^ ?^17"'* I - 2Yr6*?k) + d"s(o) : flo""rt; on using the normal equations.Recalling that f,, = - l. we have

n : -ffro.,,{t ).

(t5.211)

Equations(15.276)and (15.277)can be combinedto givethe followingmatrix equation

"'0"*(-n)

0**(-l) llo""tor I t 0 * . (0r ). " ( o ) " ' 0 " . * ( l - r ) i l ^ '

lll d""(n) l ' 6"r@- l). I *-",0, i l "

llll,:ll llll,ll

:ll

(15.278)

FILTERING

561

(d) Maximum-entrcpy filtering. In thermodynamics, entropy is a measureof the disorder (unpredictability) of molecular motion. In information theory, Shannonand Weaver(1949) regardentropyas a mesure of the unpredictability of a time series. The amount of information that can be extractedincreases with the entropy.At one extreme,a perfectlypredictable series,such as a sine wave,bearsno information and at the other extremewhite noiseis completelyunpredictableand hence potentially carries maximum infiormation. Maximum-entropyfiltering attempts to produce a filtered output that is as unpredictableas possiblewhile still having the same autocorrelation function as the original time series,that is, of all the time seriesthat havea given autocorrelationfunction, maxlmum-entropyfiltering selectsthe one that hasthe maximum unpredictability. We could regard the number of digits required to encodeinformation as a measureof entropy.For example, if we have four equally probable events,the variouspossibilities can be encodedas 00, 01, 10, I l, which requirestwo digits where 2 : -log,(ll4); for eight equally probable events we would nied three digits(3 : -log,(1i8)). In general,for equallyprobable events,the entropy is measuredby -tog,( l/p), where P is the probability of each event. When the eventsare not equally probable,the entropy S,,is the average:

s, : Ittoe(Upilt\tp): _ f, ^r r,.

(ts.2./s)

(The baseof the logarithms is arbitrary exceptwhen comparingentropieswith differentbases.)For a signal of infinite length, we definethe entropy density S as S:

lim [.S,,(r+ l)]

( r5.280)

The matrix d*, in eq. (15.218)is of Toeplitzform and Smylie,Clarke,and Ulrych (1973)deducerhe relation

Usually,operationson data assumethat $""() is zero outside the range of measurements.However, Burg (1972, 1975)suggestedthat a more reasonablechoice of the unknown valuesof d""(,1)is one that adds no information, henceadds no entropy so that S is stationary with respectto $""(k), lkl > n. Thus, using eqs. (15.282)and (15.283),we get the result

as/aoss(ft): r:

l.ltklo-(z)ldo,,

(15.284) Although O_(;) is an infinite series,eq. (15.284)impfiesthat UA_(z)is a finite seriesof the form

trc*():L,,r.

{ttlz,-yJ"'i" 1*-frn,r't,},

(1 5 . 2 8 1 ) : where Eprediction-errorpower for n : -, oN : Nyquist frequency,and O_(ro)is the spectraldensity of g,, that is,

wherewe can take o:;;'^minimum-phase, G(z-r) as maximum-phase. In additionto satisfyingeq. (15.286),O-(z)must be consistent with the known autocorrelation values. +",(.i),|t1< r. Let O , , ( : ): 6 " , ( - n ) z " + ' . . + d * " ( * l ; z ' + 0**(0)+ g"(l); + ... + Qr,@)2,, then the terms in
I./;0"-(r- s): -Esi, i: 0,l, . . .,n, (rs.287) w h e r eb j li s t h e K r o n e c k e dr e l t a( 6 i : I , t : 7 , ' 6 r : 0 , i + 7). This can be expressed in terms of /'* $** as follows:

s : j mr _ :

n lln (ton/n)+ (l/4to-)l tn Jo-1r;1dr. r_oN (15.283)

In practical problems,we work with a finite signal 9,, which we assumeto be a sampleof one memberof an ensemble.Although we have only a finite number of valuesof 0..(,r), theoreticallyan infinity of values of 0-,(J) existsoutsidethe rangeof our measurements.

+p,.

ljl
(15.288)

p; being zero for j > 0; thus, the convolution is zero forT positive,equalsE for i :0, and has unspecified valuesp, forT negative. Taking :-transforms,we get

FG)A"Q): -E + P(z\

( r 5.289)

where

o-(o): o-(;): I 0""(O,r*.(1s.282) Smylieet al. (1973),flo* ,nu,Lin infinite,

(1 5 . 2 8 5 )

BecauseO_(:) is a real function of z, ll0_(z) must also be real, henceco : Z * and therefore -a l/O-(:) : Lr,t' : G(z)G(z-'), (15.286)

ll*6o)i:-ffiI

E- : (to./n).*o

tkt>n.

P ( z :) 0 , 2 ' * . ' . * p , , z - , . Factorizationof O,,(_-) gives a,(:) : G(z)G(z') : ( g o + g , z t . . . + g , z " ) ( g , z - *" . . . * 9,,). Substitutingin eq. (15.289)gives or

F(z)G(z)G(:') : -E + P(:),

F(i)G(z) : l-E + P(z)llG(z t) I : (llgr)f-E t (p ...+ p_,2,)l ,: + x [ + ( g , l g o ) zI + . . . + ( g , l g n ) , , , 1 - ' .

s62

BACKGROUND MATHEMATICS

Therefore, (-l

+ f,z + "' + f,z')(go* g,z * "' * g,z') : - ElSo * negativepowers of z.

As the number of spikesin the outputs i, decreases. the results become simpler.Wiggins (1978) takes as a measureof "simplicity" the quantity f defined b1 the equation

Becausethe left-hand side has no negativepowersof z, it follows that (-l

+ frz + "'+ f,t')Gu* g,z* "' * g,z') : - Elso,

or r@G{z): -Elgo:

-80,

E:

then seeksa maximum of f by varying the filter coefficients,f. This leads to |y'r equationsobtained in the usual way:

Ci,

af:o:tdf,

on equatingpowersof z (includingz0).Thus,

G(i -- -s,tfQ) and

af* ( 15.290)

(15.292) :1(z): F(z)lEtt2' !C(z) g, Note that if is minimum-delay,both and F(:) can be taken as minimum-phase,iC(z-t)and F(: ') being maximum-phase. Finally, we have from eqs. (15.286)and (15.292) O-(z) : l:(*):(*

')l' : E[F(z)F(z')] |

(15.293)

Thus, to determine the maximum-entropy spectral density,we first find the prediction-errorfilter f and the associated error power E by solvingeq. (15.278) and then use eq. (15.293) to find the maximumentropy spectral density, O-(z); the methods of then enableus to find the desiredmaximum$15.5.6c entropysignal.Equation(15.278)can be solvedby a recursivemethod similar to the Levinson algorithm; details are given in Smylie et al. (1973),Andersen (1974),and Robinsonand Treitel (1980). (e) Minimum-entropy filtering. Minimum-entropy filtering (Wiggins, 1977, 1918)attemptsto find a linear filter that maximizesthe "spiky" characteristicsof a signal, thereby reducing the disorder of the signal, henceminimizing the entropy.Maximizing the spikynessof a signal is equivalentto finding the smallest number of large spikesconsistentwith the observed signal.One way of increasingthe spikynessof a signal g, is to raise the values to some positive powel for example,the fourth power, becausethis makes the difference between large and small values much greater.Becausethis criterion is especiallysensitiveto high amplitudes,it tends to focus attention on the strongest events, which we assume are reflections standingout againstthe backgroundnoise. We assumethat we haveN traces,eachcoveringthe sametime interval from / : 0 to t : nA. We write g,, for the value of the ith trace at t : j\.The filter coe f f i c i e n tasr e f , k : 1 , 2 , . . , N , , a n d t h e f i l t e r o u t p u t is ft,,.Then,

h,,:

lfos,, r

(1s.294)

a afo

: r l+rarilyrl' aLa "l\4 "t.

t@,(z): IGQ)G('')l ' : F(z)F(i t)|fi : F(z)F(zt)lE. (1 5 . 2 9 1 )

Taking 9 as the value of G as n -+ * , comparison o f e q s .( 1 5 . 2 9 1a) n d ( 1 5 . 2 8 6s) h o w st h a t

,,: lrl(+ntr)' , rrszsst

f : If,,

-.(I,'x;, )I(4,,)'lii,,

Writing u, : Z,hi, : n X (varianceof the ith output) (because(h,,)^,: 0), we have

I(,t,}

hl,- u;,r,l h,,)s,, r : o, k:1.2,

,N/

Thus,

*): I("'l his,., *), I(,, .,) 0,8,, using eq. (15.294),we find

.,s,-)]: I(,,') his,, r) I[,''.,](I "r,s, Interchangingthe order of summasion on the lefthand side gives

)t,12,:'r(L r,,,r,,,r)]

:\0''1ni's''' r)'

k:1,2,

,Nt

(1s.2e6) The summation over 7 on the left-hand side of the equationis the autocorrelationofthe ith trace,so that the expressionin squarebracketsis a weightedsum of the autocorrelationsof the observedsignals.The summationover/ on the right-hand side of the equation is a cross-correlationof the inputs and the outputs cubed,the effectofthe cubing being to give great weight to the spiky components. Becausethe filter coefficientsenter into the calculation of u,,f,, and h,,in eq.(15.296), the equationscannot be solveddirectly.However,we can assumevalues forf initially, use theseto obtain the quantitiesr.r,,f,, h,,,then solveforf . Thesevaluescan then be usedto determineu,,l ,, and h,,againand so the equationscan be solvedfor a secondfilter. Wiggins( 1978)statesthat about four to six iterationsare usually enough to determine the fllter with sufficientorecision.

----.!

PROBLEMS

563

Problems

V x Vtp:0: V'V x A,

l5.l (a) Show that the set ofhomogeneousequations b y s e t t i n bg , : 0 , i : 1 , 2 , . . . , , , i n . q . :.bjlry9 (15.3a) has nontrivial solutions when det(a) : Q. (Hint. Take n equal to a small number, for example, 3, divide through by x' solve the first two equations for x,/r., xrlx, and substitutethe solution in tire third equation,obtaining det(a) : 0. Becausex, can have any value,we havean infinite number of soiutions,all rgQurringthat det(a): 0. Generalizefor anyn.) (b) If two equationsin eq. (15.3a)are noi indepen_ dent, show that det(a) : 0; conversely,if det(a) I 0, the equations are independent.(Hint. If two equa_ tions are not independent,they must be the sameex_ cept for a multiplicative constant; apply rule no. I for determinants.) 15.2 (a) Provethe followingcorollaryof eq. (15.2): ly*,o,,M,r: o: It-l,y*ka,,Mo,, i+k. \{(bJverify eq. (15.20)usingeq. (15.2)and part (ar. 15.3 In mechanics,moment or torque M of a force F about a point O is equal to the product of the magni_ tude of the vectorr from O to the point of application ofF and the componentofF perpendicular io r. Show thatM:rxF. 15.4 Verify eq. (15.10).(Hint; Write the vectorsin terms of componentsand use the relations between vector productsof the unit vectors.) 15.5 Three vectorsA, B, and C can be multiolied togetherin threeways:(A . B)C, A . (B x C), and A x (B x C). (a) Which of thesethreeare scalarsand which vecrors. and what are the directionsof the vectors? (b) Showthat A.(B x C): B.(C x A): c.(A x B) P\

a\

a,l

: lr. b, b.l. lt, .;, ;,1 gives the volume of the parallelopipeddefinedby A, B, and C, and that changingthe cyclic order changes the sign: A.(B x C): -A.(C x B): _8.(A x C)

: -c.(B x A).

( c ) S h o wt h a t A x ( B x C ) : ( A . C ) B - ( A . B ) x C. (Hint: Use eqs.(15.8)and (15.10)to expandboth sides.) (d) Why are parenthesesnecessaryin writing A x (B xC),butnotforA.BXC? 15.6 Show the following: (a) The vector Vrf is perpendicularto the contours q, : constant. (b) V,l, is in the direction of, and equal in magnitude to, the maximum rate of increaseof rl. (c) The rate of increaseof rf in any direction is equal to the projection of Vrf in that direction. 15.7 By direct expansionusing eqs. (15.8),(15.10) and the definition of V, verify the following identities:

v x ( v x A ) : v ( v . A )_ V ' A (the latter being valid only in rectangular coordi_ nates). 15.8 Useeqs.( 15.15)to (15.l7)to verifythe followrng expressionsfor Vrf, V A, V1f for (a) cylindrical coordinates and for (b) spherical coordinates (seefig. 2.34).ln cylindrical coordinates,x : r cos 0. -y:rsin0,z:z; and

v.r,:9;Ir,*l#'.*#',, V . A :

v'(, :

! r r , 'q' r a l a A ' a 9 A . 0r

r d0

0:'

. ),:#.t::, ,i('#)

where i,, i, i. are unit vectors in the direction of in_ creasingr,0, z, andA,, Ao,l-- arecomponentsof A in the r-, 0-, z-directions.In sphericalcoordinates,x : r s r n 0 c o s6 , y : r s i n 0 s i n 6 , z : r c o s 0 , a n d au' I v' Y ,ir:tsi +la$' * Ar't raet. rsinea5l,' 'I . a I a V.A : .' ^ ( r , A , ,+ - t , l " s i nO t r-dr rslnHdt 1 A aA . !

r s i ne a 6 ' afr,atll V ,' , l r : l *

I

dl. ^ao o

12dr\ ar I r: sin o ag\ttn I d'q, ,. rinzg66t'

ag

wherei,, i,, i. are unit vectorsin the directiorrsof increasing4 0,0, and A,, An,A*are componentsof A in the r-. 0-. and g-directions. 15.9 (a) Provethat ( 2 I m 2_ t n 2 : l , where((, m, n) arc the direction cosinesof a vector. (Hint: Start with a vectorA with the direction cosines ((, m, n\ and find A,. (b) The perpendicularfrom the origin to a plane has lengthI and direction cosines({ m, n). Show that the equationof the plane is (x*my*nz:h. (Hint: The perpendicularfrom the origin O meetsthe plane at P(h{ hm, hn). lf Q(x, y, z) is any point in the plane,OP is perpendicularto PQ.) 15.10 (a) When more than two matrices are multiplied togetheqshowthat the order of multiplying adjacentpairs is arbitrary; thus

. 1,,/76: (.,/,A)5 : . n(,A6\. : 6'.4,, 4r by applying (b) Prove that (.1.8'6)' the basiclaw of matrix multiplication. (c) Show that the multiplication of partitioned matricesgivesthe sameresult as the basiclaw of multiplication by settingup matrices. d and .t9 of sizes3 x 4

564

BACKGROUND MATHEMATICS

and 4 x 5 and carrying out the multiplication for the unpartitioned matrices and for . i partitioned with a 2 X 3 matrix in the upper left corner and,ti with a 3 X 3 matrix in the upper left corner. 15.11 (a) Referringto $9.5.5,show that the error e, : h, - g, * f, can be written in matrix form as 5 : .76 - 9.7 and that the normal equationsbecome t:tr.% :0o'0"*,

(:trV)

,V:

where!9 has the sameform as, I in eq. (15.23)and .V, 6,,4 arecolumn matricesof orders(n + l) x l, (2n + 1) x l, (2n + 1) x I, respectively. (b) Showthat v in eq. (15.55)is the minimum valueof E. (c) Show that y* in eq. (15.60)is the sameas y. 1 5 . 1 2( a ) S t a r t i n g f r o m e q s . ( 1 5 . 3 7 ) t o ( 1 5 . 3 9 ) , namely, _

-t:

x3

e':l+x+'' +-'+...21 3! .

.T3

slnr:r-

3!

+

15

x7

5!

7l

y4

-ro

x:

+...-

cosx=l--' -"'-" +...2l 4t 6l

15.15 A periodic function g(r) can be representedby a finite seriesof the form

g(t): s,(r) :- 1 . + * S r , + q, sin rtonl). ,Po L \P, cosraot "best fit" (a) Show that, if S,(l) gives a least-squares to g(t), p, and q, must be equal to a, and b, in eqs. ( 1 5 . 9 9a) n d ( 1 5 . 1 0 0 ) . (b) If we calculatea,, b, up to r : 5, then decidethat we need a better approximation by extendingup to r : 8 , m u s tw e r e c a l c u l a t e a , . , b , f ro r: l , . . . 5 ? 1 5 . 1 6V e r i f ye q . ( 1 5 . 1 0 4 ) . 15.17 (a) Writing g(t) : r(t) + jx(r), Qr) : R(o) + jX(to), whereg(l) e (t,l) and r(t), x(r), R(to),and X(co) are real,derivethe following: R(or) :

t|

[rtl) cos o/ * r(1) sin t'rl] dl,

|

- x(/) cos t,lll dt.' [r(l) sin tol

J -

showthat X(al :

c o s r r : j { e : ' + e, ' ) ,

sinx :

r'l'lllll

_ t -

, tu'* 1. ,'1,,, , - formulas) ,

i.,

j(Euter's

(cos,r + j sin x) : s'

( b ) E v a l u a t |e, , 1 1 2 , , : , ;f,o r : r : 2 - 3 j , : . : 4 + 9 l ; express:, and z. in polar form and repeatthe above, verifying that the resultsare the samein both cases. (c) Verify the following formulas.(Hint. Start with the third Euler formula, forrn the sum, and then equate real and imaginary parts.) ln"v

t, '

sin

I costx + ,-y):

,=o n I

I sin(-r+ r1) :

r:o

.o, [r + ](r - l)fl.

stn

r"y sinln^v

.

',I

Stn

.in [x + ](n- l)rl.

rl

15.13 (a) Use the method of least squaresto fit the line V: Vo* az to the data in table 15.2. (b) Fit the curve Z : Vo+ az * bzl to the samedata. 15.14 Showthat eq. (15.50)has the matrix solution

./:lUU\"1

f-

r(tl:

(ll2nll

[ R ( o ) c o sa t

- X ( a ) s i nr , r l ] d o ;

J -

(cosr+jsin.t)":elr'\ : (cos nx -r j sin il"r) (de Moivre's theorem).

n I

I

f| [R(or) sin orr + X(o) cos t,lt]do. J -

x(1) = (ll2rl

(b) When g(l) is a real function, show that R(tr) :

R(- o). X(.ll: r f-

-y,

o r ) .G ( - o r ) : G ( o r ) - :

I c(/): (l/2n)RelI Ctrl.,''a, I

LJ-

I

f-

: 1l/n) | tnlrl cos t,l/ - X(r,r)sin o1l do. J,, (c) When g(l) is real and even,prove that G(o) is real and even. (d) When g(t) is realand odd, provethat G(to)is imaginary and even. (e) When g(t) is real and causal,show that

t-

R(or) : I S(r)cos tor dr, r,n

//. Table 15.2 Velocily-depth data

lll, I l, l,ll :;ll.

t :llrl x; r; . l

l ' : ll'?' xr x'i

:

:

',il ll

l

0.50km 1.00 1. 5 0

2.02km/s 2.t6 2.32

2.00km 2.50 3.00

2.50km/s 2.58 2.60

565

PROBLEMS

X(a) : -l

f-

Jn

Sft)sin or dr.

15.18 (a) Verifyeq. (15.123)by subtractingthe transforms of two stepfunctions. (b) Show that the waveshapein the time domain correspondingto a boxcarband-passfilter (fig. 15.15)is (i) the modulatedsinusoid

(c) Show that eq. (15.176)becomesfor digital functions l,:

f> o, | [g'(") g'(t + t)]' dt expandand iJentify the various integralsas S,,(0) or 0,,(l).) (b) Sameas (a) exceptfor a random function. 15.23 Two functions,X(o) and R(
"lll:ll i:ll ;; ll : l l 'tl

y, ll F i g . 1 5 . 1 5 B o x c a rf i l t e r

|

",-,t."" - llln'

15.24(a) Verify the fotio*ing palrs: t, <+ ntls,*l,

Jltl = (2lrrtl sin [j(r,r, r,lJ/]cos [j(o, * o,,)/]; (ii) the differencebetweentwo sinc functions. 15.19 Proveeq. (15.146)by finding the inversetransform of G,(co)* G.(o). in eq. (15.158) 15.20 (a) Verify the four relationships by drawingcurvesto representgr(t) and gr(t), carrying out the requiredreflectionsand translationsand comparing the results. (b) Verify the last three relationshipsof eq. (15.158) by substitutionin the integralexpressions. 15.21 Provethe secondrelationshipin eq. (15.159). 15.22 (a) Show that the autocorrelationof an aperiodic function has its greatest value for zero shift. (Hint: Startwith the identity

i]ltn)

.

Laplace transform

n : positiveinteger;

tlt

'

c o s ' 0 ,*1z \ , +

\,

s2*4u.rr (t 2)'srep(t 2) <->e ' "(5!/s6); (t - 2)'step(l - 3) e+ e-I(s2* 2s + 2)1s3.

(b) Find the inversetransformsof l(s, + 9), e a'i(s'+ 9), I ds/(s'+ 9), l/sb: + 9). J, 15.25 Solvethe following differentialequationsusing Laplacetransforms(see$15.1.9for part (b)): (a)

dl

,' + JY: )e."'

!:4atx:0;

oif

(b)

d'/*5dl

*4r:

dtl d/ d1 : U a n y d v d r ' 3

sinh 21, l 6

atl:0.

15.26 Show that sinh kr er (: sinh kL)l(2' - 2z cosh kA + l), coshkl e> (l - ; coshkA)/(22- 2: coshkA + 1). ) nd (15.187b ) e1 5 . 2 7( a ) S h o w t h a t e q s .( 1 5 . 1 3 6 a come for z-transforms g, "<+ z"G(z). ( b ) S h o wt h a t e q s .( 1 5 . 1 3 7a) n d ( 1 5 . 1 8 8b) e c o m er,e spectively, e ,,tg,<) G(ze-t,,ty,e ,,,g,G(ze .^). 15.28 Using the digital functions a, : fao,a1, et e+ ao]and b,: [bu,b, brl, work out the equivalentsof e q s .( 1 5 . 1 4 5a) n d ( 1 5 . 1 4 7i)n t e r m so f z - t r a n s f o r m s . 15.29 (a) Show that the wavelet z(2 - z) is noI minimum-phaseby consideringthe variation of the p h a s e1 a s i n $ 1 5 . 5 . 6 . (b) Generalizethis resultfor the waveletz"(c'- z),then to the waveletz"W(z),wheren : integer and W(z)is any minimum-phasewavelet. 15.30 The z-transformof a waveletis { [ 1 + ( 0 . 5 + 0 . s j ) ; ] [ 1+ ( 0 . 5 0 . s j ) ; ] ( l 0 . 5 2 ) ] ' . (a) Plot the waveshape. (b) Plot the roots with respectto the unit circle;what can you say about the phaseof the wavelet? (c) What is the zero-phasewaveletwith the samespect r u m ( s e e$ 1 5 . 5 . 6 d )P?l o t i t . (d) Plot the roots of the zero-phasewavelet.

566

BACKGROUND MATHEMATICS

15.31 Show that a zero-phasewavelethas its maximum amplitude at the origin. (Ifult: Considerthe energy spectrum.) 15.32 (a) lf (q - bz) is minimum-phase,show that ll(a - bz) is also minimum-phase. (b) Show that the convolution of two minimum-delay waveletsis also minimum-delay. (c) Given that A(z) is minimum-phase,whereasB(z) is causal but not minimum-phase,under what conditions will the sum, A(z) + BQ),be minimum-phase? (d) Writing .B*,,4r, and./9** for the column matriceson the left sidesof eqs.(15.229)and (15.230),we have .i'9*-c,./1+:,fi**. Show that.Z** is minimum-phaself .)9* is. (Hint: Note that .j/I : zk.B*.) 15.33 Showthe following for a minimum-phasewavelet of the form

fl@,-')

W(z):'=

f{ur - ')

,

a,*0,

bk+0.

(a) The initial amplitudecannot be zero. (b) The maximum amplitude is not necessarilythe first element. 15.34 If a spectrumR(z) does not have roots ; : 0, -r l, show that there is one and only one minimumphasewaveletcorrespondingto R(z) (ignore multiplicativeconstantsc, wherelcl : l). 15.35 (a) Given that the spectrum of a wavelet(assumedto be minimum-phase) is -62-2 - 5z-r * 38 (15.225) 5z 622,useeq. to find the wavelet. (b) Find the waveletusing the Levinsonalgorithm (see eqs.(15.226)to (15.230)). 1 5 . 3 6G i v e nt h e w a v e l ew t , : 1 1 4 4-, 9 6 , * 5 6 , 4 8 , l , - 6 , 1 ,0 l : (a) Show that the z-transform of the autocorrelation (the spectrum)is Rk) : 1442-6- 9602-s+ 6642 a * 72002 3 - l30l5z-' - lll00z | + 35430 - 1ll00z - l30l5z' + 720023 * 6642a- 9602s+ 14426; (b) Show that w, is minimum-delay (find the factors of

w(z)). (c) Show that the correspondingzero-phasewaveletis

02, -40, -39, ti}, -39,*40, t2).

(d) What is the earliest causal linear-phasewavelet correspondingto part (c)? 15.37 Show that when we usethe definition z : e*j'^, e q .( 1 5 . 2 3 1b) e c o m e s g" = (U2rjL){G(z)2, I dz, where the integration is along the unit circle in the counterclockwisedirection. 15.38 Show that the right-hand expressionin eq. (15.242)correspondsto g, delayedby one time unit; hence,verify eq. (15.243).(Hint: Investigatethe righthand term in eq. (15.242)graphically,then study the

summationin relation to the graph.) 15.39 (a) Verify the relationf,Q) : -f,@, t I 0 when both filters havethe samecutofffrequency,by starting from the relation/l(t) + f,(t) e f'.(,o) + F,(a). (b) Compare the values of f ,(t), f Jt), fi, andfi. 15.40 Show that the filter .F(
lt'll < coo, lrl > ro,

is identicalwith that of eq. (15.251)exceptthat/l(l) is shifted by k time units. 15.41 Obtain the digital filter correspondingto the Butterworth filter for n : 3. 15.42 Verify the transforms in eqs. (15.262) to (15.269).(Hint: ln eq. (15.263),multiply sinc (ntlZ) by boxr, (r) and useeq. (15.1a6); for eq. (15.264), note the resultin eq. (15.268);for eq. (15.269),note problem6.21a.) 15.a3 (a) Verify eq. (15.216).(Hint: Follow the same procedure as in the derivation of eq. (9.73a) except that there is no equationcorrespondingtof because it is constant.) (b) Verify eq. (15.278). References Andersen. N. O. 1974. On the calculation of filter coefficients for maximum entropy spectral analysis.Geophysics,39l69 72. Beth, M. 1974. Spectral Analysis in Geophysit's. Amsterdam: Elsevier. Bendat, J. S., and A. G. Piersol. 1966.Measurementand Analysis of Random Data. New York: Wiley. Blackman, R. B., and J. W. Tukey. 1958. The MeasurementoJ' Power Spectra. New York: Dover. Burg, J. P. 1972. The relationship between maximum entropy spectra and maximum likelihood spectra. Geophysics, 31:. 31s 6. Burg, J. P. 1975. Maximum entropy spectral analysis.Ph.D. thesis, Department of Geophysics, Stanford University, Palo Alto, California. Cassand,J., B. Damotte, A. Fontanel, G. Grau, C. Hemon, and M. Lavergne. 1971. Seismic Filtering. Tulsa: Society of Exploration Geophysicists.(Translated by N. Rothenburg from Le Filtrage en Sismique. Paris: Editions Technip, 1966.) Cheng, D. K. 1959.Analysisof Linear Systems.Reading, Mass.: Addison-Wesley. Churchill, R. V 1963. Fourier Series and Boundary Value ProbIems,2d ed. New York: McGraw-Hill. Claerbout, J. F. 1963. Digital filtering and applications to seismic detection and discrimination. M.Sc. thesis, Massachusetts Institute of Technology,Cambridge, Mass. Claerbout, J.F. 1976. Funtlamentals of Geophysical Data Processlng New York: McGraw-Hill. Claerbout, J. F., and F. Muir. 1973. Robust modeling with erratic data. Geophysics,38: 826-44. Cooley, J. W, and J. W. Tukey. 1965.Algorithm for the machine calculation of complex Fourier series. Math. Comput., 19:

297-301. Fail, J. P, and G. Grau. 1963. Les filtres en eventail. Geophys. Prosp., ll:131 63.

561

REFERENCES

Finetti, I., R. Nicolich, and S. Sancin. 1971.Review on the basic theoretical assumptions in seismic digital filtering. Geophys. Prosp., 19: 292 320. Kanasewich, E. R. 1973. Time SequenceAnalysis in Geophysics. Edmonton: University of Alberta Press. Kaplan, W. 1952. Advanced Calculus. Reading, Addison-Wesley.

Mass.:

Kulh6nek, O. 1976. Introtluction to Digital Filtering in Geophlts' lc.r Amsterdam: Elsevier. Kurita, T. 1969. Spectral analysisof seismicwaves,Part I, Data windows for the analysis oftransient waves. Spec. Contrib. Geophys. Inst., Kyoto Univ.,9:9'7 122. Lee, Y. W 1960. Statistical Theory of Communication. New York: John Wiley. Papoulis, A. 1962. The Fourier Integral and its Applications. New York: McGraw-Hill. Pipes, L. A., and L. R. Harvill. 1970. Applied MathematicsJor Engineersand Physicists,3ded. New York: McGraw-Hill. Postic, A., J. Fourmann, and J. Claerbout. 1980. Parsimonrous deconvolution. Preprint of paper presented at the SEG 50th Annual Meeting, Houston.

of Communications. lJrbana: University of Illinois Press. Silvia, M. T., and E. A. Robinson. 1979. DeconvolutionoJ Geophysical Time Series in the Explorationfor Oil and Natural Gas Amsterdam: Elsevier. Simpson, S. M., E. A. Robinson, R. A. Wiggins, and C. I. Wunsch. 1963.Studiesin optimum filtering of single and multiple stochastic processes, Science Report 7' Contract AFl9(604)7378. Cambridge, Mass.: Massachusetts lnstitute of Technology. Smylie, D. 8., C. K. G. Clarke, and T. J. Ulrych. 1973. Analysis of irregularities in the earth's rotation. In Methods in Computa' tional Physics,Vol. 13,Geophysics,B.A Bolt. ed., pp. 391 430' New York: Academic Press. Stoffa, P L., P Buhl, and G. M. Bryan. 1974. The application of homomorphic deconvolution to shallow-water marine seismology. Geophysics,39:401 26. Taylor, H. 1981.The /, norm in seismicdata distribution. In De,ilop^rnrt in Geophysical Exploration Methods-2' A. A. Fitch' ed., pp. 53 76. London: Applied SciencePublishers. Treitel, S., and E. A. Robinson. 1964.The stability of digital filters. IEEE Trans. Geosti. Electron., GE-2: 6 18.

Potter, M. C., and J. L. Goldberg. 1987.Mathematital Methods. Englewood Cliffs, N.J.: Prentice Hall.

Treitel. S., and E. A. Robinson. 1981. Maximum entropy spectral decomposition of a seismograminto its minimum entropy component plus noise. Geophysit's,46:I 108 15.

Rietsch, E. 1979. Geophone sensitivities for Chebyshev optimized arrays. Geophysics,44:ll42 3.

Treitel, S., J. L. Shanks, and C. W Frasier. 1967' Some aspects of fan filtering. Geophltsics,32:789-800.

Robinson, E. A. 1962. Random Waveletsand Cv'bernetit' S1'stezrs London: Griffin.

Ulrych, T. J. 1971.Application of homomorphic deconvolution to seismology. Geophysics,36: 650-60.

Robinson, E. A. 1967a. Multichannel Time SeriesAnalysis v'ith Digital Computer Progrums.San Francisco: Holden-Day.

Weast, R. C., ed. 1975.Hundbook oJ'Chemistryand Physir'.r,56th ed. Cleveland. CRC Press.

Robinson, E. A. 1967b.Predictive decomposition of time series with application to seismic exploratio+ Geophysits, 32: 418 84.

Wiggins, R. A. 1977. Minimum entropy deconvolution. In ProoJ- the Internationu! Symposium on Computer-Aided ,udirgt Seismit' Analysis and Distrimination, pp. 7 14. New York: IEEE Computer SocietY.

Robinson. E. A. 1967c. Statisticul Communi(dtion ttnd Detection. London: Griffin. Robinson, E. A., and S. Treitel. 1973. The Rohinson Treitel Reatler. Tulsa: Seismograph Service. Robinson, E. A., and S. Treitel. 1980. GeophysicalSignul Anuly.in. Englewood Cliffs, N.J.: Prentice Hall. Schneider.W. A., K. L. Larner, J. P Burg, and M. M. Backus. 1964. A new data-processingtechnique for the elimination of ghost arrivals on reflection seismograms. Ceophysits, 29: 783 805. Shannon. C. E., and W Weaver. 1949. The Mathematicul Theory

Wiggins, R. A. 1978. Minimum entropy deconvolution Geoexplorution, 16: 21 35. Wiggins, R. A., and E. A. Robinson. lg65 Recursive solution to ihe multichannel liltering problem. J. Ceophys Res, 70:

r 8 8 59 1 . Wylie, C. R., Jr. 1966. Advanced Engineering Mathematics, 3d ed. New York: McGraw-Hill. Yilmaz. O. 1987.Seismk'Data Protessing.Tulsa: Society of Exploration Geophysicists.

Index

Note; Pagenumbers in italics indicate definitions, in boldfaceindicate figures Abbott, H. L., 3 abbreviations,list of, 569 ABC refraction method, 433, 506 abnormal pressure(overpressure),108, l l 8 1 9 , 1 2 6 ,1 2 8 , 1 2 9 , 1 3 0 , 3 5 1 d e t e c t i o no i I l 8 1 9 , 1 2 8 ,1 2 9 , 1 3 0 effect on velocity, 108, 128, 129, 130 role in diapiric flow and faulting, 365, 368.372 absorption, 59 60, J9 coefficient of,59, 180 effect on waveshape, 60 expressionsfor, 59 60 frequency dependenceof, 59, 177, 1 8 0 ,l 8 l in LYL, 124 measurement of, 180 relative importance vs. spreading, 60, 6l values of constants, 180 acceleration-cancelinghydrophone, 223 accelerometerto measure earth motion, 7t5 ?51 used in inertial positioning, 198 accommodation, 401 acoustic impedance, 75 calculation by inversion, I 36 8, I 39 contrast, 76..7: drc to chemical changesor recrystalization, 128, l 3 l , 1 5 2 .1 5 3 a c o u s t i c( s o n i c )p o s i t i o n i n g ,1 9 7 8 o f s t r e a m e r ,1 8 , 1 9 8 ,1 9 9 u s i n g D o p p l e r - s o n a r ,1 9 5 , 1 9 7 8 using transponders, 197 activity, seismic,23 8, 29, 30 A/D (analog-to-digital) conversion, 230 Adachi's refraction method, 433 4 adaptive filtering, 299 addition of complex numbers, 523 of matrices, 520 of traces,see stacking ofvectors,518 adjoint (matrix), 520 AGC (automatic gain control), AVC, 1 5 , 1 8 , 2 2 6 , 2 2 7 - 82, 2 7 age, effect on velocity, 120 aggradation,409 alr gun: arrays, 21 1, 213, 214, 216 far-lield signature of , 214, 211 land,204,205, 218 marine, 18, 68, 21 l, 213, 214, 215, 216,217 use in profiling, 507 waveshape, 214, 216, 217 air shooting, 204,206

airwave,6,8,l47 Airy phase,54,485, 485,486, 510 365,365 alaucogen, Alford rotation,476 technique reconstruction algebraic (ART),497 alias,2'79,282 aliasfilter, 230, 233, 282 aliaslobe,248 aliasing,146,282-3,451 2 spatial,252-3,4512 ambientnoise,225,254 353 ambiguityin interpretation, AmeradaPetroleumCo.,8, l4 Ameradatree,14 200,201, ammoniumnitrate(explosive), 203 226 9.230 2,233 amplifier, analog,2279,231 digital,230,232,234 dynamicrangeof,226 gainconlrol.typesof: automatic ( A G C ,A V C ) ,1 5 ,1 8 , 2 2 6 , 2 2 87 : b r 226:instantanary.230:ganged. neousfloating-point(lFP), 230; 226; manual,I 5; preprogrammed, quaternary, 230 recordingaccuracy,226 for, 226 requirements 24-bit,226,230 amplitude,-i4 in processing, 303,313, adjustments 3I 5, 338,341,392 affectedby changewithin Fresnel zone,155 build-upnearcriticalangle,77-8, 1 6 8 ,1 7 0 on traveltime,177 dependence effectof bedthicknesson, 174,176, 118,179,464:reflectorcurvature on, 156,157;tuningon,174,116, t77,118,119,464 325 of envelope, factorsaffecting , 58 9, 177, I 80, 313 ashydrocarbonindicator,79, 81, t2t 2, 152,415 18,475,478 363,460;seealso mapping(displays), horizonslice as net thicknessmeasure,174,176' 417 preservation, 3l3, 315 2 ,7 8 , 5 3 1 , 5 3 3 spectrum of stackedsectionsvs.normalincidence,3l5 standout,145,146 303,315 surface-consistent,

515

v a r i a t i o nw i t h o f f s e t( A V O ) , 1 8 , 2 1 , 78 81,321,415,418,476 ANA, I93 analog amplifier, 227 9, 231 analog-to-digital conversion (A/D, digitizing),230 analytical Irace,325 anchored transponders(pingers), 197, 453 angle: of approach, 88, 89 of incidence,62, 63 of reflection (refraction), 62, 63 angular frequency, 34 angular wavenumber, .J4 a n i s o t r o p y .3 8 . 5 5 8 . 5 5 , 1 2 2 . 1 2 5 anticipation filter, 2 97 anticlinal trap, 351, 352 antinode, Jj antisymmetric wavelet, l8l aperture, migration, 326, 328 9, 457 aplanatic surface (curve), 428, 429' 500 a p p a r e n td i p , 8 8 , 8 9 apparent dip moveout, 248 apparent truncation, 404 apparent velocity, 88, 130 a p p a r e n t - v e l o c i t yf i l t e r i n g , 1 8 5 ,3 1 5 1 6 , 315,317,318,319 apparent wavelength,88 apparent wavenumber, 88 Aquapulse (sleeveexploder), 217. 507 Aquaseis, 204, 218 arbitrary line (3-D), 4J9 archaeologicalsurveys, 512 a r e a l a r r a y s ,2 4 7 , 2 5 0 1 , 2 5 2 Argo, 193 a r r a y - s o n i ct o o l , 1 3 3 , 1 3 4 array stack, 250 arrays,247-252 areal, 247, 250-1,252 attenuating noise, 184 basic concepts,247 causing high-frequency loss, 253 directivity of,247-51 effect on transients,250 effective length of 248,250 of geophonesvs. source points, 251 harmonic wave response of,24'7 50 lobes, 248 of marine sources, 213' 214, 216 maximum length oi 254 practical constraints on, 251 2, 254 reject region of, 248 responseof, 247-50, 247 tapered (weighted), 250, 251

516 uniform linear,247-50 use in refraction, 252 arrival time (traveltime), 2 ART (algebraic reconstruction technique), 497 artificial illumination display, 460 artillery ranging, 4 aspect ratio (pore), 122,475 asymmetric spread, 88, 243, 268 at-the-geophonedigitization, 18, 223, 224 Atlantic Refining, 8 atomic clock in radiopositioning, 192, 194,195 atomic rveight. effect on velocity, I 16, I19 attenuation: o f c o h e r e n t n o i s e b y a r r a y s ,1 8 4 , 2 5 1 compensation for, 177 o f h i g h f r e q u e n c i e s1, 8 4 , 1 8 5 , 2 5 3 of multiples, 166, 168-9, 320, 321 ofnoise by adding signals. 184-5 of noise by arrays, I 84, 250, 25 I seea/so absorption attenuation mechanisms, I 77 attic oil, 500,514 attribute, -125 complex trace, J2J display,460 inversion to velocity, see seismiclog 3-D (dip magnitude, azimuth, artific i a l i l l u m i n a t i o n ) , 1 8 ,4 6 0 , 4 6 1 , 463, plate 6 autocorrelation, 285 7. 285, 542. 543 calculation using rrratrices,521 normalized, 286 7 of random functions, J4J a u t o m a t i cg a i n c o n t r o l ( A G C ) , 1 5 , 1 8 , ))A ))1 R ))7 automatic picking, tracking, 18, 325, 460 t,460 automatic statics determination, 303 5, 306,307 Autotape, 193 AVA (amplitude variation with angle), 1 8 ,2 1 , 7 8 - 8 r , 3 2 1 ,4 1 5 ,4 t 8 , 4 7 6 AVC (automatic volume control) 15, 18, ))6 ))'7-9, )27 average(equivalent) velocity, 91, 128, 140 AVO (amplitude variation with offset), 1 8 ,2 1 , 7 8 8 1 , 3 2 1 , 4 t 5 , 4 t 8 , 4 7 6 azimuth display, 18, 460,461,463 azimuthal anisotropy, 55 azimuthal VSP,487 B/o ratio: used to determine lithology, I 16, 119 used to distinguish fluids, 122,124 backprojection, 324, 494 5, 494, 496 backreef, J8J, 3M,385 Backus filter, 284 backward branch of diffraction, 68 backward (forward) Gregory-Newton formula, 528 backward prediction filter, 560 balancing sections,370 band-passfilter, 556-7, 5J6 Barbers Hill Dome, l0 barrier reef, 384, 385 Barry's method, 439 40, 441 Barton, Donald C., 5, 10, 15 base of gas-hydratereflection, 128, 131

INDEX

baseof LVL, 124;seealsolow-velocity (weathered) layer(LVL) basemap,J57 basesurvey(time lapse),499 basementthrusts,364 5, 366 basin-floorfan, 405, 408 Bauer'smethodoffinding intervalvelocity, 140 I Bean-bagenergysource,207 beats,44 bed thicknessdetermination,173-7, 178,179 BelleIsle,5 BetsySeisgun(energysource),207,209 bilinearinterpolation,3/ / bin.451,452 binarygaincontrol,2-10 binary scale,229 binomialseriesexpansions, 522 bioherm(reef),383 Biot equations,I l2 "bird dog,"241 (S-wavesplitting),56, 57, birefringence 476,480.482 bit (binary),230 bit (drill),199 blaster,2, 201,242 blastingcap,2, 201,203 blastphone,6, S blind (hidden)zone,95,433, 437,446 block(3-D).453 Blondeaumethod,I 24, 126,2723 Body,J. 8., 5 body force,40 body wave,41, 44 9 Boomer,2l4.2l8 borehole: salt proximitysurveysfrom, 428,429, 500,s02 sonicloggingin, 500 sourcesin, 205 velocitysurveyin, 130 l. 135,14l, 142 borehole-compensated sonde,l3l 2, 133 boreholegeophone,130,488 boreholeteleviewer, 500,503 borehole-to-borehole studies.21, 497 9 bottomcable,260 boundaryconditions,47, 70 lor Lovewaves,52 lor normal-modepropagation, 483 at planeinterface,73,74, 75 for Rayleighwaves,49 for Stoneleywaves,50 lor stretchedstring,35 boxcar,28l, 535 boxcarwindow,558 principle,152 brachistochrone branchof diffraction,68 multiple,156,159 reverse, 156,159 breaks,first, 228,232 brightspot,18,363,415 examples of, 416,plates2, 3, 5, 10,12 broadsidereflectionspread,243 broadsiderefraction spread,42 7 brutestack,452 Brutusenergysource,207,209 bubbleeffect,2ll, 213-14 214,216,217 effecton waveshape, Buffalo gun (energysource),207, 209 bulk modulus,-i8

burial depth: effecton porosity,108,1l8 effecron velocity,118-20,l2l,122 buriedfocus,I 56, 157,158,160,392, 393 for long offsets,156,158 phaseshift at, 157 253 buryinggeophones, butterflyfilter,315 Butterworthfilter,557 8, JJZ cable,geophone, 2, 223,242,244,268; seea/sostreamer cable(streamer)reel,2l3, 225 cablestrumming,225 Cagniard,L., 82 camera: 233-4 electrostatic, photographic, 10,233 rastertype,234 CameronMeadowsfield(casehistory), l0 l3 cap,electricblasting,2,201,203 flooding,500 carbon-dioxide CarterOil Co., l9 CAT scanning,492 comparedwith seismictomography, 497 Cauchy,Baron,3 Cauchyprincipalvalue,544 causalwavelet(function),181,550 CDP,seecommonmidpoint(CMP) celestialnavigation,192 cement-bond log, 133 centraldifference, J28 cepstralanalysis(deconvolution), 298 9,554,5 cepstrumdomain,291J,554 chain(survey),l9l, 241 changesin waveshape: due to absorption,60, 180 dueto filtering,l8l, 182,236,237 due to ghosts,163-5,168,292-3 due to peg-legmultiples,163,168 401, in reflection-character analysis, 412 t3,415 channelwaves(normal-modepropagation, seamwaves),53, 483 6,483, 487,510,511,512 usedto locatefaultsin coal seams, 5t0l2 (erosional), channels 386,388,389,413, 415,463,509, 510,plates8, 10,15, l6 (recording), 8, 10,l l, l4 channels minimumnumberof, 254 chaoticreflections, 409 character(of event),145,146 roots(eigenvalues), 522 characteristic characteristics of events,seedistinguishing featuresof events of faulting,373,375,376 characteristics charge{explosive ). seeexplosives chargesizeand depth,202,254 chart: stacking,244-5 wavefront,91, 95,267 8, 269 Chebyshevnorm (fit), 342 Cheopspyramid,318,320 chirp filter,552 circularraypath,equationsfo1 93 circularshooting(marine3-D acquisition),453,456,plate9

I {

INDE,I-

clay, elTecton velocity, 116, 117 18 client represent ative, 241 clients/contractors, 239, 240 | climate, effect on deposition, 402 clock, aromic, 192, 194, 195 closing contour, 351 c l o s i n gl o o p s , 1 8 , 3 5 2 closure,35l CMP, see common midpoint coal: e x p l o r a t i o nf b r , 2 3 , 4 8 6 , 5 0 8 , 1 2 l o c a t i n gf a u l t s i n , 5 0 9 1 0 , 5 1 2 longwall mining of, 23, 508 properties of, 508 u s eo f i n - s e a m m e t h o d s . 5 0 9 l 2 use of surface methods, 508 9, 510 coastal onlap, 404,406 COCORP,4I8 coefficient of absorption, 59, 180 coefficient of reflection, 35, 63,76 7. 78, lt6, ll7 coefllcient of transmission, 35, 76, 77, 78 cofactor (determinant, matrix), 517. 520 coherence1 , 45 6,145 multichannel. 288 9, 290 coherent noise, 18J a t t e n u a t i o no l I 8 4 , 1 8 5 , 2 5 1 coinc ide n t-t ime curve, 412, 443, 41 7 c o l l i s i o n ( s u b d u c t i o n )z o n e , 3 6 4 5 , 3 6 8 , 418,4t9 color used in displays, 138, 234, 362, 363,460, 464, sae plates,especially p l a t e s1 4 , 1 5 C o l o r a d o S c h o o lo f M i n e s , l 0 comb (sampling lunction), 28I , 536, 539 t r a n s f o r mo f , 2 8 1 , 5 3 9 combination (composite) displays (3-D), 459-60, plate 7 c o m b i n e d{ i n t e g r a t e d )n a v i g a l i o ns y s tems,195 combining geophone and hydrophone records, 293,294 common-depth point, .ieecommonmidpoint (CMP) common-geophone gather,24J common midpoint (CMP), l3 14, 18,

t9, t83,2423,2445 common-midpoint field operations, 242 3,244-5 common-midpoinl galher, 2 45 common-midpoint multiplicity, 2zl4 common-midpoint section, relation to zero-offsetsection, 321 common-midpoint stacking, 244, 320 | e f f e c to i d i p o n , 3 1 6 1 8 ,3 1 9 , 3 2 0 common-midpoint stacking chart,

244 s common-offset galher. 245 common-source galher, 2 45 common-tangent migration method, 328 compaction: differential, 351, 367, 370, 385 effect on velocity, I 18 20 Compagnie G6n6rale de G6ophysique,

t4 in streamer,198,199,225, compasses 453 complexnumber,522 3,522 complexplane,522 complextraceanalysis,325 6 complexvelocity,60

compliance, 3Z component of dip moveout, 89 component of a vector, 518 compositing,229 compressibility,38 compressional leatures (structures), 364 5,366 compressional (P-) wave,44 velocity of, 44 computer: modeling, 391 2 used in survey planning, 241 concentric folding, 370 concordance,404 condensed seclion, 407, 4O9 conjugate complex, 523 conjugate filter, 557 Consortium of Continental Reflection P r o f i l i n g( C O C O R P ) . 4 1 8 constraints (least squares),525 6 continuous coverage(profrling), l, 244 continuous linear source, 203 4 continuous velocity log, see sonic log contouring map data, 356 contracting industry, development ol, t 3 1 4 , 2 34 , 2 6 contractors/clients,239, 240, 241 con trast: acoustic impedance, 76 7 velocity, 76 conventional well-velocity survey, 130 I, t32, t4t,l42 convergent boundary, 164, 366, 367 converted waves, Z-i energy distribution at interface, 76, 78 c o n v o l u t i o n ,2 T 9 8 1 , 5 3 8 , 5 4 0 l , 5 4 5 , 546,548 c a l c u l a t i o nu s i n g m a l r i c e s ,5 2 1 formulas for, 280 l, 548 multidimensional2 , 85,542 relation to cross-correlation,285, 541 of' sampled functions, 548 c o n v o l u t i o nt h e o r e m ,2 8 0 l , 5 3 8 , 540 1.545.546 o o n v o l u t i o n a lm o d e l , 3 7 , 1 4 1 . 2 8 3 4 correcting for: a b s o r p t i o na n d p e g - l e gm u l t i p l e s ,3 1 3 near-surfacevariations, 261 2, 263.

266.303 6, 307,338 sourcesignature,338 sphericaldivergence, 315, 492 usingfirst breaks,256,259 usingrefractionprofile,266 vtr vB,256,258 correctingreflecliondata: 262 differentialweathering, dip moveout(DMO), 18,91,306, .ll6 18.319,320:rulesfor applying,3 I 8 for geophones in betweensource points,266 normalmoveout(NMO), I9, 146, 303 specialrefractionprofile for, 266 weathering),15,l8 statics(elevation, 1 9 ,1 4 6 , 2 6 21 , 2 6 3 , 2 6 6 , 3 063, 307,338 staticsfor marinedata,457 at sourcepoints. for traveltimes 261 2,263,26 for waveletvariations,292,295.298, 299 300

correcting refraction data, 429 correlation (mathematical), 285 8, 289,

s38,s412,543 autocorrelation, 2857, 542,543 calculation using matrices, 521 cross-correlation, .reecrosscorrelation multichannel coherence,288-9 normalized, 286-7 ofVibroseis data, 208, 261,287 8, 289 correlation function, 285, 541 normalized, 287 correlation ghost, 208 correlation reflection method, 15, l8 correlation of reflections, 15, 268 a c r o s sf a u l t s , 3 7 3 , 3 1 7 with refractions, 429 corridor stack, 490, 492, 493 cosine transform, 533 cosine window, 559 costs, 26 8, 30 coupled waves,57 coverage,redundant, 242 C r a m e r ' sr u l e , 5 l 7 l 8 C r a w f o r d ,J o h n M . , l 9 creep strain, 38 crew: activity: 23, 21, 25, 26. 28. 29 organization of,239 40 criterion (norm) for goodnessof fit, 259,342,559 critical angle, 63 damping (geophone),220 distance, 9J crooked-line methods, 246, 248 oross-oorrelation,285, 287 8, 538,

s4t 2, s43 calculation using matrices, 521 cycle skip, 304 normalized, 287 of random functions, 543 relation 1o convolution, 285, 541 of sampled functions, 548 of Vibroseis recordings, 208, 261, 287 8,289 cross-correlationtheorem, 285, 538, 54t 2 cross-dip,8890 cross-energyspectrum, 542 cross-equalizetraces,295 cross-holemeasurements,497, 498, 499 orossline (3-D slice),459 c r o s s - p r o d u c(t v e c t o r s ) ,5 1 8 1 9 ,5 / 8 cross-secttons: correlating reflectionson, 268 drawing phantoms on, 268 effect of not migraIing,267 record sections, 18, 19,268,429 types of (depth, time, migrated, unmigrated),267 crossover distance.96 cross-spread,89, 243 crossline(3-D),459 crow's foot array,252 c r u s t a ls t u d i e s , 4 l 8 l 9 subsidence,40l2,403 cubic packing, 107, 108 curl, J19 curvature of reflector, see reflector curvature curve of maximum convexity. -l2Z

578 curved-raypathweatheringcorrections (Blondeaumethod),124, 126, curvedvelocity surfacesproducinglocusing,156-7,160 gutoff,229 cutoff frequency,229 cycle,eustatic,402, 404-5, 406 cycleskip: in cross-correlation, 304 in S-waveprocessing,474 in velocity logging,132 cylindrical divergence,59 cylindrical-wave simulation(Simplan section),322 4 D/A converter,233 d'Alembert's solutionof waveequalion. 34,41, damping(geophone) critical,220 electromagnetic, 15,219,221 mechanical, 218,219 oil,l5 dampingfacIor,60 dataacquisition,239 data block (on seismicsection),353,354 data display,.reecamera;displaymodes data ownership, 239 data processing, 229,275 343 apparentvelocityfiltering,185,315 1 6 ,3 1 7 , 3 1 8 , 3 1 9 automaticstaticscorrection,303 6, 307 convolution,279 8l; srz a/.roconvolution correlation,285-8 ' data reduction.reflection,261 2, 266 8; seealso data reduction,reflection deconvolution, 21, 285,292 303;see a/.sodeconvolution earlyhistoryof, 275 fieldprocessing, 261 flow chart, 341 frequencyfiltering,18,229,233,300, 301 interactiveprocessing, 340 migrationmethods,267 8,326-35. 336.337;seeaisomigration objectiveof, 276 phaseconsiderations, 290 I preservation of amplitude,313, 3l 5 sequence, typical,335,338,340,341 signatureprocessing, 297 stacking,19,244-5,316 24;seealso stacking velocityanalysis,18,306 13,314, 315;seeaisovelocityanalysis workstations, 340 data rate,223 data recording,226 34 analog,227 9, 231,232,233 digital, 18,20 1, 230,232,233,234 data reduction,reflection,261 2,266 8 drawinghorizonson sections, 268 elevationand weathering corrections. 15,261,2,266 fieldprocessing, 261 picking and gradingrelections,266 7 preparingcross-sections, 267-8 useof recordsections, 268 datum, reference,261

INDEX

De Golyer,EverettLee, 5, 8 de Moivre'stheorem,523,564 deadreckoning,I92, I97 Decca,193,194 decibel,59 conversiontable for, 571 (detachment d6collement zone),365, 367,369 deconvolution, 21, 285,292 303 analog,l8 cascading,292 choosingparameters for, 302 3 delayedspike,287 deterministic inverse-fi ltering,292 gapped,298 homomorphic(cepstral),298 9,298 multichannel,303 phasedetermination in, 291,296 predictive,166,168,298,341 spiking(whitening),295 8 time-variant(TV), 300,302 waveletshaping,298,299-300,301 deghosting, 292'3, 294 del(vectoroperator), 519 delaycap,203 delayoperator,528 delaytime,439 methods of refraction computation: Barry'smethod,439,40,44li Tarrant'smethod,440,441;Wyrobek's method,441,2,M,447 delayedspike(deconvolution), 297 demultiplexing, 230, 335 Denham'slormulafor high-frequency limit, 188,235,253 densities of minerals, I 16,I 17,I t9 of rocks,I 16,I 17,I 18,ll9; relation to mineraldensities, I16, lt9 density,relationto velocity,ll2, ll4, i l 5 , l 1 6 ,l 1 7 ,l 1 9 Gardner'srule, I 16,I 19 densitylogs,I l4 depositional energy,398 depositional models,404-5,406,407 depositionaltime line (surface), 403 4, 40s depthof burial: effecton porosity,108,l18 effecton velocity,118-20,l2l,122 depthcontroller,18,225,226,221 depthconversion factor,433,436 ,u {r P- (^r r, h . l - ur r-L^! !+r v^r .,

1f<

!4J

depthmigration, 8,326,333,4,333, 336,337 depth point, 87 depth section,267,363 4. seealso depth migration derivative operator, 528 derivative theorems, 538, 539, 545,546 desired wavelet characteristics,l8l, 182 destructive shelf concept, 388 detachment (d6collement) zones, 365, 367,369 detectable (resolvable) limit of bed thickn e s s ,1 7 3 7 , 1 7 8 , 1 7 9 detector,218 depth, 225 measuring acceleration, \98, 225, 253 measuring pressurevariations, 223-5 measuring velocity, 220 see also geophone; geophone,

moving-coil;geophones,typesof; hydrophone determinant,5l 7-18 of a squaremalrix, 520 usein solvingsimultaneous equations,5l7-18 deterministic inversefiltering,292 detonatingcord,203,204 plow for burying,204,206 detunedamplitude,464 diamondarray,250,232 diapir,shale,368,372 diapiricflow, 128,368 difference,central,528 difference display,section(time-lapse), 460,499 differenceoperator,528 differentialcompaction,351,367,370, 385 numericalsoludifferentialequations, tion ol 529-30 differentialGPS,196 differentialpressure, 108 effecton velocity,ll8, 120,128 differentialweathering corrections, 262 diffractedreflection,l6l, 188 diffraction,63-8, I 59-61, 163,164,165, 166, 161,t87 backward(forward)branches, 68 characteristics ol 159-61,163,164, 165,166,167 from bent reflector,160,165 from edges,159 60,163 from half-plane,66, 67,68, 160,164 from holein a reflectot 160,165 from part of a planesurface,64 5 from pointsource,157,159 60, 163 from step,16l,167 from wedge,160 I, 166 migrationof,327,328 phantom,16l,161 reflected, l6l,188 relationto reflection,65,66,67, 159 theoryof, 63-6, 68 time-domainsolutionfor, 65 6 wavefiontconstructionusingHuygens'principle,67, 68 migration,327--9,334 diffraction-stack diffractiontraveltimecurves,159 60, 163,164,165,167,398 effectof overlyingvelocity on, 398 digiralAGC,233 digitalamplifier,230,232,234 digitalinstrumentspecifications, 571 20 l; seeulsodata digitalprocessing, processing 18,20,229,230,232, digitalrecording, )11 '14 digital representations,229--30 digital revolution, 20, 275 digital systems,547 8, 550 digital-to-analog (D/A) converter,233 digitization: in the amplifier, 230 33 a t t h e g e o p h o n e ,1 4 , 1 8 , 2 2 , 2 2 3 , 2 2 4 in the streamer,225 dilatation, JZ, J8 dilitational (P-) wave, 44,45; seealso Pwave dim spot (hydrocarbon indicator), 415, 417 Dinoseis, 204 dip:

INDEX

apparent,88,89 calculationfrom components, 88 9 cross,88-90 effecton peg-legmultiples,317 effecton reflectingpoints,87,90-l; seealso dip-moveout(DMO) correctlon formulasfor, 87, 88-90 dip-azimuth,dip-magnitude displays(3D), 18,460,plate6 dip filtering,315 dip moveout,15,87 8,82, 146 componentof, 89 dip-moveout (DMO) correction, 18, 91,

3 0 63, 1 61 8 , 3 1 9 , 3 2103,5 , 3 4 r dipshooting, 12,15,18,19 dipole sonic log, 500 dipping reflector, 86 91,92 dipping refractor, 97-8 Dirac delta (unit impulse), 535 6, JJj direct (forward) modeling, J9t) direct recording, 229 direct wave, 86 direction of shooting, effect oi, 245,247 directional charges,202 4 directional wells, VSP in, 489 directivity: of continuous linear source, 203-4 of linear array,247 50 o l s i g n a l p l u s g h o s t , 1 6 3 5 , 1 6 8 ,1 8 7 of tapered array, 269, 270 d i s p e r s i o n , 6 01 , 6 1 , 6 2 , 6 3 ,1 2 0 I , 1 3 2 o f c h a n n e lw a v e s( n o r m a l m o d e ) , 485.512 effeot on waveshape.50. 6l importance of, 6l inversen , ormal, 6l of Love waves,52 of pseudo-Rayleighwaves,54 of Rayleigh waves,50 r e l a t i o nt o a b s o r p t i o n ,6 1 . 6 3 relation to group velocity, 6l of Stoneley waves, 54 displacement potential functions, 46 7 for derivation of Knott's equations, 15 lor P-waves only, 46 for Rayleigh waves. 49 for spherically symmetrical source,47 for Stoneley waves,50 for 2-D, 3-D solids, 46 7 for waves in fluid media, 46 for waves in transverselyisotropic media, 57 display: artificial illumination, 460 a z i m u t h , 1 8 , 4 6 0 , 4 6 1 , 4 6 3 ,p l a t e 6 c o l o r u s e d i n , 1 3 8 ,2 3 4 , 3 6 2 , 3 6 3 , 4 6 0 . 464, plates, especially 14, l5 composite (combination), 459-60. plate 7 difference, 460 dip magnitude,460,46l, plate6 dual-polarity, 417, plate 3 fault slice, 418.459,464, 465, plates 5,13 horizon slice, 459, 460 l, 463 4, p l a t e s4 , 6 , 8 , 1 0 , I I , 1 2 , 1 5 , 1 6 ,

t7 ;l j': iil

l

tri ;"r

s{: , ,vfl s1 I

#; $ji

,*4*

modes, parametersof, 233 4 residual, 460 reversed-polarity,363 sun shade,460

ofS-wavedata,475,416 4 techniques,363 of 3-D data,459-60,46 1,46.2,463, 464,465 time slice,245, 459,460,461,46.2, plate 9 variablearea,variabledensity,234 wiggly trace,234 distinguishing betweensandand shale, ttz, ll3, usingvelocity,ll2,ll3, ll4, 115, 1 1 6 l, l 7 , l l 8 featuresof events,145 6 distinguishing of diffractions,159 61, 163,164,165, 166,161 1619,l7O,l7l of multiples, of reflectedrefractions,17l,114 of reflections,146 159,160,16l, 162 of refractions,169,17l , 172,113 ofsurfacewaves,172 (remotedataunits), distributedsystems 223,2U distribution,J-16 derivativeof, 5J6 (div),519 divergence divergence of waves,cylindrical,spherical,59 for,177,313,'3t5 correction 409 divergentreflections, diversity stack.322 divingwaves,98 9, 98, 100 usedto find velocity,135 Dix'equation: for intervalvelocity,130 for rms velocity,92 Dix velocity,130,134 D M O c o r r e c t i o n1,8 , 9 1 3, 0 6 ,3 1 6 - 1 8 , 319,320,335,341 time,./-k,elc.\,276, domain(frequency, 277 43 dominantfrequency, wavelength, Dopplerformula,194,198 (positioning),195, Doppler-sonar t97 8 518 dot product(vectors), Doty, William, l9 536, double-sided exponentialdecay, 537 doubleweathering,124 downlap,404,401,409 downlapsurface,404, 407,408 downsweep,208 downwardcontinuationof seismicfield, 330 dragcables,260 drapefeatures,367 drapeovera reel 367,38J drill bit, 199 drilling equipment,18, 199,200,201' 202,242 drilling fluid (mud),199 drive(gas,water,etc.),5/J drowningof a reef,J8J dry-writecamera,233-4 dual-polaritydisplay,417,plate3 dumpingdata,335 dynamiccorrections,146 interrelationwith staticscorrections, 303 dynamic range,226 dynamicalsimilarity,390,391 dynamite,200 l, 218;seealsoexplo' SIVES

Dynasource, 20"1 ,209 earthquakeseismology,1, 28 Eckhardt,E. A., 4, 8 editingfielddata,335,338 editing sonic(continuousvelocity)logs, 133 effective array length, 248 effectiveelasticparameter,107 effectiveFresnelzone,154 35, 522 eigenstates, eigenvalues, elasticconstants(moduli),38 effecton velocity,44,107 relationsbetween.38. 68.69 valuesof, 38,71 elastic(strain)energy,38 9 elasticlimit, -18 elasticmoduli.seeelasticconstants ElasticWaveGenerator,207,2W theoryof, 33-9 elasticity, electricarc (sparker),207,214.217, 507 electricblastingcap,201,203 8, l8 electricalseismograph, electromagnetic damping,15,219,221 measuring electromagnetic distance (EDM), t9t 2,241 geophone,^reegeoelectromagnetic phone,moving-coil electrostatic camera.2l3 4 elevationcorrections,146,261 2, 263, 266. 303 5. 30,6 limestone, 4. 5, 359 Ellenberger elliptical polarizalion,46 (equivalent) wavelet,148, embedded 18t,284,29t,299 oi, l8l, 182 desiredcharacteristics end-onspread,243 398 energy,depositional, energy,velocityof travelof, 6l energydensity,58 partitioningat interface,76 relationto intensity,59 ofa trace.286,287.542 energy-frequency relationfor marine 218 sources. land,198 211,212 energysources, choiceof, 210 I I impulsivesources,198 206,207,208, 209:explosives,200-4, 206;large surfacesources,2045,206;small surfacesources,205 6, 207 206,208-10, nonimpulsivesources, 211,212:Sosie,210,212;Vibroseis, 206,208-10,211 471-4 S-wavesources. marine,213-18 energysources, air gun, 213,214,215,216,217 bubbleeffect,211,213-14,217 choiceof, 217 18 211,213 explosives, imploders,214 types,214,216,217 miscellaneous relation betweenenergyand freq u e n c y , 2 l Tl 8 of,214,216 signatures energysourcesfor VSR 488 9 engineeringsurveys,505 8 marine surveys,seemarine profiling objectivesof, 505 reflectionsurveyson land, 506 refractionsurveyson land, 505-6 505,506 usingS-waves,

580 e n h a n c e do i l r e c o v e r y( E O R ) , 1 2 1 . 4 9 9 500, 501, 513-t4, 5 14, 5t5 m o n i t o r i n g o f , l 2 l , 4 9 9 5 0 0 ,5 0 1 , 5 1 5 enhanced weight drop, 205 ensemble,J43 entropy, 299, 56 1, 562 e n v e l o p e ,1 4 6 . 3 2 5 amplirude, J25 environmental applications, 512 environmental considerations,240 E O R ( e n h a n c e do i l r e c o v e r y ) ,l 2 l , 4 9 9 5 0 0 , 5 0 1 ,5 1 3 1 4 , 5 t 4 , 5 1 5 Etitvos, Baron Roland von. 3 equalization, 315 equivalent (average)velocity, 9l, 128, 134 equivalent (embedded)wavelet, 148, l8l , 284, 29t, 299-300 ergodic function, 543 erosional Iruncation. 401 error (goodnessof fit), 295, 342. 559 error field, J42 Euler Cauchy method, J29 Euler's formulas, 564 . European Association of Exploration Geophysicists(EAEG), 29 eustasy,402 eustatic cycle, 404 5, 406 evanescentwave,43, 6J Evans, John W., 4 E v e .A . S . .2 8 events: characteristicsof, 145 6 seismic,2 Evison (pseudo-Love) waves,486, 512 exploding reflector,327, 392, 393 exploration seismology: a p p l i c a t i o n so f , l , 5 0 5 1 5 objectives, I exploralion (sparse)3-D method, 4-52 exploration with S-waves.471 6,471, 478,479,480,,181, 482 explosivegas mixture, 204,207 explosives, 2, 200 4, 206, 218, 242, 253 concentrating energy downward,

202 4 directional charges. 202 4 minimumshootingdistances, 201 minimumsizeof.254 shootingnearsurface.204.2M as a smallenergysource,207 typesof: ammoniumnitrate,200, 203;detonatingcord,204,206;'dynamite,200-.1, gas, 218;explosive 204.207 exponentialdecay,536, 537 extendedrange(XR) shoran,.19J extendedresolution.25J Denham'srule for maximumuseable frequency, 188,235,253 / k ( p x l d o m a i np, l o t , 2 5 43, 1 5 , 1 6 , 315 facies,seismic,398,409 faciesanalysis,seismic,398,401,407, 409-t2,4r3,4r4 faciesboundary(line,surface),150, 403 4,405 fan(f-k) filtering,185,315 16,J15, 317,318,3r9 fan shooting,8, 12, 18,427,428,446 Faraday's law of induction,219 far-fieldeffect,48

INDEX

Fast Fourier Transiorm(FFT), 549 50 fathometer,506 7 fault, faulting; characteristics of, 373,375,316 in coal,508,509,510,512 continuity of, 376, 460, 46.2,463 effectof velocityincreaseon, 373 evidences of, 376, 377,318,461,463 evidences on linesin differentdirections,357 on horizonslices,463,plates6, 8, 1l, l3 inversionol 363 locating,357,37l, 373, 376, 317,318, 461, 463 nomenclature,3Tl,3TS 128, relationto abnormalpressure, 368 relationto structuralstyle,364 8, 369,370,371 relationto underlying/overlying features,365,372,373,379 relationto well picks,359,361 on time slices,461,63 typesol 3& 5,367,368,369,370 verticalresolutionof, 174, 176 fault-planereflection,376 fault slice(3-D),418.459,464,46,5, plates5, l3 fault trap,351,352, laultsin coal,detection oi,509 I0,5ll, 512 faultsand log correlations, 361 Faust'sequationfor velocity,108 featheringof streamer. 236 feedbackfiltering,293, 556 Fejerkernelwindow 558 Fermat'sprinciple(of Ieasttime.brachistochrone),58, 1JZ,185 F-essenden, Reginald A., 3, 4, 8, 13,2l FFT (FastFourierTransform),549 50 fieldequipmentfor land surveys: drilling,18,199,200.201,242 recording(analog,digital),226 34 surveying, 19l 2,241 ^sec ci,roamplifier;energysources. land;geophone, movingcoil fieldequipmentfor marinesurveys: locatingthe streamer,198,199 .iteaLioenergysources, marine;hydrophone;marinepositioning;radiopositioning; streamer lield layouts,reflection,243 4,452 6 arrays,iee arrays broadside, 243 (CMP), 242, common-midpoint 244 5 continuousprohling,244 crookedline.246,248 cross-spread, 243-4 end-on,243 gappedsplit,243 in-lineoffset,243 land3-D, 453,4567 marine3-D, 452 3,454,455,456 noiseprofile(walkaway), 253,255 split-dip,243,244 swath,4JJ 245 6, 247,453 undershooting, fieldlayouts,refraction: ABC, 433,506 broadside, 427 f a n , 8 ,1 2 , 1 8 , 4 2 7 , 4 2 8

four-shot,42'/, 433,506,507 (unreversed), 425 7 in-linereversed split,426 land: field operations, conductof,240-l crew organization, 239-40,242 drilling,199,200,201 261 fieldprocessing, permitting,241 programof, 241 recording,242 3 shooting,242 l9l 2,241 surveying, marine: fieldoperations, 240 creworganization, of, 15,18 earlydevelopment generalprocedures, 258 61, 262,263, 2fJ/.,265,428, 429 profiling, .ieemarine profiling refractionsurveys,428,429 seealso marinepositioning fieldparameters, selectionof, 253 6, 251,258 fieldtesting,255 6 noiseanalysis,253 4.255 261 fieldprocessing, fieldrecords,243 fieldstatics,26l fieldtesting,255-6 15"(45")migration,331 555-6,JJj filter analysis(synthesis), filter panel,301,302 filter stability,555 filtering: by the earth,283 4 of reflectors,163,168 by a sequence by stacking,184,185,300 by a waterlayer,284-5,292,550 filters,2, l8: adaptive,299 alias,230,233,282 analogvs.digital,555 (fan,pie-slice, butapparent-velocity terfly),185,315-16,317,318,319 (high-pass, low-pass), band-pass 556 7 Butterworth,55T 8 chirp,557 conjugate,557 3 deghosting,292 inverse,292 deterministic feedback, 293, 556 frequency,18,229, 233, 300,301 inverse,284,292 Kalman,299 (Wiener),293,295,296, least-squares 559-60 matched,557 561-2 maximum-entropy, median,490 562 minimum-entropy, optimum,293,295, 296,559-62 polarization,480,483, 48,4,487 prediction-error, 560 predictive,556 292 3,556 recursive, shaping,297 time-variant(TV), 302 window(gate),558 9 seea/sodeconvolution method,527 9 finite-difference usedto evaluatederivatives,528 usedto evaluateintegrals,528-9

INDEX

usedto interpolate,528 usedto migrate,330 3, 334 usedto solvedifferentialequations, 529 30 finite-differencemigration, 330 3, 334 finite-differenceoperators,528 fire-floodtechnique(EOR),499 500, 501,514,515 first breaks,228,232 usedin near-surface correction,256, 259,266,305 fit (norm,criterion),295,342,559 fix (navigation), 195 fl,agging,24l indicator),152, flat spot(hydrocarbon plates2, 3, 5 415,416,418, flatteninghorizons,358, 359, 362, plate t4 Flexichoc, 214,217 Flexotir,2lT floating-pointamplifier,18,230, 234 floating streamer,^reestreamer flooding(surface), marine,404,408, 409 floods(gas,water,etc.),500,501,5131 4 ,5 1 5 flow chart,processing, 34l flow structures, 365,367, 368,370,372, 373,374,379,381,382,383 flowerstructure,365,368 fluid incompressibility, 38 fluids: in pore space,effecton velocity, 1 0 89 , l 2 l - 2 , r 2 4 , r 2 s waveequationin, 47 focus,seeburied focus focusrngeffects,156-7,160 fold (CMP),244 fold structures, 365,369,31O,376, 379. 380 forereef,J8J, 385;seealso reefs foresetbeds,.183 format (formatter),230 formatverification,335,341 forward branch(diffraction),68 forward (backward)Gregory-Newton formula,J28 forward (direct)modeling,-190 forward prediction-errorfi lter, J60 499 4-D (time-lapse) surveys, four-shot(refraction)method,427, 433, 506,507 Fourieranalysis,34, 43,27'l 8 Fourierintegral,532 Fourier projectiontheorem,493-4 277, 531-2 Fourierseries, multidimensional, 533 277 8 Fouriersynthesis, Fourier transform,277, 532-3,532 comparisonwith Laplacetransform, 545 last (FFT), 549-50 implementationof , 2'18-9 278,533 multidimensional, theoremson, 538 9 fracture-finder log, 133 fracturestudies,21, 56,476,480,500,

s02 freezing,effecton velocity,121,124 frequency,34 angular,34 dominant,43

effecton velocity,120-1 fundamental(natural), 35,219 frequencydomain,275 frequencyfiltering, 18,229, 233,300, 301 frequencymodulation,229 migration, 334 frequency-space frequencyspectrum,278, 531 Fresnelzone,152 for planewave,155 relationto horizontalresolution,154 relationto migration,155 lor sphericalwave,152, 154 ol 159,162 3-D effects fringe-area width (surveyplanning),452 front-endmuting, 147 frost breaks,126,127 fundamental(natural)frequency,35, 219 gain: automatic(AGC, AVC), 15,226, 227 8 binary,230 ganged,226 floating-point(IFP), instantaneous 18,230.234 preprogrammed,226 quaternary,230 Gal'perin arrangemenr,476 galvanometer,9, 10, 233 gappeddeconvolution, 298 gappedsplit (spread),243 gapsin coverage,245,246 Gardner'smethodof definingsalt domes,427-8,429 Gardner'srule relatingdensityand velocily, I 16 Garrett,L. P.,3, 8 gasaccumulations causingfocusing, I 57, t60 gaschimney, 415 gasexploder,204,218 gashydrate,128,l3f gasin porespaces: effecton amplitude,seehydrocarbon indicators effecton velocity,108 109,l2l-2, t24, t25 equationfor velocity: Gassmann's in fluid-filledporousrocks,I 10, I l2 108 in packingof spheres, gate: boxcar,28l,535 window 558-9,558 gathers, 245,316 Gauss'theorem,519 Gaussian(normal)distribution,JJ9 Gaussianwindow,559 Gazdagmigration,3J0 Geertsmaand Smit equation,112I J

gelatindynamite,200,201, 218;seealso explosives Co., 14 GeneralGeophysical generalized inversion,340,3423 generalizedRayleigh(Stoneley)waves, 3, s0, 52,53-4,133,489 generalized reciprocalmethod(GRM), 434-9 geneticalgorithm,343 Geocor,290 geologicconcepts,t50 2,352

Geologic Engineering Co., 4-5, 8 geologic history deduced from seismic data, 357,358, 359, 360 geometric similarity, 390 geometry of reflection paths: for constant velocity: dipping reflector, 86 91; horizontal reflector, 8 5 6 for linear increasein velocity with depth, 93-5 geometry of refraction paths: for dipping refractor(s), 9'l -8, 433-4 for diving waves, 98-9 for horizontal refractor(s), 95 7 for linear increase in velocity above a refractor, 99-100 geophone,digitization at the, 14, 18, 22,

223.2U geophone,horizonlal, 474 geophone, movingcoil, 15,218-23,2 I 8 couplingto ground,223 damping(overdamped,critical, undamped),220,221 damping,typesof: electromagnetic, | 5, 219,220, 221; mechanical,219; oil,15 dampingfactor,219 equationof motion,219 hum-bucking,221-2,222 measuringvelocity,220,225 naturalfrequencyof,220;relationto passband, 221 distortionless phaseshift,dependence on damping and frequency,220, 221,222;linear with frequency,220 response to driving force,220-l, 222 constant), sensitivity(transduction 220 transientresponseof , 219-20,221 geophonecable,2, 223,242,244,268: seealso slreamer geophonedelaytime, 439 geophonegroup,2,222 geophonespacing,minimum,254 geophonestrings,222 geophones, typesoft (condenser), 9, 14 capacitance seegeophone,movelectromagnetic, ing coil m e c h a n i c6a,l 7 , 8 , 9 , 1 4 ,1 8 moving movingcoil. seegeophone. coil piezoelectric,218;seealsohydrophone l5 variablereluctance, geophysicalactivity,23-8, 29, 30 futureprospectsof, 23-4 Research Corp.(GRC), Geophysical 8-9, 14 Service,Inc. (GSI), 14 Geophysical geostatistics, 502 geothermalstudies,512 ghosts,163-5,167 correlationghosts,208 deghosting,292-3,294 effecton waveshapeand directivity, 164-5,168,186,187,293,294 53940 Gibbs'phenomenon, glacialdrift weathering,124,126 Global PositioningSystem(GPS),18, 22. t9t. 195-7 differential,196 GoldenLane.Mexico.5. 8

582 goodnessof fit. criterion (norm). 259.

342,ss9 grade(ofa reflection), 267,356 gradient(grad),519 Grant Geophysics,14 gravity data, integrationwith, 390 Gray,T., 3 grazing incidence,77 Gregory Newton formulas,528 GRM (generalizedreciprocalmethod), 434-9 groundmixing,253 ground roll, 49 attenuation of, 172,316,318 characteristics of, 172 seealso Rayleigh waves ground-waterexploration,512 group,geophone(hydrophone),2, 222, 224 group interval,244, 254 group velocity,56, 60-l , 60, 62,63 of channelwaves,486 growthfault, 367,368,371 guidedwaves,seechannelwaves G u l fO i l , 6 , 8 ,1 9 gyrocompass,195 plus minusmethod, Hagedoorn's 442 3,444 Hales'method, 443-6 half-intercept time,441,446,447 half-plane,diffractionfrom, 66,67, 68, 160,164 HaliburtonGeophysical Services, l4 hammerenergysource,201,209,473 Hammingwindow,559 Hanningwindow,559 Harkins,T. I., 15 harmonicwave,43 harmonics,35 Haseman, W. P,4, 8 Hayes,HarveyC., 13 hazardousliquids,studiesinvolving,512 headwaves,l, 2, 8l-2, 8l ; seealsorefractionentries Hecker,Otto, 3 Heiland,C. A., l0 Helmholtzseparation method,40 herring-bone array,252 packing,108,109 hexagonal Hi Fix, 193 hiddendata, J02 hidden(blind) zone,95,433, 437,446 high-cut(alias)filter, 230,233, 282 high-frequency losses, causesof, 253 high-frequency solution,1l2 high-line pickup,227 high-pass nlter,556-7,JJ6 high-resolution surveys,253 highstandsystemtract, 408,409 Hilbert transforms,276,29l,543 4 hill-climbing method,342 historyof seismicexploration,3,23,24, 275 hold-downweight,208 hole fatigue,202 holenoise,mutingfor,320 hole in a reflector,diffraction from, 160, r65 homogeneous equations, J18 nontrivialsolutionoi 518,563 homomorphic(cepstral)deconvolution, 298-9,299

INDEX

Hooke'slaw, 3 for isotropicmedium,37-8,37 for transverselyisotropicmedium, 56 horizonflattening,358,359,3623, plate 14 horizonslice(3-D),459,4601,4634 examples of, plates4, 6, 8, 10, I I, 12, 1 5 ,1 6 ,1 7 horizon tracking, 325,460 horizonvelocityanalysis, 311, 313,314,

3ts

horizontal impact,472, 473 horizontalreflector,85-6 horizontal refractor.95-6 horizontalresolution,172, 177 horizontalsection(timeslice),459 horizontal vibralor, 472 3 hovercraftas Vibroseissource.210 HumbleOil Co..9 Huygens'principle,44 usedin constructing wavefronts: diffraction,67, 68; reflection,62, 63; refraction,62, 63,8l hybridmigration,334 hybridspread,253,271 Hydrapulse(energysource),207 hydratereflection,128,131 hydraulichammer(energysource),207 hydrocarboncontent,effecton velocity, t21,2,125 hydrocarbongeneration, 350-l (HCI), 18,21, hvdrocarbon indicators 79,3i, 152,401,415 t8, 464, 475 6,513p s , 3 , 1 0 ,1 2 , late2 hydrocarbonmigration,351 hydrocarbonreservoirs: delineationol 514 description of, 514 l5 surveillance of,515 Hydrodist,193 hydrophone, 223 5, 223,221 acceleration-can celing,223 comparisonof output with geophone o u t p u t , 2 2 56 , 2 2 8 respondingto acceleration, 224 5 sensitivityof, 224 seealso slreamer Hydrosein,2l4 hyperboliccoordinates, 194,195 icebergdetection,3 identificationof events,seedistinguishing featuresof events identifyingreflectionswith interfaces, 148,149,150,r5r, 356,359,361, 464,465,488,490,493 identifyingreflections with refractions, 429,4N illumination(sun-shade) display,460 imagepoint,85,166 7,l'70 imageray,333 impactblaster,203 impactor(energysource),210,212 impedance, acoustic,ZJ impedance contrast,76 impedance-matching transformerfor hydrophone,224 imploders,214,217 implosion,213 impulse,unit,279,535 6, 5.lJ impulseresponse, 279,547 incoherentnoise,183-4,-18.1 incompressibility, 38

Independent ExplorationCo., l4 induction well logs and overpressure, 128,129 inertialnavigation,198 information rheory,275 effecton wavemotion. inhomogeneities. 46 initial suppressi on, 228-9,228 in-line offsetspread,243 in-line refractionprofiling, 425-7 in-seammethodsfor coal.509 l2 innermute,320 floating-point(IFP) aminstantaneous plifier,18,230,234 325 instantaneous frequency, loweringof (HCI), 416 phase,325 instantaneous velocity,130, 132 instantaneous typicalvalues instrumentspecifications, of.57l 93 integralequationsfor wavepath, integralrelation for inverse:-transform, 554 integraltheoremsfor transforms,538, s39.545,546 276 7 integraltransforms, integratednavigation,195 integrationoperator,528, 529 intelligentinterpolation,253,324,452 intensity,J9 relationto energydensity,59 361-3,460 I interactiveinterpretation, 340,341 interactiveprocessing, intercepttime,9J relationto delaytime,439,440 usedto find depth,95, 97,98 (constructive, destructive), interference 43-5,43 InternationalAssociationof Geophysical Contractors(IAGC), 240 interpolation,282,324 intelligent,253,324,452 528 usingfinitedifferences, reflection: interpretationprocedures, collection,examinationof data, 3536 349 vs.consistency, correctness deducinggeologichistory,358,359, 360 363-4,459 60, displaytechniques, 46t.462,46.3,465 363 drawingconclusions, 353 fundamentalassumptions, integrationwith other geophysical data,389-90 integrationwith well data,359,361, 362 354 6 interpretationapproaches, mappinghorizons,355 6, 357,359 modeling,seemodeling optimisticvs.mostprobable,349 pickingreflections, 356 7 398-415 seismicstratigraphic, 3-D.460 1.463-6 340,361 3 usingworkstations, interpretationof refractiondata,geologic, 446 seealsorefractioncomputation ideal,349 interpreter, interstitial fluids,effecton velocity, 108-9,110,112-13,l t8, r2t-2, lu,t25 interstitialwater.tSl

; t I

t

t I

$ I

583

INDEX

intervalvelocity,I 30, I 31, 132,140 l, 309-10,311 Bauer'smethodof determining. 140-1 obtainedfrom reflectionamplitude, 135-8,139 obtainedfrom well survey,13l 6l inversedispersion, inversefilter, 284, 292, 550 inverseFourier transform,277, 532 inverseLaplacetransform,545 inversemodeling,390 inverseof squarematrix, J20 inversez-transform,integralrelation for, 554 inversion,,l-lJ flatteninghorizons,358' 359,362 generalized,340,342 3 joint, 502 logs,135-8,139,418 to seismic structural,363 9 tomographic,2l,492 irrotational(P-\ wave,44 isometricdisplays(3-D),460 isopachmaps,359 isopachthickening,359 isotropicmedium,37: seealso anisot' ropy; transverseisotropy joint inversion,502 jug, seegeophone jug hustlers, 240,242 Kalmanfilter,299 Karcher,J. C., 4, 8, 13 Kevs,D. A., 28 kiuematicsimilarity,390 kineticenergyof a wave,58 Kirchhoffstheorem(formula),4/ K i t e ,W C . , 5 K n o t t ,C . G . , 3 Knott'sequations, 75-6 waves, 54, 486, Krey (pseudo-Rayleigh) 5t2 (,, C.,to, f_, f,, norms,295,342,559 -17 Lam6'sconstants, lampitude,554 land seismicsurveys,seefieldequipment for land surveys;fieldlayouts, reflection;field layouts,refraction; land;refraction fieldoperations, field methods radiopositionlane(phase-comparison ing), 194 Laplacetransform,276,2'79,545-6,545 comparisonwith Fouriertransform, 545 inverse,545 theoremson, 545 6 usedin diffractiontheory,63-4, 656 Laplacian,39,519 distribution, Laplacian(exponential) 342,559 lateral velocityvariations,392-8, 399, 400 law of reflection,62 law of refraction(Snell'slaw), 62, 63,13 layingout seismicline,241 lead, structural,J63 lead(fan shooting),427,446 lead-insection,225,226

leastabsolutedeviationfit, -142 (Wiener)filter,293, 295, least-squares 296,559-60 matrix solution, 524-5;with constraints,525;with weights,525-6 multitracecase,526-7 295,523, 527 normalequations, (Wiener)fit, 342, 559 least-squares leasttime,principleof, 58' 157,185 Lee,Milford, 19 Leibnitz'rule, 539 Levinsonrecursionalgorithm,552-3, 552 liftering,299,554 (separable), 123 limit, resolvable line (3-D slice),4J9 line-endtapel 452 line filter,230 line layout,directivityof, 241 lineararray,24'750,247 in velocitywith depth, linearincrease 93s abovea refractor,99-100 wavelet,533,535,554 linear-phase linearramp,302,536 linearsource,directivity,of, 203-4 279, 546 7,546 linearsystems, in seriesand parallel,547 lithology,relationto Blrr.,475 lobe,248,249,250 198,199 locatingthe streamer, logarithmicdecrement(log dec),60,220 logisticsin land work, 242 3 long-pathmultiples,I 62, 166-9,170, 17l sonde,132 long-spaced longitudinal(P-) wave,44 "looking aheadof the bit," 488 "looking to the sideof the borehole," 488 loop method(3-D),456 Lorac,193 Loran-C,193,194 lossangle,60 L o v eA , . E.H.,3 Lovewaves,3, 52-3,52, 486 display,4l 6 low-amplitude low-cut filter, 2-10 component,I 36 low-frequency shadow(HCI), 415,416 low-frequency solution,I I l low-frequency low-passfilter,556 7,5J6 layer(LVL), low-velocity(weathered) 124 absorptionin, 124 baseof, 124 correctionsfor, 15, l8-19, 146,261-3' 266,303-5,306 determiningthicknessof and velocity in, 256,258,259,266 first-breaksrefraction,305-6 producingghosts,163 reflectioncoefficientat baseof, 77, 124 lowstandof sealevel,409 lowstandsystemtracts(fan,wedge), 405,409 low-velocitylayer LVL, .see 522 Maclaurinseriesexpansions, in streamer,198, magneticcompasses 199 magneticdata,integrationwith, 390

magnetictape recording: analog,18-19,229 digital,229,230 moveableheads,l8-19 marine,260 magnetometer, main (major,principal) lobe,248, 249 Mallet, Robert,3 map migration,335,338,339 mappingreflectinghorizons,268, 357' 359 279 mapping(transforms), marine flooding, 409 marinepositioning(navigation): acoustic,197-8 generalprinciplesand requtrements, 192,3 inertial,/98 seeradiopositioning radiopositioning. (radionavigation ) redundantsystems,193,195 satellite,194-7,196 translocation,192 3, 192,194,196 marineprofiling,235, 260,262'263, 2U,265 engineeringsurveYs,506 8 usesof, 260,506 manne surveys: methods,258-60 conventional costsof, 27 8,29,30 earlyhistorYof, 15,18, l9-20 equipment,2ll-18: seealso energY mamarine;hYdroPhone; sources, radiopositioning: rinepositioningl stfeamer profiling, .ieemarine Profiling refraction,428,429 and obstructedoperashallow-water tions,260 3-D,see3-D methods 260-l operations, transition-zone MarlandOil Co.,6,8 Marthor,473 matchedfilters,557 matchinghydrophoneand geophonerecords,2256,228 matrix,519: equationol 522 characteristic elementsof, 520 I operations,520 order oi, 520 partitioning,J20 typesof, 520 usedto calculateconvolutionand correlation,521 equations. usedto solvesimultaneous 520 I usedto solveWiener (normal) equations,52l Maud Field,9 maximizingpower of stackedtrace,305 maximumconvexitycvve, 327 maximum depth: ofburial, relationto velocity,122, r24, 126 of circular raYPenetration,99 maximum-entropyfiltering (deconvolurion),299, 561-2,561 maximumlikelihood estimate,342, 5s9 maximum-phase(-delay)wavelet,551 Maxipulse,217 Mayne,Harry, 13,19 CDP patent,13-14

INDEX

584 McCollum,Burton,4.5, 8, 10,13,14, 10

McCollumExplorationCo., 8 McCollumGeologicalExplorationCo., 8 McDermott,Eugene,8 measurementof velocity,seevelocity, methodsof determining mechanicaldamping,218,219 4, 6, 1, 8, 9, mechanicalseismograph, 1 4 ,1 8 medianfiltering, 490 mergezone,302 methanehydrateproducingreflection, l 2 8 ,1 3 1 MexicanEagle(Shell),5 effecton velocitY,I l3 microcracks, 253,z9s microspread, midpoint,8Z migratedseclion,267, 357 migration,326 7,326 aperture,326, 328 9, 331, 451 depth,326,333-4,333,336'337 of diffraction,327,328 distance,452 effectof not migrating,267,334 effecton resolution,328'333' 334 5 explodingreflectormodel,327,392, 395 methodsof, seemigrationmethods of multiples,333,334 n o i s e1, 7 7 , 3 3 4 326, relationto DMO processing, 335,340 velocity,138,340 migrationmethods,267'-8,269,326 35 beforestack(prestack),326,335 commontangent,328 depth.326,333 4,333,336,337 diffractionstack,327-9 330 3: 15"(45') aPfinitedifference, proximation,331 334 frequency-space, (Stolt),329-frequency-wavenumber 30,329 hybrid,334 map,335,338,339 omega-x,334 (Gazdag,330 phase-shifting relativemeritsoi 334 .129 slant-stack, by swingingarcs,267 3-D,457,458 335 turning-wave, usingwavefrontcharr,2678,269 migrationof petroleum,351 migrator'sequation,326 Milankovitchcycles,406 Milne, John,3 Milne'smethodfor solvingdifferential equations,529 minimaxnorm (fit), 342 minimizingrhe(,, (. norm, 295 wavelet.set'minimumminimum-delay phasewavelet filtering,562 minimum-entropy wavelet,I 81,290-1, minimum-phase 550 3 290,533,534, conversionto, 291,555 finding minimum-Phasewaveletfor givenspectrum,552 3 propertiesol 550-2 Mini-Primacord ,207, 209

Miniranger,193 Mintrop, Ludger,4,5, 7' l3 mirror formula (curvedreflector),155-6 (mistie),18,357 misclosure wavelet,291,551 mixed-phase mixing, 18,229 Mobil Oil, 18 mode,3J 73 modeconversion, at liquid solidinterface,474,415 at solid-solidinterface,73 model: of nonporousrock, 110' I l3 for reservoirsimulation,514 rock, 107 13 of sedimentary modeling,390 2 computer,391 2 forward (direct),-190 inverse,-190 physical,390I 492 in planningVSP surveys, ray-trace,337, 392,393' 394' 395,396 seismiclogs seismiclogs,^ree .reesynthetic syntheticseismogram, selsmogram modulusof rigidity,38 Mohorovi6i6,A., 8l molecularweight,effecton velocity, I 1 6 ,1 1 9 monitor record,230 56 monoclinicanisotropy, Monte Carlo melhod,342 movableheads(magnetictaPe),l8 moveout,/46 of diffractions,159 60, 163 dip, ls, 87 8, 87,89, 145,t46 moveout, normal(NMO),86,87 8, 145,146 usedin attenuatingmultiPles,166 usedin identifyingevents,145,146 moveoutliltering,185,315-16,J1J, 317,318,319 .reegeophone, moving-coilgeophone, moving-oil mud, 199 mud weight,128 288-9 multichannelcoherence, 303 deconvolution, predictionmethods,560 multidimensional: convolution,285,542 Fourierseriesand transforms,533 multiplebranches,/56, 159,16l effectof crossdip on, I 59 2, 10,222 multiplegeophones, 247,251 multiplesources, (3-D), multiplesourcesand streamers t8,21 2,212,452 multiples,1619, 161,170,171,2923, 294,298,303,320,321,322 attentuationol 166,168-9,171,298, 303 of, 161-9,170,l7l characteristics from baseof LVL, 124 ghosts,163-5,167,168,225-6,228, 292,3,294 long-path,162,166-9,170 pegJeg,163,167,168,508 short-path,162-6,162,167,168'169 in waterlayer,165-6,168-9,170, 284 5 multiplexer,230 multiplicationof matrices,520

multiplicity(cMP), 244,245,456 muskeg,effecton velocity,126 muting,147,319-20,319 effecton multiPlicitY,320 front-end,147 inner(tail),J20 147 schedule, surgical,320 Nash Dome,3, 9 natural (fundamental)frequency,35, 219 navigation.marineiseemarineposttioning navigationaccuracy,192-3,194,195' 196,197,453 navigationcomputer,259 Navstar(GPS)satellitesystem,18,22, 191,195-7 near-fieldeffecl,48 261 2. 263, corrections, near-surface 266 refraction,256,259 plot, 338,34l near-trace neper,59,7l recording,4 80,483 nine-component Nitramon,200,203 normal NMO (normalmoveout),.see moveout(NMO) NMO velocity,1-t0 nodalplanes(Lovewaves),5J node.Jj noise,183 4,18-l analysisof,253-4,255 attenuationof, 184-5'250 2 225 in marinestreamer, originsof, 183-4 4, 253' 235, profile(walkawaY),253 506 random,183 l83 repeatable, typesol 183 seealso signallnoiseratio; signal/noise ratio improvement 2I 0' 2ll nonlinearsweep(Vibroseis), partitioningat, nonnormalincidence, 778 nonporousrock velocitY,110' I l3 nonprimaryreflectionevents,character174 isticsol 159-'12,173' norm (fit, criterion),295,342,559 normal: 61 dispersion, distribution,559 295,523,527 equations, faulting,365,366,367, 368,311,315 incidencePartitioning,76-7 pressure, 108,126 strain,J6 Jj stress, seechannel normal-modepropagation, waves normalmoveout(NMO)' 86, 145,146 correctionsfor, 19, 146 pseudo,lTl removaloi 86, 87 8,89 3ll' 320 stretch,310, usein identifYingevents,86,146 velocity, 130;seealsovelocity analysis (spectrum) 286 7 normalizedcorrelations, notchfilter,230,233 296 notchin signalsPectrum,

I I

585

INDEX

NR area,18-i numericalsolution of differentialequations: Euler Cauchymethod,529 higher-order equations, 530 Milne'smethod,J29 Runge-Kuttamethod,529 30,529 Nyquist frequency,wavenumber, 282, 283 patterns,409, obliqueprogradational 413 observer,240 oceanographic surveys,27, 30,260,263 odd-arm srar array,252 offr,ap,409.412 offset,85 maximum,minimum,254 offsetspace,245 vsP,487 Omega,193,194 omega-xmigration,334 Omnipulse(energysource),207,473, 474 one-sided exponentialdistribution,5J9 onlap,404,405, 40,6,401, 409 coastal,404 marine,405 OPECcartel,23 operatorlength,302 operators: finitedifference. 527-8 vector,519 Opseissystem,223,224 optimizinga stack,289 optimumfilters,29J, 295.296,559 62 optimumwide-bandhorizontalstacking,321 OrchardDome,6 organ-pipeanalogy: normal-modepropagation, 483 4 stretchedstring,35 organizationof land (marine)crews, 239 40 orthorhombicanisotropy, 55 6, JJ (geophone), overdamping 220 overdeterminati on, | 93, 497 overpressure, seeabnormalpressure overshoot(Gibbs'phenomenon), 539 40 p r (1 p\ transform, 37,278,324 P-wave,44, 45 modulus,44 velocityof, 44 P- and S-wavesectionscompared, 415 P- and S-wavevelocitiescompared, 4 4 ,1 6 packingof spheres, 107 8, 109 paging,102,260,263 paleontological data,useof, 401,402, 409,513 (palinspastic, paleosections restoredsections),359,464 paleostructure, 359 paracycle, 402,406 parallel-line method(3-D),452 parameter, raypath,63,73,93 parameterchecking,340 parameterdetermination: in dataprocessing, 338.340 .ieea/soselectionof f eld parameters

parametricequationsfor circular raypath,93 parasequence, 409 paravane, 225,454 Parseval'stheorem,542 parsimonious 559 deconvolution, partialfractions,530 1, J.l0 particlemotion,duringpassage of P45-6 and S-waves, partitioningof energyat interface: 75 6 fixedby Knott's equations, fixed by Zoeppritz'equations,73-5 at normalincidence,76 7 77-8 at nonnormalincidence, partitioningof matrices,J20 party chief,party manager, 239,2q passiveseismicmethods,500, 502,512 patchmethod(3-D),453 patch reef,384, 385,386 patentlawsuit:TexasCo. vs. SunOil C o . ,l 3 1 4 patents: Evansand Whitney,4 F e s s e n d e3n4, , 1 3 , 2 1 Hayes,13 Karcher,l3 Mayne,13 14 McCollum,13, Mintrop,l3 patterns, l8 depositional,404 12,405,406,407, 408,412 seealso arrays peg-legmultiples,163, 167,168'508 effectof dip, 3I 7 163 effecton waveshape, periodof a wave,34 periodicsignal(function\,27'7,287, 531,532 permafrost,/26 effecton velocity,126 permeability, J51,513 permit man, 240 permitting,241 P e r r i n eI .,, 5 petroleumexploration,importanceof work in. l, 8, 26 seismic generation petroleum and migration. 350 I Petty,Dabney,9, 20 Petty,O. Scott,9 Co.,9 Engineering PettyGeophysical 1 0 ,1 4 ,1 5 , 1 61, 9 . 2 0 phantomdiffraction,161,161 phantomhorizon,| 52,268 phase: maximum,5J.l minimum,seeminimum-phase wavelet mixed,551 zero,seezero-phasewavelet phase(angle),34 (navigation)systems, phase-comparison t94, 195 290 1 phaseconsiderations, lateral,326 phasediscontinuities, phaseplots,326 phaseresponse: of geophone,220, 221,222' 228 of hydrophone,225, 228 phasereversalon reflection,77 phaseshift: at baseof LVL, 163 4

'77,484,485 beyondcriticalangle,63, geophone, 225-6, of hydrophonevs. 228 220 linearwith frequency(geophone), on passingthroughburiedfocus,157 at surfaceof ocean,293 phase-shifting(Gazdag)migration,330 phasespectrum , 278,53I , 533 phasevelocity,56,60, 130 phasing,410,416 phone,seegeophone;geophone, moving coil; geophone,typesof; hydrophone photographic recording(camera),8, 10, 233 physicalmodeling,390-1 physicallyrealizable functions,550,554 (records),266-7, pickingreflections 267,356-7,357 automatic,325 pie-slice(apparentvelocity)filtering, 185,315-16,315, 317,318,319 PierreShale,absorptionin, 180 piezoelectricenergysource,207 piezoelectric 218,223 5 hydrophone, piezoelectric transducer, 207 pilot trace,295,304 piloling, 192 pinchouttrap,351 pinger(acoustictransponder), 197,198, 453 planepolarizedwave,46 planewave,42 plane-wave simulation(Simplansectionl,322-4 plane-wave solutions to waveequation. 4t2 plasticyieldpoint, 38 plateboundaries, typesof, 364 plate-tectonic habitats,366-7 playback, 18,20,229 plottingcross-sections, 267 8, 269 plottingmachine,95 plus minusmethod,442 3,442,444 POD (propane-oxygen detonator),207 point tracking,460 Poisson, S. D., 3 Poisson's ratio,J8 ratio log, 133 Poisson's polarity,SEGstandard, l8l, 183 polarityreversal,415,416,plate2 polarization,elliptical,46 polarizationfiltering, 480,483, 48/, 487 polarizedwave(plane,elliptical),46 p o l e( o f a f u n c t i o n5t .5 1 . 5 5 5 PoncaCity,Oklahoma,6 pore-aspect ratio,122,475 porosity,107 by soniclogs,133 determination 18,513 relationto amplitude, relationto density,1l6 relationto depth,118 relationto velocity,107,108,109, 1 1 0 , 1 1 2 , 1l 13 6, ,l 1 7 1 8 , 1 2 0 Port Barresaltdome,9 positioning. marine.seemarinepositioning postingdata on maps,356 potentiallunctions,velocity,4'7;seealso potentialfunctions displacement Poultermethod(air shooting),204,206 power of a trace,287 l4 Prakla,Prakla-Seismos,

586 Pratt, Wallace,9 preamplifier, 230,233 predictiondistance(span),560 prediction error,298 filter, 560 predictionfrlter,298 predictionlag,298 predictivedeconvolution, 166,168,298, 341 predictivefiltering, 556 preliminary (brute) stack,340 preservation(recovery)of amplitude, 313,315 prestackmigration, 326,334, 335 presuppression, 228 pressure, 35 abnormal,seeabnormalpressure differential,108,118,128 effecton velocity,108,1l8-20, 121,

r22 effective,108 gradientof, 126 normal, 108,126 price of oil, gasvs. seismicactivity,24-6 Primacord,204 primary recovery,5,13 Primary Source,207,209 principal(main)lobe,248,U9 principalprocessing pass,340 principleof: leasttime (Fermat's),58, 152,185 rcciprocity,97,322 37, 147,322,546 superposition, processing flow chart,34l processing procedures, 335,338,340 processingseismicdata,seedara processing profiling: marine,seemarine profiling vertical,.reevertical seismicprofiling (VSP) progradation(deposition), 409 progrademotion,5l program,24l projection(Radontransform), 37,277, 278,324,493 Fourier projectiontheorem,493-4 propagationerrorsin radiopositioning, 194 propertiesof seismicevents,seedistinguishingfeaturesof events proximitysurveys,382,500,502 (Evison)waves,486, 512 pseudo-Love pseudo-normal moveout,l7l pseudo-random sweeps, 210 (Krey) waves,54, 486, pseudo-Rayleigh 5t2 pull-apart zone,364,365, 366 pull-up,push-down,384,385,396,400 Pulse-8,143 pulse-widthmodulation,229 Q (qualityf^ctor),60,177 3.13 Q-correction, Q-map,497 quadraturefrltet 544 quadraturetrace,325, 544 quaternary-gainamplifier,230 quefrency,554 "quick-look"methodof findinginterval velocity,140 I rada\ 193

INDEX

168-9,l7l, radialmultiplesuppression, 303 radio usedto transmittime-break,8, 9, 1 0 ,l 8 (radionavigation), 18, radiopositioning 193-4,24t, 453 earlydevelopment of, 18 frequencies used,193 integratedsystems,195 laneidentification,194 phase-comparison 194,195 systems, propagationerrorsin, 192 3,194 measuringtime, 192, systems 193-4 systemsmeasuringtime difference, 192,193,t94 useof atomicclocksin, 192,194,195 193,194 useof redundancy, 192,194,195 useof translocation, usedin land work, 241 Radon(t-p) transform(slantstack),37, 277,278,324,493 ramp (ramping),299 random functions,542-3,543 autocorrelationof, J4.l of, 543 cross-correlation randomnoise,18-i,287 effectof stackingon, 184,185 randomnumbers,tableol 570 range,192 raster,234 ray-tracemodeling,331,392.393,394, 395,396 Raydist,18,193 Rayleigh,Lord, 3 Rayleighcriterion: for resolution,173 for specularreflection,63 Rayleighwaves,49-50,49, 51,52 boundaryconditions,49 dispersionof, 50 (Stoneley), generalized 3, 50,52, 53 4, 133 potentialfunctionsfor,49 (Krey) waves,J4 pseudo-Rayleigh usein locatingsaltdomes,l0 velocityof,49,52 seea/soground roll RayleighWillis formula,217,218 raypalh,42 raypathcurvature,91 5 raypathparameter,63, 73, 93 raypathfor velocity as function of depth: integralequationsfor, 93 for linearincreaseof velocity,93-5 for sphericalshells,99 RDU (remotedataunit),223,224 read heads(read-after-write),2J0 realizable,physically,550, 554 reciprocaltime, 97, 426 reciprocity,principleof,97, 322 distinguishing recognitionof events,.see featuresof events record,seismic,6, 7, ll, 11,232 recordsection,18, 19,22, 229,268,429 recording: analog,2279,231,233 181,183,571 conventions, digital,226,229,230,231,232,233, 234 direct,229 land,241-3

on magnetictape,229 on paper,2, 6,8,224,232 18 19,229 reproducible, rectangulararray,252 recursivefiltering, 293, 556 reducedrefraction sections,429 reductionof reflectiondata: of refraction data, 429 of surveydataby computer,241 seealso dala reduction,reflection (CMP). 244.245 redundantcoverage redundantnavigationsystems,193, 260 reefs,-i-J,/,352, 382-5,386 criteria for recognitionoi, 383 5 drowningof, 383 modeloi 392-3 patch,384,385,386 213,225 reelfor streamer, referencedatum,261 reflecteddiffraction,l6l, 188 reflectedrefraction,17l, 174,429 reflectedwave,62 reflectingpoint, 87 equivalentto an area,152,154 of (DMO) for up-dipdisplacement offsetreceivers,901,92:correction for,316 18,320 reflection,62 angleof,62 at baseofgas hydrate,128,l3l at baseof LVL, seeghosts of, 146 59, 160,l6l' characteristics on, 157,159,l6l, 162:3-Deffects t62 ol -lJ,63,76 7,76,78, coefficient l16 continuityol 409 l6l,188 diffracted, at grazingangle,77 identilicationusingVSP,150,l5l' 359,464,465,488, 490,493 identificationwith refractionevent, 429,430 identificationwith specificinterface, 148,149,150,151,356,359,361, 464,465,488,490,493 proseeinterpretation interpretation. cedures(reflection) law of,62 natureof, 150 2, 153 pathsfor constantvelocity,85-91 with pathsfor velocityincreasing depth,93-5,99 100 pathsfor velocitylayering,91 3,94, 98-9, 100 phasereversal,77 polarity,ll2 records,6, 11,232 63 specular, total, 63 77-8 wide-angle, analysis,389,401, reflection-character 412-13,415 35, 63,76-7, 18' reflectioncoefficient, 116 complexvaluesof, 63 typical valuesof, 77 method,marine.seemarine reflection surveys reflectionmethods,land,2, l9l 2,199, 239 56,257,258 crew organization,23940

INDEX

fieldoperations, 241-3;drilling, 199200,242;energysources, 200-l l, 212;permitting,241;recording, 242-3; surveying, l9l-2, 241 logistics,242-3 practicalconstraints,245-6, 247, 248 recording,seerecording,land; recording seismicdata, forms of recordingconventions, 181,183,571 selectionoffield parameters,seeselection of field parameters 3-D methods.see3-D methods seealso dataprocessing;data reduction, reflection;energysources, land; field layouts,reflection reflectionpaths: for constantvelocity,85 9l for velocityincreasing linearlywith depth,93-5,99-100 for velocitylayering,91,3, 94, 98-9, 100 reflectionpatterns,404 12,405,406, 407.408.412 reflectionpoint variationwith dip, 90 l,92 correction for.316 18.320 reflectivity,35 reflectorcurvature,155-9,160,161 causingburiedfocus,156,157,158 mirror formulafor. 156 refractedwave,62 refraction(headwave),| , 2, 8l-2, 8I characteristics of, 169,l7l, 172,173 first postulatedby Mohorovidic,8l identificationwith a reflection.429. 430 paths,95 100;'seealsorefractionformulasfor raypaths reflected,171,174,429 refraction(process), 62-3,62 angleof,62 law of (Snell's). 62 refractioncomputation: ABC method.414.506 Adachi'smethod,433-4 Barry'smethod, 439-40,441 basicformulamethods,433 data reductionand processing. 429 33 delay{ime methods,439, 442 four-shotmethod, 427,433, 506 generalized reciprocalmethod (GRM),434-9 Hagedoorn'splus minus method, 442 3.444 Hales'graphicalmethod,zt43-6 Tarrant'smethod,440,4l Thornburgh'smethod,442, 4/.3, 44'7 usingsecondarrivals,426,433 wavefrontmethods,442-6 Wyrobek'smethod,441 2, M,447 refractionfield layouts,seefield layouts, refraction refractionfield methods,2 3 broadside, 427 comparisonwith reflectionmethod, 1 . 23 engineeringsurveys,505-6, 507 fan,8,18,427,428 four-shot,427 in-lineprofiling,4257 marine surveys,428,429

587 spread types, see field layouts, relraction use in defining LVL, 305 6 use in defining salt domes, 427-8 refraction fiormulas for raypaths: diving wave, 98 9 linear increasein velocity, 99 100 several dipping horizons with same strike, 433-4 several horizontal refractors, 96 7 single dipping refractor, 97-8 single horizontal refractor, 95 refraction interpretation: computation, see refraction computatlon geologic, 446 statics corrections, 429 using second arrivals, 426, 429, 43J refraction methods field layouts, see field layouts, refraction field methods, see refraction field methods formulas for, see refraction formulas for raypaths interpretation. see refraction interpretatlon marine,428,429 refraction records (early examples),7, ll refraction second arrivals, 95, 426, 433 refraction section, reduced, 429 refraction spreads, see field layouts, refraction refractor dip, 97-8 regression,404 reject region (of array),248 relation to array length, 250 relation between density and velocity, 1 1 4 ,l 1 5 , 1 1 6 ,l 1 9 relative importance of absorption and spreading, 60 remotedata unit (RDU),223,224 repeatablenoise, 183 repetitive 3-D (4-D) surveys,499 report, seismic,363,5'll 2 r e p o s i t i o n i n gd a t a . s e em i g r a t i o n l m i g r a tion methods reproducible recording, 18-19, 229 resampling, 282, 338, 341,490 reservolrs: continuity of, 514 delineation of, 464, 5 13, 514 description of, 464, 5 13, 514-15 natureof, 513 14 simulation, 514 surveillance,J1-1,515 residual display,460 resistivity, relation to velocity, 108 resolution: definition ol 122 dependenceon frequency, 173-4, 175, 176 desired wavelet for, 181 extended, 253 horizontal, 172, 177, 452 of migrated sections, 177 vertical, see vertical resolution resolvable (separable) limit, 173 retarded potential, retarded time, 4I retrograde motion, 50, 51 reverberation, 165-6, 169 reversal of phase on reflection, 77

reversebranch,156 reversedrag, 368 reversefault, 3U, 366,375 reversepolarity, 181 reverseVSP,488 reversed-polaritydisplay,363 reversedrefractionprofiling, 18,425-7 rho (rho-rho,rho-rho-rho)mode,194 Ricker.N. H..9 Ricker wavelet,18-l Rieber,Frank,9,18,21 rift (zone),364,365,366 rigiditymodulus,38 rim syncline,382 ringing: Gibbs'phenomenon, 539-40 reverberation,165-6, 169,284-5, 550 rippability,505,506 rms (root-mean-square) velocity,85, 9 1 - 2 , 1 2 8 , 1 3 01 .3 4 .1 4 0 rock models,107-13 RogersGeophysical Co., 14 roll-along(CMP) recording,244;see (CMP) a/socommon-midpoint roll-alongs\\,itch,242, 244 rollover (error in connectingcables), 268 rollover(fault block rotation),368,371 (rms)velocity,85, root-mean-square 9 t 2 , 1 2 8 , 1 3 01, 3 4 ,1 4 0 Rosaire.E. E.. 9 rotarydrilling, 18, 199,200,201,202, 242 rotationalfault, 366,375 rotationalwave,seeS-wave RPS(positioningsystem),193 Runge-Kuttamethod, 529-30,529 rupture,38 45,46 S-wave,44, advantages of, 471 (splitting),56, 57,476, birefringence 480.482 datadisplay,465,476 data interpretation. 475-6 dataprocessing,4T4 5 degreesof freedom,46 explorationwith, 18,21,471-6,477, 478,479,480,4gl, 492 generationof, 205,207,471 4, 475 recording,47l-4, 475 sectionsplotted at doublescale,475 study with vertical seismicprofiles, 556 usedin engineering studies,505,506 usedto indicatefluids in rock pores, 108-9,110,112-t3,t2l 2,tU,125 usedto indicatelithology, lll,ll2, 1 1 3 - 1 6l 1 , 7 ,l t 8 , l 1 9 usedto study fracturing, 55 6 velocity,M, 109,116,ll9, l2l-2, lU safetyconsiderations,240 salt dome,-1J1,352, 370,374, 379,382, 383 definitionof flanks,2l, 382,427 8, 429 detection oi 3, 5, 6, 8,9, 10,427, 428,488 formationof, 373,374,379,381,382 seealso salt-flowstructures salt-flowstructures,367, 368, 370,373, 374,379,382;seea/sosaltdome

588 salt foreruinner,10,ll lead,427, 428,446 proximitysurvey,382,500,502 salt-solution structure,368 sample-and-hold circuit,230,233 sampling,146,177,281 3, 547 8 function(comb),281 spatial,146,283 theorem,177,282,5478 sandline,l12 sandand shale,distinguishing basedon velocity,112,113,l14, l15, 116, l 1 7 ,l l 8 saphe,554 satellitepositioning,192,194-7 scalarproduct(vectors), 5.18 quantity,5/8 waveequation,39-40 scaler(inversion),,135 scalinglaw for sources,199 scalingtheorems,538,545 Schlumberger, l4 sealevelchanges,effectsof, 402-3, 404 9 seafloordepth variations,effectsof on velocity,392-3,397 seafloorspreading,364,365 seamwave,seechannelwaves secondarrivals(secondary refractions), 95,426.429,433 secondary recovery, 513 sectiontitleblock,353,354 sectlons: balancing,370 examinationol 353-4 sedimentbypassing, 405 sedimentstaNarion,404 sedimentary rock model,107 I I SEG (Societyof ExplorationGeophysic i s t s )1, 0 , 2 4 , 2 9 polarSEGstandard(reversed, negative) ity, 'f8l, 183 Seiscrop section(timeslice),459 seismicactivity,23,24 8,29,30 seismicamplifier,seeamplifier seismic contourmap,2, 6,12, 13,19, 357 seismiccontractors, development ol 4, s , 8 , 9 1 0 ,1 4 seismiccross-section, seecross-section seismicdata processing, seedata processing seismicevent,2, 145 6 seismicexploration,historyof, 3 23, 24-6,2'/s seismicfacies,409 seismicfaciesanalysis,398,401,4A7, 409-t2,4r3,4r4 classification, 410-l I seismichydrocarbonindicators,18,21, 79, 81, 152,401,415 18,464, 4 7 56 , 5 1 3 SeismicImmunitiesGroup, 13 seismic log (inversion), 135 8, 139,401, 412,413,415,plates1,4 seismicmethods,passive, 500,512 seismicpatentinfingementsuit, 13 seismicrecord,6, 7, ll, 11, 232 seismicreflection-character analysis, 389,401,412,13,415 seismicreflectionmethod,seereflection methods, land;marinesurveys

INDEX

seismicrefractionmethod,seerefraction entries seismicreport,163,571-2 seismicsequence analysis,401 3,401 seismicship, 2ll, 212,213 seismicspread,typesof, seefield layouts, reflection;field layouts,refraction seismicstratigraphy, 18, 398,401-15 seismictomography,21, 492 9,492 basicproblem,4923 comparedwith CAT scanning,497 solutionfor continuousray distribution,493 6 solutionusingpixels,496-7 usedto interpretcrosshole measurements,497-9 seismicvelocity,seevelocityof seismic waves seismograph, electrical,8, 18 mechanical, 4, 6, 7, 8,9, 14,l8 Seismograph ServiceCorp.(SSC),14 seismometer, 2,218; seea/sogeophone; geophone, moving-coil;hydrophone (Gesellschaft), Seismos 5 6, 8, 10, I I, t4 semblance,288 usedto optimizestack,289 sensitivity: of geophones,220-1, 222 of hydrophones, 224 separablelimit (resolvablelimit), 173 sequence,404 sequence 401 3 analysis, sequenceboundary,404, 407 sequencestratigraphy,398,401-3 (Taylor,Maclaurin,biseriesexpansions nomial),522 seriesparallelconnectionof geophones, 222 SH-wave,46 usein exploration,seeS-wave,explorationwith shadowzone,376,416 shale,distinguishing from sand,l12, 1 1 3 - 1 6 , 1 1l 7 1 ,8 shalediapir,368,312 flow (flowage),367, 368,372, 382 line,I 12 porosity,ll7 velocityvariations, lll, I l6 shale-shale reflections, I 16,150,l5l,

ts2 shapeof the seismicwavelet,181,182 shapingfilter,297 shearmodulus,-J8 sheartransit-timelog, 500 shearwave,seeS-wave shear-wave splitting,56, 57 shearingstrain,-16 shearingstress, JJ shelf-margin systemtract,409 shift theorems,538,545,546 shingling,169,l7l ship, seismic,211,212,213 shooter,240 shootingdown-dip(up-dip),97 shootinga well, 130 l, 130,132,l4l, 142 shoran,18,19-l errorsin location,193 extendedrange(XR), 193

short-pathmultiples,162 6, 162,167, 168,169 225 shortstreamer, shot, 2; seealso energysources,land shotholedrilling, 199-200,201,202,242 shotpoint.2'.seealsosourcepoint shotpoint(source)delaylime,439 shotpoint(source)traveltimecorrections,261-3 Shover,474 SI units,570*1 non-Slunits,571 sidelobe,248 side-scansonar,260, 2U,265, 507 8 side-swipe event,310 sigmoidprogradationalpattern,409, 410,412 sign-bitrecording,289-90,289,291 290 sign-bitsemblance, signfunction(sgn),533,535 signal,183 compression, 227 2.12 enhancement, recorder,231,232,505 enhancement signal/noise(S/1f),18-t effectof addingsignalswith random noise,184 (S/M) improvementby: signaVnoise adaptivefiltering,299 seedeconvolution deconvolution, 292-3,294 deghosting, inversefiltering,292 deterministic frequencyfiltering,18,229,233,300, 301 homomorphic(cepstral)deconvolution,298-9,299 (Wiener)frltering,293, least-squares 295,296,559,60 maximum-entropy deconvolution, 299, 561-2 maximum-entropy filtering,562 predictivedeconvolution, 166,298 recursive filtering,556 (whitening), spikingdeconvolution 29s.-8,299 299 time-variant(TV) deconvolulion, usingnoiseanalysis,2534,255 299-300,301, waveletprocessing, 45'7 4\q

signature,146 292 signatureprocessing, 214,216, signatures of marinesources, 211 silencedrifle (energysource),207,209 similarfolding,370 Simplanstacking,322-4 simulatedanealing,342 5.14 simulationmodel(reservoir), technique reconstruction simultaneous (srRT),497 sinc function, 281, 537 sincwindow 558 sinetransform,5-i.J singingrecords,165 6, 165,169,284-5, 343 single-foldrecording,242,244 techreconstruction SIRT (simultaneous nique),497 324 slantstack(t p transform),278, migration,329 sledgehammer(energysource),207, 209 217,507 sleeveexploder(Aquapulse),

INDEX

sleevegun, 214 slice,459 composite, 459 60, plate 7 f a u l t , 4 1 8 , 4 5 9 , 4 6 4 , 4 6 5 , p l a t e s5 , l 3 h o r i z o n , 4 5 9 , 4 6 3 - 4 , p l a t e s4 , 6 , 8 , 1 0 , 1 1 ,1 2 ,t s , 1 6 , t 7 line, arbitrary line, crossline,459 trme, 245, 389, 459, 460, 461, 462, 463, plate 9 slope fan, 405,408,409 slowness (specific transit time), 4 I , 93, tt7 smeanng: o f m a r i n e d a t a , 1 8 5 ,2 1 7 of reflecting points, 3 I 7, 335 of wavefronts, 327 smile (migration), 327, 328, 334 Snell's law, 62 3,62 generalizedform of, 63, 73 Society of Exploration Geophysicists ( s E G ) , 1 0 ,2 4 , 2 9 solutions of wave equation: planewave,41 2 spherical wave,42 3 for spherically symmetric source, 479 s o n a rp o s i t i o n i n g : D o p p l e r - s o n a r ,1 9 5 , 1 9 7 8 u s i n g t r a n s p o n d e r s1 , 97 sonde (sonic), l3l 2. l3l, 133 sonic (continuous velocity) Iogs, I g. l 3 l 4 , 1 3 5 ,t 3 6 editingof, 133 r n a c c u r a c i e isn , 1 3 2 3 matrix velocities used in interpreta_ tion, I l8 used to detect fractures, 133 used to determine poisson'sratio, I 3i used to determine porosity, I 33 used to investigatecasing bond, t33 4 uslng array-sonic tool, I 33, 134 velocities used in porosity determinatron,I l8 sonic waveform logging, 500 sonobuoy,428,429 S o n o g r a p h ,1 8 . 2 1 S o s i em e t h o d , 2 1 0 , 2 1 2 source: contlnuous linear, 203 4 d o w n h o l e ( V S P ) ,4 8 8 unit (shooting Dnit), 242, 243 lor VSP, 488 9 see also energy sources, land; energy sources,manne source delay Iime, 439 source-signaturecorrection, 292, 298, 338 source waveshape(marine), 214, 216, 217 sourcepoint (shotpoint), 2 delay time, y'J9 traveltime correction, 261 2,266 Soursile (energy source), 207 Southwestern Industrial Electronics

(srE),r5

SPlogs,ll2, I 13 span(predictiondistance), 560 SparkPak (energysource),207,209 sparker, 2 14, 217,218,507 WASSPvariation,217,2lB sparse3-D method,4J2 spatialaliasing, 252 3,451 2

589

spatial frequency, 533 spatial sampling, 146, 245, 252-3 special functions, 533, 535 6, 537 specilic heats of gas at constant pressure (volume), 47 specific transit time (slowness),4 l. 93, It7 specifications,instrument. 571 spectral density, 287, 542 spectrum, frequency.53.1 specular reflection, Rayleigh criterion for, 63 speculativedata acquisition. 239 sphericalcavity surrounding source,motion of, 48 9 s p h e r i c a ld i v e r g e n c e 5, 9 , 1 7 7 ,3 1 3 . - l l 5 c o r r e c t i o nf o r , 3 1 3 , 3 1 5 . 4 0 2 . 4 9 2 spherical wave,42 equation,42 3 s p h e r i c a lw a v e f r o n t sw i t h l i n e a r i n creasein velocity. 94. 5 spherically symmetrical source. waves generatedby,47 9 s p i k i n g d e c o n v o l u t i o n{ w h i t e n i n p . ) .

29s 8,299 delayedspike,297 8,299 in frequencydomain,296 7.298 rn timedomain,295 6 spillpoint,35/ Spindletopsaltdome,3 split-dipspread.243 splitspreadsimulation, 322 4 spread.243 typesof',243 4,425 8,429 ,vt,a/sofieldlayouts,reflection;field layouts,refraction spreadingcomparedwith absorption, 60.61 SSC(Seismograph ServiceCorp.),l4 stabilityof filters,JJ5 stackarray,250 stacksections, preliminary,340 stacking,19,244 5, 244, 316 24 to approximate normal-incidence sect i o n ,3 l 5 cMP,2445 (DMO) correction, dip-moveout 316 I 8, 319,320 diversiry.322 Simplan,322 4 vertical,185, 199,204 5, 232,242, 259.261 weighted, 321.2 staokingchart,244,245 stackingvelociry,I 30, 140,3I 0 I l, 32I relationto dip and strike,3l I relationto othervelocities,134 standardSEC polarity,181,183 standing(stationary)wave,-Jj standout,amplitude.145,146 starvationevidences. 404, 409 starvedbasin,J8J staticscorrections,l8 19,146,261 3. 266,303 5, 306,307,457 fieldstatics,261 by maximizingpowerof stacked trace,303 relationto NMO corrections. 303 for S-waves,4745 usingrefractionstatics,305 6 usingsurface-consistent model,I 8, 303 5, 306,307 stationangle,193

stationarity, 299 statlonary time series,54-l statlonary wave, JJ steam-flood monitoring, 500 s t e a mg u n , 2 1 7 steepestascent, method ot,305, 342 step function (unit step), 5JJ step response,547 stick graph (stickogram), 148 stiffness,JZ s t i l l s t a n d( o f s e a l e v e l ) ,3 2 9 , 4 0 4 Stolt migration, 329 30, J29, 334 Stoneley,R., 3 Stoneley waves,3, 50, 52,53 4. 133, 489 boundary conditions, 50 dispersion of, 54 s t r a i n ,J 6 , 3 7 normal, shearing,36 in seismicwave passage,37 strain (elastic) energy,38-9 s t r a t i g r a p h i ci n t e r p r e t a t i o n ,3 9 8 , 4 0 1 l 5 o n h o r i z o n s l i c e s , 4 6 3 4 , p l a t e s ,4 , 6 , 8 , 1 0 ,l l , 1 2 , t 5 , 1 6 l, 7 h y d r o c a r b o ni n d i c a t o r s ,4 0 1 , 4 1 5 l 8 reflection-characteranalysis,401, 4t2 13.415 seismic-faciesanalysis, 401. 409 12, 413,414 seismic-sequence analysis,401 3 s t r a t i g r a p h i ct r a p s , 3 5 1 , 3 8 8 9 , 3 9 0 classificationol 389 s t r e a mc h a n n e l s ,3 8 6 , 3 8 8 , 3 8 9 o n h o r i z o n s l i c e s ,p l a t e s8 , 1 0 , 1 5 , 1 6 ,

t7

streamtracking,46(.) streamer,18,225,226,227 arrangement for 3-D recording, 452 3,454,455,456 compliantsection, 225 deployment,258 9 depthcontroller,I 8, 225 depthdetectors, 225 digitizingdatain,225 feathering, 236 l o c a t i n g1, 8 ,1 9 8 . 1 9 9 magnetic compasses in, 198,199,225, 453 noisegenerated in, 225 snort.ll) tail buoy.198,225.226 water-break detectors, 225 stress,35 stretch(NMO),-110. 311,320 stretchedstring,wavesin, 33-5 strikeof bed,88, 89,90 strike-slipzone,364,365,366,375 stringgalvanometer, 9, l0 stringof geophones, 222 strip recorder,260 stripping,433 structurallead,363 structuralsIyle,364 70,364 relationto basement, 364,365,366 typesof, 364 70, 371,372,3j3, 374 subcropsat unconforminity, 464 subductionzone,364,365,368,4l 8, 4t9 subsaltimaging,334 subshots,242,259 subsidence, tectonic,thermal,402,403 subsurfacestackingcharI, 244,245 subsurfaceIrace,I 57, 398

590 Sun Oil Co. lawsuitwith TexasCo., l3 14 sun-shadedisplay,460 superimposedmodesof display,234 SuperiorOil Co., 18 principle,37, l4'7,322, superposition 546 supervisor,239-40 surface-consistent methods,I 8 amplitudeanalysis,303,315 with constraints, 304 deconvolution, 303 least-squares solution,304 model,JOJ staticscorrections, 303-5,306,307 waveletprocessing,303 surfaceenergysources,19,204 l0,2ll, 212 air gun, 18,204,205,208 air shooting,204,206 burieddetonatingcord, 204,206 gasgun,204,205 horizontal vibrator,472 3 impactors,207, 2O9,210,212 for ,S-wave generation, 205,4714, 475 small energysources,205-6, 207,209 Sosie,2l0,2l2 Vibroseis,.reeVibroseisrecording weightdropper,18,204,205 surfacestackingchart,244, 245 surfacewaves,49 55,49 attenuation ol 172,316,318 characteristics of, 49, 172 groundroll, 49, 172 Lovewaves,52-3,486 pseudo-Rayleigh (Krey) waves,54, 486,5t2 Rayleighwaves,.reeRayleighwaves Stoneleywaves,3, 50, 52, 53-4. 133, 489 tube waves,.seetube waves surgicalmute,320 surveyingmethods: land,l9l 2 marine,192 8, 199 strveyor,240 SZ-wave,46: seealso S-wave swathmethod(3-D),453,456 sweep(Vibroseis), 208,210,211 Syledis, 193 Sylvanshale,6 symbolsused: geological, 572,573 mathematical, xiii xv well,572 symmetrysystems, 55-6,5J symmetrytheorem,538 syntheticacoustic-impedance log,see seismiclog syntheticaperture,260 syntheticmicroseismogram, 54 syntheticseismograms, 116, 146-50, 391 2,412-13 accountingfor multiples,148 as exampleof modeling,39l madefrom deepinduction logs, I 16 manufacture of, 148-50 one-dimensional, 147,150,392 for S-waves,500 two-dimensional, 392 useof, 148,150,412 syntheticsoniclog, seeseismiclog

INDEX

Syslapmethod,474 system,linear,279, 5461,546 in parallel,series,547 systemtracts,405,408,409 'r p (p-r) transform, 37,277,278, 324 Z-A7'method,134 5, 134 tail buoy,198,225,226 tail mute,J20 tapetransport,233 raper,291,297, 298 Iaperedarray,250 directivityas a functionof dip, 269 70 Tarrant'smethod, 440,441 Taylorseriesexpansions, 522 tear fault, 365,370 tectonicsubsidence,402,403 Teledyne,l4 telemetrymethods,22, 223,224 televiewer,borehole,500,503 temperature, effecton velocity,l2l,123, 124,499500,515 TexasCo.,lawsuitwith SunOil Co., l3 t4 theoremson Fouriertransforms,538-9 on Laplacetransforms,545 6 thinbed, 174 thin-bedreflebtions, 174,176,177, 1 7 89 thinJayeranisotropy, 55 thin-skintectonics, 419 Thornburgh'swavefrontmethod,442, 443,447 three-component boreholegeophone, 488 three-component recording,21,416. 480,482,483 3-D interpretationof 2-D data,398 3-D methods: acquisition:land,453,456 7; marine, 22, 23, 452 3, 454,455,456 display,459 60. 461,462,463 exploration(sparse3-D),452 improvements over2-D, 457,458, 460,462,463.4& interpretation. 461 6: interactive (workstation), 460 l; stratigraphic, 3 8 9 , 4 6 3 - 4p,l a t e 4 s , 6 , 8 , 1 0 ,l l , 1 2 migration,457 processing, 457,459 requirements, 451 2 for reservoirstudies,459,460,464, 465,466 thrustfaulting,364-5,367,368,369, 370,3'73,375,396,4N thumper,18, 19,204,205 tidal staticscorrection,457 tie points,244 tlme:

boundary (line, surface), 150,403 break, 9, 18, 229, 232, 242 delay time, see delay time seclion, 267 significanceof reflections,403 4, 405 slice (3-D), 245, 459, 460.461,63 Iie,268 time-average equation, 1l7 time-depth function, 4-lJ time-distancecurves, ree traveltime curves

time-domainconvolution,Jeeconvolution time-invariantsystems,279, 546 (4-D) measurements, time-lapse 499500,501 time line (surface),403 4,405 time-sequential data,335 time significanceof reflections,403-4, 405 time slice(3-D), 245, 459, 460,461,43 time-variant(TV) deconvolution,299 filtering, 302 timing events,-t56 lines,232 wheels, 15,l6 toe structure,368,371 Toeplitzmatrix,520 tomography,seismic,seeseismictomography toplap,404, 406,407 Toran,193 torsionbalance,usein petroleumexploration,3, I I total excitationtransform,547 total reflection,6J trace,234 energydensity(power)of, 286.287, 542 pllot, 295 subsurface, 157,l6l 135 8, 139,412,413, traceinversion, 415 tracemodeling,.reeray-tracemodeling trace-sequential data,335 tracking,460 trademarks, list of, 469 70 (transponder), transducer 193,197,453, 501 transductionconstant(geophone), 220 transf'erfunction,547 transformboundary(plateboundary), 364,365,366 transformiault, 364 transforms, 276 7 cosine(sine),278 essential aspectol 276 7 Fourier,277 8,277, 532 3, 532 integral,276 7 inverseFourier,277,278,532 inverseLaplace,545 Laplace,seeLaplacetransform r p (Radon),3'7,277,2'78, 324 :-transform,292,548 50,548,554 transgression, 404,406 transgressive systemtract,409 219-20. transientresponse of geophone, 251 250 transients, effecton arrayresponse, transit(theodolite)and chainsurvey, t91, 241 transitsatellite,18,22, 194-5,196 transittime,41, 93, 117 transitionzone,260 operations, 260-l 192-3,193,194, 196 translocation, 35, 76 transmission coefficient, I 93, 197.453,501 transponder, typesof, 243 transportvehicles, transpose of a matrix,520 isotropy,38, 55,56 transverse Hooke'slaw for, 56 velocity,58 waveequationfor, 56

591

I\DEX

: = n : \ . r > € l S - \ w a v e ,4 4 :ial. ,:i I : i : ' 3 . o l - .3 5 1 . 3 5 2 ::apDe.i $aves. .seechannel waveS :::relirme tarrivaltime),2 : : : i e l i r m ec u r v e s : i:fractron. 159 60, 163, 164, 165, | 67. 398 ;:;ping reflector,86-7 d r r e . ' ts a r e . 8 6 :rrnzontal reflector,85 6 .er:ral horizontal refractors, 96 7 .:rgle drpprng refractor, 97 8 .r:rgle horizontal refractor, 95, 96 ::e:rch:ble bedrock, 505 :::,rrsular gindoq 559 :::pie .une'tion. 364. 365 T:r.nnder.193 1;\ *3res.515.5J,489 seneratlon of. 54 :ellectiLrncoellicient for, 54 ' i d \ e e q u a t i o nf o r , 5 3 r r n r r g e l T e c t s1. 7 4 , 1 1 5 , 1 1 6 , 1 7 7 , 1 7 8 , | ?9. -16-1 i : i r b t d t t \ c u r r e n t ,3 8 6 : ; r b r d i t l t a n s . 4 6 4 ,p l a t e l 2 ternrns \\a\e. -lJ-5,J82, 500 t\\ LrrJrmensionalconvolution, 285, i

i r

i l t e n n g . - l l - 5 1 6 ,3 1 7 ,3 1 8 F!.urler series.533 F o u n e r t r a n s f o r m s ,5 3 3 i : - D s u r r e y .- i 9 , 9

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J.{. J-

- 1 .r

O1. I 40 [ ]::::::a:3lit)n trf -i-:;r::--.::-:3i:f . 258 ta:i:,::, -i ' ,.-n .:i f 1 9 + : : ' . i 1 : . : { : : 1 . : ; ' P e st r f .r=C :: ::.1::;-:::,

L dden. .\.. .1 Jn.rlnlormity: r e t l e c t i o n .1 5 0 .3 5 7 , 3 8 6 rrap. -l-i1. 352, 390 r l p e - 1 . t y p e - 2 ,4 0 2 . 4 0 5 ,4 0 9 L n c r a c k i n gp h a s ea m b i g u i t y ,5 5 4 underdamped geophone, 220, 221 undershooting, 245 6, 245. 247 u n i l o r m l i n e a r a r r a y ,2 4 7 5 0 , 2 4 7 u n i t c o s t s2 . 7,28,30 u n i t i m p u l s e .2 7 9 , 5 3 5 6 , 5 J 5 , 5 3 7 response,279, 547 u n i t s t e p{ l u n c t i o n ) .5 J J , 5 . 1 5 response,54Z u n i t v e c t o q5 1 8 , 5 1 9 L n i t e d G e o p h y s i c a lC o . . l 4 unmigrated section,267 u p - d i p d i s p l a c e m e n ot f r e f l e c t i n gp o i n t . 87.90 I c o r r e c t i o nf o r , 1 8 ,9 1 , 3 0 6 .3 1 6 1 8 , 3 1 9 , 3 2 0 ,3 3 5 ,3 4 1 updates, 198 u p h o l e g e o p h o n e ,1 8 ,2 2 9 u p h o l e s u r v e y ,2 5 6 , 2 5 8 u p h o l e t i m e ,2 2 9 , 2 3 2 upsweep,208 upthrust flower structure, 365,368 using the ratio p/tr: t o d e t e r m i n el i t h o l o g y . I 1 6 . l l 9 to distinguish interstitial fluids, 122, 124

126

::. 4ll

:r--la-!:.;,

,.:t-::'

8. 399,400

:- ,:9i

-

lldi

from ShaleS,

. : 1 . . . : . , . : r . - - - .l l 8 \anl:li:-:

::.J j:lil,

_:i i--1 leni.;.1- {::Jli::.of R3ii:'.:51,

;::lgratiOn. -326.

'. :. 99 100 .. 1:lN-lt) of $aye

:]F: rriL\L,rr:::.e::.i \elN'rt\- :::i::.\:. . : i::crmtninc: Baue:-r ;-:ci-..r--r r.ethod. l{(l

l{t

_l<

l.

l p < r - n rr - n t , . ; . C o 1 1 \ c 1 1 I ] t rnr je. i j : ' . r : ' , e r .l l ) . l - 1 0 l . l - : : .r J 1 . l { l fiOm C:r:i-t r.l:.- l15 f r o m h e a d a a r e : _9 5 LrggnS. j." $nlc lt1! frtrm rede'-tr..narnplrtude. l3,i 8. t39 t'rom rertr.-rl 'er'rn:c profiles (VSP). l - : , r J. t - . J ! ! I .\lrneth.rl. 1-lJ 5. /-ll b) \ el!-r:rt).lependent processing.I 38 -l': I: method. :6. /,iJ. 136 velor-it1.IlF.! rri: a p p a r e n t .! ! 1 - l r l a r e r a u er e q u i r a l e n t r .9 / , 1 1 8 . 1 3 0 , 1 4 0 D i \ . 1 - ? / i l.l - l group. ir'r' grrruF \ elocit) i n s t a n t a n e . r u s1. - 1 ( 1l -. l l i n t e r ra l . \ t c l n t e r \a l \ e l o c i t ) migrali(ln. 1-?.\.-iJ0 N\{O. 1-?(/ pha>e.-16.60. 1-l(i ) ,5 , 9 l 2 , rms (root-mean-square8 1 _ ' . i 1 3 0 .l 3 + . 1 4 0 s t a c k i n g \. ( c \ t a c k i n gv e l o c i t y v e l o c i t r a n a l l s i s ( s p e c t r u m ) ,1 8 , 3 0 6 1 3 , 3 1 4 .3 1 5 abnormal pressuredetection, 128 a c c u r a c vo f . - l l 0 I l . 3 1 2

(horizonvelocity alongreflections 31l, 313,314,315 analysis), 9 conventional,306 depthof burial.findingmaximum. 122,124,126 effectof staticson, 305,306,307 from velocitypanels,309,310 horizonvelocityanalysis,3l l, 313. 3r4,315 interpretationof, 31I values,309 1l. pickingtime-velocity 312,340 rulesfor picking,309-1I usesand limitations,311 function(GRM), 4J,r velocity-analysis velocitydependence on: l l8 9, 126,128. abnormalpressure, 129,130 ageof rocks,120,123 atomicweight,I l6 109 cementation, claycontent,I 16,I l7-18 ll8-20 compaction, l14, l15, 116,l19 density, depth of burial, I 18-20,l2l, 122 118,120,128 differentialpressure, zt4,58,69 elasticconstants, formationresistivity,108 freezing,121, 124, 126 frequency, 60 1,62,63,120 I gasconrent,108-9,l2l 2, lU, lL< hydrocarbonsaturation,125 fluids,108.109,ll0. ll:interstitial 1 3 ,l t 7 , l l 8 , l 2 l 2 , 1 2 4 , 1 2 5 l i t h o l o g yl l,l , l 1 2 , I 1 3 ,l 1 4 , l l 5 . I t6,il7, ll8, l19 maximumdepthof burial. 122.I l{. 126 p o r o s i r y1,0 8 1 , 0 9 ,l l 0 , l l 2 - 1 3 . l l 6 . I t 7 1 8 ,1 2 0 pressure, 108,109,ll0, ll8 20.l2l. 1 2 2 ,t 2 6 , 1 2 81, 2 9 ,1 3 0 l2l, 123,124 temperature, velocityspectrum,.set'velocityanallsis velocityvariations: effectsof graduallateralvariations. 392 3, 39s.397,398 with depth. effectsof linearincrease 93 5, 99 100 effectsof suddenlateralvariations. 395-6,398,399,400 effectsof verticalgradients,9l-5 velocityof wavetypes: body waves,44 Lovewaves,52 P-waves,44 Rayleighwaves,49 Stoneleywaves,50 S-waves,44 tube waves,53 wavesin fluids,47 isotropicmedia. wavesin transversely 58 verticalresolution,172, 173 7, 178,119 292 criteriafor, 112.lT3 7 illustratedby faults,174,116 illustratedby wedge,174,115 thin-bedeffects,I 74, 176,177,178, 179 verticalsection(3-D),459 verticalseismicprofiling(VSP),21,23, 53, 130,487 92, 487, 493,494,495

s92 deconvolution of, 490, 492 in deviated wells, 487, 489 migration ot, 490. 492, 494 planning for,492.495 processing. 489 92, 493, 494, 495 recording, 488-9 removing multiples in, 490, 492 reverse. 488 separatingupgoing from downgoing waves,48990,491 sourcesfor.488 9 stacking upgoing waves,490, 492,493 stretching time scale of,492.495 tube-wavenoise in. 489 types and usesof. 464, 465,48'7 8, 495 VSP-Io-CRP transform, 490, 494 walkaway, 187 well geophone for, 488, 489 vertical stacking, 185. 199, 204 5. 232, 242, 25t , 2s9, 26 I vertical velocity gradient, effectsof. 91 5 vibratory plough. 204, 206 Vibroseis recording, 18, 19, 206, 208 l0.2ll analysis,208, 262, 287 8, 289 correlation in the field, 261 f i e l d t e c h n i q u e s2, 0 8 . 2 1 0 m a r i n e v e r s i o n ,2 1 7 p h a s ec o n t r o l , 2 l 0 phase ofcorrelated records, 291 sweep2 , 0 6 . 2 0 8 ,2 1 0 .2 l l : d o w n sweep,upsweep,208; linear, 206, 2 1 0 ; n o n l i n e a r ,2 l 0 , 2 l l ; p s e u d o rundom, 210 using nonlinear sweep,2 10, 2ll usrng pseudo-random sweep,210 Vines Branch. Oklahoma, 6 Viola limestone, 5. 6 v o l u m e c o n t r o l , a u t o m a t i c . 1 5 , 1 8 ,2 2 6 , 227 I voxel, 4J2 V S P ,. s t ev e r t i c a ls e i s m i cp r o f i l i n g ( V S P ) walkaway, 253 4, 253. 255 w a l k a w a yV S P , 2 8 7 Wassp.2l7.2l8 water-break detector. 225 water-depth variations, effect on velocity.392 3,397 w a t e r - f l o o d ,5 0 0 . 5 1 3 . 5 l 5 water layer as a filter, 284 5 water reverberation, 165 6, 165, 169, 284 5,343 water-reverberationfilter, 284 5, 343 water wave, 485 watergun, 214, 216,211, 507 wave, 4 l amplitude, 34 angular frequency,angular wavenumber, 34 conversion of. 73 energy density, .ieeenergy density frequency, 34 harmonic, 4J intensity, J9 kinetic energy,58 period, -14 phase (angle), 34 on stretched string, 33 5 types: body, 4/. cylindrical, 59,322. 323; elliptically polarized, 46: eva-

IN DEX

nescent, 43, 63: P-wave,44: plane, 42; plane-polarized, 46; S-wave, 44; spherical, 42; surface, 49; trfte. 53 wave, evanescenl,43, 63 w a v e e n e r g y .p a r t i l i o n i n g .r ( , ( ,p a r t i tioning of energy at interface wave equations: in fluid medium, 47 general form of, 40 one-dimensional, 34 P-wave, 40 S-wave, 40 scalar, 39-40 with source term. 40 I for transverselyisotiopic media, 56 7 vector,40 wave equations. solutions of: d ' A l e m b e r t ' s ,3 4 . 4 l plane-wave,4l 2 spherical wave,42 3 for spherically symmetrical source, 479 wave motion, separation into P- and Swaves wave path. integral equations when /:

V(:).923 waveliont. 4/ w a v e f r o n tc h a r t , 9 1 , 9 5 , 2 6 7 8 , 2 6 9 wavefront methods of refraction computation.442 6 waveguidepropagation, 53, 483 .6, 487, 510.5lt,5t2 wavelength,34 apparent, 88 dominant, 4-l wavelet: c a u s a l ,1 8 1 , 5 5 0 desired characteristicsof', l8l l i n e a r - p h a s e5. 3 3 , 5 5 4 m a x i m u m - p h a s e5, 5 1 m i n i m u m - p h a s e( - d e l a y ) ,. i e a minimum-phase wavelet mixed-phase,29l.55l p h y s i c a l l yr e a l i z a b l e5. 5 0 l . 5 5 4 R i c k e r .1 4 8 ,l 8 l , 1 8 3 ,1 8 4 zero-phase,sec zero-phasewavelet wavelet processing, 148, 299 300,299, 301,457.459 wavclel-shape v ar i a t i o n s .c o r r e c t i o n s for, 298 wavelet shaping. 298, 299 300. 301 wavenumber,J4 angular, -?4 apparent, 8B waves in stretched string, 33 -5 w a v e s h a p el,8 l , 1 8 2 ,2 0 5 , 2 0 8 , 2 1 4 , 2 1 6 , 217,283 4 changesdue to absorption, 60, 180 c h a n g e sd u e t o f i l t e r i n g , l 8 l . 1 8 2 , 236.237 c h a n g e sd u e t o g h o s t s ,1 6 3 5 , 1 6 8 . 2)5 6 a)R )q?--1 changesdue to pegJeg multiples, 163, 168 in reflection-characteranalysis,401,

4 1 2t 3 . 4 t 5 weatheredlayer,set,low-velocity (weathered) layer (LVL) w e a t h e r i n g c o r r e c t i o n s1, 5 . l 8 1 9 . 1 4 6 , 261 2.263,266 curved raypath (Blondeau) method, 1 2 4 . 1 2 6 , 2 7 23 differential, 262

weathering velocity,124,256,258.16l wedgeasexampleof verticalresolution. 174,175,176 weightdrop, 18, 19,204, 205 enhanced, 205,207 horizontal,474 weightedsIack,321-2,32I weightingtime seriesto makethemminimum phase,291,299,555 well data,integrationwith seismicdata, 359,361 well-logcorrelation,361 well velocitysurveys: continuous, seesonic(continuousvelocity)logs conventional, 10,130 l, 132,141. 142 WesternGeophysical Co. of America, t4 whacker(energysource),210,212 whitenoise,additionof,295.2967 whitening,2958,299 Whitney, Willis8.,4 wide-anglereflection,77 Wiechert, Emil,3 Wienerautocorrelation theorem,J4J Wiener(least-squares, l,) c'iterion,342, 559 Wiener(least-squares) filter,293,295, 296,559 60 solutionusingmatrices,295 Wiener Hopf equation,560 wiggly-trace displaymode,2J4 Willis,RayleighWillisformula,217. 218 windmillarray,250 l. 252,457 window(timeinterval),253,558 carpentry,540,559 filter (gate),558 9, JJ8 word,230 workstation(interactive), 18.22. 23. 340.361 3, 361,460,464 WorldWar 1,4 WorldlWarll, l8 wrap-around aliasing, J/5, 316,317, 490 wrenchfaulting,365,366,368,375 Wylie'stime-average equation,I17, I18 Wyrobek'smethod,441 2,446,447 X-array,252 Xr 7":curves,85-6, 87 Xr I: method,86,134,136 specialprofilesfor, 134 YamatsuImpactor(energysource),207 Young'smodulus,J8 :-transform, 292,54850,548,554 applicationto digitalsystems. 550 calculationof (FFT), 549 50 integralrelationfor. 554 Zacamixtlewell (Mexico),8 zero-offset VSP,487 zero-phase wave,l8l, 183,184,291. 4 5 7 , 4 5 9 , 5 3 3 , 5 3 5 , 45 5 3 zig-zagsourceline,456 Zoeppritz,Karl, 3 Zoeppritzequations, 73 5

Appendices

A

List ofabbreviations used

AAPG A/D AGC AGI AIMME API ART AVA AVO CGG CMP COCORP

CW D/A DMO EAEG EDM EOR FFT GPS GRC GRM GSA GSI HVA IAGC IEEE IFP IFP LVL NMO OPEC OTC

AmericanAssociationof PetroleumGeologists,Tulsa analog-to-digital automaticgain control AmericanGeologicalInstitute,Alexandria. Va. American Institute of Mining and Metallurgical Engineers American Petroleum Institute, Washrnston. D.C. algebraic reconstruction technique amplitude variation with angle amplitude variation with offset Compagnie Generale de G6ophysique common midpoint (method) Consortium for Continental Reflection Profiling continuous wave digital-to-analog dip moveout (processing) European Association of Exploration Geophysicists,The Hague electromagneticdistance measurement enhanced oil recovery fast Fourier transform Global positioning system Geophysical ResearchCorporation generalizedreciprocal (refraction) method Geological Society of America, Boulder, Co. Geophysical Service Inc. horizontal velocity analysis International Association of Geophysical Contractors, Houston Institute of Electrical and Electronics Engineers instantaneousfloating point Institut Frangais du P6trole low-velocity layer, also called weathering layer normal moveout Organization of Petroleum Exporting Countries Offshore Technology Conference, Richardson, Tex.

RDU rms SEG SEPM SGRM SI SIE SIRT SP SPE SSC 3-D TI TV USGE VSP B

remote data unit root mean square Society of Exploration Geophysicists, Tulsa Society of Economic Paleontologistsand Mineralogists Soci6t6G6ophysiquede RecherchesMini6res Syst6meInternational (units) Southwestern Industrial Electronic Company simultaneous reconstruction technique self (or spontaneous)potential Society of Petroleum Engineers,Dallas SeismographServiceCorporation th ree-dimension aI Texas Instruments time variant United StatesGeological Survey. Reston, Va. vertical seismicprofiling

Trademarks and proper names used

Name ANA Aquapulse

Whose ftadename

Prakla GMBH We.sternGeophysia'al90. 67 America Aquaseis Imperial Chcmitul lndust rie.s Ltd Argo Cubit' Western Data Autotape Cubic Western Data BeanBag D evelopmental Geophys it's Betsy Mapco Boomer EG & G International Dinoseis ARCO Oil and Gas Co. Elasticwavegenerator Bison Instruments Flexichoc Institute Frang'aisdu Pitole Flexotir Institute Frangais du Pttrole Gassp Shell Development Hi-fix Decca Survey Ltd Hydrapulse CMI HydraulicHammer Geco-Prakla Hydrodist Tellurometer Hydrosein Western Geophysical Co. oJ' America

APPENDICES

510 Lorac Marthor Maxipulse Miniranger Nitramon Omnipulse Opseis Primacord PrimarySource Pulse-8 Raydist RPS Seiscrop Seisloop Shover Sosie Soursile SparkPak Syledis Syslap Toran Trisponder Vaporchoc Vibroseis Wassp YumatsuImpactor

Seismograph Service Corp. Institute Frangais du Pitole Western Geophysical Co. of America Motorola Inc. E. I. Du Pont de Nemours Co. Bolt Technology Applied Automation Inc. Ensign BickJbrd Co. Shear-wave Technology Decca Survey Ltd Hastings-Raydist Motorola Inc. Geophysical Service Inc. Geophysical Service Inc. Geco-Prakla Socitte Nationale ElfAquitaine Geomichanique Geomarines Systems Geco-Prakla Compagnie Gdnirale de Giophysique SercelS.A. Motorola Compagnie Gtntrale de Gtophysique Conoco Inc. TeledyneExploration Japex GeoscienceInstitute

Governmentdevelopedrystems Global Positioning System (GPS) Loran Navstar (GPS) Omega radar shoran Transit

C Randomnumbers 20891t3007 9521109221154339488223141865'71 2013719305711480403501380795081211134806 6060597685261475137939533049832546986469 3152259282168563865531862842830869406945 42094t7446277759946663704609s15502992164 5471415832043241359742328143035823185798 8973034685570004379863121120031853862439 12049 962663188607814 To obtain a random sequencerestrictedto a given range,choosea rule and begin applying it at an arbitrary location. For example,to get values lying between +8, we read numbersin pairs and use the first to give the sign (perhapsmaking evennumberspositive and odd ones negative)and simply omit any 9's we may come to. If we wish a different sequence,we begin at a different place and perhaps omit every other number or everythird number.

D

Units

units SI (SystimeInternational) Prefixes DimensionsSymbol Name Example of use l0r8 l0'5 l0'2

10, 106 l0l

E P T G M k

exa peta tera gigahertz giga mega megawatt kilo kilometer

m p n p i

milli micro nano pico femto atto

I 10-l 10-6

10, l0 12 l0-rs l0-,8

millimeter microwatt nanosecond picosecond

Baseunits Dimensions

Symbol Unil

Length

m

Mass

kg

Time

A Current TemperatureK cd Intensity Planeangle rad Solidangle sr

Equivalencies

3.281feet, (1/0.3048) feet, 39.37inches, 10'oingstroms, 0.0006214 statute mile, (l/1609)statutemile, (1i1853.n 2a ) utical mile kilogram 2.205pounds, (1/0.4536) pounds, 0.001102shortton second year in geologicage dating,means yearsbeforethe present ampere 5:0'C k e l v i n 2 9 3 . 1K candela radian (57.30"), l/0.01745 degree steradian

meter

Derived units Equivalencies Dimensions Symbol Use square 0.0001hectare, m2 Area meter 0.002471acre, 0 . 3 8 6 1x 1 0 - 6 squaremile 0.001liter, m3 cubic Volume meter 264.17U.S. gallons,6.2898barrels, acre0.0008107 foot,2l9.97 UK gallons

51r

APPENDICES Density

kg/m3

Force

N

Pa

Pressure

Energy,work J

\

)o.wer

Frequency Velocity

Acceleration Charge Potential Resistance Capacitance Magnetic flux Magnetic field strength Inductance

Hz m/s

m/s2 C V O F Wb

0.001g/cm3, 0.06243pound (mass)/cubicfeet newton kg-m/s', 0.2248pound, 105dynes pascal N/m2,l0-s bars, x l0 3 0.1450 pound/square i n c h , 9 . 8 6 x9 1 0 - 6 atmosphere joule Btu, N-m, (1/1055) (l/4186)kilocalorie, 107ergs,0.73756 foot-pound 'lva\\ )s,\\\\)\\ horsepoweq 3.412Btu/hour cycle/s 1.942nautical miles/hours, 2.237mile (knots) /hour

hertz

T

tesla

H

henry

ingstrom foot

A ft

inch milligal

ln.

nauticalmile/hour pound pound/square inch statute mile

knot lb psi st. mile

Geophonetransductionconstant:0.25 V/cm/s Geophonenatural frequencytolerance: + 0 . 5H z Geophonedynamic range: 140dB Geophonedistortion: < 0.2Voat 2 cm/s at 1 2H z Hydrophone sensitivity:49 Ylbar : 490 p,YlPa 10 V/cm/s at 100Hz, I Vicrn)sat l0 Hz Streamernoise:< 15 pV Ground unrest: 10-ato 10-6cm/s ( b ) Digiti:erslrecording instrunrents Frequencyresponse: 3 to 750 Hz Time accuracy:0.005Vo Recordingrange:I 14 to 120dB Linearitlr t0.0\9o Distortion: 0.0120(3 to 750 Hz) Systemnoise:( 0.2 rrV Alias filtering:190to 214 dB/octave Crossfeedisolation:95 dB Channelmatching:0.17o Operatingtemperature:-50 to +75.C Operatingaltitude:to 5500m

Wb/mr, N/A-m, loa gauss, lOegamma Wb/A, V-s/A

(c) Recordingconventions Channel I is toward north or eastlif line is crooked,

0.1 0.316 0.501 0.708 1 t.4t3 1.99s 3.162 l0 101

0.01 0.1 0.251 0.501 I 1.997 3.980 l0 100 108

average

direction

tletcr-

RecordedVibroseissweepleadsthe baseplate velocity by 90'

Energy ratio 10 12 108 10-4

overall

mines(channelsenseshouldnot be changed alonga line) Upward kick on a geophoneyieldsa negative number and a downswingon monitor record (seefig. 6.49) Pressureincreaseon a hydrophoneyields a negativenumber and a downswingon monitor record

mGal

E Decibel sonvension Amplitude ratio dB _120 10-6 -80 104 -40 0.01

0 3 6 l0 20 80

(a) Geophones

10smilligals c,ou\omb A.-s volt W/A ohm V/A farad A-s/V weber V-s, 108maxwell

Non-SI units (c66reviotions ) Unit Symbol

*20 -10 - 6 -3

F Typical instrument specifications and conventions

G

A seismic report

(a) Title page List: for whom work was done, name of project or area, datesof project,name of contractor making report, individualsresponsiblefor report. (b) Enclosures List: attachments,documentsthat go with report. Figures, maps, and sectionsshould be used where they

572 are able to conveyinformation more clearly than textual description.Encloseonly relevantdata. Consider photoreducingmaps or sectionsfor inclusion. Label enclosuresso they will be identifiable if separated from report.

APPENDICES (i) Appendix L 2. 3. 4. 5.

Copy of lines processed. Statistics. Specialstudiesnot relevantto main objectives. Personnellist. References.

(c) Abstract Briefly statewhy work was done, what was done, and how results are to be used. No lonser than a halfpage. d) Introduction l . Briefly stateobjectives.If report coversonly pro-

J.

cessing,review relevant information about field operationsor previousprocessing. Describe location of work, usually with a map. Distinguishdata being discussedfrom other data shown on map. Describedata quality in generalterms and nature of problemsencountered(multiples,static problems,line misties,structureproblems,and so on).

H

Symbols used in rnapping

(a) Structuresymbols

*+*

--+-r

(J) Results l. Include copiesof sectionsand list of data processed. 2. Ltst problemsencountered,including where unable to read tapes,poor documentation,survey, elevation,uphole problems,and so on. 3. Specialproblemsobserved. (g) Conclusions Did processingmeetobjectives?How could objectives havebeenbetter met?

Anticlinal axis, reversalof dip directlon Anticline plunging to the left Synclinal axis plunging to the left Normal fault with upthrown side to the north hachures or block, on downthrown side Thrust or reversefault; barbs are on side ol upper block; contours in lower may be dashed when underneath the fault

(e) Processingproceduresand analysis L Standardprocessingsequenceused(often shown by flow chart). Parametervalues,method of datum correction.Discussprocesses by trade names and describeobjectivesand methodsthat unusual programsemploy. 2. Describetestingdone to determineprocessingsequenceand parameters.Locations of test points. Describe(often includeexampleof) displaysused to determineparametersfor muting, filtering, determining stack response,velocity, static corrections, residualstatics,and so on. 3. Experimentation done,where,and conclusions. 4. Discuss velocities; data from previous work, wells, other sources.How often were velocity analysesrun, datum used. How much velocity variation?

Apparent dip

Strike-slip fault showing senseof movement Dashed or dotted contours indicate inferred or doubtful structure or sometimesan alternate interpretation or other kind of data (perhaps outline of gravity anomaly, inferred subcrops.and so on)

-4' \ 1 O

Strike and dip ofbedding, the number indicating the amount ol dip (usually in degrees)

(b) Wellsymbols

o

o

#

Wall location

Oilwsll

Oil and gas wsll

)& ,\{' Gas well

g Abandoned oil well

JL Y Dry hole

>x Abandoned9as

(h) Recommendations Reprocessing, further testing,next work in this area.

Dry holB, with show of gas

+ Shut-inw6ll

+ Dry hole, with show of oil

l i

A P P E ND I C E S (c) Rocksymbols

Fi;6;:;Z-:1 l--;:,o,.J -

l:'"".;;.dl

-l : ::r: : : : : : l - l - -

, , ,l

I

Conglomerate

Sandstone

\

l-::--

-

-l

,,l

r \

lfrryrtr iliiil

ffii

fD(x)oasal fi

rgnoousor Metamorphicrocks

L----

--1

Shale

r 16-d-l

l-^=^!l

Siltstone

! l l l ! t ! r - t r l

I

Clay3tone

/ /

l---.IL-l LrmY Shale

Limy Sandstone

Rr--i-T-il

l-------l-

SandyShale

\ l

| -

/ /

ffi l-*rlrLl-ti Limestone

- F iz-= z m - = 4 ' ;=n ffi Oolomiiic Limeslone

f7-'-r-'-fi Dolomite

by combiningsymbols' are represented rockcompositions Note: Intermediate

W

Anhydrite

s

$nq

B 6 .F #

I

E

m i $ ; g : - * , oE e

E B

E s

q

R'E

ul J

E lrl

z o

c

>'q

P

Y -

9 ; A € y R A U Y . i E s

e v ,

< ! 9 . r . : . =

e H R o 6 Z 6 ( € ( € d

'':.uE

.9 6.: 'F tr-c;_o a '> =o F ;

';i

i6 i >E d o q >

a , E ; 6 o 0 2 .:q:ls'Y

! c F V X E E - i

;o ,F E E 'i o ? - . , 4 -

H e. ' E : E5 6

.=

-.= k h

EHEE a o O t r -; tra: b^9 6 o *i>,G o =

' :

=

_^tr q)u :

qH

E)

Eg;: 6 F.-'q. L

Q =

q 9 U E E F } I ? ; - ! o x

h

U

f , : i a f * , * q s * r * s : 5 6 ! *

t

, . , ' , , : l , : , : ' ', , i ,

'rr'*#*flsd**

o o

E

tr

3150

9250 Llnc

3300

Plate 2 Bicolor display showing stackedhydrocarbon accumulations.Note high amplitudes (bright spots),polarity reversals,flat spots,time sagsdue to the increasedtime to traversethe hydrocarbonaccumulations.(From Brown, l99l: 163.)

-fT

ll

I

WELLA

t.xs -'lr

it I

) ) )l

L ;

i , l I Ir ) ) ) t >

) l I ? )) )! )' ' ) I D l} ) ] I I ; t I t

i , i )t

ir

!t

iiriiiilliiri

I r )r , ; ) l EEEIr ) r l, T.rll pl r -t:r-:

D )

> -

l r : l ) ) ); t ) Ir I

l F I

I

l

ili

ii

I

t

tt, ; D D

t t

I

lr

I

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Plate4 A vertical section(top) from a 3-D volumeinverted to seismicJogform and a horizon slice(bottom) over a hydrocarbon accumulation.(Courtesyof CGG.)

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Plate 7. Compositedisplaysat a workstation help in understandingfeatures.(from Brown, 1991;70, 71.) (a) Portion of a time slice(top half) and vertical section(bottom half); (b) cubedisplayshowingline and crosslineon sidesand time sliceon top.

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Time slice from a circle shoot about a salt dome. (From Brown, 1991:63.)

l ' l a t c l 0 H o r i z o n s l i c c s h o w i n g a s t r e a mc h a n n c l .T h e s u p c r - i m p o s ect o l n l o u r s i l l u s t - r a t ct l l c s l n l c t r . l r c a r t d t h e b r i , s l t t n e sisn t h c h i g h p o r t i o n o f t h c c h a n n e l i n d i c a t c sh y d r o c a r b o n si l t a c h a n n e ls a n t l r e s e r v o i r ' ( F r o m B r o w n . 1 9 8 5 :1 2 3 . )

Plate 11 Horizon sliceal,ongan angularunconformity reflection.The NW-SE lineationsindicatethe subcropof differentmembersdipping to the SW,and the W-E lineationsindicatefaults. (From Brown, 1991:135.)

Plate l2 Horizon slicewith overlainstructuralcontoursshowingbright spots indicatingstratigraphic hydrocarbon-gas accumulations, in a turbiditefan. (From srown, t99l: t3b.)

Plate 13 Fault slice showing structure adjacent to the fault plane and secondary splinter faults. (From Brown. Edwards. and Howard, 1987.)

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Plate 15 Horizon slicesmade as indicatedin plate 14.An interpretationis shownin plate 16.(After Hardage,1993;courtesyof TexasBureauof EconomicGeology.)(a) Common variable-densitycolor coding; (b) biasedcolor coding that helpsdefineancientstreamdeposition.

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Plate 16 Pre- and post-3-D interpretations.(After Hardage,1993;courtesyofTexas BureauofEconomic Geology.)(a) Interpretationbasedon well control and2-D seismicdata (blue lines.;,superimposed on horizon slice;(b) interpretationbasedon 3-D horizon sliceand well data. Display parametersfor (a) and (b) are slightly different.

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