Examples of a Nonlinear Inversion Method Based on the T Matrix of ScatteringT heory:A pplicationt o Multiple Suppression

1. Examplesof a NonlinearInversionMethodBasedon the T Matrix of
ScatteringTheory:Applicationto Multiple Suppression
E M. CarvaIho? PPPG/Federal Univ. of Bahia and Petrobras,Brazil;
A. B. Wegiein,ARC0 Oil and Gas;and R. H. Stolt, Conoco Inc.
SP1.4
Multiple suppression is an important,
longstanding and only partially solved problem in
exploration seismology. A multidimensionalmethod
derived from inverse scatteringtheory (A. B. Weglein
and R. H. Stolt,1991) invertsseismicreflectiondata and
removes multiples. The proceduredoes not rely on
periodicityor differentialmoveout,nor does it requirea
modelof the multiplegeneratingreflectors. An estimate
of the sourcesignatureis required. The inversesolution
is expressedin a series. Each successive term in the
series requires a migration-inversionof progressively
more complicated data. However, in this nonlinear
methodeach migration-inversionis performedwith the
6ame constant velocity mig.@ion-inversionoperator.
That is, in contrast to iterative linear methods, the
operatorinvertedisthe sameforeachterm inthe series.
Our experienceindicatesthatconvergenceis rapidfor a
wideclassofexamples. Forvery strongreflectorsand a
homogeneous starting model, four or five terms
produced excellent results. In principle,the method
suppressesall multiples.However,in itspresentform,it
is computationallyfeasible for removing all surface
multiples.
lCiTRODUCTlON
Methodsfor removingmultiplereflectionshave a
longhistoryin explorationseismology.At the workshop
on the Suppressionor Exploitationof Multiplesat the
1990 SEG annual meeting, several excellent papers
reviewed and comparedthese techniques. Hardy and
Hobbs (1991) also presentan overviewand proposea
multiplesuppressionstrategy.
NMO-stacking,f-k and p-z filteringare examples
of moveoutbased methodsrequiringstackingvelocity
information. NMO-stackinggenerallyis effectivewhen
the moveout differences between a primary and a
multiplewiththe same zero offsettime for 10,000 ft of
offset is, after moveoutcorrection,greater than 40 ms.
Optimizedp-z filtering methods usually can separate
primary from multiple when, in the same moveout
correctedCMP gather,theydifferbyat least10 ms.
Predictivedeconvolutiontechniquesdepend on
multiplesbeing periodicreplicasof the primaryat each
receiver. They don’t dependon moveoutdifferencesOr
require stacking velocity information. However, their
shortcomingisthatthey can predictonlymultipleswhich
are periodic. Space-timemethodsare effectiveOnlyat
(or near)zero offsetand (near)verticalincidence. P- Z
methodsin principleextend periodicityto all p-values,
butin practicehavetheirownproblemsand limitations.
Modeling and subtraction methods reqUire an
accurateestimateof the sourceof multiplesand, hence,
are appropriate for water bottom multiples (Wiggins,
1988).
As expected, discontinuousreference velocity
linear inversion (e.g., Lui, 1984) suffers the same
restrictions/benefitsas wave equation modeling and
subtraction.
Ware and Aki (1969) and Verschuur,et al. (1988)
are among those using a reflectivitymatrix model of
seismicdata (in one and multidimensions,respectively)
to develop algorithms for the removal of surface
multiples.
RFAI ISTIC NQbllINFAR SFlSMlC INVFRSlDN GO&&
f!d!JtTlPr F SUPPRFSSU
Nonlinearinversionmethodsoftenare viewedas
a procedure to improve the parameter estimation
capabilityof a linearmethod.A lessambitiousand more
realistic goal for nonlinear seismic inversion is the
suppressionof multiples.
In exploration geophysics, iterative linear
inversion(in all its differentforms)is commonlythought
of as the only nonlinearapproach. In fact, there are
numerousalternatenonlinearapproaches. The T matrix
formalism of scattering theory provides a useful
frameworkfor inverse scatteringproblemsand various
linearand nonlinearapproaches.
Amongearlyworksinthisarea are Moses(1956)
Wolf (1969), Razavy (1975) and Prosser (1980).
Weglein, Boyse and Anderson (1981) and Stolt and
Jacobs (1980a, 1980b) adopted and extended this T
matrixtechnologyforthe surfaceseismicproblem.
DefineGreen functionsG, Go, wave functionsP,
PO,and differentialoperatorsL and Loin the actualand
reference media, respectively. A(w) is the source
signature.
The Lippman-Schwingerequationexpressedintermsof
P and POis
P=<+G,JfP
where
V=Lo-L
Definethe T operatoras
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2. 2 Inversion and multiple suppression
TGos VG
(1)
From this definition and the Lippmann-Schwinger
equation, one finds
T=V+VGV
T=V+TGoV
Define
A A
8 and s as projectionsonto the receiver and
source planes, respectively.
Then the seismic scattered (or reflection)data is
D=P-p,
D / A = ABGoVGAs= hgGoTGohl(3)
The goal is to determine V from measured values of D
on the surface of the earth.
Expand T and V in a power series in D/A
T=‘&“T, v = &v,
n=l
(4) ?I=1 (5)
where E is a parameter used to track the power in the
data D/A. Afterthe calculationof T and V, E is set to one.
Substitutionof (4) and (5) into (2b) and (3) and equating
equal powers of s,
In the second example (Figure 2a) the moveout
difference between the primary from the lower reflector
and the firstmultiplefrom the upper reflectoris 6 ms over
10,000 feet of offset. This is in the range (0 to 10 ms)
where differential moveout methods would have
difficulty. Figures 2b and 2c show VI and Vl+V2 for this
model. The multipleis removed in Vl+V2.
from &I CONCLUSIONS
D / A = hgGoV,G,,As
from c*
(7)
from c3
and so on.
Accordingto equations (6), (7) and (a), each successive
:erm in the series for V is found by migration-inversion
withthe same operator G
L
- _
Typically, the right hand sides of equations (6), (7) and
(8) require a volume integral over the subsurface for
each factor Go. For a homogeneous reference medium,
with a free surface, Go consists of two terms. The first
propagates directly from point (A) to point (B) in the
subsurface, whereas the second represents the
propagationfrom (A) to the free surface and then to (B).
If one ignores the first of these and retains the second,
then the right hand sides of (6), (7) and (8) become
surface integrals. (see Stolt and Jacobs, 1980b
Weglein and Stolt, 1991) This amounts to keeping
surface multiplesand ignoringinterbed multiples.
EXAMPLES
We show test results for two one-dimensional
models. In each case our data is a shot record Fourier
transformed analytically over offset. Go is the Green
function for a homogeneous medium, with a free
surface, whose properties correspond to the layer
containingthe source and receiver.
Figure 1a illustrates the first model. Figures 1b
and lc give the inversion result for the V1 and Vl+V2 +
V3 + V4 terms, respectively. As expected, the surface
multiples are suppressed and the small interbed
multiples remain.
The nonlinearity of a new wave theoretic
inversion method can be exploited for multiple
suppression. The multidimensional method does not
rely on moveout differences, periodicityor modeling.
The procedureconsistsof a series of uncham&g
migration-inversionoperations applied to a sequence of
effective data. Initial synthetic data tests indicate rapid
convergence for a wide range of models. An estimate of
the source signature is required.
-mGFMFNTS
CNPQ and Petrobras are thanked for supporting
PMC and ABW duringa sabbaticalyear in Brazil. ARC0
Oil and Gas Co. and Conoco, Inc. are thanked for
continuous support and encouragement. Doug Foster
and Tim Keho are thanked for helpful discussions and
comments.
J
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3. Inversion and multiple suppression 3
--..
PEFERENCFS
Weglein, A. B. and Stolt, R. H. (1991), I. The wave
physics of downward continuation, wavelet estimation,
and volume and surface scattering: II. Approaches to
linear and non-linear migration-inversion,“Mathematical
Frontiers in Reflection Seismology” edited by W. W.
Symes, SIAM/SEG.
Stolt,R. H. and Jacobs, B. (1980a), “Inversionof seismic
data in a laterally heterogeneous medium”, Stanford
Exploration Report No. 24, pp. 135-152, (1980b) “An
approach to the inverse seismic problem”, Stanford
ExplorationReport No. 25, pp. 121-I 34.
Weglein, A. B., Boyse, W. E. and Anderson, J. E. (1981),
Obtaining three dimensional velocity informationdirectly
from reflection seismic data: An inverse scattering
formalism: Geophysics,V. 46, no. 8.
Lui, C. Y., (1984), “Born inversion applied to reflection
seismology”, Ph.D. Thesis, U. of Tulsa, Department of
Geophysics.
Moses, H. E. (1956), Calculation of the scattering
potential from reflectioncoefficients, Phys. Rev., V. 102,
pp. 559-567.
Razavy, M. (1975), Determinationof the wave velocity in
an inhomogeneous medium from the reflection
coefficients, Journ. Acousti. Sot. Am., V. 58, pp. 956-
963.
Devaney, A. J. and Weglein, A. B. (1989), Inverse
scattering usingthe Heitler equation. Inverse Problems,
December, 1989, V. 5, No. 3, pp. 49-52.
Hardy, R. J. J. and Hobbs, R. W. (1991), A strategy for
multiple suppression, First Break, V. 9, No. 4, April,
1991.
Verschuur, D. J., Herrmann, P., Kinneging, N. A.,
Wapenaar, C.P.A. and Berkhout, A. J. (1988),
Elimination of surface-related multiply reflected and
converted waves, 58th Meeting, Society of Exploration
Geophysicists, Expanded Abstracts, pp. 1017-l 020.
Ware, J. A., and Aki, K. (1969), Continuousand discrete
inverse scattering problems in a stratified elastic
medium. 1. Plane waves at normal incidence. Journ.
Acoust. Sot. Am., V. 45, pp. 91 l-921.
Wiggins, J. W. (1988), Attenuation of CWTIpleXwater-
bottom multiples by wave-equation based prediction
and subtraction,Geophysics, V. 53, pp. 1527-1539.
REFLECTOR 0 FREE SURFACE
1 0.01 Km t
2 Km/s source/receiver
REFLECTOR1
3 Km/s
REFLECTOR2
4 Km/s
Figurela. ModelNumber1.
0.0; Km
I
t
0.072 Km
0.6
0.3
-2 0.0
-0.3
-0.6
0.0
I
1
2
10102 10202
10201 201M
1o.y "ltMJc~020,
I
~1010*02
1020102
101'
‘I ‘I
A2
'0'0102 1020201
1010101 10mm212 2010102
102 1020101 2010201
201 20lolO' 2020101
I
MULTlPLES
*
I I I I I
0.2 0.4 0.6 0.8 1.0
DEPTH (Km)
dV1Figure lb. dz forModelNumber1.
0.6
0.3
4 0.0
-0.6j--,---
0.0 0.2 0.4 0.6 0.8 1.0
DEPTH(Km)
IFigurelc.
d (VI+V,+V3’.“’ )
dz
forModelNumber1.
I -
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4. 4 Inversionand multiple suppression
REFLECTOR 0
t
FREE SURFACE
2400 m V q 3600 m/s
I
REFLECTOR1
1st INTERFACE
REFLECTOR2
2nd INTERFACE
V ~‘4100 m/s
Figure2a. ModelNumber2. Thefirstmultiplefromthefirst
reflectorhasamoveoutpatternveryclosetothemoveoutofthe
primaryfromthesecondreflector.
3600
. ...........................
2400
3 1800 -
3
2
F 1200 -
600
1
0 600 1200 1800 2400 3600
OFFSET(M)
Figure2b. MoveoutpatternsformodelinFigure2a.The
solidlineistheprimaryofthefirstreflector.Thedashedline
istheprimaryofthesecondreflectorandthefirstmultipleof
thefirstreflector.
0.4
0.2
wIa
; 0.0
SGI‘I
-0.2
-0.4
0.4
0.2
m Ia
I
g 0.0
+N
g ‘0
‘0
-0.2
-0.4
0 2 4 6 8
DEPTH (Km)
Figure2c. d”!dz forthemodelinFigure2a.
0 2 4 6 8
DEPTH (Km)
d(V +V )
Figure2d. dz1 forthemodelinFigure2a.
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