This paper presents a multidimensional method for attenuating internal multiples using an inverse scattering series, which does not rely on periodicity or a model of generating reflectors. The method is demonstrated to be effective for internal multiple attenuation, showing convergence and effectiveness even in complex scenarios involving interfering primary and multiple events. A physical interpretation allows the separation of tasks in the inversion process, enhancing the method's utility in seismic data processing.

1. H036 INTERNAL MULTIPLE ATTENUATIO N
F.V. ARAUJO', A.B. WEGLEIN2, P.M . CARVALH03 and R.H. STOLT`
' Bahia University, Brazil
2 Schlumberger Cambridge Research, Madingley Road, Cambridge CB3 OEL, UK
3 Petrobras ' Conoco Inc .
In this paper we present a multidimensional method for attenuating internat multiples that derives from
an inverse scattering se ries. The method doesn't depend on periodicity or differential moveout, nor does it
require a model for the multiple generating re flectors.
The inverse scattering series approach to medium property determination is a direct multidimensional seismic
inversion method and can be written for acoustic , elastic anisotropic and anelastic media. Its usefulness as
an inversion tool is limited by the fact that it only converges for very small contrasts in medium properties .
However, a physical interpretation is given to the terras of the series that allows a separation into subseries
that perform different tasks in the inversion process: a) surface multiple attenuation ; b) internal multiple
attenuation ; and c) spatial location of reflectors and parameter estimation. In contrast to the original
inversion series the two subseries for surface and internat multiple attenuation always converge . The idea
appears in Weglein and Stolt (1993) and the subseries for suppressing surface multiplee is developed and
described in Carvalho , Weglein and Stolt ( 1992).
The construction of the inversion series (Moses ( 1956), Prosser ( 1969), Razavy (1975)) for the scattering
potential (or model perturbation ), V, can be written as
V =Vl+V2+V3-+ . . .
where Vi is ith order in the data, Dl,
D1 = GoV1Go .
Go is the homogeneous free space Green 's function. The higher order terms are found in a recursive way
V2 = -V1GoV1
V3 = -V1GoV1GoV1 - V1GoV2 - V2GoV1 .
A multiple attenuated data, D , can be written in terms of V as follows :
D = GoVGo
= D1+D2 + D3+ . . .
where
Di =GoVGo .
In this paper, we present a subseries of the inversion series for V that performs internal multiple attenuation .
The internal multiple attenuating subseries is constructed from the pieces of the odd terms associated with
removing multiple reflections. All the even terms and pieces of the odd terms that represent the removal of
refractions are not included . The method is multidimensional, convergent, and always attenuates internal
multiples . For realistic earth contrasts, the multiples are essentially eliminated . In the examples, a plane
wave is normal incident on a 1D acoustic medium and the data, D1i is the reflection response .
The data and the data after multiple attenuation are shown for two models . The primaries are labeled P
and the multiplee M . The method is effective in the presence of interferring primary and multiple events
(example 2).
EAEG - 56th Meeting and Technical Exhibition - Vienna, Austria, 6 - 10 June 1994

2. Acknowledgement s
The authors would like to express their appreciation to UFBA, Petrobrás, CNPq, Schlumberger Cambridge
Research, ARCO and Conoco for supporting various portions of this research project .
References
Weglein, A .B . and Stolt, R.H., 1993, I. The wave physics of downward continuation, wavelet estimation, and
volume and surface scattering. II. Approaches to linear and non-linear migration-inversion, in "Mathematical
frontiers in reflection seismology", Ed . W.W. Symes, SIAM/SEG .
Carvalho, P.M., Weglein, A .B ., and Stolt, R .H., 1992, Non-linear inverse scattering for multiple suppression :
application to real data, Part I, SEG Expanded Abstracts, 1093-1095 .
Moses, H.E., 1956, Calculation of scattering potential from reflection coefficients, Phys . Rev., 102, 559-567 .
Prosser, R.T., 1969, Formal solutions of inverse scattering problems, J . Math. Phys., 10, 1819-1822.
Razavy, M ., 1975, Determination of the wave velocity in an inhomogeneous medium from reflection data, J .
Acoustic Soc . Am., 58, 956-963.
0 .08-7
~~ Íhl Í~~Ilh
.
i~ ~ l
50
p
m
0 .00
0. 4 -
7 U ~! J1MI' o
m
0.12 -
111 "Ij UIIIJBIJIJIJ
Om -------------------
2 km/s
60 m
0.16 --
0 .24 --
Fig.1 Multiple attenuation for two layer model .
0.00
I
50
0.00
1
JWIIII1WWIII0 .08-
0 .12 --
0 .16 --
14 Om
200m
4 km/ s
6 km/ s
4 km/ s
1,0
-p
Om -------------------
2 km/ s
40m
0 .24 ---{
132m
17 6m
4 .6 km/ s
2 .2 km/ s
2 .3 km/ s
Fig.2 Multiple attenuation for model with interferring primary and multiple events .