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Sigsbee data example

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 $ 2 100$ ft of offset on either side, for a total of $ 4 200$ ft of missing offset, or a gap of $ 29$ traces at the $ 150$ 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.

splitspread
splitspread
Figure 4.
Original Sigsbee data. (a): zero-offset section. (b): one shot. (c): resampled shot. The near $ 4 200$ft or $ 29$ traces of offset were excluded. All figures have amplitude scaled by $ t^{0.8}$. $ [{\bf ER}]$
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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 $ p(s,r_1=41000,r_2=41000,t)$ 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.

pseudocontrib
pseudocontrib
Figure 5.
Generation of a pseudo-primary trace for $ p(s,r_1=41000,r_2=41000,t)$. (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 $ s$. (c): The same pseudo-primary contribution gather as (b), but with the missing near offsets. The images are scaled by $ t^{0.8}$ for display purposes. $ [{\bf CR}]$
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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.

pseudocontribslice
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 $ t^{0.8}$ for display purposes. $ [{\bf CR}]$
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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 $ 4$-$ 5$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 $ 20$ higher than in the original data.

pseudo
pseudo
Figure 7.
The near $ 10 000$ ft of offsets of pseudo-primaries generated from input data missing the $ 2000$ 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 $ t^{0.8}$ for display purposes. $ [{\bf CR}]$
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Figure 7 shows the nearest $ 10 000$ 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 $ 60 000$ 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 $ 0.05$ 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.

pseudocut
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 ($ 1$) and a shot ($ 2$), and an area with crossing events and noise (3). The image is scaled by $ t^{0.8}$ for display purposes. $ [{\bf CR}]$
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Subsections
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Next: Interpolation of Sigsbee in Up: Curry and Shan: Near-offset Previous: Interpolation in frequency and

2009-04-13