Trace interpolation beyond aliasing

Spitz (1991) popularized the application of prediction-error filters to regular trace interpolation and showed how the spatial aliasing restriction can be overcome by scaling the frequencies of F-X PEFs. An analogous technique for T-X filters was developed by Claerbout (1992, 1999) and was applied for 3-D interpolation with non-stationary PEFs by Crawley (2000). The T-X technique implies stretching the filter in all directions so that its dip spectrum is preserved while the coefficients can be estimated at alternating traces. After the filter is estimated, it is scaled back and used for interpolating missing traces between the known ones. A similar method works for finite-difference plane wave destructors, only we need to take special care to avoid aliased dips at the dip estimation stage.

A simple synthetic example of interpolation beyond aliasing is shown in Figure . The input data are clearly aliased and non-stationary. To take the aliasing into account, I estimate the two dips present in the data with the slope estimation technique of the previous subsection. The first dip corresponds to the true slope, while the second dip corresponds to the aliased dip component. In this example, the true dip is non-negative everywhere and is easily distinguished from the aliased one. Throwing away the aliased dip and interpolating intermediate traces with the true dip produces the accurate interpolation result shown in the right plot of Figure .

 

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Figure 13 Synthetic example of interpolation beyond aliasing with plane-wave destructor filters. Left: input aliased data, right: interpolation output.

Figure  shows a marine 2-D shot gather from a deep water Gulf of Mexico survey before and after subsampling in the offset direction. The data are similar to those used by Crawley (2000). The shot gather has long-period multiples and complicated diffraction events caused by a salt body. The amplitudes of the hyperbolic events are not as uniformly distributed as in the synthetic case of Figure . Subsampling by a factor of two (the right plot in Figure ) causes a clearly visible aliasing in the steeply dipping events. The goal of the experiment is to interpolate the missing traces in the subsampled data and to compare the result with the original gather shown in the left plot of Figure .

 

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Figure 14 2-D marine shot gather. Left: original. Right: subsampled by a factor of two in the offset direction.

A straightforward application of the dip estimation equations (-) applied to aliased data can easily lead to erroneous aliased dip estimation. In order to avoid this problem, I chose a slightly more complex strategy. The algorithm for trace interpolation of aliased data consists of the following steps:

1.

Applying Claerbout's T-X methodology, stretch a two-dip plane-wave destructor filter and estimate the dips from decimated data.

2.

The second estimated dip will be degraded by aliasing. Ignore this initial second-dip estimate.

3.

Estimate the second dip component again by fixing the first dip component and using it as the initial estimate of the second component. This trick prevents the nonlinear estimation algorithm from picking the wrong (aliased) dip in the data.

4.

Downscale the estimated two-dip filter and use it for interpolating missing traces.

The two estimated dip components are shown in Figure . The first component contains only positive dips. The second component coincides with the first one in the areas where only a single dip is present in the data. In other areas, it picks the complementary dip, which has a negative value for back-dipping hyperbolic diffractions.

 

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Figure 15 Two components of the estimated dip field for the decimated 2-D marine shot gather.

Figure  shows the interpolation result and the difference between the interpolated traces and the original traces, plotted at the same clip value. The method succeeded in the sense that it is impossible to distinguish interpolated traces from the interpolation result alone. However, it is not ideal, because some of the original energy is missing in the output. A close-up comparison between the original and the interpolated traces in Figure  shows that imperfection in more detail. Some of the steepest events in the middle of the section are poorly interpolated, and in some of the other places, the second dip component is continued instead of the first one.

 

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Figure 16 Left: 2-D marine shot gather after trace interpolation. Right: Difference between the interpolated and the original gather.

 

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Figure 17 Close-up comparison of the interpolated (right) and the original data (left).

One could improve the interpolation result considerably by including another dimension. To achieve a better result, we can use a pair of plane-wave destructors, one predicting local plane waves in the offset direction and the other predicting local plane waves in the shot direction.