All of the perturbative approaches to multidimensional wave equation processing. for example. wave equation migration (see, e.g., Claerbout, 1971; French, 1975: Schneider, 1978; Stolt, 1978; Sattlegger et al, 1980), or Born approximation inversion (see, e.g., Cohen and Bleistein, 1979; Raz, 1981: Clayton and Stolt, 1981) require some input velocity information. In the Born approximation to inversion, a reference or background velocity is chosena nd a perturbationa boutt his velocity is determined.S imilarly, a velocity model is a required input to all wave equation migration techniques.

1. Seismic 13176
FIG.6. Reconstructionwith depthcorrection.100percentcontrast,
Cr = 2000 misec.
ity contrasts(50 and 100percent)in Figures5 and7, distortionin
thereconstructionsis quiteserious.The improvementof thedepth
correctionshownin Figures4, 6, and 8 is apparent.
Futurework
We intendto examinethe effectsof the two nonlinearcorrec-
tionson different scatteringgeometrieswhich includesingleand
multiple scatterers.When this is done successfully,we intendto
apply the inversionalgorithmto real seismicreflectiondata.
Sensitivity of Born Inversion to the Choice
of Reference Velocity: A Simple Example
A. B. Weglein, Cities Service; und S. H. Gray,
Amoco Production Research
S13.6
We examine the sensitivity of the Born model to the input
backgroundvelocity. We usea one-dimensionalanalyticexample
to point out the difference between a correctiveprocedureand
merely a perturbativeone. We examine variousaspectsof the
sensitivityissue,includingthetrade-offbetweenvelocitydetermi-
nationand mappingof reflectorlocation. Lnparticular,we show
thatonechoicefor thebackground,orreference,velocityCx leads
to an accuratedeterminationof thelocationof thefirstreflectorbut
an inaccurateestimationof the velocity below that reflector. A
secondchoice for CR can reversethis situation,accuratelyesti-
matingthe velocitybutnot thereflectorlocation.Also, thereis a
rangeof choicesfor C, for whichtheresultsof an inversionmay
actuallyyield lessaccuratevelocityestimatesthantheestimateCR
itself. Althoughthisproblemisdiscussedwithin thecontextof the
Born model, it is an issuecommon to all perturbativemethods
(e.g., migrationmethods)which transformsurfacereflectiondata
into a mapof subsurfacereflectors.
All of the perturbativeapproachesto multidimensionalwave
equationprocessing.for example. wave equationmigration(see,
e.g., Claerbout,1971;French, 1975:Schneider,1978;Stolt, 1978;
Sattleggeret al, 1980), or Born approximationinversion(see,
e.g., Cohenand Bleistein, 1979;Raz, 1981:Clayton and Stolt,
1981) requiresome input velocity information.In the Born ap-
proximationto inversion,a referenceor backgroundvelocity is
chosenanda perturbationaboutthis velocityis determined.Simi-
larly, a velocity model is a requiredinput to all wave equation
migrationtechniques.
The purposeof this paperis to examine the sensitivityof the
Born approximationto this inputinformation.One importantas-
pect of this questionis whetherthe perturbation,asgiven by the
Born inversion, will be of a corrective nature. In this context
correctivemeansthat an improvementof, or correctionto, the
estimatedbackgroundvelocitytakesplaceafterapplyingtheBorn
inversion.Specifically,onewouldlike resultsof theBornapproxi-
mationfor the velocityto be closerto the actualvelocitythanthe
backgroundvelocity is. A secondaspectof thisissueis thetrade-
off betweenvelocitydeterminationandreflectormapping.Thatis,
one would hopethat it would be possibleto determinecorrectly
both the locationof a reflectorand the velocity below that re-
flector. We illustrate,by meansof a simpleexample, thesetwo
aspectsof the sensitivityissue.
Considera I-D acousticmediumwhere the propagationof the
wave field P(z ,r) is governedby:
(1)
wherec(:) is thelocalacousticvelocity. Characterizetheacoustic
velocityc(: ) in termsof a homogeneousreferencevelocityCKand
a variationin the indexof refractionu(z) asfollows:
1
-=$I WZ)].
c*(2)
In equation(2), the numberC, is an input into the model; it is
chosen(or guessed)beforethe inversionis carriedout.
For simplicity, we assumethat an impulsivesourceis usedto
probethemedium,andthatthissourceis locatedat; = 0, within a
half-spacewherethe acousticvelocityhastheconstantvalueC,.
Thus the incident field is given by P,(;,,) = 6(t -z/C,,). The
total field is the sum of the incident field and a scatteredfield
P,(z,t). Under these circumstances,Gray and Bleistein (1980)
showedthat, within the Born approximation,the variationin the
indexof refractiona(:) can beexpressedin termsof thescattered
field PJz ,t) at ; = 0 by
I
nz:r,
a(:) = -4 P,(o,t)c~r.
0
(3)
From equations(2) and (3), the reconstructedvelocity depends
explicitly on the choiceof referencevelocity.
To simplify our examplefurther, we assumethat the medium
consistsof two half-spacesin contact,with acousticvelocitiesC0
andC , andthe interfacebetweenthem locatedat a depthZ0 (see
Figure 1). Thenthereflectedfield returnedto a receiveratz = 0 is
given by
4v(O,r)= = S(t - 2Z,,/C,J.
C, +c,,
For our examplewe set C,, = 1, C, = 917, andZ,, = 1, and find
from equation(4) that
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2. Seismic 13 177
-fz= 0
t +Z
FIG. I. Two half-spaceswith acousticvelocitiesC,, andC
P,(O,f) = * S(1~ 2) = $6(f - 2),
which is independentof the referencevelocity C,<. Now we can
calculatethe Born CX(:)from equation(3):
From this resultandequation(2). it is clear that
(,“(,) = c‘,f_
i
; cc,
v2 c,, : > c,,
The superscriptB indicatesa Born inversion velocity CH(:) =
C,( 1 + (Y)~“’ given by equation(3).
Now we examinetheeffectsofdifferentchoices,or guesses,for
CKon thequalityof the inversion.lf we chooseCK = 1, thenthe
locationof thereflectorwill becorrectlydeterminedbutthecalcu-
lated velocity in the region: > 1 will be too large. On the other
hand, for a different choiceofC,( (i.e.. C,?= 9/7,/? = ,908).
the velocity below the reflector is correctlydeterminedbut the
location of the reflector is incorrect. We see from this simple
exercisethat it is impossibleto simultaneouslyfind the correct
locationandsizeofa singlereflectorin theBornapproximation.It
is alsoimpossibleto simultaneouslypredictthecorrectvelocityin
boththe half-spaces.
Next, we showthatacertainrangeofchoicesfor CHcanleadto
velocityestimatesin the lower layer [via equation(5)] which are
actuallyfurtherfrom the true velocity thanCK is. Thus, a certain
rangeof values for C,! can lead to incersions which are not correc-
tive. Moreover,it istruein generalthat therangeof valuesfor CH
cancontainvaluesvery close to C,, (for this example the range IS
1.065< C,(). In fact, it can be shownthat the closerC, is to C,,
(i.e.. the smallerthe computed(Yis in the secondhalf-space),the
closerthis noncorrectivesetof valuesfor C,, comesto containing
the numberC,,. This is dur to the fact thatthe bdundarybetween
corrective and noncorrectivechoices for referencevelocity lies
betweenC,, andC , This itnpliesthateventhoughthetruevalueof
N canbearbitrarilysmallfor; > C,, , thereconstructedvelocity(,’
can be a less accurateestimate for : >CCKthan the reference
velocityC,<.Thisisa somewhatsurpriarngresult,in thattheamall-
nessof thecyis usuallyusedtojustify the Bornapproximation.In
this sense.the Born inverxionis extremelysensitiveto the choice
of referencevelocity.
Thus, thissimpleexampledemonstrateshow the choiceof ref-
erencevelocityin theBornmethoddependson what isconsidered
theprincipalobjectiveof the inversion.We haveshownthatif the
primaryaim is to determinethe locationof the first reflectorthen
theoptimalreferencevelocityshouldbe chosento be thevelocity
of thefirstlayer. However, we haveshownthatthis isnotthebest
referencevelocity if the principalaim is to determinethe velocity
of thelowerhalfspace.In situationswherethevelocitystructureis
morecomplex,thechoiceof referencevelocitywill againdepend
on the objective of the inversion. If, for example, the velocity
increasesordecreasessteadilywithdepth,thechoiceC,<= C,,will
leadtolessandlessaccuratemappingof reflectorswithdepth(see,
e.g., Claytonand Stolt, 1981).
In conclusion,the sensitivityof Born inversionto the input
quantityCR leads to a trade-off between reflector location and
velocity determinationin the simplest possible example. Further
workwill be requiredtoevaluatemorecomplicatedmodelswhere,
perhaps,globalcorrectivemeasuresshouldbedefined.Thechoice
of a referencevelocityto optimizesomejudiciouscombinationof
velocity determinationand reflector mapping could be useful.
These issuesalso need to be investigatedfor nonconstantback-
groundBorn inversion(seee.g., Clayton and Stolt, 1981;Weg-
lein, 1982).
Explorationistsalreadyhavea qualitativeawarenessof someof
theissuesraisedin thispaperin relationtotheanalogousmigration
methods.This simple example, usingthe Born model, helpsUS
beginto quantifythis understanding.
References
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Geophysics,v. 36.p, 467-481.
Clayton, R.W., andStolt, R. H., 1981. A Born-WKBJ inversion method
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Cohen. I. K., and Bleistein, N., 1979, Velocity inversion for acoustw
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