Examples of a Nonlinear Inversion Method Based on the T Matrix of Scattering Theory: Application to 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 multidimensional method derived from inverse scattering theory (A. B. Weglein and R. H. Stolt, 1991) inverts seismic reflection data and removes multiples. The procedure does not rely on periodicity or differential moveout, nor does it require a model of the multiple generating reflectors. An estimate of the source signature is required. The inverse solution is expressed in a series. Each successive term in the series requires a migration-inversion of progressively more complicated data. However, in this nonlinear method each migration-inversion is performed with the 6ame constant velocity mig.@ion-inversion operator. That is, in contrast to iterative linear methods, the operator inverted is the same for each term in the series. Our experience indicates that convergence is rapid for a wide class of examples. For very strong reflectors and a homogeneous starting model, four or five terms produced excellent results. In principle, the method suppresses all multiples. However, in its present form, it is computationally feasible for removing all surface multiples.

lCiTRODUCTlON

Methods for removing multiple reflections have a long history in exploration seismology. At the workshop on the Suppression or Exploitation of Multiples at the 1990 SEG annual meeting, several excellent papers reviewed and compared these techniques. Hardy and Hobbs (1991) also present an overview and propose a multiple suppression strategy.

NMO-stacking, f-k and p-z filtering are examples of moveout based methods requiring stacking velocity information. NMO-stacking generally is effective when the moveout differences between a primary and a multiple with the same zero offset time for 10,000 ft of offset is, after moveout correction, greater than 40 ms. Optimized p-z filtering methods usually can separate primary from multiple when, in the same moveout corrected CMP gather, they differ by at least 10 ms.

Predictive deconvolution techniques depend on multiples being periodic replicas of the primary at each receiver. They dont depend on moveout differences Or require stacking velocity information. However, their shortcoming is that they can predict only multiples which are periodic. Space-time methods are effective Only at (or near) zero offset and (near) vertical incidence. P- Z methods in principle extend periodicity to all p-values, but in practice have their own problems and limitations.

Modeling and subtraction methods reqUire an accurate estimate of the source of multiples and, hence,

are appropriate for water bottom multiples (Wiggins, 1988).

As expected, discontinuous reference velocity linear inversion (e.g., Lui, 1984) suffers the same restrictions/benefits as wave equation modeling and subtraction.

Ware and Aki (1969) and Verschuur, et al. (1988) are among those using a reflectivity matrix model of seismic data (in one and multidimensions, respectively) to develop algorithms for the removal of surface multiples.

RFAI ISTIC NQbll INFAR SFlSMlC INVFRSlDN GO&& f!d!Jt TlPr F SUPPRFSSU

Nonlinear inversion methods often are viewed as a procedure to improve the parameter estimation capability of a linear method. A less ambitious and more realistic goal for nonlinear seismic inversion is the suppression of multiples.

In exploration geophysics, iterative linear inversion (in all its different forms) is commonly thought of as the only nonlinear approach. In fact, there are numerous alternate nonlinear approaches. The T matrix formalism of scattering theory provides a useful framework for inverse scattering problems and various linear and nonlinear approaches.

Among early works in this area 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 matrix technology for the surface seismic problem.

Define Green functions G, Go, wave functions P, PO, and differential operators L and Lo in the actual and reference media, respectively. A(w) is the source signature.

The Lippman-Schwinger equation expressed in terms of P and PO is

P=

2 Inversion and multiple suppression

TGo s 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 projections onto 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. After the calculation of T and V, E is set to one.

Substitution of (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 first multiple from the upper reflector is 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 multiple is removed in Vl+V2.

from &I CONCLUSIONS

D / A = hgGoV,G,,As

from c*

(7) from c3

and so on.

According to equations (6), (7) and (a), each successive :erm in the series for V is found by migration-inversion

with the 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 propagation from (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 multiples and ignoring interbed 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 containing the source and receiver.

Figure 1 a illustrates the first model. Figures 1 b 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, periodicity or modeling.

The procedure consists of a series of uncham&g migration-inversion operations 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 during a sabbatical year 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.

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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), Inversion of seismic data in a laterally heterogeneous medium, Stanford Exploration Report No. 24, pp. 135-152, (1980b) An approach to the inverse seismic problem, Stanford Exploration Report No. 25, pp. 121-I 34.

Weglein, A. B., Boyse, W. E. and Anderson, J. E. (1981), Obtaining three dimensional velocity information directly 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 reflection coefficients, Phys. Rev., V. 102, pp. 559-567.

Razavy, M. (1975), Determination of 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 using the 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), Continuous and 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 CWTIpleX water- 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

REFLECTOR 1

3 Km/s

REFLECTOR 2

4 Km/s

Figure la. Model Number 1.

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

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102 1020101 2010201 201 20lolO' 2020101

I MULTlPLES

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I I I I I 0.2 0.4 0.6 0.8 1.0

DEPTH (Km) dV1 Figure lb. dz for Model Number 1.

0.6

0.3

4 0.0

-0.6j--,--- 0.0 0.2 0.4 0.6 0.8 1.0

DEPTH (Km)

I Figure lc. d (VI +V,+V3. ) dz for Model Number 1. I

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4 Inversion and multiple suppression

REFLECTOR 0

t FREE SURFACE

2400 m V q 3600 m/s

I

REFLECTOR 1 1st INTERFACE

REFLECTOR 2 2nd INTERFACE

V ~4100 m/s

Figure 2a. Model Number 2. The first multiple from the first reflector has a moveout pattern very close to the moveout of the primary from the second reflector.

3600

. ..\....\..........\...........

2400

3 1800 - 3

2 F 1200 -

600

1

0 600 1200 1800 2400 3600 OFFSET(M)

Figure 2b. Moveout patterns for model in Figure 2a. The solid line is the primary of the first reflector. The dashed line is the primary of the second reflector and the first multiple of the first reflector.

0.4

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-0.4

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Figure 2c. d! dz for the model in Figure 2a.

0 2 4 6 8 DEPTH (Km)

d(V +V ) Figure 2d. dz 1 for the model in Figure 2a.

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