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 (1992a, 1999) and
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 in between the known ones. A very
similar method works for finite-difference plane wave destructors,
only we need to take a special care to avoid aliased dips at the dip
estimation stage.

Figure 13 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. Subsampling by a factor of two (the right plot in Figure 13) causes a clearly visible aliasing in the steeply dipping events. The goal of my first experiment was 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 13.

Figure 13

A straightforward application of the dip estimation equations (16-18) 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 infected by aliasing. Ignore this initial 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.
- Down-scale the estimated two-dip filter and use it for interpolating missing traces.

Figure 14

Figure 15 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 perfect in the sense that some of the original energy is missing in the output. A closeup comparison between the original and the interpolated traces in Figure 16 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.

Figure 15

Figure 16

The interpolation result can be considerably improved 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.

9/5/2000