next up previous [pdf]

Next: Interpolation of Sigsbee in Up: Sigsbee data example Previous: Sigsbee data example

Interpolation of Sigsbee in time and offset

A single output shot record of the pseudo-primaries ( $ p(r_1,r_2,t)$) generated in Figure 7 is used as the training data ($ { \bf d}$) in equation 3 to generate a nonstationary PEF, $ { \bf f}_{\textrm{ns}}$. The regularization operator in equation 3, $ { \bf R}_{\textrm{ns}}$, is a two-dimensional Laplacian that operates over time and offset for each local filter lag. Once estimated, the PEF is then used to interpolate the missing near offsets of the sampled data, $ { \bf m}_{\textrm{unknown}}$, using equation 4. This process is repeated for all of the shots of both the pseudo-primary data and the original sampled data.

Figure 9 was generated by interpolating the near-offset gap using a $ t$-$ h$ 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 $ 76 000$ free coefficients for each shot. We applied $ 400$ iterations of a conjugate-direction solver to solve both the filter-estimation and interpolation problems.

There are several interesting things to see in the $ t$-$ h$ 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 $ t$-$ h$ 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 $ t$-$ h$ interpolated result.

txinterp
txinterp
Figure 9.
Interpolation with pseudo-primaries and a $ t$-$ h$ PEF. The front panel is an interpolated constant-offset section, the right panel is a single shot record, and the top panel a time slice. The spurious event in (1) and (2) is not present, but the region with conflicting slopes (3) is only partially interpolated, with an incorrect slope present. The image is scaled by $ t^{0.8}$ for display purposes. $ [{\bf CR}]$
[pdf] [png]

txyinterp
txyinterp
Figure 10.
Interpolation with pseudo-primaries and a 3D $ t$-$ h$-$ s$ PEF. The front panel is an interpolated constant-offset section, the right panel is a single shot record, and the top panel a time slice. The spurious event at ($ 1$) and ($ 2$) is not present and the region with crossing slopes ($ 3$) is better interpolated than in the 2D example. The water-bottom is more variable in amplitude, but the constant-offset section is more consistent from shot to shot. The image is scaled by $ t^{0.8}$ for display purposes. $ [{\bf CR}]$
[pdf] [png]

While the $ t$-$ h$ approach gives a reasonable result when viewing a single shot, the extremely choppy constant-offset section shows one limitation of the $ t$-$ h$ approach. We next use a 3D nonstationary $ t$-$ h$-$ s$ PEF that is $ 10\times5\times5$ elements and varies every $ 10$ elements on the time axis, $ 3$ elements on the offset axis, and $ 4$ elements on the source axis. We solve for this PEF using $ 100$ iterations of a conjugate-direction solver. Once estimated, we use this PEF, with $ 121$ million coefficients, to interpolate the missing data, using $ 200$ iterations of a conjugate-direction solver on equation 4, using a starting model for the unknown near offsets of NMO-corrected copies of the nearest recorded offset for that midpoint.

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 $ t$-$ h$-$ s$ PEF, but with a much lower computational cost and memory requirement.


next up previous [pdf]

Next: Interpolation of Sigsbee in Up: Sigsbee data example Previous: Sigsbee data example

2009-04-13