Fundamentals Of Seismic Tomography
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GEOPHYSICAL
MONOGRAPH
SERIES
David V. Fitterman, Series Editor
Larry R. Lines,Volume Editor
NUMBER
6
FUNDAMENTALS
SEISMIC
OF
TOMOGRAPHY
By Tien-when Lo and Philip L. Inderwiesen
SOCIETY
OF EXPLORATION
GEOPHYSICISTS
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Lo, Tien-when, 1957-
Fundamentals of seismictomography/by Tien-whenLo and Philip L. Inderwiesen.
p. cm. (Geophysicalmonographseries;no. 6) Includesbibliographicalreferencesand index. ISBN 978-1-56080-028-6:$22.00
1. Seismictomography.I. Inderwiesen,PhilipL., 1953II.
Title.
III.
QE538.5.L6
Series.
1994
551.2' 2' 0287--dc20
94-23818 CIP
ISBN 978-0-931830-56-3 (Series) ISBN 978-1-56080-028-6 (Volume)
Societyof ExplorationGeophysicists P.O. Box 702740
Tulsa, OK 74170-2740
¸ 1994by the Societyof ExplorationGeophysicists All rightsreserved.This bookor portionshereof may not be reproduced in anyform withoutpermission in writing from the publisher. Published
1994
Reprinted2000 Reprinted2004 Reprinted 2006 Reprinted2008 Printed in the United States of America
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Contents
Preface I
2
vii
Introduction
1.1 The Concept of SeismicTomography .............
1
1.2
3
Applications ...........................
1.3 Ray vs. Diffraction Tomography ............... 1.4 Suggestionsfor Further Reading ...............
5 6
Seismic Ray Tomography
9
2.1
Introduction
9
2.2
Transform
2.2.1
2.3
2.4
3
...........................
Methods
.......................
10
Projection Slice Theorem ...............
10
2.2.2 Direct-TransformRay Tomography .......... 2.2.3 BackprojectionRay Tomography ........... SeriesExpansion Methods ................... 2.3.1 The Forward Modeling Problem ...........
16 20 22 23
2.3.2
Kaczmarz'
26
2.3.3
ART
...................
.....................
33
............................
39
2.5 Suggestionsfor Further Reading ...............
42
Seismic Diffraction
45
3.1
Summary
Method
and SIRT
Introduction
Tomography
...........................
45
3.2
Acoustic Wave Scattering ................... 3.2.1 The Lippmann-SchwingerEquation ......... 3.2.2 The Born Approximation ............... 3.2.3 The Rytov Approximation ............... 3.2.4 Born vs. Rytov Approximation ............ 3.3 Generalized Projection Slice Theorem ............ 3.3.1 CrosswellConfiguration ................ iii
46 47 51 52 56 58 59
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4
3.3.2 Vertical SeismicProfile Configuration ........ 3.3.3 Surface Reflection Configuration ........... 3.4 Acoustic Diffraction Tomography ............... 3.4.1 Direct-Transform Diffraction Tomography ...... 3.4.2 BackpropagationDiffraction Tomography ...... 3.5 Summary ............................ 3.6 Suggestionsfor Further Reading ...............
72 78 82 84 87 90 •3
Case
95
Studies
4.1
Introduction
4.2
Steam-Flood EOR Operation ................. 4.2.1 CrosswellSeismicData Acquisition .......... 4.2.2
4.3
4.4
4.5
...........................
Traveltime
95
Parameter
Measurements
.........
B
C
99
4.2.3 Image Reconstruction ................. 4.2.4 Tomogram Interpretation ............... Imaging a Fault System .................... 4.3.1 CrosswellSeismicData Acquisition ..........
108 111 125 125
4.3.2
.........
130
4.3.3 Image Reconstruction ................. 4.3.4 Tomogram Interpretation ............... Imaging Salt Sills ........................ 4.4.1 Assumptions and Preprocessing............ 4.4.2 Data Acquisition .................... 4.4.3 Diffraction Tomography Processing.......... 4.4.4 Tomogram Interpretation ............... Suggestionsfor Further Reading ...............
133 135 137 137 139 141 146 150
Traveltime
Parameter Measurements
A Frequency and Wavenumber A.1 Frequency ............................ A.2
95 96
Wavenumber
153 153
..........................
154
The Fourier Transform B.1 Fourier Series ..........................
157 158
B.2 Exponential Fourier Series ...................
159
B.3 FourierTransform- Continuousf(x) ............. B.4 FourierTransform-Sampledf(x) ...............
160 161
B.5
162
Uses of Fourier Transforms
Green's
..................
Function
167
C.1 Filter Theory .......................... C.2 PDE's as Linear Operators .................. C.3 Green's Function Example ................... iv
168 170 173
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C.4 Suggestionsfor Further Reading ...............
INDEX
174
175
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Preface
Motivated by the successfulimplementation of medical tomography in the early 1980s, geophysicistsand production engineershave attempted analogousmethodsusingseismicenergyfor hydrocarbonexploration,reservoir characterization, and production engineering.The theoretical methods and field techniquesemployed are broadly classifiedas "seismictomography" and were fundamentally developedin the late 1980s. Today, seismic tomography is conductedon a commercial basis with its theory anchored on a solid base, and its strengths and limitations known. Field applications have demonstrated that seismictomography can provide valuable services in upstream operations,suchas mapping subsurfacestructures,delineating reservoirs,and monitoring enhancedoil recoveryprocesses. Our book developsthe fundamentalsof seismictomography at the level of a tutorial or practical guide. Considerableeffort has gone into making the book self-containedso that any reader who has had calculuscan easily follow the material. Referencesfor further reading on specifictopics are given at the end of each chapter. We give a short statementfollowingeach referencedetailingits significance as a supplementto this book. In doingso we hope the reader will not feel the referencesmust be read to fully understand a given concept. We use appendicesto review physicalterminology and mathematics required to understand the theoretical presentations. We present various tomographicmethods in a logical and straightforward manner. Unlike many other books on tomography, we use standard notation for variableswhich span the variousmethods,enablingthe reader to easily contrast differences. Also, mathematical steps glossed-overby most research
articles
are filled-in
for our readers.
Sometimes
we deviate
from well-known derivations to provide a deeper physical understanding. However,for completeness,our derivationsare followed-upwith references to the "well-known" derivations at the end of the chapter. In addition, we discussthe limitations of seismictomographyand illustrate successes and pitfalls with casestudies. Our ultimate intent is that after reading this presentation, the reader will exhibit both a greater understandingand appreciation for seismictomography articles presentedin the literature. Chapter I is introductory and summarizes the developmentof seismic tomography and describeshow this new technologycan benefit the oil industry at both the exploration and producing stages. Chapters 2 and 3 are tutorials on the theoretical fundamentals of seismic ray tomography and seismicdiffraction tomography,respectively. Chapter 4 presentsthe vii
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data acquisition,processing,and tomogram interpretation for three seismic tomographycasestudies. Each casestudy has its own unique data acquisition, data processing,and interpretation challenges.They provide useful insight into designingand conductingfuture tomographystudies. The authors thank Texaco for releasing data and results for the case studies published in Chapter 4. We also acknowledgethe releaseof the McKittrick data by Texaco'sjoint partner, Chevron, for that project. Both Eike Rietsch and Bob Tatham of Texaco have encouragedthe authors to pursue this project and have provided support during its progress. We also acknowledgeour fellow boreholeseismologyteam membersat Texaco: Danny Melton, Don Howlett, Ron Jackson, and Stan Zimmer for their contributionsto this field throughout the past few years. In addition, David Fitterman, the SEG monographsserieseditor, and Larry Lines, the volume editor for this book, have been patient with our progressand have provided valuable guidance. Finally, the authors thank Texaco for permission to publish this book.
Philip L. Inderwiesen Tien-when
Lo
E•tP TechnologyDepartment Texaco
Inc.
Houston, Texas
viii
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Chapter
I
Introduction
1.1
The Concept of Seismic Tomography
We definetomography as an imagingtechniquewhichgeneratesa crosssectionalpicture (a tomogram)of an object by utilizing the object'sresponseto the nondestructive,probingenergyof an external source.Seismic tomographymakesuseof sourcesthat generateseismicwaveswhich probe a geologicaltarget of interest.
Figurel(a) is an exampleconfiguration for crosswell seismic tomography. A seismicsourceis placedin one well and a seismicreceiversystem
in a nearbywell. Seismicwavesgeneratedat a sourceposition(soliddot) probe a target containing a heavy oil reservoirsituated between the two wells. The reservoir's responseto the seismicenergyis recordedby detec-
tors (open circles)deployedat differentdepthsin the receiverwell. The reservoiris probedin many directionsby recordingseismicenergywith the samereceiverconfigurationfor different sourcelocations.Thus, we obtain a networkof seismicraypathswhich travel throughthe reservoir. The measured responseof the reservoir to the seismic wave is called
the projectiondata. Tomographyimagereconstruction methodsoperateon
the projectiondata to createa tomogram suchasthe onein Figurel(b). In this casewe usedprojectiondata consistingof direct-arrivaltraveltimes and seismicray tomographyto obtain a P-wavevelocitytomogram.Generally,differentcolorsor shadesof gray in a tomogramrepresentlithology
with differentproperties.The high P-wavevelocities(dark gray/black)in the tomogramin Figurel(b) are associated with reservoirrockof high oil saturation.
Seismictomographyhas a solid theoreticalfoundation. Many seismic
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2
CHAPTER 1. INTRODUCTION
Crosswell Seismic Configuration
o
source
o
receiver
/'-waveVelocity Tomogram
heavy oil (a)
(b)
F•G. 1. (a) Geometry forcrosswell seismic tomography example. P-wave energy traveling alongraypaths probethe geological target. (b) P-wave
velocitytomogramreconstructed from observedtraveltimedata. Different shadesof gray correspond to differentP-wavevelocities.
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1.2.
APPLICATIONS
3
tomography techniqueshave closeties to more familiar seismicimaging methods such as traveltime inversion, Kirchhoff migration, and Born inversion. For example,seismicray tomographyused to determinelithologic velocity is essentially a form of traveltime inversion and seismic diffraction tomographyis closelyrelated to Born inversionand seismicmigration. Thus, seismictomography may actually be more familiar to you at this point than you might think sinceit is just another aspectof the subsurface imaging techniquesgeophysicistshave been using for years. 1.2
Applications
Seismictomography is applicable to a wide range of problems in the oil industry, ranging from exploration to developmentto production. The casestudies presentedin Chapter 4 demonstrate that seismictomography can complement conventional seismicmethods and provide unique, previously unavailable subsurface information. Tomography applied to surface seismicdata can generatesubsurfacevelocity modelsfor explorationproblems. These velocity models can in turn be used as soft information in the geostatisticalinterpolation of well-log data between wells. Seismictomography applied to developmentand production problems is generally implemented by a crosswellconfiguration, as shown in Fig-
ure l(a). Figure 2 illustratesthe benefitof crosswellseismictomography for reservoir
characterization
over conventional
reservoir
characterization
tools. Figure 2(a) representsthe true geologybetweentwo wellsin a producing field where the producing formation is a tar sand layer overlaid by
a thinner, lesspermeablebed (shadedinterval). The heavy oil in suchtar sandsis somewhatimmobile unlessheated using the enhancedoil recovery techniqueof steam flooding. A production engineerplanning to steam flood a tar sand interval needsto know whether the lesspermeablebed is capable of confiningthe steam to the tar sand. In our cartoon the lesspermeable bed is breached by a small fault. Well logging is a conventionalreservoir characterization tool that provides information about the reservoironly a small distancefrom the bore-
holeasdepictedin Figure2(b). Thus,no hardgeological informationabout the unprobed reservoir between the wells can be extracted from conven-
tional well-log data. The well-log data will only show that the low permeability layer exists between 500 and 600 feet in well A and between 400 and 500 feet in well B. Based upon the relative formation dips in each well, the engineermay decidethe low permeability layer is continuousand interpret
the well-logdata usinglinear interpolationas shownin Figure2(c). The small fault is thereforenot detectedand unexpectedsteam flood results will
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CHAPTER 1. INTRODUCTION
(a) true geology
(b) well logging
(c) well logging
interpre[ation
200 ft 4OO ft 600 ft
steam-' 8OO ff
A
B
A
B
(d) tomography
A
B
(e) _tomography Interpretati__on
I logging tool I
source
D receiver
A
B
A
B
FIG.2. (a)Truegeology wewishtoknow.(b)Welllogs sample onlyashort distance intothereservoir, requiring sometypeof interpolation between wells asdepicted in(c).(d)Crosswell seismic records theearth's response toseismic energy between wells thereby permitting animage reconstruction
of thegeology asshownin (e).
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1.3.
RAY
V$.
DIFFRACTION
TOMOGRAPHY
5
occur.
On the other hand, crosswellseismictomography can directly probe the reservoirbetweenthe two wellsas shownin Figure 2(d). A downhole seismicsourcein well A generatesseismicwavespowerful enoughto travel through the reservoirand to be recordedby sensitivedetectorsin well B. After applyingtomographyprocessingto the projectiondata, a tomographic
interpretationliketheonein Figure2(e) mightbe obtained.Thus,the engineer will be aware of the small fault and can make the necessaryalterations to the steam flood operation. Although just a cartoon, Figure 2 illustrates how crosswellseismictomography is a more reliable tool for delineating the reservoir between wells than any interpolation method between well logs. However, crosswellseismictomography becomesan even more significant tool for reservoir characterizationwhen used in conjunctionwith well-log information and core data as is demonstrated in Chapter 4.
1.3
Ray rs. Diffraction
Tomography
To do seismictomographywe must model the seismicwavetraveling through the subsurface. Both ray and diffraction theoretical models are available to us for describingseismicwave phenomena. Which model we use dependsupon the relative sizesof the seismic wavelength and the target we wish to image. A judicial choiceof theoretical model for a given seismicwave and target becomesimportant to the successof the seismic tomography application. If the target's size is much larger than the seismicwavelength,then we may model the propagationof seismicwavesas rays usingray theory. This is similar to using geometrical optics to describe light wave propagation throughlenses.Seismictomographybasedupon the ray theoreticalmodel is discussedin Chapter 2 under the title Seismic Ray Tomography. We subdivide the topic into "transform methods" and "seriesexpansion methods." The transform methods are commonlyused in medical tomography experimentswhile the seriesexpansionmethodsseemuchusein seismictomographyapplications.Currently seismicray tomographyis very popular becauseit is simple to implement under a variety of situations, is computationally fast, and givesgood results. When the size of the target is comparableto the seismicwavelength, then we model the propagationof seismicwavesas scatteredenergy using diffraction theory. Such a target scatters the seismic wave in many directions and only diffraction theory can properly model this response. Seismic tomography based upon the diffraction theoretical model is discussedin Chapter 3 under the title Seismic Diffraction Tomography.As
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6
CHAPTER
1.
INTRODUCTION
you will see,seismicdiffraction tomographypresentedin its simplestform requiresrestrictionson the source-receiver geometry,ignoresmultiplescattering of energy, places limits on the sizes and velocity contrastsof targets, and is computationally intensive. Becauseof these restrictions the method is currently applied only to a few select situations. However, re-
centdevelopments, whichwe list under "suggestions for further reading"in Chapter 3, are overcomingsomeof these restrictions. Our presentationof seismicdiffractiontornographyin its simplestform shouldgive you a solid foundationfor understandingand appreciatingthesedevelopments. Severalcasestudiesare presentedin Chapter 4 to illustrate the application of theory to varioussituations. We emphasizethe need to assimilate as much data as possiblefrom other sources,such as from well logs and core samplesin the crosswelltomographyexamples. Only by integrating all information availablewith the tomogram can one make an optimum assessment
about
the reservoir.
' As a final note, we have crisply divided the application of seismictomographyinto ray and diffractiontomography,dependingupon the relative sizesof the seismicwavelengthand target. However,in reality the probing seismicwave is usually a broad-bandsignal consistingof a large range of wavelengths,and the subsurfacecontainspotential targetswith relative sizesranging from small to large. Thus, this suggestsa blend of seismicray tornographyand seismicdiffractiontornographybe used to optirnallyimage all possibletargets. Although interesting,we pursuethis possibilityno further as it is more of a researchmatter at this point in time. In this book we will concentrateon presentingthe fundamentalsof seismictomography.
1.4
Suggestions for Further Reading Aki, K., and Richards, P., 1980, Quantitative seismology:Theory and methods, Vol. II: W. H. Freeman & Co. Section 13.3.5 o.f Chapter 13 presents a classificationscheme based upon seismic wavelengthand target size which will give you a goodidea when to use ray theory or diffraction theory.
Anderson,D. L., and Dziewonski,A.M., 1984, Seismictomography: ScientificAmerican, October, 60-68. Popular arlicle on seismic ray tomographyapplied to imaging the earth's mantle.
Lines, L., 1991, Applications of tomography to borehole and reflectionseismology:Geophysics:The Leading Edge, 10, 11-17. Overview of seismic ray tomographyapplications.
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1.4.
SUGGESTIONS
FOR
FURTHER
READING
Menke, W., 1984, Geophysical data analysis: Discrete inverse theory: AcademicPress,Inc. Our bookaddressesonly those topics in inverse theory requiredto understand the basicsin seismic tomography. Menke's bookprovides a goodintroduction to inverse theory.
Tarantola, A., 1987, Inverseproblem theory: Methods for data fitting and model parameter estimation: Elsevier. A comprehensivebookon inversetheorywhichincludesmanyprob-
7
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Chapter
2
Seismic Ray Tomography 2.1
Introduction
We begin the study of seismictomographywith image reconstruction methods based on ray theory. We assumethat the sourceproducesseismic wave energy with wavelengthsmuch smaller than the size of the inhomogeneitiesencounteredin the medium. Only when this assumptionis obeyed can the propagation of the seismicwave energy be properly modeled by rays. Otherwise, the seismicdiffraction tomography in Chapter 3 must be applied to solve the problem. Two groups of image reconstruction methods exist for doing seismic ray tomography. The transform methodsin Section 2.2 comprisethe first group. Applicationsof transform methodshave their roots in astronomical and medical imaging problems. They are very limiting as far as seismic imaging problemsare concernedsince straight raypath propagation and full-scan aperture are generally assumed.However, the transform methods make an excellentintroduction to the principlesof tomographybecauseof their simplicity and serve as a bridge between applications of tomography in other fields with applicationsin seismology.Also, the developmentof seismicdiffraction tomography has a closerelationship with the transform methods. The series expansionmethodsin Section 2.3 comprisethe second groupof imagereconstructionmethods. Out of all the methodspresentedin this book the seriesexpansionmethodspresently seethe most usein seismic tomography. Therefore, a large part of Chapter 2 is spent addressingthe seriesexpansionmethods. Beforeproceedingfurther one shouldhave a good graspof the Fourier transform conceptsto understandthe material in Section 2.2. Appendix B
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10
CHAPTER
2. SEISMIC
RAY TOMOGRAPttY
.---i'""• Ox-ray transmitte -
r
FIG. 3. Setup for a medical tomography experiment. X-ray scansare taken
in differentdirectionsabout a person'shead by rotating the transmitterdetector assembly.
presentsa review of the Fourier transform.
2.2
Transform
Methods
The projection slice theorem is presentedfirst in this sectionsinceit providesthe theoreticalfoundationfor the transformmethods. Then, two transform methods are derived from the projection slice theorem' directtransformray tomographyand backprojectionray tomography.
2.2.1
Projection Slice Theorem
The derivationof the projectionslicetheoremis illustratedby a typical medicaltomographyexperiment. Figure 3 showsthe setup for medicaltomography.A donut-shapedx-ray transmitter-detectorassemblysurrounds the target, a person'shead in this example. X-ray intensityis measuredfor a fixed orientationof the assembly.Then the assemblyis rotated about the personso that x-rays pass through the head in a different direction. The experimentis completedwhen the person'shead is scannedin all directions. The objective of the transform methods is to use attenuation information
from the measuredx-ray intensitiesto reconstructa cross-sectional image
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2.2.
'fRANSFORM
METHODS
11
of the person'shead in the z - z plane which containsthe transmitters and detectors. Thus, a brain tumor which attenuates x-rays differently than normal tissuesmay be readily "seen"by the radiologist.
Figure 4 depictsa cross-section of a person'shead. The varying contrasts within the target representnonuniformx-ray attenuation associated with a tumor, normal tissues,and the skull. For the purposeof deriving
the projectionslicetheorem,we definethe modelfunctionM(z, z) • as the spatial distribution of the attenuation. In general, the model function representsthe unknown distribution in spaceof some physical property of the
target medium which affectsthe propagatingenergy in some observable manner. A typical model function usedin seismictomographyis the reciprocal compressional-wavevelocity, or slowness,which has a direct influence on the observedtraveltime of the propagatingenergy. The projectionslicetheoremrequiresthat observationsof the propagating energy be taken along a given projection which is perpendicular to the raypaths. Figure 4 illustrates a singleprojection in the medical tomography experiment. X-rays emitted by the transmitters travel along the parallel rays and are recordedby detectorspositionedalongthe u-axis. The rotated
spatialcoordinatesystem(u,v) is introducedto describeall of the possible orientationsfor the transmitter-detectorassemblyabout the target. The v-axis is defined parallel to the direction of x-ray propagationand the uaxis, defined perpendicular to the v-axis, is the direction along which the x-ray intensity is measured. If the u- v coordinatesystemsharesthe same origin as the z- z coordinate system, then the relationship between the two coordinatesystemswhen one is rotated through an angle 0 relative to the other
is
z
sin 0
cos 0
u]
ß
(1)
For a givenray in Figure 4 we defineP(u, O) as the decimalpercent drop in x-ray intensity,
P(u, o) =
[.'o-
0)l/o,
where I(u,O) is the intensity measuredby the detector at (u,O) and Io is the x-ray intensity at the transmitter. We refer to P(u, O) as the data 1Other literature on tomographymight refer to the model function as an image function or as an object function. We chose "model function" to be consistent with inverse problem terminology.
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12
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
x X-RAY
SOURCES
v
P(., O)
Z
U
FIG. 4. Cross section of a person's head where varying contrasts repre-
sentnonuniformx-ray attenuation.The projectionP(u, t•) is the decimalpercent drop in x-ray intensity measuredalong the rotated coordinateaxis, u. The u-axis is perpendicularto the v-axiswhich alwaysparallelsthe x-ray
propagationdirection. The model function M(x, z) providesa numerical value
for the attenuation
and is an unknown
from the observedprojectionsP(u, •).
which
must
be determined
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2.2.
TRANSFORM
METHODS
13
function2. If the attenuationis small overthe raypath, then data function P(u, O) is linearlyrelatedto the attenuationM(z, y) as the line integral,
e(,,, o)-
(2)
ay
taken overthe raypath.s Note that eachobservationof the data functiona P(u, O)providesan empiricalsolutionto equation(2) alongthe givenraypath without actually knowingthe model function M(x, z). If there is negligiblex-ray attenuation outside the target, then equation (2) givesthe same measuredprojectionfor any transmitter-detector separationas long as each transmitter and detector remainsoutsideof the
target. We usethis assumptionto rewriteequation(9•)with infinitelimits; a mathematical step which will be taken advantageof later in this section. Thus,
0)- f::
z)a.
(a)
To obtain a simpler and more meaningful relationship between the
modelfunctionM(x, z) and the data functionP(u, 0), we transformeach into the Fourier
domain.
The 2-D Fourier
transform
of the model function
M(z, z) is
117I(k•, k,.)- f;: ];: M(x, z)e-J(k•x +k,.z)dxdz ' (4) where k• and kz are spatial frequenciesalong the x- and z-axes, respectively. Spatial frequency,or, wavenumber,is definedas k = 2•r/A where is wavelength. Figure 5 representsthe 2-D Fourier transform's amplitude
spectrum • ofa hypothetical model function M(x, z). Notethatif •(k•, 2P(u, O)is a projection in the ray tomo•aphyproblem,but is c•ed a data]unction in inv•e
theory te•nology.
The,
we c•
the v•iable P a "data f•ction"
sine re.on we c•ed M(x,y) the "modelf•ctioff' data f•ction
with the v•iable
P to re.rid
for the
e•Ser. However,we representthe
you that the me•ed
data •e projections.
awe o•y considerthe 5ne• inv•se problem in t•s book. Thus, the data f•ction will •waya be •ne•ly related to the model f•ction, even if • approximation is req•red to force the •ne• relatio•p. The •amption of am• x-ray attenuation is req•red for the x-ray tomo•aphy problem.
•Although the actuMobservation is the x-ray intensityl(u,O) in t•a c•e, we wi• frequently refer to "the observation of the data f•ction" from
the observed
since it is in•rectly
obt•ned
data.
•Althoughthe2-D Fo•ier tryfore of themodelf•ction •(x,z) h• bothmp•tude •d ph•e spectra,we representthe 2-D Fo•ier tryafore M(kx,k•) with o•y the
mpftude spect•
component, desi•ated • ]•(k,,
k,) I.
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14
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
K FIG. 5. The essenceof the projection slice theorem is represented. The
2-D Fouriertransformof a hypotheticalmodelfunctionM(x, z) produces the amplitudespectrumI M(ks, k,) [. The contoursin the ks - k, plane connectequal valuesof amplitude. The amplitudespectrum[ P(fi, d) [ from the 1-D Fouriertransformof the data functionP(u, t•) representsa
sliceof [ J17/(ks, k,) [ alongthefl-axisin theks- k, plane.
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2.2.
TRANSFORM
METHODS
15
is known,then the unknownmodelfunctionM(z, z) can be foundby the 2-D inverse Fourier transform
M(x,z) = 1fj: fj: 11•'I(k• k,)eJ(k•x +k,Z)dk•dk, (5) 4•r2
,
ß
Nowlet•5(f•,0) represent the1-DFourier transform ofthedatafunction P(u, O) alongthe u-axis,shownin Figure4. The 1-D Fouriertransformis written
0)- f/: where f/ is spatial frequencyalongthe u-axis. Substitutingequation(3) into equation(6) gives
We nowwishto put equation(7) entirelyin termsof z and z. The variable • is replacedwith z and z usingthe inverseof equation(1) givenby v
- sin 0
cos 0
z
'
Usingequation(8) andreplacing dvduwith dzdzin equation(7) weget,
•5(f•, O)-- f:: f:: M(x, z)e-Jf•( xcos 0+zsin O)dxdz = f:: f:: M(x,z)e-j[(f•cosO)x +(ftsinO)z]dx ' (9) Comparingthe integrands of equation(9) andequation(4) weseethat equation(9) is simplythe 2-D Fouriertransform of M(x, z) wherekx and k, are restricted to the Q-axis by setting k•
=
•cos0, and
k,
=
f/sin0.
(10)
This relationship is evident in Figure 5.
Substitutingequation(10) into equation(9) we write
•5(fi, O)- /:: /:: M(x,z)e-J(k:•x +k,Z)dxdz, (11)
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16
C•APTER
2.
SEISMIC
RAY
TOMOGRAPHY
whereks and kz are definedby equation(10). Comparingthe integrands of equations(11) and (4) showsthat we haveachieveda simplerelationship betweenthe data functionP(u, O) and the modelfunctionM(x, z) in the spatial frequencydomain,
P(•, O) -
A7/(k,,, kz).
(12)
In words,equation(12) statesthat the 1-D Fouriertransformof the pro-
jectionrepresented bythedatafunction P(f],_0)isequalto onesliceofthe 2-D Fouriertransformof the modelfunctionM(kz, k•) definedon the loci: kz = f] cos0 and k, = f]sin 0. Equation(12) is calledthe projectionslice theorem.
The projection slice theorem givesonly one slice of the model function per projection as shown in Figure 5. We will now show how many projections at different angles of 0 are used to reconstruct the entire model function via the projection slice theorem. The two techniquespresented are the direct-transform ray tomography method and the backprojection
ray tomographymethod. In Chapter 3 we will define an analogoustheorem for the reconstructionmethods in diffraction tomography called the generalizedprojectionslice theorem.
2.2.2
Direct-Transform Ray Tomography
Direct-transform ray tomography utilizes the projection slice theorem in a straightforward manner. We showedin the previous section that the application of the projection slice theorem to the 1-D Fourier transform
of a singleprojection represented bythedatafunction /5(12, 0) determines onlyonesliceofthemodel function A•(k•- 12cos0,k• - f] sin0). Figure 5 illustrates such a slice through the model function. To recoverthe entire model function, the target must be probed from many different directions. Figure 6 showsthree directions along which x-rays probe the head of our make-believepatient. The observeddata functionsfor thesethree pro-
jectionsare P(u, 0•), P(u, 02), and P(u, 0a). After applyingthe projection slice theorem to the 1-D Fourier transformsof these data functions,we obtain the three slicesthrough the model function's amplitude spectrum shownin Figure 7. Now the 2-D Fourier transform of the model function M(k•, k,) is better definedthan by the singlesliceshownin Figure5, but is still inadequatefor imagereconstruction.We must probe the target with x-rays from all directionsletting 0 range from 0 degreesto 180 degrees. Only then will the k•- kz plane in Figure 7 be completelycoveredby slices
M(f] cos0,f]sin0), where0 rangesfrom 0 degreesto 180 degrees.After such an experiment, the complete 2-D Fourier transform of the unknown
modelfunctionM(x, z) is determinedalongradial linesin the k•-k, plane.
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2.2.
TRANSFORM
METHODS
17
•X
Z
02 FXG. 6. The cranium of our make-believepatient is probed with x-rays in three different directions: 0•, 09.,and 03. Application of the projection slice
theorem to the data functions resulting from thesc projectionsyields the slicesdepicted in Figure 7.
ToobtainM(z, z) from.g/(f•cosO,f• sin0), analgorithm employing the direct-transformray tomography method must first interpolate the data
from a polar grid (f•cosO,f•sinO) onto a Cartesiangrid (k•,,kz) in the k• - kz plane, or
AT/(f•cosO, f•sinO)in,•r__•ot.,• A•(k•,kz).
(13)
One must exercisecaution in performingthe interpolation sincelarge errors introduced by the operation could obscurethe true solution.
Lastly, a 2-D inverseFouriertransformof M(k•,,k;•) is performedto obtain M(x, z),
M(z, z) = 4•rI 2 '•,•
,•
.•l(k• k,)eJ(k•z 4-k•Z)dk•,dk ' ß (14) '
Thus, the image reconstructionis completedand the technicianmay give the tomogram of the patient's head to the radiologistfor interpretation. The direct-transformray tomographymethod is easilysummarizedin five steps:
Step 1: Acquirethe data functionP(u, O) of the target with the projectiondirection,0, rangingfrom 0 degreesto 180 de-
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18
CHAPTER
2. SEISMIC RAY TOMOGRAPHY
M(•cosO 3, • sin63 )1
• sin61 )
K
Kz F•(•. 7. Plot showing slicesthroughthe unknownmodelfunction'samplitudespectrumI M(k=, k•) I foundby applyingthe projectionslicetheorem to the data functionsfoundfor the x-ray projectiondirections 01, 0•, and 03 shownin Figure 6.
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2.2.
TRANSFORM
METHODS
19
grees.Rememberthat the unknownmodelfunctionM(z, z) correspondingto the target representsa physicalproperty which affectspropagatingenergyin somemanner(e.g., attenuation which affectspropagatingx-ray intensity in medical tomographyor slownesswhich affectsseismicwavetrav-
eltimesin seismictomography). Thus, a data functionis just the line integral of the model function along each ray, or
Step 2: Perform a 1-D Fourier transform along the u-axis for each data function given by
P(n, o)- f:: o)-J Step 3: Use the projection slice theorem to obtain slicesof the 2-D
Fourier
transform
of the model
function.
Each slice is
defined by
cos0,iqsin0) -- J5(i2,0). Step 4: Convert the 2-D Fourier transform of the model func-
tion in the k• -kz planefrom a polar grid (f• cos0, f•sin 0) to a Cartesiangrid (k•, kz),
Step 5: Perform a 2-D inverseFouriertransformon M(k•,, k•) to obtain M(x,z), the reconstructedimage of the target. The inversetransform is given by
47i-2
The direct-transformray tomographymethod would be quick to implement if it were not for the fourth step above requiring interpolation of the model function in the frequency domain. In the next section we present backprojectionray tomographywhich obviates the need for interpolation resultingin a faster and more accuratealgorithm.
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CHAPTER
2.2.3
Backprojection
2.
SEISMIC
RAY
TOMOGRAPtIY
Ray Tomography
Backprojection • ray tomography usesthe samedata acquisition scheme asthe direct-transform ray tomography:recordthe data functionP(u, O)by experimentallymeasuringthe line integral of the unknown mode] function
M(:r,z) along differentraypathsor projections. The differencebetween backprojectionray tomographyand direct-transformray tomography is how we compute the model function from the data function. To derive the backprojectionray tomography method, we first write
downthe2-D inverse Fouriertransform of themodelfunction•(k•, k,),
•(• ' z) = 4•2•
•
•(• ' •)•j(•+•z)••
(•5)
ß
Next we makea changeof variablesin equation(15) by replacingk• with • cos0 and k• with •sin 0, and by changingthe integration from dk•dk,
to ] • l d•dO.• This gives,
M(x,z) = 4•2 M(•cos0,•sin0) • ej•(xcosO + zsin 0)I•ld•dO.
(1•)
Integrationwith respectto 0 in equation(16) can be rewritten • two integrals,
1 • M(r,z) = 4=•
• (• cos 0,• sin 0)
• ej•(xcos 0+ zsin0) I•ld•dO +•
• '
•[• •o•(0 +•), • •in(0 +•)]
• •j•[• •o•(0+ •) + z•in(0 + •)] I• [•0.
(17)
Using the fundamentaltrigonometricangl•sum relations,cos(0+ •) = - cos0 andsin(0+ •) = - sin0, we may rewritethe secondmodelfunction •The te•
"b•mjection"
imp•es the inve•e problemwherewe st•t with the pr•
jection •d •lve for the model f•ction. Here the projections •e t•en Mong raypat•. • Section 3.4.2 the me•ed projectio• of scattered energy •e described by the wave equation •d we •e • •Mogo• te•, "backpropagation."
•The ch•ge • inte•ation is •Mogo• coor•n•es is the r•M
to compute the •ea of a •sk. •st•ce
from
c•e to prese•e the si• negative vMu•.
the •sk's
to goingfrom C•tesi•
coorSnatesto pol•
For a •sk we replace dxdz by rdOdr, where r
center.
The
absolute
vMue
of •
is t•en
of the •fferentiM •ea when we will shortly •ow
in o•
• to t•e
on
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2.2.
TRANSFORM
21
METHODS
on the right-handsidein equation(17) as,
.g/[f•cos(0 + •r),f•sin(0 + •r)] = .tfl(-f•cosO,-f• sinO).
(18)
The secondsetof integralson the right-handsidein equation(17) is rewritten by replacingthe modelfunctionwith equation(18), applyingthe anglesum relationsusedin obtainingequation(18) to the exponential,setting f• - -f• and dfl - -dfl, and reversingthe direction of integration with respectto fl. With theseoperationsequation(17) is written,
M(x z) = 4•21 • ,
•(fl cos 0,flsin 0)
x ej•(xcos 0+ zsin0)
+•
•(•
cos•, • sin•)
• •j•(• •o•• • • •i. •) I • I ••. Combiningthe integralswith respectto the variable• we get, '
4•
'
• d•(• •o•• • • •i. •) I • I ••.
(•)
Usingthe projection slicetheorem,wereplaceM(• cos•,•sin •) in equastruction
formula
ß
,
4•2
'
We cansummarizethe backprojection ray tomographyreconstruction method in just three steps:
Step 1: Data acquisition.Let the modelhnction M(x, z) representthe unknownparameter(such• seismicwaveslow-
ness)at position(x, z). Experimentally determinethe line integralof the modelfunctionalongeachray whichyieldsa setof datahnctions(such• seismic wavetraveltime),
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22
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPttY
Step 2: Take the 1-D Fourier transform of each data function
along the u axis.
P(fl, O)- /:: P(u, O)e-Jfl udu. Step 3: Use the backprojectionformulaequation(20) to compute the unknownmodelfunctionM(x, z), or
=
4•.2
x
'
+
[ I
Unlike direct-transformray tomography,backprojectionray tomography does not require a 2-D interpolation in the wavenumberdomain, and therefore,is in generalfaster and more accuratethan direct-transformray tomography. It shouldbe mentionedthat mostcommercialCAT (ComputerizedAxial Tomography)scannersuse the backprojectionray tomographyor its modificationas their image reconstructionalgorithm. 2.3
Series Expansion Methods Seriesexpansionmethods comprisea group of computation algorithms
which,like the transformmethods,determinethe modelfunctionM(x, z) of the target area. However,unlike the transformmethods,thesealgorithms easily allow curved raypath trajectories through the target area and are therefore well suited for applicationsin seismictomography. As before, we
restrictthe discussion to a 2-D problemsothat the modelfunctionM(z, z) is determined in a plane which cuts through the target and containsall of the sources and receivers.
Our discussionof the seriesexpansionmethods is divided up into three subsections. Section 2.3.1 presentsthe forward modeling problem which permits us to predict the tomographydata in terms of a system of linear equationswhich explicitly contain an estimate of the true model function. Section2.3.2 showshow the true model function is determinedusing Kaczmarz' method. The method devised by Kaczmarz in 1937 is iterative and determinesan approximate solution to the true model function. Drawing an analogy,the Kaczmarz method is to the seriesexpansionmethod as the
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2.3.
SERIES
EXPANSION
METHODS
23
projection slice theorem is to the transform methods. We exploit Kaczmarz' method in Section 2.3.3 to derive two seriesexpansionalgorithms:
the algebraicreconstruction technique(ART) and the simultaneous iterative reconstruction technique(SIRT).
2.3.1
The Forward Modeling Problem
As will be shown shortly, the seriesexpansionmethod iteratively up-
datesan estimatedmodelfunctionM e't so that it converges towarda true model function M true. The updates are found by comparingthe observed data functionpolo with a predicteddata functionppre. Forwardmodeling is required to determine the predicted data function and is the subject of this section.
Equation (2) in Section2.2 definesthe experimentalprocessfor the transformmethodsas the line integral of the function M(a:,z) along a straight raypath in the v-axis direction. Accordingto the abovenotation equation(2) couldbe written,
Pøbø(u, O)-- i Mt"u•(z' z)dv' ay
We did not requirea forwardmodelingprocedurein Section2.2 becausethe raypathswere straight and the projectionslicetheoremcouldbe employed
directlyto determinethe true modelfunctionMtrue(x,z). For the seriesexpansionmethodswe wish to include curvedraypaths.
Equation(2) is easilytransformedto accommodate curvedraypathsby rewriting the model function in terms of a position vector r. Thus, for a
givensource-receiver pair the line integralof the modelfunctionM(r) over the raypath is
pobo = f• MtrU•(r)dr, ay
wherethe observed projectiongivenby the datafunctionpolo represents the measured lineintegral(observed tomography data) and MtrUe(r) is the true model function which remains to be determined. The last equation is used to formulate the forward modeling by setting
P- i M(r)dr, ay
(21)
where P is now the predicteddata function and M(r) is the estimated
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CHAPTER
24
2.
SEISMIC
RAY
MI
M2
M3
M4
M5
M7
M8
M9
M10 Mll
TOMOGRAPtIY
M6 M12
M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24 FIG. $. The seriesexpansionmethodsusea discretemodelfunctionMj, j - 1,..., J, whereMj is the averagevalueof the continuous modelfunction M(r) within the jth cell. Here J - 24.
modelfunctions. Thus, forwardmodelingis definedas determiningthe predicted data function from the line integral along the raypath through a known, but estimated, model function. Just as was done with the transform methods, the model function in the seriesexpansionray tomographyis discretizedto allow computationby digital computer. Figure 8 showsan image area of a target divided into many small cells. Each cell is assignedthe averagevalue of the physical parame-
ter (e.g., x-ray attenuation,slowness, etc.) represented by the continuous modelfunctionM(r) within that cell. The modelfunctionin Figure 8 is divided into 24 cells and is written discretely as M•, where j - 1,..., 24.
Thus, Mj represents the averagevalueof M(r) within the jth cell. Figure 9 depicts a single ray traveling through the discretizedmodel function. Equation(21) is rewritten in discreteform, to describeray travel through the discretemodel function, as J
Pj=l
whereMj is theestimatedmodelfunctionfor thejth cell,,5' i is the raypath SHere we will symbolize the predicted data function as P and the estimated model function a• M for brevity. Thesesymbolswill be changedto ppre and M est, respectively, in the following section.
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2.3.
SERIES
EXPANSION
•
Z
METHODS
25
r
7
6
..•_1.. ••
source
receiver • •._•..1•.. ••15 M16M17IV116 M19 M20 M21 M22 M23 M24 FIG. 9. Ray travel through a discretemodel function. The resulting data function, determinedby the line integral through the discretemodel function, is definedby equation(22).
length of the ray within the jth cell, and J is the total number of cells in the gridded target. The example in Figure 9 has J - 24 cells, but the ray
penetratesonly sevencells(j - 12, 11, 10, 16, 15, 14, and 13). To keep equation(22) consistentwith equation(21) we set Sj - 0 for all cellsnot penetratedby the ray. After all, the ray'spath lengthSj for the jth cell is obviously zero if the ray did not traverse that cell. Figure 9 shows17 cellsfor which we don't have information becausethe singleraypath did not traversethem. By addingmore sourcesand receivers around the unknowntarget region, differentrays samplethe 17 unsampled cellsin addition to someof the cellsalready sampled. The addition of extra rays is depicted in Figure 10. Now all of the cells are interrogatedby this network of rays.
We must modify the index notationof equation(22) to includea projection value for every ray. If Pi representsthe projection, or line integral,
predictedfor the ith ray, then equation(22) is rewritten, J
Pi = E Mj$ij,for/- 1,...,1,
(23)
whereI is the total numberof rays,$ij is the path lengthof the ith ray throughthe jth cell, and, asbefore,Mj is the discreteestimateof the model functionfor the jth cell and J is the total numberof cells.Equation(23) is
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26
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
source
ray I
receiver ray 2
!
ray I
FIG. 10. Generally, a single ray does not provide information on all of the model function's cells. However, by using more source and receiver locations around the target all of the cells can eventually be sufficiently interrogated. I rays were found sufficienthere.
the formulation of the "forward modeling problem" usedin seriesexpansion ray tomography.
Equation (23) can effectivelymodel the data acquisitionprocessif we let the projectionsPi, i - 1,..., I, be the observeddata (i.e., traveltime or decimalpercentdecreasein x-ray intensity)and the modelfunction j: 1,..., J, be the true, but unknown model function, or J
Piø•' = • M]"•'*Sii, fori- 1,...,I.
(24)
j=l
Kaczmarz' method providesthe theoreticalframeworkfor indirectly solving equation(24) for the true modelfunction. 2.3.2
Kaczmarz'
Method
In this section we introduce Kaczmarz' method to indirectly solve
equation (24)forthetruemodelfunction M]ru*,j = 1,..., J, whichisthe tomogram. As stated in the previoussection,forward modelingis required to determine the true model function. Thus, before proceedingwe will reformulateequation(23) into a matrix form to simplifythe mathematical discussion.Sinceequation(23) is discreteits elementsare easily put into matrices.In matrix form equation(23) becomes,
P
-
SM,
(25)
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2.3.
SERIES
EXPANSION
METHODS
27
where the predicted projections in data vector P are, P1
P
-
.
,
Pl
the discrete estimated model function values in model vector M are, ml
m
-
.
,
Ma
and the raypath lengthsfor I rays and J cells in S are, S• S•.•
s
S•. S•
-
... ...
S• S•
. ß
S•i
S•2
-..
Note that S in equation (25) can be thought of as a linear operator that operates on the estimated model vector M producing the predicted data vector
P.
We couldalsoformulateequation(24) in matrix form as
po•, = SM'•.
(29)
Althoughwe will not directlysolveequation(29), we wouldwant to determinethe true modelvectorM truegivenpob•and S. The problembecomes oneof findinga generalized inverseoperatorS-•.9 Then wecouldapplythe generalizedinverseoperatorS-• to both sidesof equation(29) to determine the true model vector, or S-ap oh, = m
S-aSM '•e M true .
Theoretically the last equation is true, but in practice it is very often difficult to determine S-a for two reasons. First, S is usually quite large and 9We write
S-g
rather
squaxe and because S-9S
tha•n S -1
as in usual matrix
notation
is not always the identity matrix.
since S-g
need not be
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28
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
sparse,which makes computation of S-• costly. Second, S is usually "ill conditioned"whichmakescomputationof S-• very unstable.iø Kaczmarz' method circumvents the problems associated with the inversion of a large and sparsematrix and provides an efficient means for determiningan approximatesolutionto equation (29) using an iterative procedure. Figure 11 presentsa flowchart outlining the method in which there are three basicstepsto the iterative part of the algorithm. An initial
estimateof the modelvectorM init is input to the iterativeloopof the algorithmandservesasthe first "currentestimate"M •st of the true solution M true. For now we assume that the initial model vector M init is known.
However,morewill be saidon the selectionof the initial modelvectorM in the casehistoriespresentedin Chapter 4.
With the current estimate of the model vector M est known, the first step is to usethe forwardmodelingproblemdefinedby equation(25) to determinea predicteddata vectorppre. This stepis carriedout by applying the linearoperatorS (determinedby someray tracingtechniqueof personal choice)definedby equation(28) to the estimatedmodelvectorM •s•,
ppr• = SM•t.
(30)
In the second step the predicted data vector PPr• is compared with
the observed data vectorpob, by takingthe difference betweenthe two. A small differenceor good agreementbetweenthe predicted and observeddata vectorsimpliesgood agreementbetweenthe estimated model vector M • and the true model vector M t•"•. Thus, if the differenceis smaller than a specifiedtolerance, then the current estimate of the model vector M •t is
output as the solutionto equation(29) in the final step of the algorithm. The selection of a suitable tolerance for the difference is discussed with the
case histories in Chapter 4. The third step of the iterative portion of Kaczmarz' method comesinto play when the difference between the predicted and observed data vec-
tors is larger than the specifiedtolerance. This important step essentially
makesuseof the differenceinformation,pob•_ pp,.e,to updatethe current estimated model vector M •
with a new estimate of the model vector
M("ew)• whichhopefullyis closerto the true modelvectorM •r"•. This third step is written in equation form as
M ("•w)•t = M e'•+ A/M, fori
-
(31)
1,...,I,
Ill conditioned meanssmallchanges in S producelargechangesin the modelfunction true or in S-g howeveryou wishto look at it
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2.3.
SERIES
EXPANSION
METHODS
29
Initial estimate
Minit
currenf estimafe est
Step
1
predicted
observed
data vector
data vector
Ppre Step
2
Step
3
pObS
FIG. 11. Flow chart for Kaczmarz' method. M i"i' is the initial estimate
of the model vector; M e'• is the current updated estimate of the model vector; ppre is the predicted data vector from the forward modelinggiven
by equation(30); andpobois the observed data vector.M eø•is iteratively updateduntil ppre matchespoboto within a specifiedtolerance.
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CttAPTER
3O
2.
SEISMIC
RAY
TOMOGRAPttY
whereAiM represents the incrementalupdate to the currentestimateof the modelvectorand superscripti meansapplyingequation(31) whenthe ith row of the pob• and ppre vectorsare compared. The new estimate M (new)e•t is then taken as the current estimate for the next iteration. As
you will soonsee,equation(31) bringsthe current modelestimatetoward the true solution, at least in theory.
The methodof computing AiM in equation(31) is obviously a crucial factor
in the success of Kaczmarz
'method.
Kaczmarz' method computes
AiM with the equation' AiM1 A i M2
-
.
,
(32)
ß
AiMj where
pi.øb s __piPre
(33) Note that the summation in the numerator is just the predicted data vector
ppre found in the forward modelingin equation(30). Now we will geometricallyderiveequation(33) and showthe convergenceof Kaczmarz'methodthrougha simpleexample.Let two rays(I = 2) travel througha two-cellmodel(J = 2) so that the vectorequation(29) for the problem can be written as
p•,b, = Sll M1 + S12M2,for ray 1, and p•b, __ S21 M1+ S22M2,forray2,
(34) (35)
whereP•'b•and P•*• are observed data and M1 and M2 are unknown TM. Rememberthat $i1 is just the ith ray's path lengththroughthe jth cell and is generallyknownfrom the forwardmodeling.We plot equations(34) and (35)in Figure12 as lines(hyperplanes) on a 2-D spacewith axesM1 and M2 .12 The solutionto the equationsoccursat point X where the two • • Here M• and M2 are unknown and therefore defined as independent variables. Only
whenthe solutionis foundare theyreferredto as M[ •ue andM• •ue as in equation(29). •2If I = 3 and J = 3, then we would be looking for the intersection point of three planes in a 3-D model space. For situations where J > 3, equation (29) representsa J-dimensionM space and we would look for the solution at the intersection of I = J
hyperplanes where a hyperplane has J - 1 dimensions. Note that we must have at least I = J hyperplanesto solve equation (29) and generally I > J.
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2.3.
SERIES
M2
EXPANSION
METHODS
31
obs
M2= P1 o\ ,,....__/ S12
•
obs
H
12M2
FIG. 12. For two rays (I = 2) in a two-cellmodel (d = 2), possiblevalues
for M• andM2 aredefined by [hetwolines(hyperplanes). p•,b,and are [he observed data from [he two rays. Point X represents[he desired
modelvalueswhichlie at the intersection of the twohyperplanes.A• M• and A•M2 for ray 1 are geometrically derivedsothat point B is the projection of point A ontothe hyperplanedefinedby equation(34). The resultof this geometricalderivationis equation(33).
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32
CHAPTER
2. SEISMIC
RAY TOMOGRAPItY
lines(hyperplanes) intersect. Sincethe solutionat point X with coordinates(M•"•'e,M•"•'e) is unknown, we must start with an initial estimate at point A given by the coordinates (M1A, M•A). Thisinitialestimatebecomes the currentestimate as shownin the flow chart for Kaczmarz'methodin Figure 11. The geometricalstepin Figure 12 is to find the perpendicularprojectionof point A
ontothehyperplane •:•defined byequation (34) at pointB. Mathematically this step is given by,
s = M+AM,and
(36)
whereA•M• and A•M2 are the corrections soughtgeometrically and defined by equation(33) for i- 1 and j- 1,2. The first geometricalrelationshipto note in Figure 12 is the similarity of trianglesAABC, AFED, and AGEH. Using these similaritieswe can immediately write
AIM• = AB cosc•= DFcos c• =
EF cos2 c•
=
__GH • EF '•E:Z'
=
EF
A • M2 = AB sinc•= DF sinc•
--GH
and
(38)
cos c• sin c•
EH
= EFG---•• .
(39)
Our task now is to determineEF, GH/•--•, and EH/GE in equations (38) and (39).
On the line segmentEF, point E is located at
(M• = P•'b'/sI•,M2 = 0). PointF is the intersection with the M•-axisof a linewhichis parallelto equation(34) andcontains the point(MIA,M•). The equation for this line is
S11M• A + S•2M•A - S••M1 + S1:•
(40)
Fromequation(40) we determinethe coordinateof point F alongthe M•-
axisas (M• - (SI•M1A+ S•2M•A)/S•I,M2 - 0). Thus,the lengthof the line segment EF is given by 1
(41) laAlthoughequations(34) and (35) both representlines,we will continueto refer to them as hyperplanes since that is what they are called when J > 3.
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2.3.
SERIES
EXPANSION
METHODS
33
The ratio GH/GE is simply, GH GE
(42)
ß
Similarly, EH/GE
is givenby,
EH
P•ø/Szz +
=
S12 .
(43)
+
results in the model corrections due to the first ray,
p•b•
I:•p re
i and AiM1-- •'!1•_--' S121 .•.S122 p•bs I•Prß
- -• AiM2- $1•$•---•+$• ,
(44)
(45)
where,rre _ SzzMi• + $z•M• t ß Equations(44) and (45) are the sameas a I
equation (33) wheni = 1 andj = 1,2. Thus,weseethat equation (33) simplydeterminesthe projectionof a modelestimateonto oneof the hyperplanesdefinedby equation(29). Carryingthisexampleonestepfurther,wecandeterminepoint I in Figure 12, the projectionof point B ontohyperplane2 definedby equation(35) for the secondray usingthe indicesi - 2, j - 1, 2 in equation(33). If we alternateprojectionsof the modelestimatesbetweenthe two hyperplanes, then the updatedmodelestimates(step3 in Figure 11) must convergeon point X as depictedin Figure 13. Thus, Kaczmarz'methodwill converge to the solutionof equation(29). 2.3.3
ART
and
SIRT
The algebraicreconstruction technique(ART) and the simultaneous iterativereconstruction technique(SIRT) are the two commonimplementations of Kaczmarz' method in seismicray tomography. This sectiondescribesthe basic features of both algorithms.
ART is a computational algorithmfor solvingequation(29) that directly uses Kaczmarz' method. Thus, the ART algorithm is comprisedof the
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34
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
M2
I
Hyperplane2
:-M I Hyperplane
I
FIG. 13. By applyingequation(31) to alternatinghyperplanes,the model estimateof (M•, M2), starting at point A, convergestowardsthe solution for equations(34) and (35) at point X. The iterative updating of the model estimate correspondsto the loop in Kaczmarz' method shown in the flow chart in Figure 11.
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2.3.
SERIES
EXPANSION
35
METHODS
stepsshownin Figure11. Wefirstsettheestimated modelfunction M• 't,
j-
1,..., or, to theinitialmodelestimate Mj"i' j - 1,
or Thenthe
followingthree stepsare iterated cyclicallyfrom one hyperplaneto the
next until the observed data p/ob,matches the predicteddata pfre, for i=
1,...,I.
Step 1: Conductforwardmodeling(ray tracing)for the ith ray usingequation(23) or equation(25), restatedfor reference here as, J
Only one ray is traced out of a total of I rays sincewe are determining the projection of the current model estimate onto only one hyperplane. Note that if our model consists
of slownesses, thenthe predicteddata P•'reare calculated traveltimes fromtheforwardmodeling andp/oh,areobserved traveltimes.
Step 2: Subtractthe predictedith ray data p•0•, from the observed ith data p/o•,,anduseequation(33) to findcorrectionsfor all of the J cellscomprisingthe model function estimateTM,
=
'
Step 3: Apply the correctionsto the model estimaterecommended by the ith ray to all orcells,
SIRT differsfrom ART in that all I rays are traced throughthe model
sothat all AiMj corrections determined forthe I hyperplanes areknown. 14Note that the model adjustment dependsupon the discrepancybetween the predicted aaadobserveddata values and the raypath length through the cells for the ith ray.
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36
CHAPTER 2. SEISMIC RAY TOMOGRAPtIY
Thenan average of AiMj withrespect to indexi is takenfor eachmodel
cellj - I , ..., J,togetnew model estimates
,j-1,...,J.
As
withART,themodel estimates M[•t areupdated untilthepredicted data Pf•ecompares favorably withtheobserved datap/oh,, i- 1,..., I. After settingthe currentmodelfunctionestimateequalto the initial
model function, orMf øt- M'. i"itforj = I
are iterated to update the model estimates:
J thefollowing threesteps
Step1: Conductforwardmodeling (ray tracing)usingequation (23)or equation(25),
for all raysi
-
Step 2: Findthecorrection foreachcellbyexamining therays cut throughthat celland averaging the corrections recommendedby eachray. Thisoperation is definedfor the jth cell by, I
= W,..= 1•AiM 1 _
forj
-
1
I
BlObS 1 s,s • - EsS=
(46)
i=1
1,...,J.
The weightWj is the numberof raysintersecting the jth cellor someothersuitableraydensityweightusedto obtain an averagecorrectionAMj.
Step 3: Determine thenewmodelestimate fromthe average modelcorrections AMj, or
M«"ew)e" = M;"+AM./,j - 1,...,J. Figure14illustrates howequation (46)makes SIRTdifferent fromART.
Asin Figure 12weuseonlytworays(orI - 2 hyperplanes) andtwomodel cells(or J - 2 modelspace) in orderto visualize theproblem.TheART algorithm isshown inFigure 14(a)andtheSIRTalgorithm inFigure 14(b).
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2.3.
SERIES
EXPANSION
M2
37
M2
E
.
D A
METHODS
•
C B Hyperplane I
(a)
(b)
FIG. 14. Comparisonof ART andSIRT algorithmsfor the two-ray(or I 2 hyperplanes) and two-cellmodel(or J - 2 modelspace)examplegiven in Figure 12' a) Convergence of model estimatesfor the ART algorithm startingwith an initial modelestimateat pointA. b) Convergence of model estimatesfor the SIRT algorithm starting with an initial model estimate at point A. The iteratively updated model estimates, determined from the
averagecorrections AMj in equation(46), are alongthe solidline defined by points A, B, C, D, and E. Each estimated point is the average of the same letter's primed and double primed projection points located on hyperplanesI and 2, respectively.
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CHAPTER
38
2.
SEISMIC
RAY
T¸M¸GRAPHY
The ART algorithmin Figure 14(a) finds the solutionby alternatelyprojecting the current model estimate onto each hyperplane. The model estimate moves along the solid line from the initial model estimate at A to B, to C, to D, and so on. On the other hand, the SIRT algorithm finds point A's projections on both hyperplanes,points B t and B •t, then moves the initial estimate from point A to point B, the midpoint between B t and Btt. For the next iteration the SIRT algorithm finds point B's projections on both hyperplanes, points C t and C", then moves the current estimate
from point B to the midpoint betweenC t and C t•, or point C. Starting with the initial model estimate at point A, SIRT convergestowardsthe solution along the solid line from point A to B, to C, to D, to E, etc. If a true model solution exists and is unique, then both ART and SIRT will convergeto that solution. However,one shouldnote that the convergence of ART depends upon the ordering of the hyperplane projections while the convergence of SIRT doesnot. You may seethis by projecting point A onto hyperplane2 first in Figure 14(a). The ART and SIRT methods, as we have stated, are intended to solve
linear systemsof equationslike thoserepresentedby equation(29) which explicitly relate the model function to the data function. But just because
equation(29) explicitly relatesthe modelfunctionto the data functionin a linear form does not imply a linear relationship for all types of model and data functions.
Take
for instance
a model
function
of slowness and
a data function of observeddirect-arrivaltraveltime. Equation (29) does not provide a linear relationshipin this casebecausethe raypat.h lengths in S are also dependent upon the slownessesdefined in the model function. Thus, we do not know the true raypath lengthsin S until the true slowness field is also known.
To solvethe nonlinearproblemin practice we computeestimatedraypath lengthsusing the estimatedslownesses in the model function and use the estimatedraypath lengthsin the ART or SIRT algorithm. This is called an iterative linear approach to solving a nonlinear problem. We can use Figure 12 to visualizewhat happensto the hyperplaneswhen solvinga nonlinear problemby an iterative linear approach. The two hyperplanesshown in Figure 12 are the true hyperplaneswhen we know the raypath lengthsin S. When we useestimated raypath lengthsin S the estimated hyperplanes will not be coincident with the true hyperplanes. The resulting projection will be different from the projection shown in Figure 12 as point A is projected onto an estimated hyperplane. Each time we update the model function slownesses,using either the ART or SIRT method, the new estimated hyperplanes will be located differently in the model space since we also have new estimated raypath lengths in S. What we hope happens is that the estimated hyperplanes will not be wildly repositioned to a new
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2.4.
SUMMARY
39
location each time the model slownessesare updated. Then as we iterate further towards the true model function slownessesthe estimated hyperplanes should become more coincident with the true hyperplanessince the estimated raypath lengths will be approachingthe true raypath lengths. Thus, we can get convergenceto a solution even though the problem is nonlinear.
2.4
Summary
1. Seismicray tomography attempts to solvethe inverseproblem formulated by the line integral equation, p
__
•.yM(r)dr,
taken over the raypath. P is called the data function and represents
the observeddata. M(r) is called the model function and representsthe spatial distribution of somephysicalproperty of the medium
whichaffectsthe propagatingenergyin someobservablemanner. The model function is unknown and the goal of seismicray tomography is to determine an estimated model function M e•t of the true model function M tr•e.
2. Transform methodsin seismicray tomography are of limited usesince straight raypaths and full scan aperturesare generallyassumed.However, they serve to introduce the tomography concept and terminology,and provideinsightinto seismicdiffractiontomographypresented in Chapter 3.
3. The projectionslicetheoremis the basisfor the transformmethods. The theorem states that the 1-D Fourier transform of the data func-
tion/5(f•,0) provides a sliceof information in thek• - k, wavenumberplaneof themodelfunctionA74(k•,kz) defined ontheloci'k• = f• cos0 and k, - f•sin 0 as shownin Figures4 and 5. Equation(12) definesthe projection slice theorem as,
4. Direct-transformray tomographyappliesthe projectionslicetheorem
tomany projections ofthedatafunction/5(f•, 0•for0 degrees _(0 _( 180 degrees.The resultis the modelfunctionM(f•cosO, f•sinO) definedon a polar grid. Interpolationof the modelon the polar grid onto a rectangularks - k, grid is requiredto take the 2-D inverseFourier
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CHAPTER
40
2.
SEISMIC
RAY
TOMOGRAPHY
transformof the modelfunctionwhichyieldsM(x, z). Interpolation error may distort the resultingestimated model function. 5. Backprojection ray tomography is another transform method which utilizes the projectionslice theorem. However,by making a change of variableswe were able to do the image reconstructionwithout the interpolationstep requiredby direct-transformray tomography.The
reconstruction formulagivenby equation(20) is
M(x,z) = 4•r• , x ej•(xcosO + zsin0)]•]dC2da. Backprojectionray tomographyor its modificationis usedas the image reconstructionalgorithm in computerizedaxial tomographybecause it is both accurate and fast.
6. Seriesexpansionmethods are the most frequently used seismictomographymethods. The model function is divided up into small cells where each cell is assignedan averagevalue of the continuousmodel function within that cell. Thus, the ith observation of the data function is related to the discretemodel function by the equation, J
Piø•' -- Z$ijM]•"•,i- l,...,I, j=l
where I is the total number of rays or observations, J is the total
numberof cellsin the discretemodel function, and $ij is the path length of the ith ray in the jth cell. The inverseproblem is to deter-
mineanestimated modelfunction M• '• of thetruemodelfunction
M]ruegiven theobserved datafunction Piø•s. 7. Kaczmarz' method iteratively solvesthe system of equations defined for the seriesexpansionmethods for the estimated model function
Mf s•. Theiterativepartof thealgorithm consists of threestepsas shownin Figure11andaninitialestimateof themodelfunctionY init must be input.
Step i requiresthat the current estimate of the model function be
usedin forwardmodeling(i.e., ray tracing) to get a predicteddata functionP/P• for the ith ray. The forwardmodelingis definedby the equation, J
j=l
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2.4.
SUMMARY
41
Step2 compares the predicteddata functionpfre with the observed datafunctionPiøb•.If theobserved andpredicted datafunctions agree
to withina specified error,thentheestimated mode] function Mf • istakena.sa goodestimate ofthetruemodelfunction M]rue. Step3 updates thecurrent estimate ofthemodel function M•• if the observedand predicteddata functionsdo not comparefavorably.The updatecorrectionis determinedby projectingthe currentestimateof the model function onto the hyperplane defined by the ith ray. The correctionis given by,
pfbz__pfre 1
The corrections to the model estimate recommended by the ith ray are applied to all J cells,
j
-
1,...,J.
Then, theupdated estimated model function M«"ew)e• becomes the current estimated model function back in step 1.
8. The arithmeticreconstruction technique(AP•T) is a seriesexpansion method which directly usesKaczmarz' algorithm.
9. The simultaneousiterative reconstructiontechnique(SIP•T) usesa modified Kaczmarz' method. Instead of updating the model after
tracing each ray in the forward modelingstep a.sis done in AP•T, all raysare tracedthroughthe currentestimatedmodeland a model correctionfound for eachray. Then the correctionsto eachmodel cell are averagedaccordingto the equation, I
i=1
forj
-
1,...,J.
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CHAPTER
42
2.
SEISMIC
RAY
TOMOGRAPHY
The weightWj is the numberof raysintersecting the jth cellor some other suitableray densityweightusedto obtain an averagecorrection
AM 1. The averagecorrectionis appliedto the currentestimated model function to get an updated or new estimated model function.
2.5
Suggestions for Further Reading Anderson, A. H., and Kak, A. C., 1982, Digital ray tracing in two-dimensional refractive fields: J. Acoust. Soc. Am., 72, 1593-1606. Addressesray tracing througha griddedrefractiveindex model commonly used in ultrasound computerizedtomography.
Berryman,J. G., 1990,Stable iterative reconstructionalgorithm for nonlinear traveltime tomography: Inverse Problems, 6, 21-42. Discussion of the nonlinear problem associatedwith traveltime tomography.
Bishop,T. N., Bube, K. P., Cutler, R. T., Langan, R. T., Love, P. L., Resnick, J. R., Shuey, R. T., Spintiler, D. A., and Wyld, H. W., 1985, Tomographicdetermination of velocity and depth in laterally varying media: Geophysics,50,903923. Traveltime tomographyapplied to reflection seisinology. Chapman, C. H., and Pratt, R. G., 1992, Traveltime tomography in anisotropicmedia- I. Theory: Geophys. J. Int., 109, 1-19. Expandstraveltime tornographyapplicationsfrom isotropic media to anisotropic media, a subject which has much current interest. A companionpaper immediately follows this paper on the applicationsof the theory. Dines, K. A., and Lytle, R. J., 1979, Computerizedgeophysical tomography: Proc. IEEE, 67, 1065-1073. First application of ray tornographyto subsurfaceimaging. Discusses both ART
and SIRT.
Langan, R. T., Lerche, I., and Cutler, R. T., 1985, Tracing of rays through heterogeneousmedia: An accurate and efficient procedure: Geophysics,50, 1456-1465. We do not elaborateon how to determine the raypath lengths through a griddedvelocitymodelfor the matrix S. This referenceis more than adequatein addressingthe problemas appliedto direct arrivals in seismic ray tomography.
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2.5.
SUGGESTIONS
FOR
FURTHER
READING
Moser, T. J., 1991, Shortest path calculation of seismicrays: Geophysics, 56, 59-67. Utilizes network theory to get raypaths and traveltimes of first arrivals. Very robustfor defining the matrix S, whether the first arrival is a direct arrival or a head wave.
Peterson, J., Paulsson, B., and McEvilly, T., 1985, Application of algebraic reconstruction techniquesto crossholeseismic data: Geophysics,50, 1566-1580. ART applied to crosswell seismic
data.
Phillips, W. S., and Fehler, M. C., 1991, Traveltime tomography: A comparisonof popular methods: Geophysics,56, 1639-1649. Comparison of various linear inversion methods.
Vidale, J. E., 1988, Finite-difference calculation of traveltimes: Bull. Sets. Soc. Am., 78, 2062-2076. Approximates the eikonal equation using finite differences to compute traveltimes for first arrivals.
43
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Chapter
3
Seismic
Diffraction
Tomography 3.1
Introduction
Seismicdiffractiontomographyis usefulfor reconstructingimagesof subsurface inhomogeneities whichfall into two categories.The first categoryincludesinhomogeneities that aresmallerin sizethanthe seismicwavelength and have a large velocitycontrastwith respectto the surrounding medium.Imagingtheseinhomogeneities with the seismicray tomography methodspresentedin Chapter 2 is generallyout of the question.The second categoryincludesinhomogeneities that are muchlargerin sizethan the seismicwavelengthand have a very small velocity contrastwith the surroundingmedium. Although seismicray tomographyis valid for imaging theseinhomogeneities, it worksbest when the velocitycontrastsare large. Note that both categories of inhomogeneity are capableof producingmeasurablescatteredwavefieldsof similar power. The large velocitycontrast of the first categoryinhomogeneity offsetsits smallsizewhilethe largesizeof the secondcategoryinhomogeneity makesup for its smallvelocitycontrast. The outlinefor this chaptercloselyparallelsthat of Chapter 2. First, in Section3.2 we review acousticwavescatteringtheory and derivetwo independentlinear relationshipsbetweendata functionsrepresenting scattered energyand the modelfunctionM(r). The modelfunctionM(r) usedin this chapteris a measureof the velocityperturbationcausedby an inhomogeneityat vectorpositionr from a constantbackground velocity.Second, usingeither of the linear relationshipsbetweena data function and the modelfunctionM(r), the generalized projectionslicetheoremis derived 45
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46
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
in Section 3.3 which servesas the foundation for the image reconstruction algorithms used in seismic diffraction tomography. Finally, two seismic diffraction tomography image reconstruction algorithms are presented in Section3.4' direct-transformdiffraction tomographyand backpropagation diffraction tomography. Before proceeding further one should have a good understanding of
the followingmathematicalconcepts: frequencyand wavenumber(AppendixA); Fouriertransform(AppendixB); Dirac deltafunctionandGreen's function (Appendix C). Also, sincewe presentseismicdiffractiontomography for inhomogeneitiesembedded in a constant velocity medium, the implementation of diffraction tomography is actually very similar to the transform methods for ray tomography. Thus, a review of Section 2.2 on ray tomography'stransformmethodsmight be of somebenefit.
3.2
Acoustic Wave Scattering
The propagationof an acousticwavefieldP(r,t) through a medium consistingof a variablevelocityC(r) and constantdensityis modeledby the acoustic wave equation,
1 O2P(r,t) V•P(r,t)C•(r) at• _ - 0,
(47)
where r is a vector position within the model and t is time. The Laplacian
operator X7• is definedin terms of the vector operator •7 which in the Cartesian coordinatesystem is given by
v -
+ 0
where•, j, and• aremutuallyorthogonal unitvectors. We use the Helmholtz form of the acoustic wave equation to describe acousticwave scattering. The Helmholtz acousticwave equation, found by
taking the temporalFouriertransformof equation(47), is
VaP(r,w)+ k•(r,w)P(r,w) = 0.
(48)
The variablek(r, w) is the magnitudeof the wavenumberat positionr and is defined by
k(r,w)= C(r)'
(49)
Notethat equation(48) dependsuponthe valuesetfor angularfrequencyw. Henceforth,for the sakeof brevity,we will write both P(r,w) and k(r,w)
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3.2.
ACOUSTIC
WAVE
SCATTERING
47
as P(r) and k(r) with the understandingthat there is a dependenceon angular frequency w. Two nonlinear integral-equation solutionsfor the scalar Helmholtz wave
equationdefinedby equation(48) arederivedin thissection.The LippmannSchwinger integral equation is one such solution which, because of its prominencein quantum mechanicalscattering, is presentedby itself in Section 3.2.1. The Lippmann-Schwinger equation nonlinearly relates the
data function P0(r), calledthe scatteredwavefield,to the modelfunction M(r). The Born approximationin Section3.2.2 linearizesthe LippmannSchwingerequation. Another nonlinear integral-equation solution is formulated in Section 3.2.3 using exponentials. Although the definition of the model function M(r) remainsthe same, the data function is different from the scatteredwavefieldP0(r). The Rytov approximationis used to linearize this nonlinear integral-equation solution. Except for the data functions, both linearized integral-equationsolutionsare identical in form. Section 3.2.4 provides a comparison between the Born and Rytov linear integral solutions.Either solutionenablesus to derive the generalizedprojection slice theorem in Section 3.3 which serves as the foundation for the diffraction tomographyimage reconstructionalgorithmsin Section3.4.
3.2.1
The Lippmann-Schwinger Equation
We begin the formulation of the acousticwave scatteringproblem with Figure 15. The acousticwave velocity is representedby C(r), where r is the vector position of a point within the model. The shaded region in Figure 15 depictsan inhomogeneityimbeddedin an otherwisehomogeneous medium. The acoustic velocity of the inhomogeneity varies spatially and can be thought of as a velocity perturbation from the constantbackground velocity Co of the homogeneousmedium.
An incidentwavefieldPi(r) is initiated by an acousticsourceand propagatesoutward in the homogeneous medium. No scatteringof the incident wavefield takes place until the inhomogeneityis reached. At that point any velocity contrast as a result of the inhomogeneitycausesthe creation of a secondwavefieldcalled the scatteredwavefieldP•(r). Each point in the inhomogeneitymay be consideredas a secondarysourceof seismic acousticenergy. Note that once acousticenergyis scatteredfrom one inhomogeneity,then that scatteredenergymay be scatteredagain from another inhomogeneitywhich leads to higher order sourcesof seismicacousticenergy. As you will see, we ignore multiple scatterings in Section 3.2.2 to linearize the Lippmann-Schwingerequation and assumethat the scattered
wavefieldP•(r) arisesonlyfromscatteringthe incidentwavefieldP•(r) from the source as depicted in Figure 15. Therefore, for layered media we as-
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48
CHAPTER
3.
SEISMIC
DIFFRACTION
Receiver
TOMOGRAPIIY
Source
C(r) = Co
FIG. 15. Acousticwavescatteringproblem.The incidentwavePi(r) propagatesfrom the sourceat the constantbackgroundvelocityCo. The velocity inhomogeneity, depictedby the shadedarea, actsas a secondarysourceand
scattersthe incidentwavefield.The scatteredwavefieldPj(r) travelsaway from the inhomogeneous regionat the backgroundvelocityCo whereit is recordedby the receiver.
sume that multiple reflectionsare negligiblewhen comparedto primary reflections.
The wavefield recorded by a receiver in the model consistsof both the
incidentwavefieldPi(r) and the scatteredwavefieldP,(r) whichwe call the total wavefieldPt(r) , or
P•(r) -
Pi(r) -[-P,(r).
(50)
For a constantdensitymodel,equation(48) describesthe propagationof the total wavefieldPt(r) throughan inhomogeneous medium,or
[V'• + k•(r)]Pt(r) = O.
(51)
At thispointwereformulate k2(r) in equation(51) asa perturbation to a constantko • for the homogeneous background mediumwherethe magnitude of ko is given by
=
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3.2.
ACOUSTIC
WAVE
SCATTERING
49
We begin by writing
-
+
[k•(r)1].
(53)
Substitutingequations(49) and (52) into the bracketedterm on the righthand sideof equation(53) for k(r) and ko, respectively, gives
ke(r)
C• 1] - ko •+k•[C•.(r) = ko 2-ko 2[1 C2(r ½;] ).
(54)
If we definethe modelfunctionM(r) asthe bracketedterm in equation(54),
thenwe get the desiredreformulation of k2(r),
k2(r) -
k• - ko•M(r),
(55)
whereM(r) is definedby
M(r)- 1 C•(r C•).
(56)
The model function M(r) in equation(55) definesa perturbationto an
otherwiseconstantko • of the background medium. When C(r) - Go in equation(56), M(r) - 0 and thereis no perturbationof ko • (i.e., by equation (55), k•(r)We now want to establish a relationship between the scattered wave-
field P,(r) and the modelfunctionM(r) usingequation(51). First, equations(50) and(55) aresubstituted intoequation(51) for Pt(r) andk2(r), respectively, giving
[•7• + k•- k•M(r)][Pi(r)+ Po(r)] -
O.
(57)
We rearrangeequation(57) so that terms involvingsourcesof scattered energyt are on the right-handside. Thus,
[V2+ ko2]Pi(r)+[V2+ ko•]Po(r)-
ko2M(r)[Pi(r)+Po(r)]. (58)
Note that the term ko•M(r)[Pi(r) + P,(r)] is the sourceof the scattered wavefieldP0(r) and, as previouslystated, dependsupon both the incident 1Termscontainingthe modelfunctionM(r) are sourcesof scatteredenergy.
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5O
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPItY
wavefieldand scatteredwavefieldat r. The left-handsideof equation(58) describes the propagationof the incidentwavefieldPi(r) and the scattered wavefieldP,(r) in the backgroundmedium, both of which travel at a constant velocity Co.
The incidentwavefieldPi(r) is generatedby a sourcein the homogeneousbackgroundand containsno scatteredenergy. Therefore, the incident wavefieldtravelsthrough the model at the backgroundacousticvelocityCo and contributesto the scattered wavefieldthrough the scatter sourceterm
in equation(58) when M(r) •- 0. Thus, the Helmholtzacousticwave equation for a constant velocity medium describesthe propagationof the incidentwavefieldPi(r) and is givenby
[V'2+ko•]Pi(r) -
O.
(59)
Equation(59) permitsus to reduceequation(58) to
[v +
-
(60)
Equation (60) describesthe propagationof the scatteredwavefieldat the backgroundvelocity when inhomogeneitiesoccur which scatter both the
incidentwavefieldPi(r) and existingscatteredenergyPs(r). Note that if no inhomogeneities occur, then M(r) = 0 and the right-hand side of equation(60) is zero,implyingthereis no sourcefor the scatteredwavefield and P•(r) = 0. Solvingequation(60) directlyfor P,(r) is difficult. A simpleapproachis to formulatean integralsolutionusingthe propertiesof the Green'sfunction developedin Appendix C. For the Helmholtz equation the Green'sfunction is the responseof the differential equation to a negative impulse source
function. 2 Thus,theGreen'sfunctionbecomes thesolutionto equation(60) if we replacethe sourceterm with a negativeimpulsesourcefunction-5(rr t) • or
[V2+ ko•]a(r[r')
-
-5(r - r').
(61)
The Green'sfunctionG(r Jr') givesthe solutionat positionr for a negative impulse at r' which correspondsto the location of a point scatterer.
The solutionto equation(61) for a 2-D spacecontainingan infinite-line scattererat r' and infinite-line receiverat r, where both lines are perpendicular to the plane containingr • and r, is
a(rIr') -- j4 Ho(X)(ko Ir- r' l)'
(6:2)
• Traditionally the Green'sfunction solutionfor the Helmholtz equation is determined for a negativesourcedensity on the right-haxtdside, or X72P + ksP = -p. In order for the Dirac delta function to represent a source density impulse, the sign in front of the Dirac delta function must also be negative.
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3.2.
ACOUSTIC
WAVE
SCATTERING
51
where Ho (•) isthezero-order Hankel function ofthefirstkind.In a 3-Dspace containingpoint scatterersand field points, the solution to equation (61) is
G(rlr')= 4•rlr-r' I '
(63)
With the Green'sfunction for equation(61) known, the solution to equation(60) is foundby multiplyingthe Green'sfunctionby the negative of the sourceterm in equation(60) and integratingover all spacewhere
M(r •) •- 0, or 3
P,(r)- -ko •/ G(r [r')M(r')[P•(r') +P•(r')]dr'. (64) Equation (64) is called the Lippmann-Schwinger equation,which is the desiredintegral solutionfor the acousticwave-scatteringproblem. We point out in Appendix C that suchan integral is analogousto obtaining an output from a filter systemwhenthe impulseresponse of the filter (i.e., the Green's
function)and the input (i.e., sourceterm fromequation(60)) are known. Equation(64) is just a convolutionintegral,whichis easilyseenif the Green'sfunctionfrom either equation(62) or (63) are substitutedin for I,?). The Lippmann-Schwingerequationnonlinearlyrelatesthe model func-
tion M(r) to the data function(scatteredwavefield)P,(r). The nonlinearity is a result of the scatteredwavefieldP,(r) insidethe integrandof
equation(64) whosevaluedepends on the modelfunctionM(r). Because of this nonlinearity,it is difficult to use equation(64) to perform either forwardmodeling(computeP,(r) from M(r)) withoutresortingto computationally extensiveapproachessuchas finite differencemethodsor to derive diffractiontomographyimage reconstructionalgorithms(compute M(r) from P,(r)). One way to get aroundthis problemis to linearize equation(64) by makinga simplifyingapproximation calledthe Born approximation.
3.2.2
The Born Approximation
The Born approximationlinearizesequation(64) by assumingthat the
scattered wavefield P,(r) is muchweakerthanthe incidentwavefield Pi(r), 3In this book we use special integration notation which should not be confused with
a line integral. Equation(64) couldbe either an integrationovera plane (2-D space) or a volme (3-D space)dependingupon which Green'sfunctionis used. Thus, to
remain general, if weareintegrating overa plane, thenf dr'• f dx'dz'; andif weare integrating overa voltune, thenf dr'::•f dx'dy'dz'.
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52
CHAPTER
3.
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TOMOGRAPHY
or
P,(r) • P•(r).
(65)
If the conditiongivenby equation(65) is true, then the Born approximation states that
Pi(r) + P,(r)
•
Pi(r).
(66)
The Lippmann-Schwinger equationis linearizedby substitutingequation(66) into equation(64),
P,(r) • -ko 2/ G(r [r•)M(r•)Pi(r•)dr •. (67) Note that the integrandno longercontainsthe scatteredwavefieldP•(r) and the data function P•(r) and mode]function M(r) are now linearly related. If the primary acoustic source is a negative impulse located at vector position r,, then, by the definition of a Green's function, we can
directlyrepresentthe incidentwavefieldP•(r') in equation(67) by a Green's function,
Pi(r')
:
G(r' [ rs).
(68)
If a pressure-sensitive receiver(e.g.,hydrophone) is locatedat positionr = rp, thenby substituting equation(68) intoequation(67) for P•(r') wefind,
P,(r,,rp) • --k• 2/ M(r')G(r' lr,)G(r pIr')dr', (69) whereP,(r,, rp) is the scattered wavefield observed at positionrp whenthe negativeimpulsesourceis located at positionr•. Both Green'sfunctions are definedby either equation(62) or (63).
Equation(69) is the Lippmann-Schwinger equationlinearizedby the Born approximation. This equation establishesthe linear relationship be-
tweenthe data function(scatteredwavefield)h(r,,rp) and the unknown modelfunctionM(r) requiredby the diffractiontomographyproblem.Note that sincewe exploitedthe Born approximationin derivingequation(69) that the modelfunctionM(r) must be a weak scatterer. Only then will the scattered
wavefield
be much weaker than the incident
wavefield
and the
conditionspecifiedby equation(65) be satisfied.
3.2.3
The Rytov Approximation
In this sectionwe derivea nonlinearintegral-equationsolutionto equa-
tion (48) usingexponentials. Althoughthe modelfunctionM(r) is defined
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3.2.
ACOUSTIC
WAVE
SCATTERING
53
the same as for the Lippmann-Schwingersolution, the data function representingthe observedscatteredwavefieldis different. The Rytov approximation establishesa linear relationship between the data function and the model function. The resulting linearized integral solution strongly resemblesequation(69) for the Born approximation. We begin the derivationby returning to equation(51) which describes the propagationof the total wavefieldP•(r) through a constantdensity, variable-velocity medium. Usingequation(55) andequation(56) asthe def-
initionsfor k2(r) andthemodelfunctionM(r), respectively, equation(51) is rewritten
as
Iv • + • - •4(•)]P,(•)
-
0.
(70)
Exceptfor settingPt(r) = Pi(r)+ P,(r), equation(70) is the sameas equation(57). We deviatefrom the earlierderivationof the LippmannSchwingerequationby representingthe total wavefieldwith the exponential equation,
P•(r)-
e•,(r),
(71)
where•b•(r) is calledthe "complextotal phasefunction." Note that, as in previous sections, variables which are a function of the position vector r
carryan impliedfrequencydependence. We wish to substituteequation(71) into equation(70) to obtain a differentialequationin termsof •,(r). The Laplacianof P,(r) mustbe taken to achieve the this result.
We start with
v•P,(•)
= v. [vP,(•)].
Substitutingequation(71) in for Pt(r) and performingthe differentialoperationsgives,
v•P,(•) = = =
v. b•,(•)v•,(•)], e•'(•)v-v•,(•)+ Ve•'(•).v•,(•), •,(•)v•,(•) + •,(•)v•,(•). v•,(•), e•'(r)[v•,(r)+V•,(r).V•,(r)].
(72)
Equations (71) and(72) aresubstituted intoequation(70)for V2P,(r) and P,(r) giving,
•,(•)[v•,(•) + v•,(•). v•,(•)]+ [• - •(•)1•,(•)
- 0. (73)
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54
CHAPTER
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Dividing through byeqb'(r), equation (73)becomes, V•6 (r) + [V6t(r) V6t(r)]+ ko • ko•M(r)-
0,
(74)
whichis the desireddifferential equationin termsof •bt(r). At this pointweintroducethe incidentwavefield Pi(r) expressed
Pi(r)-
e0i(r),
(75)
where$•(r) is calledthe "complex incidentph•e function."The complex incidentph•e function•i(r) is relatedto the complextotal ph•e function •t(r) throughthe "complex ph•e difference function,"definedby
0a(r) =
0t(r)- 0,(r).
(76)
Sincethetotalwavefield P•(r) variesfromthe incidentwavefield P•(r) only whenscatteringoccurs,we can surmisethat the complexph•e difference function$a(r) is a meansof accounting for scatteredenergy.
Wecontinue thederivation byreplacing 0,(r)in equation(74) by •a(r) • definedby equation(76). Carryingthis operationout yieldsa differentialequationin termsof 0,(r) and
V•0i(r)+ V•0•(r) + [V0i(r). V0i(r)]+ 2[V0i(r). +[V$•(r). V&a(r)]+ k• - k•M(r) - 0.
(77)
The termsin equation(77) are rearrangedin a form whichwill proveconvenient later on,
=
+
(78)
The termsinsidethe squarebracketson the left-handsideof equation (78) are all relatedto the incidentwavefield and havea sumequal to zero. This is e•ily showntrue by usingequation(59) whichdescribes the propagation of the incidentwavefieldPi(r) throughthe background medium.Equation(75) defining Pi(r)is substituted intoequation (59)to get the differentialequationin termsof •i(r). Followingthe sameprocedurewhichgavethe Laplacianof Pt(r) in equation(72), the Laplacianof
V•P•(r)- e0•(r)[v•0•(r) + V0•(r).V0,(r)].
(79)
Substituting equations (79) and (75)into equation(59) for V•P•(r) and P•(r), respectively, gives
e•i(r)[v2•i(r) + V•i(r)-V•i(r)] + k•e•i(r)- O.
(80)
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3.2.
ACOUSTIC
WAVE
SCATTERING
55
Dividing equation (80)bye•i(r)results inanequation which demonstrates that the bracketedterms in equation(78) havea zerosum,or
V2•i(r) + V•i(r).V•i(r)
+ ko • -
O.
(81)
+ ••(•).
(8•)
By equation(81), equation(78)is rewritten •
•v•,(•). v•(•)
+ v•(•)
- -v•(•).
v•(•)
Equation(82) providesa crucialrelationshipwhichwe will uselater on in this derivation. Now we turn our attention to the important product
P•(r)•(r)
which will becomethe data functionfor this derivationor a
me•ure of the scattered wavefield. For purposesof the derivation the Laplacian is taken of this product. Performingthe differentiationwe have,
V•[P,(•)•,(•)]
- V. [•,(•)VP,(•) + P,(•)V•,(•)] = •,(•)V•(•) + •V•(•). V•,(•) +P,(r)V2•d(r).
(83)
Rearrangingthe termsin equation(83) and makinguseof the fact that
V•P•(r)-
-k•P•(r) by equation(59), weget the following,
2VP,(r). V•(r)+
P,(r)V•(r)
- V•[Pi(r)•(r)]- •(r)V•P,(r) = V2[pi(r)•d(r)]+ •d(r)koP•(•) 2 = [V• + k•]P,(r)•d(r). (84)
Switchingthe left- and right-handsidesof equation(84) and using the definitionof P•(r) givenby equation(75), we can write the following,
Iv • + •]P,(•)o,(•)
-
•v•,(•). v•,(•) + P,(•)v•o,(•)
= •(•)vo•(•). v•,(•) + p•(•)v•o,(•) = •P,(•)v•,(•). v•,(•) + P•(•)v•,(•) = p,(•)[•v•,(•). vo,(•) + V•d(•)].
(SS)
The quantity inside the square brackets on the right-hand side of equation (85) is definedby the earlier "crucial"relationshipgiven by equa-
tion (82). Substitutingequation(82) into equation(85) gives,
IV• + •]P,(•)o,(•)
-
-P,(•)[v•,(•).
vo,(•)-
•(•)].
(86)
Just • w• donefor equation(60), equation(86) can be solvedfor P•(r)•d(r) in termsof an integralequationby exploitingthe propertiesof the Green'sfunction. The resultingsolutionis
P,(•)o,(•) - f P,W)[vo,(•'). v•,(•')•(•')]a(• I•')d•',(87)
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CHAPTER
56
3.
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TOMOGRAPHY
where G(r Ir') is a Green'sfunctiongivenby either equation(62) or equation(63). The data functionP•(r)da(r) is nonlinearlyrelatedto the modelfunction m(r) sinceda(r) appearsinsidethe integrandof equation(87). Both forward modeling(computingP•(r)•a(r) from M(r)) and image reconstruction(computingm(r) from P•(r)&a(r)) are difficultbecauseof this nonlinearity. At this point in the derivation the Rytov approximation is formulated to remedy this situation. The Rytov approximation linearizes
equation(87) by assuming the conditionV•a(r) <( 1. When V•a(r)is smallthe quantityX7•a(r'). V•a(r') in equation(87) canbe neglected and we write the approximation,
-f
(ss)
As with the Born approximation,the incidentwavefieldPi(r') can be representedby a Green's function,
P•(r') =
G(r'}r,),
(89)
for a negativeimpulsesourceat vectorpositionrs. If a receiveris located
at positionr - rr, then by substituting equation(89) into equation(88) we have,
P•(r•,rv)•Sa(r,,rv) • -ko •f m(r')G(r' lr,)O(r vIr')dr'. (90) The Green'sfunctionsaresatisfiedby eitherequation(62) or equation(63). However,to useequation(90) properly,the gradientof the phasedifference functionV•Sa(r),mustbe smallas requiredby the Rytov approximation.
3.2.4
Born vs. Rytov Approximation
Except for the data functions, the linearization of the Lippmann-
Schwinger equationusingthe Bornapproximation in equation(69) is identical to that of the Rytov approximationin equation(90). Herewe demonstrate that the data functionPi(r,,rr)•Sa(r,,rr) associated with the Rytov approximationreducesto the data function associatedwith the Born
approximation, the scattered wavefield P•(rs,rp), whenthe complex phase difference functiond•(rs,rr) issmall.Next,weshowthat forcing to be smallis the sameasrequiringa weakscattered wavefield P,(r,,rr) for the Born approximation.Finally, we state the limitationsfor applying either the Born or Rytov approximation.
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3.2.
ACOUSTIC
First
WAVE
SCATTERING
57
we wish to show that
Pi(r0,rr)•ba(r0,rr) •
P0(r0,rr),
(91)
when4a(r0,r•) is small. This resultis achieved by usingthe Taylorseries
expansion ofe4a(r0, r•) which isgiven by,
•*•(•0,•)- 1+,•(•,'•)+ *](•"•) 2] +
3] +.... (92)
When •(r•, r•) • 1, we canneglectthe secondand higherordertermsin the Taylor seriesexpansionand write
•(•,,•)
• •(•',•)
- 1.
(•)
The approximationof the data function •sociated with the Rytov approximation is found by multiplying equation (93) by the incident wavefield
Pi(r•, rp), giving,
P,(r,,rr)•(r,,rr) • Pi(r•,rr)[e•(r•'rr) - 1].
(94)
This approximation is e•ily shownto be the scatteredwavefieldP•(r•, rr)
byusing therelationships, Pi(r,,rr)- e•i(r•,rr), P•(r•,rr)- e•'(r•,rr) andP•(r,, rr) - P•(r•, rr)-Pi(r,, rr) , fromequations (75), (71), and(50), respectively.Hence,equation(94) is rewritten:
•,(•. •),,(•. •) • •,(•. •)[•,(•. •) - 1]
• •,(•,, r•) _ •,(•,,r•) • •
•,(r•, •) - •,(r•, r•) •(r•, r•).
(•)
Thus,the data functionsfor equations(69) and (90) are approximately the samewhen•(r•, r r) is small. Now we wishto showthat setting•(r•,rr) • 1 is the same• the Bornapproximation requirement of a weakscatteredfield(i.e., P•(r,, rr) • P•(r•, rr) ). We beginby explicitlywritingthe total and incidentwavefields
P•(r)-
A,(r)eJ•b,(r),
(96)
Pi(r) - Ai(r)eJ•bi(r),
(97)
and
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58
CHAPTER
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respectively, where;bt(r)and;b,(r)aretherealphases andAt(r) andA,(r) arethe amplitudes.Equation(76) givesthe definitionof the complexphase difference
function
as
4a(r) = 4,(r)- 4,(r).
(98)
Solvingequations (71) and(75) for dr(r) anddi(r), thensubstituting the resultsinto the last equationgives
da(r) = lnPt(r) - lnP•(r).
(99)
Usingthe explicitrelationsfor Pt(r) and Pi(r) givenby equations (96) and (97), equation(99) becomes,
da(r)- In[At(r)1 q-j[•bt(r)•bi(r)] '
(100)
The complexphasedifference functionrid(r) in equation(100)is small whenIn ta,(r)/Z,(r)l << I and [;bt(r)- ;b•(r)]<< 1. This is the comparableto requiringthat the difference betweenPt(r) •na g(r) be small
for the Bornapproximation. The In ta,(r)/a,(•)l termin equation (100) demonstrates that the Bornapproximation is a weakscatteringapproximation. Also,the accumulative phasedifference, [;bt(r)- ;b•(r)l, maybecome significantif the regionwhere M(r) • 0 is large relative to the seismic wavelength.Thus, in addition to requiringweak scatterers,the size of the
inhomogeneities maybecome an importantfactorto consider whenusing the Born approximation.
On the otherhand,the Rytovapproximation doesnot requirerid(r) to be small,only that the gradientof rid(r) be small(i.e., the changeof rid(r) withina wavelength be small). The Rytovapproximation requires
onlya smooth modelfunction andplaces norestrictions onthestrength of the scatterersor their size. Thus, the Rytov approximationis a smooth scatteringor smoothperturbationapproximation. 3.3
Genera'ized Projection
Slice Theorem
Either equation(69) or (90) in Section3.2 definea linearintegralrelationshipbetweena data functionrepresenting scatteredenergyand the modelfunctionM(r). In thissectionwe simplifythis linearintegralrelationshipby taking the spatial Fourier transform of the data functionover
the sourceandreceiverprofilesandby takingthe 2-D spatialFouriertransform of the modelfunction. The result is the generalizedprojectionslice
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3.3.
GENERALIZED
PROJECTION
SLICE
theorem which serves as the theoretical
THEOREM
foundation
59
for the seismic diffrac-
tion tomography image reconstruction algorithms presented in Section 3.4.
We let P(r,, rr) for the remainingpart of thischapterrepresent the data
function foreitherP,(r,, rr) oftheBornapproximation or Pi(rs,rr)•b,(rs,rr) of the Rytov approximation. Thus, the generic equation,
-}o
(xox)
definesthe linear integral relationshipbetweenthe data function and model
functiongivenby eitherequation(69) or (90).. Also,sincemostsourceand receivergeometriesare for 2-D problems,we will exclusivelyusethe Green's
functiondefinedby equation(62). We find it instructive to derive the generalizedprojection slicetheorem for three typical source-receiverconfigurations: the crosswellprofile, the
vertical seismicprofileor VSP, and the surfacereflectionprofile. One may justify this approach by noting that the spatial Fourier transform of the
data functionP(r•, rr) mustbe takenoverthe sourceand receiverprofiles. However, by the end of this section the reader will find that the theorem is actually independentof the source-receiverconfiguration.
3.3.1
Crosswell Configuration
We begin the derivation of the generalized projection slice theorem for the crosswellconfiguration by replacing the position vectors in equation (101) with coordinatesfrom the x-z plane. Figure 16 showsa crosswell configuration with sourcelocations representedby solid circlesand receiver locations identified by open circles. We assume both source and receiver wells are vertical
so that the x-coordinates
of all sources and receivers are
given by the constants,ds and dr, respectively.The z-coordinates of the sourceand receiverlocationsare z• and zr. Thus, the positionvectorsfor
the source locationr• andreceiver locationrr' canbe expressed in terms of their respective coordinates (d•, z•) and (dr, zr). Substitutingthesecoordinatesinto equation(101) for the positionvectorswe get,
x O(z, z I ds,zs)O(dv, zv [ z, z)dxdz, (102) for the crosswellconfiguration.
The Green'sfunction G(z,z I a,,•,)
i, equation(102)is chosento
representa cylindrical wave emitted from an infinite line sourcelocated a•
the coordinates(d,, z,). Equation(62) is the appropriateGreen'sfunction
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60
CHAPTER
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..X
Source Receiver
ds
dp
z
FIG. 16. Source-receivergeometry for the crosswelltomography experiment. The sourcelocations are representedby solid circles at the constant horizontal location d,. The receivers are represented by open circles at the
constanthorizontallocationdr . The orientationof the line sourcesand line receiversis perpendicularto the page. We restrict our reconstructionof the
modelfunctionto d, < z < dr.
for this problem. We rewrite the position vectors in the Green's function
in termsof x-z coordinates using(x, z) for r, (d,, %) for r', and I r - r ' I-
Vf(x-d•) 2+ (z- %)2. Makingthesesubstitutions in equation (62)gives, J
G(x,z Id,,z,)- •Ho(•)(koV/(X - d,)•+ (z- z,)•).
(103)
The Green'sfunctionin equation(103) is awkwardto use becauseof
thezero-order Hankelfunction Ho (•). TheGreen's function represents a cylindricalwaveat the point (x, z) propagatingawayfrom a line sourceat (d,, z,) with vectorwavenumber ko. Fortunately,it turnsout that the zeroorder Hankel function can be mathematically thought of as the summation
(integration)at (x, z) of an infinitenumberof planewaveswhosewavefronts are tangentto the cylindricalwavefront.This plane wavedecomposition of the cylindrical wave is defined by
+
-
1/_•1eJ[kl(z-z,)+?llx-d, I]dkl (104)
•r
oo 7•
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
61
wherethe wavenumberfor a givenplanewavefor the crosswellconfiguration
has the components k• alongthe z-axis and 7• alongthe x-axis.4 The
absolute valueof (x -ds) is required because 71 - +v/ko • -k•, whichis alwaysgreaterthan zero. The absolutevalueofx-ds insures7•(x-do) > O, regardlessof the value of x.
Figure17 illustratesthe conceptbehindequation(104). Planewave•1 is perpendicularto the cylindricalwavefrontat (x,z). This plane wave has the samevectorwavenumberko as the cylindricalwavefrontat (x, z) and has the wavenumbercomponents(7•,kl). As previouslymentioned we require the • componentfor all plane waves be positive, defined by
7• - +v/ko2 -k•.
This definitionof 7• ensures that all planewavesare
tangent to the cylindrical wavefront with a wavenumbermagnitude of ko.
The directionofpropagation for planewave#1 is a = tan-• (7•/k•). Plane wave #2 is traveling in the +x directionwith vectorwavenumberko•: and components (7• = ko,k• = 0). Note that the summationof planewave#1
by equation(104) at (x,z) is at the onsetof the plane wavewhile the summationof plane wave #2 at (x,z) occurslater on in its wave train. The waveformof any planewavewill be summedby equation(104) at a later point in its wave train if the plane waveis tangent to the cylindrical wavefrontat a point other than (x, z). Now we apply plane wave decompositionto the Green's function in
equation (103). This is achievedby substitutingthe plane wave decompositionof the zero-orderHankelfunctiondefinedby equation(104) into equation(103) giving,
+
-
where the absolutevalue can be droppedby requiringz > ds. Note that this Green's function is adequatefor the crosswellconfigurationshown in Figure 16 since we are usingequation (102) to image only betweenthe
sourceline and receiverline or d0 < z < dv. We are effectivelysaying M(z, z) = 0 outsideof the imagingarea. The Green'sfunctionG(dv,zr [ z,z) in equation(102) is chosento representa cylindricalwaverecordedat (dv,zv) scatteredby an infinite line scattererlocatedat (z,z). As with the line source,equation(62) is the appropriate Green's function for the line scatterer. After substituting 4In derivationsof the generalized projectionslicetheoremfor differentsource-receiver configurations, the kl component is always taken along the line of sourcesand -yl is perpendic,,lar to k• in a right-handed coordinate system sense. Here the sourceline just happens to be along the z-axis. The reason for doing this will become clear in a few pages.
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62
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
#2
#1
z
FIG. 17. Plane wave decomposition of a cylindricalwavefrontat (x,z). Two plane waves, labeled #1 and •2 out of an infinite number of plane waves tangent to the cylindrical wavefront, illustrate the concept behind equation(104).
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
63
coordinatesfor positionvectorsin equation(62) and performinga plane wave decompositionon the Green's function we get,
G(dr, z•[x,z) - 4•----•• fj l eJ[k•(zr - z)-l?•(dr - x)]dk• (106) for x ( dr. Here a given plane wave has the wavenumbercomponents k• along the z-axis (parallel to the receiverline) and 7• alongthe x-axis (perpendicularto the receiverline), analogousto the kx and 7• reference directionswith respect to the sourceline. The direction of propagation
for the scatteredplanewaveis tan-• (72/k2). As with equation(105),this Green's function for the scattered wavefieldis adequate since the imaging
areahasthe limits d0 < x < dp. One final relationship is required before we continuethe derivation. We must write down the equation for taking the Fourier transform of the data
functionP(d,,zo; dr, zr) in equation(102) alongthe sourceline (z,) and the receiverline (zr). This relationshipis foundby first takingthe Fourier transformof P(d,, z,; dr, zr) with respectto z, followedby a secondFourier transformwith respectto zr whichproducesthe doubleintegralFourier transform,
x e-jk•z•e-Jkrzr dz•dzr '
(107)
wherek• and kr are the wavenumbers of the Fouriertransformsalongthe sourceline and the receiverline, respectively.Note that the directionstaken for kl and k2 in the two Grecn's functions are oriented the same as ks and
kr, a "lucky" coincidence as far as the derivationis concerned. Now we may resume the derivation of the generalized projection slice theorem for the crosswellconfigurationby obtaining an integral relation-
ship betweenthe data functionPo(d•,k•;dr, kr) and the modelfunction M(x, z). We beginby substitutingthe linearintegralrelationshipbetween the data functionP(d•, z•; dr, zr) andthe modelfunctionM(x, z), defined by equation(102) into equation(107). This substitutiongivesthe relationship,
-k o 2
x
x
M(x,z)
G(x,z [ d,,zo)½-Jk, z,dzo
[
(os)
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64
CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPttY
between thedatafunction ]5(d,, k,;dr,kr)andthemodel function M(x,z). Notethatwehavechanged theorderof integration in equation (108). Next,wesubstitute theGreen's functions fromequations (105)and(106) intoequation (108),regroup termswithsimilarintegration variables, and interchange the orderof integration for the twoinnermost integrals over
eachGreen'sfunction,givingthe equation,
•5(d, Iq,) = k• ' ko' ' d•,, 4 1 •
M(x ' z}
(109)
{•• 1eJ[-•2z+'2(dp-z)]• • -J(•p-'2)Zpdzpd The integralsin equation(109) with respectto the variablesz, and
zr areeasilyevaluated in termsof Diracdeltafunctions asshownin AppendixB. The followingare the integralsolutions: 5
:'ø e-j(k• +k,)z, dz,
(110)
•øe-j(k •,- ka)zp dzp 2•rS(kr- k,).
(111)
and
Substituting theintegral solutions givenbyequations (110)and(111) intoequation(109)andevaluating theintegrals withrespectto thevariables
k, andk=(remembering that7• - ko • - k• and7• - ko • - k•) gives, '
;
-
--
4
%%
•
M(z,z)
(112) where
'rr- Vko =-/%=
(113)
7,- 7ko • - k, •.
(114)
and
5Wechose kl andksin thesamedirection asksandkp,respectively, sothatsimplifying relationships llke equations(110) and (111) couldbe used.
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
65
Both7pand% arewavenumber components perpendicular to the wavenumber componentskp and k0 which parallel the receiverand sourcelines, respectively 6.
The doubleintegralin equation (112_) isjust a 2-D Fouriertransform of the modelfunctionM(x, z) definingM(k:,, k,) alongcirculararcsin the
k• - k, plane,the placement of the arc depending uponk0 andk•?. At this point we must make a brief digressionto further explain this concept.
Figure 18(a) showsfour down-holesourcepositionslabeled $•, $2, and $4 in one well and four down-hole receiver positions labeled R•,
Rs, and R4 in another well. A line diffractorlies at the location (x, z). If
] represents the directionof a planewavepropagating froma sourceto the point(x, z), thenby equation(105) the associated wavenumber components (7•, k•) can be written,
koi -
7•:•+ k•.,
(115)
where • and •. are unit vectors in the positive x-direction and z-direction, respectively.Now, as a mathematicalabstraction,let • point in the opposite directionof a planewavetravelingawayfrom the sourceto the point (x, z),
or • - -{. By equation (115)andfromthefactthat kx- -ks and7• - 7: by equation(110), we may write the components of koõas (116) Figure 18(a) showsthe unit vector g for eachof the four sourceslabeled with subscriptsas gx, •2, •a, and g4. Now let •) representthe direction of a plane wave propagating from the
point(x, z) towardoneof the receivers in Figure18(a). By equation(106) and by the fact that equation(111) definesk•. = kr and 72 = 7r, we may write the componentsof ko•) as
(117)
Figure 18(a) showsthe unit vector• for eachof the four receiverslabeled with subscriptsas •)x, •2, •)a, and •4.
Figure18(b) showsthe wavenumber domainrepresentation of the source wavenumbervectorsko•x, ko.q2,ko.qa,and ko.•,•. Similarly,Figure 18(c) showsthe wavenumberdomain representationof the plane wavestravel-
ing from the point (x,z) to the four receiversas the vectorsko•l, ko•., 6Refer back to equation (107) for the definitionsof kr axtdk•.
7Equation(112) is analogousto equation(12) definingthe projectionslicetheorem for ray tomography. Instead of arcs, the projection slice theorem defines straight lines in the k: - k: plane as shown in Figure 5.
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66
CHAPTER 3. SEISMIC DIFFRACTION
TOMOGRAPtIY
•x
s1
R1
Zs2^ • ß
p
•^ R2
^2
S4
R4
(a)
Kz
Kz
(b)
(c)
FIG. 18. The wavenumber components (k,,%) and (kr,7r) in equation (112) dependuponthe sourceand receiver locations, respectively, relativeto thelinediffractor in themodel.(a) Foursources (soliddots) and four receivers(open dots) are shown. The unit vectorsradiatefrom a line diffractorand point towardtheir respectivesourceor receiver.The unit vectorsfor the sourcespoint oppositeto the unit vectorsof the incidentwaves. (b) The wavenumbervectorsfor eachsourceare shownon
the k• - kz planeas terminatingalonga dashedcirclewith radiusko. If sources are deployed from +oo to -oo in the boreholedirection,thenthe
entiredashed circleterminus is defined.Equation(116)defines thecomponents(k•, %) for thesewavenumber vectors.(c) The wavenumber vectors for eachreceiver areshown onthe k•- kz planeasterminating alonga dashedcirclewith radiusko. As for the source,the entire dashedcircle terminusis definedif receiversare placedfrom +c• to -c•.
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3.3.
GENERALIZED
PROJECTION
ß
SLICE
,ks
THEOREM
67
kP •K x
p,i•-Kx ß
ß
o
Kz (a)
(b)
F•c. 19. Wavenumber vector componentsfor the crosswellconfiguration. (a) The componentsof a sourcewavenumbervector kog are shownas pro-
jectionsontothe k•-axis and kz-axisasdefinedby equation(116). (b) Similarly, the components of a receiver wavenumber vector kot5are shown as
definedby equation(117).
kof)3, and koO4. If the sourcescan be deployedfrom +c• to -c• along the source well, then the wavenumber domain representation of the source wavenumbervectors ko.• are vectors pointing to the dashedsemicirclein
Figure 18(b). Likewise,if the receiverscan be deployedfrom +c• to -c• in the receiver well, then the wavenumber domain representation of the plane wavestraveling to these receiversare vectors pointing to the dashed
semicirclein Figure 18(c). The components of ko•,and ko• definedby equations(116) and (117) are showngraphicallyin Figures 19(a) and 19(b), respectively.At this point we add the two vectorskof,and koO which givesthe vector equation,
=
(7r -7s)• + (kr + k•)•..
(118)
The dot productbetweenko(g+ •) and positionvectorr pointingto the line diffractorat (x, z) in the modelis
ko( +
. ,: -
+
.
+
=
k•z +
=
(7r - 70)x 4-(kr 4- ko)z.
(119)
At this point we note that equation(119) multipliedby -j formsthe terms
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68
CHAPTER
3. SEISMIC
DIFFRACTION
TOMOGRAPttY
for the exponentials in the integrandof equation(112). Substituting equation (119) intoequation(112) resultsin the moreinterpretable equation,
tS(d•,k•;dp,kp)
(120)
where/•/(k•, kz)isthe2-DFourier transform ofthemodel function M(z,z), or
=
+ +
(121)
Usingthe last result,equation(112) may be written morebrieflyas
P(do ko kp)-- --kø2 eJ(7pdp - %d,) ~ + f>)]. ; dp, M[ko(f, ' 4 %7p
(122)
The integrationlimits in eitherthis equationor in equation(120) must obeythe spatiallimitationsset by equations(105) and (106). That is, the imagingareais restrictedto ds< x < dp. We do thisby settingM(z, z) to zeroin the integrandwhenthe integrationis outsideof the imagingarea, whichis the sameasrestrictingthe integrationto only the imagingarea.
Equation (122)establishes a relationship between /5(ds, ks;dp, kr),the doubleintegralFouriertransformof the data functionalongthe sourceand
receiver profiles, andg•[ko(•+ •)], the2-D Fouriertransform ofthemodel function for the diffraction tomographyproblem. Considerthe crosswell
configuration in Figure20(a). The unit vector• pointstowardthe single sourcefrom the linescattererlocatedat (x, z) whilereceiversdeployedfrom +cx>to -cx>alongthe receiverwell causethe unit vector• to havethe range within the dashedsemicircle.The locusof the quantityko(• + •) in the wavenumberdomainis a semicirclewith radius ko centeredon the point (-ko, 0), asshownin Figure20(b). Note that the coverage in the wavenumber domainchanges asthe source is moved.For example,Figure20(c) showsthe unit vector• for the source
at -cx>.Sincethe receivers areat thesamelocations asin Figure20(a)the coverage of • remainsthe same.The locusof the quantityko(.•q-•) in the wavenumberdomainis shownin Figure 20(d) as a semicirclewith radius
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
69
,.
ß ource -
source
-
at
O0
-
',
,'
•o• .ource
at
+oo
(a)
{e)
(c)
gx
gx
Kz
(b)
Kz
Kz
(d)
FI6. 20. Coverage providedthe modelfunctionM(k=, k,) bythreedifferent sourcelocationsrelative to the receiverboreholewith receiversranging in depth from +cx>to -cx>. The wavenumbervector ko• is shownin (a), (c), and (e) for sourcesat the samedepth as the line scatterer,at -cxv, and at +oo, respectively. The termini for the receiver wavenumbervectors are shownas the dashed circles. For the fixed receiverlocations, the solid circle
arcsin (b), (d), and (f) are the coverages providedof the modelfunction
37/(k=, kz)bythesource locations in (a), (c),and(d),respectively. Thesolid arcsare a resultof the vectorsum, koi + kolS,as definedin equation(122) on the right-hand side.
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70
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPItY
ko centeredon the point (0,-ko). Figure20(e) showsthe unit vector• for the sourceat -]-•x)and the same receivercoverage. Now the locusof the quantity ko(• + •) in the wavenumberdomainis the semicirclecenteredon
(0, +ko) shownin Figure20(f). We can use equation(122) in the diffractiontomographyproblemto compute the Fourier transform of the unknown model function M on the
semicircular locidefined by ko(•+ •)in Figures 20(_b), 20(d),and20(f). The loci definedby ko(O+ •) in the evaluationof M are calledthe slice of the 2-D
Fourier
transform
of the model function.
The
data function
P(d,, z,; dr, zr) representing the observed scatteredwavefieldalongthe receiverline is alsocalledthe projection.Equation(122) links the slicesof M with the projections, resulting in its name, the generalizedprojection
slicetheorem. Thetheorem states that15,(d,, k,;dr,kr), thedouble integral Fourier transformof the data function along the sourceand receiver lines, is equal to M[ko(g + •))], the 2-D Fourier transformof the model function evaluated along the semicircularslice, multiplied by the quantity
k• eJ(7pdr We may extend the situation shown in Figure 20 by deployingmany
sources from+½x•to -½x•in the sourcewellasshownin Figure21(a). Using more sources results in more slices in the wavenumber
domain as illustrated
in Figure 21(b). The rangeof all possiblekoõis depictedby the dashed
line. We useequation (122)to compute the modelfunctionAYl[ko(õ + •)] along the solid slicesin the k, - kz plane. For sucha multisourcemultireceiver configuration,the 2-D Fourier transform of the model function
can be recoveredonly within the portionsof the two disk-shapedregionsof the k• - kz plane coveredby the solid semicircles.
Most multisource-multireceiverconfigurationsprovide only a partial coverageof M(ka,,kz) leaving parts of the model uncertain. Information from sliceson the entire k•- k• plane is required to uniquely de-
termineM(x, z), whichis computedfrom the inverseFouriertransformof M(k•,, k•). Most of the informationprovidedby the crosswell configuration in Figure 21 is in the directionof the vertical wavenumberkz. Only longer wavelengthinformation(smallerwavenumbervalues)about the modelis provided in the horizontal wavenumberdirection k•. Thus, with the crosswell configurationwe expect good vertical resolution and poor horizontal resolution. Also, the resultingmodel is nonuniquebecausewe may arbitrar-
ily define a modelspectrum to "fill-in"theundefined partsof/l•/(k•,k,).
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
71
•x
()
()
()
() )
ß
8ource
locus of ko 8
0
receiver
•o•.. o• • o(• + •)
(a)
(b)
Fla. 21. (a) The situationin Figure20 is expandedby placingsourcesin depthfrom+oo to -oo. (b) The dashedcircleis the terminusof the vector ko• for the sourcesin (a). The solid circlesare the resultingvectorsum
ko(•q-I5).Nowthemodelfunction/f4(k=, k,) iswelldefined withinthetwo circular regionsof solid line coverageusing the generalizedprojection slice
theoremfor the crosswellconfigurationdefinedby equation(122).
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72
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
ds
dp
z
.
source
o receiver
FIG. 22. Source-receivergeometry for the VSP experiment.
We restrict
our reconstruction of the modelfunctionto x < dr and z > d,.
3.3.2
Vertical Seismic Profile Configuration
Now we derivethe generalizedprojectionslicetheoremfor the vertical seismicprofile (VSP) source-receiver configurationin Figure 22. Sources (soliddots)are deployedalonga line parallelto the x-axis at a constant vertical location z, = d,. Receivers(open dots) are deployedinsidea receiverwell parallelto the z-axisat a constanthorizontallocationxr = dr. Thus, the positionvectorsfor the sourcelocation r• and receiverlocation
rr canbe expressed in termsof theirrespective coordinates (x•,d•) and (dr,zr). Weshallassume thatthedomainof themodelfunction M(x, z) is restrictedto x < dr andz > d•. The derivationfor the VSP configuration followscloselythe derivationof the crosswellconfigurationof the last section. In fact, we end up with the sameform of the generalizedprojection
slicetheoremgivenby equation(122), but with its applicationbasedon the VSP geometry.
Substitutingthe VSP coordinatesinto equation(101) for the position
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
73
vectorsrs and r r gives
P(zs,ds;dr,zr) -
-ko•
M(z,z)
(123)
Again we chooseequation (62) to representthe Green'sfunctionsin a 2D medium so that the acoustic sourcesare infinite line sources,the diffractors in the model are infinite line diffractors, and the receiversare infinite
line receivers. Thus, we are once again dealing with cylindrical acoustic waves in a 2-D medium.
The developmentof the Green's functions here closelyfollows the discussionof the previous section and will not be presented in detail. The
Green'sfunctionG(z,z I z,,d,) for the infinite line sourceat (z,,d,)is found by substitutingthe appropriatecoordinatesinto equation(62) for position vectorsand performing a plane wave decompositionon the zer•
orderHankel function H••) ofthefirstkind.Theresulting integral formof the Green's function
is
Jff• •eJ[k•(xx,)+7•(z - d,)]dk•, (124) G(x, zlx•,d,) = 4• 7• for z > ds. Note that, as before, the wavenumber component k• is taken
along the sourceprofile which for the VSP geometryis in the direction of the x-axis. The wavenumbercomponent7x is alongthe z-axis and is defined
as7• - v/ko • - k•. A similardevelopment oftheGreen's function G(dr,zr I z, z) for an infinite line diffractorat (z, z) whoseenergyis recordedby a receiverlocatedat (dr, zr) givesthe equation,
j f: 1eJ[k•.(z r_z)+
- z)lak2, (125)
for z < dr. Herethe wavenumber componentkg.is takenalongthe receiver profile which for the VSP geometryis in the direction of the positive z-axis. The wavenumber79. is in the positive x-axis direction and is defined as
Finally, we must write down the equation for taking the Fourier trans-
formof the datafunctionP(xs, ds;dr, zs)in equation(123)alongthesource line (xs) and the receiverline (zr). This relationship is foundby first taking the Fouriertransformof P(xo,ds;dr, zs) with respectto zs followedby a secondFouriertransformwith respectto zr whichproducesthe double integral Fourier transform,
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CHAPTER
74
3.
P(•:, , d, ;ct,, , •:,,)
SEISMIC
DIFFRACTION
TOMOGRAPHY
/_:,o /_•P(z,, d,; dp, zp) x e-jk'x' e-jkpzp dxsdzp,
(126)
wherek, and kp are the wavenumbers of the Fouriertransformalongthe source line and the receiver line, respectively. Note that k• is in the same
directionas k, and k= is in the samedirectionas We continue the derivation of the generalized projection slice theo-
rem for the VSP configurationby substitutingequation(123) into equa-
tion(126)to geta relationship between thedatafunction P(k,,d,;dr, and the modelfunctionM(x, z). Carrying-outthis substitutiongivesthe equation,
where we have changedthe order of integration. Next we substitutethe Green'sfunctionsdefinedby equations(124) and (125) into equation(127), regrouptermswith similar integrationvariables, and interchangethe order of integration for the two innermost integrals over each Green's function, yielding the equation,
•5(k, dp, kp) =k• 1 /_• /_©M(a:, 'd,; 4 (2a')' oo z)
{/_
(128)
}
• • d[-•z+•(d.- •)]
•-i(• - •)Z•z.d• d•dz
The integrals in equation (128)withrespect to the variables x• andzr are easily evaluated in terms of Dirac delta functions as
'øe-J(k• +k,)Xsdx '
(129)
_=ø e-J(k p- k2)zp dz r
(130)
and
2•r5(kp- k:•).
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
75
Substitutingequations(129) and (130)into equation(128) and evaluating the integralswith respect to the variableskl and k2 (rememberingthat • - •o• -•i •na • - •o• -•) give•,
'
ß
'
=
--
4
7•7p
•
•
M(•, •)
• •-J(•. +7•)•-J(•-%)•&,
(•s•)
where7, and7p are definedby equations(113) and (114). Rememberthat the wavenumbercomponentsk, and kp parallel the sourceand receiver profiles, respectively. As we saw in the previoussection,the exponential terms in the integrand
of equation(131) may be rewritten• the dot productof the vectorsum, •o(• + •)
: = :
(-•i - 7•) + (7•i + •) (•.i- %•) + (•i + %•) (•. +•)i+(%-%)• k•+k•,
(•3a)
with the positionvector r pointing to (x,z). The unit vector • points in the oppositedirectionof a plane wave traveling from a sourceto (x, z) and the unit vector • points in the direction of a plane wave propagating kom (x, z) to a receiver.The components of ko• and ko• are showngraphically in Figures23(a) and 23(b), respectively.Sourcesare deployedkom +• to -• alongthe sourceline parallel to the x-axis and receiversare deployed kom +• to -• alongthe receiverline parallel to the z-axis. This geometry resultsin vectorspointing to the d•hed semicirclesin Figure 23. Performingthe dot productbetweenequation(132) for the VSP configuration and the position vector r gives,
•o(a + •).•
-
(•i
+ •).
(•a + •)
=
k• x + k• z
=
(k, + 7p)x+ (kp- 7,)z.
(133)
Note that equation(133) multipliedby -j is just the term in the exponential of the integrandin equation(131). Substitutingequation(133) into equation(131) resultsin the equation,
ß
-
•
x e-J(k•x + k•z)dxdz
M(•, •)
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76
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
kp
:
ks
:Kx
Kz (a)
(b)
FIG. 23. Wavenumber vector componentsfor the VSP configurationas de-
finedby equation(132). (a) The components of a sourcewavenumber vector kogare shownas projectionsonto the k•-axis and kz-axis;and likewise,(b) the componentsof a receiverwavenumbervector koI3.
(134)
where37/(k•,kz)isthe2-DFourier transform ofthemodelfunction M(x, z) takenoverthe imagearearestrictedto x < dr and z > d,. Notethat equation (•a4) is the sameas equation(122) for the crosswell configuration exceptfor the directionsover which the doubleintegral Fourier transforms
of the data functionP(x,, z,; zr, zp) are taken.We requirethat the transforms be taken along the sourceand receiverprofiles,which for the VSP configurationare differentfrom the crosswellconfiguration. Equation(134) is the generalizedprojectionslicetheoremfor the VSP
configuration. It statesthat P,(k,,d,;dp,kr), thedoubleintegralFourier transformof the data function, taken along the sourceline and the receiver
line, is equal to M[ko(,• + 15)],the 2-D Fouriertransformof the model function, evaluatedalong the semicircularslicemultiplied by the quantity -Y
%%
'
Figure 24(a) showssourcesand receiversdeployedfrom +oo to -oo alongtheir respectiveprofilesfor the VSP configuration.Figure24(b) is a representationof the model coverageprovidedby the VSP configuration
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
77
z
Kz ^
ß source
locus of ko 8
o receiver
locusof/•• ( • + I•)
(a)
(b)
FIG. 24. (a) Sourcesand receiversare extendedfrom +cx>to -cx> along their respectiveprofiles.(b) The dashedcircleis the terminusof the vector ko• for the sourcesin (a). The solidcirclesare the resultingvectorsum
ko(•+/3). Themodelfunction iff(k•,k,) is welldefined withinthezone of solidline coverageusingthe generalizedprojectionslicetheoremfor the VSP configuration definedby equation(134).
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78
CHAPTER
3.
SEISMIC
DIFFRACTION
0
TOMOGRAPIIY
.X
ß
source
o receiver
FIG. 25. Source-receivergeometry for the surfacereflection experiment, where in the marine data acquisition case the depths are measured from sea level.
We restrict
our reconstruction
of the model function
for the
geometryshownhere to z • dp.
wherethe model reconstructiondefinedby equation(134) is restrictedto
pointsx • dpandz • d,. The rangeof all possible ko• is depictedby the dashedline in Figure24(b). We useequation(134) to computethe model
function]l•[ko(•+ •)] alongthe solidslicesin the k• - kz plane.Note that theVSPconfiguration provides a different coverage of ]17f(kx, kz) than the crosswellconfigurationin Figure 21. Thus, we shouldexpect different estimatedmodel functionsM(x, z) from each configurationeven though the true model is the same. As with the crosswellconfiguration, resolution
varieswith directionand the resultingmodel is nonunique.
3.3.3
Surface Reflection Configuration
The last source-receivergeometry we will derive the generalizedprojection slicetheoremfor is the surfacereflectionconfigurationin Figure 25.
The sources (soliddots)aredeployed alonga lineparallelto the x-axisat a constantverticallocationz• - d•. Similarly,the receivers(open dots) are deployedalong a line parallel to the x-axis at a constantvertical location
zp= dp. Thus,the positionvectorsfor the sourcelocationr• andreceiver locationrp canbe writtenin termsof theirrespective coordinates (xs,ds) and(xp,dp).Weshallrestrictthedomainof themodelto depthsbelowthe deeperof the two, d• or dp. As with the previoustwo configurations, the derivationheregeneratesa similar form for the generalizedprojectionslice theorem, but with its applicationbasedon the surfacereflectiongeometry.
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3.3.
GENERALIZED
PROJECTION
79
SLICE THEOREM
We beginby substitutingthe surfacereflectioncoordinates for the positionvectorsr• andrr intoequation(101)whichgives
x
zI
4, I
z)aaz.
We assume a 2-D mediumandchoose equation(62) to represent theGreen's functionsfor the infinite-lineacousticsources,infinite line diffractors,and the infinite line receivers.Thus, all acousticwaveswill be cylindrical.
The Green'sfunctionG(x,z I x•, d•) for the sourceis the sameas the VSP'sGreen'sfunctiondefinedin equation(124) sinceboth configurations have the same sourcegeometry.Thus,
J/5 1eJ[k•(xx)+?•(zd•)ldk•, (136) for z > d,. The wavenumber component k• is takenalongthe sourceprofile which is in the direction of the x-axis. The wavenumber component 7• is
alongthez-axisandis defined 7• - V/ko 2 - k•2. Sincethe receivergeometryis similarto the sourcegeometryweusethe same form of the Green's function as for the sourcesgiving
3__'/5 1eJ[k9.(x r_x)-72(d r- Z)]dk 2(137) wherek2 is the wavenumbercomponentparallelto the receiverline and 72
is alongthe z-axis,where72 - V/ko • -k•. Notethat -?2(dr - z) is used sincewe removedthe absolutevalue in the plane wave decompositionof the Hankel functionfor z > dr.
Proceeding with the derivation, wewritedownthe equation for taking theFouriertransform ofthedatafunction P(x•, d•;xr, dr) in equation (135) alongthe sourceline(x•) andreceiver line(xr). The relationship is found by first takingthe Fouriertransform of P(x•, d•;xr, dr) with respectto xs, followed by a second Fouriertransformwith respectto xr whichproduces the doubleintegral Fourier transform, P(zo,d•;zr,
x e-Jz•koe-Jxrkrdx•dxr,
(138)
wherek, andkr arethewavenumbers of theFouriertransforms alongthe sourceline and the receiverline, respectively.As with previousconfigurations,k• is in the samedirectionas k, and k2 is in the samedirectionas kr ß
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CItAPTER
8O
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
Next, we substituteequation(135) into equation(138), giving
P(k,,d,;kp,dr) -
-k•
M(x,z)
(139)
x
•(•, • I •., a.)•-j•'•'
x
6(•:,,,4, [ •:,z)•-j•'"
where we have changedthe order of integration. Next, we substitutethe Green'sfunctionsdefinedby equations(136)
and (137) into equation(139), regrouptermswith similarintegrationvariables, and interchangethe order of integration for the two innermostintegralsovereachGreen'sfunction,yieldingthe equation,
15(k '' d,;kr, dr) = k•o 1 • 4 (2•r)' x
x
M(z' z)
(140)
{f_•1•'It" +'•'(' - •')1 j'_"-'(•'+t')' } --
e
--
sdzsdk•
e-j(kr - k=)zPdzrdk= dzdz.
The integralsin equation(140), with respectto the variablesx, and are easily evaluated in terms of Dirac delta functions as
øø e-j(k• +k,)Z,dx ' 27rS(k•+ k,),
(141)
j e-j(kp - k:•)%, 2•rS(k•,- k2).
(142)
and
Substitutingequations(141) and (142)into equation(140) andevaluating the integralswith respectto the variablesk• and k= (rememberingthat 7• - ko • -k• and7• - ko • -k•) gives,
ko •ej(-%dp - -/,d,)oo M(x, z) 4 7,% oo oo (143)
where7, and 7p are perpendicular to k, and kp.
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3.3.
GENERALIZED
PROJECTION
ß
SLICE
THEOREM
t•s':-Kx
81
t•p •Kx
•z
•:z
(a)
(b)
FIG. 26. Wavenumber vector componentsfor the surfacereflectionconfigu-
rationasdefinedby equation(144). The dashedsemicircles arethe termini of the wavenumber vectorsfor sourcesand receiverspositionedfrom -oo to
+cx•alongthe x-axis. (a) The components of a sourcewavenumber vector kogare shownas projectionsontothe kx-axisand kz-axis;and likewise,(b) the componentsof a receiver wavenumber vector koP.
As we saw for the earlier configurations,the exponential terms in the
integrandof equation(143) may be rewritten as the dot productof the vector sum,
•o(• + •)
=
(-• - 7•.) + (•.• - 7•.•.) (•0• - •0•.) + (• - •.)
= =
(•o + •)• + (--r• --r•)• kx• + kz•,
(144)
with the positionvectorr pointingto (x,z). The unit vector.qpointsin the oppositedirectionof a planewavetravelingfrom a sourceto (x, z) and the unit vector f) points in the directionof a plane wavepropagatingfrom (x, z) to a receiver.The components of koõand ko•)are showngraphically in Figures26(a) and 26(b), respectively.Sourcesare deployedfrom +oo to -c• along the sourceline parallel to the x-axis and receiversare deployed from +c• to -oo along the receiverline also parallel to the x-axis. This geometryresultsin vectorspointing to the dashedsemicirclesin Figure 26. Performingthe dot product betweenequation(144) and the position vector r gives,
•o(• + 0).,,
-
(•
+ •,•)-(•
+ z•)
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82
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
=
k•z + kzz
=
(k, + kr)z + (-7, -7r) z.
(145)
Notethat equation(145) multipliedby -j isjust the term in the exponential of the integrandin equation(143). Substitutingequation(145) into equation(143) resultsin the equation,
_ ko •ej(-7rdr - %d,)oo M(x, z) 4 %%0 oo x e-J(k•x + k•z)dxdz =
_ 4
=
-4
- %d,)_
• •j(-•dp - •)
.
U[ko(• + •)],
(146)
where 217/(k•, k.) isthe2-DFourier transform ofthemodel function M(x, z) takenoverthe imagearearestrictedto the greaterof z > d, or z >dp. Equation(146) is the generalized projectionslicetheoremfor the surface
reflection configuration. It states that['•(k,,d•;kr,dp),thedouble integral Fourier transform of the data function, taken along the source line and
the receiver line,is equalto 20[ko(S + iO)l,the 2-D Fouriertransform of the model function, evaluatedalong the semicircularslicemultiplied by the
quantity --•eJ(-7pdp %%0- 7sds) ' Figure 27(a)shows sourcesand receiversdeployedfrom alongtheir respectiveprofilesfor the surfacereflectionconfiguration.The rangeof all possibleko• is depictedby the dashedline in Figure 27(b). We useequation(14{5)to computeM[ko(g+ •)] alongthe solidslicesin the k: - kz plane. The surfacereflectionconfigurationprovidesa different coverage of M(kz, k,) than the crosswell and VSP configurations andtherefore we shouldexpect differentestimatedmodelsfrom each configuration. Once again,each configurationprovidesa differentresolutionand degreeof nonuniqueness dependingupon how the configuration"fills-in" the model spectrum and how much of the model spectrum the configurationleaves undefined.
3.4
Acoustic Diffraction Tomography
The generalizedprojection slice theorem was derived in the previous sectionfor the crosswell,VSP, and surfacereflection configurationswhich
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3.4.
ACOUSTIC
DIFFRACTION
•
TOMOGRAPttY
83
x
source
receiver
........
locus of/• o s
locus of/• o (s + p)
(a)
(b)
F•G. 27. (a) Sourcesand receiversat the samedepth are extendedfrom +c• to -c• in the directionof the x-axis. (b) The dashedcircle is the terminusof the vectorkoõfor the sourcesin (a). The solidcirclesare the
resulting vector samko(g+I5). Themodel function A7/(kx, k,) iswelldefined within the zoneof solid line coverageon the -k• half plane.
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CHAPTER
$4
3.
SEISMIC
DIFFRACTION
TOMOGRAPtIY
correspond to equations(122), (134), and (146), respectively.Foreachconfigurationthe theoremestablishesa relationshipbetweenthe doubleFourier transformof the data functions takenalongthe sourceand receiverprofiles and the 2-D spatial Fourier transform of the model function. Further analysisof the generalizedprojection slicetheorem showsthat each sourcelocation definesthe model function along a circular arc in the ks - k• domain, as depicted in Figures 21, 24, and 25 for each source-receiverconfiguration. To obtain the model function M(x, z) we must take the inverseFourier
transform of.•/[ko(•+•)] whichisnotreadilycarried-out sinceAY/[ko(•+•)] is defined by circular arcs. Thus, in this section we develop two image reconstructionalgorithmsto handle this problem called the direct-transform diffraction tomography and the backpropagationdiffraction tomography.
Thedirecttransform methodinvolves estimating thevalues of •[ko(•,+ P)] on the k•- k• coordinate plane, which is analogousto the direct-transform ray tomography method presentedin Section 2.2.2. On the other hand, the backpropagationmethod takes a more elegant mathematical approach to achieve the same result. The backpropagation method presented here is analogousto the backprojection ray tomography presentedin Section2.2.3.
3.4.1
Direct-Transform
Diffraction Tomography
The generalized projection slice theorem enables us to compute the 2-D Fouriertransformof the model function M[ko(• + •)] from s•tte•ea wavefielddata. Therefore,the unknownmodel functionM(•, z) can be ob-
tainedaslongaswecanperform aninverse Fourier transform onAYl[ko(• + •)]. However,most inverseFourier transformalgorithmsuse Cartesian coordinates • which meansthey can handle the inverseFourier transform
from?l•f(k•,k•) to M(z,z), butnotfrom?l•f[ko(• + •)j to M(z,z). Figure 28 illustratesthe differencebetweenthe loci of M[ko(• + •)l (solid
dot•)•nd thelociof •(k,, k•) (opendot•)for a crosswell configuration. Clearlythe lociof the tworepresentations of AY/do not coincide andwe cannot directly make use of the inverseFourier transform.
An obvious,brute forcesolutionis to estimatevaluesof M(k•, k•) on a
rectangular kz - k• gridfromvalues of AY/[(• + •)] obtained alongcircular arcs using the generalizedprojection slice theorem. As mentionedin Section 2.2.2, one must exercise caution in performing an interpolation since the operation may introduce error which can obscure the true solution. However, once the values of M are found on a k• - k• rectangular grid,
we can find the unknownmodel function M(x, z) through a 2-D inverse sKeep in mind that in diffraction tomographythe data functionconsistsof measured projections of scattered energy.
•Namely, (x,z) in the spacedomainand (k•, k,) in the wavenumberdomain.
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3.4. ACOUSTIC DIFFRACTION TOMOGRAPHY
o
o
o
85
o
o
o
o
o
K,x o
o
o
o
o
o
o
0 Iocu• of (k,•, k•.) grid
FIG. 28. The generalizedprojectionslicetheoremdefinesthe modelfunc-
tionM(k•,,kz) at thesoliddotsalongcircular arcsin theks- kzplane.To gettheestimated modelfunctionM(x, z) the2-D inverse Fouriertransform
mustbe takenof M(kr,k,) defined at points(opendots)on a Cartesian grid. Direct-transform diffraction tomography definesthe opendotsfrom the solid dots by interpolation.
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86
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
Fourier transform. This is called direct-transform diffraction tomography and can be summarized in five steps' Step 1: Acquire tomography data by probing the target with acoustic waves and record the scattered
wavefield
informa-
tion fromthe target, represented by the data functionP(x, z). Step 2: Take the double Fourier transform of the data function
P(x, z) along the sourceand receiverlines, which can be representedby
=
,of_,o
't,+
'
(147)
Here is and ir are the distancesalongthe sourceand receiver lines, respectively. Equation (147) correspondsto equation(107) for the crosswellconfigurationwhen is - zs and ir = zr; to equation(126)for the VSP configuration when I• -- x• and ir = zr; and to equation(138) for the surfacereflectionconfiguration whenl• = x• and lr - xr. Step 3: Compute the 2-D Fourier transform of the unknown
modelfunction A7l[ko(k 4-•)] alongcircular arcsusingthe generalizedprojection slicetheorem,
ll•l[ko(• -t-•)] = 47o%, pdp4-'y,d,)•5(k,, kp). k• ø e- j (:l:'y
(148)
Equation(148) corresponds to equation(122) for the crosswellconfiguration, equation(134) for the VSP configuration, and equation(146) for the surfacereflectionconfiguration. Step 4: Perform a 2-D interpolation on the values of M from
thegeneralized projection slicetheorem to determine AT/on the rectangular k,-
kz grid, (149)
Step 5: Take the 2-D inverseFouriertransformof M(k•,kz) to obtain M (x, z),
M(zz)= I
"
1171(k• k,)e j(k•z+k'Z)dk•dk, (150)
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3.4.
ACOUSTIC
3.4.2
DIFFRACTION
TOMOGRAPttY
87
Backpropagation Diffraction Tomography
We saw that direct-transformdiffractiontomographyestimatedvalues
ofthemodelfunction •[ko(õ + •))] ona rectangular k, - k, gridenabling usto taketh• inverse 2-D Fouriertransform of • to getthe estimated modelfunctionM(x, z). The interpolationsteprequiredis computationally slow and the possibilityof introducingerrors into the interpolatedmodel function is very real. In this section we introduce the more mathemat-
ically elegant method of backpropagationdiffraction tomography. With this methodthe requiredcoordinatetransformationfrom M[ko(• + •)] definedon semicircles to M(kx, k,) definedon a rectangulargrid is performed while taking the 2-D inverseFourier transform thereby obviatingthe need for interpolation.
The derivation begins with the 2-D inverseFourier transform of the modelfunctionM(k•, k•) givenby
1• •ß ß•(k•,k,)eJ(k•z +k,Z)dk•dk, M(x,z) = 4• . (151) Since•[ko(• + •)] is defined in termsof thewavenumbers (k•, k•), wewill performa changeof variablesfrom (k•, k,) to (k•, kr) on the right-hand sideof equation(151). This will permit us to directlytake the 2-D inverse
Fouriertransform of the modelfunction•[ko(• + •)] withouthavingto perform an interpolation • required by the direct-transform diffraction tomography.Equation(151) in the form for this changeof variablesis
M(x z) = 4•21 •• ' x I
•[ko(• +•)]eJ[k•(k•, kr)z +k,(ks, kr)z ] I
(152)
where•[ko(i + •)] is themodelfunction fromthegeneralized projection slicetheorem,k•(k,,kr) and k,(k,,kr) are the wavenumbers expressed in termsof the variables(k•, kr) , and J(kr, k, [ k•, kr) is the JacobJan relating dk,dkr with dk•dk,. The JacobJan can be expressed in termsof the determinant,
J(k• k,]k,,kr) ' Ok:Ok,
Ok•Ok:
= Ok: Ok r Ok r Ok,'
(153)
The form of equation(152) is • far • we can go without selectinga particular source-receiver configuration.In the followingderivationswe shall
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88
CHAPTER
3. SEISMIC DIFFRACTION
TOMOGRAPttY
complete thechange of variables for thethreesource-receiver configurations presented in the previous section.The resultingequationfor eachconfiguration is in a form which permits use of a 2-D inverseFourier transform
algorithmandthe resultis backpropagation diffraction tomography. Crosswell Configuration.--The modelfunctionfor the crosswell configurationisdefined bythegeneralized projection slicetheoremin equation (122) which may be rewritten as
.l17f[ko(• +f))]= 4%7r r - %d•)P(d• ko • e-j(%,d ,k•;dr,kr) . (154) From equation (119) we see that the coordinatesystem transformation
between(k•, kz) and (ks,kr)is givenby
Substituting this coordinate systemtransformation into equation(153) givesthe Jacobianfor the crosswellconfiguration,
J(k•,kz Ik,,kr)= k,%,+k•,%.
(156)
Substituting equations (154),(155),and(156)intoequation(152)gives,
M(z 1/••'/_• ,z)= 4•.2 •'4%7r •-o•e-j(7rdr_%d•) -
+
+
x [ks 7r+krYs [dks dkp, and rearrangingtermsgivesthe equationfor backpropagation diffraction tomographyfor the crosswellconfiguration,
(157)
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3.4.
ACOUSTIC
DIFFRACTION
TOMOGRAPttY
89
VSP Configuration.-- The model function for the vertical seismicprofile configurationis definedby the generalizedprojectionslicetheoremin equation (134) which may be rewritten as
•[•o(•+•)] = 4,0,• ko • e_j(,• _,,•,)p(•,,• ;• ,•) . (•S8) From equation (133) we see that the coordinatesystem transformation
between(k•, k•) and (k•, kv)is givenby
k•(k,, k•) =
k, + 7•
= •+•-•,•na = kv- •k•- k•.
(159)
Substitutingthis coordinatesystemtransformationinto equation(153) givesthe Jacobianfor the VSP configuration,
Substitutingequations(158), (159) and (160)into equation(152) gives,
M(x, z) = 4•• ß ß • e x [k•k v+%7v ]dk•dkv, and rearrangingterms gives the equationfor backpropagationdiffraction tomographyfor the VSP configuration,
M(•,•) = •
ß ß
k•
'
x eJ[(k, + 7v)x + (kv- 7,)z]ak, akv.
(161)
Surface Reflection Configuration.-- The model function for the surface fiection configurationis definedby the generalizedprojection slice theorem in equation(146) whichmay be rewritten •
•[ko(• +P)] = 47•7• e-J(-7•d• - 7•d•)p(k• d,'k•,dp). (162)
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CHAPTER
9O
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
From equation (144) we see that the coordinatesystem transformation between(k•, kz) and (ks,kp) is givenby
--
ks+ kr, and
=
-%
- 7r
(163) Substitutingthis coordinatesystem transformationinto equation (153) givesthe JacobJanfor the surfacereflectionconfiguration,
s(•,• I k•,•) = kr•'• - k•'•r.
(164)
Substitutingequations(162), (163) and (164)into equation(152) gives,
and upon rearrangingterms we get the equationfor performingbackpropagationdiffractiontomographyfor a surfacereflectionconfiguration,
1f_•/:I•'r,-/•,•'• Ip(/• •) (165) x eJ[(ks + kp)x + (-% - 7p)Z]dk,dk•, '
Equations(157), (161), and (165) are usedin backpropagation diffraction tomographyimage reconstructionalgorithmsfor the crosswell,VSP, and surfacereflectionconfigurations,respectively.
3.5
Summary
1. The model used for acousticwave scattering was the Helmholtz form of the acousticwaveequationfor a constant-densitymedium of vari-
able velocityC(r) givenby
IV • + •:=(•.)]•,(•.) -
o,
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3.5.
SUMMARY
91
where Pt represents the total wavefield which contains both the incident wavefield from the source and scattered
wavefields from inho-
mogeneitiesembeddedin a constant-velocitybackgroundmedium. 2. We treat the inhomogeneitiesas secondary sourcesand write the acousticwaveequation as the inhomogeneous differentialwaveequation,
IV +
-
where ko is the constant wavenumbermagnitude of the background
mediumand M(r) is the modelfunctionfor diffractiontomography defined as a perturbation from the constant backgroundvelocity Co given by,
.
We found two integral equation solutions to our formulation of the acousticwave scattering problem. The first was found by letting the
total wavefieldPt(r) be the sum of the incidentwavefieldPi(r) and scatteredwavefieldP,(r). We formulateda Green'sfunctionintegral solution to the inhomogeneousdifferential wave equation in terms of the scattered wavefield P, resulting in the nonlinear LippmannSchwingerequation. We linearizedthe Lippmann-Schwingerequation
by assumingthe Born Approximation(i.e., P•(r) << Pi(r)) resulting in a linearrelationshipbetweenthe data functionPs(r) and the model functionM (r),
I
I r')dr',
whereP.(r.,rp) becomesthe data functionobservedat positionrp when the negative impulse sourceis located at position r•.
The
Green'sfunctionsare definedby eitherequation(62) or (63). 4. The secondintegral equation solution was found by representingthe
totalwavefield Pt(r)bytheexponential equation eqSt(r), where the complextotal phasefunctionqbt(r)was set equal to the sum of the complexincidentphasefunctionq5i(r)and the complexphasedifference function qba(r). After lengthymanipulationsof thesephases, formulating a Green's function solution to the resulting inhomoge-
neouswaveequationwhichoperatedon the quantity Pi(r)c)a(r), and
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92
CHAPTER
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TOMOGRAPttY
applyingthe Rytov approximation (i.e., Vdd(r) << 1), we found,
P•(r,,r•)•a(r,,r•) • -ko •/M(r')G(r' Ir,)G(r• Ir')dr'. Herethe data functionis P•(r,, r•)qba(r,,r•), insteadof P,(r), which is linearlyrelated to the modelfunctionM(r). 5. The Born approximation is valid for weak scatterers of limited size while the Rytov approximation requires only a smooth model function.
6. Letting P(r,,r•) representthe data functionfor either the Born or Rytov approximationintegralsolutions,we simplifiedthe integral equationsolutionsto the generalizedprojection slicetheoremfor three sourcereceiverconfigurations.The generalizedprojection slice the• rem for the crosswellconfigurationis
P(d, k,ßdp, kp)= --k• eJ(%dp7,d,)_ ' ' 4 Thisequationestablishes a relationship betweenP(d•, k•;d•, kp),the doubleintegralFouriertransformof the data functionalongthesource andreceiverprofilesin the crosswell configuration, and M[ko(• + •)], the 2-D Fouriertransformof the modelfunctionalongcurvesin the k• - k, plane. The curvesare definedby the sum of the vector ko•
whichpointsfrom the point (•, z) in the modelto the sourceandthe terminusof the vectorko• whichpointsfrom (•, z) to eachreceiver. 7. The generalizedprojectionslicetheoremfoundfor the VSP configuration
is
'
ß '
=
-4
+
8. The generalizedpro•ection slice theorem found for the surMce reflection configuration is
P(k, d,'kp, dp)= k• eJ(-7p½ - 7,d,)•[ko(g +•)1. • • 4 7s7p 9. Thecoverage of •(kz, kz)in thekz- kzplaneisdependent uponthe sourcereceiverconfigurationand alwaysincompleteto somedegree. This resultsin a nonuniquelyestimatedmodel M(x, z) that h• inherentlypoor resolutionin directions•sociated with the incomplete coveragein the kz - k, plane.
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3.6.
SUGGESTIONS
FOR FURTHER
READING
93
10. In Section3.4 the generalized projectionslicetheoremcouldnot be directlyappliedto estimatingM(z,z). This occursbecausethe 2D Fourier transformrequiresdata points on a rectangulargrid in the kx- k; plane and the generalizedprojectionslice theorem defines the data points on semicircles.The direct-transformdiffraction tomographyjust performsan interpolationto solvethe grid differences, or
before taking the inverse2-D Fourier transform to get the image
M(x, z). However,the potentialfor interpolationerrorshowingup in M(x, z) is very real. 11. The backpropagationdiffractiontomographymethod accomplishes thesameresultasthedirect-transform methodby replacingthe (kx, k,)
variablesin the 2-D inverseFouriertransformby (k0,kr), the variablesusedin the generalizedprojectionslice theorem. The resulting coordinatetransformationis dependentupon the source-receiverge-
ometry,but wecanfind M(z, z) directlythrougha 2-D inverseFourier transform without the need for interpolation.
3.6
Suggestions for Further Reading Devaney,A. J., 1984,Geophysicaldiffractiontomography:IEEE trans., ClE-22, 3-13. First proposalof seismicdiffractiontomography.
Esmersoy, C., Oristaglio,M. L., and Levy, B.C., 1985,MultidimensionalBorn velocityinversion'singlewidebandpoint source: J. Acoust. Soc. Am., 78, 1052-1057. Reformulares the Born inversion problem so that the data function becomes the field extrapolatedthroughthe wave equation.A variable backgroundvelocity is permitted.
Morse, P.M., and Feshbach,H., 1953, Methods of theoretical physics:McGraw-Hill. We just throw equation(10•), the plane-wave decompositionof a cylindrical wave, right at you withoutproof. Section 7.2 of this referencederivesthe equation in detail with the resultingequationfound on page 8œ3.
•zbeck,A., and Levy,B.C., 1991,Simultaneous linearized inversion of velocity and density profiles for multidimensional acoustic media: J. Acoust. Soc. Am., 89, 1737-1748.
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94
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPIIY
This paper includesthe formulation for simultaneouslyreconstructing both the velocity and density using the Born approximation. Pratt, R. G., and Worthington, M. H., 1990, Inversetheory applied to multi-source cross-holetomography. Part 1: Acoustic wave-equation method' Geophys. Prosp., 38, 287-310. We have presenteddiffraction tomographyin terms of a homogeneousbackgroundmedium with velocityCo. This paper presentsa nonlinear inversiontechniquein the frequencyspacedomain which essentiallypermits inhomogeneous acoustic backgroundmedia. A companionpaper treats the same subjectwith respectto elastic media. Rajan, S. D., and Frisk, G. V., 1989, A comparisonbetweenthe Born and Rytov approximationsfor the inversebackscattering problem: Geophysics,54, 864-871. Includes numerical comparisonsof the two approximations.
Wu, R. S., and ToksSz,M. N., 1987, Diffraction tomographyand multi-sourceholography applied to seismic imaging: Geophysics, 52, 11-25. Modified Devaney's plane wave seismic diffractiontomographyusingline sourcesandformulated the backpropagationreconstructionalgorithmsfor crosswell, VSP, and surface seismic geometries.
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Chapter
Case 4.1
4
St udies
Introduction
The purposeof this chapter is two-fold. First, through casestudies,we illustrate the proceduresfor implementingthe theory presentedin Chapters 2 and 3. Second,the casestudieshighlight some of the benefits of using seismictomographyin the oil industry. The first two casestudiesutilize crosswellseismicdata in conjunction
with the simultaneous iterativereconstruction technique(SIP•T)presented in Chapter 2 on seismicray tomography. The first casestudy addresses the production problem of monitoring the progressof a steam-floodenhancedoil recovery(EOR) program.The secondcasestudy involvesmore of a developmentproblemin which the structural interpretationof a faultcontrolledreservoirmust be better understoodfor in-fill drilling. The third casestudy usesthe seismicdiffractiontomographypresentedin Chapter 3 to image two salt sills using marine surface seismicdata. We selected this problemto illustratethe seismicdiffractiontomographymethodologyand limitations rather than to solvean exploration problem. 4.2
Steam-Flood
EOR Operation
Somereservoirsencounteredby the oil industry containpetroleumresemblinga heavy,tar-like substancerather than a low-viscosity fluid. To produce such reservoirs,steam is injected into the reservoirwith the intent of heating-upthe petroleummaking it lessviscousso that it may flow. A steam-floodenhancedoil recovery(EOR) operationis expensive and may be adverselyaffectedby reservoirinhomogeneities whichchannel 95
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96
CHAPTER
4.
CASE
STUDIES
steam away from parts of the reservoirthe production engineerwishesto heat. This casestudy demonstrateshow crosswellseismicray tomography can identify reservoirinhomogeneitiesbefore initiating steam flooding and monitor the EOR operation during steam injection. The Potter B1 tar-sand reservoir in the Midway Sunset Field, California, producesheavy oil using steam-flood EOR operations. We conductedtwo crosswellseismicexperimentsat a steam injection site in the Midway Sunset Field with the intent of aiding decisionmaking by the productionengineers on the steam-flood EOR operation. A presteam injection crosswellseismic survey was run to tomographically image reservoir inhomogeneitieswhich
might affect the operation and to provide a "base-line"P-wave velocity tomogram for comparisonwith a later poststeam injection P-wave tomogram. The P-wave velocity in this tar-sand reservoirdrops dramatically when the reservoiris heated. Thus, a secondpoststeam injection crosswell
seismicsurveyprovidesa P-wavevelocity tomogramwhich readily identifies the heated parts of the reservoirwhen comparedwith the "base-line" tomogram. The productionengineermay alter the chosensteam injection program if steam by-passesparts of the reservoir.
4.2.1
Crosswell Seismic Data Acquisition
The data acquisitiongeometryis depictedby the cross-section and map view in Figure 29. The seismicsourcewas located in the temperature observation well T02 which is deviated 99 ft to the south. The production well 183 served as the receiver well and is offset 283 ft from the TO2 well at
the surfaceand 184 ft at the TD of well TO2. The steam injector well is sit-
uateddowndipfrom the TO2-183 profile(dip directionis to the NNE) and the steam is expected to travel updip where it will intersect the crosswell profile. Bolt TechnologyCorporation's downholeair gun shownin Figure 30 was used as the seismicsourcein this experiment. The air gun has an 80 cubic inch chamber and is pressurizedto 2000 psi above the ambient pressurein the borehole. Geosource'sthree-componentgeophoneVSP tool was usedto detect the seismicenergy. A DSS-10 systemrecordedthe seismicsignalsat a sample interval of 0.5 ms with a one secondrecord length. The downhole air gun was fired at 40 depth stations from 1360 ft to 1750 ft at 10 ft intervals, while the VSP tool was set at fixed station depths. The source wasfired four timesat eachstation so that better signal-to-noiseratio could be attained in the stackedsignal. The 40 sourcestationswere repeatedfor each of the 40 receiverstations,which ranged from 1630 ft to 1240 ft at 10 ft intervals, giving a total of 1600 unique source-receiverpair locations. The common-receiver gather in Figure 31 was recordedby the vertical
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4.2.
STEAM-FLOOD
EOR
OPERATION
97
Midway Sunset Crosswell Experiment Cross-section
TO2 183
283
- 0
ft
- 200
- 400 _ 600 - 800
Potter
-
1000
-
1200
Depth
A B1
- 1400 -
1600
Antelope Shale -
1800
- 2000
<--
18411
Map View TO2
99fti TI•
"• 400 DIpI•i
:
+ Steam Injector ß 134R
18411"'-.. .
ß
183
FIG. 29. Well configurationfor the Midway Sunset crosswellexperiment shownin cross-sectionand map view. Solid circlesin the map view indicate well surface locations. The air gun source was placed in the temperature observationwell TO2 which is deviated 99 ft along the dashed line. A three-componentVSP tool was located in the vertically drilled production well 183. The dotted line indicates the profile defined by connectingthe
total depth (TD) positionsof the two wellswhich corresponds to a 184 ft horizontal
offset.
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98
CHAPTER4.
CASE STUDIES
FIG. 30. Bolt employeesare shownpreparing an 80 cubic inch air gun at the TO2 well. The umbilical to the air gun containsa pressureline and a wireline. Photo courtesy of Don Howlett, Texaco.
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4.2.
STEAM-FLOOD
EOR
OPERATION
99
component of the VSP tool fixed at a depth of 1500 ft while the source moved through its 40 stations. The P-wave first arrivals are easily identified and have a dominant frequency of approximately 200 Hz. The large amplitude event immediately followingthe P-wavefirst arrival is the S-wave first arrival.
Time
zero occurs at 60 ms on the record which was monitored
by a pressuresensormountednear the air gun. The large amplitude events trailing the S-wavearrival are from upward traveling tube waves. Steam injection commenced shortly after completion of the presteam injection crosswellseismicsurvey. The steam flood operation continuedfor approximatelyone year beforewe ran the poststeaminjection crosswellseismic survey. The secondsurvey was acquired along the sameTO2-183 profile with data acquisition parameters similar to the first survey. Tomographic imageswere processedusingtraveltime ray tomographyfor both data sets in an attempt to provide the productionengineerswith information on the performance of the steam flood operation. 4.2.2
Traveltime
Parameter
Measurements
The crosswellseismicdata in this casestudy were processedusing the
simultaneousiterative reconstructiontechnique(SIRT) describedin Section 2.3.3. A three-step recipe was given in that sectionfor reconstructing tomographicimages using SiRT. Here we explain how the traveltime parameters usedin the SIRT recipe were measured,first for a P-wave velocity tomogram and then for an S-wavevelocity tomogram. The traveltime parameters required by SIRT are the sourceand receiver locations and measured traveltimes. The first step in determining source and receiver locations is to run a borehole survey. At Midway Sunset we
usedGyroData'stool to obtainthe surfacelocations(x-y positions)for variousdepths(z-positions)in both the T02 and 183 wells. With the borehole geometriesknownall that remainsis to providegoodmeasureddepthvalues to the sourceand receiver. Depth gaugeson the vehiclesusedto deploy the sourceand receiver in the boreholesprovided the measureddepth values in
this experiment. The starting positionof the sourcewas recalibratedeach time a sourcepasswas completed. We havesinceattached collar locatorsto the sourceand receiver to calibrate the measureddepth estimates. Gamma ray loggingtools can be usedin place of collar locatorsto achievesimilar calibrations.
The most critical and tediousstep of ray tomography processingis selecting the traveltimesfrom the observeddata. The observeddata, given
by p/ob•forthe ith source-receiver pairin equation(46),shouldbeselected from data preprocessed for maximum signal-to-noiseratio. To provide this optimum environmentfor pickingP-wave traveltimes,the Midway Sunset
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100
CHAPTER
4.
CASE
STUDIES
Common-Receiver Gather (1500 ft) Source Depth (ft) 1400
P -Wave
1500
100-
1600
1700
,
•
,
,
;
$ -Wave
2ooI:
Time (ms)
Tube Wave
I
I
300 -
I I.
.i ii i
'
, 400 -
_•
,
,
F•(•. 31. Common-receivergather for the VSP tool at a depth of 1500 ft. The 40 traces correspondto the sourcestations.
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4.2.
STEAM-FLOOD
Source
EOR
OPERATION
Well
101
Receiver
Well
Low Velocity
High Velocity
Direct arrlval's ray path ....
Head wave's ray path FIc. 32. This cartoon depicts the raypaths for a direct arrival with traveltime Td and a head wave with traveltime Th • Td.
data required only a minimum-phasetrapezoidal bandpassfrequencyfilter of 50-100-200-300
Hz.
Generally one selects first arrivals on crosswellseismic data when Pwave traveltimes are desired. That is, the first arrival catchesour eye as being the first significant signal. However, in many casesthe first arrival is not a direct arrival, but a head wave which travels along an interface as shownby the solid raypath in Figure 32. The head-wavetraveltime pick createsa problem when the forward modeling method used in step I of the SIRT algorithm determinesonly direct arrival raypaths, such as the one
depictedin Figure 32. Obviously,a computedraypath giving the predicted
traveltimep•re in equation(46) whichis not consistent with the event's observed traveltimep/oh,will causewrongcellsin themodelto beupdated at incorrect
velocities.
The remedy for this dilemma can take two avenues. The avenuewith the
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102
CHAPTER
4.
CASE
STUDIES
potholes is to stick with the ray tracing schemewhich models only direct arrivals. Here we are forced into trying to identify when the first arrival is a head wave and then, whether we should attempt to pick a later, more noisy,event as the direct arrival or to make no traveltime pick at all. This can be a very frustrating processwhen the subsurfaceis prone to many head waves.
The secondavenueis to use a ray tracing schemewhich modelsthe first arrival's raypath, whether it be a direct arrival or head wave. Some of these schemesare referencedat the end of Chapter 2. A possibleproblem with these schemesis that a raypath may be found which gives a first arrival correspondingto an event with negligible amplitude. Thus, we end up selecting the wrong observedtraveltime once again. Our experience showsthat when head wavesare prevalent, this secondavenueseemsto be the more successfulof the two and the least frustrating. However, for the Midway Sunset data we ray traced for predicted direct-arrival traveltimes since head waveswere not a problem. The direct arrival versusfirst arrival problem is an important consideration in selectingmeasuredtraveltimes. However, another concernjust as important is the radiation patterns of the source and receivers,sincethese determinewhich polarity one shouldpick. The air gun is an explosive-type
sourcewith a positive(outwards)particle motion in the first half-cycleof the signalwith greateststrengthdirectedhorizontallyas shownby the radiation pattern in Figure 33. For a constant velocity medium the rays betweensourceand receiversare straight. Figure33 showsthe vertical(V) and radial (R) geophonecomponents.For a positive particle motion the vertical componentproducesa positivekick whenthe wavestrikesthe geophone from above, and the horizontal componentproducesa positive kick as long as the wave is traveling radially outwards. The vertical componentof the geophoneremains fixed along the borehole axis so that any changeof polarity in the signal can be attributed to the up or down propagation of the P-wave. If the radial componentwere indeedfixed at all receiverlevels,then the polarity would alwaysbe positive except for under very extreme subsurfacegeologicconditions. However,the two horizontalcomponentsin a VSP tool are not fixed and rotate randomly
throughoutthe survey.The goodnewsis that the P-wave'sfirst half-cycle should always be positive on the radial component. Thus, we can numerically rotate the horizontal componentsso that we always have a positive first kick on the radial component. The even better news is that with the air gun sourcethe receiverremainsfixed over each sourcepass(common-
receivergather). This meanswe needto determinethe numericalrotation only once per receiver level. For the Midway Sunset survey we only were required to determine40 rotation anglesfor the entire survey! To summa-
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4.2. STEAM-FLOOD EOR OPERATION
103
Air gun Source/ Geophone Receiver Source
Receiver V
R
P-Wave
FIG. 33. Thiscartoondepicts theP-waveradiationpatternwitha positive polarityduringthefirsthalf-cycle of motion(explosive source). Thevertically(V) andradially(R) recorded firsthalf-cycles areshown ontheright asflagswith the expectedpolaritiesandsignalstrengths represented. The verticalcomponent changes polaritywith receiverdepthwhilethe radial component'spolarity remains positive.
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104
CHAPTER
4.
CASE STUDIES
rize Figure 33, we must keep in mind that the polarity will changeon the vertical component, but not on the radial component. Thus, its best to cross-reference one componentwith the other during traveltimepicking.
The Bolt air gun alsoemitsan SV-wave(verticallypolarizedS-wave). The radiation pattern for the SV-wave is shownin Figure 34 and resembles a four-leaf clover. The particle motion polarities of the first half-cycleare shownas small arrowsperpendicularto the raypaths. For the rays shown, positive polarity correspondsto an arrow pointing in the counter-clockwise direction and negative polarity correspondsto an arrow pointing in the clockwisedirection. The resulting first half-cyclesof the S-wave are shown
as flagsfor the vertically (V) and radially (R) recordedsignals. Similar to the P-wave, the $V-wave exhibits a phase reversal on the vertical com-
ponent as the receiverchangesdepth from above the sourceto below the source. The radial component does not change polarity. Thus, we must be cautiousin selectingtraveltimesfor the S-wavearrivals,making sureto pick the correct polarity. Again, selectingthe correct polarity is best done by viewingboth componentswhile traveltime picking. The next step in obtainingquality traveltime picksis to insureconsistent
picksfor the selectedevent (direct arrival here) over the entire crosswell data set. With surfaceseismicdata we tie the selectedevent around loops definedby intersectingstrike and dip lines. An analogoustechniquemay
be applied to crosswelldata providedthat the same sourceand receiver stations are used throughoutthe survey,or at least over large portionsof the survey.
The Midway Sunsetdata acquisitionprogram calledfor common-receiver
gathersas depictedby the cross-section in Figure 35(a). The sourcewas movedthroughits set of fixedstationsasthe receiverremainedfixed. These locationsplot on a vertical line at receiverdepth R4 in the source-receiver depth plane on the right in Figure 35(a). Locationsof other commonreceivergathers are representedby vertical lines correspondingto other fixed receiverstations. If we keep the same sourcestations for each receiver
(whichwe did), thenthe data canbe sortedinto common-source gathersas shownin Figure35(b). The source-receiver depth locationsplot alonghorizontal lines for common-sourcegathers, such as the one at sourcedepth S3.
Finally, Figure 35(c) showsthat common-offset gathersare alsopossible where the offset is a depth differencebetween sourceand receiver.
Figure 36 indicateshowthe common-receiver and common-source gathers are used to tie time picks around loops. The sourcesand receivers correspondingto straight rays within the box in the cross-section on the left definea squareloop in the source-receiverdepth plane on the right. We
beginwith the common-source gather at station S2 and selecttraveltimes from receiverstations R2 to R5. Next we take the common-receivergather
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4.2.
STEAM-FLOOD
EOR
OPERATION
105
Air gun Source ! Geophone Receiver Source
Sl/-Wave
Receiver V
•
o
R
o
FIG. 34. This cartoon depicts the $V-wave radiation pattern with the upgoing $V-waves exhibiting a positive polarity on the first half-cycle while
the downgoing SV-wavesexhibita negativepolarity.The vertically(V) and radially (R) recordedfirst half-cyclesare shownon the right as flagswith the expectedpolarities and signalstrengthsrepresented.The vertical component changespolarity with receiver depth while the radial component's polarity remainsnegative. No S-wavesignalis receiveddirectly acrossfrom the sourcesince the radiation pattern is zero horizontally.
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106
CHAPTER
Cross-section
4.
CASE STUDIES
Source/ ReceiverDepth Plane
SOURCE
RRCEIVER
WELL
WELL
Commor•-Receiver
-
Receiver Deptl•
RI RI
R2 R.3 R4 RS R6 •
S1-
Gather
I
$2. I I
Depths $4, I
$5,
R6
Cross-section SOURCE
Source/ ReceiverDepth Plane WELL
Common-Source
:'_'____:. s4
' CC .. -.-•
RI
Sl
R2
S2.
R.3
83
R4
85.
R6
86.
Source/ ReceiverDepth Plane RECEIVER
WELL
(c)
Receiver Dept!•
WELL
Gommon-Off•
Gather
_-•, R•
S2
R2
S5
R.3
S4
R4
S6
Deptlm 84
R5
Cross-section SOURCK
SI
I
RECEIVER
WELL
Sl
(b)
$6
SI-
Dept!• S4-
R5
85-
R6
S6-
FIG. 35. Possiblecrosswellseismicgathersshownby raypath correlations
on the left andin the source-receiver depthplaneon the right for: (a) receiver,(b)source,and (c) offsetgathers.
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4.2.
STEAM-FLOOD
EOR
OPERATION
107
Tying Data Using Source / Receiver Gathers Cross-section
Source / Receiver Depth Plane
souac•
Rccclvcr Dcptlu
WELL S1
S2
RI
WELL
R2 R3 R4 R•
R6
R1
.........
R2 !
S3
R3
S4
R4
S•
R•
S6
R6
S3 • I
Depths
i
I
S6 •
I <-
FIo. 36. The sourcesand receivers within the range of the box of the cross-sectionare used to tie traveltime picks around the loop shown in the source-receiverdepth plane. Two sourcegathers and two receivergathers are required to complete one loop.
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108
CHAPTER4.
CASE
STUDIES
at R5 and pick traveltimesoff of tracescorrespondingto sourcestationsS2 to S5. Then the common-sourcegather at S5 is used and time picks taken from receiverstations R5 to R2. Finally, the loop is completedby picking times for tracesS5 to S2 on the common-receivergather at R2. Sincethe S2 trace on common-receivergather R2 is the same trace R2 on common-source gather S2, the traveltime picksmust be the same. If the picksappear to be different, then a mispick must have been made somewherealong the loop and should be rectified before you continue. All picks in the data set can and should be tied in this manner to provide consistency. Generally, on the first pass, loops are made as large as the data quality will allow with in-fill loops checkedlater. Besidesproviding consistentpicks, this method also lets one identify poor data regions in the source-receiverdepth plane and to developstrategiesfor tying time picks around "not useable" data regions. Finally, a recommendedpractice is to plot the computed traveltimes from the convergedimage reconstructionon the data recordswith the observedtraveltime picks. Since the image reconstructionis basedupon both good and bad picks,many times the bad pickswill "stand out" when compared with the computed traveltimes; assumingmost of your traveltime picks are good. Also, traveltime picksleft out of the inversionprocessbecauseof uncertainty are more clearly identified by the computed traveltimes permitting you to add them to the inversionprocess.
4.2.3 With
Image Reconstruction the P-wave
direct
arrival
traveltimes
selected and the associ-
ated source/receiverlocationsdetermined,we proceedby establishingthe
gridded modelwhichrepresents theinitialmodelfunction Mj"it, where j = 1,..., J and J is the total number of cells in the gridded model. We chosesquare cells 5 ft on a side giving 42 cells horizontally and 84 cells
verticallyfor a total of J = 3528 cellsin the model.• Figure 37 showsthe initial estimate of the P-wave velocity profile between wells TO2 and 183. We chose a two-layer initial model since the velocity contrast between the Potter B1 tar-sand and Antelope shale in
Figure 29 wassignificantand the averagelayer velocitiesand interfacedip were known from well logs in the TO2 and 183 wells. The model in Fig1We originallystarted with 10 ft squarecellsbecausethe sourceand receiverspacings were 10 ft. Five foot square cells worked just a.s well, but with more resolution. Lesser size cells did not work well. Note that all of the tomograms shown in this section are
interpolated to 1] ft square cellsfordisplay.
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4.2.
STEAM-FLOOD
EOR OPERATION
Initial TO2 200
109
Guess 183
Offset (ft) 150
100
50
0
Velocity (fUs)
1400
..':..... 6650 ...
..........6900 1450
.•!::?:ii:•ii::?:i!ii[ 6970 '::;;:;:::::.::; '•:•:•;•:•:•:•*• 7015 1500
-:::.>:+•-'..::•
•:;•?• 7070
Depth (ft)
i•':J-'-"•';•:: 7120
;/•i=iii• 7176
1550
.::::::::::'./.::.:
724O
7350
1600
7550 8100
1650
'['
FIG. 37. Two-layermodel usedas the initial estimatein the SIl•T algorithm.
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11o
CHAPTER
4.
CASE STUDIES
ure37isthereciprocal 2 oftheinitialmodel function Mj'•i' which becomes thefirstmodelestimate Mf •t in theSIRTalgorithm. Step I of the SIRT algorithm is to perform ray tracing through the
estimated modelMf stforeachsource-receiver pairwithanobserved traveltime,Piø•'s.As mentioned earlier,we usea "Snell'sLaw" type of ray tracing for this data set, which modelsonly direct arrivals. Thus, sincewe selected1600 observedtraveltimes, we end up with 1600 predicted direct-
arrivaltraveltimesp•,re,wherethe subscripti is theindexfor the ith sourcereceiverpair. Besidesgiving the predicteddirect-arrivaltraveltimes,the ray tracing also providesthe raypath length through each cell as required by equation(46) in step 2 of the SIRT algorithm. The raypath length for the
ith ray in the jth modelcell is represented by the variable Step 2 of the SIRT algorithmusesequation(46) to determinethe model
corrections AM1 totheestimated modelMf st.Thepredicted direct-arrival traveltimes p•,reandraypathlengths$ij fromstep1 alongwith the associatedobserved traveltimesPiø•'sare usedto computeeachterm in the summationin equation(46). The weightI/V1 for the jth cell is simplythe number of rays which intersect the jth cell, frequently called the ray density. Ray densityis easily determinedduring ray tracing when computing the raypathlength Oncethe corrections AM 1 for all J = 3528modelcellshavebeendetermined in step 2, we apply thosecorrectionsto the estimated model function
M[stinstep 3,giving anew estimated model function MJnew)est. However,
becauseof the nonlinearnature of seismicray tomographywe apply only a fractionof the updategivenby equation(46), say60 percent. This prevents instabilities and is most important during the first few iterations when the correctionscan be quite large. The model updates are also smoothedwith a small spatial filter before they are applied to help reduce instabilities. These extra stepsare necessarybecausein seismicray tomographySIRT is a linear inversion scheme used in an iterative
manner to solve a nonlinear
problem, as discussedin Chapter 2. Steps I through 3 are repeatedusingthe previousiteration's new model
estimate M«n•w)•tasthecurrent estimated model Mfst.Thesteps are
iterated until we are satisfiedthat the estimated model has convergedto an acceptablesolution. Figures 38 through 42 show the estimated P-wave models in terms of velocity for the 1st, 5th, 10th, 20th, and 44th iterations of the SIRT algorithm, respectively.The first 10 iterations showthe development of large scalefeatures in the P-wave tomograrnssuch as the high
:ZRemember that the SIRT algorithmusesslowness for themodelfunctionMj instead of velocityVj, wherefor the jth cell, the two are relatedby Mj = 1/Vj. We displayVj because most people are more familiar looking at velocity rather than slowness.
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4.2.
STEAM-FLOOD
EOR
OPERATION
111
velocityfeature at 1475ft and low velocityfeature at 1550ft near the TO2 well. After 10 iterations,onlysmall-scale adjustmentsaremadeto the large features defined in the early iterations. There is not a lot of difference between the 20th iteration
and the 44th
iteration P-wave tomograms. A natural questionto ask is, "When do we stop the iterations?" One method used to answerthis questioninvolves plotting traveltime residualsas a function of iteration, as shownin Figure 43 for the P-wave data.
We see that the traveltime
residual decreases
significantlyduring the first 10 iterations in which large-scalefeatureson the P-wave tomogram are determined. After the 10th iteration the traveltime residual curve begins to toe-out as small-scalefeatures are added to the large-scalefeatures. By the 44th iteration the traveltime residual curvehad approachedthe horizontal,implying further iterationscould not improvethe convergence and the P-wavetomogramin Figure42 is the best possibleestimate of the true velocity profile. This method assumesthat as the computed traveltimes approachthe observedtraveltimes, the estimated model approachesthe true model. Observed
traveltimes
for shear wave direct arrivals were also selected.
However, because the S-waves are imbedded in other arrivals we could select
only 745 of the 1600possibleS-wavearrivals. Utilizing a similarprocedure as for the P-wavetomogram,we constructedthe S-wavetomogramin Figure 44. The noisy appearanceof the S-wavetomogramis a direct result of the small number of rays utilized by the SIRT method. Approximately one year after the presteam injection crosswellseismic experiment we ran a poststeam injection experiment to evaluate the successof the EOR steam flood program. We processedthe secondcrosswell
seismicdata set for a poststeaminjectionP-wavetomogramusingthe same processingproceduresas for the presteaminjection P-wave tomogram in Figure 42. Figure 45 comparesthe poststeaminjection tomogramwith the presteaminjection tomogram, both sharingthe samevelocityscale. Heated parts of the reservoir resulted in a reduced P-wave velocity as expected based upon core study results.
4.2.4
Tomogram Interpretation
The presteam injection P-wave tomogram in Figure 42 is usedto delineate reservoirinhomogeneities.We made the lithology/porosityinterpretation shownin Figure 46 assuminga direct correlationbetweenthe P-wave velocity and lithology/porositydeterminedfrom cores.The Potter A sand, Potter B1 sand, and Antelope shale are delineated. The Potter B1 sand is the reservoirin which steam was to be injected. Using core information we interpreted the high and low velocity zones within the Potter B1 sand
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112
CHAPTER 4. CASESTUDIES
Iteration TO2 200
1400
I
Offset (ft) 150
100
183 50
0
Velocity (ft/s) 6650 .
1450
ß
:':..... s•oo ............ '6970
ß :•:•.:•:•: 7015 1500
ß •!•-'-'•...:i•:• 7070 .:.>•:.:.-.'•.-:::
Depth (ft)
ß •:•[•:-..:•i 7120 1550
•:....:.....•.•:.:• 7176 .:::::•::-...-:
7240 1600
7350
7550 8100 1650
FIG. 38. P-wavevelocitytomogramafter 1st iteration.
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4.2.
STEAM-FLOOD
EOR OPERATION
Iteration TO2 200
113
5 183
Offset (ft) 150
100
50
0
Velocity (fUs)
1400
:?:iii•iii::.': 6650 1450
:':':"•!! 7015 1500
!•:..._.•. • 7070 !-"'i!=•i 'i 7120
Depth (ft)
7176
1550
7240 7350
1600
7550 8100 1650
_
FIC. 39. P-wave velocity tomogram after 5th iteration.
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114
CHAPTER4.
Iteration TO2 200
1400
10
Offset (ft) 150
100
CASESTUDIES
183 50
0
Velocity (fUs)
.i.:!i!i::::iiii:: 6650
. :.....:.: :.:..
':............ '6900 1450
'.:::::::::::::::;: .'::•:::::::::::.: .........
'::..::•::.':::::.:
:iiiiii?:?:?:ii?:ii 6970
1500
Depth
.....•......• 7OlS •?//.7070 ::.:•;•... -. 7120
(ft) 1550
•i•! •':'•:• 7176 7240
1600
7350 7550
8100
FIG. 40. P-wave velocity tomogram after 10th iteration.
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4.2.
STEAM-FLOOD
EOR OPERATION
Iteration ,i'O2 200 1400
115
20
Offset (ft) 150
100
183
5O
0
Velocity (fus) .::i:'•!:•:!:... 6650 ..::::::•i¾:: .... 6900
1450 ......... .........
................... 6970 :.::::.'::::::::;:;: .:.:.:.:.:.:..-.:.:
................. 7015 1500
•:...•:.: 7070 ?::.•: 7120
Depth (ft) 1550
-:i:::::•:.-.'•'..-:-'•:: 724O
1600
7350
7550 8100 1650
FIG. 41. P-wave velocity tomogram after 20th iteration.
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116
CHAPTER
Iteration TO2 2OO
44
Offset (ft) 150
lOO
4. CASE STUDIES
183 50
0
1400
Velocity (fUs)
1450
'i:!:i:i:i:i:i:!:!: :.:.:.:.:.:.:.:.:.: ..........
ii.•! 7015
::• /::
1500
:•-.;.•:.•.' 7070
Depth
••."-'."";• 7120
(ft)
•i!i•i!':."i.?iL::::•?,:.•i!:!.. ''
•:i:•:•:•:'::•,• 7176
..
7240 1600
735O
7550 8100 1650
FIG. 42. P-wave velocity tomogram after 44th iteration.
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4.2.
STEAM-FLOOD
EOR
OPERATION
117
Iteration
0
10
2O
3O
4O
Traveltime Residual
(xl(• 4 s2)
2
I
Traveltime
FIG.
Residual
43. Plot of traveltime
=i= •,•i:•iire 'Piobs )
residuals as a function
of iteration
for P-wave
velocity tomograms. Traveltime residual is defined as the sum of the squaresof all differencesbetweenpredicted and observedtraveltimes.
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118
CHAPTER
Iteration TO2 200
1400
50
Offset (ft) 150
100
4. CASE STUDIES
183
50
0
Velocity (fUs) ......
2634
•:•'•.i-'!!•:'
•:'•:•*•"•": 3041 1450
iiiiii!1111!111i 3113 1500
:'• '3242 , .......
Depth (ft)
i.11:...i::.• 32• ....:.:..-..-.:..-.
1550
3385 3508
3645 1600 3898
4300
1650
FIG. 44. S-wave tomogram processedin a similar fashion as the P-wave tomogram in Figure 42.
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4.2.
STEAM-FLOOD
EOR
OPERATION
Presteam Injection Tomogram Oei•h TO2 (ft)
200
Offset (ft) 150
100
183 5•) i
119
Poststeam Injection Tomogram Dei•h TO2
0
•)
• 1400
::::::::::::::::::::::::::::::::::::.:.:..'.::
-:i::i::?..-:.. •....¾...!...,..•:• .......•.... :.............
2o0 !
Off#t (ft) •so !
•oQ I
183 so i
1450 ::::::::::::::::::::::::: ::::::::::::::::::::::!.•:::!:::!:: :::.:.;,:.:.:.:.. ß -
1550 _
' .....
. :..
6293
i.:•.•.'..i::!:!i!i!•ii:i::..' ......" i:i:::i:i:i".• S705 1500 :i .".:?.!:!:i:i ..... •.......... ::....... .:.:.:.:.:.:.:.:.:.:.>;.:.6936
..]i•.!i•i:i:i:i:i:i:i:[3•":' _ 1550 ß-.-.-••,.-,.,.L...-.'-'.'-'.
i
:::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::,: ..:::::::::::::::::::::::::: :.::::::::::::::::::::::::::::::::':".--.• :-:-:-:-:-:.:' ============================================= '.:.: ?'•::!:i:i:i:i:!:i:!:i:!:i:i:i:!:. :.:•-.•!i::::::' ::!:::!:!:i::::':":"!":'"' :":'::i:i:!:!:!.'.:.':
•.i:•;.:=•.?.•i•.,..%..•.;:!:!;:!ii::.;..!... ' •....ß ..... •04• :....,•.•½.,x.•;:::::::,,:.:.::::.::::::..: .
:.:•.•.,?.,.. .......... ..
1500
o
.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ===================== Velocity
i•x.•!ii:i:!:i:i.. ".......:-:.:.:.:.... .....!:::i.i:i:i.i:i:i:!:!:i:i•:. ::.•`•:!:i:!:!:!:;:•:i•:•:•::i•::•:;:i:!:;:!:i:i.!•!:i:•:i•:::•::i:•:i:i:!:•:•:;:i:i:i:: 7200 :.:.',•,•.:.:.:.:.:.:.:.:.;.:.:.:.;...;.:.;.:.:.:.:.;.:.:.:...:.: ;.:.:.:.:.;.:.:•:.',-.•: '•:::.':.;:.•: ::::. ================================================= -1.•.:.:..:::::..-.. '"::: -::•:,'•-• .......... '......... ::.'•.-•.•-'..:.: ...... :.•,.,.:•,•::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :2::•::.:•i:.,x.'
:-.:,-.:. -:.-.-...:.:..............-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.....-..S; '.-.x.¾4::.... 790I !::•..'•i.';:.>-."-!'-::!:!:!:!:!:!::: :':....... "::!:i:!:!:!:::i..x%.•i•:-.:..•:!.
:::::::::::::::::::::::::: ..... ::::::::::::::::::::::::::::::::: 6000
F](•. 45. •'he presLeeLm injecLionP-weLve LomogreLm on LhelefL forms Lhe baseline LomogreLm for observingreservoir changesas eLresulLof sLeeLm injecLion. The posLs[eeLm injection P-wave [omogreLm on the right Laken one year leLterindirectly indiceLtes a significeLnt portion of the reservoirwas healed. P-wave velociL¾in LheheeLv¾ oil reservoirrock is reducedwhen Lhe reservoir
is healed.
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120
CHAPTER
4.
CASE STUDIES
Lithology/Porosity Interpretation TO2 200 1400
Offset (ft) 150
100
183 50
0
Velocity (rt/s) 6650 ß 6900
1450
.....
::i.!:i:i:!.i:!:
1500
ii•."a•'.:i•11• 7070
Depth
(ft)
1550
'-':•l•i•..*.." 7120 .•i!•! 7176 7240 735O
lSOO
',i',i',i',i',iii',i?, 7sso 8100
1650
FIG. 46. Lithology/porosityinterpretationof the presteaminjection Pwave tomogram. The Potter A sand, Potter B1 sand, and Antelope shale are shown. Core data showedthat high and low velocity zoneswithin the Potter B1 sand correspondto low and high porosities,respectively.
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4.2.
STEAM-FLOOD
EOR
OPERATION
121
as low and high porosities, respectively. Thus, the reservoir is certainly not homogeneousas far as porosity is concerned. At the top of the Potter B1 sand are high velocityclay stringers.The productionengineersrely on these clay stringers to confinesteam to the Potter B1 sand. However, the clay stringers do not appear continuous,and actually drop in velocity towards the center of the tomogram indicating a possiblebreach through which steam may flow acrossinto the Potter A sand. Thus, the production engineer might anticipate some steam lossinto the Potter A sand. We can also estimate a porosity tomogram of the Potter B1 tar sand reservoirusing the P-wave velocity tomogram and Wyllie's time-average equation in the form,
whereV! is the fluid velocity,V,• is the matrix velocity,and V is the P-wave tomogram velocity. The measuredaveragematrix velocity and pore-fluid velocityfor the Potter B1 sandare Vm = 10000ft/s and VI = 4500 ft/s, respectively.The P-wave tomogramin Figure 42 providesthe valuesfor V, giving the porosity tomogram shownin Figure 47. This porosity tomogram is valid only for the Potter B1 tar sand for which the averagevaluesof Vm
and V! weredetermined.The porosityvaluesfor the Potter B1 sandrange from 33 percent to 44 percent accordingto the porosity tomogram. These porosity valuesare higher than the porositiesdeterminedfrom coresamples which have high values of 32 percent. This discrepancyimplies that the Wyllie's time-average equation does not take into account all petrophysical properties which affect the P-wave velocity, such as clay content. However, we do believe that the relative porosity information for the Potter B1 sand is meaningful. Such porosity information is helpful to engineerswho need to model the production of a reservoir. The secondobjective of the Midway Sunset field tomography project was to monitor the EOR steam-injection project. To monitor the steam flood progressin the Potter B1 reservoir using seismictomography, we must know the effect of heated Potter B1 reservoirrock on seismicvelocity. Coresfrom a well located 300 ft east of the 183 well provided Potter B1 sand samplesfor which P-wave velocitiescould be measuredin the laboratory at various temperatures. Figure 48 displays the laboratory measured P-wave velocities for a Potter B1 sand core at temperatures of 25øC, 55øC, 90øC,
and 125øC, under confiningpressuresof 500 psi, 1000 psi, 1500 psi, and 2000 psi. Clearly, we will expect the P-wave velocity to decreaseas the temperature of the reservoir increasesas a result of steam injection. The core results provide us with the temperature-velocity relationship required to interpret the presteam and poststeam injection tomograms in
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122
CHAPTER
4.
CASE STUDIES
Porosity Tomogram TO2 200
Offset (ft) 150
100
183 50
0
Porosity 20
..:::.-:
1400
:..-:.:
25 30
1450
33 35 38 41
Depth (ft)
45
1550
.•,.
1600
1650
FIG. 47. Porosity tomogram determined from the P-wave velocity totoogram in Figure 42 and Wyllie's time-average equation. The lithology]porosityinterpretationfrom Figure 46 is superimposedfor comparison.
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4.2.
STEAM-FLOOD
EOR OPERATION
Fwave
123
Velocity vs. Temperature
8500
8000
75OO
Velocity
7OOO
2000 psi 15o0 psi 6500
1000 psi 500 psi 6000
20
40
60
80
100
120
Temperature (deg C)
FIc. 48. Measured P-wave velocitiesfor a Potter B1 sand core at temperatures of 25øC, 55øC, 90øC, and 125øC, under confiningpressuresof 500 psi, 1000 psi, 1500 psi, and 2000 psi.
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124
CHAPTER
Sonic Log at TO2 Well
4.
CASE STUDIES
Tomogram-Sonic Log Comparison
1400
400
1500
5O0
•-•Smoothed Sonic Log
Depth (ft)
P-wave Tomogram 1600
600
6000
7500
10000
Velocity (if/s)
12500
7000
7500
8000
8500
Velocity (if/s)
FI(•. 49. The original soniclog from the TO2 well is shownon the left. On the right, a smoothedversionof the soniclog is comparedwith a tomogram profile parallel to and offset 15 ft from the TO2 well in Figure 42.
Figure 45. The seismicP-wave velocity over parts of the Potter B1 tar sand dropped by 15 percent to 20 percent after one year of steam injection. We alsoseethat the clay stringersin the upper Potter B1 sand restrict the steam flood to the Potter B1 down dip. However, further up dip the clay stringersare breachedby the steam and somesteam is lost to the Potter A sand, which exhibits a similar velocity-temperature relationship as the Potter B1 tar sand reservoir. Figure 45 indicatesthat the Potter B1 sand was not uniformly heated and gave the production engineersinformation for modifying the steam injection project at this site. Finally, we checkedthe reliability of the presteam injection tomogram by comparinga tomogram profile taken about 15 ft in and parallel to the TO2 well in Figure 42 with the soniclog in the TO2 well. Figure 49 shows the original soniclog velocity on the left and providesa comparisonof the smoothedsonic velocity log with the tomogram profile on the right. The sonic log and tomogram profile are in good agreementas to overall trend. Velocity differencescorrespondingto the clay stringersmay be attributed to averaging-outof the high velocitiesby the tomographyinversion. The Potter B1 reservoir,correspondingto depths below 1500 ft, show the sonic
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4.3.
IMAGING
A FAULT
SYSTEM
125
log and tomogram profile agree to within 5 percent. Thus, analysis of the tomogramsin this portion of the Potter B1 tar sand are made with some justified confidence. Since similar data acquisition and processing techniques were applied for the poststeam injection tomogram, we also assumethis reliability analysisextends to the poststeam injection results as well.
We make the following conclusionsregarding crosswellseismictomography basedupon this casestudy' 1) Crosswelltomographyas a reservoir characterizationtool is usefulfor determiningreservoirlithologyand porosity inhomogeneitieswhich is not possiblewith only well log information; and 2) Crosswelltomographyis usefulfor monitoringnonuniformheating of reservoirrock betweenwells as a result of steam flooding.
4.3
Imaging
a Fault System
The McKittrick Field in California is located near the Midway Sunset Field and produces from the Potter sand, a massive unconsolidated conglomeratewith permeabilitiesranging from 1 to 10 darcys. Figure 50 shows
the well-logbasedreservoirgeologyinterpretationprior to runningcrosswell seismictomography betweenthe 806 and 429 wells. The McKittrick Thrust placed Miocene age diatomite over PleistoceneTulare sand. A subthrust fault developedsubsequentto the McKittrick Thrust which intersectsthe 806 well at a depth of 800 ft. The Potter sandreservoircontainsa heavyoil whichis subjectto gravity drainage. Basedon data from the 806 well, the subthrust is believedto act as a sealingfault which preventsfurther downwardmigration of the heavy oil. The well data show that the Potter sand above 800 ft is saturated
with
up to 50 percent oil while the Potter sand below 800 ft is desaturatedwith lessthan 30 percent oil. The subthrust lies at the boundary betweenthese two
zones.
The estimated oil reservesin this reservoir and optimum development of the field depend upon proper positioningof the subthrust. Thus, the objective of this crosswelltomography study is to image the faults associated with
the reservoir
desaturated
Potter
4.3.1
and define the boundaries
of the saturated
and
sand.
Crosswell Seismic Data Acquisition
The clamped-vibratorsourceshown in Figure 51 was provided by Chevron for this experiment. The sourceis coupled to the borehole wall through the clamp located at the top of the tool. One of the smaller lines
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126
CHAPTER
4.
CASE
STUDIES
Well-log Based Interpretation wel
well
8O6
429 Oft
315
ff
20O ft
D!atomite
McKIttrlck
.
..••
_ 40Oft _
60Oft
_
8OO ft
1 ooo ft
FIG. 50. The reservoir geologyinterpretation before the crosswellseismic survey was run between the 806 and 429 wells. The interpretation was basedmainly upon well-log data and somesurfacegeology.
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4.3.
IMAGING
A FAULT
SYSTEM
127
q
FIG. 51. Chevron'sclamped-vibratorsourcein preparation for deployment. The primary componentsfrom top to bottom are the clamp for source-toboreholecoupling, the hydraulic servovalveand actuator module, and the reaction mass. Photo courtesyof Don Howlett, Texaco.
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128
CHAPTER
4.
CASE STUDIES
Data Acquisition Geometry well
806
well
429 Oft
315 ft
:
.___ Shallowest Receiver --
Source
2O0 ft
40O ff
--
$00ft
Deepest Receiver --
80Oft
FIG. 52. Data acquisitiongeometry involved keepinga 90-degreereceiver aperture bisectedby the horizontal through the source. The aperture was formed using thirty-three receiver levels spaced at 20 ft intervals about each sourcelocation. The shallowestand deepestreceiver positionsfor the sourceat a depth of 400 ft is depicted.
provides air pressureto the clamp for its activation and deactivation. The other small line is a wire line which both supportsthe tool and provideselec-
tricity to the hydraulicservovalveand actuator which controlthe sweeping action of the vibrator. A pump at the surface pressurizedhydraulic fluid which flows to and from the sourcethrough the two larger diameter hoses. This pressuredrives a hydraulic cylinder which is attached to a 50 pound reaction masslocated at the bottom of the tool. An axial driving motion imparts a vertical stresson the boreholewall at the clamp which introduces the seismicenergy into the formation. Figure 52 showsthe data acquisitiongeometry. The clamped vibrator was deployed in the 806 well while a three-componentgeophonereceiver
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4.3.
IMAGING
A FAULT
SYSTEM
129
Common-Source Gather (660 ft) Time
(ms) 1100
Receiver Depth (ft) 1000
900
800
700
600
500
400
300
200
5O
lOO
150
200
FIG. 53. A common-sourcegather from the radial componentfor a source depth of 660 ft and receiver depths ranging from 160 ft to 1150 ft at 10 ft intervals.
was deployed in the 429 well. Thirty-seven common-sourcegathers were collectedwith the sourcerangingin depth from 200 ft to 920 ft at 20 ft intervals. Common-sourcegathers were collected, rather than commonreceivergathers,becauseof the unwieldynature of the four linesattached to the sourcewhich had to be clampedtogether at regular intervalsas the source was lowered into the well. For most source levels 33 receiver levels
were recorded at 20 ft intervals in such a manner that a 90-degree angle receiveraperture bisectedby the horizontalat the sourcewas acquired,as shown in Figure 52.
A linear sweepwas applied to the vibrator from 10 Hz to 360 Hz with a sweeplength of 14 s and a listen time of 2 s. Four sweepsper source level were requiredto get a goodsignal-to-noise ratio. Figure 53 showsa common-sourcerecord for the clamped vibrator at a depth of 660 ft and receiverstationsrangingfrom 160 ft to 1150ft at 10 ft intervals. The record is from the radial componentand showsboth first-arrival and reflected Pwave
events.
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CHAPTER
130
4.3.2
Traveltime
Parameter
4.
CASE STUDIES
Measurements
The trove]time p•r•meter measurementsfollow • procedure similar to the one for the Midway Sunset Field. Chevron provided desweptdata with the horizontal componentsdecomposedinto r•dial and transversecomponents. We applied a zero-phasebandpassfrequencyfilter to further increase the signal-to-noiseratio for optimum traveltime picking. Becauseof the "blocky" nature of the subsurfacewe chose to select direct-arrivM P-wave tr•veltimes, similar to wh•t we did with the Midway Sunset dat•. The only exceptionMdifferencehas to do with what polarity to chosefor the direct-arrival traveltime. The clamped vibrator sourcehas a radiation pattern significantlydifferent from the air gun. Figure 54 showsthe Pw•ve r•di•tion pattern for the clamped vibrator sourcein a homogeneous
mediumwith the signalrecordedby geophonereceiverswith vertical(V)
andradiM(R) components. s Theclamped vibratorisessentially a vertically directed dipole which has P-wave motion directed towardsthe sourcein the lower lobe while, at the same instant, the P-wave motion in the upper lobe is •way from the source.For a homogeneous mediumthis radiation pattern results in no polarity changeon the vertical geophonecomponentwhile polarity changeson the radial component.Theoretically,no P-waveenergy should be observedat the same depth as the clamped-vibrator sourcein a homogeneousmedium.
Figure 55 depictsthe S-waveradiation pa•tern for •he clampedvibrator which may be compared with the S-w•ve r•diafion pattern for the air
gun sourcein Figure 34. The clampedvibrator shouldproduceexcellent S-w•ves for use in crosswelltomography since the S-wave radiation pattern's maximum strength is directed horizontally. As with Figure 54, in a homogeneous medium we expect no polarity changeof the direct arrival on the vertical componentwhile a polarity change is expected on the radial component.
At first glance the dipole nature of the damped vibrator appears to produce recordswhich should be easy to interpret. After all, there are no polarity reversalsseen on the verticM componentfor either P-wave or S-wavedirect arrivalsin Figures54 or 55, respectively.However,now introduce • velocityfield suchthat a P-waveraypath initially traveling upwards gets refracted so that it is • downgoingray at the receiver. The P-wave event you are selectingappears to shift • cycle on the vertical component. Thus, youmustmakea decision:1) to pickon the reversedpolaritybecause
you suspectthe ray turnedup-to-down(or down-to-up),or 2) to pickthe same polarity at the shifted time becauseyou suspecta velocity change. For complicatedsubsurfacevelocities this decisionprocessis tedious and 3CompaxeFigure54 for the clampedvibrator with Figure33 for the •ir gun.
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4.3.
IMAGING
A FAULT
SYSTEM
131
Clamped Vibrator Source ! Geophone Receiver Source
Receiver
V
R
'
P-Wave
o
-I-
o
FIG. 54. This cartoon depicts the clamped vibrator's P-wave radiation pattern where particle motion in the upper lobe is away from the source while in the lower lobe the particle motion is towards the source at the sameinstant (dipolesource).The vertically(V) and radially (R) recorded signalsare shownon the right as flags with the expectedpolarities and signalstrengthsrepresented.The verticalcomponentremainsnegativewith receiverdepth while the radial component'spolarity changessign.
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132
CHAPTER
4.
CASE
STUDIES
Clamped Vibrator Source / Geophone Receiver Source
Receiver
V
R
S-Wave
FIG. 55. This cartoon depicts the •qV-wave radiation pattern for the clamped vibrator with all $Vowaves exhibiting the same polarity as indi-
catedby the smallarrowsperpendicularto the raypaths. The vertically(V) and radially (R) recordedsignalsare shownon the right as flagswith the expectedpolarities and signalstrengthsrepresented.
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4.3.
IMAGING
A FAULT
SYSTEM
133
requires the use of properly oriented horizontal components. The horizontal geophonecomponentsare free to take on random directions when the receiver tool is moved from level to level. Frequently the relative amplitudes on the horizontal components are used for numerical
rotation to get the radial and transversecomponentsdesiredin traveltime picking. However, the polarity of the incoming P-wave or S-wavemust be assumedto selectthe proper rotation. Figures54 and 55 showthat both Pwaveand S-wavepolarities changeon the horizontal componentwith depth, thus complicatingthe rotation analysisprocess.Add a complexsubsurface and the processof orienting horizontal componentsbased on relative amplitudes becomesnontrivial. The best bet is to run an orientation device with the receiverpackagewhen using a clamped-vibratorsource. Properly oriented horizontal componentsare a definite help in picking both P-wave and S-wave arrivals from a dipole sourcewhich travel through a complex medium.
Also, since the clamped vibrator required common-sourcedata acquisition, every source-receiverpair in the survey required a unique rotation analysis to orient the horizontal components. A common-receiverdata acquisitiontechniqueis more desirableif an orientation deviceis not available. Then, one only needs to perform one rotation analysis per receiver level since the receiver remains fixed for all source levels as was done with
the Midway Sunset data.
4.3.3
Image Reconstruction
We used the same image reconstructionmethod here as for the Mid-
waySunset datain theprevious section. Theinitialmodel function Mjni• had square cells 10 ft on a side which is half of the source-to-sourceand receiver-to-receiverspacings. Ten-foot square cells required 32 horizontal cells and 60 vertical
cells for a total of J = 1920 cells in the model.
The
initial model function valueswere given a constantvelocityof 5300 ft/s. Just as with the Midway Sunset image reconstruction,step i of the SIRT algorithm utilized a "Snell's Law" type of ray tracing to model the direct arrivals for both raypath and computed traveltime. Fifty iterations were requiredfor the estimatedmodel to convergeto the true model. The resulting P-wave velocity tomogram is shown in Figure 56, resampledto 2.5 ft square cellsfor display. One part of the image reconstructionmethodologynot mentionedin the previoussectionis how to selecta color(or gray) scale. One school of thought is to use fixed intervals of velocity by assigningmany colors (gray levels)at one time whichhighlightall featuresincludingartifacts. Although this techniquemay be great for quality control purposes,it is
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!•4
CHAPTER
4. CASE STUDIES
/'-Wave Velocity Tomogram Well 806
Offset (ft)
0
100
Well 429
200
300
300 .......................... I.......................... I......
Velocity (ft/s) 4000 :i:::":?:i::'•:'
ß.•,'.:: 5546 i•-.....
.."..•:. -.-.'...• ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
................ 5794
.::i:i:•:i:i:i::.:.'.•::.•:i:: '" ============================================================ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
•:i:i:i:i:i.'i:i:i:i:!:i::•.•i;.;:." -.'"":':':':':':':':':':':':':::'!:i:!:i:i:i:M'
.........6500
::::::::::::::::::::::::::::::::::::::::::::: •'-. ================= ß"•':' 7206 9000 .......:-'..:.'-.:-:."-:i:i:i.'!:i:'•' :':':' ....'•.... L•:'•:•..... :i:•:::::i.%-'.•:•.'•:.•.•s::'.!.•.?.:'. .............
..;....:. ........ ..:.::..::..!:-...:.;::.:.
700 . Saturated Potter "--.............. ' "-..':
.!?,!•'
FIG. 56.
.'.
.....
P-wave velocity tomogramreconstructionof the McKittrick
Thrust and subthrust.
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4.3.
IMAGING
A FAULT
SYSTEM
135
not very useful for displaying the results for which the tomographic data were acquired. Instead, one should highlight the tomogram's features for
which we have an interest. This generallyrequiresthat the persondoing the image reconstructioneither possess,or work with another personwho possesses, some interpretation skills. With this study we knew that the diatomite, Tulare sand, saturated Potter sand, and desaturatedPotter sand were targets of interest. Well data provided the information that the McKittrick Thrust lies at the boundary of the diatomite and Tulare sand while the subthrust at a depth of 800 ft in the 806 well separates the saturated Potter sand from the unsaturated Potter sand. Using this combination of well information with what the P-
wave velocity tomogramwas telling us, we chosea gray scaleto highlight the desiredfeatures. Only sevenshadesof gray were neededto fulfill the objectivesfor which this tomogramwas run. The resultingtomogramdisplay is in reality a lateral extensionof the well-loginformation obtained at the 806-well. If a velocity scale at fixed intervals had been used, the well information correlationwith the tomogramwould not be apparent and the resulting tomogram would be lessof a benefit to the end user, or maybe evenconfusing. 4
4.3.4
Tomogram Interpretation
The important objective of this tomogram was to image the saturated Potter sand sealed-offfrom further gravity drainageby the subthrustintersecting well 806 at 800 ft. Figure 57 is our interpretation of the tomogram in Figure 56, which when comparedwith the well-log based interpretation in Figure 50, showsthat the lateral extent of the saturated Potter sand is muchlongerthan previouslythought and doesnot extend as far in the vertical direction. This interpretation is basedupon the correlationthat, for the Potter sand, seismic P-wave velocity increasesas oil saturation increases. The tomogram in Figure 56 delineatesthe McKittrick Thrust which placed the Miocene age diatomite above the Pleistoceneage Tulare. The tomogram alsosuggeststhe presenceof two smaller scalesubthrustswhich penetrated the McKittrick Thrust fault plane after it was in place. The two subthrusts appear to have throws as small as 40 ft. Overall, the crosswellseismic tomogram showed a more complex fault systemthan previouslythought and substantiallyredefinedthe boundaries of the saturated
Potter
sand.
Such information
which determines
and ver-
4Even though the tomogram in Figure 56 was resstapledto 2.5 ft square cells for display, the image remains "sharp" because only seven gray levels were used in the display.
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136
CHAPTER
4.
CASE STUDIES
Tomogram-Based Interpretation well
806
well
429 Oft
315
ft
Dlatomlte Tulare
200 ft
Sand
McKIttrlck Thrust
400 ft
600 ft
800 ft
1000 ft
FIG. 57. Our interpretationof the P-wavevelocitytomogramin Figure56.
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4.4.
IMAGING
SALT
SILLS
137
ifies the reservoirconfigurationenablesthe developmentengineersto optimally develop the field. 4.4
Imaging Salt Sills
We chosea shallow salt sill problem to illustrate the seismicdiffraction tomography imaging techniqueusing the Born approximation presentedin
Chapter 3.5 The imagingtechniquerequiresa constantvelocitymedium with imbedded objects of finite extent which scatter seismicenergy. This requirement,along with other simplifications,is only partially honoredby the salt sill problem. Thus, the resulting tomogram is instructive as to the limitations of the method as presented in this book. Extensions of the methodologyto complexsituationsare discussedin referenceslisted at the end of Chapter 3.
4.4.1
Assumptions and Preprocessing
This casestudy usesa marine seismicdata set collectedover two shallow salt sills as depicted in Figure 58. Application of the seismicdiffraction tomography methodologyusing the Born approximation in Chapter 3 requiresmaking someassumptionsand preprocessingthe data. We require the object under investigationto have a finite extent. Fig-
ure 15 showsa finite-extentobject whichhas a velocityperturbationC(r) confinedwithin the gray area. The regionsurroundingthe gray area is the backgroundmedium with constantvelocityCo. The finite-extent object in this casestudy consistsof the shallowsalt sills and the surroundingsedimentary layers. The overlying sea water is taken as the constant velocity backgroundmedium. Collectively, the salt sills and surroundingsedimentsdo not satisfy the assumptionsof the Born approximation asstated at the end of Section3.2.4. However, the scatterers comprisingthe water bottom do meet the requirements of the Born approximation and shouldimage properly. The velocity contrast between the salt and sedimentsprobably violate the weak scattering approximation. The salt sill tops are closeenough to the water layer that they can be consideredpart of a "finite-extent" object and, with the exception of possiblyviolating the weak scattering approximation, should image properly. The salt sill bases,especiallythe large salt sill base, are SFindingthe data functionPi(r)qbd(r)for the Rytov approximation(Section3.2.3) is considerablymore involved than determiningthe data function Ps(r) for the Born approximation (Section 3.2.2). Thus, we use only the Born approximationin this caae study.
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138
CHAPTER
Salt
Sill
4.
CASE
STUDIES
Problem Surface
5000 ft/s
4000 It
Water Bottom • .........
-....... .....
•.:.....:.:.:.:.:.:.:.:.:.:.:.:.: •
FIG. 58. The two salt sills and surroundingsedimentary layers form the
finite-extentobjectwhichhavevelocitiesrepresented by C(r). The velocity
ofthewaterlayeris(7•- 5000ft/s. comprisedof scatterersquite removedfrom the constantbackgroundwater layer. The accumulative phase differencebetween the total and incident
wavefields(seeSection3.2.4) is likely to be quite large for thesescatterers and a proper velocity image for the salt sill basesis unlikely.
Besidesthe assumptions stated above,we also assume:(1) constant densityin all media,(2) 2-D wavepropagation,and (3) multiple freedata. We know beforehandthat none of our assumptionsis one-hundredpercent valid. However, we continue so that we may see the result and learn. The wavefield recorded by a source-receiverpair in a marine seismic
recordrepresents the total wavefieldPt(r), whichconsistsof the incident wavefieldPi(r) and scatteredwavefieldP•(r) as definedin equation(50). Figure 59 showsa source-receiver pair for the marine seismicdata case.The recordedincident wavefieldtravels directly betweenthe sourceand receiver while the recordedscattered wavefieldtravels over a considerablylonger raypath to get to the receiver. Thus, the scatteredwavefield arriveslater than the incident wavefieldbecauseof the large water bottom depth.
The scatteredwavefieldP,(rs, rp) is the data functionusedin diffraction tomographywhenequation(101) is represented throughthe Bornapproximation. Thus, a required preprocessingis to extract the scattered wavefield from the marine
seismic records.
The water bottom
in this case
study is at a depth of 4000 ft or greater. The time lag betweenthe recorded incident wave and the wavefield scattered from the water bottom at the near
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4.4.
IMAGING
SALT
SILLS
139
Incident Wavefield vs. Scattered Wavefield Raypath Source
Receiver
•
%, •
,/'
/
Surface
Scattered
Wave
CO= 5000IVs
\
4000It
o.o. ..............
:i•:i:•:•:i:i*i:•i:ii•iii•i•iii?:iii!i::iiiiiii::i?:•ii::i:•!::iii::iiii:•-:•i!i::i::i•:::i::i•i::•ii::!::::::?:?:::::•:::::::• ............... ?:i::iiii? ................ '"'!::11"'•i•?:•, ------- -
FIc. 59.. The raypath for the incident wavefieldwill always be much shorter than any raypath for the scattered wavefieldbecauseof the large water layer thickness.
source-receiveroffset is a large 1.6 seconds. Thus, the data function is simply determined by muting the incident wavefield from the total wavefield for each marine
seismic record.
The diffraction tomographymethod alsoutilizes infinite-linesourcesand scatterersas definedby the Green'sfunction in equation(62). Infinite-line sourceshave a cylindrical divergence in which amplitude decreasesin a
constantvelocitymediaby r-«, wherer is distancetraveled.However, the observed data are acquired in a 3-D medium with point sourcesand scattererswhich have a sphericaldivergencein which amplitude decreases
by r -1. Thus, to makethe observeddata "mimic"a cylindricaldivergence
wemultiplythe dataoneachtraceby r« = (Cot)«(Cois theconstant backgroundvelocity) beforeperformingthe tomographicprocessing.As an alternative we could have applied diffraction tomography using the 3-D
Green'sfunctionin equation(63) in a 3-D modelwhichhasa 2-D geometry. Thus, the sourcesand scatterers would exhibit spherical divergencein our calculations, but at the cost of significantly greater computation time.
4.4.2
Data Acquisition
The data for this casestudy came from a preexisting2-D marine seismic survey. The experimental geometry in Figure 60 showsthe marine streamer
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140
CHAPTER
4.
CASE STUDIES
Data Acquisition Geometry Re.
vets
I
824
v
v
3
120
2
;[
246;[ Next
Source
Receivers
................_V V V I
2
3
V 120
FI,_q.60. Experimental geometry for the marine seismicsurvey.
cable with 120 hydrophones spaced at 82 ft intervals with a near offset of 824 ft. The air gun sourcewas fired at 246 ft intervals. We chosebackpropagation diffraction tomography for the surface reflection configurationgiven in Section 3.4.2 to do the image reconstruction.
Equation (165) showsthat we must take the Fouriertransformof the data
function P(rs,rp) alongthesource andreceiver profiles to get/5(k,,kp). Since we require that the spatial frequency content for the receiver profile be the same as the source profile, only every third hydrophonetrace was used. Thus, the effective receiverspacingwas the same as the sourcespacing of 246 ft. We used 271 sourcesout of the survey line to provide an adequateaperture for imaging the salt sills.
Five representative sourcerecordstaken at equalsourcepoint (SP) intervalsfrom alongthe marine seismicsurveyare shownin Figure 61. Source records$P 823 and $P 995 are taken from over the smaller and larger salt sills, respectively.Sourcerecord $P 1081 lies at the edgeof the larger salt sill. Several events of interest can be identified on source record $P 995 at the near offsets. The water bottom reflection is found at 1.80 sec-
onds while the top-of-salt and bottom-of-salt reflections lie at 2.36 seconds and 2.92 seconds,respectively. Most of the reflectionsbetween 3.60 seconds and 5.30 secondsare identified as multiples and even more multiples are found beyond 5.80 seconds. Figure 27 showsthe hypothetical coverageof a model function in the k= - kz plane for the surfacereflectionconfigurationif sourcesand receivers
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4.4.
IMAGING
SALT
Time (s)
SILLS
Marine
•/ SP1167
081
141
Seismic SP 995
Records SP 909
SP 823
F..... !'
, 't'.
I
•'•
ß
•:•I!,i:... '"ß. ß ' !1tl•..'.•..•'ß .•-' >•
F ,' ß.•-•'. :'1,i•:•....•>.-
....
,,.>•-.•';•.. ß. .
•..,•./:,.:,,-..• .' . • ,•.•••..-._..-..•.-.•!••>-.•-<•-";'•'--:.-I-'"'•/':•-:-'.-.-•"'"--: : "- -" -••'
4 '•":•"'.: '• '• .... ..
. :: .•_ ,:-.":,•>•
• • . •....•:•'•.• ß •-• .•-..
'"•' 2- 'i•"!•i•'•.•'-•.ff•--:•"
1::::'•/'"-.:L•?-': ':'
.;..• •..•..
......•
.....•-:•.•.•.•
F•o. 61. Five marine seismicrecordsextracted at equal sourcepoint intervals from the marine survey.
couldbe placedalongthe surfacein both directionsout to infinity. In this study we are limited to a streamer cable towed to one side of the seismic
sourceas shownat the top of Figure 62. The resultingcoverageof the model function in the kx - kz plane for scattererslocated at a depth of 8000 ft is shownas solid arcs. Each solid arc corresponds to a separate sourceand streamerlocation. The dashedline representsthe locusof ko.q. The associateddotted radial lines point to the sourcelocationsused to constructthe coverage.Obviouslywe can expectlessresolutionand more nonuniqueness for the imagedmodelfunctionusinga streamercablethan from the ideal situationin Figure 27.
4.4.3
Diffraction Tomography Processing
Figure 63 illustratesthe flow chart for implementingbackpropagation diffractiontomographyfor a surfacereflectionconfiguration as givenby
equation(165). The total wavefield P•(xo,xr,t) wasrecorded on the field records, wheret istraveltime,andx• andxr arethedistances froma fixed referenceto the sourceand receiveras illustratedin Figure 64. The 2D Cartesiancoordinate systemin Figure64 hasits originat sealevelwith
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142
CHAPTER
sourcs
streamer
4.
CASE STUDIES
cable
!
locus of k o s
locus of ko( s +'• ) F•c•. 62. Coveragein the k= - k• plane for the model function at 8000 ft depth using the sourceand streamer cable configurationfrom this study. The solid arcs represent the coverage.
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4.4.
IMAGING
SALT
SILLS
143
Diffraction Tomography Procedure Pt(Xs,Xp, t) Step I
Mute
direct
arrivals
Ps(x s, Xp,t) Step 2
Step 3
Step 4
Fourier transform along time axis, t
Fourier transform along x
s andXp
Backpropagation, equation (3.119)
•(x,z) FIG. 63. Flow chart for implementing backpropagationdiffraction tomography in the salt sill case study.
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144
CHAPTER
4.
CASE
STUDIES
Cartesian Coordinate System
• source
•Y
•7
•7
X'
0
receivers
FIG. 64. Cartesian coordinate system for the salt sill case study. The
sourceand receiveroffsets,x• and xp, are measuredfrom a commonorigin.
the z-direction positive downwards. Both sourceand receiverdepthswere set to sealevel for implementationof backpropagationdiffractiontomography.
StepI (in Figure63) is the preprocessing stepdiscussed in the previous sectionin which the recordedincidentwavefieldPi(xs,xr,t ) was muted from the total wavefield. Thus, after step 1 the seismicrecordsare assumed
to containonly the scatteredwavefieldP•(xs,xp,t), whichis the desired data function for the Born approximation. In Step 2 we take a 1-D Fourier transform of each trace of the scattered
wavefieldP•(x•,xp,t) with respectto time, t. Thus, we transformthe datafunctionfromits space-time domainrepresentation P•(x•, xp,t) to the space-frequency domainrepresentation P•(x•,xp,w), wherew represents the angular frequency.
Figure 65 showsthe data volumesbefore and after this 1-D Fourier transform. Beforethe Fouriertransformthe x•-axis, xr-axis, and /-axis span the data volume. After the 1-D Fourier transform the data volumeis
spannedby the x•-axis, xp-axis,andthew-axis. We haveeffectivelyreduced the original scatteredwavefieldinto many single-frequencyscatteredwavefield data sets as depictedin Figure 65. We seethat each single-frequency scattered wavefielddata set is representedby a plane perpendicularto the
w-axiswithinthe P•(x•,xp,w) data volume.The backpropagation diffraction tomographyformulain equation(165) can be appliedto oneor all of
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4.4.
IMAGING
SALT
SILLS
145
Ps(Xs, Xp, t) x
Space-time domain representation
,I
Fourier transform along
x
$
esxs'xp' Space-frequency domain representation
Single frequency scattered
wavefield
data
FI(•. 65. The Fouriertransformof the scatteredwavefieldPo(zo,zr,l ) is taken with respect to time t resulting in single-frequencyscattered wavefield
data Po(z,, zr,• ).
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146
CHAPTER
4.
CASE STUDIES
thesesinglefrequencydata setsin reconstructing an image.• Step 3 is to apply a 2-D Fouriertransformto P•(x•, xp,w) alongthe xj-axis and the xp-axis. This operationtransformsthe scatteredwavefieldPj(x•,•cp,w)in the space-frequency domainto the scatteredwavefield Po(ko,kp,w)in the wavenumber-frequency domain,whichis requiredby
equation(165). The 2-D Fouriertransformis expressed by equation(138) whichdefinesk• and kpasthe wavenumbers of the Fouriertransformalong the sourceline and receiverline, respectively.Note that the depthvaluesof
the sourced• andreceiverdr areset to zeroin equation(138) asindicated in Figure64. We applyequation(138) to P•(x•,xp,w) onefrequency w at a time.
Step 4 is to actually apply equation (165) to the transformeddata
functionor scatteredwavefield,P•(k•, kp,w). Equation(165) is numerically evaluatedat eachpoint on the x - z plane to get the modelfunction
M(x, z). A separatemodelfunctionM(x, z)is determined eachtimeequation (165) is appliedto a differentangularfrequencyw. Again,both d• and dp are set to zero. The wavenumberko in the backgroundmediumis set equalto w/Co, whereCo is the water velocity(5000ft/s) and w is the fre-
quency.Thewavenumbers k• andkprangefrom-koto ko. Foreach(k•, pair, the corresponding verticalwavenumbers, % and7p,are computed by
7•- V/ko • - k]and 'rp- y/ko • - kp •,respectively. We obtaineda sequence of modelfunctionsM(x, z) by applyingequation (165) at frequencies from 12 Hz to 26 Hz. Thesemodelfunctionswere then stackedto form one multifrequencymodel function. The model function was then convertedto velocityC(x, z) usingequation(56) and the resultingdiffractiontomogramis shownin Figure 66.
4.4.4
Tomogram Interpretation
The velocity tomogram in Figure 66 clearly depicts the water bottom
andthe presence of twosalt sillsalongthe surveyline. The largersaltsill is about 31,800 ft wide while the smaller salt sill is about 8,860 ft wide. An apparent 20 percent velocity variation exists within each salt sill. The
layer boundariesof the sedimentarylayers surroundingthe salt sills are not resolvedand the gray scalewas selectedto emphasizethe salt sills and water bottom for analysis.
For comparison sake,the samedata set wasprestackdepth migrated, with the resultsshownin Figure 67. The velocitymodel for the depth •Equation (165) doesnot showan explicit frequencydependence.That is because we stopped writing the angular frequency dependence of the acoustic wavefield and
wavenumberafter equation (49) in Chapter 3 for the "sakeof brevity."
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4.4.
IMAGING
SALT
SILLS
147
Salt Sill Diffraction Tomogram Surface Location (S. P.) 0
1219 ' ,,, t ............
1119 I ............
1019 •.............
919 •............
819 t ............
Water Layer
719 -
:
ß ...::-;.;.:.;.;.:::.:.:.:.:.:.:.:.:.:-L-'-'-'..-'.-'.;.-'.',;.. :.:..'.......:.:-:-:-;.:.:-:-:.:.:.:."•:;::::-.'-:.:.: .-" :. ...:.:..•-:-:.:.:?•:;::..'::-:-:-:.: '..;-:-;.:-:..'.:-.'•'.... .: ;.:-:.:.:.;.:.:-:-:-:-:.:-:..'.:-:, ...... :::::::::::::::::::::::::::::::: ..................... ::........ :................ ::: ....... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::•.....`• -.'•....'••i• "•.- --.'-:-.'-.'-'. - :::::::::::::::::::::::::::::::::::::::
i:!:i:!:i:i:!:i:!:!:!:!:!:!:!:i:•.)k.%•i:..•::$:::::.:.:.....•:::!:i:!:!:!:!:!:!:!:i:!:!:!:i:!:!:!:!.i:i:::!:i:!:::!:!:!:!:!:!::•!•:::..i:!:!:i:i•.:.•:i::.:::2i: :"-,•....'-:•:•......•...•..'.•.•:..-:.'c•'::::i:i:!:!•: :':'..'::• i:i:!:i:!:!:!:i:!:!:!:!:i
:!:!:!:!:i:!:!:!:i:!:!:!:!:!:!:•::.:..'•. :.!:!:!:!:!:!:!:!:!•'.--,•:::'•:•i::;•:•'•' ..•:...•:•:!:!:!•....•:•::!•:•.:i•::•:::•.::..:•::.::...::..•:::..::•.•::.::.::::•:!:!:i:!:!:!:::!•.•:!...•!:! ..•:::•::::::)' ======================= ::i:!:: :i:! :i:::•: :::::: ::i:::::i: i:i:!: i:i:i:::::!:i:i :!:i:i:!:!:i::::::: i.:'..%•....:'.'!:!:!:!:! :!:!:!: i:::::::::!:!:i :i:!:::!:!:! :i:i:!:i:::! :!:!:!: i:::!:i:::i :i:::!:'.-'.'•:'.!:i:i:: :::i:i :!:!:::::: :::::::::::::::i:i: ::i:i: ::i:::! :::i:: :i:i:i::•:i::."."::: ".'::i :::::: ::::::::::::::::::::::::::::-':.'" .•
Depth
(kft)
Velocity Scale 5000
1
8198
10000
ft
11347
12271 12649
(ft/s)
13027 13992 15000
Fio. 66. The diffraction tomogramshowingvelocitiesfor the water bottom and salt sills along the survey line.
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148
CHAPTER
4.
CASE
STUDIES
Prestack Depth Migrated Section Surface Location (S. P.) 1219
0---•
1119
.........
-•.•••••• .......
Depth(•)
t
"' ' ....
............. ••
1019
•
919
819
I
SALT SlLL•..•,•.•.:".-' "' "'-' ' ' '"•' -."•'•;'•-"•-'•-'• ..... •'- ....
•, ........ .•.•..•.•.•• .•-••
719
•
•-•
• ....•-
•-••
•
-•-:'
10000ft
FIG. 67. The prestackdepth migrated sectioncorrespondingto the velocity tomogram in Figure 66. Processingand display courtesyof Guy Purnell, Texaco.
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4.4.
IMAGING
SALT
SILLS
149
migration was derived using an interactive prestack migration focusinganalysis approach. The salt sills are labeled and their boundariescorrespond to the strong reflectionevents surroundingthe labels. The water bottom is also clearly visible. The water bottoms defined in both the diffraction tomogram and the depth-migrated section are in good agreement as to location and shape. The salt sill tops, which are visible on the seismic section, are not well
definedin the tomogram. This is possiblybecauseof the low frequencies (12 to 26 Hz) usedto constructthe tomogram,whichmay not providethe needed resolution. However, the widths of the salt sills are comparable. The shapesof the salt sill basesare in very poor agreement. Obviously the depth-migratedsectionportrays a more correctpicture than the tomogram for the salt sill bases.The invalid assumptionof a finite-extentobject for the deeper salt sill basesis causing their images to be both mispositioned and degradedon the tomogram. This can be shownusingthe first line of equation(146) in a simplecomputation. First, we focus our attention on an infinitesimally small part of the base of salt reflector
where the salt-sediment
interface
is horizontal.
This
isolatesa singlescattererof the incidentwavefieldat location(zo, zo), which we definein termsof a modelfunctionas M(z, z) = 6(z- zo)6(z - zo), where the deltas are Dirac delta functions. Next we place the source and
receiverdirectly abovethe scattererwith the coordinates(x0 = Xo,do= O)
and (ze = zo, de = 0), respectively. In doingso we haveinsuredthat the scatteredenergyfrom the selectedscattererand the specularreflectionfrom the horizontal interfaceapproximatelyshare the samevertical raypath and traveltime.
Second, given the set-up from the last paragraph we must evaluate the other parameterswhichgo into equation(146). For vertical raypaths Figure 26 indicates that the sourcewavenumbervector has components (7, = ko, ko = 0) and the receiverwavenumbervectorhas components
(7e - ko, ke = 0). Substitutingthesecomponents into equation(144) gives the wavenumber componentskr = 0 and k• = -2ko required by
equation(146). Third, we substitute the model function for the selectedscattereralong with the informationderivedin the previousparagraphinto equation(146) and integrate. The result is
- - o,o;t,,, e(t,o - o,o) - I
(2),
where 2Zois the round-trip raypath length betweenthe selectedscatterer and the zero-offsetsource-receiverpair. We restate the last equation by
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150
CHAPTER
4.
CASE STUDIES
substitutingin equation(52) for k0 whichgives,
- o.o; - o.0)
lej(2zo/C'o )
ß
Here2zo/Coisjust the traveltimeto of the associated diffractionevent(or specularreflectionevent) on the seismictrace, or to = 2zo/Co. It states that the incident energy travels to the selectedscatterer and the scattered energytravelsto the receiverat the constantbackgroundvelocity Co. For a finite-extent object this propagation model holds, but the salt sill bases obviouslycannotbe consideredas part of a finite-extentobject sincethey
are so distantfrom the water layer (our constantvelocitybackground). Replacingthe propagationvelocityof salt with that of water cannotyield a properly imaged model function as we will now demonstrate. SP 995 in Figure 61 showsthe specular reflection from the base of the largesalt sill at time to = 2.92 seconds.Even thoughFigure 67 showssome reflectordip at SP 995, we will assumethe scatteredenergy'straveltimeand the specular reflection's traveltime are so close as to be the same and that
the raypathsare nearlyvertical. Usingto = 2.92 s and Co = 5000 ft/s we get an imagedepthon the tomogramof Zo= 7300ft. The depth-migrated sectionin Figure 67 showsthe largesalt sill's baseat SP 995 at a depth of 10000ft, a 2700 ft differencewhich is easily greater than can be accounted
for by reflectordip. The diffractiontomogramin Figure66 showsa high velocity anomalynear zo = 7300 ft which is probably the image locationof the large salt sill's base. Thus, this example demonstratesthat scatterers outsideof an acceptablefinite-extent object are mispositionedand result in a deteriorated image. Finally, we ask, "What can be done to get a correcttomographicimage?" The immediate answeris to use a variablebackgroundvelocityinstead of the constant velocity assumedin Chapter 3. We would want a variablebackgroundvelocity which doesn'tscatter a significantamountof energy and is capableof properly imaging the model function in space. Then the higherfrequencyperturbationsto the variablebackgroundvelocity wouldbe smallerin sizeand magnitudethan for a constantbackground velocity, making the Born approximation more acceptable. Several of the referencesat the end of Chapter 3 address the application of a variable
background velocity. 4.5
Suggestions for Further Reading Gibson, Jr., R. L., 1994, Radiation from seismicsourcesin cased and cemented boreholes: Geophysics, 59, 518-533. Theo-
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4.5.
SUGGESTIONS
FOR FURTHER
READING
151
retical investigationinto the radiated energy by volumetric
(air gun), radial stress,and axial stress(clampedvibrator) sources located in cased and cemented boreholes.
The volu-
metric sourceis the only sourcewhich doesnot have a nonlinear frequencydependentradiation pattern.
Howlett, D. L., 1991, Comparison of borehole seismicsources under consistent field conditions: Expanded Abstracts for the Society of Exploration Geophysicists'Sixty-first Annual International Meeting and Exposition, Nov. 10-14, Houston, Texas. Comparesdata from explosive,clamped-vibrator,air glun,and cylindrical-bender-crgstalsourcestaken at Tezaco's geophysicaltest facility in Humble, Texas.
Meredith, J. A., ToksSz, M. N., and Cheng, C. H., 1993, Secondary shear wavesfrom sourceboreholes:Geophys. Prosp., 41,287-312. Borehole sourcessubmergedwithin a liquid can create tube waves. When the tube wave velocity is greater
than the formarion'sS-wave velocity,a reach wave (convertedS-wave)is generatedby the tubewavewhichmay interfere with other events recordedin the receiver well.
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Appendix
A
Frequency
and
Wave
er
numb
Frequencyand wavenumberare two basicterms usedextensivelyin this book to describewave propagation. This appendix is intended to clarify their definitions.
A.1
Frequency
Assumea sourceemits the sinusoidalsignal in Figure A.1. The particle
displacementB(t) of the mediumat the sourceis shownas a function of time, t. The signal repeatsitself every 4 ms with the sameamplitude after a lapseof time T calledthe "period." Another measureof the signal'scycle is the frequencyf whichgivesthe numberof cyclesthe signalgoesthrough per unit time, or 1
f = •,
(A-l)
where the appropriate unit is hertz, abbreviated "Hz", which stands for "cycles per second."
Many times we will representa sinusoidalsignal in terms of the projection of a rotating vector onto some axis. Thus, we might represent the
particle displacementin Figure A.1 by the equation,
B(t)-[B[sin(-•t-•b), (A-2) 153
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154
APPENDIX
A. FREQUENCY AND WAVENUMBER
I B(t) ß
I
I
L*
FIG. A.1. Particledisplacement B(t) is plottedas a functionof time,t, for a sinusoidalsignal with a period T- 4 ms. The associatedfrequencyis, f - 1/T- 250 Hz. Angularfrequencyis definedasco- 2•rf - 2•r/T.
whereI B I is the magnitudeof the rotating vectorB(t) and •bis the phaseof
thesignalat timet - 0.• Sincethe vectorB(t) rotatesthrough2•rradians everyperiodT, we call 2•r/T in equation(A-2) the "angularfrequency" which is written as2
co- •- = 2•'f.
(A-3)
In Appendix B you will see that actual signalscan be thought of as a sum of many sinusoidalsignalsof differentfrequency,where each can have a different magnitude and initial phase. The Fourier transform is usedto
decompose an actualsignalinto its angularfrequencycomponents I B(w) I and •b(w). A.2
Wavenumber
When we definedangularfrequencycothe observationpoint of the signal waskept fixed in space(at the sourcein Figure A.1) and we studied the signal'spropertiesin time. In this sectionwe arrive at similar conceptsby observingthe signalthroughoutspaceat a fixed time. Figure A.2 showsa sinusoidalsignal traveling in either the +X of-X direction with a wavelength,,•, the distancerequired for the waveformto repeat itself. 1The phaseof the signal in Figure A.1 is zero at time t = 0. 2Both w and f are commonlyreferredto as "frequency."The contextof the equation will tell you which type of frequency.
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A.2.
WAVENUMBER
155
I B(x) I
I
Lø
FIG. A.2. Particle displacementB(x) is plotted as a functionof location, x, for a sinusoidalsignal traveling either in the +X or -X direction with
a wavelengthof A = 4 m. The associated wavenumberis, 1/A• - .25/m. The angularwavenumberis definedas k• -- -l-2•r/Awherethe signchosen is decidedby the direction of wave propagation.
Wavenumberis analogousto frequencyf, but is definedin terms of the pe-
riodicdistanceof the signalA asq-1/A. The signchosenfor the wavenumber is decidedby the directionof wave propagation. Most of the time we work with angular wavenumber, 2•r
k• - +-•-.
(A-q)
We generallyrefer to the angular wavenumberasjust the wavenumberand let the context of the equation define the type of wavenumber. Unlike time, wave propagation in spacecan be in many directions and actually must be defined in terms of a vector instead of a scalar. In higher
order dimensions of spaceequation(A-4) must be written as a vector, 27r,,
k = Tg,
(A-5)
where • is a unit vector pointing in the direction of propagation. Figure A.3(a) showsa sinusoidalwavepropagatingalongthe •, directionin a 2-D spacewith wavelengthX. We plot the wavenumberfor this wave in the
k• - k, planein FigureA.3(b). If ko = 2•r/A represents the magnitudeof the wave'swavenumber,then k in equation(A-5) can be decomposed into its vector components,
k
-
kog - k•,i+k,i,
(A-6)
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156
APPENDIX
A. FREQUENCY AND WAVENUMBER
(a)
(b)
FI•. A.3. (a) A sinusoidalwavepropagatesalongthe •; directionin a 2D spacewith wavelength A. (b) The wavenumber of the propagatingwave in FigureA.3(a)is k = (2•r/A)•;. The components of the wavenumber are k• and k•, both of positive value in this case.
where the valuesof k', and kz have signs determined by the direction of
wavepropagation;both positivein FigureA.3(b). Lastly,a singlefrequencywavewill travel onewavelengthin oneperiod. We may determinea specialvelocityof propagationfor this wavecalledthe "phasevelocity" defined by,
=
= Ikl =
(A-7)
Note that the phasevelocityis definedfor a singlefrequency.If a signal is composedof sinusoids of many differentfrequencies, then the velocityof propagationof that signalmay not be the sameas the phasevelocity,but a different velocity called the group velocity, definedas
V = Ok,, •:+ • •'
(A-S)
In this book we useo.,- Ck so that V - C and we assumeno dispersion.
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Appendix The
B
Fourier
Transform
Seismictomographyfrequently utilizes Fourier transformsof functions. Therefore, an understandingof the conceptis important to understandthe material presentedin this book. This appendix is intended to refreshyour memory of the Fourier transform. You might want to review Appendix A since we make use of the concepts of frequency and wavenumber in our discussion.
If h(t) is a function of time, then we representits temporal Fourier transform by,
FT[h(t)] :• •(w),
(B-i)
where w is the temporal frequency.The inverseFourier transform is represented by,
We refer to h(t) and h(w) as a Fourier-transform pair. Similarly,we may take a spatial Fourier transformof a function g(x), where x is a spatial coordinate, and represent the operation by,
FT[g(x)]
•
•(k•),
(B-3)
where k• is the spatial frequency,or angularwavenumber •, alongthe xdirection. As before, the inverseFourier transform operation is represented by,
rT-•[O(k=)] • • Most of the time the n•es ened to just frequency •d
g(x).
temporM frequency•d
wavenmber,
respectively. 157
•g•
(B-4) waven•ber •e short-
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158
APPENDIX
B.
THE FOURIER
TRANSFORM
Beforepresentingthe actual Fouriertransformequationsusedto compute the operationsimpliedby the aboveequations,we give a short review of the Fourierseriesand exponentialFourierseriesto providea transition; sincemost everyoneis familiar with the Fourierseries. Then, the Fourier transformof continuousfunctionsis discussed followedby the Fouriertransform of sampledfunctions.The last sectiongivessomespecialtransforms used in this book.
Fourier
B.1
Series
A periodicfunctionf(x) of length2L canbe writtenin termsof a series of cosinesand sinesprovided it contains a finite number of discontinuities and a finite number of maximum and minimum values. The series is called
a Fourier seriesand is representedby,
The seriescoefficientsao, an, and b• are given by,
1/_L
ao = -L •;f(x)dx,
1/_ œ
a, = --L Lf(z) cos\ L dx'
(B-6)
(B-7)
and,
-
fix)sin (-•-/dx
(B-8)
respectively. Equation(B-6) showsthat ao/2 is just the averagevalueof
f(x) overtheintervalI-L, L], commonly calledtheDCshift. Equation (B7) givesthesamedefinition asequation(B-6)for aowhenn = 0 andwill be usedhenceforth. Also,equation(B-8) requires bo= 0 for anyf(x). For our purposes, equations(B-5), (B-7), and (B-8) are best rewritten in terms of spatial frequency,or wavenumber 2, definedas 2•r radians
per wavelength.The longestwavelength for f(x) is 2L; determinedby 2Notethat wecouldequallyas well presentthis discussion in termsof a time coordinate t and its frequency w .
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B.2.
EXPONENTIAL
FOURIER
159
SERIES
settingn = 1 in equation(B-5). The corresponding wavenumber or fundamentalfrequencyis 7r/L. Higher valuedwavenumbers (shorterwavelengths)are foundby multiplyingthe fundamentalfrequencyby n where n = 1, 2, 3, ...;,n = i givesthe fundamentalfrequency.Thus, we now write the wavenumberalongthe x-axis as kn-- n7r/L and the Fourierseriesin terms of wavenumber as,
f(x) - -•-+
anosknx+bnsinknx
where the seriescoefficientsanand bn are defined by,
(In -
cos
(B-10)
-
f(x)sinknxdx.
(B-11)
Exponential
Fourier Series
•
L
and,
B.2
Here we rewrite the Fourier seriesof the previous section in terms of
exponentials,thusgettingusonestepcloserto the Fouriertransformwhich also usesexponentials. The key equation permitting this step is Euler's formula,
eJknx= cosknx +jsinknx,
(B-12)
wherej - x/Z-1. We recognize that the cosineis an evenfunctionand the sineis an odd functionand useequation(B-12) to write, cosknx =
sinknx =
ejknx + e-jkn x 2
-, and
ejknx _ e-jkn x
(B-13)
, respectively.
(B-14)
Substitutingequations(B-13) and (B-14) into equation(B-9) and regroupingthe terms with respectto the signof the exponentialgives, -jknx ß (B- 15)
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160
APPENDIX
B.
THE
FOURIER
TRANSFORM
Equations(B-10) and (B-11) showthat a, - a_, and b, - -b_,, respectively. Applying theserelationshipsto the secondserieson the right-hand sidein equation(B-15) and redefiningthe summationfrom n - -1 to -c• givesthe exponentialFourier series,
f(x) - Z c,•eJk,-,x,
(n-16)
wherec,.,- (a•- jb•)/2 and k• - n•/L. Substitutingequations (B10) and (B-11) for a• and b• in the definitionof c•, alongwith using equation(B-12), givesthe integralequationfor the seriescoe•cientsca,
C• -- • for n - 0,•1,•2,
B.3
•3,...,
Lf(z)e-Jk•xdz,
(B-17)
•c•.
Fourier Transform-
Continuous f(x)
The Fourier transform is intended to operate on nonperiodic functions
over an infinite range. Thus, we can no longerrestrict x to the range
I-L, L] asdefined fortheexponential Fourierseries.However, equations (B16) and (B-17) can be utilized in extendingz to infinite limits. First, we substituteequation(B-17)into equation(B-16) resultingin,
or
.•(k,•)eJk'• x__1 where 2L'
=
f(:rt)e _t, _jk,., X,dx •. L
The frequencyinterval betweensuccessive k• is given by,
Ak k•+•k• (n+1)• •r •r We rewrite this equation in a convenientform, 1
Ak
2L
2•r
(B-18) (B-19)
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B.4. FOURIER TRANSFORM- SAMPLED F(X)
161
Taking the limit of equation(B-20) as L--• c• allowsus to write,
lim (•L)- dk
œ--,oo
271'
(B-21)
and to write kn as a continuousvariable k. Now, taking the limits of equa-
tions(B-18) and (B-19) as L -• oo and usingthe resultfrom equation(B21) yields the Fourier transformequationsfor a nonperiodiccontinuous function,
I
f(:) -
I•(k)eJkzdk oo '
(B-22)
f(x)e-Jkxdx.
(B-2a)
and
i>(k) -
Equations(B-22) and (B-23) enableusto computeoneof the Fouriertransform pairsgiventhe other. Equation(B-22) is the inverseFouriertransform operationrepresented by equation(B-4) and equation(B-23) is the Fourier transformoperationrepresented by equation(B-3). B.4
Fourier Transform-
Sampled f(x)
The use of digital computersto carry out computationsrequiresthat we samplethe continuousfunctionf(x). We may take a samplefrom the continuousfunction f(x) at intervalsof Ax giving the discretesamples: ß.., f-2, f- •, f0, f•, f2,. ß., where the subscriptsindicate the sample number. If the sample number is n, then the positionof the sampleis x - nAx.
However,a sampledfunctioncan no longercontainwavenumbers (or frequencies)out to 4-00 as demonstratedby the integrallimits for the continuousfunction f(x) in equation(B-22). The shortestwavelengthrepresented by a sampled function is 2Ax which gives the highest possible wavenumber(or frequency),
knyq= Az'
(B-24)
referredto as the Nyquistfrequency. a Note that this alsoappliesto temporally sampled functions. aThe Nyquist frequencyhas a wavelengthor period which is sampled by two points; or we may state that the shortest wavelength or period in a sampled signal must contain at least two sample points.
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162
APPENDIX
B.
THE FOURIER
TRANSFORM
In sectionsof this book where we are referring to a sampledfunction, you will see the limits on the inverse Fourier transform equationswritten
as :t:•r. Equation (B-24) showsthat thesefrequencylimits correspondto a unit sample interval, Az = 1. A unit sample interval is commonly usedin
digital computationsto avoidmultiplyingby Az, whichbecomesnothing more than a scalefactor. Thus, you will seethe Fourier transformequations written as 4
f(x)- •1 , p(k)eJkXdk ,
(B-25)
P(k) -
(B-26)
and
f(x)e-Jkx dz,
whenf(x) and•(k) aresampled functions. Again,theclueto f(x) being sampledis the noninfinite frequencylimits. Of course,in actual computer
computations we usuallydo not useequations(B-25) and (B-26) directly. Instead, the Fast FourierTransform(FFT) method is utilized. B.5
Uses
of Fourier
Transforms
FromequaLion (B-26) weseethat F(k) is generallya complexvalued function, called the complex spectrum, which may be written as,
P(k) -
•Re{P(k)} + jlm{P(k)},
(B-27)
whereRe and Im designatesthe operationof taking the real and imaginary
partsof P(k), respectively. FigureB.1shows thelocation of onepointof the complexspectrumin the complexplane;so calledbecausethe real part of the complexfunctionis plottedon oneaxisand the associatedimaginary part on the other axis. The figure demonstratesthat the complexspectrum can be also describedin terms of polar coordinates.In polar form we can write the complexspectrum as,
P(•) - Ip(•)leJ•(•),
(s-28)
whereI P(•) I iscalledtheamplitude spectrum givenby,
]#(•)] - V/ee{#(•)} •+t,•{#(•)}•, 4Mar•y times geophysicists cha•ngethe sign in the exponentialof the forward and inverse Fourier transforms
when the transform
involves time.
This is done so that we
stay consistentwith the physics, that is, a wave traveling in the +x direction is described
byA(w,kx)eJ(kxxwt)andnotA(w,kx)eJ( kxx+ wt).
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B.5.
USES OF FOURIER
TRANSFORMS
163
Complex Plane
Im {F(k)}l (Re {.•(k)} ,lm {•(k)} ) > Re {F(k)}
FIG. B.1.
Complex plane with an arrow directed to the point
(Re{17'(k)},Im{17'(k)}). Thepointin polarcoordinates is represented by the magnitudeI F(k) I definedby equation(B-29) and the phase(I)(k) definedby equation(B-30). The magnitudeis calledthe amplitudespectrum and the phaseis called the phasespectrumwhen eachis plotted separately as a function
of k.
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164
APPENDIX
B.
THE
FOURIER
TRANSFORM
and (I)(k) is calledthe phasespectrumgivenby,
(I)(k)- tan -1 Im{F(k)} •
ß
(B-30)
The amplitude spectrumgivesthe magnitudeof a particular sinusoidat frequencyk while the phasespectrumgivesits shift in spaceor time. Most often we are interested in the frequency content of a signal and a plot of the amplitude spectrum servesthis purpose. Throughout this book we work in tl•e Fourier domain when it simplifies derivations. In addition to the above concepts of amplitude and phase
spectra, we encounterthree other applicationsof the Fourier transform' the 2-D Fourier transform, the Fourier transform of the time derivative,
andthe Fouriertransform of eJkox. Supposewe wish to take the Fourier transform of a function f(x,z) with respect to the spatial variables x and z. This operation is called a 2-D Fouriertransform. We beginby taking the Fouriertransformof jr(x, z) with respect to x, or
P(k•,z) -
f(z,z)e-Jk•zdz,
(B-31)
wherethe wavenumberalongthe x-axis is denotedby ks. Next we apply a Fouriertransformto equation(B-31) with respectto z, or
•(k,,,kz) -
•(k,:,z)e-JkzZdz,
(B-32)
where the wavenumberalong the z-axis is denoted by kz. Substituting
equation(B-31) into equation(B-32) givesthe definitionof a 2-D Fourier transform,
•(k•,k,) -
f(x,z)e-J(k•x + k•Z)dxdz ' (B-33)
Similarly, the inverse2-D Fourier transform may be definedas,
f(x,z)
/_•o /_••(k•, k•)eJ(k•x +k•Z)dk•dk• ' (B-34)
471-2
Time derivativesare simpler in the Fourier domain which is one reason many differentialequationsare solvedin the frequencydomain. Look at the inverse Fourier transform,
= 1
f'(w)e-Jwt dw,
(B-35)
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B.5.
USES OF FOURIER
TRANSFORMS
165
wheret is time andw is frequency.The first time derivativeof equation(B35) is, ef(t)
= -• • /_•o j.•(•)• _
dt
a•, -J•t
= FT-•[-j•P(•)].
(B-36)
A second-orderderivative is found by taking a time derivative of equation (B-36) or,
dt 2
(B-37) Thus, an nth order derivative is defined,
rr-•[(-i•)-p(•)].
dtn
(B-38)
WesimplymultiplyP(w) bytheappropriate factor s (-jw)" andtakethe inverse Fourier
transform.
ThelastitemtonoteistheFourier transform oftheexponential e-Jkox, which comesup many timesin diffractiontomography.We determinethis in a roundabout way. We use the Dirac delta function defined in Appendix C
in thisderivation. Taketheinverse Fouriertransform of .b(k)- 5(k- ko), wherethe Diracdeltafunctionis locatedat wavenumber koin the spatial frequencydomain. By equation(B-22) we find that, f(x)
= -2•r
5(k- ko)eJkxdk,
= l,j•o•. 2•r
Thus, theFourier transform ofeJkoxmust be2•rS(k-ko).
SUse(+jkx) n and (+jk,) n for spatialderivatives.
(B-39)
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Appendix
C
Greens
Function
Equation(60) in Chapter3 on seismicdiffractiontomographydescribes the propagationof the scatteredwavefieldP0(r) at a constantbackground velocitywhen inhomogeneities scatter both the incidentwavefieldPi(r) and existingscatteredenergyP0(r). We rewrite equation(60) here for your reference as
Iv • + •o•]•,(•) -
•o•U(•)[•,(•)+ •,(•)1.
(C-l)
Through the use of Green'sfunctionswe end up with an integral solution to thisequationgivenby equation(64), the Lippmann-Schwinger equation, which we rewrite here for your referenceas,
?o(,,) - -• f •(,,I•")•(,?)[?,(,?) +•0(•')]a•'. (c-•) G(rl rt) is the Green'sfunctionand the integralis takenovera planein 2-D space or a volume in 3-D space.
The intent of this appendixis to provideyou with an intuitive feelingfor Green's functions. With this understandingyou will know how to immediately write down an integral solution to any inhomogeneousdifferential equation with constant coefficients,such as the solution to the partial dif-
ferentialequation(C-l) givenby the integralequation(C-2). We do NOT go overtechniques for determiningthe Green'sfunctionG(r I r•) whichare readily found in many texts with chaptersdevoted to the subject, such as referencedat the end of this appendix. However, an example problem is
workedlater on to giveyousomeideaof howonemay solvefor G(r I r•) . The approachwe take for conveyingthe ideas of Green's functions is throughthe conceptsinvolvedwith filter theory. We do this for two reasons: 167
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168
APPENDIX
C.
GREEN'S
FUNCTION
Gt LINEAR OPERATOR
FIG. C.1. Gt is the impulseresponseof the linear operator giventhe input discrete impulse 50 at time sample t - 0.
(1) its easyto see, and (2) many of you are alreadyfamiliar with filter theory,especiallythe conceptof convolution.We will look at the discretely sampledcasefirst and then extrapolate to the analogcase.
C.1
Filter Theory
In filter theory we can input a discrete impulse 5• of unit height at samplet into a linear operator or systemwhich givesan output, calledthe impulse responseGr. Figure C.1 showsa plausibleimpulse responseof a linear operatorwhen a unit impulse5o is input at sample! - 0.• A linear systemis time invariant when the impulse responseG• is found at the same time lag relative to the time sample of the input impulse. For example, Figure C.1 showsan input impulse50 with an impulseresponseGt starting at time sample0. If we had usedan impulse5•, then the impulseresponse G• would have begun at time sample5 which may be written as Gt-•. The impulseresponseG• in Figure C.1 can also be scaled. We may multiply the unit impulseinput 5oby a scalarf0 whichresultsin the impulse response G•-0 beingscaledby f0 asshownin FigureC.2(a). The t-0 in the subscript of G•-0 indicates the delay of the impulse responseGt because of a delay in the input. Here no delay occurs and the output foG• is the responseto the input fo5o, where f0 ---1. We can scale the impulse input 51 by a scalar fl - 1 which results in the linear operatorresponseflG•-i - G•-i in Figure C.2(b). The subscript t-
I indicates that G• is delayed by one time unit in accordancewith the
time invarianceprinciplestatedearlier.Similarly,FigureC.2(c) showsthe impulseinput 52 scaledby f2 - -1/2 resultsin a linear operatorresponse 1We assumea unit interval, At = 1, betweensamples.
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C. 1.
FILTER
THEORY
'
169
fog t.o
-1 •o o OPERATOI•.
t
(a)
+
1Gt.1
1•1 LINEAR
0
!
OPERATOI•
o
t
•) +
-1/2õ2
f2 Gt.2 LINEAR OPF•ATO•
(c)
01--1 ft llU-
LINEA• OPF•ATOI•
!øi
Pt
(d)
FIG. C.2. Illustration of the principle of superpositionfor a linear operator. The input consistsof a seriesof scaled, time-shifted impulsesgiven by
ft -- j•050-}'f151 -}'j•252,wherefo = -1 in (a), fl ----1 in (b), and/2 = -1/2 in (c). The resultingoutput to f• is the sum of the responses in (a), (b), and (c): P• - foG,-o 4-fxG,_x + f2G,-:• shownin (d). This is just an example of convolutionof an input signal ft with an impulseresponseof a linear filter
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APPENDIX
170
C.
GREEN'S
FUNCTION
FigureC.2(d)is thesumof the inputsandoutputsfromFiguresC.2(ac). The discretizedinput is thoughtof as a sum of scaledimpulsesgiven by,
ft -- fo•o q-fl•l q-f2•2-- (-1)50 + (1)51 q-(-•1)5:•.
(C-3)
The linear operator'sresponseto the input function ft is given by, ---- foGt-o q- flGt-1 q- f2Gt-2
= (-1)G,_0 q-(1)G,_l q-(-•
(c-4)
Equations(C-3) and (CL4)combinedillustratethe principleofsuperposition for a linear operator. That is, the same output results whether a linear
operatoractson the entire input as depictedin Figure C.2(d), or on each individual componentof the input as shownin Figures C.2(a-c) and the results summed.
Generalizingequation(C-4) we have,
This equationis the well-knownconvolutionequationfor two discretized signals. Here the convolutionis between the input function .It and the impulse responseof the linear operator Gt which givesthe responseof the linear operator Pt. If the above functions are continuous,then the sum in equation(C-5) becomesan integralor
P(t)- • f(t')G(t - t')dt',
(C-6)
wheredr' is explicitlywritten in placeof At' in equation(C-5) whichwas assumedequal to one.
C.2
PDE's as Linear Operators The partial differentialequation(C-l) can be cast in the samelight as
thelinearoperatorin filtertheory.Thesourceterm,ko•M(r)[Pi(r)+P,(r)], is analogousto the input to the linear operator. The linear operator is the
bracketedpart of the functionon the left-handside,IX72+ ko•], and the solutionto the differentialequation,P,(r), canbe thoughtof asthe output from the linear operator.
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C.2.
PDE'S
AS LINEAR
-6 (r-r')
OPERATORS
•
171
•2+k2 ] • G(rlr' oI )
- -5(r-r') [•2+k2o] G(rlr') FIG. C.3. The Green's function is the impulse responseof a linear differential equationoperatorto a sourceterm (input) givenby a negativeDirac delta function. Comparethis with the impulseresponsein filter theory depictedin Figure C.1.
As in the previous section, the first step is to determine the impulse responseof the linear operator to an impulsive input. For the partial dif-
ferentialequationwe usea negativeDirac deltafunction 2 -5(r-
r •) for
the impulsiveinput and the Green'sfunctionG(r- r •) for the impulseresponseto the negative Dirac delta function. Thus, using the analogiesin the previousparagraph,equation(C-1) is rewritten as,
IV2+ ko•lG(r I r') -
-5(r-r'),
to get the impulseresponse,or Green'sfunction,G(r I r') = •(r-
(C-7) r').
Note that the space vector r is the independent variable here instead oftime used in the previoussectionon filter theory. Figure C.3 summarizes the analogy made between the differential equation operator and the filter theory operator in Figure C.1.
The Green'sfunctionsolutionG(r I r •) to equation(C-7) is the impulse responseto the negative Dirac delta function. Just as with filter theory,
we may weight the negativeDirac delta function,say -f(r')5(rr'), on the right-handsideof equation(C-7) whichresultsin the solution(output) f(r')G(r Jr'). Using the principleof superpositionshownin Figure C.2, the resultingresponse isjust the integration(sum) overall outputresponse 2The Dirac delta function6(x) has ax•infiaitesi•
width, infiniteheight,and an
area equal toone atx = 0. It isdefined asf_+•S(x)dx = 1. When combined with another function, f_+•!(z)a(z-Xo)= l(Zo),where theDirac delta function islocated at 3• '-' 3•o.
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172
APPENDIX
C.
GREEN'S
FUNCTION
componentsfrom the linear operator, or
: - fr,
(c-s)
wherethe negativesignin front of the integralis presentbecausethe Green's function is the impulse responseto a negative Dirac delta function. Equa-
tion (C-8) is similarto equation(C-6) exceptfor the negativesignandthe independent variable.
Settingf(r') in equation(C-8) to the right-handsideof equation(C-l),
becomesthe sourceterm (input to the linear operator) in equation(C8) and we get the integralsolutiongivenby equation(C-2). Thus, the integralsolutionto equation(C-l) isjust the negativeof the convolution of the sourceterm (input) f(r) with the Green'sfunction(impulseresponse), analogousto filter theory. The r- r • part of the Green'sfunction(see equations(C-9) and (C-10) below)givesthe "lag"in the impulseresponse as a result of the weightednegativeDirac delta function being located at a position other than r • = 0. This is a space-invariantproperty equivalentto the time-invariant property of the linear operator in the previous section on filter theory. Chapter 3 on seismic diffraction tomography makes extensiveuse of Green's functions. However, closeinspection reveals that only two Green's
functionsare actuallyevermentioned,both aresolutionsto equation(C-7). The 2-D solutionto equation(C-7) is exclusivelyusedin Chapter 3 and is given by,
[r - r' I), a(rl') - JHo(1)(ko whereHo (1)is thezero-order Hankelfunction of thefirstkind. Herethe negative Dirac delta function representsan infinite-line seismicsourceor an infinite-line scatterer at r' which causesa cylindrically shaped seismic disturbance
that is determined
at an infinite-line
field location
r. Both the
infinite-line source and infinite-line field location are perpendicular to the
plane representingthe 2-D space. A 3-D spacesolutionto equation(C-7) is givenby,
ejkolr-r'l
G(rlr')= 4•rlr-r']'
(C-10)
This equation is not usedin Chapter 3 sincemost data acquisitionschemes, suchas crosswellseismic,are gearedto 2-D spaceor cross-sectional studies
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C. 3.
GREEN'S
FUNCTION
EXAMPLE
173
rather than volumetric investigations.The 3-D spacesolution represents point seismicsourcesor point scatterersat r t which causea sphericalseismic disturbancethat is determined at a point field location r.
C.3
Green's Function Example
In this section we find the integral solution of a simple 1-D differential equation to reinforcethe ideas presentedin the last section. The method of determining the Green's function here is just one example of many and we refer you to the referencesat the end of this appendix for other methods. The differential equation is,
[d-••a2]p(x) - f(x), for -cx> <x. (C-11) The first step is to find the impulseresponse,or Green'sfunction,G(z) to a negative Dirac delta function sourceterm located at x - 0. The above equationis rewritten,a
dx - aV']G(x) - -5(x).
(C-12)
We solvefor G(x) by taking the Fouriertransformof eachterm in equation (C-12) as discussed in Appendix B which gives,
[(jk)2 - aV']0(k) -
-1,
(C-13)
wherek is the wavenumber andj - v/Z'-•. Solvingfor (•(k) givesthe Green's function in the wavenumberdomain,
=
+1
(C-14)
Taking the inverseFourier transformof the last equation givesthe Green's function for a negative impulsive sourceterm at x - 0,
G(x) -
2a
(C-15)
If the negative Dirac delta function werelocated at x - x', then the Green's function would be written with a spatial lag,
G(xI x') =
2a
'
(C-16)
3Assumeno lag, x t = O, in the Dirac delta function at this point, and introduce the lag when doing the convolution(sum of weightedoutput from the linear operator) betweenf(x) and G(x).
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174
APPENDIX
C.
GREEN 'S FUNCTION
whichis usedin the integralsolutionin equation(C-17). We knowthe integralsolutionto equation(C-12) isjust the negativeof the convolution betweenthe Green'sfunctionG(x) and the sourcefunction f(x) or,
p(x) = -/-+2 f(x')G(z Ix')dz ',
(C-17)
wherethe x - x' in the Green'sfunctionis just the spatial lag as a result of the weightednegative Dirac delta function being located at x = x'. Substitutingequation(C-16) intoequation(C-17) givesthe integralsolution to our example,
2a
dx'.
(C-18)
We can now solvefor p(x) by integratingequation(C-18) oncethe source term f(z) is definedin equation((3-11). As a quick checktry a negative impulsivesourcelocatedat x'= 0 givenby f(x')= -5(x•). Your solution shouldbe equation(C-15). C.4
Suggestions for Further Reading The following referencescover the computation of Green's functionsin
detail.
Arfken, G., 1970, Mathematical methodsfor physicists,2nd edition: Academic Press, Inc. Courant, R., and Hilbert, D., 1953, Methods of mathematical
physics:JohnWiley and Sons(Interscience). Morse, P.M., and Feshbach,H., 1953, Methods of theoretical physics: McGraw-Hill.
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Index acoustic wave equation 46 Helmholtz acoustic
form
wave field
data acquisition crosswell configuration 2
46
data
46
function
11
algebraic reconstruction technique 33 amplitude spectrum 162 angular frequency 154
observed pobs
ari thmeticreconstructiontechnique(see Aa)
predicted P or prre
related
to true
model
function
M true 23 related
to estimated
model
func-
tion M or M est 23
ART 33, 41 compared with SIRT 37 nonlinear aspect 38
vector DC
shift
form
27
158
diffraction tomography 5 background velocity C'o 48 backprojection ray tomography 20, 40 CAT
formula
function
50
function
51
data
21
function
Born approximation 52, 138 Rytov approximation 56
sununary 21
backpropagatlon diffraction tomography
genericdata/model relationship59
87
illustrated
crosswell configuration 88 Jacobian
87
model function 45, 49
scatteredwavefieldPs(r) 47
surface reflection configuration 90 vsp configuration 89 borehole survey 99 Born Approximation 51 Rytov approximation comparison
total
wavefield
Born approximation 48 Rytov approximation 53 use with ray tomography 6 wavelength consideration 5
56
when Dirac
cell size 108
delta
to use 45 function
171
direct arrival 101, 101 direct-traxmform diffraction tomography
color scale 133
85
complex spectrum 162 polar form 162 computerized axial tomography 22 convolution equation 170 continuous
137
incident wavefieldPi(r) 47
procedure for 143
crosswell
Green's
Green's
constant density assumed 48
scan 22
reconstruction
2-D
3-D
form
steps for implementing 86 direct-transform ray tomography 16, 17, 39
170
seismic
Enhanced oil recovery 95
advantage over well logs 4
EOR
benefit
estimated model vector Euler's formula 159
of 3
crosswellseismic gathers 106 175
95 28
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176 first
INDEX arrival
101
advantages of 28
forward modeling defined direct
flow chart
arrival
102
discrete formulation first as'rival 102
formulated
25
in a continuous
domain
23
matrix
form
26
time invariance
in terms of wavenumber Fourier transform 2-D 164
159
nonperiodic continuous functions 161
of e-J køx 165 operation symbols 157 representation 161
sampled function 162 derivatives
164
transform pairs 157 frequency 153 generalized projection slice theorem 58
crosswellconfiguration68 statement
of 70
surface reflection configuration 82 vsp configuration 76 gray scale 133 function
171
2-D solution to wave equation 172 3-D solution to wave equation 172 example 173 group velocity 156 head wave hertz 153
168
Lippmann-Schwinger equation 51 Born approximation 51 linearized
52
nonlinearity 51 lithology interpretation of P-wave tomogram 120 magnitude 154 McKittrick
Field
125
medical tomography configuration 10 data function
11
Midway Sunset Field 96 model function 11, 49 cell size 108
diffraction tomography45, 49 discrete
24
estimated
M or M est
related to predicted data function P or ppre
23
incremental updateAiM•' 30 initial estimate 28, 108, 109 new estimate M(new) est 28 related
to observed
tion pobs 23
101
vector
form
27
nonunique 70
image function 11 impulse response 168 wavefield
28
.true M true
hyperplane 30
incident
matrix
principle of superposition170 scaling 168
series 158
exponential form 160
Green's
27
linear system
need for 23
time
Laplacian operator 46 linear inverse problem 13 generalized inverse operator ill-conditioned
raypath lengths 27 Fourier
29
24
47
Nyquist frequency 161 object function 11
integration notation 51
observed data vector 28
Jacobian
partial differential equation linear operator 170 period 153 phase 154 phase spectrum 164 phase velocity 156
87
crosswellconfiguration 88 surface reflection configuration 90 vsp configuration 89 Kaczmarz' method 26, 40
data
func-
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INDEX
177
plane wave decomposition 60 porosity tomograrn 122 predicted data vector 28, 28 projection 13 projection data '1 projection slice theorem defined
ray 5
series expansion method 9, 22, 40 estimated model function, M ½st23 forward modeling 26 Kaczmarz'
simultaneous
direct-transform
method
17
wavelength 9 wavelength consideration5 with diffraction tomography 6 ray tracing direct arrival 102 first arrival 102 information
traveltime slowness
136
70
Rytov approximation 56 Born approximation comparison 56
complex incident phase function
P-wave
air gun 103 clamped vibrator 131 S-wave
air gun 105 clamped vibrator 132 spatial frequency 13 steam flood enhanced oil recovery 95 tomogram 1 base line 96
lithology interpretation 120 McKittrick
function
137
observed seismic
119
52
seismic
total
downhole air gun 100 downhole clamped vibrator 129 marine
true model
surface
seismic
diffraction
5
function
26
tomography 1
data
seismicray tomography 39 seismic tomography 1
134
core study 121,123 reliability check 124 selecting a scale 133
salt sill problem 137 diffraction tomograxn 147 prestack depth migrated section 148 47
thrust
meaning of colors 1 porosity estimation 122 presteam vs. poststeam injection
complex total phase function 53 Rytov-data function 55
wavefield
117
source radiation pattern
complex phase difference function
scattered
residual
11
air gun 98 downhole clamped vibrator 127
based on well logs 126
on data
tech-
source
110
orienting horizontal components air gun source 102 clamped vibrator 133 reservoir interpa•tation based on tomogram a•ad well in-
note
reconstruction
SIRT 35, 41, 99 compared with ART 37 example 110 handling nonlinear aspect 110 nonlinear aspect 38 ray density weight 36, 42 terminating iterations 111
receiver
formation
iterative
nique (seeSIRT)
ray density 110 ray tomography 5 assumptions 9 backprojection 21
resolution
26
predicted data function, prr½ 23 true model function, M true 23
16
essence of 14
SIRT
method
observeddata function,pobs23
141
1
wavefield
Born approximation 48 Rytov approximation 53 transform methods 9, 39 traveltime parameters 99
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178
INDEX
traveltime pick consistency crosswell seismic data
107
mispick 108 poor data 108 surface seismic data 104
traveltime picking 99 direct m'rival 102 first arrival 102
polarity change
air gun (P-wave)103 air gun (S-wave)105 claxnped vibrator (P-wave)131 clampedvibrator (S-wave)132 radiation pattern 102 tying crosswell data 107 using computed traveltimes 108 traveltime residual 111, 117 true model
vector
28
velocity perturbation 47
wavelength 154 wavenumber 13, 155 angular 155 vector components
155
well logging disadvantages3
Wyllie's time averageequation 121
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GEOPHYSICAL
MONOGRAPH
SERIES
David V. Fitterman, Series Editor
Larry R. Lines,Volume Editor
NUMBER
6
FUNDAMENTALS
SEISMIC
OF
TOMOGRAPHY
By Tien-when Lo and Philip L. Inderwiesen
SOCIETY
OF EXPLORATION
GEOPHYSICISTS
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Lo, Tien-when, 1957-
Fundamentals of seismictomography/by Tien-whenLo and Philip L. Inderwiesen.
p. cm. (Geophysicalmonographseries;no. 6) Includesbibliographicalreferencesand index. ISBN 978-1-56080-028-6:$22.00
1. Seismictomography.I. Inderwiesen,PhilipL., 1953II.
Title.
III.
QE538.5.L6
Series.
1994
551.2' 2' 0287--dc20
94-23818 CIP
ISBN 978-0-931830-56-3 (Series) ISBN 978-1-56080-028-6 (Volume)
Societyof ExplorationGeophysicists P.O. Box 702740
Tulsa, OK 74170-2740
¸ 1994by the Societyof ExplorationGeophysicists All rightsreserved.This bookor portionshereof may not be reproduced in anyform withoutpermission in writing from the publisher. Published
1994
Reprinted2000 Reprinted2004 Reprinted 2006 Reprinted2008 Printed in the United States of America
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Contents
Preface I
2
vii
Introduction
1.1 The Concept of SeismicTomography .............
1
1.2
3
Applications ...........................
1.3 Ray vs. Diffraction Tomography ............... 1.4 Suggestionsfor Further Reading ...............
5 6
Seismic Ray Tomography
9
2.1
Introduction
9
2.2
Transform
2.2.1
2.3
2.4
3
...........................
Methods
.......................
10
Projection Slice Theorem ...............
10
2.2.2 Direct-TransformRay Tomography .......... 2.2.3 BackprojectionRay Tomography ........... SeriesExpansion Methods ................... 2.3.1 The Forward Modeling Problem ...........
16 20 22 23
2.3.2
Kaczmarz'
26
2.3.3
ART
...................
.....................
33
............................
39
2.5 Suggestionsfor Further Reading ...............
42
Seismic Diffraction
45
3.1
Summary
Method
and SIRT
Introduction
Tomography
...........................
45
3.2
Acoustic Wave Scattering ................... 3.2.1 The Lippmann-SchwingerEquation ......... 3.2.2 The Born Approximation ............... 3.2.3 The Rytov Approximation ............... 3.2.4 Born vs. Rytov Approximation ............ 3.3 Generalized Projection Slice Theorem ............ 3.3.1 CrosswellConfiguration ................ iii
46 47 51 52 56 58 59
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4
3.3.2 Vertical SeismicProfile Configuration ........ 3.3.3 Surface Reflection Configuration ........... 3.4 Acoustic Diffraction Tomography ............... 3.4.1 Direct-Transform Diffraction Tomography ...... 3.4.2 BackpropagationDiffraction Tomography ...... 3.5 Summary ............................ 3.6 Suggestionsfor Further Reading ...............
72 78 82 84 87 90 •3
Case
95
Studies
4.1
Introduction
4.2
Steam-Flood EOR Operation ................. 4.2.1 CrosswellSeismicData Acquisition .......... 4.2.2
4.3
4.4
4.5
...........................
Traveltime
95
Parameter
Measurements
.........
B
C
99
4.2.3 Image Reconstruction ................. 4.2.4 Tomogram Interpretation ............... Imaging a Fault System .................... 4.3.1 CrosswellSeismicData Acquisition ..........
108 111 125 125
4.3.2
.........
130
4.3.3 Image Reconstruction ................. 4.3.4 Tomogram Interpretation ............... Imaging Salt Sills ........................ 4.4.1 Assumptions and Preprocessing............ 4.4.2 Data Acquisition .................... 4.4.3 Diffraction Tomography Processing.......... 4.4.4 Tomogram Interpretation ............... Suggestionsfor Further Reading ...............
133 135 137 137 139 141 146 150
Traveltime
Parameter Measurements
A Frequency and Wavenumber A.1 Frequency ............................ A.2
95 96
Wavenumber
153 153
..........................
154
The Fourier Transform B.1 Fourier Series ..........................
157 158
B.2 Exponential Fourier Series ...................
159
B.3 FourierTransform- Continuousf(x) ............. B.4 FourierTransform-Sampledf(x) ...............
160 161
B.5
162
Uses of Fourier Transforms
Green's
..................
Function
167
C.1 Filter Theory .......................... C.2 PDE's as Linear Operators .................. C.3 Green's Function Example ................... iv
168 170 173
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C.4 Suggestionsfor Further Reading ...............
INDEX
174
175
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This page has been intentionally left blank
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Preface
Motivated by the successfulimplementation of medical tomography in the early 1980s, geophysicistsand production engineershave attempted analogousmethodsusingseismicenergyfor hydrocarbonexploration,reservoir characterization, and production engineering.The theoretical methods and field techniquesemployed are broadly classifiedas "seismictomography" and were fundamentally developedin the late 1980s. Today, seismic tomography is conductedon a commercial basis with its theory anchored on a solid base, and its strengths and limitations known. Field applications have demonstrated that seismictomography can provide valuable services in upstream operations,suchas mapping subsurfacestructures,delineating reservoirs,and monitoring enhancedoil recoveryprocesses. Our book developsthe fundamentalsof seismictomography at the level of a tutorial or practical guide. Considerableeffort has gone into making the book self-containedso that any reader who has had calculuscan easily follow the material. Referencesfor further reading on specifictopics are given at the end of each chapter. We give a short statementfollowingeach referencedetailingits significance as a supplementto this book. In doingso we hope the reader will not feel the referencesmust be read to fully understand a given concept. We use appendicesto review physicalterminology and mathematics required to understand the theoretical presentations. We present various tomographicmethods in a logical and straightforward manner. Unlike many other books on tomography, we use standard notation for variableswhich span the variousmethods,enablingthe reader to easily contrast differences. Also, mathematical steps glossed-overby most research
articles
are filled-in
for our readers.
Sometimes
we deviate
from well-known derivations to provide a deeper physical understanding. However,for completeness,our derivationsare followed-upwith references to the "well-known" derivations at the end of the chapter. In addition, we discussthe limitations of seismictomographyand illustrate successes and pitfalls with casestudies. Our ultimate intent is that after reading this presentation, the reader will exhibit both a greater understandingand appreciation for seismictomography articles presentedin the literature. Chapter I is introductory and summarizes the developmentof seismic tomography and describeshow this new technologycan benefit the oil industry at both the exploration and producing stages. Chapters 2 and 3 are tutorials on the theoretical fundamentals of seismic ray tomography and seismicdiffraction tomography,respectively. Chapter 4 presentsthe vii
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data acquisition,processing,and tomogram interpretation for three seismic tomographycasestudies. Each casestudy has its own unique data acquisition, data processing,and interpretation challenges.They provide useful insight into designingand conductingfuture tomographystudies. The authors thank Texaco for releasing data and results for the case studies published in Chapter 4. We also acknowledgethe releaseof the McKittrick data by Texaco'sjoint partner, Chevron, for that project. Both Eike Rietsch and Bob Tatham of Texaco have encouragedthe authors to pursue this project and have provided support during its progress. We also acknowledgeour fellow boreholeseismologyteam membersat Texaco: Danny Melton, Don Howlett, Ron Jackson, and Stan Zimmer for their contributionsto this field throughout the past few years. In addition, David Fitterman, the SEG monographsserieseditor, and Larry Lines, the volume editor for this book, have been patient with our progressand have provided valuable guidance. Finally, the authors thank Texaco for permission to publish this book.
Philip L. Inderwiesen Tien-when
Lo
E•tP TechnologyDepartment Texaco
Inc.
Houston, Texas
viii
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Chapter
I
Introduction
1.1
The Concept of Seismic Tomography
We definetomography as an imagingtechniquewhichgeneratesa crosssectionalpicture (a tomogram)of an object by utilizing the object'sresponseto the nondestructive,probingenergyof an external source.Seismic tomographymakesuseof sourcesthat generateseismicwaveswhich probe a geologicaltarget of interest.
Figurel(a) is an exampleconfiguration for crosswell seismic tomography. A seismicsourceis placedin one well and a seismicreceiversystem
in a nearbywell. Seismicwavesgeneratedat a sourceposition(soliddot) probe a target containing a heavy oil reservoirsituated between the two wells. The reservoir's responseto the seismicenergyis recordedby detec-
tors (open circles)deployedat differentdepthsin the receiverwell. The reservoiris probedin many directionsby recordingseismicenergywith the samereceiverconfigurationfor different sourcelocations.Thus, we obtain a networkof seismicraypathswhich travel throughthe reservoir. The measured responseof the reservoir to the seismic wave is called
the projectiondata. Tomographyimagereconstruction methodsoperateon
the projectiondata to createa tomogram suchasthe onein Figurel(b). In this casewe usedprojectiondata consistingof direct-arrivaltraveltimes and seismicray tomographyto obtain a P-wavevelocitytomogram.Generally,differentcolorsor shadesof gray in a tomogramrepresentlithology
with differentproperties.The high P-wavevelocities(dark gray/black)in the tomogramin Figurel(b) are associated with reservoirrockof high oil saturation.
Seismictomographyhas a solid theoreticalfoundation. Many seismic
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2
CHAPTER 1. INTRODUCTION
Crosswell Seismic Configuration
o
source
o
receiver
/'-waveVelocity Tomogram
heavy oil (a)
(b)
F•G. 1. (a) Geometry forcrosswell seismic tomography example. P-wave energy traveling alongraypaths probethe geological target. (b) P-wave
velocitytomogramreconstructed from observedtraveltimedata. Different shadesof gray correspond to differentP-wavevelocities.
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1.2.
APPLICATIONS
3
tomography techniqueshave closeties to more familiar seismicimaging methods such as traveltime inversion, Kirchhoff migration, and Born inversion. For example,seismicray tomographyused to determinelithologic velocity is essentially a form of traveltime inversion and seismic diffraction tomographyis closelyrelated to Born inversionand seismicmigration. Thus, seismictomography may actually be more familiar to you at this point than you might think sinceit is just another aspectof the subsurface imaging techniquesgeophysicistshave been using for years. 1.2
Applications
Seismictomography is applicable to a wide range of problems in the oil industry, ranging from exploration to developmentto production. The casestudies presentedin Chapter 4 demonstrate that seismictomography can complement conventional seismicmethods and provide unique, previously unavailable subsurface information. Tomography applied to surface seismicdata can generatesubsurfacevelocity modelsfor explorationproblems. These velocity models can in turn be used as soft information in the geostatisticalinterpolation of well-log data between wells. Seismictomography applied to developmentand production problems is generally implemented by a crosswellconfiguration, as shown in Fig-
ure l(a). Figure 2 illustratesthe benefitof crosswellseismictomography for reservoir
characterization
over conventional
reservoir
characterization
tools. Figure 2(a) representsthe true geologybetweentwo wellsin a producing field where the producing formation is a tar sand layer overlaid by
a thinner, lesspermeablebed (shadedinterval). The heavy oil in suchtar sandsis somewhatimmobile unlessheated using the enhancedoil recovery techniqueof steam flooding. A production engineerplanning to steam flood a tar sand interval needsto know whether the lesspermeablebed is capable of confiningthe steam to the tar sand. In our cartoon the lesspermeable bed is breached by a small fault. Well logging is a conventionalreservoir characterization tool that provides information about the reservoironly a small distancefrom the bore-
holeasdepictedin Figure2(b). Thus,no hardgeological informationabout the unprobed reservoir between the wells can be extracted from conven-
tional well-log data. The well-log data will only show that the low permeability layer exists between 500 and 600 feet in well A and between 400 and 500 feet in well B. Based upon the relative formation dips in each well, the engineermay decidethe low permeability layer is continuousand interpret
the well-logdata usinglinear interpolationas shownin Figure2(c). The small fault is thereforenot detectedand unexpectedsteam flood results will
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CHAPTER 1. INTRODUCTION
(a) true geology
(b) well logging
(c) well logging
interpre[ation
200 ft 4OO ft 600 ft
steam-' 8OO ff
A
B
A
B
(d) tomography
A
B
(e) _tomography Interpretati__on
I logging tool I
source
D receiver
A
B
A
B
FIG.2. (a)Truegeology wewishtoknow.(b)Welllogs sample onlyashort distance intothereservoir, requiring sometypeof interpolation between wells asdepicted in(c).(d)Crosswell seismic records theearth's response toseismic energy between wells thereby permitting animage reconstruction
of thegeology asshownin (e).
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1.3.
RAY
V$.
DIFFRACTION
TOMOGRAPHY
5
occur.
On the other hand, crosswellseismictomography can directly probe the reservoirbetweenthe two wellsas shownin Figure 2(d). A downhole seismicsourcein well A generatesseismicwavespowerful enoughto travel through the reservoirand to be recordedby sensitivedetectorsin well B. After applyingtomographyprocessingto the projectiondata, a tomographic
interpretationliketheonein Figure2(e) mightbe obtained.Thus,the engineer will be aware of the small fault and can make the necessaryalterations to the steam flood operation. Although just a cartoon, Figure 2 illustrates how crosswellseismictomography is a more reliable tool for delineating the reservoir between wells than any interpolation method between well logs. However, crosswellseismictomography becomesan even more significant tool for reservoir characterizationwhen used in conjunctionwith well-log information and core data as is demonstrated in Chapter 4.
1.3
Ray rs. Diffraction
Tomography
To do seismictomographywe must model the seismicwavetraveling through the subsurface. Both ray and diffraction theoretical models are available to us for describingseismicwave phenomena. Which model we use dependsupon the relative sizesof the seismic wavelength and the target we wish to image. A judicial choiceof theoretical model for a given seismicwave and target becomesimportant to the successof the seismic tomography application. If the target's size is much larger than the seismicwavelength,then we may model the propagationof seismicwavesas rays usingray theory. This is similar to using geometrical optics to describe light wave propagation throughlenses.Seismictomographybasedupon the ray theoreticalmodel is discussedin Chapter 2 under the title Seismic Ray Tomography. We subdivide the topic into "transform methods" and "seriesexpansion methods." The transform methods are commonlyused in medical tomography experimentswhile the seriesexpansionmethodsseemuchusein seismictomographyapplications.Currently seismicray tomographyis very popular becauseit is simple to implement under a variety of situations, is computationally fast, and givesgood results. When the size of the target is comparableto the seismicwavelength, then we model the propagationof seismicwavesas scatteredenergy using diffraction theory. Such a target scatters the seismic wave in many directions and only diffraction theory can properly model this response. Seismic tomography based upon the diffraction theoretical model is discussedin Chapter 3 under the title Seismic Diffraction Tomography.As
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6
CHAPTER
1.
INTRODUCTION
you will see,seismicdiffraction tomographypresentedin its simplestform requiresrestrictionson the source-receiver geometry,ignoresmultiplescattering of energy, places limits on the sizes and velocity contrastsof targets, and is computationally intensive. Becauseof these restrictions the method is currently applied only to a few select situations. However, re-
centdevelopments, whichwe list under "suggestions for further reading"in Chapter 3, are overcomingsomeof these restrictions. Our presentationof seismicdiffractiontornographyin its simplestform shouldgive you a solid foundationfor understandingand appreciatingthesedevelopments. Severalcasestudiesare presentedin Chapter 4 to illustrate the application of theory to varioussituations. We emphasizethe need to assimilate as much data as possiblefrom other sources,such as from well logs and core samplesin the crosswelltomographyexamples. Only by integrating all information availablewith the tomogram can one make an optimum assessment
about
the reservoir.
' As a final note, we have crisply divided the application of seismictomographyinto ray and diffractiontomography,dependingupon the relative sizesof the seismicwavelengthand target. However,in reality the probing seismicwave is usually a broad-bandsignal consistingof a large range of wavelengths,and the subsurfacecontainspotential targetswith relative sizesranging from small to large. Thus, this suggestsa blend of seismicray tornographyand seismicdiffractiontornographybe used to optirnallyimage all possibletargets. Although interesting,we pursuethis possibilityno further as it is more of a researchmatter at this point in time. In this book we will concentrateon presentingthe fundamentalsof seismictomography.
1.4
Suggestions for Further Reading Aki, K., and Richards, P., 1980, Quantitative seismology:Theory and methods, Vol. II: W. H. Freeman & Co. Section 13.3.5 o.f Chapter 13 presents a classificationscheme based upon seismic wavelengthand target size which will give you a goodidea when to use ray theory or diffraction theory.
Anderson,D. L., and Dziewonski,A.M., 1984, Seismictomography: ScientificAmerican, October, 60-68. Popular arlicle on seismic ray tomographyapplied to imaging the earth's mantle.
Lines, L., 1991, Applications of tomography to borehole and reflectionseismology:Geophysics:The Leading Edge, 10, 11-17. Overview of seismic ray tomographyapplications.
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1.4.
SUGGESTIONS
FOR
FURTHER
READING
Menke, W., 1984, Geophysical data analysis: Discrete inverse theory: AcademicPress,Inc. Our bookaddressesonly those topics in inverse theory requiredto understand the basicsin seismic tomography. Menke's bookprovides a goodintroduction to inverse theory.
Tarantola, A., 1987, Inverseproblem theory: Methods for data fitting and model parameter estimation: Elsevier. A comprehensivebookon inversetheorywhichincludesmanyprob-
7
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This page has been intentionally left blank
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Chapter
2
Seismic Ray Tomography 2.1
Introduction
We begin the study of seismictomographywith image reconstruction methods based on ray theory. We assumethat the sourceproducesseismic wave energy with wavelengthsmuch smaller than the size of the inhomogeneitiesencounteredin the medium. Only when this assumptionis obeyed can the propagation of the seismicwave energy be properly modeled by rays. Otherwise, the seismicdiffraction tomography in Chapter 3 must be applied to solve the problem. Two groups of image reconstruction methods exist for doing seismic ray tomography. The transform methodsin Section 2.2 comprisethe first group. Applicationsof transform methodshave their roots in astronomical and medical imaging problems. They are very limiting as far as seismic imaging problemsare concernedsince straight raypath propagation and full-scan aperture are generally assumed.However, the transform methods make an excellentintroduction to the principlesof tomographybecauseof their simplicity and serve as a bridge between applications of tomography in other fields with applicationsin seismology.Also, the developmentof seismicdiffraction tomography has a closerelationship with the transform methods. The series expansionmethodsin Section 2.3 comprisethe second groupof imagereconstructionmethods. Out of all the methodspresentedin this book the seriesexpansionmethodspresently seethe most usein seismic tomography. Therefore, a large part of Chapter 2 is spent addressingthe seriesexpansionmethods. Beforeproceedingfurther one shouldhave a good graspof the Fourier transform conceptsto understandthe material in Section 2.2. Appendix B
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10
CHAPTER
2. SEISMIC
RAY TOMOGRAPttY
.---i'""• Ox-ray transmitte -
r
FIG. 3. Setup for a medical tomography experiment. X-ray scansare taken
in differentdirectionsabout a person'shead by rotating the transmitterdetector assembly.
presentsa review of the Fourier transform.
2.2
Transform
Methods
The projection slice theorem is presentedfirst in this sectionsinceit providesthe theoreticalfoundationfor the transformmethods. Then, two transform methods are derived from the projection slice theorem' directtransformray tomographyand backprojectionray tomography.
2.2.1
Projection Slice Theorem
The derivationof the projectionslicetheoremis illustratedby a typical medicaltomographyexperiment. Figure 3 showsthe setup for medicaltomography.A donut-shapedx-ray transmitter-detectorassemblysurrounds the target, a person'shead in this example. X-ray intensityis measuredfor a fixed orientationof the assembly.Then the assemblyis rotated about the personso that x-rays pass through the head in a different direction. The experimentis completedwhen the person'shead is scannedin all directions. The objective of the transform methods is to use attenuation information
from the measuredx-ray intensitiesto reconstructa cross-sectional image
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2.2.
'fRANSFORM
METHODS
11
of the person'shead in the z - z plane which containsthe transmitters and detectors. Thus, a brain tumor which attenuates x-rays differently than normal tissuesmay be readily "seen"by the radiologist.
Figure 4 depictsa cross-section of a person'shead. The varying contrasts within the target representnonuniformx-ray attenuation associated with a tumor, normal tissues,and the skull. For the purposeof deriving
the projectionslicetheorem,we definethe modelfunctionM(z, z) • as the spatial distribution of the attenuation. In general, the model function representsthe unknown distribution in spaceof some physical property of the
target medium which affectsthe propagatingenergy in some observable manner. A typical model function usedin seismictomographyis the reciprocal compressional-wavevelocity, or slowness,which has a direct influence on the observedtraveltime of the propagatingenergy. The projectionslicetheoremrequiresthat observationsof the propagating energy be taken along a given projection which is perpendicular to the raypaths. Figure 4 illustrates a singleprojection in the medical tomography experiment. X-rays emitted by the transmitters travel along the parallel rays and are recordedby detectorspositionedalongthe u-axis. The rotated
spatialcoordinatesystem(u,v) is introducedto describeall of the possible orientationsfor the transmitter-detectorassemblyabout the target. The v-axis is defined parallel to the direction of x-ray propagationand the uaxis, defined perpendicular to the v-axis, is the direction along which the x-ray intensity is measured. If the u- v coordinatesystemsharesthe same origin as the z- z coordinate system, then the relationship between the two coordinatesystemswhen one is rotated through an angle 0 relative to the other
is
z
sin 0
cos 0
u]
ß
(1)
For a givenray in Figure 4 we defineP(u, O) as the decimalpercent drop in x-ray intensity,
P(u, o) =
[.'o-
0)l/o,
where I(u,O) is the intensity measuredby the detector at (u,O) and Io is the x-ray intensity at the transmitter. We refer to P(u, O) as the data 1Other literature on tomographymight refer to the model function as an image function or as an object function. We chose "model function" to be consistent with inverse problem terminology.
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12
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
x X-RAY
SOURCES
v
P(., O)
Z
U
FIG. 4. Cross section of a person's head where varying contrasts repre-
sentnonuniformx-ray attenuation.The projectionP(u, t•) is the decimalpercent drop in x-ray intensity measuredalong the rotated coordinateaxis, u. The u-axis is perpendicularto the v-axiswhich alwaysparallelsthe x-ray
propagationdirection. The model function M(x, z) providesa numerical value
for the attenuation
and is an unknown
from the observedprojectionsP(u, •).
which
must
be determined
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2.2.
TRANSFORM
METHODS
13
function2. If the attenuationis small overthe raypath, then data function P(u, O) is linearlyrelatedto the attenuationM(z, y) as the line integral,
e(,,, o)-
(2)
ay
taken overthe raypath.s Note that eachobservationof the data functiona P(u, O)providesan empiricalsolutionto equation(2) alongthe givenraypath without actually knowingthe model function M(x, z). If there is negligiblex-ray attenuation outside the target, then equation (2) givesthe same measuredprojectionfor any transmitter-detector separationas long as each transmitter and detector remainsoutsideof the
target. We usethis assumptionto rewriteequation(9•)with infinitelimits; a mathematical step which will be taken advantageof later in this section. Thus,
0)- f::
z)a.
(a)
To obtain a simpler and more meaningful relationship between the
modelfunctionM(x, z) and the data functionP(u, 0), we transformeach into the Fourier
domain.
The 2-D Fourier
transform
of the model function
M(z, z) is
117I(k•, k,.)- f;: ];: M(x, z)e-J(k•x +k,.z)dxdz ' (4) where k• and kz are spatial frequenciesalong the x- and z-axes, respectively. Spatial frequency,or, wavenumber,is definedas k = 2•r/A where is wavelength. Figure 5 representsthe 2-D Fourier transform's amplitude
spectrum • ofa hypothetical model function M(x, z). Notethatif •(k•, 2P(u, O)is a projection in the ray tomo•aphyproblem,but is c•ed a data]unction in inv•e
theory te•nology.
The,
we c•
the v•iable P a "data f•ction"
sine re.on we c•ed M(x,y) the "modelf•ctioff' data f•ction
with the v•iable
P to re.rid
for the
e•Ser. However,we representthe
you that the me•ed
data •e projections.
awe o•y considerthe 5ne• inv•se problem in t•s book. Thus, the data f•ction will •waya be •ne•ly related to the model f•ction, even if • approximation is req•red to force the •ne• relatio•p. The •amption of am• x-ray attenuation is req•red for the x-ray tomo•aphy problem.
•Although the actuMobservation is the x-ray intensityl(u,O) in t•a c•e, we wi• frequently refer to "the observation of the data f•ction" from
the observed
since it is in•rectly
obt•ned
data.
•Althoughthe2-D Fo•ier tryfore of themodelf•ction •(x,z) h• bothmp•tude •d ph•e spectra,we representthe 2-D Fo•ier tryafore M(kx,k•) with o•y the
mpftude spect•
component, desi•ated • ]•(k,,
k,) I.
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14
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
K FIG. 5. The essenceof the projection slice theorem is represented. The
2-D Fouriertransformof a hypotheticalmodelfunctionM(x, z) produces the amplitudespectrumI M(ks, k,) [. The contoursin the ks - k, plane connectequal valuesof amplitude. The amplitudespectrum[ P(fi, d) [ from the 1-D Fouriertransformof the data functionP(u, t•) representsa
sliceof [ J17/(ks, k,) [ alongthefl-axisin theks- k, plane.
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2.2.
TRANSFORM
METHODS
15
is known,then the unknownmodelfunctionM(z, z) can be foundby the 2-D inverse Fourier transform
M(x,z) = 1fj: fj: 11•'I(k• k,)eJ(k•x +k,Z)dk•dk, (5) 4•r2
,
ß
Nowlet•5(f•,0) represent the1-DFourier transform ofthedatafunction P(u, O) alongthe u-axis,shownin Figure4. The 1-D Fouriertransformis written
0)- f/: where f/ is spatial frequencyalongthe u-axis. Substitutingequation(3) into equation(6) gives
We nowwishto put equation(7) entirelyin termsof z and z. The variable • is replacedwith z and z usingthe inverseof equation(1) givenby v
- sin 0
cos 0
z
'
Usingequation(8) andreplacing dvduwith dzdzin equation(7) weget,
•5(f•, O)-- f:: f:: M(x, z)e-Jf•( xcos 0+zsin O)dxdz = f:: f:: M(x,z)e-j[(f•cosO)x +(ftsinO)z]dx ' (9) Comparingthe integrands of equation(9) andequation(4) weseethat equation(9) is simplythe 2-D Fouriertransform of M(x, z) wherekx and k, are restricted to the Q-axis by setting k•
=
•cos0, and
k,
=
f/sin0.
(10)
This relationship is evident in Figure 5.
Substitutingequation(10) into equation(9) we write
•5(fi, O)- /:: /:: M(x,z)e-J(k:•x +k,Z)dxdz, (11)
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16
C•APTER
2.
SEISMIC
RAY
TOMOGRAPHY
whereks and kz are definedby equation(10). Comparingthe integrands of equations(11) and (4) showsthat we haveachieveda simplerelationship betweenthe data functionP(u, O) and the modelfunctionM(x, z) in the spatial frequencydomain,
P(•, O) -
A7/(k,,, kz).
(12)
In words,equation(12) statesthat the 1-D Fouriertransformof the pro-
jectionrepresented bythedatafunction P(f],_0)isequalto onesliceofthe 2-D Fouriertransformof the modelfunctionM(kz, k•) definedon the loci: kz = f] cos0 and k, = f]sin 0. Equation(12) is calledthe projectionslice theorem.
The projection slice theorem givesonly one slice of the model function per projection as shown in Figure 5. We will now show how many projections at different angles of 0 are used to reconstruct the entire model function via the projection slice theorem. The two techniquespresented are the direct-transform ray tomography method and the backprojection
ray tomographymethod. In Chapter 3 we will define an analogoustheorem for the reconstructionmethods in diffraction tomography called the generalizedprojectionslice theorem.
2.2.2
Direct-Transform Ray Tomography
Direct-transform ray tomography utilizes the projection slice theorem in a straightforward manner. We showedin the previous section that the application of the projection slice theorem to the 1-D Fourier transform
of a singleprojection represented bythedatafunction /5(12, 0) determines onlyonesliceofthemodel function A•(k•- 12cos0,k• - f] sin0). Figure 5 illustrates such a slice through the model function. To recoverthe entire model function, the target must be probed from many different directions. Figure 6 showsthree directions along which x-rays probe the head of our make-believepatient. The observeddata functionsfor thesethree pro-
jectionsare P(u, 0•), P(u, 02), and P(u, 0a). After applyingthe projection slice theorem to the 1-D Fourier transformsof these data functions,we obtain the three slicesthrough the model function's amplitude spectrum shownin Figure 7. Now the 2-D Fourier transform of the model function M(k•, k,) is better definedthan by the singlesliceshownin Figure5, but is still inadequatefor imagereconstruction.We must probe the target with x-rays from all directionsletting 0 range from 0 degreesto 180 degrees. Only then will the k•- kz plane in Figure 7 be completelycoveredby slices
M(f] cos0,f]sin0), where0 rangesfrom 0 degreesto 180 degrees.After such an experiment, the complete 2-D Fourier transform of the unknown
modelfunctionM(x, z) is determinedalongradial linesin the k•-k, plane.
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2.2.
TRANSFORM
METHODS
17
•X
Z
02 FXG. 6. The cranium of our make-believepatient is probed with x-rays in three different directions: 0•, 09.,and 03. Application of the projection slice
theorem to the data functions resulting from thesc projectionsyields the slicesdepicted in Figure 7.
ToobtainM(z, z) from.g/(f•cosO,f• sin0), analgorithm employing the direct-transformray tomography method must first interpolate the data
from a polar grid (f•cosO,f•sinO) onto a Cartesiangrid (k•,,kz) in the k• - kz plane, or
AT/(f•cosO, f•sinO)in,•r__•ot.,• A•(k•,kz).
(13)
One must exercisecaution in performingthe interpolation sincelarge errors introduced by the operation could obscurethe true solution.
Lastly, a 2-D inverseFouriertransformof M(k•,,k;•) is performedto obtain M(x, z),
M(z, z) = 4•rI 2 '•,•
,•
.•l(k• k,)eJ(k•z 4-k•Z)dk•,dk ' ß (14) '
Thus, the image reconstructionis completedand the technicianmay give the tomogram of the patient's head to the radiologistfor interpretation. The direct-transformray tomographymethod is easilysummarizedin five steps:
Step 1: Acquirethe data functionP(u, O) of the target with the projectiondirection,0, rangingfrom 0 degreesto 180 de-
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18
CHAPTER
2. SEISMIC RAY TOMOGRAPHY
M(•cosO 3, • sin63 )1
• sin61 )
K
Kz F•(•. 7. Plot showing slicesthroughthe unknownmodelfunction'samplitudespectrumI M(k=, k•) I foundby applyingthe projectionslicetheorem to the data functionsfoundfor the x-ray projectiondirections 01, 0•, and 03 shownin Figure 6.
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2.2.
TRANSFORM
METHODS
19
grees.Rememberthat the unknownmodelfunctionM(z, z) correspondingto the target representsa physicalproperty which affectspropagatingenergyin somemanner(e.g., attenuation which affectspropagatingx-ray intensity in medical tomographyor slownesswhich affectsseismicwavetrav-
eltimesin seismictomography). Thus, a data functionis just the line integral of the model function along each ray, or
Step 2: Perform a 1-D Fourier transform along the u-axis for each data function given by
P(n, o)- f:: o)-J Step 3: Use the projection slice theorem to obtain slicesof the 2-D
Fourier
transform
of the model
function.
Each slice is
defined by
cos0,iqsin0) -- J5(i2,0). Step 4: Convert the 2-D Fourier transform of the model func-
tion in the k• -kz planefrom a polar grid (f• cos0, f•sin 0) to a Cartesiangrid (k•, kz),
Step 5: Perform a 2-D inverseFouriertransformon M(k•,, k•) to obtain M(x,z), the reconstructedimage of the target. The inversetransform is given by
47i-2
The direct-transformray tomographymethod would be quick to implement if it were not for the fourth step above requiring interpolation of the model function in the frequency domain. In the next section we present backprojectionray tomographywhich obviates the need for interpolation resultingin a faster and more accuratealgorithm.
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CHAPTER
2.2.3
Backprojection
2.
SEISMIC
RAY
TOMOGRAPtIY
Ray Tomography
Backprojection • ray tomography usesthe samedata acquisition scheme asthe direct-transform ray tomography:recordthe data functionP(u, O)by experimentallymeasuringthe line integral of the unknown mode] function
M(:r,z) along differentraypathsor projections. The differencebetween backprojectionray tomographyand direct-transformray tomography is how we compute the model function from the data function. To derive the backprojectionray tomography method, we first write
downthe2-D inverse Fouriertransform of themodelfunction•(k•, k,),
•(• ' z) = 4•2•
•
•(• ' •)•j(•+•z)••
(•5)
ß
Next we makea changeof variablesin equation(15) by replacingk• with • cos0 and k• with •sin 0, and by changingthe integration from dk•dk,
to ] • l d•dO.• This gives,
M(x,z) = 4•2 M(•cos0,•sin0) • ej•(xcosO + zsin 0)I•ld•dO.
(1•)
Integrationwith respectto 0 in equation(16) can be rewritten • two integrals,
1 • M(r,z) = 4=•
• (• cos 0,• sin 0)
• ej•(xcos 0+ zsin0) I•ld•dO +•
• '
•[• •o•(0 +•), • •in(0 +•)]
• •j•[• •o•(0+ •) + z•in(0 + •)] I• [•0.
(17)
Using the fundamentaltrigonometricangl•sum relations,cos(0+ •) = - cos0 andsin(0+ •) = - sin0, we may rewritethe secondmodelfunction •The te•
"b•mjection"
imp•es the inve•e problemwherewe st•t with the pr•
jection •d •lve for the model f•ction. Here the projections •e t•en Mong raypat•. • Section 3.4.2 the me•ed projectio• of scattered energy •e described by the wave equation •d we •e • •Mogo• te•, "backpropagation."
•The ch•ge • inte•ation is •Mogo• coor•n•es is the r•M
to compute the •ea of a •sk. •st•ce
from
c•e to prese•e the si• negative vMu•.
the •sk's
to goingfrom C•tesi•
coorSnatesto pol•
For a •sk we replace dxdz by rdOdr, where r
center.
The
absolute
vMue
of •
is t•en
of the •fferentiM •ea when we will shortly •ow
in o•
• to t•e
on
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2.2.
TRANSFORM
21
METHODS
on the right-handsidein equation(17) as,
.g/[f•cos(0 + •r),f•sin(0 + •r)] = .tfl(-f•cosO,-f• sinO).
(18)
The secondsetof integralson the right-handsidein equation(17) is rewritten by replacingthe modelfunctionwith equation(18), applyingthe anglesum relationsusedin obtainingequation(18) to the exponential,setting f• - -f• and dfl - -dfl, and reversingthe direction of integration with respectto fl. With theseoperationsequation(17) is written,
M(x z) = 4•21 • ,
•(fl cos 0,flsin 0)
x ej•(xcos 0+ zsin0)
+•
•(•
cos•, • sin•)
• •j•(• •o•• • • •i. •) I • I ••. Combiningthe integralswith respectto the variable• we get, '
4•
'
• d•(• •o•• • • •i. •) I • I ••.
(•)
Usingthe projection slicetheorem,wereplaceM(• cos•,•sin •) in equastruction
formula
ß
,
4•2
'
We cansummarizethe backprojection ray tomographyreconstruction method in just three steps:
Step 1: Data acquisition.Let the modelhnction M(x, z) representthe unknownparameter(such• seismicwaveslow-
ness)at position(x, z). Experimentally determinethe line integralof the modelfunctionalongeachray whichyieldsa setof datahnctions(such• seismic wavetraveltime),
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22
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPttY
Step 2: Take the 1-D Fourier transform of each data function
along the u axis.
P(fl, O)- /:: P(u, O)e-Jfl udu. Step 3: Use the backprojectionformulaequation(20) to compute the unknownmodelfunctionM(x, z), or
=
4•.2
x
'
+
[ I
Unlike direct-transformray tomography,backprojectionray tomography does not require a 2-D interpolation in the wavenumberdomain, and therefore,is in generalfaster and more accuratethan direct-transformray tomography. It shouldbe mentionedthat mostcommercialCAT (ComputerizedAxial Tomography)scannersuse the backprojectionray tomographyor its modificationas their image reconstructionalgorithm. 2.3
Series Expansion Methods Seriesexpansionmethods comprisea group of computation algorithms
which,like the transformmethods,determinethe modelfunctionM(x, z) of the target area. However,unlike the transformmethods,thesealgorithms easily allow curved raypath trajectories through the target area and are therefore well suited for applicationsin seismictomography. As before, we
restrictthe discussion to a 2-D problemsothat the modelfunctionM(z, z) is determined in a plane which cuts through the target and containsall of the sources and receivers.
Our discussionof the seriesexpansionmethods is divided up into three subsections. Section 2.3.1 presentsthe forward modeling problem which permits us to predict the tomographydata in terms of a system of linear equationswhich explicitly contain an estimate of the true model function. Section2.3.2 showshow the true model function is determinedusing Kaczmarz' method. The method devised by Kaczmarz in 1937 is iterative and determinesan approximate solution to the true model function. Drawing an analogy,the Kaczmarz method is to the seriesexpansionmethod as the
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2.3.
SERIES
EXPANSION
METHODS
23
projection slice theorem is to the transform methods. We exploit Kaczmarz' method in Section 2.3.3 to derive two seriesexpansionalgorithms:
the algebraicreconstruction technique(ART) and the simultaneous iterative reconstruction technique(SIRT).
2.3.1
The Forward Modeling Problem
As will be shown shortly, the seriesexpansionmethod iteratively up-
datesan estimatedmodelfunctionM e't so that it converges towarda true model function M true. The updates are found by comparingthe observed data functionpolo with a predicteddata functionppre. Forwardmodeling is required to determine the predicted data function and is the subject of this section.
Equation (2) in Section2.2 definesthe experimentalprocessfor the transformmethodsas the line integral of the function M(a:,z) along a straight raypath in the v-axis direction. Accordingto the abovenotation equation(2) couldbe written,
Pøbø(u, O)-- i Mt"u•(z' z)dv' ay
We did not requirea forwardmodelingprocedurein Section2.2 becausethe raypathswere straight and the projectionslicetheoremcouldbe employed
directlyto determinethe true modelfunctionMtrue(x,z). For the seriesexpansionmethodswe wish to include curvedraypaths.
Equation(2) is easilytransformedto accommodate curvedraypathsby rewriting the model function in terms of a position vector r. Thus, for a
givensource-receiver pair the line integralof the modelfunctionM(r) over the raypath is
pobo = f• MtrU•(r)dr, ay
wherethe observed projectiongivenby the datafunctionpolo represents the measured lineintegral(observed tomography data) and MtrUe(r) is the true model function which remains to be determined. The last equation is used to formulate the forward modeling by setting
P- i M(r)dr, ay
(21)
where P is now the predicteddata function and M(r) is the estimated
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CHAPTER
24
2.
SEISMIC
RAY
MI
M2
M3
M4
M5
M7
M8
M9
M10 Mll
TOMOGRAPtIY
M6 M12
M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24 FIG. $. The seriesexpansionmethodsusea discretemodelfunctionMj, j - 1,..., J, whereMj is the averagevalueof the continuous modelfunction M(r) within the jth cell. Here J - 24.
modelfunctions. Thus, forwardmodelingis definedas determiningthe predicted data function from the line integral along the raypath through a known, but estimated, model function. Just as was done with the transform methods, the model function in the seriesexpansionray tomographyis discretizedto allow computationby digital computer. Figure 8 showsan image area of a target divided into many small cells. Each cell is assignedthe averagevalue of the physical parame-
ter (e.g., x-ray attenuation,slowness, etc.) represented by the continuous modelfunctionM(r) within that cell. The modelfunctionin Figure 8 is divided into 24 cells and is written discretely as M•, where j - 1,..., 24.
Thus, Mj represents the averagevalueof M(r) within the jth cell. Figure 9 depicts a single ray traveling through the discretizedmodel function. Equation(21) is rewritten in discreteform, to describeray travel through the discretemodel function, as J
Pj=l
whereMj is theestimatedmodelfunctionfor thejth cell,,5' i is the raypath SHere we will symbolize the predicted data function as P and the estimated model function a• M for brevity. Thesesymbolswill be changedto ppre and M est, respectively, in the following section.
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2.3.
SERIES
EXPANSION
•
Z
METHODS
25
r
7
6
..•_1.. ••
source
receiver • •._•..1•.. ••15 M16M17IV116 M19 M20 M21 M22 M23 M24 FIG. 9. Ray travel through a discretemodel function. The resulting data function, determinedby the line integral through the discretemodel function, is definedby equation(22).
length of the ray within the jth cell, and J is the total number of cells in the gridded target. The example in Figure 9 has J - 24 cells, but the ray
penetratesonly sevencells(j - 12, 11, 10, 16, 15, 14, and 13). To keep equation(22) consistentwith equation(21) we set Sj - 0 for all cellsnot penetratedby the ray. After all, the ray'spath lengthSj for the jth cell is obviously zero if the ray did not traverse that cell. Figure 9 shows17 cellsfor which we don't have information becausethe singleraypath did not traversethem. By addingmore sourcesand receivers around the unknowntarget region, differentrays samplethe 17 unsampled cellsin addition to someof the cellsalready sampled. The addition of extra rays is depicted in Figure 10. Now all of the cells are interrogatedby this network of rays.
We must modify the index notationof equation(22) to includea projection value for every ray. If Pi representsthe projection, or line integral,
predictedfor the ith ray, then equation(22) is rewritten, J
Pi = E Mj$ij,for/- 1,...,1,
(23)
whereI is the total numberof rays,$ij is the path lengthof the ith ray throughthe jth cell, and, asbefore,Mj is the discreteestimateof the model functionfor the jth cell and J is the total numberof cells.Equation(23) is
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26
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
source
ray I
receiver ray 2
!
ray I
FIG. 10. Generally, a single ray does not provide information on all of the model function's cells. However, by using more source and receiver locations around the target all of the cells can eventually be sufficiently interrogated. I rays were found sufficienthere.
the formulation of the "forward modeling problem" usedin seriesexpansion ray tomography.
Equation (23) can effectivelymodel the data acquisitionprocessif we let the projectionsPi, i - 1,..., I, be the observeddata (i.e., traveltime or decimalpercentdecreasein x-ray intensity)and the modelfunction j: 1,..., J, be the true, but unknown model function, or J
Piø•' = • M]"•'*Sii, fori- 1,...,I.
(24)
j=l
Kaczmarz' method providesthe theoreticalframeworkfor indirectly solving equation(24) for the true modelfunction. 2.3.2
Kaczmarz'
Method
In this section we introduce Kaczmarz' method to indirectly solve
equation (24)forthetruemodelfunction M]ru*,j = 1,..., J, whichisthe tomogram. As stated in the previoussection,forward modelingis required to determine the true model function. Thus, before proceedingwe will reformulateequation(23) into a matrix form to simplifythe mathematical discussion.Sinceequation(23) is discreteits elementsare easily put into matrices.In matrix form equation(23) becomes,
P
-
SM,
(25)
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2.3.
SERIES
EXPANSION
METHODS
27
where the predicted projections in data vector P are, P1
P
-
.
,
Pl
the discrete estimated model function values in model vector M are, ml
m
-
.
,
Ma
and the raypath lengthsfor I rays and J cells in S are, S• S•.•
s
S•. S•
-
... ...
S• S•
. ß
S•i
S•2
-..
Note that S in equation (25) can be thought of as a linear operator that operates on the estimated model vector M producing the predicted data vector
P.
We couldalsoformulateequation(24) in matrix form as
po•, = SM'•.
(29)
Althoughwe will not directlysolveequation(29), we wouldwant to determinethe true modelvectorM truegivenpob•and S. The problembecomes oneof findinga generalized inverseoperatorS-•.9 Then wecouldapplythe generalizedinverseoperatorS-• to both sidesof equation(29) to determine the true model vector, or S-ap oh, = m
S-aSM '•e M true .
Theoretically the last equation is true, but in practice it is very often difficult to determine S-a for two reasons. First, S is usually quite large and 9We write
S-g
rather
squaxe and because S-9S
tha•n S -1
as in usual matrix
notation
is not always the identity matrix.
since S-g
need not be
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28
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
sparse,which makes computation of S-• costly. Second, S is usually "ill conditioned"whichmakescomputationof S-• very unstable.iø Kaczmarz' method circumvents the problems associated with the inversion of a large and sparsematrix and provides an efficient means for determiningan approximatesolutionto equation (29) using an iterative procedure. Figure 11 presentsa flowchart outlining the method in which there are three basicstepsto the iterative part of the algorithm. An initial
estimateof the modelvectorM init is input to the iterativeloopof the algorithmandservesasthe first "currentestimate"M •st of the true solution M true. For now we assume that the initial model vector M init is known.
However,morewill be saidon the selectionof the initial modelvectorM in the casehistoriespresentedin Chapter 4.
With the current estimate of the model vector M est known, the first step is to usethe forwardmodelingproblemdefinedby equation(25) to determinea predicteddata vectorppre. This stepis carriedout by applying the linearoperatorS (determinedby someray tracingtechniqueof personal choice)definedby equation(28) to the estimatedmodelvectorM •s•,
ppr• = SM•t.
(30)
In the second step the predicted data vector PPr• is compared with
the observed data vectorpob, by takingthe difference betweenthe two. A small differenceor good agreementbetweenthe predicted and observeddata vectorsimpliesgood agreementbetweenthe estimated model vector M • and the true model vector M t•"•. Thus, if the differenceis smaller than a specifiedtolerance, then the current estimate of the model vector M •t is
output as the solutionto equation(29) in the final step of the algorithm. The selection of a suitable tolerance for the difference is discussed with the
case histories in Chapter 4. The third step of the iterative portion of Kaczmarz' method comesinto play when the difference between the predicted and observed data vec-
tors is larger than the specifiedtolerance. This important step essentially
makesuseof the differenceinformation,pob•_ pp,.e,to updatethe current estimated model vector M •
with a new estimate of the model vector
M("ew)• whichhopefullyis closerto the true modelvectorM •r"•. This third step is written in equation form as
M ("•w)•t = M e'•+ A/M, fori
-
(31)
1,...,I,
Ill conditioned meanssmallchanges in S producelargechangesin the modelfunction true or in S-g howeveryou wishto look at it
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2.3.
SERIES
EXPANSION
METHODS
29
Initial estimate
Minit
currenf estimafe est
Step
1
predicted
observed
data vector
data vector
Ppre Step
2
Step
3
pObS
FIG. 11. Flow chart for Kaczmarz' method. M i"i' is the initial estimate
of the model vector; M e'• is the current updated estimate of the model vector; ppre is the predicted data vector from the forward modelinggiven
by equation(30); andpobois the observed data vector.M eø•is iteratively updateduntil ppre matchespoboto within a specifiedtolerance.
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CttAPTER
3O
2.
SEISMIC
RAY
TOMOGRAPttY
whereAiM represents the incrementalupdate to the currentestimateof the modelvectorand superscripti meansapplyingequation(31) whenthe ith row of the pob• and ppre vectorsare compared. The new estimate M (new)e•t is then taken as the current estimate for the next iteration. As
you will soonsee,equation(31) bringsthe current modelestimatetoward the true solution, at least in theory.
The methodof computing AiM in equation(31) is obviously a crucial factor
in the success of Kaczmarz
'method.
Kaczmarz' method computes
AiM with the equation' AiM1 A i M2
-
.
,
(32)
ß
AiMj where
pi.øb s __piPre
(33) Note that the summation in the numerator is just the predicted data vector
ppre found in the forward modelingin equation(30). Now we will geometricallyderiveequation(33) and showthe convergenceof Kaczmarz'methodthrougha simpleexample.Let two rays(I = 2) travel througha two-cellmodel(J = 2) so that the vectorequation(29) for the problem can be written as
p•,b, = Sll M1 + S12M2,for ray 1, and p•b, __ S21 M1+ S22M2,forray2,
(34) (35)
whereP•'b•and P•*• are observed data and M1 and M2 are unknown TM. Rememberthat $i1 is just the ith ray's path lengththroughthe jth cell and is generallyknownfrom the forwardmodeling.We plot equations(34) and (35)in Figure12 as lines(hyperplanes) on a 2-D spacewith axesM1 and M2 .12 The solutionto the equationsoccursat point X where the two • • Here M• and M2 are unknown and therefore defined as independent variables. Only
whenthe solutionis foundare theyreferredto as M[ •ue andM• •ue as in equation(29). •2If I = 3 and J = 3, then we would be looking for the intersection point of three planes in a 3-D model space. For situations where J > 3, equation (29) representsa J-dimensionM space and we would look for the solution at the intersection of I = J
hyperplanes where a hyperplane has J - 1 dimensions. Note that we must have at least I = J hyperplanesto solve equation (29) and generally I > J.
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2.3.
SERIES
M2
EXPANSION
METHODS
31
obs
M2= P1 o\ ,,....__/ S12
•
obs
H
12M2
FIG. 12. For two rays (I = 2) in a two-cellmodel (d = 2), possiblevalues
for M• andM2 aredefined by [hetwolines(hyperplanes). p•,b,and are [he observed data from [he two rays. Point X represents[he desired
modelvalueswhichlie at the intersection of the twohyperplanes.A• M• and A•M2 for ray 1 are geometrically derivedsothat point B is the projection of point A ontothe hyperplanedefinedby equation(34). The resultof this geometricalderivationis equation(33).
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32
CHAPTER
2. SEISMIC
RAY TOMOGRAPItY
lines(hyperplanes) intersect. Sincethe solutionat point X with coordinates(M•"•'e,M•"•'e) is unknown, we must start with an initial estimate at point A given by the coordinates (M1A, M•A). Thisinitialestimatebecomes the currentestimate as shownin the flow chart for Kaczmarz'methodin Figure 11. The geometricalstepin Figure 12 is to find the perpendicularprojectionof point A
ontothehyperplane •:•defined byequation (34) at pointB. Mathematically this step is given by,
s = M+AM,and
(36)
whereA•M• and A•M2 are the corrections soughtgeometrically and defined by equation(33) for i- 1 and j- 1,2. The first geometricalrelationshipto note in Figure 12 is the similarity of trianglesAABC, AFED, and AGEH. Using these similaritieswe can immediately write
AIM• = AB cosc•= DFcos c• =
EF cos2 c•
=
__GH • EF '•E:Z'
=
EF
A • M2 = AB sinc•= DF sinc•
--GH
and
(38)
cos c• sin c•
EH
= EFG---•• .
(39)
Our task now is to determineEF, GH/•--•, and EH/GE in equations (38) and (39).
On the line segmentEF, point E is located at
(M• = P•'b'/sI•,M2 = 0). PointF is the intersection with the M•-axisof a linewhichis parallelto equation(34) andcontains the point(MIA,M•). The equation for this line is
S11M• A + S•2M•A - S••M1 + S1:•
(40)
Fromequation(40) we determinethe coordinateof point F alongthe M•-
axisas (M• - (SI•M1A+ S•2M•A)/S•I,M2 - 0). Thus,the lengthof the line segment EF is given by 1
(41) laAlthoughequations(34) and (35) both representlines,we will continueto refer to them as hyperplanes since that is what they are called when J > 3.
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2.3.
SERIES
EXPANSION
METHODS
33
The ratio GH/GE is simply, GH GE
(42)
ß
Similarly, EH/GE
is givenby,
EH
P•ø/Szz +
=
S12 .
(43)
+
results in the model corrections due to the first ray,
p•b•
I:•p re
i and AiM1-- •'!1•_--' S121 .•.S122 p•bs I•Prß
- -• AiM2- $1•$•---•+$• ,
(44)
(45)
where,rre _ SzzMi• + $z•M• t ß Equations(44) and (45) are the sameas a I
equation (33) wheni = 1 andj = 1,2. Thus,weseethat equation (33) simplydeterminesthe projectionof a modelestimateonto oneof the hyperplanesdefinedby equation(29). Carryingthisexampleonestepfurther,wecandeterminepoint I in Figure 12, the projectionof point B ontohyperplane2 definedby equation(35) for the secondray usingthe indicesi - 2, j - 1, 2 in equation(33). If we alternateprojectionsof the modelestimatesbetweenthe two hyperplanes, then the updatedmodelestimates(step3 in Figure 11) must convergeon point X as depictedin Figure 13. Thus, Kaczmarz'methodwill converge to the solutionof equation(29). 2.3.3
ART
and
SIRT
The algebraicreconstruction technique(ART) and the simultaneous iterativereconstruction technique(SIRT) are the two commonimplementations of Kaczmarz' method in seismicray tomography. This sectiondescribesthe basic features of both algorithms.
ART is a computational algorithmfor solvingequation(29) that directly uses Kaczmarz' method. Thus, the ART algorithm is comprisedof the
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34
CHAPTER
2.
SEISMIC
RAY
TOMOGRAPHY
M2
I
Hyperplane2
:-M I Hyperplane
I
FIG. 13. By applyingequation(31) to alternatinghyperplanes,the model estimateof (M•, M2), starting at point A, convergestowardsthe solution for equations(34) and (35) at point X. The iterative updating of the model estimate correspondsto the loop in Kaczmarz' method shown in the flow chart in Figure 11.
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2.3.
SERIES
EXPANSION
35
METHODS
stepsshownin Figure11. Wefirstsettheestimated modelfunction M• 't,
j-
1,..., or, to theinitialmodelestimate Mj"i' j - 1,
or Thenthe
followingthree stepsare iterated cyclicallyfrom one hyperplaneto the
next until the observed data p/ob,matches the predicteddata pfre, for i=
1,...,I.
Step 1: Conductforwardmodeling(ray tracing)for the ith ray usingequation(23) or equation(25), restatedfor reference here as, J
Only one ray is traced out of a total of I rays sincewe are determining the projection of the current model estimate onto only one hyperplane. Note that if our model consists
of slownesses, thenthe predicteddata P•'reare calculated traveltimes fromtheforwardmodeling andp/oh,areobserved traveltimes.
Step 2: Subtractthe predictedith ray data p•0•, from the observed ith data p/o•,,anduseequation(33) to findcorrectionsfor all of the J cellscomprisingthe model function estimateTM,
=
'
Step 3: Apply the correctionsto the model estimaterecommended by the ith ray to all orcells,
SIRT differsfrom ART in that all I rays are traced throughthe model
sothat all AiMj corrections determined forthe I hyperplanes areknown. 14Note that the model adjustment dependsupon the discrepancybetween the predicted aaadobserveddata values and the raypath length through the cells for the ith ray.
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36
CHAPTER 2. SEISMIC RAY TOMOGRAPtIY
Thenan average of AiMj withrespect to indexi is takenfor eachmodel
cellj - I , ..., J,togetnew model estimates
,j-1,...,J.
As
withART,themodel estimates M[•t areupdated untilthepredicted data Pf•ecompares favorably withtheobserved datap/oh,, i- 1,..., I. After settingthe currentmodelfunctionestimateequalto the initial
model function, orMf øt- M'. i"itforj = I
are iterated to update the model estimates:
J thefollowing threesteps
Step1: Conductforwardmodeling (ray tracing)usingequation (23)or equation(25),
for all raysi
-
Step 2: Findthecorrection foreachcellbyexamining therays cut throughthat celland averaging the corrections recommendedby eachray. Thisoperation is definedfor the jth cell by, I
= W,..= 1•AiM 1 _
forj
-
1
I
BlObS 1 s,s • - EsS=
(46)
i=1
1,...,J.
The weightWj is the numberof raysintersecting the jth cellor someothersuitableraydensityweightusedto obtain an averagecorrectionAMj.
Step 3: Determine thenewmodelestimate fromthe average modelcorrections AMj, or
M«"ew)e" = M;"+AM./,j - 1,...,J. Figure14illustrates howequation (46)makes SIRTdifferent fromART.
Asin Figure 12weuseonlytworays(orI - 2 hyperplanes) andtwomodel cells(or J - 2 modelspace) in orderto visualize theproblem.TheART algorithm isshown inFigure 14(a)andtheSIRTalgorithm inFigure 14(b).
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2.3.
SERIES
EXPANSION
M2
37
M2
E
.
D A
METHODS
•
C B Hyperplane I
(a)
(b)
FIG. 14. Comparisonof ART andSIRT algorithmsfor the two-ray(or I 2 hyperplanes) and two-cellmodel(or J - 2 modelspace)examplegiven in Figure 12' a) Convergence of model estimatesfor the ART algorithm startingwith an initial modelestimateat pointA. b) Convergence of model estimatesfor the SIRT algorithm starting with an initial model estimate at point A. The iteratively updated model estimates, determined from the
averagecorrections AMj in equation(46), are alongthe solidline defined by points A, B, C, D, and E. Each estimated point is the average of the same letter's primed and double primed projection points located on hyperplanesI and 2, respectively.
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CHAPTER
38
2.
SEISMIC
RAY
T¸M¸GRAPHY
The ART algorithmin Figure 14(a) finds the solutionby alternatelyprojecting the current model estimate onto each hyperplane. The model estimate moves along the solid line from the initial model estimate at A to B, to C, to D, and so on. On the other hand, the SIRT algorithm finds point A's projections on both hyperplanes,points B t and B •t, then moves the initial estimate from point A to point B, the midpoint between B t and Btt. For the next iteration the SIRT algorithm finds point B's projections on both hyperplanes, points C t and C", then moves the current estimate
from point B to the midpoint betweenC t and C t•, or point C. Starting with the initial model estimate at point A, SIRT convergestowardsthe solution along the solid line from point A to B, to C, to D, to E, etc. If a true model solution exists and is unique, then both ART and SIRT will convergeto that solution. However,one shouldnote that the convergence of ART depends upon the ordering of the hyperplane projections while the convergence of SIRT doesnot. You may seethis by projecting point A onto hyperplane2 first in Figure 14(a). The ART and SIRT methods, as we have stated, are intended to solve
linear systemsof equationslike thoserepresentedby equation(29) which explicitly relate the model function to the data function. But just because
equation(29) explicitly relatesthe modelfunctionto the data functionin a linear form does not imply a linear relationship for all types of model and data functions.
Take
for instance
a model
function
of slowness and
a data function of observeddirect-arrivaltraveltime. Equation (29) does not provide a linear relationshipin this casebecausethe raypat.h lengths in S are also dependent upon the slownessesdefined in the model function. Thus, we do not know the true raypath lengthsin S until the true slowness field is also known.
To solvethe nonlinearproblemin practice we computeestimatedraypath lengthsusing the estimatedslownesses in the model function and use the estimatedraypath lengthsin the ART or SIRT algorithm. This is called an iterative linear approach to solving a nonlinear problem. We can use Figure 12 to visualizewhat happensto the hyperplaneswhen solvinga nonlinear problemby an iterative linear approach. The two hyperplanesshown in Figure 12 are the true hyperplaneswhen we know the raypath lengthsin S. When we useestimated raypath lengthsin S the estimated hyperplanes will not be coincident with the true hyperplanes. The resulting projection will be different from the projection shown in Figure 12 as point A is projected onto an estimated hyperplane. Each time we update the model function slownesses,using either the ART or SIRT method, the new estimated hyperplanes will be located differently in the model space since we also have new estimated raypath lengths in S. What we hope happens is that the estimated hyperplanes will not be wildly repositioned to a new
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2.4.
SUMMARY
39
location each time the model slownessesare updated. Then as we iterate further towards the true model function slownessesthe estimated hyperplanes should become more coincident with the true hyperplanessince the estimated raypath lengths will be approachingthe true raypath lengths. Thus, we can get convergenceto a solution even though the problem is nonlinear.
2.4
Summary
1. Seismicray tomography attempts to solvethe inverseproblem formulated by the line integral equation, p
__
•.yM(r)dr,
taken over the raypath. P is called the data function and represents
the observeddata. M(r) is called the model function and representsthe spatial distribution of somephysicalproperty of the medium
whichaffectsthe propagatingenergyin someobservablemanner. The model function is unknown and the goal of seismicray tomography is to determine an estimated model function M e•t of the true model function M tr•e.
2. Transform methodsin seismicray tomography are of limited usesince straight raypaths and full scan aperturesare generallyassumed.However, they serve to introduce the tomography concept and terminology,and provideinsightinto seismicdiffractiontomographypresented in Chapter 3.
3. The projectionslicetheoremis the basisfor the transformmethods. The theorem states that the 1-D Fourier transform of the data func-
tion/5(f•,0) provides a sliceof information in thek• - k, wavenumberplaneof themodelfunctionA74(k•,kz) defined ontheloci'k• = f• cos0 and k, - f•sin 0 as shownin Figures4 and 5. Equation(12) definesthe projection slice theorem as,
4. Direct-transformray tomographyappliesthe projectionslicetheorem
tomany projections ofthedatafunction/5(f•, 0•for0 degrees _(0 _( 180 degrees.The resultis the modelfunctionM(f•cosO, f•sinO) definedon a polar grid. Interpolationof the modelon the polar grid onto a rectangularks - k, grid is requiredto take the 2-D inverseFourier
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CHAPTER
40
2.
SEISMIC
RAY
TOMOGRAPHY
transformof the modelfunctionwhichyieldsM(x, z). Interpolation error may distort the resultingestimated model function. 5. Backprojection ray tomography is another transform method which utilizes the projectionslice theorem. However,by making a change of variableswe were able to do the image reconstructionwithout the interpolationstep requiredby direct-transformray tomography.The
reconstruction formulagivenby equation(20) is
M(x,z) = 4•r• , x ej•(xcosO + zsin0)]•]dC2da. Backprojectionray tomographyor its modificationis usedas the image reconstructionalgorithm in computerizedaxial tomographybecause it is both accurate and fast.
6. Seriesexpansionmethods are the most frequently used seismictomographymethods. The model function is divided up into small cells where each cell is assignedan averagevalue of the continuousmodel function within that cell. Thus, the ith observation of the data function is related to the discretemodel function by the equation, J
Piø•' -- Z$ijM]•"•,i- l,...,I, j=l
where I is the total number of rays or observations, J is the total
numberof cellsin the discretemodel function, and $ij is the path length of the ith ray in the jth cell. The inverseproblem is to deter-
mineanestimated modelfunction M• '• of thetruemodelfunction
M]ruegiven theobserved datafunction Piø•s. 7. Kaczmarz' method iteratively solvesthe system of equations defined for the seriesexpansionmethods for the estimated model function
Mf s•. Theiterativepartof thealgorithm consists of threestepsas shownin Figure11andaninitialestimateof themodelfunctionY init must be input.
Step i requiresthat the current estimate of the model function be
usedin forwardmodeling(i.e., ray tracing) to get a predicteddata functionP/P• for the ith ray. The forwardmodelingis definedby the equation, J
j=l
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2.4.
SUMMARY
41
Step2 compares the predicteddata functionpfre with the observed datafunctionPiøb•.If theobserved andpredicted datafunctions agree
to withina specified error,thentheestimated mode] function Mf • istakena.sa goodestimate ofthetruemodelfunction M]rue. Step3 updates thecurrent estimate ofthemodel function M•• if the observedand predicteddata functionsdo not comparefavorably.The updatecorrectionis determinedby projectingthe currentestimateof the model function onto the hyperplane defined by the ith ray. The correctionis given by,
pfbz__pfre 1
The corrections to the model estimate recommended by the ith ray are applied to all J cells,
j
-
1,...,J.
Then, theupdated estimated model function M«"ew)e• becomes the current estimated model function back in step 1.
8. The arithmeticreconstruction technique(AP•T) is a seriesexpansion method which directly usesKaczmarz' algorithm.
9. The simultaneousiterative reconstructiontechnique(SIP•T) usesa modified Kaczmarz' method. Instead of updating the model after
tracing each ray in the forward modelingstep a.sis done in AP•T, all raysare tracedthroughthe currentestimatedmodeland a model correctionfound for eachray. Then the correctionsto eachmodel cell are averagedaccordingto the equation, I
i=1
forj
-
1,...,J.
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CHAPTER
42
2.
SEISMIC
RAY
TOMOGRAPHY
The weightWj is the numberof raysintersecting the jth cellor some other suitableray densityweightusedto obtain an averagecorrection
AM 1. The averagecorrectionis appliedto the currentestimated model function to get an updated or new estimated model function.
2.5
Suggestions for Further Reading Anderson, A. H., and Kak, A. C., 1982, Digital ray tracing in two-dimensional refractive fields: J. Acoust. Soc. Am., 72, 1593-1606. Addressesray tracing througha griddedrefractiveindex model commonly used in ultrasound computerizedtomography.
Berryman,J. G., 1990,Stable iterative reconstructionalgorithm for nonlinear traveltime tomography: Inverse Problems, 6, 21-42. Discussion of the nonlinear problem associatedwith traveltime tomography.
Bishop,T. N., Bube, K. P., Cutler, R. T., Langan, R. T., Love, P. L., Resnick, J. R., Shuey, R. T., Spintiler, D. A., and Wyld, H. W., 1985, Tomographicdetermination of velocity and depth in laterally varying media: Geophysics,50,903923. Traveltime tomographyapplied to reflection seisinology. Chapman, C. H., and Pratt, R. G., 1992, Traveltime tomography in anisotropicmedia- I. Theory: Geophys. J. Int., 109, 1-19. Expandstraveltime tornographyapplicationsfrom isotropic media to anisotropic media, a subject which has much current interest. A companionpaper immediately follows this paper on the applicationsof the theory. Dines, K. A., and Lytle, R. J., 1979, Computerizedgeophysical tomography: Proc. IEEE, 67, 1065-1073. First application of ray tornographyto subsurfaceimaging. Discusses both ART
and SIRT.
Langan, R. T., Lerche, I., and Cutler, R. T., 1985, Tracing of rays through heterogeneousmedia: An accurate and efficient procedure: Geophysics,50, 1456-1465. We do not elaborateon how to determine the raypath lengths through a griddedvelocitymodelfor the matrix S. This referenceis more than adequatein addressingthe problemas appliedto direct arrivals in seismic ray tomography.
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2.5.
SUGGESTIONS
FOR
FURTHER
READING
Moser, T. J., 1991, Shortest path calculation of seismicrays: Geophysics, 56, 59-67. Utilizes network theory to get raypaths and traveltimes of first arrivals. Very robustfor defining the matrix S, whether the first arrival is a direct arrival or a head wave.
Peterson, J., Paulsson, B., and McEvilly, T., 1985, Application of algebraic reconstruction techniquesto crossholeseismic data: Geophysics,50, 1566-1580. ART applied to crosswell seismic
data.
Phillips, W. S., and Fehler, M. C., 1991, Traveltime tomography: A comparisonof popular methods: Geophysics,56, 1639-1649. Comparison of various linear inversion methods.
Vidale, J. E., 1988, Finite-difference calculation of traveltimes: Bull. Sets. Soc. Am., 78, 2062-2076. Approximates the eikonal equation using finite differences to compute traveltimes for first arrivals.
43
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Chapter
3
Seismic
Diffraction
Tomography 3.1
Introduction
Seismicdiffractiontomographyis usefulfor reconstructingimagesof subsurface inhomogeneities whichfall into two categories.The first categoryincludesinhomogeneities that aresmallerin sizethanthe seismicwavelength and have a large velocitycontrastwith respectto the surrounding medium.Imagingtheseinhomogeneities with the seismicray tomography methodspresentedin Chapter 2 is generallyout of the question.The second categoryincludesinhomogeneities that are muchlargerin sizethan the seismicwavelengthand have a very small velocity contrastwith the surroundingmedium. Although seismicray tomographyis valid for imaging theseinhomogeneities, it worksbest when the velocitycontrastsare large. Note that both categories of inhomogeneity are capableof producingmeasurablescatteredwavefieldsof similar power. The large velocitycontrast of the first categoryinhomogeneity offsetsits smallsizewhilethe largesizeof the secondcategoryinhomogeneity makesup for its smallvelocitycontrast. The outlinefor this chaptercloselyparallelsthat of Chapter 2. First, in Section3.2 we review acousticwavescatteringtheory and derivetwo independentlinear relationshipsbetweendata functionsrepresenting scattered energyand the modelfunctionM(r). The modelfunctionM(r) usedin this chapteris a measureof the velocityperturbationcausedby an inhomogeneityat vectorpositionr from a constantbackground velocity.Second, usingeither of the linear relationshipsbetweena data function and the modelfunctionM(r), the generalized projectionslicetheoremis derived 45
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46
CHAPTER
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in Section 3.3 which servesas the foundation for the image reconstruction algorithms used in seismic diffraction tomography. Finally, two seismic diffraction tomography image reconstruction algorithms are presented in Section3.4' direct-transformdiffraction tomographyand backpropagation diffraction tomography. Before proceeding further one should have a good understanding of
the followingmathematicalconcepts: frequencyand wavenumber(AppendixA); Fouriertransform(AppendixB); Dirac deltafunctionandGreen's function (Appendix C). Also, sincewe presentseismicdiffractiontomography for inhomogeneitiesembedded in a constant velocity medium, the implementation of diffraction tomography is actually very similar to the transform methods for ray tomography. Thus, a review of Section 2.2 on ray tomography'stransformmethodsmight be of somebenefit.
3.2
Acoustic Wave Scattering
The propagationof an acousticwavefieldP(r,t) through a medium consistingof a variablevelocityC(r) and constantdensityis modeledby the acoustic wave equation,
1 O2P(r,t) V•P(r,t)C•(r) at• _ - 0,
(47)
where r is a vector position within the model and t is time. The Laplacian
operator X7• is definedin terms of the vector operator •7 which in the Cartesian coordinatesystem is given by
v -
+ 0
where•, j, and• aremutuallyorthogonal unitvectors. We use the Helmholtz form of the acoustic wave equation to describe acousticwave scattering. The Helmholtz acousticwave equation, found by
taking the temporalFouriertransformof equation(47), is
VaP(r,w)+ k•(r,w)P(r,w) = 0.
(48)
The variablek(r, w) is the magnitudeof the wavenumberat positionr and is defined by
k(r,w)= C(r)'
(49)
Notethat equation(48) dependsuponthe valuesetfor angularfrequencyw. Henceforth,for the sakeof brevity,we will write both P(r,w) and k(r,w)
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3.2.
ACOUSTIC
WAVE
SCATTERING
47
as P(r) and k(r) with the understandingthat there is a dependenceon angular frequency w. Two nonlinear integral-equation solutionsfor the scalar Helmholtz wave
equationdefinedby equation(48) arederivedin thissection.The LippmannSchwinger integral equation is one such solution which, because of its prominencein quantum mechanicalscattering, is presentedby itself in Section 3.2.1. The Lippmann-Schwinger equation nonlinearly relates the
data function P0(r), calledthe scatteredwavefield,to the modelfunction M(r). The Born approximationin Section3.2.2 linearizesthe LippmannSchwingerequation. Another nonlinear integral-equation solution is formulated in Section 3.2.3 using exponentials. Although the definition of the model function M(r) remainsthe same, the data function is different from the scatteredwavefieldP0(r). The Rytov approximationis used to linearize this nonlinear integral-equation solution. Except for the data functions, both linearized integral-equationsolutionsare identical in form. Section 3.2.4 provides a comparison between the Born and Rytov linear integral solutions.Either solutionenablesus to derive the generalizedprojection slice theorem in Section 3.3 which serves as the foundation for the diffraction tomographyimage reconstructionalgorithmsin Section3.4.
3.2.1
The Lippmann-Schwinger Equation
We begin the formulation of the acousticwave scatteringproblem with Figure 15. The acousticwave velocity is representedby C(r), where r is the vector position of a point within the model. The shaded region in Figure 15 depictsan inhomogeneityimbeddedin an otherwisehomogeneous medium. The acoustic velocity of the inhomogeneity varies spatially and can be thought of as a velocity perturbation from the constantbackground velocity Co of the homogeneousmedium.
An incidentwavefieldPi(r) is initiated by an acousticsourceand propagatesoutward in the homogeneous medium. No scatteringof the incident wavefield takes place until the inhomogeneityis reached. At that point any velocity contrast as a result of the inhomogeneitycausesthe creation of a secondwavefieldcalled the scatteredwavefieldP•(r). Each point in the inhomogeneitymay be consideredas a secondarysourceof seismic acousticenergy. Note that once acousticenergyis scatteredfrom one inhomogeneity,then that scatteredenergymay be scatteredagain from another inhomogeneitywhich leads to higher order sourcesof seismicacousticenergy. As you will see, we ignore multiple scatterings in Section 3.2.2 to linearize the Lippmann-Schwingerequation and assumethat the scattered
wavefieldP•(r) arisesonlyfromscatteringthe incidentwavefieldP•(r) from the source as depicted in Figure 15. Therefore, for layered media we as-
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48
CHAPTER
3.
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DIFFRACTION
Receiver
TOMOGRAPIIY
Source
C(r) = Co
FIG. 15. Acousticwavescatteringproblem.The incidentwavePi(r) propagatesfrom the sourceat the constantbackgroundvelocityCo. The velocity inhomogeneity, depictedby the shadedarea, actsas a secondarysourceand
scattersthe incidentwavefield.The scatteredwavefieldPj(r) travelsaway from the inhomogeneous regionat the backgroundvelocityCo whereit is recordedby the receiver.
sume that multiple reflectionsare negligiblewhen comparedto primary reflections.
The wavefield recorded by a receiver in the model consistsof both the
incidentwavefieldPi(r) and the scatteredwavefieldP,(r) whichwe call the total wavefieldPt(r) , or
P•(r) -
Pi(r) -[-P,(r).
(50)
For a constantdensitymodel,equation(48) describesthe propagationof the total wavefieldPt(r) throughan inhomogeneous medium,or
[V'• + k•(r)]Pt(r) = O.
(51)
At thispointwereformulate k2(r) in equation(51) asa perturbation to a constantko • for the homogeneous background mediumwherethe magnitude of ko is given by
=
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3.2.
ACOUSTIC
WAVE
SCATTERING
49
We begin by writing
-
+
[k•(r)1].
(53)
Substitutingequations(49) and (52) into the bracketedterm on the righthand sideof equation(53) for k(r) and ko, respectively, gives
ke(r)
C• 1] - ko •+k•[C•.(r) = ko 2-ko 2[1 C2(r ½;] ).
(54)
If we definethe modelfunctionM(r) asthe bracketedterm in equation(54),
thenwe get the desiredreformulation of k2(r),
k2(r) -
k• - ko•M(r),
(55)
whereM(r) is definedby
M(r)- 1 C•(r C•).
(56)
The model function M(r) in equation(55) definesa perturbationto an
otherwiseconstantko • of the background medium. When C(r) - Go in equation(56), M(r) - 0 and thereis no perturbationof ko • (i.e., by equation (55), k•(r)We now want to establish a relationship between the scattered wave-
field P,(r) and the modelfunctionM(r) usingequation(51). First, equations(50) and(55) aresubstituted intoequation(51) for Pt(r) andk2(r), respectively, giving
[•7• + k•- k•M(r)][Pi(r)+ Po(r)] -
O.
(57)
We rearrangeequation(57) so that terms involvingsourcesof scattered energyt are on the right-handside. Thus,
[V2+ ko2]Pi(r)+[V2+ ko•]Po(r)-
ko2M(r)[Pi(r)+Po(r)]. (58)
Note that the term ko•M(r)[Pi(r) + P,(r)] is the sourceof the scattered wavefieldP0(r) and, as previouslystated, dependsupon both the incident 1Termscontainingthe modelfunctionM(r) are sourcesof scatteredenergy.
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5O
CHAPTER
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SEISMIC
DIFFRACTION
TOMOGRAPItY
wavefieldand scatteredwavefieldat r. The left-handsideof equation(58) describes the propagationof the incidentwavefieldPi(r) and the scattered wavefieldP,(r) in the backgroundmedium, both of which travel at a constant velocity Co.
The incidentwavefieldPi(r) is generatedby a sourcein the homogeneousbackgroundand containsno scatteredenergy. Therefore, the incident wavefieldtravelsthrough the model at the backgroundacousticvelocityCo and contributesto the scattered wavefieldthrough the scatter sourceterm
in equation(58) when M(r) •- 0. Thus, the Helmholtzacousticwave equation for a constant velocity medium describesthe propagationof the incidentwavefieldPi(r) and is givenby
[V'2+ko•]Pi(r) -
O.
(59)
Equation(59) permitsus to reduceequation(58) to
[v +
-
(60)
Equation (60) describesthe propagationof the scatteredwavefieldat the backgroundvelocity when inhomogeneitiesoccur which scatter both the
incidentwavefieldPi(r) and existingscatteredenergyPs(r). Note that if no inhomogeneities occur, then M(r) = 0 and the right-hand side of equation(60) is zero,implyingthereis no sourcefor the scatteredwavefield and P•(r) = 0. Solvingequation(60) directlyfor P,(r) is difficult. A simpleapproachis to formulatean integralsolutionusingthe propertiesof the Green'sfunction developedin Appendix C. For the Helmholtz equation the Green'sfunction is the responseof the differential equation to a negative impulse source
function. 2 Thus,theGreen'sfunctionbecomes thesolutionto equation(60) if we replacethe sourceterm with a negativeimpulsesourcefunction-5(rr t) • or
[V2+ ko•]a(r[r')
-
-5(r - r').
(61)
The Green'sfunctionG(r Jr') givesthe solutionat positionr for a negative impulse at r' which correspondsto the location of a point scatterer.
The solutionto equation(61) for a 2-D spacecontainingan infinite-line scattererat r' and infinite-line receiverat r, where both lines are perpendicular to the plane containingr • and r, is
a(rIr') -- j4 Ho(X)(ko Ir- r' l)'
(6:2)
• Traditionally the Green'sfunction solutionfor the Helmholtz equation is determined for a negativesourcedensity on the right-haxtdside, or X72P + ksP = -p. In order for the Dirac delta function to represent a source density impulse, the sign in front of the Dirac delta function must also be negative.
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3.2.
ACOUSTIC
WAVE
SCATTERING
51
where Ho (•) isthezero-order Hankel function ofthefirstkind.In a 3-Dspace containingpoint scatterersand field points, the solution to equation (61) is
G(rlr')= 4•rlr-r' I '
(63)
With the Green'sfunction for equation(61) known, the solution to equation(60) is foundby multiplyingthe Green'sfunctionby the negative of the sourceterm in equation(60) and integratingover all spacewhere
M(r •) •- 0, or 3
P,(r)- -ko •/ G(r [r')M(r')[P•(r') +P•(r')]dr'. (64) Equation (64) is called the Lippmann-Schwinger equation,which is the desiredintegral solutionfor the acousticwave-scatteringproblem. We point out in Appendix C that suchan integral is analogousto obtaining an output from a filter systemwhenthe impulseresponse of the filter (i.e., the Green's
function)and the input (i.e., sourceterm fromequation(60)) are known. Equation(64) is just a convolutionintegral,whichis easilyseenif the Green'sfunctionfrom either equation(62) or (63) are substitutedin for I,?). The Lippmann-Schwingerequationnonlinearlyrelatesthe model func-
tion M(r) to the data function(scatteredwavefield)P,(r). The nonlinearity is a result of the scatteredwavefieldP,(r) insidethe integrandof
equation(64) whosevaluedepends on the modelfunctionM(r). Because of this nonlinearity,it is difficult to use equation(64) to perform either forwardmodeling(computeP,(r) from M(r)) withoutresortingto computationally extensiveapproachessuchas finite differencemethodsor to derive diffractiontomographyimage reconstructionalgorithms(compute M(r) from P,(r)). One way to get aroundthis problemis to linearize equation(64) by makinga simplifyingapproximation calledthe Born approximation.
3.2.2
The Born Approximation
The Born approximationlinearizesequation(64) by assumingthat the
scattered wavefield P,(r) is muchweakerthanthe incidentwavefield Pi(r), 3In this book we use special integration notation which should not be confused with
a line integral. Equation(64) couldbe either an integrationovera plane (2-D space) or a volme (3-D space)dependingupon which Green'sfunctionis used. Thus, to
remain general, if weareintegrating overa plane, thenf dr'• f dx'dz'; andif weare integrating overa voltune, thenf dr'::•f dx'dy'dz'.
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52
CHAPTER
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or
P,(r) • P•(r).
(65)
If the conditiongivenby equation(65) is true, then the Born approximation states that
Pi(r) + P,(r)
•
Pi(r).
(66)
The Lippmann-Schwinger equationis linearizedby substitutingequation(66) into equation(64),
P,(r) • -ko 2/ G(r [r•)M(r•)Pi(r•)dr •. (67) Note that the integrandno longercontainsthe scatteredwavefieldP•(r) and the data function P•(r) and mode]function M(r) are now linearly related. If the primary acoustic source is a negative impulse located at vector position r,, then, by the definition of a Green's function, we can
directlyrepresentthe incidentwavefieldP•(r') in equation(67) by a Green's function,
Pi(r')
:
G(r' [ rs).
(68)
If a pressure-sensitive receiver(e.g.,hydrophone) is locatedat positionr = rp, thenby substituting equation(68) intoequation(67) for P•(r') wefind,
P,(r,,rp) • --k• 2/ M(r')G(r' lr,)G(r pIr')dr', (69) whereP,(r,, rp) is the scattered wavefield observed at positionrp whenthe negativeimpulsesourceis located at positionr•. Both Green'sfunctions are definedby either equation(62) or (63).
Equation(69) is the Lippmann-Schwinger equationlinearizedby the Born approximation. This equation establishesthe linear relationship be-
tweenthe data function(scatteredwavefield)h(r,,rp) and the unknown modelfunctionM(r) requiredby the diffractiontomographyproblem.Note that sincewe exploitedthe Born approximationin derivingequation(69) that the modelfunctionM(r) must be a weak scatterer. Only then will the scattered
wavefield
be much weaker than the incident
wavefield
and the
conditionspecifiedby equation(65) be satisfied.
3.2.3
The Rytov Approximation
In this sectionwe derivea nonlinearintegral-equationsolutionto equa-
tion (48) usingexponentials. Althoughthe modelfunctionM(r) is defined
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3.2.
ACOUSTIC
WAVE
SCATTERING
53
the same as for the Lippmann-Schwingersolution, the data function representingthe observedscatteredwavefieldis different. The Rytov approximation establishesa linear relationship between the data function and the model function. The resulting linearized integral solution strongly resemblesequation(69) for the Born approximation. We begin the derivationby returning to equation(51) which describes the propagationof the total wavefieldP•(r) through a constantdensity, variable-velocity medium. Usingequation(55) andequation(56) asthe def-
initionsfor k2(r) andthemodelfunctionM(r), respectively, equation(51) is rewritten
as
Iv • + • - •4(•)]P,(•)
-
0.
(70)
Exceptfor settingPt(r) = Pi(r)+ P,(r), equation(70) is the sameas equation(57). We deviatefrom the earlierderivationof the LippmannSchwingerequationby representingthe total wavefieldwith the exponential equation,
P•(r)-
e•,(r),
(71)
where•b•(r) is calledthe "complextotal phasefunction." Note that, as in previous sections, variables which are a function of the position vector r
carryan impliedfrequencydependence. We wish to substituteequation(71) into equation(70) to obtain a differentialequationin termsof •,(r). The Laplacianof P,(r) mustbe taken to achieve the this result.
We start with
v•P,(•)
= v. [vP,(•)].
Substitutingequation(71) in for Pt(r) and performingthe differentialoperationsgives,
v•P,(•) = = =
v. b•,(•)v•,(•)], e•'(•)v-v•,(•)+ Ve•'(•).v•,(•), •,(•)v•,(•) + •,(•)v•,(•). v•,(•), e•'(r)[v•,(r)+V•,(r).V•,(r)].
(72)
Equations (71) and(72) aresubstituted intoequation(70)for V2P,(r) and P,(r) giving,
•,(•)[v•,(•) + v•,(•). v•,(•)]+ [• - •(•)1•,(•)
- 0. (73)
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54
CHAPTER
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Dividing through byeqb'(r), equation (73)becomes, V•6 (r) + [V6t(r) V6t(r)]+ ko • ko•M(r)-
0,
(74)
whichis the desireddifferential equationin termsof •bt(r). At this pointweintroducethe incidentwavefield Pi(r) expressed
Pi(r)-
e0i(r),
(75)
where$•(r) is calledthe "complex incidentph•e function."The complex incidentph•e function•i(r) is relatedto the complextotal ph•e function •t(r) throughthe "complex ph•e difference function,"definedby
0a(r) =
0t(r)- 0,(r).
(76)
Sincethetotalwavefield P•(r) variesfromthe incidentwavefield P•(r) only whenscatteringoccurs,we can surmisethat the complexph•e difference function$a(r) is a meansof accounting for scatteredenergy.
Wecontinue thederivation byreplacing 0,(r)in equation(74) by •a(r) • definedby equation(76). Carryingthis operationout yieldsa differentialequationin termsof 0,(r) and
V•0i(r)+ V•0•(r) + [V0i(r). V0i(r)]+ 2[V0i(r). +[V$•(r). V&a(r)]+ k• - k•M(r) - 0.
(77)
The termsin equation(77) are rearrangedin a form whichwill proveconvenient later on,
=
+
(78)
The termsinsidethe squarebracketson the left-handsideof equation (78) are all relatedto the incidentwavefield and havea sumequal to zero. This is e•ily showntrue by usingequation(59) whichdescribes the propagation of the incidentwavefieldPi(r) throughthe background medium.Equation(75) defining Pi(r)is substituted intoequation (59)to get the differentialequationin termsof •i(r). Followingthe sameprocedurewhichgavethe Laplacianof Pt(r) in equation(72), the Laplacianof
V•P•(r)- e0•(r)[v•0•(r) + V0•(r).V0,(r)].
(79)
Substituting equations (79) and (75)into equation(59) for V•P•(r) and P•(r), respectively, gives
e•i(r)[v2•i(r) + V•i(r)-V•i(r)] + k•e•i(r)- O.
(80)
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3.2.
ACOUSTIC
WAVE
SCATTERING
55
Dividing equation (80)bye•i(r)results inanequation which demonstrates that the bracketedterms in equation(78) havea zerosum,or
V2•i(r) + V•i(r).V•i(r)
+ ko • -
O.
(81)
+ ••(•).
(8•)
By equation(81), equation(78)is rewritten •
•v•,(•). v•(•)
+ v•(•)
- -v•(•).
v•(•)
Equation(82) providesa crucialrelationshipwhichwe will uselater on in this derivation. Now we turn our attention to the important product
P•(r)•(r)
which will becomethe data functionfor this derivationor a
me•ure of the scattered wavefield. For purposesof the derivation the Laplacian is taken of this product. Performingthe differentiationwe have,
V•[P,(•)•,(•)]
- V. [•,(•)VP,(•) + P,(•)V•,(•)] = •,(•)V•(•) + •V•(•). V•,(•) +P,(r)V2•d(r).
(83)
Rearrangingthe termsin equation(83) and makinguseof the fact that
V•P•(r)-
-k•P•(r) by equation(59), weget the following,
2VP,(r). V•(r)+
P,(r)V•(r)
- V•[Pi(r)•(r)]- •(r)V•P,(r) = V2[pi(r)•d(r)]+ •d(r)koP•(•) 2 = [V• + k•]P,(r)•d(r). (84)
Switchingthe left- and right-handsidesof equation(84) and using the definitionof P•(r) givenby equation(75), we can write the following,
Iv • + •]P,(•)o,(•)
-
•v•,(•). v•,(•) + P,(•)v•o,(•)
= •(•)vo•(•). v•,(•) + p•(•)v•o,(•) = •P,(•)v•,(•). v•,(•) + P•(•)v•,(•) = p,(•)[•v•,(•). vo,(•) + V•d(•)].
(SS)
The quantity inside the square brackets on the right-hand side of equation (85) is definedby the earlier "crucial"relationshipgiven by equa-
tion (82). Substitutingequation(82) into equation(85) gives,
IV• + •]P,(•)o,(•)
-
-P,(•)[v•,(•).
vo,(•)-
•(•)].
(86)
Just • w• donefor equation(60), equation(86) can be solvedfor P•(r)•d(r) in termsof an integralequationby exploitingthe propertiesof the Green'sfunction. The resultingsolutionis
P,(•)o,(•) - f P,W)[vo,(•'). v•,(•')•(•')]a(• I•')d•',(87)
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CHAPTER
56
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
where G(r Ir') is a Green'sfunctiongivenby either equation(62) or equation(63). The data functionP•(r)da(r) is nonlinearlyrelatedto the modelfunction m(r) sinceda(r) appearsinsidethe integrandof equation(87). Both forward modeling(computingP•(r)•a(r) from M(r)) and image reconstruction(computingm(r) from P•(r)&a(r)) are difficultbecauseof this nonlinearity. At this point in the derivation the Rytov approximation is formulated to remedy this situation. The Rytov approximation linearizes
equation(87) by assuming the conditionV•a(r) <( 1. When V•a(r)is smallthe quantityX7•a(r'). V•a(r') in equation(87) canbe neglected and we write the approximation,
-f
(ss)
As with the Born approximation,the incidentwavefieldPi(r') can be representedby a Green's function,
P•(r') =
G(r'}r,),
(89)
for a negativeimpulsesourceat vectorpositionrs. If a receiveris located
at positionr - rr, then by substituting equation(89) into equation(88) we have,
P•(r•,rv)•Sa(r,,rv) • -ko •f m(r')G(r' lr,)O(r vIr')dr'. (90) The Green'sfunctionsaresatisfiedby eitherequation(62) or equation(63). However,to useequation(90) properly,the gradientof the phasedifference functionV•Sa(r),mustbe smallas requiredby the Rytov approximation.
3.2.4
Born vs. Rytov Approximation
Except for the data functions, the linearization of the Lippmann-
Schwinger equationusingthe Bornapproximation in equation(69) is identical to that of the Rytov approximationin equation(90). Herewe demonstrate that the data functionPi(r,,rr)•Sa(r,,rr) associated with the Rytov approximationreducesto the data function associatedwith the Born
approximation, the scattered wavefield P•(rs,rp), whenthe complex phase difference functiond•(rs,rr) issmall.Next,weshowthat forcing to be smallis the sameasrequiringa weakscattered wavefield P,(r,,rr) for the Born approximation.Finally, we state the limitationsfor applying either the Born or Rytov approximation.
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3.2.
ACOUSTIC
First
WAVE
SCATTERING
57
we wish to show that
Pi(r0,rr)•ba(r0,rr) •
P0(r0,rr),
(91)
when4a(r0,r•) is small. This resultis achieved by usingthe Taylorseries
expansion ofe4a(r0, r•) which isgiven by,
•*•(•0,•)- 1+,•(•,'•)+ *](•"•) 2] +
3] +.... (92)
When •(r•, r•) • 1, we canneglectthe secondand higherordertermsin the Taylor seriesexpansionand write
•(•,,•)
• •(•',•)
- 1.
(•)
The approximationof the data function •sociated with the Rytov approximation is found by multiplying equation (93) by the incident wavefield
Pi(r•, rp), giving,
P,(r,,rr)•(r,,rr) • Pi(r•,rr)[e•(r•'rr) - 1].
(94)
This approximation is e•ily shownto be the scatteredwavefieldP•(r•, rr)
byusing therelationships, Pi(r,,rr)- e•i(r•,rr), P•(r•,rr)- e•'(r•,rr) andP•(r,, rr) - P•(r•, rr)-Pi(r,, rr) , fromequations (75), (71), and(50), respectively.Hence,equation(94) is rewritten:
•,(•. •),,(•. •) • •,(•. •)[•,(•. •) - 1]
• •,(•,, r•) _ •,(•,,r•) • •
•,(r•, •) - •,(r•, r•) •(r•, r•).
(•)
Thus,the data functionsfor equations(69) and (90) are approximately the samewhen•(r•, r r) is small. Now we wishto showthat setting•(r•,rr) • 1 is the same• the Bornapproximation requirement of a weakscatteredfield(i.e., P•(r,, rr) • P•(r•, rr) ). We beginby explicitlywritingthe total and incidentwavefields
P•(r)-
A,(r)eJ•b,(r),
(96)
Pi(r) - Ai(r)eJ•bi(r),
(97)
and
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58
CHAPTER
3. SEISMIC
DIFFRACTION
TOMOGRAPHY
respectively, where;bt(r)and;b,(r)aretherealphases andAt(r) andA,(r) arethe amplitudes.Equation(76) givesthe definitionof the complexphase difference
function
as
4a(r) = 4,(r)- 4,(r).
(98)
Solvingequations (71) and(75) for dr(r) anddi(r), thensubstituting the resultsinto the last equationgives
da(r) = lnPt(r) - lnP•(r).
(99)
Usingthe explicitrelationsfor Pt(r) and Pi(r) givenby equations (96) and (97), equation(99) becomes,
da(r)- In[At(r)1 q-j[•bt(r)•bi(r)] '
(100)
The complexphasedifference functionrid(r) in equation(100)is small whenIn ta,(r)/Z,(r)l << I and [;bt(r)- ;b•(r)]<< 1. This is the comparableto requiringthat the difference betweenPt(r) •na g(r) be small
for the Bornapproximation. The In ta,(r)/a,(•)l termin equation (100) demonstrates that the Bornapproximation is a weakscatteringapproximation. Also,the accumulative phasedifference, [;bt(r)- ;b•(r)l, maybecome significantif the regionwhere M(r) • 0 is large relative to the seismic wavelength.Thus, in addition to requiringweak scatterers,the size of the
inhomogeneities maybecome an importantfactorto consider whenusing the Born approximation.
On the otherhand,the Rytovapproximation doesnot requirerid(r) to be small,only that the gradientof rid(r) be small(i.e., the changeof rid(r) withina wavelength be small). The Rytovapproximation requires
onlya smooth modelfunction andplaces norestrictions onthestrength of the scatterersor their size. Thus, the Rytov approximationis a smooth scatteringor smoothperturbationapproximation. 3.3
Genera'ized Projection
Slice Theorem
Either equation(69) or (90) in Section3.2 definea linearintegralrelationshipbetweena data functionrepresenting scatteredenergyand the modelfunctionM(r). In thissectionwe simplifythis linearintegralrelationshipby taking the spatial Fourier transform of the data functionover
the sourceandreceiverprofilesandby takingthe 2-D spatialFouriertransform of the modelfunction. The result is the generalizedprojectionslice
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3.3.
GENERALIZED
PROJECTION
SLICE
theorem which serves as the theoretical
THEOREM
foundation
59
for the seismic diffrac-
tion tomography image reconstruction algorithms presented in Section 3.4.
We let P(r,, rr) for the remainingpart of thischapterrepresent the data
function foreitherP,(r,, rr) oftheBornapproximation or Pi(rs,rr)•b,(rs,rr) of the Rytov approximation. Thus, the generic equation,
-}o
(xox)
definesthe linear integral relationshipbetweenthe data function and model
functiongivenby eitherequation(69) or (90).. Also,sincemostsourceand receivergeometriesare for 2-D problems,we will exclusivelyusethe Green's
functiondefinedby equation(62). We find it instructive to derive the generalizedprojection slicetheorem for three typical source-receiverconfigurations: the crosswellprofile, the
vertical seismicprofileor VSP, and the surfacereflectionprofile. One may justify this approach by noting that the spatial Fourier transform of the
data functionP(r•, rr) mustbe takenoverthe sourceand receiverprofiles. However, by the end of this section the reader will find that the theorem is actually independentof the source-receiverconfiguration.
3.3.1
Crosswell Configuration
We begin the derivation of the generalized projection slice theorem for the crosswellconfiguration by replacing the position vectors in equation (101) with coordinatesfrom the x-z plane. Figure 16 showsa crosswell configuration with sourcelocations representedby solid circlesand receiver locations identified by open circles. We assume both source and receiver wells are vertical
so that the x-coordinates
of all sources and receivers are
given by the constants,ds and dr, respectively.The z-coordinates of the sourceand receiverlocationsare z• and zr. Thus, the positionvectorsfor
the source locationr• andreceiver locationrr' canbe expressed in terms of their respective coordinates (d•, z•) and (dr, zr). Substitutingthesecoordinatesinto equation(101) for the positionvectorswe get,
x O(z, z I ds,zs)O(dv, zv [ z, z)dxdz, (102) for the crosswellconfiguration.
The Green'sfunction G(z,z I a,,•,)
i, equation(102)is chosento
representa cylindrical wave emitted from an infinite line sourcelocated a•
the coordinates(d,, z,). Equation(62) is the appropriateGreen'sfunction
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60
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
..X
Source Receiver
ds
dp
z
FIG. 16. Source-receivergeometry for the crosswelltomography experiment. The sourcelocations are representedby solid circles at the constant horizontal location d,. The receivers are represented by open circles at the
constanthorizontallocationdr . The orientationof the line sourcesand line receiversis perpendicularto the page. We restrict our reconstructionof the
modelfunctionto d, < z < dr.
for this problem. We rewrite the position vectors in the Green's function
in termsof x-z coordinates using(x, z) for r, (d,, %) for r', and I r - r ' I-
Vf(x-d•) 2+ (z- %)2. Makingthesesubstitutions in equation (62)gives, J
G(x,z Id,,z,)- •Ho(•)(koV/(X - d,)•+ (z- z,)•).
(103)
The Green'sfunctionin equation(103) is awkwardto use becauseof
thezero-order Hankelfunction Ho (•). TheGreen's function represents a cylindricalwaveat the point (x, z) propagatingawayfrom a line sourceat (d,, z,) with vectorwavenumber ko. Fortunately,it turnsout that the zeroorder Hankel function can be mathematically thought of as the summation
(integration)at (x, z) of an infinitenumberof planewaveswhosewavefronts are tangentto the cylindricalwavefront.This plane wavedecomposition of the cylindrical wave is defined by
+
-
1/_•1eJ[kl(z-z,)+?llx-d, I]dkl (104)
•r
oo 7•
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
61
wherethe wavenumberfor a givenplanewavefor the crosswellconfiguration
has the components k• alongthe z-axis and 7• alongthe x-axis.4 The
absolute valueof (x -ds) is required because 71 - +v/ko • -k•, whichis alwaysgreaterthan zero. The absolutevalueofx-ds insures7•(x-do) > O, regardlessof the value of x.
Figure17 illustratesthe conceptbehindequation(104). Planewave•1 is perpendicularto the cylindricalwavefrontat (x,z). This plane wave has the samevectorwavenumberko as the cylindricalwavefrontat (x, z) and has the wavenumbercomponents(7•,kl). As previouslymentioned we require the • componentfor all plane waves be positive, defined by
7• - +v/ko2 -k•.
This definitionof 7• ensures that all planewavesare
tangent to the cylindrical wavefront with a wavenumbermagnitude of ko.
The directionofpropagation for planewave#1 is a = tan-• (7•/k•). Plane wave #2 is traveling in the +x directionwith vectorwavenumberko•: and components (7• = ko,k• = 0). Note that the summationof planewave#1
by equation(104) at (x,z) is at the onsetof the plane wavewhile the summationof plane wave #2 at (x,z) occurslater on in its wave train. The waveformof any planewavewill be summedby equation(104) at a later point in its wave train if the plane waveis tangent to the cylindrical wavefrontat a point other than (x, z). Now we apply plane wave decompositionto the Green's function in
equation (103). This is achievedby substitutingthe plane wave decompositionof the zero-orderHankelfunctiondefinedby equation(104) into equation(103) giving,
+
-
where the absolutevalue can be droppedby requiringz > ds. Note that this Green's function is adequatefor the crosswellconfigurationshown in Figure 16 since we are usingequation (102) to image only betweenthe
sourceline and receiverline or d0 < z < dv. We are effectivelysaying M(z, z) = 0 outsideof the imagingarea. The Green'sfunctionG(dv,zr [ z,z) in equation(102) is chosento representa cylindricalwaverecordedat (dv,zv) scatteredby an infinite line scattererlocatedat (z,z). As with the line source,equation(62) is the appropriate Green's function for the line scatterer. After substituting 4In derivationsof the generalized projectionslicetheoremfor differentsource-receiver configurations, the kl component is always taken along the line of sourcesand -yl is perpendic,,lar to k• in a right-handed coordinate system sense. Here the sourceline just happens to be along the z-axis. The reason for doing this will become clear in a few pages.
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62
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
#2
#1
z
FIG. 17. Plane wave decomposition of a cylindricalwavefrontat (x,z). Two plane waves, labeled #1 and •2 out of an infinite number of plane waves tangent to the cylindrical wavefront, illustrate the concept behind equation(104).
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
63
coordinatesfor positionvectorsin equation(62) and performinga plane wave decompositionon the Green's function we get,
G(dr, z•[x,z) - 4•----•• fj l eJ[k•(zr - z)-l?•(dr - x)]dk• (106) for x ( dr. Here a given plane wave has the wavenumbercomponents k• along the z-axis (parallel to the receiverline) and 7• alongthe x-axis (perpendicularto the receiverline), analogousto the kx and 7• reference directionswith respect to the sourceline. The direction of propagation
for the scatteredplanewaveis tan-• (72/k2). As with equation(105),this Green's function for the scattered wavefieldis adequate since the imaging
areahasthe limits d0 < x < dp. One final relationship is required before we continuethe derivation. We must write down the equation for taking the Fourier transform of the data
functionP(d,,zo; dr, zr) in equation(102) alongthe sourceline (z,) and the receiverline (zr). This relationshipis foundby first takingthe Fourier transformof P(d,, z,; dr, zr) with respectto z, followedby a secondFourier transformwith respectto zr whichproducesthe doubleintegralFourier transform,
x e-jk•z•e-Jkrzr dz•dzr '
(107)
wherek• and kr are the wavenumbers of the Fouriertransformsalongthe sourceline and the receiverline, respectively.Note that the directionstaken for kl and k2 in the two Grecn's functions are oriented the same as ks and
kr, a "lucky" coincidence as far as the derivationis concerned. Now we may resume the derivation of the generalized projection slice theorem for the crosswellconfigurationby obtaining an integral relation-
ship betweenthe data functionPo(d•,k•;dr, kr) and the modelfunction M(x, z). We beginby substitutingthe linearintegralrelationshipbetween the data functionP(d•, z•; dr, zr) andthe modelfunctionM(x, z), defined by equation(102) into equation(107). This substitutiongivesthe relationship,
-k o 2
x
x
M(x,z)
G(x,z [ d,,zo)½-Jk, z,dzo
[
(os)
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64
CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPttY
between thedatafunction ]5(d,, k,;dr,kr)andthemodel function M(x,z). Notethatwehavechanged theorderof integration in equation (108). Next,wesubstitute theGreen's functions fromequations (105)and(106) intoequation (108),regroup termswithsimilarintegration variables, and interchange the orderof integration for the twoinnermost integrals over
eachGreen'sfunction,givingthe equation,
•5(d, Iq,) = k• ' ko' ' d•,, 4 1 •
M(x ' z}
(109)
{•• 1eJ[-•2z+'2(dp-z)]• • -J(•p-'2)Zpdzpd The integralsin equation(109) with respectto the variablesz, and
zr areeasilyevaluated in termsof Diracdeltafunctions asshownin AppendixB. The followingare the integralsolutions: 5
:'ø e-j(k• +k,)z, dz,
(110)
•øe-j(k •,- ka)zp dzp 2•rS(kr- k,).
(111)
and
Substituting theintegral solutions givenbyequations (110)and(111) intoequation(109)andevaluating theintegrals withrespectto thevariables
k, andk=(remembering that7• - ko • - k• and7• - ko • - k•) gives, '
;
-
--
4
%%
•
M(z,z)
(112) where
'rr- Vko =-/%=
(113)
7,- 7ko • - k, •.
(114)
and
5Wechose kl andksin thesamedirection asksandkp,respectively, sothatsimplifying relationships llke equations(110) and (111) couldbe used.
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
65
Both7pand% arewavenumber components perpendicular to the wavenumber componentskp and k0 which parallel the receiverand sourcelines, respectively 6.
The doubleintegralin equation (112_) isjust a 2-D Fouriertransform of the modelfunctionM(x, z) definingM(k:,, k,) alongcirculararcsin the
k• - k, plane,the placement of the arc depending uponk0 andk•?. At this point we must make a brief digressionto further explain this concept.
Figure 18(a) showsfour down-holesourcepositionslabeled $•, $2, and $4 in one well and four down-hole receiver positions labeled R•,
Rs, and R4 in another well. A line diffractorlies at the location (x, z). If
] represents the directionof a planewavepropagating froma sourceto the point(x, z), thenby equation(105) the associated wavenumber components (7•, k•) can be written,
koi -
7•:•+ k•.,
(115)
where • and •. are unit vectors in the positive x-direction and z-direction, respectively.Now, as a mathematicalabstraction,let • point in the opposite directionof a planewavetravelingawayfrom the sourceto the point (x, z),
or • - -{. By equation (115)andfromthefactthat kx- -ks and7• - 7: by equation(110), we may write the components of koõas (116) Figure 18(a) showsthe unit vector g for eachof the four sourceslabeled with subscriptsas gx, •2, •a, and g4. Now let •) representthe direction of a plane wave propagating from the
point(x, z) towardoneof the receivers in Figure18(a). By equation(106) and by the fact that equation(111) definesk•. = kr and 72 = 7r, we may write the componentsof ko•) as
(117)
Figure 18(a) showsthe unit vector• for eachof the four receiverslabeled with subscriptsas •)x, •2, •)a, and •4.
Figure18(b) showsthe wavenumber domainrepresentation of the source wavenumbervectorsko•x, ko.q2,ko.qa,and ko.•,•. Similarly,Figure 18(c) showsthe wavenumberdomain representationof the plane wavestravel-
ing from the point (x,z) to the four receiversas the vectorsko•l, ko•., 6Refer back to equation (107) for the definitionsof kr axtdk•.
7Equation(112) is analogousto equation(12) definingthe projectionslicetheorem for ray tomography. Instead of arcs, the projection slice theorem defines straight lines in the k: - k: plane as shown in Figure 5.
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66
CHAPTER 3. SEISMIC DIFFRACTION
TOMOGRAPtIY
•x
s1
R1
Zs2^ • ß
p
•^ R2
^2
S4
R4
(a)
Kz
Kz
(b)
(c)
FIG. 18. The wavenumber components (k,,%) and (kr,7r) in equation (112) dependuponthe sourceand receiver locations, respectively, relativeto thelinediffractor in themodel.(a) Foursources (soliddots) and four receivers(open dots) are shown. The unit vectorsradiatefrom a line diffractorand point towardtheir respectivesourceor receiver.The unit vectorsfor the sourcespoint oppositeto the unit vectorsof the incidentwaves. (b) The wavenumbervectorsfor eachsourceare shownon
the k• - kz planeas terminatingalonga dashedcirclewith radiusko. If sources are deployed from +oo to -oo in the boreholedirection,thenthe
entiredashed circleterminus is defined.Equation(116)defines thecomponents(k•, %) for thesewavenumber vectors.(c) The wavenumber vectors for eachreceiver areshown onthe k•- kz planeasterminating alonga dashedcirclewith radiusko. As for the source,the entire dashedcircle terminusis definedif receiversare placedfrom +c• to -c•.
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3.3.
GENERALIZED
PROJECTION
ß
SLICE
,ks
THEOREM
67
kP •K x
p,i•-Kx ß
ß
o
Kz (a)
(b)
F•c. 19. Wavenumber vector componentsfor the crosswellconfiguration. (a) The componentsof a sourcewavenumbervector kog are shownas pro-
jectionsontothe k•-axis and kz-axisasdefinedby equation(116). (b) Similarly, the components of a receiver wavenumber vector kot5are shown as
definedby equation(117).
kof)3, and koO4. If the sourcescan be deployedfrom +c• to -c• along the source well, then the wavenumber domain representation of the source wavenumbervectors ko.• are vectors pointing to the dashedsemicirclein
Figure 18(b). Likewise,if the receiverscan be deployedfrom +c• to -c• in the receiver well, then the wavenumber domain representation of the plane wavestraveling to these receiversare vectors pointing to the dashed
semicirclein Figure 18(c). The components of ko•,and ko• definedby equations(116) and (117) are showngraphicallyin Figures 19(a) and 19(b), respectively.At this point we add the two vectorskof,and koO which givesthe vector equation,
=
(7r -7s)• + (kr + k•)•..
(118)
The dot productbetweenko(g+ •) and positionvectorr pointingto the line diffractorat (x, z) in the modelis
ko( +
. ,: -
+
.
+
=
k•z +
=
(7r - 70)x 4-(kr 4- ko)z.
(119)
At this point we note that equation(119) multipliedby -j formsthe terms
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68
CHAPTER
3. SEISMIC
DIFFRACTION
TOMOGRAPttY
for the exponentials in the integrandof equation(112). Substituting equation (119) intoequation(112) resultsin the moreinterpretable equation,
tS(d•,k•;dp,kp)
(120)
where/•/(k•, kz)isthe2-DFourier transform ofthemodel function M(z,z), or
=
+ +
(121)
Usingthe last result,equation(112) may be written morebrieflyas
P(do ko kp)-- --kø2 eJ(7pdp - %d,) ~ + f>)]. ; dp, M[ko(f, ' 4 %7p
(122)
The integrationlimits in eitherthis equationor in equation(120) must obeythe spatiallimitationsset by equations(105) and (106). That is, the imagingareais restrictedto ds< x < dp. We do thisby settingM(z, z) to zeroin the integrandwhenthe integrationis outsideof the imagingarea, whichis the sameasrestrictingthe integrationto only the imagingarea.
Equation (122)establishes a relationship between /5(ds, ks;dp, kr),the doubleintegralFouriertransformof the data functionalongthe sourceand
receiver profiles, andg•[ko(•+ •)], the2-D Fouriertransform ofthemodel function for the diffraction tomographyproblem. Considerthe crosswell
configuration in Figure20(a). The unit vector• pointstowardthe single sourcefrom the linescattererlocatedat (x, z) whilereceiversdeployedfrom +cx>to -cx>alongthe receiverwell causethe unit vector• to havethe range within the dashedsemicircle.The locusof the quantityko(• + •) in the wavenumberdomainis a semicirclewith radius ko centeredon the point (-ko, 0), asshownin Figure20(b). Note that the coverage in the wavenumber domainchanges asthe source is moved.For example,Figure20(c) showsthe unit vector• for the source
at -cx>.Sincethe receivers areat thesamelocations asin Figure20(a)the coverage of • remainsthe same.The locusof the quantityko(.•q-•) in the wavenumberdomainis shownin Figure 20(d) as a semicirclewith radius
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
69
,.
ß ource -
source
-
at
O0
-
',
,'
•o• .ource
at
+oo
(a)
{e)
(c)
gx
gx
Kz
(b)
Kz
Kz
(d)
FI6. 20. Coverage providedthe modelfunctionM(k=, k,) bythreedifferent sourcelocationsrelative to the receiverboreholewith receiversranging in depth from +cx>to -cx>. The wavenumbervector ko• is shownin (a), (c), and (e) for sourcesat the samedepth as the line scatterer,at -cxv, and at +oo, respectively. The termini for the receiver wavenumbervectors are shownas the dashed circles. For the fixed receiverlocations, the solid circle
arcsin (b), (d), and (f) are the coverages providedof the modelfunction
37/(k=, kz)bythesource locations in (a), (c),and(d),respectively. Thesolid arcsare a resultof the vectorsum, koi + kolS,as definedin equation(122) on the right-hand side.
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70
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPItY
ko centeredon the point (0,-ko). Figure20(e) showsthe unit vector• for the sourceat -]-•x)and the same receivercoverage. Now the locusof the quantity ko(• + •) in the wavenumberdomainis the semicirclecenteredon
(0, +ko) shownin Figure20(f). We can use equation(122) in the diffractiontomographyproblemto compute the Fourier transform of the unknown model function M on the
semicircular locidefined by ko(•+ •)in Figures 20(_b), 20(d),and20(f). The loci definedby ko(O+ •) in the evaluationof M are calledthe slice of the 2-D
Fourier
transform
of the model function.
The
data function
P(d,, z,; dr, zr) representing the observed scatteredwavefieldalongthe receiverline is alsocalledthe projection.Equation(122) links the slicesof M with the projections, resulting in its name, the generalizedprojection
slicetheorem. Thetheorem states that15,(d,, k,;dr,kr), thedouble integral Fourier transformof the data function along the sourceand receiver lines, is equal to M[ko(g + •))], the 2-D Fourier transformof the model function evaluated along the semicircularslice, multiplied by the quantity
k• eJ(7pdr We may extend the situation shown in Figure 20 by deployingmany
sources from+½x•to -½x•in the sourcewellasshownin Figure21(a). Using more sources results in more slices in the wavenumber
domain as illustrated
in Figure 21(b). The rangeof all possiblekoõis depictedby the dashed
line. We useequation (122)to compute the modelfunctionAYl[ko(õ + •)] along the solid slicesin the k, - kz plane. For sucha multisourcemultireceiver configuration,the 2-D Fourier transform of the model function
can be recoveredonly within the portionsof the two disk-shapedregionsof the k• - kz plane coveredby the solid semicircles.
Most multisource-multireceiverconfigurationsprovide only a partial coverageof M(ka,,kz) leaving parts of the model uncertain. Information from sliceson the entire k•- k• plane is required to uniquely de-
termineM(x, z), whichis computedfrom the inverseFouriertransformof M(k•,, k•). Most of the informationprovidedby the crosswell configuration in Figure 21 is in the directionof the vertical wavenumberkz. Only longer wavelengthinformation(smallerwavenumbervalues)about the modelis provided in the horizontal wavenumberdirection k•. Thus, with the crosswell configurationwe expect good vertical resolution and poor horizontal resolution. Also, the resultingmodel is nonuniquebecausewe may arbitrar-
ily define a modelspectrum to "fill-in"theundefined partsof/l•/(k•,k,).
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
71
•x
()
()
()
() )
ß
8ource
locus of ko 8
0
receiver
•o•.. o• • o(• + •)
(a)
(b)
Fla. 21. (a) The situationin Figure20 is expandedby placingsourcesin depthfrom+oo to -oo. (b) The dashedcircleis the terminusof the vector ko• for the sourcesin (a). The solid circlesare the resultingvectorsum
ko(•q-I5).Nowthemodelfunction/f4(k=, k,) iswelldefined withinthetwo circular regionsof solid line coverageusing the generalizedprojection slice
theoremfor the crosswellconfigurationdefinedby equation(122).
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72
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
ds
dp
z
.
source
o receiver
FIG. 22. Source-receivergeometry for the VSP experiment.
We restrict
our reconstruction of the modelfunctionto x < dr and z > d,.
3.3.2
Vertical Seismic Profile Configuration
Now we derivethe generalizedprojectionslicetheoremfor the vertical seismicprofile (VSP) source-receiver configurationin Figure 22. Sources (soliddots)are deployedalonga line parallelto the x-axis at a constant vertical location z, = d,. Receivers(open dots) are deployedinsidea receiverwell parallelto the z-axisat a constanthorizontallocationxr = dr. Thus, the positionvectorsfor the sourcelocation r• and receiverlocation
rr canbe expressed in termsof theirrespective coordinates (x•,d•) and (dr,zr). Weshallassume thatthedomainof themodelfunction M(x, z) is restrictedto x < dr andz > d•. The derivationfor the VSP configuration followscloselythe derivationof the crosswellconfigurationof the last section. In fact, we end up with the sameform of the generalizedprojection
slicetheoremgivenby equation(122), but with its applicationbasedon the VSP geometry.
Substitutingthe VSP coordinatesinto equation(101) for the position
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
73
vectorsrs and r r gives
P(zs,ds;dr,zr) -
-ko•
M(z,z)
(123)
Again we chooseequation (62) to representthe Green'sfunctionsin a 2D medium so that the acoustic sourcesare infinite line sources,the diffractors in the model are infinite line diffractors, and the receiversare infinite
line receivers. Thus, we are once again dealing with cylindrical acoustic waves in a 2-D medium.
The developmentof the Green's functions here closelyfollows the discussionof the previous section and will not be presented in detail. The
Green'sfunctionG(z,z I z,,d,) for the infinite line sourceat (z,,d,)is found by substitutingthe appropriatecoordinatesinto equation(62) for position vectorsand performing a plane wave decompositionon the zer•
orderHankel function H••) ofthefirstkind.Theresulting integral formof the Green's function
is
Jff• •eJ[k•(xx,)+7•(z - d,)]dk•, (124) G(x, zlx•,d,) = 4• 7• for z > ds. Note that, as before, the wavenumber component k• is taken
along the sourceprofile which for the VSP geometryis in the direction of the x-axis. The wavenumbercomponent7x is alongthe z-axis and is defined
as7• - v/ko • - k•. A similardevelopment oftheGreen's function G(dr,zr I z, z) for an infinite line diffractorat (z, z) whoseenergyis recordedby a receiverlocatedat (dr, zr) givesthe equation,
j f: 1eJ[k•.(z r_z)+
- z)lak2, (125)
for z < dr. Herethe wavenumber componentkg.is takenalongthe receiver profile which for the VSP geometryis in the direction of the positive z-axis. The wavenumber79. is in the positive x-axis direction and is defined as
Finally, we must write down the equation for taking the Fourier trans-
formof the datafunctionP(xs, ds;dr, zs)in equation(123)alongthesource line (xs) and the receiverline (zr). This relationship is foundby first taking the Fouriertransformof P(xo,ds;dr, zs) with respectto zs followedby a secondFouriertransformwith respectto zr whichproducesthe double integral Fourier transform,
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CHAPTER
74
3.
P(•:, , d, ;ct,, , •:,,)
SEISMIC
DIFFRACTION
TOMOGRAPHY
/_:,o /_•P(z,, d,; dp, zp) x e-jk'x' e-jkpzp dxsdzp,
(126)
wherek, and kp are the wavenumbers of the Fouriertransformalongthe source line and the receiver line, respectively. Note that k• is in the same
directionas k, and k= is in the samedirectionas We continue the derivation of the generalized projection slice theo-
rem for the VSP configurationby substitutingequation(123) into equa-
tion(126)to geta relationship between thedatafunction P(k,,d,;dr, and the modelfunctionM(x, z). Carrying-outthis substitutiongivesthe equation,
where we have changedthe order of integration. Next we substitutethe Green'sfunctionsdefinedby equations(124) and (125) into equation(127), regrouptermswith similar integrationvariables, and interchangethe order of integration for the two innermost integrals over each Green's function, yielding the equation,
•5(k, dp, kp) =k• 1 /_• /_©M(a:, 'd,; 4 (2a')' oo z)
{/_
(128)
}
• • d[-•z+•(d.- •)]
•-i(• - •)Z•z.d• d•dz
The integrals in equation (128)withrespect to the variables x• andzr are easily evaluated in terms of Dirac delta functions as
'øe-J(k• +k,)Xsdx '
(129)
_=ø e-J(k p- k2)zp dz r
(130)
and
2•r5(kp- k:•).
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
75
Substitutingequations(129) and (130)into equation(128) and evaluating the integralswith respect to the variableskl and k2 (rememberingthat • - •o• -•i •na • - •o• -•) give•,
'
ß
'
=
--
4
7•7p
•
•
M(•, •)
• •-J(•. +7•)•-J(•-%)•&,
(•s•)
where7, and7p are definedby equations(113) and (114). Rememberthat the wavenumbercomponentsk, and kp parallel the sourceand receiver profiles, respectively. As we saw in the previoussection,the exponential terms in the integrand
of equation(131) may be rewritten• the dot productof the vectorsum, •o(• + •)
: = :
(-•i - 7•) + (7•i + •) (•.i- %•) + (•i + %•) (•. +•)i+(%-%)• k•+k•,
(•3a)
with the positionvector r pointing to (x,z). The unit vector • points in the oppositedirectionof a plane wave traveling from a sourceto (x, z) and the unit vector • points in the direction of a plane wave propagating kom (x, z) to a receiver.The components of ko• and ko• are showngraphically in Figures23(a) and 23(b), respectively.Sourcesare deployedkom +• to -• alongthe sourceline parallel to the x-axis and receiversare deployed kom +• to -• alongthe receiverline parallel to the z-axis. This geometry resultsin vectorspointing to the d•hed semicirclesin Figure 23. Performingthe dot productbetweenequation(132) for the VSP configuration and the position vector r gives,
•o(a + •).•
-
(•i
+ •).
(•a + •)
=
k• x + k• z
=
(k, + 7p)x+ (kp- 7,)z.
(133)
Note that equation(133) multipliedby -j is just the term in the exponential of the integrandin equation(131). Substitutingequation(133) into equation(131) resultsin the equation,
ß
-
•
x e-J(k•x + k•z)dxdz
M(•, •)
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76
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
kp
:
ks
:Kx
Kz (a)
(b)
FIG. 23. Wavenumber vector componentsfor the VSP configurationas de-
finedby equation(132). (a) The components of a sourcewavenumber vector kogare shownas projectionsonto the k•-axis and kz-axis;and likewise,(b) the componentsof a receiverwavenumbervector koI3.
(134)
where37/(k•,kz)isthe2-DFourier transform ofthemodelfunction M(x, z) takenoverthe imagearearestrictedto x < dr and z > d,. Notethat equation (•a4) is the sameas equation(122) for the crosswell configuration exceptfor the directionsover which the doubleintegral Fourier transforms
of the data functionP(x,, z,; zr, zp) are taken.We requirethat the transforms be taken along the sourceand receiverprofiles,which for the VSP configurationare differentfrom the crosswellconfiguration. Equation(134) is the generalizedprojectionslicetheoremfor the VSP
configuration. It statesthat P,(k,,d,;dp,kr), thedoubleintegralFourier transformof the data function, taken along the sourceline and the receiver
line, is equal to M[ko(,• + 15)],the 2-D Fouriertransformof the model function, evaluatedalong the semicircularslicemultiplied by the quantity -Y
%%
'
Figure 24(a) showssourcesand receiversdeployedfrom +oo to -oo alongtheir respectiveprofilesfor the VSP configuration.Figure24(b) is a representationof the model coverageprovidedby the VSP configuration
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3.3.
GENERALIZED
PROJECTION
SLICE
THEOREM
77
z
Kz ^
ß source
locus of ko 8
o receiver
locusof/•• ( • + I•)
(a)
(b)
FIG. 24. (a) Sourcesand receiversare extendedfrom +cx>to -cx> along their respectiveprofiles.(b) The dashedcircleis the terminusof the vector ko• for the sourcesin (a). The solidcirclesare the resultingvectorsum
ko(•+/3). Themodelfunction iff(k•,k,) is welldefined withinthezone of solidline coverageusingthe generalizedprojectionslicetheoremfor the VSP configuration definedby equation(134).
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78
CHAPTER
3.
SEISMIC
DIFFRACTION
0
TOMOGRAPIIY
.X
ß
source
o receiver
FIG. 25. Source-receivergeometry for the surfacereflection experiment, where in the marine data acquisition case the depths are measured from sea level.
We restrict
our reconstruction
of the model function
for the
geometryshownhere to z • dp.
wherethe model reconstructiondefinedby equation(134) is restrictedto
pointsx • dpandz • d,. The rangeof all possible ko• is depictedby the dashedline in Figure24(b). We useequation(134) to computethe model
function]l•[ko(•+ •)] alongthe solidslicesin the k• - kz plane.Note that theVSPconfiguration provides a different coverage of ]17f(kx, kz) than the crosswellconfigurationin Figure 21. Thus, we shouldexpect different estimatedmodel functionsM(x, z) from each configurationeven though the true model is the same. As with the crosswellconfiguration, resolution
varieswith directionand the resultingmodel is nonunique.
3.3.3
Surface Reflection Configuration
The last source-receivergeometry we will derive the generalizedprojection slicetheoremfor is the surfacereflectionconfigurationin Figure 25.
The sources (soliddots)aredeployed alonga lineparallelto the x-axisat a constantverticallocationz• - d•. Similarly,the receivers(open dots) are deployedalong a line parallel to the x-axis at a constantvertical location
zp= dp. Thus,the positionvectorsfor the sourcelocationr• andreceiver locationrp canbe writtenin termsof theirrespective coordinates (xs,ds) and(xp,dp).Weshallrestrictthedomainof themodelto depthsbelowthe deeperof the two, d• or dp. As with the previoustwo configurations, the derivationheregeneratesa similar form for the generalizedprojectionslice theorem, but with its applicationbasedon the surfacereflectiongeometry.
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3.3.
GENERALIZED
PROJECTION
79
SLICE THEOREM
We beginby substitutingthe surfacereflectioncoordinates for the positionvectorsr• andrr intoequation(101)whichgives
x
zI
4, I
z)aaz.
We assume a 2-D mediumandchoose equation(62) to represent theGreen's functionsfor the infinite-lineacousticsources,infinite line diffractors,and the infinite line receivers.Thus, all acousticwaveswill be cylindrical.
The Green'sfunctionG(x,z I x•, d•) for the sourceis the sameas the VSP'sGreen'sfunctiondefinedin equation(124) sinceboth configurations have the same sourcegeometry.Thus,
J/5 1eJ[k•(xx)+?•(zd•)ldk•, (136) for z > d,. The wavenumber component k• is takenalongthe sourceprofile which is in the direction of the x-axis. The wavenumber component 7• is
alongthez-axisandis defined 7• - V/ko 2 - k•2. Sincethe receivergeometryis similarto the sourcegeometryweusethe same form of the Green's function as for the sourcesgiving
3__'/5 1eJ[k9.(x r_x)-72(d r- Z)]dk 2(137) wherek2 is the wavenumbercomponentparallelto the receiverline and 72
is alongthe z-axis,where72 - V/ko • -k•. Notethat -?2(dr - z) is used sincewe removedthe absolutevalue in the plane wave decompositionof the Hankel functionfor z > dr.
Proceeding with the derivation, wewritedownthe equation for taking theFouriertransform ofthedatafunction P(x•, d•;xr, dr) in equation (135) alongthe sourceline(x•) andreceiver line(xr). The relationship is found by first takingthe Fouriertransform of P(x•, d•;xr, dr) with respectto xs, followed by a second Fouriertransformwith respectto xr whichproduces the doubleintegral Fourier transform, P(zo,d•;zr,
x e-Jz•koe-Jxrkrdx•dxr,
(138)
wherek, andkr arethewavenumbers of theFouriertransforms alongthe sourceline and the receiverline, respectively.As with previousconfigurations,k• is in the samedirectionas k, and k2 is in the samedirectionas kr ß
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CItAPTER
8O
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
Next, we substituteequation(135) into equation(138), giving
P(k,,d,;kp,dr) -
-k•
M(x,z)
(139)
x
•(•, • I •., a.)•-j•'•'
x
6(•:,,,4, [ •:,z)•-j•'"
where we have changedthe order of integration. Next, we substitutethe Green'sfunctionsdefinedby equations(136)
and (137) into equation(139), regrouptermswith similarintegrationvariables, and interchangethe order of integration for the two innermostintegralsovereachGreen'sfunction,yieldingthe equation,
15(k '' d,;kr, dr) = k•o 1 • 4 (2•r)' x
x
M(z' z)
(140)
{f_•1•'It" +'•'(' - •')1 j'_"-'(•'+t')' } --
e
--
sdzsdk•
e-j(kr - k=)zPdzrdk= dzdz.
The integralsin equation(140), with respectto the variablesx, and are easily evaluated in terms of Dirac delta functions as
øø e-j(k• +k,)Z,dx ' 27rS(k•+ k,),
(141)
j e-j(kp - k:•)%, 2•rS(k•,- k2).
(142)
and
Substitutingequations(141) and (142)into equation(140) andevaluating the integralswith respectto the variablesk• and k= (rememberingthat 7• - ko • -k• and7• - ko • -k•) gives,
ko •ej(-%dp - -/,d,)oo M(x, z) 4 7,% oo oo (143)
where7, and 7p are perpendicular to k, and kp.
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3.3.
GENERALIZED
PROJECTION
ß
SLICE
THEOREM
t•s':-Kx
81
t•p •Kx
•z
•:z
(a)
(b)
FIG. 26. Wavenumber vector componentsfor the surfacereflectionconfigu-
rationasdefinedby equation(144). The dashedsemicircles arethe termini of the wavenumber vectorsfor sourcesand receiverspositionedfrom -oo to
+cx•alongthe x-axis. (a) The components of a sourcewavenumber vector kogare shownas projectionsontothe kx-axisand kz-axis;and likewise,(b) the componentsof a receiver wavenumber vector koP.
As we saw for the earlier configurations,the exponential terms in the
integrandof equation(143) may be rewritten as the dot productof the vector sum,
•o(• + •)
=
(-• - 7•.) + (•.• - 7•.•.) (•0• - •0•.) + (• - •.)
= =
(•o + •)• + (--r• --r•)• kx• + kz•,
(144)
with the positionvectorr pointingto (x,z). The unit vector.qpointsin the oppositedirectionof a planewavetravelingfrom a sourceto (x, z) and the unit vector f) points in the directionof a plane wavepropagatingfrom (x, z) to a receiver.The components of koõand ko•)are showngraphically in Figures26(a) and 26(b), respectively.Sourcesare deployedfrom +oo to -c• along the sourceline parallel to the x-axis and receiversare deployed from +c• to -oo along the receiverline also parallel to the x-axis. This geometryresultsin vectorspointing to the dashedsemicirclesin Figure 26. Performingthe dot product betweenequation(144) and the position vector r gives,
•o(• + 0).,,
-
(•
+ •,•)-(•
+ z•)
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82
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
=
k•z + kzz
=
(k, + kr)z + (-7, -7r) z.
(145)
Notethat equation(145) multipliedby -j isjust the term in the exponential of the integrandin equation(143). Substitutingequation(145) into equation(143) resultsin the equation,
_ ko •ej(-7rdr - %d,)oo M(x, z) 4 %%0 oo x e-J(k•x + k•z)dxdz =
_ 4
=
-4
- %d,)_
• •j(-•dp - •)
.
U[ko(• + •)],
(146)
where 217/(k•, k.) isthe2-DFourier transform ofthemodel function M(x, z) takenoverthe imagearearestrictedto the greaterof z > d, or z >dp. Equation(146) is the generalized projectionslicetheoremfor the surface
reflection configuration. It states that['•(k,,d•;kr,dp),thedouble integral Fourier transform of the data function, taken along the source line and
the receiver line,is equalto 20[ko(S + iO)l,the 2-D Fouriertransform of the model function, evaluatedalong the semicircularslicemultiplied by the
quantity --•eJ(-7pdp %%0- 7sds) ' Figure 27(a)shows sourcesand receiversdeployedfrom alongtheir respectiveprofilesfor the surfacereflectionconfiguration.The rangeof all possibleko• is depictedby the dashedline in Figure 27(b). We useequation(14{5)to computeM[ko(g+ •)] alongthe solidslicesin the k: - kz plane. The surfacereflectionconfigurationprovidesa different coverage of M(kz, k,) than the crosswell and VSP configurations andtherefore we shouldexpect differentestimatedmodelsfrom each configuration. Once again,each configurationprovidesa differentresolutionand degreeof nonuniqueness dependingupon how the configuration"fills-in" the model spectrum and how much of the model spectrum the configurationleaves undefined.
3.4
Acoustic Diffraction Tomography
The generalizedprojection slice theorem was derived in the previous sectionfor the crosswell,VSP, and surfacereflection configurationswhich
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3.4.
ACOUSTIC
DIFFRACTION
•
TOMOGRAPttY
83
x
source
receiver
........
locus of/• o s
locus of/• o (s + p)
(a)
(b)
F•G. 27. (a) Sourcesand receiversat the samedepth are extendedfrom +c• to -c• in the directionof the x-axis. (b) The dashedcircle is the terminusof the vectorkoõfor the sourcesin (a). The solidcirclesare the
resulting vector samko(g+I5). Themodel function A7/(kx, k,) iswelldefined within the zoneof solid line coverageon the -k• half plane.
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CHAPTER
$4
3.
SEISMIC
DIFFRACTION
TOMOGRAPtIY
correspond to equations(122), (134), and (146), respectively.Foreachconfigurationthe theoremestablishesa relationshipbetweenthe doubleFourier transformof the data functions takenalongthe sourceand receiverprofiles and the 2-D spatial Fourier transform of the model function. Further analysisof the generalizedprojection slicetheorem showsthat each sourcelocation definesthe model function along a circular arc in the ks - k• domain, as depicted in Figures 21, 24, and 25 for each source-receiverconfiguration. To obtain the model function M(x, z) we must take the inverseFourier
transform of.•/[ko(•+•)] whichisnotreadilycarried-out sinceAY/[ko(•+•)] is defined by circular arcs. Thus, in this section we develop two image reconstructionalgorithmsto handle this problem called the direct-transform diffraction tomography and the backpropagationdiffraction tomography.
Thedirecttransform methodinvolves estimating thevalues of •[ko(•,+ P)] on the k•- k• coordinate plane, which is analogousto the direct-transform ray tomography method presentedin Section 2.2.2. On the other hand, the backpropagationmethod takes a more elegant mathematical approach to achieve the same result. The backpropagation method presented here is analogousto the backprojection ray tomography presentedin Section2.2.3.
3.4.1
Direct-Transform
Diffraction Tomography
The generalized projection slice theorem enables us to compute the 2-D Fouriertransformof the model function M[ko(• + •)] from s•tte•ea wavefielddata. Therefore,the unknownmodel functionM(•, z) can be ob-
tainedaslongaswecanperform aninverse Fourier transform onAYl[ko(• + •)]. However,most inverseFourier transformalgorithmsuse Cartesian coordinates • which meansthey can handle the inverseFourier transform
from?l•f(k•,k•) to M(z,z), butnotfrom?l•f[ko(• + •)j to M(z,z). Figure 28 illustratesthe differencebetweenthe loci of M[ko(• + •)l (solid
dot•)•nd thelociof •(k,, k•) (opendot•)for a crosswell configuration. Clearlythe lociof the tworepresentations of AY/do not coincide andwe cannot directly make use of the inverseFourier transform.
An obvious,brute forcesolutionis to estimatevaluesof M(k•, k•) on a
rectangular kz - k• gridfromvalues of AY/[(• + •)] obtained alongcircular arcs using the generalizedprojection slice theorem. As mentionedin Section 2.2.2, one must exercise caution in performing an interpolation since the operation may introduce error which can obscure the true solution. However, once the values of M are found on a k• - k• rectangular grid,
we can find the unknownmodel function M(x, z) through a 2-D inverse sKeep in mind that in diffraction tomographythe data functionconsistsof measured projections of scattered energy.
•Namely, (x,z) in the spacedomainand (k•, k,) in the wavenumberdomain.
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3.4. ACOUSTIC DIFFRACTION TOMOGRAPHY
o
o
o
85
o
o
o
o
o
K,x o
o
o
o
o
o
o
0 Iocu• of (k,•, k•.) grid
FIG. 28. The generalizedprojectionslicetheoremdefinesthe modelfunc-
tionM(k•,,kz) at thesoliddotsalongcircular arcsin theks- kzplane.To gettheestimated modelfunctionM(x, z) the2-D inverse Fouriertransform
mustbe takenof M(kr,k,) defined at points(opendots)on a Cartesian grid. Direct-transform diffraction tomography definesthe opendotsfrom the solid dots by interpolation.
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86
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPttY
Fourier transform. This is called direct-transform diffraction tomography and can be summarized in five steps' Step 1: Acquire tomography data by probing the target with acoustic waves and record the scattered
wavefield
informa-
tion fromthe target, represented by the data functionP(x, z). Step 2: Take the double Fourier transform of the data function
P(x, z) along the sourceand receiverlines, which can be representedby
=
,of_,o
't,+
'
(147)
Here is and ir are the distancesalongthe sourceand receiver lines, respectively. Equation (147) correspondsto equation(107) for the crosswellconfigurationwhen is - zs and ir = zr; to equation(126)for the VSP configuration when I• -- x• and ir = zr; and to equation(138) for the surfacereflectionconfiguration whenl• = x• and lr - xr. Step 3: Compute the 2-D Fourier transform of the unknown
modelfunction A7l[ko(k 4-•)] alongcircular arcsusingthe generalizedprojection slicetheorem,
ll•l[ko(• -t-•)] = 47o%, pdp4-'y,d,)•5(k,, kp). k• ø e- j (:l:'y
(148)
Equation(148) corresponds to equation(122) for the crosswellconfiguration, equation(134) for the VSP configuration, and equation(146) for the surfacereflectionconfiguration. Step 4: Perform a 2-D interpolation on the values of M from
thegeneralized projection slicetheorem to determine AT/on the rectangular k,-
kz grid, (149)
Step 5: Take the 2-D inverseFouriertransformof M(k•,kz) to obtain M (x, z),
M(zz)= I
"
1171(k• k,)e j(k•z+k'Z)dk•dk, (150)
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3.4.
ACOUSTIC
3.4.2
DIFFRACTION
TOMOGRAPttY
87
Backpropagation Diffraction Tomography
We saw that direct-transformdiffractiontomographyestimatedvalues
ofthemodelfunction •[ko(õ + •))] ona rectangular k, - k, gridenabling usto taketh• inverse 2-D Fouriertransform of • to getthe estimated modelfunctionM(x, z). The interpolationsteprequiredis computationally slow and the possibilityof introducingerrors into the interpolatedmodel function is very real. In this section we introduce the more mathemat-
ically elegant method of backpropagationdiffraction tomography. With this methodthe requiredcoordinatetransformationfrom M[ko(• + •)] definedon semicircles to M(kx, k,) definedon a rectangulargrid is performed while taking the 2-D inverseFourier transform thereby obviatingthe need for interpolation.
The derivation begins with the 2-D inverseFourier transform of the modelfunctionM(k•, k•) givenby
1• •ß ß•(k•,k,)eJ(k•z +k,Z)dk•dk, M(x,z) = 4• . (151) Since•[ko(• + •)] is defined in termsof thewavenumbers (k•, k•), wewill performa changeof variablesfrom (k•, k,) to (k•, kr) on the right-hand sideof equation(151). This will permit us to directlytake the 2-D inverse
Fouriertransform of the modelfunction•[ko(• + •)] withouthavingto perform an interpolation • required by the direct-transform diffraction tomography.Equation(151) in the form for this changeof variablesis
M(x z) = 4•21 •• ' x I
•[ko(• +•)]eJ[k•(k•, kr)z +k,(ks, kr)z ] I
(152)
where•[ko(i + •)] is themodelfunction fromthegeneralized projection slicetheorem,k•(k,,kr) and k,(k,,kr) are the wavenumbers expressed in termsof the variables(k•, kr) , and J(kr, k, [ k•, kr) is the JacobJan relating dk,dkr with dk•dk,. The JacobJan can be expressed in termsof the determinant,
J(k• k,]k,,kr) ' Ok:Ok,
Ok•Ok:
= Ok: Ok r Ok r Ok,'
(153)
The form of equation(152) is • far • we can go without selectinga particular source-receiver configuration.In the followingderivationswe shall
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88
CHAPTER
3. SEISMIC DIFFRACTION
TOMOGRAPttY
complete thechange of variables for thethreesource-receiver configurations presented in the previous section.The resultingequationfor eachconfiguration is in a form which permits use of a 2-D inverseFourier transform
algorithmandthe resultis backpropagation diffraction tomography. Crosswell Configuration.--The modelfunctionfor the crosswell configurationisdefined bythegeneralized projection slicetheoremin equation (122) which may be rewritten as
.l17f[ko(• +f))]= 4%7r r - %d•)P(d• ko • e-j(%,d ,k•;dr,kr) . (154) From equation (119) we see that the coordinatesystem transformation
between(k•, kz) and (ks,kr)is givenby
Substituting this coordinate systemtransformation into equation(153) givesthe Jacobianfor the crosswellconfiguration,
J(k•,kz Ik,,kr)= k,%,+k•,%.
(156)
Substituting equations (154),(155),and(156)intoequation(152)gives,
M(z 1/••'/_• ,z)= 4•.2 •'4%7r •-o•e-j(7rdr_%d•) -
+
+
x [ks 7r+krYs [dks dkp, and rearrangingtermsgivesthe equationfor backpropagation diffraction tomographyfor the crosswellconfiguration,
(157)
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3.4.
ACOUSTIC
DIFFRACTION
TOMOGRAPttY
89
VSP Configuration.-- The model function for the vertical seismicprofile configurationis definedby the generalizedprojectionslicetheoremin equation (134) which may be rewritten as
•[•o(•+•)] = 4,0,• ko • e_j(,• _,,•,)p(•,,• ;• ,•) . (•S8) From equation (133) we see that the coordinatesystem transformation
between(k•, k•) and (k•, kv)is givenby
k•(k,, k•) =
k, + 7•
= •+•-•,•na = kv- •k•- k•.
(159)
Substitutingthis coordinatesystemtransformationinto equation(153) givesthe Jacobianfor the VSP configuration,
Substitutingequations(158), (159) and (160)into equation(152) gives,
M(x, z) = 4•• ß ß • e x [k•k v+%7v ]dk•dkv, and rearrangingterms gives the equationfor backpropagationdiffraction tomographyfor the VSP configuration,
M(•,•) = •
ß ß
k•
'
x eJ[(k, + 7v)x + (kv- 7,)z]ak, akv.
(161)
Surface Reflection Configuration.-- The model function for the surface fiection configurationis definedby the generalizedprojection slice theorem in equation(146) whichmay be rewritten •
•[ko(• +P)] = 47•7• e-J(-7•d• - 7•d•)p(k• d,'k•,dp). (162)
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CHAPTER
9O
3.
SEISMIC
DIFFRACTION
TOMOGRAPHY
From equation (144) we see that the coordinatesystem transformation between(k•, kz) and (ks,kp) is givenby
--
ks+ kr, and
=
-%
- 7r
(163) Substitutingthis coordinatesystem transformationinto equation (153) givesthe JacobJanfor the surfacereflectionconfiguration,
s(•,• I k•,•) = kr•'• - k•'•r.
(164)
Substitutingequations(162), (163) and (164)into equation(152) gives,
and upon rearrangingterms we get the equationfor performingbackpropagationdiffractiontomographyfor a surfacereflectionconfiguration,
1f_•/:I•'r,-/•,•'• Ip(/• •) (165) x eJ[(ks + kp)x + (-% - 7p)Z]dk,dk•, '
Equations(157), (161), and (165) are usedin backpropagation diffraction tomographyimage reconstructionalgorithmsfor the crosswell,VSP, and surfacereflectionconfigurations,respectively.
3.5
Summary
1. The model used for acousticwave scattering was the Helmholtz form of the acousticwaveequationfor a constant-densitymedium of vari-
able velocityC(r) givenby
IV • + •:=(•.)]•,(•.) -
o,
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3.5.
SUMMARY
91
where Pt represents the total wavefield which contains both the incident wavefield from the source and scattered
wavefields from inho-
mogeneitiesembeddedin a constant-velocitybackgroundmedium. 2. We treat the inhomogeneitiesas secondary sourcesand write the acousticwaveequation as the inhomogeneous differentialwaveequation,
IV +
-
where ko is the constant wavenumbermagnitude of the background
mediumand M(r) is the modelfunctionfor diffractiontomography defined as a perturbation from the constant backgroundvelocity Co given by,
.
We found two integral equation solutions to our formulation of the acousticwave scattering problem. The first was found by letting the
total wavefieldPt(r) be the sum of the incidentwavefieldPi(r) and scatteredwavefieldP,(r). We formulateda Green'sfunctionintegral solution to the inhomogeneousdifferential wave equation in terms of the scattered wavefield P, resulting in the nonlinear LippmannSchwingerequation. We linearizedthe Lippmann-Schwingerequation
by assumingthe Born Approximation(i.e., P•(r) << Pi(r)) resulting in a linearrelationshipbetweenthe data functionPs(r) and the model functionM (r),
I
I r')dr',
whereP.(r.,rp) becomesthe data functionobservedat positionrp when the negative impulse sourceis located at position r•.
The
Green'sfunctionsare definedby eitherequation(62) or (63). 4. The secondintegral equation solution was found by representingthe
totalwavefield Pt(r)bytheexponential equation eqSt(r), where the complextotal phasefunctionqbt(r)was set equal to the sum of the complexincidentphasefunctionq5i(r)and the complexphasedifference function qba(r). After lengthymanipulationsof thesephases, formulating a Green's function solution to the resulting inhomoge-
neouswaveequationwhichoperatedon the quantity Pi(r)c)a(r), and
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92
CHAPTER
3. SEISMIC DIFFRACTION
TOMOGRAPttY
applyingthe Rytov approximation (i.e., Vdd(r) << 1), we found,
P•(r,,r•)•a(r,,r•) • -ko •/M(r')G(r' Ir,)G(r• Ir')dr'. Herethe data functionis P•(r,, r•)qba(r,,r•), insteadof P,(r), which is linearlyrelated to the modelfunctionM(r). 5. The Born approximation is valid for weak scatterers of limited size while the Rytov approximation requires only a smooth model function.
6. Letting P(r,,r•) representthe data functionfor either the Born or Rytov approximationintegralsolutions,we simplifiedthe integral equationsolutionsto the generalizedprojection slicetheoremfor three sourcereceiverconfigurations.The generalizedprojection slice the• rem for the crosswellconfigurationis
P(d, k,ßdp, kp)= --k• eJ(%dp7,d,)_ ' ' 4 Thisequationestablishes a relationship betweenP(d•, k•;d•, kp),the doubleintegralFouriertransformof the data functionalongthesource andreceiverprofilesin the crosswell configuration, and M[ko(• + •)], the 2-D Fouriertransformof the modelfunctionalongcurvesin the k• - k, plane. The curvesare definedby the sum of the vector ko•
whichpointsfrom the point (•, z) in the modelto the sourceandthe terminusof the vectorko• whichpointsfrom (•, z) to eachreceiver. 7. The generalizedprojectionslicetheoremfoundfor the VSP configuration
is
'
ß '
=
-4
+
8. The generalizedpro•ection slice theorem found for the surMce reflection configuration is
P(k, d,'kp, dp)= k• eJ(-7p½ - 7,d,)•[ko(g +•)1. • • 4 7s7p 9. Thecoverage of •(kz, kz)in thekz- kzplaneisdependent uponthe sourcereceiverconfigurationand alwaysincompleteto somedegree. This resultsin a nonuniquelyestimatedmodel M(x, z) that h• inherentlypoor resolutionin directions•sociated with the incomplete coveragein the kz - k, plane.
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3.6.
SUGGESTIONS
FOR FURTHER
READING
93
10. In Section3.4 the generalized projectionslicetheoremcouldnot be directlyappliedto estimatingM(z,z). This occursbecausethe 2D Fourier transformrequiresdata points on a rectangulargrid in the kx- k; plane and the generalizedprojectionslice theorem defines the data points on semicircles.The direct-transformdiffraction tomographyjust performsan interpolationto solvethe grid differences, or
before taking the inverse2-D Fourier transform to get the image
M(x, z). However,the potentialfor interpolationerrorshowingup in M(x, z) is very real. 11. The backpropagationdiffractiontomographymethod accomplishes thesameresultasthedirect-transform methodby replacingthe (kx, k,)
variablesin the 2-D inverseFouriertransformby (k0,kr), the variablesusedin the generalizedprojectionslice theorem. The resulting coordinatetransformationis dependentupon the source-receiverge-
ometry,but wecanfind M(z, z) directlythrougha 2-D inverseFourier transform without the need for interpolation.
3.6
Suggestions for Further Reading Devaney,A. J., 1984,Geophysicaldiffractiontomography:IEEE trans., ClE-22, 3-13. First proposalof seismicdiffractiontomography.
Esmersoy, C., Oristaglio,M. L., and Levy, B.C., 1985,MultidimensionalBorn velocityinversion'singlewidebandpoint source: J. Acoust. Soc. Am., 78, 1052-1057. Reformulares the Born inversion problem so that the data function becomes the field extrapolatedthroughthe wave equation.A variable backgroundvelocity is permitted.
Morse, P.M., and Feshbach,H., 1953, Methods of theoretical physics:McGraw-Hill. We just throw equation(10•), the plane-wave decompositionof a cylindrical wave, right at you withoutproof. Section 7.2 of this referencederivesthe equation in detail with the resultingequationfound on page 8œ3.
•zbeck,A., and Levy,B.C., 1991,Simultaneous linearized inversion of velocity and density profiles for multidimensional acoustic media: J. Acoust. Soc. Am., 89, 1737-1748.
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94
CHAPTER
3.
SEISMIC
DIFFRACTION
TOMOGRAPIIY
This paper includesthe formulation for simultaneouslyreconstructing both the velocity and density using the Born approximation. Pratt, R. G., and Worthington, M. H., 1990, Inversetheory applied to multi-source cross-holetomography. Part 1: Acoustic wave-equation method' Geophys. Prosp., 38, 287-310. We have presenteddiffraction tomographyin terms of a homogeneousbackgroundmedium with velocityCo. This paper presentsa nonlinear inversiontechniquein the frequencyspacedomain which essentiallypermits inhomogeneous acoustic backgroundmedia. A companionpaper treats the same subjectwith respectto elastic media. Rajan, S. D., and Frisk, G. V., 1989, A comparisonbetweenthe Born and Rytov approximationsfor the inversebackscattering problem: Geophysics,54, 864-871. Includes numerical comparisonsof the two approximations.
Wu, R. S., and ToksSz,M. N., 1987, Diffraction tomographyand multi-sourceholography applied to seismic imaging: Geophysics, 52, 11-25. Modified Devaney's plane wave seismic diffractiontomographyusingline sourcesandformulated the backpropagationreconstructionalgorithmsfor crosswell, VSP, and surface seismic geometries.
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Chapter
Case 4.1
4
St udies
Introduction
The purposeof this chapter is two-fold. First, through casestudies,we illustrate the proceduresfor implementingthe theory presentedin Chapters 2 and 3. Second,the casestudieshighlight some of the benefits of using seismictomographyin the oil industry. The first two casestudiesutilize crosswellseismicdata in conjunction
with the simultaneous iterativereconstruction technique(SIP•T)presented in Chapter 2 on seismicray tomography. The first casestudy addresses the production problem of monitoring the progressof a steam-floodenhancedoil recovery(EOR) program.The secondcasestudy involvesmore of a developmentproblemin which the structural interpretationof a faultcontrolledreservoirmust be better understoodfor in-fill drilling. The third casestudy usesthe seismicdiffractiontomographypresentedin Chapter 3 to image two salt sills using marine surface seismicdata. We selected this problemto illustratethe seismicdiffractiontomographymethodologyand limitations rather than to solvean exploration problem. 4.2
Steam-Flood
EOR Operation
Somereservoirsencounteredby the oil industry containpetroleumresemblinga heavy,tar-like substancerather than a low-viscosity fluid. To produce such reservoirs,steam is injected into the reservoirwith the intent of heating-upthe petroleummaking it lessviscousso that it may flow. A steam-floodenhancedoil recovery(EOR) operationis expensive and may be adverselyaffectedby reservoirinhomogeneities whichchannel 95
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96
CHAPTER
4.
CASE
STUDIES
steam away from parts of the reservoirthe production engineerwishesto heat. This casestudy demonstrateshow crosswellseismicray tomography can identify reservoirinhomogeneitiesbefore initiating steam flooding and monitor the EOR operation during steam injection. The Potter B1 tar-sand reservoir in the Midway Sunset Field, California, producesheavy oil using steam-flood EOR operations. We conductedtwo crosswellseismicexperimentsat a steam injection site in the Midway Sunset Field with the intent of aiding decisionmaking by the productionengineers on the steam-flood EOR operation. A presteam injection crosswellseismic survey was run to tomographically image reservoir inhomogeneitieswhich
might affect the operation and to provide a "base-line"P-wave velocity tomogram for comparisonwith a later poststeam injection P-wave tomogram. The P-wave velocity in this tar-sand reservoirdrops dramatically when the reservoiris heated. Thus, a secondpoststeam injection crosswell
seismicsurveyprovidesa P-wavevelocity tomogramwhich readily identifies the heated parts of the reservoirwhen comparedwith the "base-line" tomogram. The productionengineermay alter the chosensteam injection program if steam by-passesparts of the reservoir.
4.2.1
Crosswell Seismic Data Acquisition
The data acquisitiongeometryis depictedby the cross-section and map view in Figure 29. The seismicsourcewas located in the temperature observation well T02 which is deviated 99 ft to the south. The production well 183 served as the receiver well and is offset 283 ft from the TO2 well at
the surfaceand 184 ft at the TD of well TO2. The steam injector well is sit-
uateddowndipfrom the TO2-183 profile(dip directionis to the NNE) and the steam is expected to travel updip where it will intersect the crosswell profile. Bolt TechnologyCorporation's downholeair gun shownin Figure 30 was used as the seismicsourcein this experiment. The air gun has an 80 cubic inch chamber and is pressurizedto 2000 psi above the ambient pressurein the borehole. Geosource'sthree-componentgeophoneVSP tool was usedto detect the seismicenergy. A DSS-10 systemrecordedthe seismicsignalsat a sample interval of 0.5 ms with a one secondrecord length. The downhole air gun was fired at 40 depth stations from 1360 ft to 1750 ft at 10 ft intervals, while the VSP tool was set at fixed station depths. The source wasfired four timesat eachstation so that better signal-to-noiseratio could be attained in the stackedsignal. The 40 sourcestationswere repeatedfor each of the 40 receiverstations,which ranged from 1630 ft to 1240 ft at 10 ft intervals, giving a total of 1600 unique source-receiverpair locations. The common-receiver gather in Figure 31 was recordedby the vertical
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4.2.
STEAM-FLOOD
EOR
OPERATION
97
Midway Sunset Crosswell Experiment Cross-section
TO2 183
283
- 0
ft
- 200
- 400 _ 600 - 800
Potter
-
1000
-
1200
Depth
A B1
- 1400 -
1600
Antelope Shale -
1800
- 2000
<--
18411
Map View TO2
99fti TI•
"• 400 DIpI•i
:
+ Steam Injector ß 134R
18411"'-.. .
ß
183
FIG. 29. Well configurationfor the Midway Sunset crosswellexperiment shownin cross-sectionand map view. Solid circlesin the map view indicate well surface locations. The air gun source was placed in the temperature observationwell TO2 which is deviated 99 ft along the dashed line. A three-componentVSP tool was located in the vertically drilled production well 183. The dotted line indicates the profile defined by connectingthe
total depth (TD) positionsof the two wellswhich corresponds to a 184 ft horizontal
offset.
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98
CHAPTER4.
CASE STUDIES
FIG. 30. Bolt employeesare shownpreparing an 80 cubic inch air gun at the TO2 well. The umbilical to the air gun containsa pressureline and a wireline. Photo courtesy of Don Howlett, Texaco.
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4.2.
STEAM-FLOOD
EOR
OPERATION
99
component of the VSP tool fixed at a depth of 1500 ft while the source moved through its 40 stations. The P-wave first arrivals are easily identified and have a dominant frequency of approximately 200 Hz. The large amplitude event immediately followingthe P-wavefirst arrival is the S-wave first arrival.
Time
zero occurs at 60 ms on the record which was monitored
by a pressuresensormountednear the air gun. The large amplitude events trailing the S-wavearrival are from upward traveling tube waves. Steam injection commenced shortly after completion of the presteam injection crosswellseismicsurvey. The steam flood operation continuedfor approximatelyone year beforewe ran the poststeaminjection crosswellseismic survey. The secondsurvey was acquired along the sameTO2-183 profile with data acquisition parameters similar to the first survey. Tomographic imageswere processedusingtraveltime ray tomographyfor both data sets in an attempt to provide the productionengineerswith information on the performance of the steam flood operation. 4.2.2
Traveltime
Parameter
Measurements
The crosswellseismicdata in this casestudy were processedusing the
simultaneousiterative reconstructiontechnique(SIRT) describedin Section 2.3.3. A three-step recipe was given in that sectionfor reconstructing tomographicimages using SiRT. Here we explain how the traveltime parameters usedin the SIRT recipe were measured,first for a P-wave velocity tomogram and then for an S-wavevelocity tomogram. The traveltime parameters required by SIRT are the sourceand receiver locations and measured traveltimes. The first step in determining source and receiver locations is to run a borehole survey. At Midway Sunset we
usedGyroData'stool to obtainthe surfacelocations(x-y positions)for variousdepths(z-positions)in both the T02 and 183 wells. With the borehole geometriesknownall that remainsis to providegoodmeasureddepthvalues to the sourceand receiver. Depth gaugeson the vehiclesusedto deploy the sourceand receiver in the boreholesprovided the measureddepth values in
this experiment. The starting positionof the sourcewas recalibratedeach time a sourcepasswas completed. We havesinceattached collar locatorsto the sourceand receiver to calibrate the measureddepth estimates. Gamma ray loggingtools can be usedin place of collar locatorsto achievesimilar calibrations.
The most critical and tediousstep of ray tomography processingis selecting the traveltimesfrom the observeddata. The observeddata, given
by p/ob•forthe ith source-receiver pairin equation(46),shouldbeselected from data preprocessed for maximum signal-to-noiseratio. To provide this optimum environmentfor pickingP-wave traveltimes,the Midway Sunset
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100
CHAPTER
4.
CASE
STUDIES
Common-Receiver Gather (1500 ft) Source Depth (ft) 1400
P -Wave
1500
100-
1600
1700
,
•
,
,
;
$ -Wave
2ooI:
Time (ms)
Tube Wave
I
I
300 -
I I.
.i ii i
'
, 400 -
_•
,
,
F•(•. 31. Common-receivergather for the VSP tool at a depth of 1500 ft. The 40 traces correspondto the sourcestations.
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4.2.
STEAM-FLOOD
Source
EOR
OPERATION
Well
101
Receiver
Well
Low Velocity
High Velocity
Direct arrlval's ray path ....
Head wave's ray path FIc. 32. This cartoon depicts the raypaths for a direct arrival with traveltime Td and a head wave with traveltime Th • Td.
data required only a minimum-phasetrapezoidal bandpassfrequencyfilter of 50-100-200-300
Hz.
Generally one selects first arrivals on crosswellseismic data when Pwave traveltimes are desired. That is, the first arrival catchesour eye as being the first significant signal. However, in many casesthe first arrival is not a direct arrival, but a head wave which travels along an interface as shownby the solid raypath in Figure 32. The head-wavetraveltime pick createsa problem when the forward modeling method used in step I of the SIRT algorithm determinesonly direct arrival raypaths, such as the one
depictedin Figure 32. Obviously,a computedraypath giving the predicted
traveltimep•re in equation(46) whichis not consistent with the event's observed traveltimep/oh,will causewrongcellsin themodelto beupdated at incorrect
velocities.
The remedy for this dilemma can take two avenues. The avenuewith the
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102
CHAPTER
4.
CASE
STUDIES
potholes is to stick with the ray tracing schemewhich models only direct arrivals. Here we are forced into trying to identify when the first arrival is a head wave and then, whether we should attempt to pick a later, more noisy,event as the direct arrival or to make no traveltime pick at all. This can be a very frustrating processwhen the subsurfaceis prone to many head waves.
The secondavenueis to use a ray tracing schemewhich modelsthe first arrival's raypath, whether it be a direct arrival or head wave. Some of these schemesare referencedat the end of Chapter 2. A possibleproblem with these schemesis that a raypath may be found which gives a first arrival correspondingto an event with negligible amplitude. Thus, we end up selecting the wrong observedtraveltime once again. Our experience showsthat when head wavesare prevalent, this secondavenueseemsto be the more successfulof the two and the least frustrating. However, for the Midway Sunset data we ray traced for predicted direct-arrival traveltimes since head waveswere not a problem. The direct arrival versusfirst arrival problem is an important consideration in selectingmeasuredtraveltimes. However, another concernjust as important is the radiation patterns of the source and receivers,sincethese determinewhich polarity one shouldpick. The air gun is an explosive-type
sourcewith a positive(outwards)particle motion in the first half-cycleof the signalwith greateststrengthdirectedhorizontallyas shownby the radiation pattern in Figure 33. For a constant velocity medium the rays betweensourceand receiversare straight. Figure33 showsthe vertical(V) and radial (R) geophonecomponents.For a positive particle motion the vertical componentproducesa positivekick whenthe wavestrikesthe geophone from above, and the horizontal componentproducesa positive kick as long as the wave is traveling radially outwards. The vertical componentof the geophoneremains fixed along the borehole axis so that any changeof polarity in the signal can be attributed to the up or down propagation of the P-wave. If the radial componentwere indeedfixed at all receiverlevels,then the polarity would alwaysbe positive except for under very extreme subsurfacegeologicconditions. However,the two horizontalcomponentsin a VSP tool are not fixed and rotate randomly
throughoutthe survey.The goodnewsis that the P-wave'sfirst half-cycle should always be positive on the radial component. Thus, we can numerically rotate the horizontal componentsso that we always have a positive first kick on the radial component. The even better news is that with the air gun sourcethe receiverremainsfixed over each sourcepass(common-
receivergather). This meanswe needto determinethe numericalrotation only once per receiver level. For the Midway Sunset survey we only were required to determine40 rotation anglesfor the entire survey! To summa-
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4.2. STEAM-FLOOD EOR OPERATION
103
Air gun Source/ Geophone Receiver Source
Receiver V
R
P-Wave
FIG. 33. Thiscartoondepicts theP-waveradiationpatternwitha positive polarityduringthefirsthalf-cycle of motion(explosive source). Thevertically(V) andradially(R) recorded firsthalf-cycles areshown ontheright asflagswith the expectedpolaritiesandsignalstrengths represented. The verticalcomponent changes polaritywith receiverdepthwhilethe radial component'spolarity remains positive.
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104
CHAPTER
4.
CASE STUDIES
rize Figure 33, we must keep in mind that the polarity will changeon the vertical component, but not on the radial component. Thus, its best to cross-reference one componentwith the other during traveltimepicking.
The Bolt air gun alsoemitsan SV-wave(verticallypolarizedS-wave). The radiation pattern for the SV-wave is shownin Figure 34 and resembles a four-leaf clover. The particle motion polarities of the first half-cycleare shownas small arrowsperpendicularto the raypaths. For the rays shown, positive polarity correspondsto an arrow pointing in the counter-clockwise direction and negative polarity correspondsto an arrow pointing in the clockwisedirection. The resulting first half-cyclesof the S-wave are shown
as flagsfor the vertically (V) and radially (R) recordedsignals. Similar to the P-wave, the $V-wave exhibits a phase reversal on the vertical com-
ponent as the receiverchangesdepth from above the sourceto below the source. The radial component does not change polarity. Thus, we must be cautiousin selectingtraveltimesfor the S-wavearrivals,making sureto pick the correct polarity. Again, selectingthe correct polarity is best done by viewingboth componentswhile traveltime picking. The next step in obtainingquality traveltime picksis to insureconsistent
picksfor the selectedevent (direct arrival here) over the entire crosswell data set. With surfaceseismicdata we tie the selectedevent around loops definedby intersectingstrike and dip lines. An analogoustechniquemay
be applied to crosswelldata providedthat the same sourceand receiver stations are used throughoutthe survey,or at least over large portionsof the survey.
The Midway Sunsetdata acquisitionprogram calledfor common-receiver
gathersas depictedby the cross-section in Figure 35(a). The sourcewas movedthroughits set of fixedstationsasthe receiverremainedfixed. These locationsplot on a vertical line at receiverdepth R4 in the source-receiver depth plane on the right in Figure 35(a). Locationsof other commonreceivergathers are representedby vertical lines correspondingto other fixed receiverstations. If we keep the same sourcestations for each receiver
(whichwe did), thenthe data canbe sortedinto common-source gathersas shownin Figure35(b). The source-receiver depth locationsplot alonghorizontal lines for common-sourcegathers, such as the one at sourcedepth S3.
Finally, Figure 35(c) showsthat common-offset gathersare alsopossible where the offset is a depth differencebetween sourceand receiver.
Figure 36 indicateshowthe common-receiver and common-source gathers are used to tie time picks around loops. The sourcesand receivers correspondingto straight rays within the box in the cross-section on the left definea squareloop in the source-receiverdepth plane on the right. We
beginwith the common-source gather at station S2 and selecttraveltimes from receiverstations R2 to R5. Next we take the common-receivergather
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4.2.
STEAM-FLOOD
EOR
OPERATION
105
Air gun Source ! Geophone Receiver Source
Sl/-Wave
Receiver V
•
o
R
o
FIG. 34. This cartoon depicts the $V-wave radiation pattern with the upgoing $V-waves exhibiting a positive polarity on the first half-cycle while
the downgoing SV-wavesexhibita negativepolarity.The vertically(V) and radially (R) recordedfirst half-cyclesare shownon the right as flagswith the expectedpolarities and signalstrengthsrepresented.The vertical component changespolarity with receiver depth while the radial component's polarity remainsnegative. No S-wavesignalis receiveddirectly acrossfrom the sourcesince the radiation pattern is zero horizontally.
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106
CHAPTER
Cross-section
4.
CASE STUDIES
Source/ ReceiverDepth Plane
SOURCE
RRCEIVER
WELL
WELL
Commor•-Receiver
-
Receiver Deptl•
RI RI
R2 R.3 R4 RS R6 •
S1-
Gather
I
$2. I I
Depths $4, I
$5,
R6
Cross-section SOURCE
Source/ ReceiverDepth Plane WELL
Common-Source
:'_'____:. s4
' CC .. -.-•
RI
Sl
R2
S2.
R.3
83
R4
85.
R6
86.
Source/ ReceiverDepth Plane RECEIVER
WELL
(c)
Receiver Dept!•
WELL
Gommon-Off•
Gather
_-•, R•
S2
R2
S5
R.3
S4
R4
S6
Deptlm 84
R5
Cross-section SOURCK
SI
I
RECEIVER
WELL
Sl
(b)
$6
SI-
Dept!• S4-
R5
85-
R6
S6-
FIG. 35. Possiblecrosswellseismicgathersshownby raypath correlations
on the left andin the source-receiver depthplaneon the right for: (a) receiver,(b)source,and (c) offsetgathers.
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4.2.
STEAM-FLOOD
EOR
OPERATION
107
Tying Data Using Source / Receiver Gathers Cross-section
Source / Receiver Depth Plane
souac•
Rccclvcr Dcptlu
WELL S1
S2
RI
WELL
R2 R3 R4 R•
R6
R1
.........
R2 !
S3
R3
S4
R4
S•
R•
S6
R6
S3 • I
Depths
i
I
S6 •
I <-
FIo. 36. The sourcesand receivers within the range of the box of the cross-sectionare used to tie traveltime picks around the loop shown in the source-receiverdepth plane. Two sourcegathers and two receivergathers are required to complete one loop.
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108
CHAPTER4.
CASE
STUDIES
at R5 and pick traveltimesoff of tracescorrespondingto sourcestationsS2 to S5. Then the common-sourcegather at S5 is used and time picks taken from receiverstations R5 to R2. Finally, the loop is completedby picking times for tracesS5 to S2 on the common-receivergather at R2. Sincethe S2 trace on common-receivergather R2 is the same trace R2 on common-source gather S2, the traveltime picksmust be the same. If the picksappear to be different, then a mispick must have been made somewherealong the loop and should be rectified before you continue. All picks in the data set can and should be tied in this manner to provide consistency. Generally, on the first pass, loops are made as large as the data quality will allow with in-fill loops checkedlater. Besidesproviding consistentpicks, this method also lets one identify poor data regions in the source-receiverdepth plane and to developstrategiesfor tying time picks around "not useable" data regions. Finally, a recommendedpractice is to plot the computed traveltimes from the convergedimage reconstructionon the data recordswith the observedtraveltime picks. Since the image reconstructionis basedupon both good and bad picks,many times the bad pickswill "stand out" when compared with the computed traveltimes; assumingmost of your traveltime picks are good. Also, traveltime picksleft out of the inversionprocessbecauseof uncertainty are more clearly identified by the computed traveltimes permitting you to add them to the inversionprocess.
4.2.3 With
Image Reconstruction the P-wave
direct
arrival
traveltimes
selected and the associ-
ated source/receiverlocationsdetermined,we proceedby establishingthe
gridded modelwhichrepresents theinitialmodelfunction Mj"it, where j = 1,..., J and J is the total number of cells in the gridded model. We chosesquare cells 5 ft on a side giving 42 cells horizontally and 84 cells
verticallyfor a total of J = 3528 cellsin the model.• Figure 37 showsthe initial estimate of the P-wave velocity profile between wells TO2 and 183. We chose a two-layer initial model since the velocity contrast between the Potter B1 tar-sand and Antelope shale in
Figure 29 wassignificantand the averagelayer velocitiesand interfacedip were known from well logs in the TO2 and 183 wells. The model in Fig1We originallystarted with 10 ft squarecellsbecausethe sourceand receiverspacings were 10 ft. Five foot square cells worked just a.s well, but with more resolution. Lesser size cells did not work well. Note that all of the tomograms shown in this section are
interpolated to 1] ft square cellsfordisplay.
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4.2.
STEAM-FLOOD
EOR OPERATION
Initial TO2 200
109
Guess 183
Offset (ft) 150
100
50
0
Velocity (fUs)
1400
..':..... 6650 ...
..........6900 1450
.•!::?:ii:•ii::?:i!ii[ 6970 '::;;:;:::::.::; '•:•:•;•:•:•:•*• 7015 1500
-:::.>:+•-'..::•
•:;•?• 7070
Depth (ft)
i•':J-'-"•';•:: 7120
;/•i=iii• 7176
1550
.::::::::::'./.::.:
724O
7350
1600
7550 8100
1650
'['
FIG. 37. Two-layermodel usedas the initial estimatein the SIl•T algorithm.
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11o
CHAPTER
4.
CASE STUDIES
ure37isthereciprocal 2 oftheinitialmodel function Mj'•i' which becomes thefirstmodelestimate Mf •t in theSIRTalgorithm. Step I of the SIRT algorithm is to perform ray tracing through the
estimated modelMf stforeachsource-receiver pairwithanobserved traveltime,Piø•'s.As mentioned earlier,we usea "Snell'sLaw" type of ray tracing for this data set, which modelsonly direct arrivals. Thus, sincewe selected1600 observedtraveltimes, we end up with 1600 predicted direct-
arrivaltraveltimesp•,re,wherethe subscripti is theindexfor the ith sourcereceiverpair. Besidesgiving the predicteddirect-arrivaltraveltimes,the ray tracing also providesthe raypath length through each cell as required by equation(46) in step 2 of the SIRT algorithm. The raypath length for the
ith ray in the jth modelcell is represented by the variable Step 2 of the SIRT algorithmusesequation(46) to determinethe model
corrections AM1 totheestimated modelMf st.Thepredicted direct-arrival traveltimes p•,reandraypathlengths$ij fromstep1 alongwith the associatedobserved traveltimesPiø•'sare usedto computeeachterm in the summationin equation(46). The weightI/V1 for the jth cell is simplythe number of rays which intersect the jth cell, frequently called the ray density. Ray densityis easily determinedduring ray tracing when computing the raypathlength Oncethe corrections AM 1 for all J = 3528modelcellshavebeendetermined in step 2, we apply thosecorrectionsto the estimated model function
M[stinstep 3,giving anew estimated model function MJnew)est. However,
becauseof the nonlinearnature of seismicray tomographywe apply only a fractionof the updategivenby equation(46), say60 percent. This prevents instabilities and is most important during the first few iterations when the correctionscan be quite large. The model updates are also smoothedwith a small spatial filter before they are applied to help reduce instabilities. These extra stepsare necessarybecausein seismicray tomographySIRT is a linear inversion scheme used in an iterative
manner to solve a nonlinear
problem, as discussedin Chapter 2. Steps I through 3 are repeatedusingthe previousiteration's new model
estimate M«n•w)•tasthecurrent estimated model Mfst.Thesteps are
iterated until we are satisfiedthat the estimated model has convergedto an acceptablesolution. Figures 38 through 42 show the estimated P-wave models in terms of velocity for the 1st, 5th, 10th, 20th, and 44th iterations of the SIRT algorithm, respectively.The first 10 iterations showthe development of large scalefeatures in the P-wave tomograrnssuch as the high
:ZRemember that the SIRT algorithmusesslowness for themodelfunctionMj instead of velocityVj, wherefor the jth cell, the two are relatedby Mj = 1/Vj. We displayVj because most people are more familiar looking at velocity rather than slowness.
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4.2.
STEAM-FLOOD
EOR
OPERATION
111
velocityfeature at 1475ft and low velocityfeature at 1550ft near the TO2 well. After 10 iterations,onlysmall-scale adjustmentsaremadeto the large features defined in the early iterations. There is not a lot of difference between the 20th iteration
and the 44th
iteration P-wave tomograms. A natural questionto ask is, "When do we stop the iterations?" One method used to answerthis questioninvolves plotting traveltime residualsas a function of iteration, as shownin Figure 43 for the P-wave data.
We see that the traveltime
residual decreases
significantlyduring the first 10 iterations in which large-scalefeatureson the P-wave tomogram are determined. After the 10th iteration the traveltime residual curve begins to toe-out as small-scalefeatures are added to the large-scalefeatures. By the 44th iteration the traveltime residual curvehad approachedthe horizontal,implying further iterationscould not improvethe convergence and the P-wavetomogramin Figure42 is the best possibleestimate of the true velocity profile. This method assumesthat as the computed traveltimes approachthe observedtraveltimes, the estimated model approachesthe true model. Observed
traveltimes
for shear wave direct arrivals were also selected.
However, because the S-waves are imbedded in other arrivals we could select
only 745 of the 1600possibleS-wavearrivals. Utilizing a similarprocedure as for the P-wavetomogram,we constructedthe S-wavetomogramin Figure 44. The noisy appearanceof the S-wavetomogramis a direct result of the small number of rays utilized by the SIRT method. Approximately one year after the presteam injection crosswellseismic experiment we ran a poststeam injection experiment to evaluate the successof the EOR steam flood program. We processedthe secondcrosswell
seismicdata set for a poststeaminjectionP-wavetomogramusingthe same processingproceduresas for the presteaminjection P-wave tomogram in Figure 42. Figure 45 comparesthe poststeaminjection tomogramwith the presteaminjection tomogram, both sharingthe samevelocityscale. Heated parts of the reservoir resulted in a reduced P-wave velocity as expected based upon core study results.
4.2.4
Tomogram Interpretation
The presteam injection P-wave tomogram in Figure 42 is usedto delineate reservoirinhomogeneities.We made the lithology/porosityinterpretation shownin Figure 46 assuminga direct correlationbetweenthe P-wave velocity and lithology/porositydeterminedfrom cores.The Potter A sand, Potter B1 sand, and Antelope shale are delineated. The Potter B1 sand is the reservoirin which steam was to be injected. Using core information we interpreted the high and low velocity zones within the Potter B1 sand
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112
CHAPTER 4. CASESTUDIES
Iteration TO2 200
1400
I
Offset (ft) 150
100
183 50
0
Velocity (ft/s) 6650 .
1450
ß
:':..... s•oo ............ '6970
ß :•:•.:•:•: 7015 1500
ß •!•-'-'•...:i•:• 7070 .:.>•:.:.-.'•.-:::
Depth (ft)
ß •:•[•:-..:•i 7120 1550
•:....:.....•.•:.:• 7176 .:::::•::-...-:
7240 1600
7350
7550 8100 1650
FIG. 38. P-wavevelocitytomogramafter 1st iteration.
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4.2.
STEAM-FLOOD
EOR OPERATION
Iteration TO2 200
113
5 183
Offset (ft) 150
100
50
0
Velocity (fUs)
1400
:?:iii•iii::.': 6650 1450
:':':"•!! 7015 1500
!•:..._.•. • 7070 !-"'i!=•i 'i 7120
Depth (ft)
7176
1550
7240 7350
1600
7550 8100 1650
_
FIC. 39. P-wave velocity tomogram after 5th iteration.
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114
CHAPTER4.
Iteration TO2 200
1400
10
Offset (ft) 150
100
CASESTUDIES
183 50
0
Velocity (fUs)
.i.:!i!i::::iiii:: 6650
. :.....:.: :.:..
':............ '6900 1450
'.:::::::::::::::;: .'::•:::::::::::.: .........
'::..::•::.':::::.:
:iiiiii?:?:?:ii?:ii 6970
1500
Depth
.....•......• 7OlS •?//.7070 ::.:•;•... -. 7120
(ft) 1550
•i•! •':'•:• 7176 7240
1600
7350 7550
8100
FIG. 40. P-wave velocity tomogram after 10th iteration.
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4.2.
STEAM-FLOOD
EOR OPERATION
Iteration ,i'O2 200 1400
115
20
Offset (ft) 150
100
183
5O
0
Velocity (fus) .::i:'•!:•:!:... 6650 ..::::::•i¾:: .... 6900
1450 ......... .........
................... 6970 :.::::.'::::::::;:;: .:.:.:.:.:.:..-.:.:
................. 7015 1500
•:...•:.: 7070 ?::.•: 7120
Depth (ft) 1550
-:i:::::•:.-.'•'..-:-'•:: 724O
1600
7350
7550 8100 1650
FIG. 41. P-wave velocity tomogram after 20th iteration.
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116
CHAPTER
Iteration TO2 2OO
44
Offset (ft) 150
lOO
4. CASE STUDIES
183 50
0
1400
Velocity (fUs)
1450
'i:!:i:i:i:i:i:!:!: :.:.:.:.:.:.:.:.:.: ..........
ii.•! 7015
::• /::
1500
:•-.;.•:.•.' 7070
Depth
••."-'."";• 7120
(ft)
•i!i•i!':."i.?iL::::•?,:.•i!:!.. ''
•:i:•:•:•:'::•,• 7176
..
7240 1600
735O
7550 8100 1650
FIG. 42. P-wave velocity tomogram after 44th iteration.
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4.2.
STEAM-FLOOD
EOR
OPERATION
117
Iteration
0
10
2O
3O
4O
Traveltime Residual
(xl(• 4 s2)
2
I
Traveltime
FIG.
Residual
43. Plot of traveltime
=i= •,•i:•iire 'Piobs )
residuals as a function
of iteration
for P-wave
velocity tomograms. Traveltime residual is defined as the sum of the squaresof all differencesbetweenpredicted and observedtraveltimes.
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118
CHAPTER
Iteration TO2 200
1400
50
Offset (ft) 150
100
4. CASE STUDIES
183
50
0
Velocity (fUs) ......
2634
•:•'•.i-'!!•:'
•:'•:•*•"•": 3041 1450
iiiiii!1111!111i 3113 1500
:'• '3242 , .......
Depth (ft)
i.11:...i::.• 32• ....:.:..-..-.:..-.
1550
3385 3508
3645 1600 3898
4300
1650
FIG. 44. S-wave tomogram processedin a similar fashion as the P-wave tomogram in Figure 42.
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4.2.
STEAM-FLOOD
EOR
OPERATION
Presteam Injection Tomogram Oei•h TO2 (ft)
200
Offset (ft) 150
100
183 5•) i
119
Poststeam Injection Tomogram Dei•h TO2
0
•)
• 1400
::::::::::::::::::::::::::::::::::::.:.:..'.::
-:i::i::?..-:.. •....¾...!...,..•:• .......•.... :.............
2o0 !
Off#t (ft) •so !
•oQ I
183 so i
1450 ::::::::::::::::::::::::: ::::::::::::::::::::::!.•:::!:::!:: :::.:.;,:.:.:.:.. ß -
1550 _
' .....
. :..
6293
i.:•.•.'..i::!:!i!i!•ii:i::..' ......" i:i:::i:i:i".• S705 1500 :i .".:?.!:!:i:i ..... •.......... ::....... .:.:.:.:.:.:.:.:.:.:.>;.:.6936
..]i•.!i•i:i:i:i:i:i:i:[3•":' _ 1550 ß-.-.-••,.-,.,.L...-.'-'.'-'.
i
:::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::,: ..:::::::::::::::::::::::::: :.::::::::::::::::::::::::::::::::':".--.• :-:-:-:-:-:.:' ============================================= '.:.: ?'•::!:i:i:i:i:!:i:!:i:!:i:i:i:!:. :.:•-.•!i::::::' ::!:::!:!:i::::':":"!":'"' :":'::i:i:!:!:!.'.:.':
•.i:•;.:=•.?.•i•.,..%..•.;:!:!;:!ii::.;..!... ' •....ß ..... •04• :....,•.•½.,x.•;:::::::,,:.:.::::.::::::..: .
:.:•.•.,?.,.. .......... ..
1500
o
.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ===================== Velocity
i•x.•!ii:i:!:i:i.. ".......:-:.:.:.:.... .....!:::i.i:i:i.i:i:i:!:!:i:i•:. ::.•`•:!:i:!:!:!:;:•:i•:•:•::i•::•:;:i:!:;:!:i:i.!•!:i:•:i•:::•::i:•:i:i:!:•:•:;:i:i:i:: 7200 :.:.',•,•.:.:.:.:.:.:.:.:.;.:.:.:.;...;.:.;.:.:.:.:.;.:.:.:...:.: ;.:.:.:.:.;.:.:•:.',-.•: '•:::.':.;:.•: ::::. ================================================= -1.•.:.:..:::::..-.. '"::: -::•:,'•-• .......... '......... ::.'•.-•.•-'..:.: ...... :.•,.,.:•,•::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :2::•::.:•i:.,x.'
:-.:,-.:. -:.-.-...:.:..............-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.....-..S; '.-.x.¾4::.... 790I !::•..'•i.';:.>-."-!'-::!:!:!:!:!:!::: :':....... "::!:i:!:!:!:::i..x%.•i•:-.:..•:!.
:::::::::::::::::::::::::: ..... ::::::::::::::::::::::::::::::::: 6000
F](•. 45. •'he presLeeLm injecLionP-weLve LomogreLm on LhelefL forms Lhe baseline LomogreLm for observingreservoir changesas eLresulLof sLeeLm injecLion. The posLs[eeLm injection P-wave [omogreLm on the right Laken one year leLterindirectly indiceLtes a significeLnt portion of the reservoirwas healed. P-wave velociL¾in LheheeLv¾ oil reservoirrock is reducedwhen Lhe reservoir
is healed.
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120
CHAPTER
4.
CASE STUDIES
Lithology/Porosity Interpretation TO2 200 1400
Offset (ft) 150
100
183 50
0
Velocity (rt/s) 6650 ß 6900
1450
.....
::i.!:i:i:!.i:!:
1500
ii•."a•'.:i•11• 7070
Depth
(ft)
1550
'-':•l•i•..*.." 7120 .•i!•! 7176 7240 735O
lSOO
',i',i',i',i',iii',i?, 7sso 8100
1650
FIG. 46. Lithology/porosityinterpretationof the presteaminjection Pwave tomogram. The Potter A sand, Potter B1 sand, and Antelope shale are shown. Core data showedthat high and low velocity zoneswithin the Potter B1 sand correspondto low and high porosities,respectively.
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4.2.
STEAM-FLOOD
EOR
OPERATION
121
as low and high porosities, respectively. Thus, the reservoir is certainly not homogeneousas far as porosity is concerned. At the top of the Potter B1 sand are high velocityclay stringers.The productionengineersrely on these clay stringers to confinesteam to the Potter B1 sand. However, the clay stringers do not appear continuous,and actually drop in velocity towards the center of the tomogram indicating a possiblebreach through which steam may flow acrossinto the Potter A sand. Thus, the production engineer might anticipate some steam lossinto the Potter A sand. We can also estimate a porosity tomogram of the Potter B1 tar sand reservoirusing the P-wave velocity tomogram and Wyllie's time-average equation in the form,
whereV! is the fluid velocity,V,• is the matrix velocity,and V is the P-wave tomogram velocity. The measuredaveragematrix velocity and pore-fluid velocityfor the Potter B1 sandare Vm = 10000ft/s and VI = 4500 ft/s, respectively.The P-wave tomogramin Figure 42 providesthe valuesfor V, giving the porosity tomogram shownin Figure 47. This porosity tomogram is valid only for the Potter B1 tar sand for which the averagevaluesof Vm
and V! weredetermined.The porosityvaluesfor the Potter B1 sandrange from 33 percent to 44 percent accordingto the porosity tomogram. These porosity valuesare higher than the porositiesdeterminedfrom coresamples which have high values of 32 percent. This discrepancyimplies that the Wyllie's time-average equation does not take into account all petrophysical properties which affect the P-wave velocity, such as clay content. However, we do believe that the relative porosity information for the Potter B1 sand is meaningful. Such porosity information is helpful to engineerswho need to model the production of a reservoir. The secondobjective of the Midway Sunset field tomography project was to monitor the EOR steam-injection project. To monitor the steam flood progressin the Potter B1 reservoir using seismictomography, we must know the effect of heated Potter B1 reservoirrock on seismicvelocity. Coresfrom a well located 300 ft east of the 183 well provided Potter B1 sand samplesfor which P-wave velocitiescould be measuredin the laboratory at various temperatures. Figure 48 displays the laboratory measured P-wave velocities for a Potter B1 sand core at temperatures of 25øC, 55øC, 90øC,
and 125øC, under confiningpressuresof 500 psi, 1000 psi, 1500 psi, and 2000 psi. Clearly, we will expect the P-wave velocity to decreaseas the temperature of the reservoir increasesas a result of steam injection. The core results provide us with the temperature-velocity relationship required to interpret the presteam and poststeam injection tomograms in
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122
CHAPTER
4.
CASE STUDIES
Porosity Tomogram TO2 200
Offset (ft) 150
100
183 50
0
Porosity 20
..:::.-:
1400
:..-:.:
25 30
1450
33 35 38 41
Depth (ft)
45
1550
.•,.
1600
1650
FIG. 47. Porosity tomogram determined from the P-wave velocity totoogram in Figure 42 and Wyllie's time-average equation. The lithology]porosityinterpretationfrom Figure 46 is superimposedfor comparison.
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4.2.
STEAM-FLOOD
EOR OPERATION
Fwave
123
Velocity vs. Temperature
8500
8000
75OO
Velocity
7OOO
2000 psi 15o0 psi 6500
1000 psi 500 psi 6000
20
40
60
80
100
120
Temperature (deg C)
FIc. 48. Measured P-wave velocitiesfor a Potter B1 sand core at temperatures of 25øC, 55øC, 90øC, and 125øC, under confiningpressuresof 500 psi, 1000 psi, 1500 psi, and 2000 psi.
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124
CHAPTER
Sonic Log at TO2 Well
4.
CASE STUDIES
Tomogram-Sonic Log Comparison
1400
400
1500
5O0
•-•Smoothed Sonic Log
Depth (ft)
P-wave Tomogram 1600
600
6000
7500
10000
Velocity (if/s)
12500
7000
7500
8000
8500
Velocity (if/s)
FI(•. 49. The original soniclog from the TO2 well is shownon the left. On the right, a smoothedversionof the soniclog is comparedwith a tomogram profile parallel to and offset 15 ft from the TO2 well in Figure 42.
Figure 45. The seismicP-wave velocity over parts of the Potter B1 tar sand dropped by 15 percent to 20 percent after one year of steam injection. We alsoseethat the clay stringersin the upper Potter B1 sand restrict the steam flood to the Potter B1 down dip. However, further up dip the clay stringersare breachedby the steam and somesteam is lost to the Potter A sand, which exhibits a similar velocity-temperature relationship as the Potter B1 tar sand reservoir. Figure 45 indicatesthat the Potter B1 sand was not uniformly heated and gave the production engineersinformation for modifying the steam injection project at this site. Finally, we checkedthe reliability of the presteam injection tomogram by comparinga tomogram profile taken about 15 ft in and parallel to the TO2 well in Figure 42 with the soniclog in the TO2 well. Figure 49 shows the original soniclog velocity on the left and providesa comparisonof the smoothedsonic velocity log with the tomogram profile on the right. The sonic log and tomogram profile are in good agreementas to overall trend. Velocity differencescorrespondingto the clay stringersmay be attributed to averaging-outof the high velocitiesby the tomographyinversion. The Potter B1 reservoir,correspondingto depths below 1500 ft, show the sonic
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4.3.
IMAGING
A FAULT
SYSTEM
125
log and tomogram profile agree to within 5 percent. Thus, analysis of the tomogramsin this portion of the Potter B1 tar sand are made with some justified confidence. Since similar data acquisition and processing techniques were applied for the poststeam injection tomogram, we also assumethis reliability analysisextends to the poststeam injection results as well.
We make the following conclusionsregarding crosswellseismictomography basedupon this casestudy' 1) Crosswelltomographyas a reservoir characterizationtool is usefulfor determiningreservoirlithologyand porosity inhomogeneitieswhich is not possiblewith only well log information; and 2) Crosswelltomographyis usefulfor monitoringnonuniformheating of reservoirrock betweenwells as a result of steam flooding.
4.3
Imaging
a Fault System
The McKittrick Field in California is located near the Midway Sunset Field and produces from the Potter sand, a massive unconsolidated conglomeratewith permeabilitiesranging from 1 to 10 darcys. Figure 50 shows
the well-logbasedreservoirgeologyinterpretationprior to runningcrosswell seismictomography betweenthe 806 and 429 wells. The McKittrick Thrust placed Miocene age diatomite over PleistoceneTulare sand. A subthrust fault developedsubsequentto the McKittrick Thrust which intersectsthe 806 well at a depth of 800 ft. The Potter sandreservoircontainsa heavyoil whichis subjectto gravity drainage. Basedon data from the 806 well, the subthrust is believedto act as a sealingfault which preventsfurther downwardmigration of the heavy oil. The well data show that the Potter sand above 800 ft is saturated
with
up to 50 percent oil while the Potter sand below 800 ft is desaturatedwith lessthan 30 percent oil. The subthrust lies at the boundary betweenthese two
zones.
The estimated oil reservesin this reservoir and optimum development of the field depend upon proper positioningof the subthrust. Thus, the objective of this crosswelltomography study is to image the faults associated with
the reservoir
desaturated
Potter
4.3.1
and define the boundaries
of the saturated
and
sand.
Crosswell Seismic Data Acquisition
The clamped-vibratorsourceshown in Figure 51 was provided by Chevron for this experiment. The sourceis coupled to the borehole wall through the clamp located at the top of the tool. One of the smaller lines
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126
CHAPTER
4.
CASE
STUDIES
Well-log Based Interpretation wel
well
8O6
429 Oft
315
ff
20O ft
D!atomite
McKIttrlck
.
..••
_ 40Oft _
60Oft
_
8OO ft
1 ooo ft
FIG. 50. The reservoir geologyinterpretation before the crosswellseismic survey was run between the 806 and 429 wells. The interpretation was basedmainly upon well-log data and somesurfacegeology.
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4.3.
IMAGING
A FAULT
SYSTEM
127
q
FIG. 51. Chevron'sclamped-vibratorsourcein preparation for deployment. The primary componentsfrom top to bottom are the clamp for source-toboreholecoupling, the hydraulic servovalveand actuator module, and the reaction mass. Photo courtesyof Don Howlett, Texaco.
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128
CHAPTER
4.
CASE STUDIES
Data Acquisition Geometry well
806
well
429 Oft
315 ft
:
.___ Shallowest Receiver --
Source
2O0 ft
40O ff
--
$00ft
Deepest Receiver --
80Oft
FIG. 52. Data acquisitiongeometry involved keepinga 90-degreereceiver aperture bisectedby the horizontal through the source. The aperture was formed using thirty-three receiver levels spaced at 20 ft intervals about each sourcelocation. The shallowestand deepestreceiver positionsfor the sourceat a depth of 400 ft is depicted.
provides air pressureto the clamp for its activation and deactivation. The other small line is a wire line which both supportsthe tool and provideselec-
tricity to the hydraulicservovalveand actuator which controlthe sweeping action of the vibrator. A pump at the surface pressurizedhydraulic fluid which flows to and from the sourcethrough the two larger diameter hoses. This pressuredrives a hydraulic cylinder which is attached to a 50 pound reaction masslocated at the bottom of the tool. An axial driving motion imparts a vertical stresson the boreholewall at the clamp which introduces the seismicenergy into the formation. Figure 52 showsthe data acquisitiongeometry. The clamped vibrator was deployed in the 806 well while a three-componentgeophonereceiver
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4.3.
IMAGING
A FAULT
SYSTEM
129
Common-Source Gather (660 ft) Time
(ms) 1100
Receiver Depth (ft) 1000
900
800
700
600
500
400
300
200
5O
lOO
150
200
FIG. 53. A common-sourcegather from the radial componentfor a source depth of 660 ft and receiver depths ranging from 160 ft to 1150 ft at 10 ft intervals.
was deployed in the 429 well. Thirty-seven common-sourcegathers were collectedwith the sourcerangingin depth from 200 ft to 920 ft at 20 ft intervals. Common-sourcegathers were collected, rather than commonreceivergathers,becauseof the unwieldynature of the four linesattached to the sourcewhich had to be clampedtogether at regular intervalsas the source was lowered into the well. For most source levels 33 receiver levels
were recorded at 20 ft intervals in such a manner that a 90-degree angle receiveraperture bisectedby the horizontalat the sourcewas acquired,as shown in Figure 52.
A linear sweepwas applied to the vibrator from 10 Hz to 360 Hz with a sweeplength of 14 s and a listen time of 2 s. Four sweepsper source level were requiredto get a goodsignal-to-noise ratio. Figure 53 showsa common-sourcerecord for the clamped vibrator at a depth of 660 ft and receiverstationsrangingfrom 160 ft to 1150ft at 10 ft intervals. The record is from the radial componentand showsboth first-arrival and reflected Pwave
events.
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CHAPTER
130
4.3.2
Traveltime
Parameter
4.
CASE STUDIES
Measurements
The trove]time p•r•meter measurementsfollow • procedure similar to the one for the Midway Sunset Field. Chevron provided desweptdata with the horizontal componentsdecomposedinto r•dial and transversecomponents. We applied a zero-phasebandpassfrequencyfilter to further increase the signal-to-noiseratio for optimum traveltime picking. Becauseof the "blocky" nature of the subsurfacewe chose to select direct-arrivM P-wave tr•veltimes, similar to wh•t we did with the Midway Sunset dat•. The only exceptionMdifferencehas to do with what polarity to chosefor the direct-arrival traveltime. The clamped vibrator sourcehas a radiation pattern significantlydifferent from the air gun. Figure 54 showsthe Pw•ve r•di•tion pattern for the clamped vibrator sourcein a homogeneous
mediumwith the signalrecordedby geophonereceiverswith vertical(V)
andradiM(R) components. s Theclamped vibratorisessentially a vertically directed dipole which has P-wave motion directed towardsthe sourcein the lower lobe while, at the same instant, the P-wave motion in the upper lobe is •way from the source.For a homogeneous mediumthis radiation pattern results in no polarity changeon the vertical geophonecomponentwhile polarity changeson the radial component.Theoretically,no P-waveenergy should be observedat the same depth as the clamped-vibrator sourcein a homogeneousmedium.
Figure 55 depictsthe S-waveradiation pa•tern for •he clampedvibrator which may be compared with the S-w•ve r•diafion pattern for the air
gun sourcein Figure 34. The clampedvibrator shouldproduceexcellent S-w•ves for use in crosswelltomography since the S-wave radiation pattern's maximum strength is directed horizontally. As with Figure 54, in a homogeneous medium we expect no polarity changeof the direct arrival on the vertical componentwhile a polarity change is expected on the radial component.
At first glance the dipole nature of the damped vibrator appears to produce recordswhich should be easy to interpret. After all, there are no polarity reversalsseen on the verticM componentfor either P-wave or S-wavedirect arrivalsin Figures54 or 55, respectively.However,now introduce • velocityfield suchthat a P-waveraypath initially traveling upwards gets refracted so that it is • downgoingray at the receiver. The P-wave event you are selectingappears to shift • cycle on the vertical component. Thus, youmustmakea decision:1) to pickon the reversedpolaritybecause
you suspectthe ray turnedup-to-down(or down-to-up),or 2) to pickthe same polarity at the shifted time becauseyou suspecta velocity change. For complicatedsubsurfacevelocities this decisionprocessis tedious and 3CompaxeFigure54 for the clampedvibrator with Figure33 for the •ir gun.
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4.3.
IMAGING
A FAULT
SYSTEM
131
Clamped Vibrator Source ! Geophone Receiver Source
Receiver
V
R
'
P-Wave
o
-I-
o
FIG. 54. This cartoon depicts the clamped vibrator's P-wave radiation pattern where particle motion in the upper lobe is away from the source while in the lower lobe the particle motion is towards the source at the sameinstant (dipolesource).The vertically(V) and radially (R) recorded signalsare shownon the right as flags with the expectedpolarities and signalstrengthsrepresented.The verticalcomponentremainsnegativewith receiverdepth while the radial component'spolarity changessign.
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132
CHAPTER
4.
CASE
STUDIES
Clamped Vibrator Source / Geophone Receiver Source
Receiver
V
R
S-Wave
FIG. 55. This cartoon depicts the •qV-wave radiation pattern for the clamped vibrator with all $Vowaves exhibiting the same polarity as indi-
catedby the smallarrowsperpendicularto the raypaths. The vertically(V) and radially (R) recordedsignalsare shownon the right as flagswith the expectedpolarities and signalstrengthsrepresented.
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4.3.
IMAGING
A FAULT
SYSTEM
133
requires the use of properly oriented horizontal components. The horizontal geophonecomponentsare free to take on random directions when the receiver tool is moved from level to level. Frequently the relative amplitudes on the horizontal components are used for numerical
rotation to get the radial and transversecomponentsdesiredin traveltime picking. However, the polarity of the incoming P-wave or S-wavemust be assumedto selectthe proper rotation. Figures54 and 55 showthat both Pwaveand S-wavepolarities changeon the horizontal componentwith depth, thus complicatingthe rotation analysisprocess.Add a complexsubsurface and the processof orienting horizontal componentsbased on relative amplitudes becomesnontrivial. The best bet is to run an orientation device with the receiverpackagewhen using a clamped-vibratorsource. Properly oriented horizontal componentsare a definite help in picking both P-wave and S-wave arrivals from a dipole sourcewhich travel through a complex medium.
Also, since the clamped vibrator required common-sourcedata acquisition, every source-receiverpair in the survey required a unique rotation analysis to orient the horizontal components. A common-receiverdata acquisitiontechniqueis more desirableif an orientation deviceis not available. Then, one only needs to perform one rotation analysis per receiver level since the receiver remains fixed for all source levels as was done with
the Midway Sunset data.
4.3.3
Image Reconstruction
We used the same image reconstructionmethod here as for the Mid-
waySunset datain theprevious section. Theinitialmodel function Mjni• had square cells 10 ft on a side which is half of the source-to-sourceand receiver-to-receiverspacings. Ten-foot square cells required 32 horizontal cells and 60 vertical
cells for a total of J = 1920 cells in the model.
The
initial model function valueswere given a constantvelocityof 5300 ft/s. Just as with the Midway Sunset image reconstruction,step i of the SIRT algorithm utilized a "Snell's Law" type of ray tracing to model the direct arrivals for both raypath and computed traveltime. Fifty iterations were requiredfor the estimatedmodel to convergeto the true model. The resulting P-wave velocity tomogram is shown in Figure 56, resampledto 2.5 ft square cellsfor display. One part of the image reconstructionmethodologynot mentionedin the previoussectionis how to selecta color(or gray) scale. One school of thought is to use fixed intervals of velocity by assigningmany colors (gray levels)at one time whichhighlightall featuresincludingartifacts. Although this techniquemay be great for quality control purposes,it is
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!•4
CHAPTER
4. CASE STUDIES
/'-Wave Velocity Tomogram Well 806
Offset (ft)
0
100
Well 429
200
300
300 .......................... I.......................... I......
Velocity (ft/s) 4000 :i:::":?:i::'•:'
ß.•,'.:: 5546 i•-.....
.."..•:. -.-.'...• ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
................ 5794
.::i:i:•:i:i:i::.:.'.•::.•:i:: '" ============================================================ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
•:i:i:i:i:i.'i:i:i:i:!:i::•.•i;.;:." -.'"":':':':':':':':':':':':':::'!:i:!:i:i:i:M'
.........6500
::::::::::::::::::::::::::::::::::::::::::::: •'-. ================= ß"•':' 7206 9000 .......:-'..:.'-.:-:."-:i:i:i.'!:i:'•' :':':' ....'•.... L•:'•:•..... :i:•:::::i.%-'.•:•.'•:.•.•s::'.!.•.?.:'. .............
..;....:. ........ ..:.::..::..!:-...:.;::.:.
700 . Saturated Potter "--.............. ' "-..':
.!?,!•'
FIG. 56.
.'.
.....
P-wave velocity tomogramreconstructionof the McKittrick
Thrust and subthrust.
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4.3.
IMAGING
A FAULT
SYSTEM
135
not very useful for displaying the results for which the tomographic data were acquired. Instead, one should highlight the tomogram's features for
which we have an interest. This generallyrequiresthat the persondoing the image reconstructioneither possess,or work with another personwho possesses, some interpretation skills. With this study we knew that the diatomite, Tulare sand, saturated Potter sand, and desaturatedPotter sand were targets of interest. Well data provided the information that the McKittrick Thrust lies at the boundary of the diatomite and Tulare sand while the subthrust at a depth of 800 ft in the 806 well separates the saturated Potter sand from the unsaturated Potter sand. Using this combination of well information with what the P-
wave velocity tomogramwas telling us, we chosea gray scaleto highlight the desiredfeatures. Only sevenshadesof gray were neededto fulfill the objectivesfor which this tomogramwas run. The resultingtomogramdisplay is in reality a lateral extensionof the well-loginformation obtained at the 806-well. If a velocity scale at fixed intervals had been used, the well information correlationwith the tomogramwould not be apparent and the resulting tomogram would be lessof a benefit to the end user, or maybe evenconfusing. 4
4.3.4
Tomogram Interpretation
The important objective of this tomogram was to image the saturated Potter sand sealed-offfrom further gravity drainageby the subthrustintersecting well 806 at 800 ft. Figure 57 is our interpretation of the tomogram in Figure 56, which when comparedwith the well-log based interpretation in Figure 50, showsthat the lateral extent of the saturated Potter sand is muchlongerthan previouslythought and doesnot extend as far in the vertical direction. This interpretation is basedupon the correlationthat, for the Potter sand, seismic P-wave velocity increasesas oil saturation increases. The tomogram in Figure 56 delineatesthe McKittrick Thrust which placed the Miocene age diatomite above the Pleistoceneage Tulare. The tomogram alsosuggeststhe presenceof two smaller scalesubthrustswhich penetrated the McKittrick Thrust fault plane after it was in place. The two subthrusts appear to have throws as small as 40 ft. Overall, the crosswellseismic tomogram showed a more complex fault systemthan previouslythought and substantiallyredefinedthe boundaries of the saturated
Potter
sand.
Such information
which determines
and ver-
4Even though the tomogram in Figure 56 was resstapledto 2.5 ft square cells for display, the image remains "sharp" because only seven gray levels were used in the display.
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136
CHAPTER
4.
CASE STUDIES
Tomogram-Based Interpretation well
806
well
429 Oft
315
ft
Dlatomlte Tulare
200 ft
Sand
McKIttrlck Thrust
400 ft
600 ft
800 ft
1000 ft
FIG. 57. Our interpretationof the P-wavevelocitytomogramin Figure56.
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4.4.
IMAGING
SALT
SILLS
137
ifies the reservoirconfigurationenablesthe developmentengineersto optimally develop the field. 4.4
Imaging Salt Sills
We chosea shallow salt sill problem to illustrate the seismicdiffraction tomography imaging techniqueusing the Born approximation presentedin
Chapter 3.5 The imagingtechniquerequiresa constantvelocitymedium with imbedded objects of finite extent which scatter seismicenergy. This requirement,along with other simplifications,is only partially honoredby the salt sill problem. Thus, the resulting tomogram is instructive as to the limitations of the method as presented in this book. Extensions of the methodologyto complexsituationsare discussedin referenceslisted at the end of Chapter 3.
4.4.1
Assumptions and Preprocessing
This casestudy usesa marine seismicdata set collectedover two shallow salt sills as depicted in Figure 58. Application of the seismicdiffraction tomography methodologyusing the Born approximation in Chapter 3 requiresmaking someassumptionsand preprocessingthe data. We require the object under investigationto have a finite extent. Fig-
ure 15 showsa finite-extentobject whichhas a velocityperturbationC(r) confinedwithin the gray area. The regionsurroundingthe gray area is the backgroundmedium with constantvelocityCo. The finite-extent object in this casestudy consistsof the shallowsalt sills and the surroundingsedimentary layers. The overlying sea water is taken as the constant velocity backgroundmedium. Collectively, the salt sills and surroundingsedimentsdo not satisfy the assumptionsof the Born approximation asstated at the end of Section3.2.4. However, the scatterers comprisingthe water bottom do meet the requirements of the Born approximation and shouldimage properly. The velocity contrast between the salt and sedimentsprobably violate the weak scattering approximation. The salt sill tops are closeenough to the water layer that they can be consideredpart of a "finite-extent" object and, with the exception of possiblyviolating the weak scattering approximation, should image properly. The salt sill bases,especiallythe large salt sill base, are SFindingthe data functionPi(r)qbd(r)for the Rytov approximation(Section3.2.3) is considerablymore involved than determiningthe data function Ps(r) for the Born approximation (Section 3.2.2). Thus, we use only the Born approximationin this caae study.
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138
CHAPTER
Salt
Sill
4.
CASE
STUDIES
Problem Surface
5000 ft/s
4000 It
Water Bottom • .........
-....... .....
•.:.....:.:.:.:.:.:.:.:.:.:.:.:.: •
FIG. 58. The two salt sills and surroundingsedimentary layers form the
finite-extentobjectwhichhavevelocitiesrepresented by C(r). The velocity
ofthewaterlayeris(7•- 5000ft/s. comprisedof scatterersquite removedfrom the constantbackgroundwater layer. The accumulative phase differencebetween the total and incident
wavefields(seeSection3.2.4) is likely to be quite large for thesescatterers and a proper velocity image for the salt sill basesis unlikely.
Besidesthe assumptions stated above,we also assume:(1) constant densityin all media,(2) 2-D wavepropagation,and (3) multiple freedata. We know beforehandthat none of our assumptionsis one-hundredpercent valid. However, we continue so that we may see the result and learn. The wavefield recorded by a source-receiverpair in a marine seismic
recordrepresents the total wavefieldPt(r), whichconsistsof the incident wavefieldPi(r) and scatteredwavefieldP•(r) as definedin equation(50). Figure 59 showsa source-receiver pair for the marine seismicdata case.The recordedincident wavefieldtravels directly betweenthe sourceand receiver while the recordedscattered wavefieldtravels over a considerablylonger raypath to get to the receiver. Thus, the scatteredwavefield arriveslater than the incident wavefieldbecauseof the large water bottom depth.
The scatteredwavefieldP,(rs, rp) is the data functionusedin diffraction tomographywhenequation(101) is represented throughthe Bornapproximation. Thus, a required preprocessingis to extract the scattered wavefield from the marine
seismic records.
The water bottom
in this case
study is at a depth of 4000 ft or greater. The time lag betweenthe recorded incident wave and the wavefield scattered from the water bottom at the near
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4.4.
IMAGING
SALT
SILLS
139
Incident Wavefield vs. Scattered Wavefield Raypath Source
Receiver
•
%, •
,/'
/
Surface
Scattered
Wave
CO= 5000IVs
\
4000It
o.o. ..............
:i•:i:•:•:i:i*i:•i:ii•iii•i•iii?:iii!i::iiiiiii::i?:•ii::i:•!::iii::iiii:•-:•i!i::i::i•:::i::i•i::•ii::!::::::?:?:::::•:::::::• ............... ?:i::iiii? ................ '"'!::11"'•i•?:•, ------- -
FIc. 59.. The raypath for the incident wavefieldwill always be much shorter than any raypath for the scattered wavefieldbecauseof the large water layer thickness.
source-receiveroffset is a large 1.6 seconds. Thus, the data function is simply determined by muting the incident wavefield from the total wavefield for each marine
seismic record.
The diffraction tomographymethod alsoutilizes infinite-linesourcesand scatterersas definedby the Green'sfunction in equation(62). Infinite-line sourceshave a cylindrical divergence in which amplitude decreasesin a
constantvelocitymediaby r-«, wherer is distancetraveled.However, the observed data are acquired in a 3-D medium with point sourcesand scattererswhich have a sphericaldivergencein which amplitude decreases
by r -1. Thus, to makethe observeddata "mimic"a cylindricaldivergence
wemultiplythe dataoneachtraceby r« = (Cot)«(Cois theconstant backgroundvelocity) beforeperformingthe tomographicprocessing.As an alternative we could have applied diffraction tomography using the 3-D
Green'sfunctionin equation(63) in a 3-D modelwhichhasa 2-D geometry. Thus, the sourcesand scatterers would exhibit spherical divergencein our calculations, but at the cost of significantly greater computation time.
4.4.2
Data Acquisition
The data for this casestudy came from a preexisting2-D marine seismic survey. The experimental geometry in Figure 60 showsthe marine streamer
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140
CHAPTER
4.
CASE STUDIES
Data Acquisition Geometry Re.
vets
I
824
v
v
3
120
2
;[
246;[ Next
Source
Receivers
................_V V V I
2
3
V 120
FI,_q.60. Experimental geometry for the marine seismicsurvey.
cable with 120 hydrophones spaced at 82 ft intervals with a near offset of 824 ft. The air gun sourcewas fired at 246 ft intervals. We chosebackpropagation diffraction tomography for the surface reflection configurationgiven in Section 3.4.2 to do the image reconstruction.
Equation (165) showsthat we must take the Fouriertransformof the data
function P(rs,rp) alongthesource andreceiver profiles to get/5(k,,kp). Since we require that the spatial frequency content for the receiver profile be the same as the source profile, only every third hydrophonetrace was used. Thus, the effective receiverspacingwas the same as the sourcespacing of 246 ft. We used 271 sourcesout of the survey line to provide an adequateaperture for imaging the salt sills.
Five representative sourcerecordstaken at equalsourcepoint (SP) intervalsfrom alongthe marine seismicsurveyare shownin Figure 61. Source records$P 823 and $P 995 are taken from over the smaller and larger salt sills, respectively.Sourcerecord $P 1081 lies at the edgeof the larger salt sill. Several events of interest can be identified on source record $P 995 at the near offsets. The water bottom reflection is found at 1.80 sec-
onds while the top-of-salt and bottom-of-salt reflections lie at 2.36 seconds and 2.92 seconds,respectively. Most of the reflectionsbetween 3.60 seconds and 5.30 secondsare identified as multiples and even more multiples are found beyond 5.80 seconds. Figure 27 showsthe hypothetical coverageof a model function in the k= - kz plane for the surfacereflectionconfigurationif sourcesand receivers
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4.4.
IMAGING
SALT
Time (s)
SILLS
Marine
•/ SP1167
081
141
Seismic SP 995
Records SP 909
SP 823
F..... !'
, 't'.
I
•'•
ß
•:•I!,i:... '"ß. ß ' !1tl•..'.•..•'ß .•-' >•
F ,' ß.•-•'. :'1,i•:•....•>.-
....
,,.>•-.•';•.. ß. .
•..,•./:,.:,,-..• .' . • ,•.•••..-._..-..•.-.•!••>-.•-<•-";'•'--:.-I-'"'•/':•-:-'.-.-•"'"--: : "- -" -••'
4 '•":•"'.: '• '• .... ..
. :: .•_ ,:-.":,•>•
• • . •....•:•'•.• ß •-• .•-..
'"•' 2- 'i•"!•i•'•.•'-•.ff•--:•"
1::::'•/'"-.:L•?-': ':'
.;..• •..•..
......•
.....•-:•.•.•.•
F•o. 61. Five marine seismicrecordsextracted at equal sourcepoint intervals from the marine survey.
couldbe placedalongthe surfacein both directionsout to infinity. In this study we are limited to a streamer cable towed to one side of the seismic
sourceas shownat the top of Figure 62. The resultingcoverageof the model function in the kx - kz plane for scattererslocated at a depth of 8000 ft is shownas solid arcs. Each solid arc corresponds to a separate sourceand streamerlocation. The dashedline representsthe locusof ko.q. The associateddotted radial lines point to the sourcelocationsused to constructthe coverage.Obviouslywe can expectlessresolutionand more nonuniqueness for the imagedmodelfunctionusinga streamercablethan from the ideal situationin Figure 27.
4.4.3
Diffraction Tomography Processing
Figure 63 illustratesthe flow chart for implementingbackpropagation diffractiontomographyfor a surfacereflectionconfiguration as givenby
equation(165). The total wavefield P•(xo,xr,t) wasrecorded on the field records, wheret istraveltime,andx• andxr arethedistances froma fixed referenceto the sourceand receiveras illustratedin Figure 64. The 2D Cartesiancoordinate systemin Figure64 hasits originat sealevelwith
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142
CHAPTER
sourcs
streamer
4.
CASE STUDIES
cable
!
locus of k o s
locus of ko( s +'• ) F•c•. 62. Coveragein the k= - k• plane for the model function at 8000 ft depth using the sourceand streamer cable configurationfrom this study. The solid arcs represent the coverage.
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4.4.
IMAGING
SALT
SILLS
143
Diffraction Tomography Procedure Pt(Xs,Xp, t) Step I
Mute
direct
arrivals
Ps(x s, Xp,t) Step 2
Step 3
Step 4
Fourier transform along time axis, t
Fourier transform along x
s andXp
Backpropagation, equation (3.119)
•(x,z) FIG. 63. Flow chart for implementing backpropagationdiffraction tomography in the salt sill case study.
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144
CHAPTER
4.
CASE
STUDIES
Cartesian Coordinate System
• source
•Y
•7
•7
X'
0
receivers
FIG. 64. Cartesian coordinate system for the salt sill case study. The
sourceand receiveroffsets,x• and xp, are measuredfrom a commonorigin.
the z-direction positive downwards. Both sourceand receiverdepthswere set to sealevel for implementationof backpropagationdiffractiontomography.
StepI (in Figure63) is the preprocessing stepdiscussed in the previous sectionin which the recordedincidentwavefieldPi(xs,xr,t ) was muted from the total wavefield. Thus, after step 1 the seismicrecordsare assumed
to containonly the scatteredwavefieldP•(xs,xp,t), whichis the desired data function for the Born approximation. In Step 2 we take a 1-D Fourier transform of each trace of the scattered
wavefieldP•(x•,xp,t) with respectto time, t. Thus, we transformthe datafunctionfromits space-time domainrepresentation P•(x•, xp,t) to the space-frequency domainrepresentation P•(x•,xp,w), wherew represents the angular frequency.
Figure 65 showsthe data volumesbefore and after this 1-D Fourier transform. Beforethe Fouriertransformthe x•-axis, xr-axis, and /-axis span the data volume. After the 1-D Fourier transform the data volumeis
spannedby the x•-axis, xp-axis,andthew-axis. We haveeffectivelyreduced the original scatteredwavefieldinto many single-frequencyscatteredwavefield data sets as depictedin Figure 65. We seethat each single-frequency scattered wavefielddata set is representedby a plane perpendicularto the
w-axiswithinthe P•(x•,xp,w) data volume.The backpropagation diffraction tomographyformulain equation(165) can be appliedto oneor all of
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4.4.
IMAGING
SALT
SILLS
145
Ps(Xs, Xp, t) x
Space-time domain representation
,I
Fourier transform along
x
$
esxs'xp' Space-frequency domain representation
Single frequency scattered
wavefield
data
FI(•. 65. The Fouriertransformof the scatteredwavefieldPo(zo,zr,l ) is taken with respect to time t resulting in single-frequencyscattered wavefield
data Po(z,, zr,• ).
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146
CHAPTER
4.
CASE STUDIES
thesesinglefrequencydata setsin reconstructing an image.• Step 3 is to apply a 2-D Fouriertransformto P•(x•, xp,w) alongthe xj-axis and the xp-axis. This operationtransformsthe scatteredwavefieldPj(x•,•cp,w)in the space-frequency domainto the scatteredwavefield Po(ko,kp,w)in the wavenumber-frequency domain,whichis requiredby
equation(165). The 2-D Fouriertransformis expressed by equation(138) whichdefinesk• and kpasthe wavenumbers of the Fouriertransformalong the sourceline and receiverline, respectively.Note that the depthvaluesof
the sourced• andreceiverdr areset to zeroin equation(138) asindicated in Figure64. We applyequation(138) to P•(x•,xp,w) onefrequency w at a time.
Step 4 is to actually apply equation (165) to the transformeddata
functionor scatteredwavefield,P•(k•, kp,w). Equation(165) is numerically evaluatedat eachpoint on the x - z plane to get the modelfunction
M(x, z). A separatemodelfunctionM(x, z)is determined eachtimeequation (165) is appliedto a differentangularfrequencyw. Again,both d• and dp are set to zero. The wavenumberko in the backgroundmediumis set equalto w/Co, whereCo is the water velocity(5000ft/s) and w is the fre-
quency.Thewavenumbers k• andkprangefrom-koto ko. Foreach(k•, pair, the corresponding verticalwavenumbers, % and7p,are computed by
7•- V/ko • - k]and 'rp- y/ko • - kp •,respectively. We obtaineda sequence of modelfunctionsM(x, z) by applyingequation (165) at frequencies from 12 Hz to 26 Hz. Thesemodelfunctionswere then stackedto form one multifrequencymodel function. The model function was then convertedto velocityC(x, z) usingequation(56) and the resultingdiffractiontomogramis shownin Figure 66.
4.4.4
Tomogram Interpretation
The velocity tomogram in Figure 66 clearly depicts the water bottom
andthe presence of twosalt sillsalongthe surveyline. The largersaltsill is about 31,800 ft wide while the smaller salt sill is about 8,860 ft wide. An apparent 20 percent velocity variation exists within each salt sill. The
layer boundariesof the sedimentarylayers surroundingthe salt sills are not resolvedand the gray scalewas selectedto emphasizethe salt sills and water bottom for analysis.
For comparison sake,the samedata set wasprestackdepth migrated, with the resultsshownin Figure 67. The velocitymodel for the depth •Equation (165) doesnot showan explicit frequencydependence.That is because we stopped writing the angular frequency dependence of the acoustic wavefield and
wavenumberafter equation (49) in Chapter 3 for the "sakeof brevity."
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4.4.
IMAGING
SALT
SILLS
147
Salt Sill Diffraction Tomogram Surface Location (S. P.) 0
1219 ' ,,, t ............
1119 I ............
1019 •.............
919 •............
819 t ............
Water Layer
719 -
:
ß ...::-;.;.:.;.;.:::.:.:.:.:.:.:.:.:.:-L-'-'-'..-'.-'.;.-'.',;.. :.:..'.......:.:-:-:-;.:.:-:-:.:.:.:."•:;::::-.'-:.:.: .-" :. ...:.:..•-:-:.:.:?•:;::..'::-:-:-:.: '..;-:-;.:-:..'.:-.'•'.... .: ;.:-:.:.:.;.:.:-:-:-:-:.:-:..'.:-:, ...... :::::::::::::::::::::::::::::::: ..................... ::........ :................ ::: ....... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::•.....`• -.'•....'••i• "•.- --.'-:-.'-.'-'. - :::::::::::::::::::::::::::::::::::::::
i:!:i:!:i:i:!:i:!:!:!:!:!:!:!:i:•.)k.%•i:..•::$:::::.:.:.....•:::!:i:!:!:!:!:!:!:!:i:!:!:!:i:!:!:!:!.i:i:::!:i:!:::!:!:!:!:!:!::•!•:::..i:!:!:i:i•.:.•:i::.:::2i: :"-,•....'-:•:•......•...•..'.•.•:..-:.'c•'::::i:i:!:!•: :':'..'::• i:i:!:i:!:!:!:i:!:!:!:!:i
:!:!:!:!:i:!:!:!:i:!:!:!:!:!:!:•::.:..'•. :.!:!:!:!:!:!:!:!:!•'.--,•:::'•:•i::;•:•'•' ..•:...•:•:!:!:!•....•:•::!•:•.:i•::•:::•.::..:•::.::...::..•:::..::•.•::.::.::::•:!:!:i:!:!:!:::!•.•:!...•!:! ..•:::•::::::)' ======================= ::i:!:: :i:! :i:::•: :::::: ::i:::::i: i:i:!: i:i:i:::::!:i:i :!:i:i:!:!:i::::::: i.:'..%•....:'.'!:!:!:!:! :!:!:!: i:::::::::!:!:i :i:!:::!:!:! :i:i:!:i:::! :!:!:!: i:::!:i:::i :i:::!:'.-'.'•:'.!:i:i:: :::i:i :!:!:::::: :::::::::::::::i:i: ::i:i: ::i:::! :::i:: :i:i:i::•:i::."."::: ".'::i :::::: ::::::::::::::::::::::::::::-':.'" .•
Depth
(kft)
Velocity Scale 5000
1
8198
10000
ft
11347
12271 12649
(ft/s)
13027 13992 15000
Fio. 66. The diffraction tomogramshowingvelocitiesfor the water bottom and salt sills along the survey line.
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148
CHAPTER
4.
CASE
STUDIES
Prestack Depth Migrated Section Surface Location (S. P.) 1219
0---•
1119
.........
-•.•••••• .......
Depth(•)
t
"' ' ....
............. ••
1019
•
919
819
I
SALT SlLL•..•,•.•.:".-' "' "'-' ' ' '"•' -."•'•;'•-"•-'•-'• ..... •'- ....
•, ........ .•.•..•.•.•• .•-••
719
•
•-•
• ....•-
•-••
•
-•-:'
10000ft
FIG. 67. The prestackdepth migrated sectioncorrespondingto the velocity tomogram in Figure 66. Processingand display courtesyof Guy Purnell, Texaco.
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4.4.
IMAGING
SALT
SILLS
149
migration was derived using an interactive prestack migration focusinganalysis approach. The salt sills are labeled and their boundariescorrespond to the strong reflectionevents surroundingthe labels. The water bottom is also clearly visible. The water bottoms defined in both the diffraction tomogram and the depth-migrated section are in good agreement as to location and shape. The salt sill tops, which are visible on the seismic section, are not well
definedin the tomogram. This is possiblybecauseof the low frequencies (12 to 26 Hz) usedto constructthe tomogram,whichmay not providethe needed resolution. However, the widths of the salt sills are comparable. The shapesof the salt sill basesare in very poor agreement. Obviously the depth-migratedsectionportrays a more correctpicture than the tomogram for the salt sill bases.The invalid assumptionof a finite-extentobject for the deeper salt sill basesis causing their images to be both mispositioned and degradedon the tomogram. This can be shownusingthe first line of equation(146) in a simplecomputation. First, we focus our attention on an infinitesimally small part of the base of salt reflector
where the salt-sediment
interface
is horizontal.
This
isolatesa singlescattererof the incidentwavefieldat location(zo, zo), which we definein termsof a modelfunctionas M(z, z) = 6(z- zo)6(z - zo), where the deltas are Dirac delta functions. Next we place the source and
receiverdirectly abovethe scattererwith the coordinates(x0 = Xo,do= O)
and (ze = zo, de = 0), respectively. In doingso we haveinsuredthat the scatteredenergyfrom the selectedscattererand the specularreflectionfrom the horizontal interfaceapproximatelyshare the samevertical raypath and traveltime.
Second, given the set-up from the last paragraph we must evaluate the other parameterswhichgo into equation(146). For vertical raypaths Figure 26 indicates that the sourcewavenumbervector has components (7, = ko, ko = 0) and the receiverwavenumbervectorhas components
(7e - ko, ke = 0). Substitutingthesecomponents into equation(144) gives the wavenumber componentskr = 0 and k• = -2ko required by
equation(146). Third, we substitute the model function for the selectedscattereralong with the informationderivedin the previousparagraphinto equation(146) and integrate. The result is
- - o,o;t,,, e(t,o - o,o) - I
(2),
where 2Zois the round-trip raypath length betweenthe selectedscatterer and the zero-offsetsource-receiverpair. We restate the last equation by
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150
CHAPTER
4.
CASE STUDIES
substitutingin equation(52) for k0 whichgives,
- o.o; - o.0)
lej(2zo/C'o )
ß
Here2zo/Coisjust the traveltimeto of the associated diffractionevent(or specularreflectionevent) on the seismictrace, or to = 2zo/Co. It states that the incident energy travels to the selectedscatterer and the scattered energytravelsto the receiverat the constantbackgroundvelocity Co. For a finite-extent object this propagation model holds, but the salt sill bases obviouslycannotbe consideredas part of a finite-extentobject sincethey
are so distantfrom the water layer (our constantvelocitybackground). Replacingthe propagationvelocityof salt with that of water cannotyield a properly imaged model function as we will now demonstrate. SP 995 in Figure 61 showsthe specular reflection from the base of the largesalt sill at time to = 2.92 seconds.Even thoughFigure 67 showssome reflectordip at SP 995, we will assumethe scatteredenergy'straveltimeand the specular reflection's traveltime are so close as to be the same and that
the raypathsare nearlyvertical. Usingto = 2.92 s and Co = 5000 ft/s we get an imagedepthon the tomogramof Zo= 7300ft. The depth-migrated sectionin Figure 67 showsthe largesalt sill's baseat SP 995 at a depth of 10000ft, a 2700 ft differencewhich is easily greater than can be accounted
for by reflectordip. The diffractiontomogramin Figure66 showsa high velocity anomalynear zo = 7300 ft which is probably the image locationof the large salt sill's base. Thus, this example demonstratesthat scatterers outsideof an acceptablefinite-extent object are mispositionedand result in a deteriorated image. Finally, we ask, "What can be done to get a correcttomographicimage?" The immediate answeris to use a variablebackgroundvelocityinstead of the constant velocity assumedin Chapter 3. We would want a variablebackgroundvelocity which doesn'tscatter a significantamountof energy and is capableof properly imaging the model function in space. Then the higherfrequencyperturbationsto the variablebackgroundvelocity wouldbe smallerin sizeand magnitudethan for a constantbackground velocity, making the Born approximation more acceptable. Several of the referencesat the end of Chapter 3 address the application of a variable
background velocity. 4.5
Suggestions for Further Reading Gibson, Jr., R. L., 1994, Radiation from seismicsourcesin cased and cemented boreholes: Geophysics, 59, 518-533. Theo-
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4.5.
SUGGESTIONS
FOR FURTHER
READING
151
retical investigationinto the radiated energy by volumetric
(air gun), radial stress,and axial stress(clampedvibrator) sources located in cased and cemented boreholes.
The volu-
metric sourceis the only sourcewhich doesnot have a nonlinear frequencydependentradiation pattern.
Howlett, D. L., 1991, Comparison of borehole seismicsources under consistent field conditions: Expanded Abstracts for the Society of Exploration Geophysicists'Sixty-first Annual International Meeting and Exposition, Nov. 10-14, Houston, Texas. Comparesdata from explosive,clamped-vibrator,air glun,and cylindrical-bender-crgstalsourcestaken at Tezaco's geophysicaltest facility in Humble, Texas.
Meredith, J. A., ToksSz, M. N., and Cheng, C. H., 1993, Secondary shear wavesfrom sourceboreholes:Geophys. Prosp., 41,287-312. Borehole sourcessubmergedwithin a liquid can create tube waves. When the tube wave velocity is greater
than the formarion'sS-wave velocity,a reach wave (convertedS-wave)is generatedby the tubewavewhichmay interfere with other events recordedin the receiver well.
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This page has been intentionally left blank
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Appendix
A
Frequency
and
Wave
er
numb
Frequencyand wavenumberare two basicterms usedextensivelyin this book to describewave propagation. This appendix is intended to clarify their definitions.
A.1
Frequency
Assumea sourceemits the sinusoidalsignal in Figure A.1. The particle
displacementB(t) of the mediumat the sourceis shownas a function of time, t. The signal repeatsitself every 4 ms with the sameamplitude after a lapseof time T calledthe "period." Another measureof the signal'scycle is the frequencyf whichgivesthe numberof cyclesthe signalgoesthrough per unit time, or 1
f = •,
(A-l)
where the appropriate unit is hertz, abbreviated "Hz", which stands for "cycles per second."
Many times we will representa sinusoidalsignal in terms of the projection of a rotating vector onto some axis. Thus, we might represent the
particle displacementin Figure A.1 by the equation,
B(t)-[B[sin(-•t-•b), (A-2) 153
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154
APPENDIX
A. FREQUENCY AND WAVENUMBER
I B(t) ß
I
I
L*
FIG. A.1. Particledisplacement B(t) is plottedas a functionof time,t, for a sinusoidalsignal with a period T- 4 ms. The associatedfrequencyis, f - 1/T- 250 Hz. Angularfrequencyis definedasco- 2•rf - 2•r/T.
whereI B I is the magnitudeof the rotating vectorB(t) and •bis the phaseof
thesignalat timet - 0.• Sincethe vectorB(t) rotatesthrough2•rradians everyperiodT, we call 2•r/T in equation(A-2) the "angularfrequency" which is written as2
co- •- = 2•'f.
(A-3)
In Appendix B you will see that actual signalscan be thought of as a sum of many sinusoidalsignalsof differentfrequency,where each can have a different magnitude and initial phase. The Fourier transform is usedto
decompose an actualsignalinto its angularfrequencycomponents I B(w) I and •b(w). A.2
Wavenumber
When we definedangularfrequencycothe observationpoint of the signal waskept fixed in space(at the sourcein Figure A.1) and we studied the signal'spropertiesin time. In this sectionwe arrive at similar conceptsby observingthe signalthroughoutspaceat a fixed time. Figure A.2 showsa sinusoidalsignal traveling in either the +X of-X direction with a wavelength,,•, the distancerequired for the waveformto repeat itself. 1The phaseof the signal in Figure A.1 is zero at time t = 0. 2Both w and f are commonlyreferredto as "frequency."The contextof the equation will tell you which type of frequency.
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A.2.
WAVENUMBER
155
I B(x) I
I
Lø
FIG. A.2. Particle displacementB(x) is plotted as a functionof location, x, for a sinusoidalsignal traveling either in the +X or -X direction with
a wavelengthof A = 4 m. The associated wavenumberis, 1/A• - .25/m. The angularwavenumberis definedas k• -- -l-2•r/Awherethe signchosen is decidedby the direction of wave propagation.
Wavenumberis analogousto frequencyf, but is definedin terms of the pe-
riodicdistanceof the signalA asq-1/A. The signchosenfor the wavenumber is decidedby the directionof wave propagation. Most of the time we work with angular wavenumber, 2•r
k• - +-•-.
(A-q)
We generallyrefer to the angular wavenumberasjust the wavenumberand let the context of the equation define the type of wavenumber. Unlike time, wave propagation in spacecan be in many directions and actually must be defined in terms of a vector instead of a scalar. In higher
order dimensions of spaceequation(A-4) must be written as a vector, 27r,,
k = Tg,
(A-5)
where • is a unit vector pointing in the direction of propagation. Figure A.3(a) showsa sinusoidalwavepropagatingalongthe •, directionin a 2-D spacewith wavelengthX. We plot the wavenumberfor this wave in the
k• - k, planein FigureA.3(b). If ko = 2•r/A represents the magnitudeof the wave'swavenumber,then k in equation(A-5) can be decomposed into its vector components,
k
-
kog - k•,i+k,i,
(A-6)
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156
APPENDIX
A. FREQUENCY AND WAVENUMBER
(a)
(b)
FI•. A.3. (a) A sinusoidalwavepropagatesalongthe •; directionin a 2D spacewith wavelength A. (b) The wavenumber of the propagatingwave in FigureA.3(a)is k = (2•r/A)•;. The components of the wavenumber are k• and k•, both of positive value in this case.
where the valuesof k', and kz have signs determined by the direction of
wavepropagation;both positivein FigureA.3(b). Lastly,a singlefrequencywavewill travel onewavelengthin oneperiod. We may determinea specialvelocityof propagationfor this wavecalledthe "phasevelocity" defined by,
=
= Ikl =
(A-7)
Note that the phasevelocityis definedfor a singlefrequency.If a signal is composedof sinusoids of many differentfrequencies, then the velocityof propagationof that signalmay not be the sameas the phasevelocity,but a different velocity called the group velocity, definedas
V = Ok,, •:+ • •'
(A-S)
In this book we useo.,- Ck so that V - C and we assumeno dispersion.
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Appendix The
B
Fourier
Transform
Seismictomographyfrequently utilizes Fourier transformsof functions. Therefore, an understandingof the conceptis important to understandthe material presentedin this book. This appendix is intended to refreshyour memory of the Fourier transform. You might want to review Appendix A since we make use of the concepts of frequency and wavenumber in our discussion.
If h(t) is a function of time, then we representits temporal Fourier transform by,
FT[h(t)] :• •(w),
(B-i)
where w is the temporal frequency.The inverseFourier transform is represented by,
We refer to h(t) and h(w) as a Fourier-transform pair. Similarly,we may take a spatial Fourier transformof a function g(x), where x is a spatial coordinate, and represent the operation by,
FT[g(x)]
•
•(k•),
(B-3)
where k• is the spatial frequency,or angularwavenumber •, alongthe xdirection. As before, the inverseFourier transform operation is represented by,
rT-•[O(k=)] • • Most of the time the n•es ened to just frequency •d
g(x).
temporM frequency•d
wavenmber,
respectively. 157
•g•
(B-4) waven•ber •e short-
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158
APPENDIX
B.
THE FOURIER
TRANSFORM
Beforepresentingthe actual Fouriertransformequationsusedto compute the operationsimpliedby the aboveequations,we give a short review of the Fourierseriesand exponentialFourierseriesto providea transition; sincemost everyoneis familiar with the Fourierseries. Then, the Fourier transformof continuousfunctionsis discussed followedby the Fouriertransform of sampledfunctions.The last sectiongivessomespecialtransforms used in this book.
Fourier
B.1
Series
A periodicfunctionf(x) of length2L canbe writtenin termsof a series of cosinesand sinesprovided it contains a finite number of discontinuities and a finite number of maximum and minimum values. The series is called
a Fourier seriesand is representedby,
The seriescoefficientsao, an, and b• are given by,
1/_L
ao = -L •;f(x)dx,
1/_ œ
a, = --L Lf(z) cos\ L dx'
(B-6)
(B-7)
and,
-
fix)sin (-•-/dx
(B-8)
respectively. Equation(B-6) showsthat ao/2 is just the averagevalueof
f(x) overtheintervalI-L, L], commonly calledtheDCshift. Equation (B7) givesthesamedefinition asequation(B-6)for aowhenn = 0 andwill be usedhenceforth. Also,equation(B-8) requires bo= 0 for anyf(x). For our purposes, equations(B-5), (B-7), and (B-8) are best rewritten in terms of spatial frequency,or wavenumber 2, definedas 2•r radians
per wavelength.The longestwavelength for f(x) is 2L; determinedby 2Notethat wecouldequallyas well presentthis discussion in termsof a time coordinate t and its frequency w .
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B.2.
EXPONENTIAL
FOURIER
159
SERIES
settingn = 1 in equation(B-5). The corresponding wavenumber or fundamentalfrequencyis 7r/L. Higher valuedwavenumbers (shorterwavelengths)are foundby multiplyingthe fundamentalfrequencyby n where n = 1, 2, 3, ...;,n = i givesthe fundamentalfrequency.Thus, we now write the wavenumberalongthe x-axis as kn-- n7r/L and the Fourierseriesin terms of wavenumber as,
f(x) - -•-+
anosknx+bnsinknx
where the seriescoefficientsanand bn are defined by,
(In -
cos
(B-10)
-
f(x)sinknxdx.
(B-11)
Exponential
Fourier Series
•
L
and,
B.2
Here we rewrite the Fourier seriesof the previous section in terms of
exponentials,thusgettingusonestepcloserto the Fouriertransformwhich also usesexponentials. The key equation permitting this step is Euler's formula,
eJknx= cosknx +jsinknx,
(B-12)
wherej - x/Z-1. We recognize that the cosineis an evenfunctionand the sineis an odd functionand useequation(B-12) to write, cosknx =
sinknx =
ejknx + e-jkn x 2
-, and
ejknx _ e-jkn x
(B-13)
, respectively.
(B-14)
Substitutingequations(B-13) and (B-14) into equation(B-9) and regroupingthe terms with respectto the signof the exponentialgives, -jknx ß (B- 15)
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160
APPENDIX
B.
THE
FOURIER
TRANSFORM
Equations(B-10) and (B-11) showthat a, - a_, and b, - -b_,, respectively. Applying theserelationshipsto the secondserieson the right-hand sidein equation(B-15) and redefiningthe summationfrom n - -1 to -c• givesthe exponentialFourier series,
f(x) - Z c,•eJk,-,x,
(n-16)
wherec,.,- (a•- jb•)/2 and k• - n•/L. Substitutingequations (B10) and (B-11) for a• and b• in the definitionof c•, alongwith using equation(B-12), givesthe integralequationfor the seriescoe•cientsca,
C• -- • for n - 0,•1,•2,
B.3
•3,...,
Lf(z)e-Jk•xdz,
(B-17)
•c•.
Fourier Transform-
Continuous f(x)
The Fourier transform is intended to operate on nonperiodic functions
over an infinite range. Thus, we can no longerrestrict x to the range
I-L, L] asdefined fortheexponential Fourierseries.However, equations (B16) and (B-17) can be utilized in extendingz to infinite limits. First, we substituteequation(B-17)into equation(B-16) resultingin,
or
.•(k,•)eJk'• x__1 where 2L'
=
f(:rt)e _t, _jk,., X,dx •. L
The frequencyinterval betweensuccessive k• is given by,
Ak k•+•k• (n+1)• •r •r We rewrite this equation in a convenientform, 1
Ak
2L
2•r
(B-18) (B-19)
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B.4. FOURIER TRANSFORM- SAMPLED F(X)
161
Taking the limit of equation(B-20) as L--• c• allowsus to write,
lim (•L)- dk
œ--,oo
271'
(B-21)
and to write kn as a continuousvariable k. Now, taking the limits of equa-
tions(B-18) and (B-19) as L -• oo and usingthe resultfrom equation(B21) yields the Fourier transformequationsfor a nonperiodiccontinuous function,
I
f(:) -
I•(k)eJkzdk oo '
(B-22)
f(x)e-Jkxdx.
(B-2a)
and
i>(k) -
Equations(B-22) and (B-23) enableusto computeoneof the Fouriertransform pairsgiventhe other. Equation(B-22) is the inverseFouriertransform operationrepresented by equation(B-4) and equation(B-23) is the Fourier transformoperationrepresented by equation(B-3). B.4
Fourier Transform-
Sampled f(x)
The use of digital computersto carry out computationsrequiresthat we samplethe continuousfunctionf(x). We may take a samplefrom the continuousfunction f(x) at intervalsof Ax giving the discretesamples: ß.., f-2, f- •, f0, f•, f2,. ß., where the subscriptsindicate the sample number. If the sample number is n, then the positionof the sampleis x - nAx.
However,a sampledfunctioncan no longercontainwavenumbers (or frequencies)out to 4-00 as demonstratedby the integrallimits for the continuousfunction f(x) in equation(B-22). The shortestwavelengthrepresented by a sampled function is 2Ax which gives the highest possible wavenumber(or frequency),
knyq= Az'
(B-24)
referredto as the Nyquistfrequency. a Note that this alsoappliesto temporally sampled functions. aThe Nyquist frequencyhas a wavelengthor period which is sampled by two points; or we may state that the shortest wavelength or period in a sampled signal must contain at least two sample points.
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162
APPENDIX
B.
THE FOURIER
TRANSFORM
In sectionsof this book where we are referring to a sampledfunction, you will see the limits on the inverse Fourier transform equationswritten
as :t:•r. Equation (B-24) showsthat thesefrequencylimits correspondto a unit sample interval, Az = 1. A unit sample interval is commonly usedin
digital computationsto avoidmultiplyingby Az, whichbecomesnothing more than a scalefactor. Thus, you will seethe Fourier transformequations written as 4
f(x)- •1 , p(k)eJkXdk ,
(B-25)
P(k) -
(B-26)
and
f(x)e-Jkx dz,
whenf(x) and•(k) aresampled functions. Again,theclueto f(x) being sampledis the noninfinite frequencylimits. Of course,in actual computer
computations we usuallydo not useequations(B-25) and (B-26) directly. Instead, the Fast FourierTransform(FFT) method is utilized. B.5
Uses
of Fourier
Transforms
FromequaLion (B-26) weseethat F(k) is generallya complexvalued function, called the complex spectrum, which may be written as,
P(k) -
•Re{P(k)} + jlm{P(k)},
(B-27)
whereRe and Im designatesthe operationof taking the real and imaginary
partsof P(k), respectively. FigureB.1shows thelocation of onepointof the complexspectrumin the complexplane;so calledbecausethe real part of the complexfunctionis plottedon oneaxisand the associatedimaginary part on the other axis. The figure demonstratesthat the complexspectrum can be also describedin terms of polar coordinates.In polar form we can write the complexspectrum as,
P(•) - Ip(•)leJ•(•),
(s-28)
whereI P(•) I iscalledtheamplitude spectrum givenby,
]#(•)] - V/ee{#(•)} •+t,•{#(•)}•, 4Mar•y times geophysicists cha•ngethe sign in the exponentialof the forward and inverse Fourier transforms
when the transform
involves time.
This is done so that we
stay consistentwith the physics, that is, a wave traveling in the +x direction is described
byA(w,kx)eJ(kxxwt)andnotA(w,kx)eJ( kxx+ wt).
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B.5.
USES OF FOURIER
TRANSFORMS
163
Complex Plane
Im {F(k)}l (Re {.•(k)} ,lm {•(k)} ) > Re {F(k)}
FIG. B.1.
Complex plane with an arrow directed to the point
(Re{17'(k)},Im{17'(k)}). Thepointin polarcoordinates is represented by the magnitudeI F(k) I definedby equation(B-29) and the phase(I)(k) definedby equation(B-30). The magnitudeis calledthe amplitudespectrum and the phaseis called the phasespectrumwhen eachis plotted separately as a function
of k.
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164
APPENDIX
B.
THE
FOURIER
TRANSFORM
and (I)(k) is calledthe phasespectrumgivenby,
(I)(k)- tan -1 Im{F(k)} •
ß
(B-30)
The amplitude spectrumgivesthe magnitudeof a particular sinusoidat frequencyk while the phasespectrumgivesits shift in spaceor time. Most often we are interested in the frequency content of a signal and a plot of the amplitude spectrum servesthis purpose. Throughout this book we work in tl•e Fourier domain when it simplifies derivations. In addition to the above concepts of amplitude and phase
spectra, we encounterthree other applicationsof the Fourier transform' the 2-D Fourier transform, the Fourier transform of the time derivative,
andthe Fouriertransform of eJkox. Supposewe wish to take the Fourier transform of a function f(x,z) with respect to the spatial variables x and z. This operation is called a 2-D Fouriertransform. We beginby taking the Fouriertransformof jr(x, z) with respect to x, or
P(k•,z) -
f(z,z)e-Jk•zdz,
(B-31)
wherethe wavenumberalongthe x-axis is denotedby ks. Next we apply a Fouriertransformto equation(B-31) with respectto z, or
•(k,,,kz) -
•(k,:,z)e-JkzZdz,
(B-32)
where the wavenumberalong the z-axis is denoted by kz. Substituting
equation(B-31) into equation(B-32) givesthe definitionof a 2-D Fourier transform,
•(k•,k,) -
f(x,z)e-J(k•x + k•Z)dxdz ' (B-33)
Similarly, the inverse2-D Fourier transform may be definedas,
f(x,z)
/_•o /_••(k•, k•)eJ(k•x +k•Z)dk•dk• ' (B-34)
471-2
Time derivativesare simpler in the Fourier domain which is one reason many differentialequationsare solvedin the frequencydomain. Look at the inverse Fourier transform,
= 1
f'(w)e-Jwt dw,
(B-35)
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B.5.
USES OF FOURIER
TRANSFORMS
165
wheret is time andw is frequency.The first time derivativeof equation(B35) is, ef(t)
= -• • /_•o j.•(•)• _
dt
a•, -J•t
= FT-•[-j•P(•)].
(B-36)
A second-orderderivative is found by taking a time derivative of equation (B-36) or,
dt 2
(B-37) Thus, an nth order derivative is defined,
rr-•[(-i•)-p(•)].
dtn
(B-38)
WesimplymultiplyP(w) bytheappropriate factor s (-jw)" andtakethe inverse Fourier
transform.
ThelastitemtonoteistheFourier transform oftheexponential e-Jkox, which comesup many timesin diffractiontomography.We determinethis in a roundabout way. We use the Dirac delta function defined in Appendix C
in thisderivation. Taketheinverse Fouriertransform of .b(k)- 5(k- ko), wherethe Diracdeltafunctionis locatedat wavenumber koin the spatial frequencydomain. By equation(B-22) we find that, f(x)
= -2•r
5(k- ko)eJkxdk,
= l,j•o•. 2•r
Thus, theFourier transform ofeJkoxmust be2•rS(k-ko).
SUse(+jkx) n and (+jk,) n for spatialderivatives.
(B-39)
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Appendix
C
Greens
Function
Equation(60) in Chapter3 on seismicdiffractiontomographydescribes the propagationof the scatteredwavefieldP0(r) at a constantbackground velocitywhen inhomogeneities scatter both the incidentwavefieldPi(r) and existingscatteredenergyP0(r). We rewrite equation(60) here for your reference as
Iv • + •o•]•,(•) -
•o•U(•)[•,(•)+ •,(•)1.
(C-l)
Through the use of Green'sfunctionswe end up with an integral solution to thisequationgivenby equation(64), the Lippmann-Schwinger equation, which we rewrite here for your referenceas,
?o(,,) - -• f •(,,I•")•(,?)[?,(,?) +•0(•')]a•'. (c-•) G(rl rt) is the Green'sfunctionand the integralis takenovera planein 2-D space or a volume in 3-D space.
The intent of this appendixis to provideyou with an intuitive feelingfor Green's functions. With this understandingyou will know how to immediately write down an integral solution to any inhomogeneousdifferential equation with constant coefficients,such as the solution to the partial dif-
ferentialequation(C-l) givenby the integralequation(C-2). We do NOT go overtechniques for determiningthe Green'sfunctionG(r I r•) whichare readily found in many texts with chaptersdevoted to the subject, such as referencedat the end of this appendix. However, an example problem is
workedlater on to giveyousomeideaof howonemay solvefor G(r I r•) . The approachwe take for conveyingthe ideas of Green's functions is throughthe conceptsinvolvedwith filter theory. We do this for two reasons: 167
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168
APPENDIX
C.
GREEN'S
FUNCTION
Gt LINEAR OPERATOR
FIG. C.1. Gt is the impulseresponseof the linear operator giventhe input discrete impulse 50 at time sample t - 0.
(1) its easyto see, and (2) many of you are alreadyfamiliar with filter theory,especiallythe conceptof convolution.We will look at the discretely sampledcasefirst and then extrapolate to the analogcase.
C.1
Filter Theory
In filter theory we can input a discrete impulse 5• of unit height at samplet into a linear operator or systemwhich givesan output, calledthe impulse responseGr. Figure C.1 showsa plausibleimpulse responseof a linear operatorwhen a unit impulse5o is input at sample! - 0.• A linear systemis time invariant when the impulse responseG• is found at the same time lag relative to the time sample of the input impulse. For example, Figure C.1 showsan input impulse50 with an impulseresponseGt starting at time sample0. If we had usedan impulse5•, then the impulseresponse G• would have begun at time sample5 which may be written as Gt-•. The impulseresponseG• in Figure C.1 can also be scaled. We may multiply the unit impulseinput 5oby a scalarf0 whichresultsin the impulse response G•-0 beingscaledby f0 asshownin FigureC.2(a). The t-0 in the subscript of G•-0 indicates the delay of the impulse responseGt because of a delay in the input. Here no delay occurs and the output foG• is the responseto the input fo5o, where f0 ---1. We can scale the impulse input 51 by a scalar fl - 1 which results in the linear operatorresponseflG•-i - G•-i in Figure C.2(b). The subscript t-
I indicates that G• is delayed by one time unit in accordancewith the
time invarianceprinciplestatedearlier.Similarly,FigureC.2(c) showsthe impulseinput 52 scaledby f2 - -1/2 resultsin a linear operatorresponse 1We assumea unit interval, At = 1, betweensamples.
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C. 1.
FILTER
THEORY
'
169
fog t.o
-1 •o o OPERATOI•.
t
(a)
+
1Gt.1
1•1 LINEAR
0
!
OPERATOI•
o
t
•) +
-1/2õ2
f2 Gt.2 LINEAR OPF•ATO•
(c)
01--1 ft llU-
LINEA• OPF•ATOI•
!øi
Pt
(d)
FIG. C.2. Illustration of the principle of superpositionfor a linear operator. The input consistsof a seriesof scaled, time-shifted impulsesgiven by
ft -- j•050-}'f151 -}'j•252,wherefo = -1 in (a), fl ----1 in (b), and/2 = -1/2 in (c). The resultingoutput to f• is the sum of the responses in (a), (b), and (c): P• - foG,-o 4-fxG,_x + f2G,-:• shownin (d). This is just an example of convolutionof an input signal ft with an impulseresponseof a linear filter
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APPENDIX
170
C.
GREEN'S
FUNCTION
FigureC.2(d)is thesumof the inputsandoutputsfromFiguresC.2(ac). The discretizedinput is thoughtof as a sum of scaledimpulsesgiven by,
ft -- fo•o q-fl•l q-f2•2-- (-1)50 + (1)51 q-(-•1)5:•.
(C-3)
The linear operator'sresponseto the input function ft is given by, ---- foGt-o q- flGt-1 q- f2Gt-2
= (-1)G,_0 q-(1)G,_l q-(-•
(c-4)
Equations(C-3) and (CL4)combinedillustratethe principleofsuperposition for a linear operator. That is, the same output results whether a linear
operatoractson the entire input as depictedin Figure C.2(d), or on each individual componentof the input as shownin Figures C.2(a-c) and the results summed.
Generalizingequation(C-4) we have,
This equationis the well-knownconvolutionequationfor two discretized signals. Here the convolutionis between the input function .It and the impulse responseof the linear operator Gt which givesthe responseof the linear operator Pt. If the above functions are continuous,then the sum in equation(C-5) becomesan integralor
P(t)- • f(t')G(t - t')dt',
(C-6)
wheredr' is explicitlywritten in placeof At' in equation(C-5) whichwas assumedequal to one.
C.2
PDE's as Linear Operators The partial differentialequation(C-l) can be cast in the samelight as
thelinearoperatorin filtertheory.Thesourceterm,ko•M(r)[Pi(r)+P,(r)], is analogousto the input to the linear operator. The linear operator is the
bracketedpart of the functionon the left-handside,IX72+ ko•], and the solutionto the differentialequation,P,(r), canbe thoughtof asthe output from the linear operator.
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C.2.
PDE'S
AS LINEAR
-6 (r-r')
OPERATORS
•
171
•2+k2 ] • G(rlr' oI )
- -5(r-r') [•2+k2o] G(rlr') FIG. C.3. The Green's function is the impulse responseof a linear differential equationoperatorto a sourceterm (input) givenby a negativeDirac delta function. Comparethis with the impulseresponsein filter theory depictedin Figure C.1.
As in the previous section, the first step is to determine the impulse responseof the linear operator to an impulsive input. For the partial dif-
ferentialequationwe usea negativeDirac deltafunction 2 -5(r-
r •) for
the impulsiveinput and the Green'sfunctionG(r- r •) for the impulseresponseto the negative Dirac delta function. Thus, using the analogiesin the previousparagraph,equation(C-1) is rewritten as,
IV2+ ko•lG(r I r') -
-5(r-r'),
to get the impulseresponse,or Green'sfunction,G(r I r') = •(r-
(C-7) r').
Note that the space vector r is the independent variable here instead oftime used in the previoussectionon filter theory. Figure C.3 summarizes the analogy made between the differential equation operator and the filter theory operator in Figure C.1.
The Green'sfunctionsolutionG(r I r •) to equation(C-7) is the impulse responseto the negative Dirac delta function. Just as with filter theory,
we may weight the negativeDirac delta function,say -f(r')5(rr'), on the right-handsideof equation(C-7) whichresultsin the solution(output) f(r')G(r Jr'). Using the principleof superpositionshownin Figure C.2, the resultingresponse isjust the integration(sum) overall outputresponse 2The Dirac delta function6(x) has ax•infiaitesi•
width, infiniteheight,and an
area equal toone atx = 0. It isdefined asf_+•S(x)dx = 1. When combined with another function, f_+•!(z)a(z-Xo)= l(Zo),where theDirac delta function islocated at 3• '-' 3•o.
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172
APPENDIX
C.
GREEN'S
FUNCTION
componentsfrom the linear operator, or
: - fr,
(c-s)
wherethe negativesignin front of the integralis presentbecausethe Green's function is the impulse responseto a negative Dirac delta function. Equa-
tion (C-8) is similarto equation(C-6) exceptfor the negativesignandthe independent variable.
Settingf(r') in equation(C-8) to the right-handsideof equation(C-l),
becomesthe sourceterm (input to the linear operator) in equation(C8) and we get the integralsolutiongivenby equation(C-2). Thus, the integralsolutionto equation(C-l) isjust the negativeof the convolution of the sourceterm (input) f(r) with the Green'sfunction(impulseresponse), analogousto filter theory. The r- r • part of the Green'sfunction(see equations(C-9) and (C-10) below)givesthe "lag"in the impulseresponse as a result of the weightednegativeDirac delta function being located at a position other than r • = 0. This is a space-invariantproperty equivalentto the time-invariant property of the linear operator in the previous section on filter theory. Chapter 3 on seismic diffraction tomography makes extensiveuse of Green's functions. However, closeinspection reveals that only two Green's
functionsare actuallyevermentioned,both aresolutionsto equation(C-7). The 2-D solutionto equation(C-7) is exclusivelyusedin Chapter 3 and is given by,
[r - r' I), a(rl') - JHo(1)(ko whereHo (1)is thezero-order Hankelfunction of thefirstkind. Herethe negative Dirac delta function representsan infinite-line seismicsourceor an infinite-line scatterer at r' which causesa cylindrically shaped seismic disturbance
that is determined
at an infinite-line
field location
r. Both the
infinite-line source and infinite-line field location are perpendicular to the
plane representingthe 2-D space. A 3-D spacesolutionto equation(C-7) is givenby,
ejkolr-r'l
G(rlr')= 4•rlr-r']'
(C-10)
This equation is not usedin Chapter 3 sincemost data acquisitionschemes, suchas crosswellseismic,are gearedto 2-D spaceor cross-sectional studies
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C. 3.
GREEN'S
FUNCTION
EXAMPLE
173
rather than volumetric investigations.The 3-D spacesolution represents point seismicsourcesor point scatterersat r t which causea sphericalseismic disturbancethat is determined at a point field location r.
C.3
Green's Function Example
In this section we find the integral solution of a simple 1-D differential equation to reinforcethe ideas presentedin the last section. The method of determining the Green's function here is just one example of many and we refer you to the referencesat the end of this appendix for other methods. The differential equation is,
[d-••a2]p(x) - f(x), for -cx> <x
dx - aV']G(x) - -5(x).
(C-12)
We solvefor G(x) by taking the Fouriertransformof eachterm in equation (C-12) as discussed in Appendix B which gives,
[(jk)2 - aV']0(k) -
-1,
(C-13)
wherek is the wavenumber andj - v/Z'-•. Solvingfor (•(k) givesthe Green's function in the wavenumberdomain,
=
+1
(C-14)
Taking the inverseFourier transformof the last equation givesthe Green's function for a negative impulsive sourceterm at x - 0,
G(x) -
2a
(C-15)
If the negative Dirac delta function werelocated at x - x', then the Green's function would be written with a spatial lag,
G(xI x') =
2a
'
(C-16)
3Assumeno lag, x t = O, in the Dirac delta function at this point, and introduce the lag when doing the convolution(sum of weightedoutput from the linear operator) betweenf(x) and G(x).
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174
APPENDIX
C.
GREEN 'S FUNCTION
whichis usedin the integralsolutionin equation(C-17). We knowthe integralsolutionto equation(C-12) isjust the negativeof the convolution betweenthe Green'sfunctionG(x) and the sourcefunction f(x) or,
p(x) = -/-+2 f(x')G(z Ix')dz ',
(C-17)
wherethe x - x' in the Green'sfunctionis just the spatial lag as a result of the weightednegative Dirac delta function being located at x = x'. Substitutingequation(C-16) intoequation(C-17) givesthe integralsolution to our example,
2a
dx'.
(C-18)
We can now solvefor p(x) by integratingequation(C-18) oncethe source term f(z) is definedin equation((3-11). As a quick checktry a negative impulsivesourcelocatedat x'= 0 givenby f(x')= -5(x•). Your solution shouldbe equation(C-15). C.4
Suggestions for Further Reading The following referencescover the computation of Green's functionsin
detail.
Arfken, G., 1970, Mathematical methodsfor physicists,2nd edition: Academic Press, Inc. Courant, R., and Hilbert, D., 1953, Methods of mathematical
physics:JohnWiley and Sons(Interscience). Morse, P.M., and Feshbach,H., 1953, Methods of theoretical physics: McGraw-Hill.
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Index acoustic wave equation 46 Helmholtz acoustic
form
wave field
data acquisition crosswell configuration 2
46
data
46
function
11
algebraic reconstruction technique 33 amplitude spectrum 162 angular frequency 154
observed pobs
ari thmeticreconstructiontechnique(see Aa)
predicted P or prre
related
to true
model
function
M true 23 related
to estimated
model
func-
tion M or M est 23
ART 33, 41 compared with SIRT 37 nonlinear aspect 38
vector DC
shift
form
27
158
diffraction tomography 5 background velocity C'o 48 backprojection ray tomography 20, 40 CAT
formula
function
50
function
51
data
21
function
Born approximation 52, 138 Rytov approximation 56
sununary 21
backpropagatlon diffraction tomography
genericdata/model relationship59
87
illustrated
crosswell configuration 88 Jacobian
87
model function 45, 49
scatteredwavefieldPs(r) 47
surface reflection configuration 90 vsp configuration 89 borehole survey 99 Born Approximation 51 Rytov approximation comparison
total
wavefield
Born approximation 48 Rytov approximation 53 use with ray tomography 6 wavelength consideration 5
56
when Dirac
cell size 108
delta
to use 45 function
171
direct arrival 101, 101 direct-traxmform diffraction tomography
color scale 133
85
complex spectrum 162 polar form 162 computerized axial tomography 22 convolution equation 170 continuous
137
incident wavefieldPi(r) 47
procedure for 143
crosswell
Green's
Green's
constant density assumed 48
scan 22
reconstruction
2-D
3-D
form
steps for implementing 86 direct-transform ray tomography 16, 17, 39
170
seismic
Enhanced oil recovery 95
advantage over well logs 4
EOR
benefit
estimated model vector Euler's formula 159
of 3
crosswellseismic gathers 106 175
95 28
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176 first
INDEX arrival
101
advantages of 28
forward modeling defined direct
flow chart
arrival
102
discrete formulation first as'rival 102
formulated
25
in a continuous
domain
23
matrix
form
26
time invariance
in terms of wavenumber Fourier transform 2-D 164
159
nonperiodic continuous functions 161
of e-J køx 165 operation symbols 157 representation 161
sampled function 162 derivatives
164
transform pairs 157 frequency 153 generalized projection slice theorem 58
crosswellconfiguration68 statement
of 70
surface reflection configuration 82 vsp configuration 76 gray scale 133 function
171
2-D solution to wave equation 172 3-D solution to wave equation 172 example 173 group velocity 156 head wave hertz 153
168
Lippmann-Schwinger equation 51 Born approximation 51 linearized
52
nonlinearity 51 lithology interpretation of P-wave tomogram 120 magnitude 154 McKittrick
Field
125
medical tomography configuration 10 data function
11
Midway Sunset Field 96 model function 11, 49 cell size 108
diffraction tomography45, 49 discrete
24
estimated
M or M est
related to predicted data function P or ppre
23
incremental updateAiM•' 30 initial estimate 28, 108, 109 new estimate M(new) est 28 related
to observed
tion pobs 23
101
vector
form
27
nonunique 70
image function 11 impulse response 168 wavefield
28
.true M true
hyperplane 30
incident
matrix
principle of superposition170 scaling 168
series 158
exponential form 160
Green's
27
linear system
need for 23
time
Laplacian operator 46 linear inverse problem 13 generalized inverse operator ill-conditioned
raypath lengths 27 Fourier
29
24
47
Nyquist frequency 161 object function 11
integration notation 51
observed data vector 28
Jacobian
partial differential equation linear operator 170 period 153 phase 154 phase spectrum 164 phase velocity 156
87
crosswellconfiguration 88 surface reflection configuration 90 vsp configuration 89 Kaczmarz' method 26, 40
data
func-
Downloaded 05/05/14 to 192.159.106.200. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
INDEX
177
plane wave decomposition 60 porosity tomograrn 122 predicted data vector 28, 28 projection 13 projection data '1 projection slice theorem defined
ray 5
series expansion method 9, 22, 40 estimated model function, M ½st23 forward modeling 26 Kaczmarz'
simultaneous
direct-transform
method
17
wavelength 9 wavelength consideration5 with diffraction tomography 6 ray tracing direct arrival 102 first arrival 102 information
traveltime slowness
136
70
Rytov approximation 56 Born approximation comparison 56
complex incident phase function
P-wave
air gun 103 clamped vibrator 131 S-wave
air gun 105 clamped vibrator 132 spatial frequency 13 steam flood enhanced oil recovery 95 tomogram 1 base line 96
lithology interpretation 120 McKittrick
function
137
observed seismic
119
52
seismic
total
downhole air gun 100 downhole clamped vibrator 129 marine
true model
surface
seismic
diffraction
5
function
26
tomography 1
data
seismicray tomography 39 seismic tomography 1
134
core study 121,123 reliability check 124 selecting a scale 133
salt sill problem 137 diffraction tomograxn 147 prestack depth migrated section 148 47
thrust
meaning of colors 1 porosity estimation 122 presteam vs. poststeam injection
complex total phase function 53 Rytov-data function 55
wavefield
117
source radiation pattern
complex phase difference function
scattered
residual
11
air gun 98 downhole clamped vibrator 127
based on well logs 126
on data
tech-
source
110
orienting horizontal components air gun source 102 clamped vibrator 133 reservoir interpa•tation based on tomogram a•ad well in-
note
reconstruction
SIRT 35, 41, 99 compared with ART 37 example 110 handling nonlinear aspect 110 nonlinear aspect 38 ray density weight 36, 42 terminating iterations 111
receiver
formation
iterative
nique (seeSIRT)
ray density 110 ray tomography 5 assumptions 9 backprojection 21
resolution
26
predicted data function, prr½ 23 true model function, M true 23
16
essence of 14
SIRT
method
observeddata function,pobs23
141
1
wavefield
Born approximation 48 Rytov approximation 53 transform methods 9, 39 traveltime parameters 99
Downloaded 05/05/14 to 192.159.106.200. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
178
INDEX
traveltime pick consistency crosswell seismic data
107
mispick 108 poor data 108 surface seismic data 104
traveltime picking 99 direct m'rival 102 first arrival 102
polarity change
air gun (P-wave)103 air gun (S-wave)105 claxnped vibrator (P-wave)131 clampedvibrator (S-wave)132 radiation pattern 102 tying crosswell data 107 using computed traveltimes 108 traveltime residual 111, 117 true model
vector
28
velocity perturbation 47
wavelength 154 wavenumber 13, 155 angular 155 vector components
155
well logging disadvantages3
Wyllie's time averageequation 121
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