Results

To increase the sampling by an integer factor, a PEF is typically estimated on the input data with some sort of change in sampling. In this example with a large gap this will not suffice. Instead, we estimate the PEF on the pseudo-primaries generated by equation 1 using equation 3 and then use that PEF to interpolate the recorded data with equation 4. The results of this experiment (using a non-stationary t-x filter) are shown in Figure .

The near offset gap is 4000 feet or 26 traces, as shown in Figure . A non-stationary PEF in the t-x domain is then independently estimated on each shot in Figure , and that PEF is then used to fill in the missing data in the input and the result is shown in Figure .

 

txinterp
Figure 5 Near offsets interpolated with a non-stationary PEF trained on the pseudo-primaries in the t,offset domain.

The result in Figure looks very good. The interpolated portion of the shot on the right side of Figure looks very good, with diffractions and crossing events correctly interpolated. Most of the crosstalk present in the pseudo-primaries in Figure is gone. However, there is a sight change in the wavelet, which can be explained by the squaring of the wavelet in the cross-correlation and the t-x domain PEF capturing spectral information.

Now looking at the front panel of Figure that corresponds to a constant-offset section, another problem becomes more apparent. The section looks somewhat jagged from one shot to the next. This is largely because this problem was solved on a shot-by-shot basis so that there is no guarantee of lateral continuity between shots. This could be remedied by using a 3D PEF in t,offset, and shot so that correlations between shots would be taken into account. Another less expensive method would be to use a non-stationary f-x-y domain PEF Curry (2007), described next.

Instead of interpolating in t-x, we can transform the data into the frequency domain and perform the interpolation individually for each frequency. This approach would largely use the same machinery as the t-x approach, except that all numbers are now complex and the filter would be 2D in shot and offset space. Since this filter does not operate along time or frequency no spectral information is captured, which should eliminate the wavelet issue in the t-x example and the added shot axis (due to the much lower memory requirements of the method) should reduce the jitteriness across the shot axis in the result.

 

fxinterp
Figure 6 Near offsets interpolated with a non-stationary PEF trained on the pseudo-primaries in the f,offset,and shot domain.

The f-x-y interpolation result, shown in Figure does reduce both of these problems. The wavelet now remains consistent between the recorded data and the interpolated data. The interpolation result also seems less jagged along the shot axis as the PEF is also estimated along this axis. However, the problem of what appears to be crosstalk is much worse. This is due to assumed stationarity in time. Since the data were Fourier Transformed as a whole and the problem is solved in the frequency domain, all dips for a given spatial location appear to be present at all times. This problem could hopefully be addressed by breaking the input data into small time windows that are more stationary in time.