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 | Interpolation of near offsets using multiples and prediction-error filters |  |
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In order to test the effectiveness of this pseudo-primary-based
method, we use a synthetic example where we already know the answer.
To create a challenging test case, we take a split-spread version of
the Sigsbee2B synthetic data set, with a zero-offset section and a
shot gather shown in Figures 4a and
4b, respectively, and remove the nearest
ft of offset on either side, for a total of
ft of missing
offset, or a gap of
traces at the
ft sampling in offset,
shown in Figure 4c. While this gap is overly
large for a single boat, a two-boat undershooting of an offshore
platform could have gaps this large.
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splitspread
Figure 4. Original Sigsbee data.
(a): zero-offset section. (b): one shot. (c): resampled shot. The
near ft or traces of offset were excluded. All figures
have amplitude scaled by .
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Let us first examine the cross-correlated pseudo-primaries prior to
the summation in equation 2 for a single output trace
from one receiver pair for all shots, both using the fully sampled
input data, shown in Figure 5b, and using the
data missing the near offsets, shown in Figure
5c. This would correspond to
in equation 1. We refer to
these images as pseudo-primary-contribution gathers, as they fulfill a
role similar to multiple-contribution gathers in the SRME algorithm
(Dragoset and Jericevic, 1998). The summations of these gathers, the
predicted pseudo-primary traces, are shown in Figure
5a, plotted alongside the original trace. We
see that the traces in Figure 5a look quite
similar, especially the two pseudo-primary traces, so the missing
sources at near-offsets did not significantly detract from the
result.
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pseudocontrib
Figure 5. Generation of a
pseudo-primary trace for
. (a):
comparison of traces of (left to right) original data,
pseudo-primary data generated from fully-sampled input data, and
pseudo-primary data generated from data with missing near offsets.
(b): A pseudo-primary contribution gather, where the horizontal axis
is shot location . (c): The same pseudo-primary contribution
gather as (b), but with the missing near offsets. The images are
scaled by for display purposes.
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Moving up from the single output trace in Figure
5, we now look at an entire zero-offset section
in Figure 6. Figure
6a shows the original zero-offset section
while 6b shows pseudo-primaries generated
using fully-sampled data, and 6c shows
pseudo-primaries generated from data with missing near offsets.
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pseudocontribslice
Figure 6. A
comparison of zero-offset sections: (a) original data, (b)
pseudo-primaries generated using all offsets, (c) pseudo-primaries
generated using all offsets other than the missing near offsets.
The quality of the pseudo-primaries degrades slightly with missing
near offsets. The images are scaled by for display
purposes.
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Note four important differences between Figures
6a and 6c.
First, the pseudo-primaries have different illumination than do the
true primaries, so the relative amplitudes within each image differ.
The amplitude of the water-bottom reflection in the pseudo-primary
image with limited offsets in Figure 6c is
more variable than is that in the original image in Figure
6a, or even that in the pseudo-primaries
generated with all of the offsets in Figure
6b. Also, the subtle reflections below the
water bottom on the left side of the image at
-
s are much less
pronounced when the input offsets to the pseudo-primary generation are
limited. Second, the pseudo-primaries compared to the original data
exhibit cross-talk. The cross-talk is composed of both coherent
events such as those above the water bottom and water-bottom multiple
in Figure 6c, and more random noise. This
cross-talk increases only slightly when the offsets are limited in the
input to the pseudo-primary generation. Third, the wavelet of the
pseudo-primary data differs from that in the original data as a result
of the cross-correlation of the primary wavelet and the multiple
wavelet. Finally, the amplitude scale of the pseudo-primary images
(6b-c) is roughly a factor of
higher
than in the original data.
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pseudo
Figure 7. The near ft of
offsets of pseudo-primaries generated from input data missing the
ft of near offsets. The front panel is a constant-offset
section, the right panel is a single shot, and the top panel a time
slice. The image is scaled by for display
purposes.
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Figure 7 shows the nearest
ft of offset of the
pseudo-primaries generated from all of the split-spread shots from
Figure 4c that are missing the near offsets. The
common-offset section on the left panel is at zero offset. Looking at
the shot gather on the right panel, we see that the water-bottom
reflection is the strongest event at early times, but a considerable
amount of cross-talk is present before the water-bottom reflection.
The time slice shown on the top panel of Figure 7 shows
that in addition to the desired reflections and diffractions extracted
from the multiple reflections, there are also strong spurious events
that are not easily identifiable as cross-talk; in particular, note
the event at roughly
ft and near offsets.
The most straightforward way to use these pseudo-primaries to
interpolate the missing near offsets would be to replace the missing
traces with the pseudo-primaries, shown in Figure 8.
The pseudo-primaries were scaled by a factor of
to match the
mean amplitude of the sampled data. The cross-talk and squaring of
the wavelet are both obvious in this Figure 8,
making the region of interpolated data easily distinguishable, as do
the spurious events seen at (1) and (2), and the extra slope in the
more complex areas like (3). As we see next, we can improve greatly
on this result by using the pseudo-primaries as training data for a
prediction-error filter.
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pseudocut
Figure 8. Original data missing the
near offsets, with pseudo-primaries spliced into the locations where
traces were missing. Note three points of interest, a spurious
event caused by the correlation of primaries with other primaries,
seen in both the constant-offset section ( ) and a shot ( ), and
an area with crossing events and noise (3). The image is scaled by
for display purposes.
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Subsections
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 | Interpolation of near offsets using multiples and prediction-error filters |  |
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Next: Interpolation of Sigsbee in
Up: Curry and Shan: Near-offset
Previous: Interpolation in frequency and
2009-04-13