A multiple attenuation method derived from an inverse scattering series is described. The inversion series approach allows a separation of multiple attenuation subseries from the full series. The surface multiple attenuation subseries was described and illustrated in Carvalho et al. (1991, 1992). The internal multiple attenuation method consists of selecting the parts of the odd terms that are associated with removing only multiply reflected energy. The method, for both types of multiples, is multidimensional and does not rely on periodicity or differential moveout, nor does it require a model of the reflectors generating the multiples. An example with internal and surface multiples will be presented.

1. Inverse scattering series for multiple attenuation: An example with surface
and internal multiples
Fernanda V. PPPG/Federal University of Bahia; Arthur B. Weglein, Schlumberger
Cambridge Research; Paulo Marcus Carvalho, SA; and R.H. Stolt, Conoco S14.4
SUMMARY
A multiple attenuation method derived from an inverse scatter-
ing series is described. The inversion series approach allows a
separation of multiple attenuation subseries from the full series.
The surface multiple attenuation subseries was described and il-
lustrated in Carvalho et al. (1991, 1992). The internal multiple
attenuation method consists of selecting the parts of the odd
terms that are associated with removing only multiply reflected
energy. The method, for both types of multiples, is multidimen-
sional and does not rely on periodicity or differential moveout,
nor does it require a model of the reflectors generating the mul-
tiples. An example with internal and surface multiples will be
presented.
INTRODUCTION
Multiple suppression is a long-standing problem in exploration
seismology. The conventional techniques used today for attenu-
ating multiples are moveout based methods such as NMO-stack,
f-k and p-r filtering, predictive methods based on periodicity as-
sumptions, wave equation modeling and subtraction methods,
and surface multiple removal methods. There are many cases
where these procedures are effective. However, there are also
many instances where multiples remain a serious problem. The
reason is that the available procedures make assumptions about
the nature of the earth that are often violated in practice. Those
methods based on moveout and on periodicity assume that the
earth is one dimensional with horizontal uniform layers’. Curved
or dipping reflectors and variations in the overburden can cause
serious problems for these 1D methods. Wave equation modeling
methods and surface removal methods can accomodate a multi-
dimensional earth. However, the method based on wave equation
modeling requires precise knowledge of the reflectors causing the
upward and downward reflections (Wiggins, 1988). The surface
removal method requires knowledge of the reflector causing the
downward reflection. It removes all downward reflections that
occur at the referred surface (e.g., Riley and Claerbout, 1976,
Dragoset, 1992, Verschuur et al., 1992).
The internal multiple attenuation method that we describe is de-
rived from an inverse scattering series. In contrast with methods
for attenuating internal multiples that require a repeated applica-
tion of surface removal and downward continuation, this inverse
scattering series approach does not require knowledge of the re-
flectors that generate the multiples, nor does it require a model
of the medium properties needed for downward continuation.
INVERSE SCATTERING SERIES
The scattered field, can be written in terms of the Lippmann-
Schwinger equation as
I
=
where is the Green function for the homogeneous half-space
and V is the model perturbation. G represents the total field
and can be written as G = G,. Substituting the latter
expression in the L-S equation you obtain the series:
= (I
+ . . . .
Expanding V as a power series in the measured data as V =
(Moses, 1956, Prosser, 1969, Razavy, 1975),
=
=
= .
The linear term is obtained directly from the measured values
of G,. The terms of higher orders and so on, are obtained
recursively. A data free of multiples is given by
D =
= . . . .
In the process of going from data to model properties, multiples
are removed and primaries are mapped from data-like events in
time to model properties in space. The full inversion series, whose
purpose is to determine model properties in space, only converges
for very small contrasts in medium properties. An idea presented
in Weglein and Stolt (1993) was to separate this full inversion se-
ries into subseries that perform separately the task of surface and
internal multiple suppression from the task of primary alteration.
The Green function for a homogeneous background with a free-
surface can be written as a sum, where is the free
space Green function and is the added term due to the free-
surface, Figure 1. A free-surface multiple suppression method
was developed by Carvalho (1992) and brief reports of that work
appear in Carvalho et al. (1991, 1992). The method was derived
by ignoring the direct interactions and substituting in the full
series by This method will eliminate all events with at
least one reflection at the free-surface. All other events, including
internal multiples, will remain in the data.
1039
Downloaded03/20/15to129.7.0.94.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/

2. 2 Multiple attenuation
Internal Multiple Attenuation
A brief description of an internal multiple attenuation subseries
is presented in Araujo et al. (1994a) with a more complete de-
velopment given in Araujo (1994b) and Araujo et al. (1994c). In
that work, it is assumed that either free-surface multiples have
been removed or that they do not represent a problem in the
data. is substituted for in the full series. The problem
is that the series with is responsible not only for attenuating
internal multiples but also for performing the alteration of the
primaries. The separation between these two tasks is done by
selecting from the series only portions of the odd terms that are
responsible for removing multiply reflected energy, Weglein and
Stolt (1993). These pieces of the odd terms will form a subseries
for attenuating internal multiples. All other terms and pieces of
terms are omitted.
Results
The reflection data shown in the examples are for a plane wave
normal incident on a acoustic medium.
Figure 2 shows the combined free-surface multiple suppression
and the internal multiple attenuation methods. Figure 2a presents
the results for the free-surface multiple suppression (Carvalho et
al., 1991) and Figure 2b uses the output generated by the exam-
ple in Figure 2a as the input for the internal multiple attenuation
method. Trace 1 is the data and trace 2 is the data after mul-
tiple attenuation. The primaries are labeled P, the free-surface
multiples SM, and the internal multiples IM.
Figure 3 shows that the internal multiple attenuation method
works well with band-limited data. Trace 1 is the data (gen-
erated without free-surface multiples), traces 2, 3 and 4 are the
multiple attenuation operators, and trace 5 is the data after mul-
tiple attenuation, given by the sum of traces 1 to 4.
CONCLUSIONS
A multidimensional internal multiple attenuation method based
on an inverse scattering series is described and examples pre-
sented. The method does not depend on periodicity, differential
nor does it require previous knowledge of the subsurface
characteristics. The subseries for internal multiple attenuation is
rapidly convergent for all contrasts in medium properties. This
is in contrast to the full inversion series that only converges for
small changes in medium properties. The identified subseries for
internal multiple attenuation has, for the tests, resulted in a
slight increase in the amplitude of the primaries.
ACKNOWLEDGEMENTS
The authors thank
berger Cambridge Research, ARCO, and Conoco for supporting
different aspects of this project.
REFERENCES
Araujo, F.V., Weglein, A.B., Carvalho, P.M., and Stolt, R.H.,
1994a , Internal multiple attenuation, EAEG Abstracts.
Araujo, F.V., Linear and non-linear methods derived from
scattering theory: Backscattered tomography and internal mul-
tiple attenuation: Ph.D. Thesis, Universidade Federal da Bahia
(in Portuguese).
Araujo, F.V., Weglein, A.B., Carvalho, P.M., and Stolt , R.H.,
Inverse scattering series approach to internal multiple at-
tenuation, in preparation, to be submitted to Geophysics.
Carvalho, P.M., 1992, Free-surface multiple reflection elimination
method based on non-linear inversion of seismic data: Ph.D. The-
sis, Universidade Federal da Bahia (in Portuguese).
Carvalho, P.M., Weglein, A.B., and Stolt, R.H., 1991, Exam-
ples of a non-linear inversion method based on the T matrix of
scattering theory: application to multiple suppression, Expanded
abstracts SEG, v.2.
Carvalho, P.M., Weglein, A.B., and Stolt, R.H., 1992, Non-linear
inverse scattering for multiple suppression: application to real
data, Part I: Expanded abstracts SEG, v.2, 1093-1095.
Dragoset, W.H., 1992, Surface multiple attenuation theory,
practical issues, examples, EAEG Abstracts, B027.
Moses, H.E., 1956, Calculation of scattering potential from re-
flection coefficients: Phys. Rev., 102, 559-567.
Prosser, R.T., 1969, Formal solutions of inverse scattering prob-
lems, J. Math. Phys., 10, 1819-1822.
Razavy, M., 1975, Determination of the wave velocity in an
mogeneous medium from reflection data: J. Acoustic Am.,
58, 956-963.
Riley, D.C. and Claerbout, J.F., 1976, 2D multiple reflections:
Geophysics, 41, 592-620.
Verschuur, D.J., Berkhout, A.J. and Wapenaar, C.P.A., 1992,
Adapt ative surface-related multiple elimination: Geophysics, 57,
1166-1177.
Weglein, A.B. and Stolt, R.H., 1993, I. The wave physics of down-
ward continuation, estimation, and volume and surface
scattering. II. Approaches to linear and non-linear
inversion: Mathematical frontiers in reflection seismology, Ed.
W.W. Symes, SIAM/SEG.
Wiggins, J.W., 1988, Attenuation of complex water-bottom mul-
tiples by wave-equation based prediction and subtraction: Geo-
physics, 53, 15271539.
Downloaded03/20/15to129.7.0.94.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/

3. Multiple attenuation 3
Fig.1 - Green function for a homogeneous medium with
a free-surface.
Fig.3 - Internal multiple attenuation with band-limited data.
Model: cl = = = =
= h2 = = 240m.
Band-pass filter (Hz): [10, 20, 90, 100].
Fig.2 - Combined a) free-surface and b) internal multiple attenuation
Model: cl = = =
hl = 80m, h2 = 72m. .
1041
Downloaded03/20/15to129.7.0.94.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/