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Interpolation of Sigsbee in frequency, offset, and shot

Figure 11 shows the result of near-offset interpolation using 2D PEFs in the $ f$-$ h$-$ s$ domain. Here the training and sampled data were first NMO-corrected at water velocity, then were broken into overlapping windows along the time axis. Both sets of these time windows were then Fourier transformed to frequency. Each frequency slice of each window is treated independently, where we estimate a nonstationary PEF on a frequency slice of the pseudo-primaries, solving equation 3 with the frequency slice of the pseudo-primaries serving as training data ($ { \bf d}$) to generate a nonstationary complex-valued PEF $ { \bf f}_{\textrm{ns}}$. This PEF is then used in equation 4 as $ { \bf
F}_{\textrm{ns}}$, where $ { \bf
m}_{\textrm{known}}$ is the corresponding frequency slice of the recorded data, and $ { \bf m}_{\textrm{unknown}}$ are the missing near offsets of this frequency slice.

In Figure 11 the time axis ($ 1 500$ samples long) of both the input sampled data and the pseudo-primaries were broken into $ 40$ overlapping windows of $ 64$ points each. Both sets of windows were then Fourier transformed to produce $ 2 560$ 2D frequency slices. Each frequency slice is treated as a separate problem, with a 2D nonstationary PEF estimated on the pseudo-primary frequency slice; the 2D PEF was four samples long on both the offset and shot axes, and the filter varied every four points on each axis, for a total of $ 67 456$ filter coefficients for each $ 496$-shot by $ 136$-offset frequency slice, estimated with $ 100$ iterations of a conjugate-direction solver. The data were then interpolated using $ 200$ iterations of a conjugate-direction solver on equation 4, again with an initial model of the missing data an NMO-corrected nearest-offset-trace from the same midpoint.

fxinterp
fxinterp
Figure 11.
Interpolation with pseudo-primaries with 2D $ f$-$ h$-$ s$ PEFs. The result is more consistent from shot-to-shot, but still contains some cross-talk (around the water bottom) from the pseudo-primary data. The spurious event ($ 1$ & $ 2$) has been removed and the crossing slopes ($ 3$) are believably interpolated. The image is scaled by $ t^{0.8}$ for display purposes. $ [{\bf CR}]$
[pdf] [png]

The frequency-domain interpolation in Figure 11 differs in several respects from the $ t$-$ h$ domain interpolation in Figure 9. First, the $ f$-$ h$-$ s$ result still has some crosstalk present, while the $ t$-$ h$ and $ t$-$ h$-$ s$ results have almost no crosstalk from spurious events. This is most visible prior to the water-bottom reflection. We believe this is in part due to the stationarity assumption within each time window. However, the large spurious event at a ($ 1$) and ($ 2$) is not present in the result in either case. Second, the $ f$-$ h$-$ s$ result shows less shot-to-shot variation than does the $ t$-$ h$ result, and less amplitude variability than the $ t$-$ h$-$ s$ result. This is because in the $ t$-$ h$ result each shot is interpolated separately as an independent problem, while the $ f$-$ h$-$ s$ interpolation estimates a PEF that spans both the offset and source axes, minimizing this jitter. Finally, the $ f$-$ h$-$ s$ result appears to contain more detail in the result than does the $ t$-$ h$ result. In particular, events below the water-bottom in the shot that were not interpolated in the $ t$-$ h$ case, in the right panel of Figure 9, were interpolated in the $ f$-$ h$-$ s$ case on the right panel of Figure 11.

This synthetic data example produced promising results for interpolation of a large gap using information only from the recorded data. The $ f$-$ h$-$ s$ approach improved results and was still multiple times faster than the $ t$-$ h$ or $ t$-$ h$-$ s$ interpolation methods. The synthetic data were noise-free, and only contained the desired primaries and multiples, with an idealized 2D geometry. Next, we examine how this method works on field data.


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

2009-04-13