Figure shows the acquisition
geometry for one particular sail line. The first streamer (shown as 1)
is extracted and used for this field data example.
The ship is moving from right to left. Surface currents
generate strong cable feathering around 642 km in the inline
direction. The receiver positions are displayed for nine shots
only every eight kilometers in Figure
.
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A near offset section for streamer 1 in Figure
is shown in Figure
. Salt intrusions make the S/N ratio very low
below three seconds. Two orders of surface-related multiples (WBM1 and
WBM2) are present in the data. The trace spacing is 75 m. The goal is
to interpolate the shots every 25 m and to recover the near
offset information before 2-D SRMP.
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The shot gathers are first sorted into CMP gathers. The binning
parameters are illustrated in Figure . The bin size
is 25
12.5 m. The goal of the interpolation is to
fill-up the empty bins shown in Figure
. From the
CMP gathers, a masking operator
is built. This masking
operator is set to zero where traces are missing and to one where
traces are present.
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From the CMP gathers and the masking operator, a model is
estimated by minimizing equations (
) and
(
). Figures
c and
d show
when the sparseness constraint is or is not applied, respectively.
Most of the aliasing artifacts caused by the missing traces in Figure
are well attenuated when the Cauchy regularization is
used. Note that the remaining artifacts in Figure
c could
be attenuated by increasing
in equation
(
) with the possible effect of damaging some
useful signal, e.g., below 5 s in Figure
a.
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Once is estimated, the interpolated CMP gathers can be
created with equation (
). Figure
illustrates the interpolation result for one CMP gather. The
sparseness constraint (Figure
b)
gives a cleaner result and preserve the steep dips better than
the radon transform without regularization (Figure
c). Note that the reconstructed traces are
less noisy than the known data and that adding some white
noise might be needed Gulunay (2003). Because the shots
are interpolated for multiple prediction only, no processing is applied
to correct for this defect.
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To better understand the effect of the Cauchy regularization on the
steep dips, Figures a,
b, and
c show the F-K spectra of the CMP gathers in Figure
for the
input data, the reconstructed data with sparseness constraint and the
reconstructed data without regularization, respectively. Most of the
steep dips are attenuated when no regularization is applied during the
inversion. This effect is clearly shown in the 15-25 Hz band where the
sparse interpolation shows some aliased energy for events going slower
than 1700 m/s (Figure
b). The same events are
attenuated when no regularization is applied in Figure
c.
Therefore, inversion with Cauchy regularization is preferred
for data interpolation in the CMP domain.
Note that the aliased energy could be attenuated in the CMP domain
by sampling the offset axis on a thinner grid. However, because SRMP is
performed in the shot domain, it is difficult to anticipate the effects
of this aliasing on the multiple prediction result. In addition, by
resampling the CMP offset axis, more CMP gathers would be needed
to maintain a uniform grid in the shot domain for the multiple prediction.
This extra-requirement would increase the total cost and
would add more strain on the interpolation technique.
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Finally, the interpolated CMP gathers are resorted into shot gathers.
Figure displays four shot gathers after
interpolation. Shots one and four are original shots from the survey
while shots two and three are new. Note that the near offset traces
have been interpolated for all the shots (known and interpolated).
Similar to what we observed in the CMP domain, the Cauchy
regularization helped to preserve the steep dips better in Figure
a. In addition, the noise level is quite high
for the interpolation result without regularization in Figure
b.
Below 5 s, the interpolation without sparseness constraint gives
better results. However, for the multiple prediction, these events
are of little interest. Overall, despite a fairly coarse acquisition geometry,
the radon-based interpolation yields accurate results for a successful
multiple prediction.
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