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rsBN 0-521-46282-7
CANNBRIDGE UNIYERSITY
PRESS
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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
+J
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
t)
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
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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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, tl
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I
ri.'.,,,*, t
9 r
'sl
Oo/o
.r-"?" .t?'
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1980
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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.
!:B At A l
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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 ($5'3'4)'Laboratory measurements(fig. 5.18) show that velocity is .rr.niiully constant when the overburdenand fluid p..rr,r... u.. changed,provided the differentialpresiure remainsconstant.Abnormal fluid pressurecon-
119
EXPERIMENTAL DATA ON VELOCITY
'
400
'/ ./
./ ^
6
N --
-Y
€
3 250
\
/
E .r'
o
,'
,,',
r, .r'^^^r(rr. .-to"'rt'
,l
,'
,''
1&
,,
,''
/
-j,, , '.r,
,'
.r{1t r}}'l,r(t
t', l t
.t'
{;,
7t' ,,' .r'
i
,'
, l ' - ,t .r-'-
,,(Fri',:p"'," ,'/n)"
,'
. ,?: .n(
/
t .h9o'
P'wavc v.loci(y, d(kft/3) t4 18 t6
/
,1'ot ,'
A'Lt't2'
t2
,/
,' ," l;**,t'',,",r'(':." .-
J
ut
7-
,{' d"'7',' y'''' y''/ ;i'y'' /
,"' .""'1","",'("',"t'
-,t
>
F O
I2a+.
20
066
F i g . 5 . 1 4 V e l o c i t ya t l0 kilobars versusdensity for silicatesand oxides. The numbers refer to mean atomic weishts. (From B i r c h .1 9 6 1 . )
3 060
i rr" I
s ?
@ 050
8 50
40
Velocity (kft/s) 12 15
60
P.wrv€ vclocily. a(km/s)
(b)
2.8
Fig. 5.12 Relation between S- and P-wave velocities (/, and (,) for various lithologies. (a) Cross-plot of laboratory measurements (after Pickett, 1963). (b) Use of S- and P-wave velocity r a t i o ( p / c r )a s a n i n d i c a t o r o f l i t h o l o g y ( f r o m S h e r i f f ,1 9 8 9 13 8 8 ) .
l
t
t
t
1 ^
d
2.O
t
Sdl ond olluvio
3.0 4.0 5 Velocity(km/s)
Sondstonas Shola5
Fig. 5. I 5 P-wave velocity density relationship lor different lit h o l o g i e s( t h e s c a l ei s l o g , l o g ) .T h e d o t t e d l i n e s h o w se q . ( 5 . 1 5 ) and the dashed lines show constant acoustic impedance ( k g / s . m r x 1 0 6 ) .A f t e r G a r d n e r e t a l . 1 9 7 4 : a n d M e c k e l a n d Nath. 1977.)
Limasiona
l
l
l
r
\l
a
tE
!
\
I
t
I
I
o
s5
I I
I I
\
z
I I
r.4
1.6
'.8
2.O
2.2
2.4
2.6
2.a
3.0
Densig (g/cc) Fig. 5.13 Histogram of density values tabulated in Birch (1942) for different lithologies. (From Grant and West, 1965.)
stitutesa severehazard in drilling wells and one use is in predictingsuch of seismicvelocitymeasurements z o n e s( s e e$ 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-
Transit time (ps/ft)
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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)
|c)
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
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1.100 1. 7 8 6 1.935 2.250
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R E F E RE N C ES
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
t.2 I.-)
(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
I4l
Offset (m)
Ofiset (m)
'F
i
(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
pv +
l
t
t+
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(d
(b)
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! = =
2.0
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';
+x !!
-
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t
se
lll
.- sE qI
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ilp =E E
l l ! :
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lll
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o. o
i l
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--
il
LI]
EE
t t
i ' -
l l
d ^
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A
r vt
.<
'
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U-o^F
3 H; = " : :< q o
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n < < ! = . -
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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 ($9.5.7)is generallyineffectivein attenuatingsuch multiplesbecauseit assumesthat, for any given trace, the reflection coefficientsat the reflectors involved are constant.However,if sea-floordip is small,the reflection anglesfor the simple reflection at offset x will be
EVENTS OTHER THAN PRIMARY REFLECTIONS
r69
jlf'#rl]tililllf [T, nllil,4H l',"1s",>1.1) I I I I I I I l,t'rli'l',1',)1'!,".P.)
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, iiili;i.,,x,riir,ii,,f3:::::l|HI::IA
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rtr, ttr))) t r, rrDr)f rr>rr D))rrrrrrfm) _ll_Jf. i.i" ,:"Dr'!.>D)r!.t.t }},lt b)ilrrDr D)Df rr., ) r , , .,., I I i1 --llJ,:?ffD)'t"li.)rt.)tr-r,r'.rx'ri>)' r',r,.r..rrDiil..l j jrr")';1 ..,,i.r.r>;,p-, r i Ip I I ;,r ; I I I I r"r,r^".---l;,f
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{
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7.,,,.'
,1ii; irr' r r'*r or*rr; ipir .+ii ;,3::l:ffit;
+i::ififiHr f*:: :*llil:sr ' jill,iii*l: **ifr n:riii ;:l :;; 1 1:L :l :::,,i i;l;';;.H;l'ffi ::Ii:1' ::X:l 1,,,':::ll' i-l'ilr:;1.:
li{f il'+1+r-';7;r;i#r.5;n-iiiffi ; 5*;lr i',,_: -br:l ':: ') r'i"lijJJ;;, 'fllt iii
jii::
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ii.i:i;ill[. ill:l:i:j :.j ri::::ll:i,'^l::
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,o-ii':;l :::fi-i ..;::i:-:::l l:i:i:: l::,:.-'*e.-b"E'd
\a)
Fig.6.32 Seismic recordshowing (Courtesy singing. of HaliburtonGeophysical (a)Fieldrecord. Services.) and(b)thesame the sameas for the first multiple at offset2r (fig. t5.35) and for the next multiple at offset 3x, and so on, so that the primary on one trace sometimescan be used to predict and compensatefor the multiple seenon anothertraceat anotheroffset.This providesthe basis for radial multiplesuppression (99.5.13). 6.3.3Refractions The onsetof headwavesis often followedby a number of parallel alignments,that is, they seemto involve a
after singing has been removed by deconvolution processing
($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
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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
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E
E
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)l
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I
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F-is.6.41
Reflection from a faulted reflector, with the fault throw indicated as fractions of the dominant wavelength.
@ 6
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o E 3
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(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
(a)
187
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Fig. 6.53 Directivity of a harmonic source located at various depths below a free surface. (After Waters, 1987.) Depth/wavelength : 0.1, 0.5,and 1.0in parts (a), (b), and (c), respectively.
s..l
Fig. 6.54
Two models that give identical first-arrival curves. (After Pautsch, 1928.)
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
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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
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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-
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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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(c) Waveforms observed on the sea floor. ((a) From Barr Fig.7.4l and Sanders, 1989;(b and c) from Barr et al., 1989.)(a) Waveform observed by a pressure detector in various water depths
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
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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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Response of seismic filters in typical
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Inputs from n channels(n usually 24 to l2O)
each channelin sequence
Analog-to-digital (A I D) conv erter l4 bits plus sign bit
Tape transport lread-after-writg
al-to-analog((D/A; converter
Paperrecord
ry
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
RodnrLn
sh(rrer
Juni(r oh\cf!rl
( 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
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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-
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246
REFLECTION FIELD METHODS
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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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Wavelront approaching a linear arrav.
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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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
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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
The Explo.alion Problem
Nois ---lAmbi€nt
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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<