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![]() | Interpolation of near offsets using multiples and prediction-error filters | ![]() |
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Figure 9 was generated by interpolating the
near-offset gap using a -
nonstationary PEF estimated on each
shot record of the pseudo-primary data. Each 2D PEF has ten points in
time and six points in offset, and varies every four points on each
axis for a total of
free coefficients for each shot. We
applied
iterations of a conjugate-direction solver to solve both
the filter-estimation and interpolation problems.
There are several interesting things to see in the -
interpolated result in Figure 9. First, little
cross-talk is present in the interpolation, such as the spurious event
at (1) and (2). This is one of the benefits of using the
-
PEF-based approach, since low amplitude at the edges of the known data
results in no large residuals in equation 4,
regardless of the crosstalk present in the training data for the
PEF. Second, the events present in the original data in a single shot
shown in the right panel are interpolated, in most cases with good
results, although the quality of the interpolation degrades deeper in
the section. This can largely be traced to the quality of the input
pseudo-primaries. It also appears that problems with the relative
amplitudes in the pseudo-primaries are amplified in this result.
Third, the common-offset section shows significant differences from
shot to shot. This is because a 2D PEF is used and each shot is
interpolated independently. This is different when a 3D PEF in time,
offset and shot is used, where correlations between shots are
used and the inconsistencies between shots penalized, discussed next.
Finally, the wavelet issues in the pseudo-primaries appear to be
mostly removed in the
-
interpolated result.
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txinterp
Figure 9. Interpolation with pseudo-primaries and a ![]() ![]() ![]() ![]() |
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txyinterp
Figure 10. Interpolation with pseudo-primaries and a 3D ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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This 3D result, shown in Figure 10, is in part an
improvement over the 2D approach. The roughness of the 2D approach
along the source axis is gone, with a smoother result that contains
slightly more noise. The spurious event present at (1) and (2) is
still gone, and the the diffractions in the constant-offset section
are more continuous, although the water-bottom amplitude is more
variable. Next, we show that using a 2D PEF on frequency slices
provides most of the benefits of using a 3D -
-
PEF, but with
a much lower computational cost and memory requirement.
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![]() | Interpolation of near offsets using multiples and prediction-error filters | ![]() |
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