Solution steering with space-variant filters

SHOT-GATHER BASED INTERPOLATION

Another possible application for using recursive steering filters is to interpolate seismic data. As an initial test we chose to interpolate a shot gather. We used a $v(z)$ velocity function to construct hyperbolic trajectories, which in turn were used to construct our dip field (similar to the seismic dips used in the previous section).

For a first test we created a synthetic shot gather using a $v(z)=a+bz$ model as input to a finite difference code. We then cut a hole in this shot gather and attempted to recover the removed values. As Figure 7 shows we did a good job recovering the amplitude within a few iterations.

combo
Figure 7.

Left, synthetic shot gather; center, holes cut out of shot gather; right, inversion result after 15 iterations.

To see how the method reacted when it was given data that did not fit its model (in this case hyperbolic moveout) we used a dataset with significant noise problems (ground roll, bad traces, etc.). Using the same technique as in Figure 7 we ended up with a result which did a fairly decent job fitting portions of the data where noise content was low, but a poor job elsewhere (Figure 8). Even where the method did the best job of reconstructing the data, it still left a visible footprint. A more esthetically pleasing result can be achieved by using the above method followed a more traditional interpolation problem using the operator $\mathbf A$ and the fitting goal

$$\mathbf A \mathbf m \approx 0 ,$$ (19)

where $\mathbf m$ is initialized with the result of our previous inversion problem and not allowed to change at locations where we have data. The bottom right panel in Figure 8 shows the result of applying a few iterations of fitting goal (19) to the bottom left result in Figure 8. By using both methodologies the interpolated data does a much better job blending into its surroundings but still is a poor interpolation result.

wz-combo
Figure 8. Top left, original shot gather; top right, gather with holes (input); bottom left, result applying equation 18, bottom right, result after applying equation (18) followed by (19).

Solution steering with space-variant filters