Internal multiple attenuation using inverse scattering: Results from prestack 1 & 2D acoustic and elastic synthetics R. T. Coates*, Schlumberger Cambridge Research, A. B. Weglein, Arco Exploration and Production Technology Summary The attenuation of internal multiples in a multidimensional earth is an important and longstanding problem in exploration seismics. In this paper we report the results of applying an attenuation algorithm based on the inverse scattering series to synthetic prestack data sets generated in on and two dimensional earth models. The attenuation algorithm requires no information about the subsurface structure or the velocity field. However, detailed information about the source wavelet is a prerequisite. An attractive feature of: the attenuation algorithm is the preservation of the amplitude (and phase) of primary events in the data; thus allowing for subsequent AVO and other true amplitude processing.

1. PR 4.2
Internal multiple attenuation using inverse scattering: Results from prestack 1 & 2D acoustic and
elastic synthetics
R. T. Coates*, Schlumberger Cambridge Research, A. B. Weglein, Arco Exploration and Production Technology
Summary
The attenuation of internal multiples in a multidimensional
earth is an important and longstanding problem in explo-
ration seismics. In this paper we report the results of ap-
plying an attenuation algorithm based on the inverse scat-
tering series to synthetic prestack data sets generated in on
and two dimensional earth models. The attenuation algo-
rithm requires no information about the subsurface structure
or the velocity field. However, detailed information about
the source wavelet is a prerequisite. An attractive feature of:
the attenuation algorithm is the preservation of the amplitude
(and phase) of primary events in the data; thus allowing for
subsequent AVO and other true amplitude processing.
Introduction
Seismic processing typically assumes that reflection data
consists of primaries, i.e., that a single upward reflection has
occured between source and receiver. Signals which do not
conform to this model are usually regarded as noise to be
attenuated. Multiples have two or more upward, and one
or more downward reflectons between source and receiver,
figure 1, and thus are regarded as noise in seismic data. Mul-
tiples may be divided into two groups: surface multiples
where one or more of the downward reflections occur at the
free surface, and internal multiples where all downward re-
flections occur below the free surface. Here we concentrate
solely on internal multiples, assuming that all free surface
multiples have already been removed from the data.
Multiple attenuation is a classic and only partially solved
problem in exploration seismics. Existing attenuation meth-
ods generally make assumptions about the earth, e.g. that
it is flat layered with a white reflection series, or about the
character of the primary and multiple signals, e.g. that they
have significantly different moveouts. In many cases these
assumptions are violated and the effectiveness of the attenu-
ation or the preservation of the primary signal are degraded.
In this paper we consider an attenuation method for internal
multiples based on inverse scattering theory. The derivation
does not assume that the earth is 1D, indeed the theory re-
quires no information about the subsurface structure or ve-
locity field. It works by predicting and subtracting internal
multiples directly from the data. However, the method does
require an accurate knowledge of the source wavelet.
Theory
The internal multiple attenuation algorithm tested here is
presented in detail in Weglein and Araujo (1994), Araujo
(1994) and Weglein et al. (1996); here we provide only the
briefest summary. In the forward problem the scattered field,
Figure 1: A schematic illustration of primaries (solid)
and multiples (dash).
is given by the Lippmann-Schwinger equation, viz
(1)
where is the Green’s function in a homogeneous refer-
ence medium, G is the Green’s function in the actual medium
and the perturbation V is the difference between the wave
operators in the actual and homogeneous medium. This
equation may be expanded in powers of the perturbation,
(2)
Similarly, if we define the data, D, as the scattered field
recorded at the surface we can write the perturbation, V as a
series in the data, i.e. we write V as
(3)
where Vn is the portion of the perturbation that is order
in the data. Substituting (3) in (2) and equating orders of the
data we obtain
(4)
(5)
(6)
Data, D, is input and the model perturbation, V, is output.
One of the tasks of inversion is the elimination of multiples.
Since is linear in D, and the latter consists of primaries
and multiples, the multiple removal must be carried out by
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2. the higher terms. In fact, the first contribution to the multiple
removal series, fur multiples of a given order, is determined
by the number of changes in direction of vertical propaga-
tion of the multiples, e.g. first-order multiples have three
reversals of vertical propagation direction and the first con-
tribution comes from part of the third term in equation (6).
The multiple attenuation series we consider consists of the
leading order attenuation term for each order of multiple.
For first-order internal multiples. the portion of the third
term chosen is determined by restricting the limits of inte-
gration. These restricted limits allow us to focus only on sig-
nals which interact with V1 on the first and third occasions at
points lower in the earth (or later in time) than on the second
occasion, thus satisfying our definition of an internal multi-
ple event of first-order.
For a 2D earth the first attenuation term may be written ex-
plicitly in the source and receiver slowness, and do-
mains as
is a small time interval which ensure that multiple scat-
tering of an event with itself is excluded fmm the multiple
attenuation series. For infinite bandwidth data may be a
single time sample, for bandlimited data should be greater
than the wavelet duration. M1 when added to the data at-
tenuates all first order internal multiples at a single step, Al-
though, equation (7) may be extended in a straight forward
way to higher-order multiples (see references above), it has
been out experience on synthetic data that these higher order
terms are rarely required due to the rapid decay with order in
the amplitude of internal multiples.
In a 1D earth each slowness propagates independently thus
the data in the slowness domain has a delta function depen-
dence on p1 – p2. i.e. t)
and equation (7) simplifies accordingly.
Example: 1D Prestack Acoustic Synthetics
To demonstrate the method in a simple 1D acoustic model
finite difference synthetics were calculated for a model con-
sisting of a 250 m thick layer = 2000 = 2.25
g/cc) separating two half-spaces (V, = 1500 m/s, = 1
g/cc). The source and receivers were located 125 m above
the top of the layer with offsets from 0 m. There
was was free surface. The synthetics are shown in figure 2.
The two primaries and the first order multiple are clearly vis-
ible in the data with the second order multiple less so.
Internal multiple attenuation
Figure 2: The ID prestack acoustic synthetics.
Figure 3 shows a detail of the synthetics after the calcula-
tion and addition of the first order mulitple attenuation term.
The first order multiple at 0.7 s has been significantly attenu-
ated. Although not shown on this time window the primaries
remain untouched. Note that the second order multiple at
0.95 s also experiences a reduction in amplitude; an addi-
tional degree of attenuation will be achieved by calculating
the second order attenuation term.
Example: 2D Acoustic Synthetics
To demonstrate the action of the 2D algorithm. equation (7),
we generate data directly in the plane-wave domain. This is
done for a simple wedge model = 4000 m/s. p = 1 g/cc)
separating two half-spaces = 1500 = 1 g/cc) by
illuminating it with a single plane wavefront with a variety of
different slownesses, see figure 4. Again the synthetics were
generated using finite differences. A single incident plane
wave generates two primaries with distinct slownesses and
a series of multiples (only one is shown) also with distinct
slownesses. figure 5.
If the ID single slowness algorithm was applied to each
slowness component independently. then the result would be
a zero multiple attenuation term since each slowness trace
exhibits only a single event. Figure 6 shows the result of ap-
plying the 1D - single slowness algorithm before transforma-
tion of the reflected wavefield into the slownesses domain at
the receiver and the result of applying the full 2D algorithm,
equation (7). to the data shown in figure 5.
The result of applying the single slowness algorithm shows
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3. Internal multiple attenuation
Figure 3: A detail of the 1D prestack acoustic synthet-
ics before (solid line) and after (filled trace) first order
multiple attenuation.
Figure 4: The 2D acoustic model, showing the incident
and reflected plane waves schematically.
the multiple has been amplified for both incident slownesses.
The single slowness multiple attenuation term has incor-
rectly predicted the phase of the multiple and hence when
added to the data it has amplified not attenuated it. In con-
trast the 2D algorithm, equation (7), has correctly predicted
the phase of the multiple, and hence, when added to the data
has significantly attenuated the multiple signal.
Example: 1D Elastic Synthetics
Finally we show the results of applying the inverse scattering
multiple attenuation algorithm to synthetics from an elastic
model. The model is shown in figure 7 and consists of an
acoustic half-space overlying an elastic layer above an elas-
tic half-space. Again the model is illuminated by a plane
wave. The data, figure 8 now consists of four primaries, an
event from the top interface and three events from the bottom
interface with different modes of propagation (PP, PS and SP
together and SS), as well as a variety of multiple events with
a mixture of P- and S-wave legs. The central panel show the
first order multiple attenuation term and the right hand panel
a comparison of the data before and after multiple attenua-
tion.
The multiple events consisting of only P-wave legs are sig-
nificantly attenuated; this is not surprising since the form
of the inverse scattering algorithm we are using assumes
an acoustic reference wave propagation and a P-wave defi-
nition of the multiple. More surprising is the fact that the
events with one or more S-waves are also attenuated, if only
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Figure 5: The reflected plane wave primaries and first
order multiples for two distinct incident plane waves.
slightly. Extending the inverse scattering multiple attenua-
tion to an elastic reference medium, which we might expect
to attenuate S-wave events more effectively, is the subject of
further research.
Discussion
We have presented the results of testing an inverse scattering
series internal multiple attenuation algorithm on 1- and 2D
acoustic and 1D elastic media. The method attenuates all
multiples of a given order at a single step and does so without
affecting primary signals.
The method requires no information about the subsurface
structure or velocity field. It predicts multiples directly from
the data. A prerequiste of the method is a detailed knowl-
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4. Figure 6: (Left and center panels) Applying the 1D
algorithm to each slowness amplifies the multiple.
(Right panel) The 2D algorithm attenuates tbe multiple
(solid line) compared to the input data (dashed line).
edge of the wavelet. The results from the acoustic models are
very encouraging showing significant attenuation in both 1D
and ZD. The algorithm is equally effective for elastic models
in attenuating multiple events of an entirely P-wave history.
Events with S-wave legs are not as well attenuated, this mo-
tivates our elastic reference medium internal multiple atten-
uation research.
References
Araújo, F.V., 1994. Linear and non-linear methods derived
from scattering theory: back scattering tomography and inter-
nal multiple attenuation: Ph.D Thesis, Universidade Federal
da Bahia. Brazil (in Portuguese).
Weglein, A.B., and Araújo, F.V., 1994. Processing reflection
data, Patent Application No. GB94/O2246.
Weglein, A.B.. Gasparotto. F.V., Carvalo, P.M. and Stolt
R.H.. 1996. An inverse scattering series method for attenu-
ating multiples in seismic reflection data, submitted to Geo-
physics.
Figure 7: The 1D elastic model illuminated by a single
plane wave.
Figure 8: Internal multiples in an elastic model illu-
minated by a single plane wave. Data and first order
multiples (left panel), first order multiple attenuation
term (center) and a comparison of the data before and
after attenuation (right) (where tbe later has been sifted
slightly for visiblity). Multiple events consisting of
only P-wave legs have been significantly attenuated.
Events with one or more S-wave legs are less well at-
tenuated.
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