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, UK3 Petrobras ' Conoco Inc .

In this paper we present a multidimensional method for attenuating internat multiples that derives froman 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 anddescribed 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 withremoving 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 internalmultiples . 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

Acknowledgement s

The authors would like to express their appreciation to UFBA, Petrobrás, CNPq, Schlumberger CambridgeResearch, 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, andvolume and surface scattering. II. Approaches to linear and non-linear migration-inversion, in "Mathematicalfrontiers 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

~ ~ Íh l Í~ ~ Ilh .i~ ~ l

50

pm

0 .00

0. 4 - 7 U ~! J1MI' om

0.12 -111 "Ij UIIIJBIJIJIJ

Om -------------------2 km/s

60 m0.16 --

0 .24 --

Fig .1 Multiple attenuation for two layer model .

0 .00I

50

0.00

1

JWIIII1WWII I0 .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 .