Summary

  In this thesis, I present a method of interpolating aliased seismic data using time- and space-domain prediction-error filters (PEFs). The method works in two steps. First, PEFs are calculated from the aliased input data. The spectrum of a PEF is the inverse of the input data spectrum. If the data are aliased, then the PEF also has an aliased spectrum. The aliased data spectrum contained in a PEF can be ``unwrapped'' easily by scaling and rescaling the PEF's axes. This dealiases the PEF spectrum. The second step is then to calculate new data, using the PEFs and the existing data. Where in the first step the PEF takes on the inverse of the aliased data spectrum, now the data takes on the inverse of the dealiased PEF spectrum. The result is a dealiased data set.

The theory for PEFs assumes that the data are wide-sense stationary. Seismic data are not stationary. The dips of events change in time and space. How to deal with nonstationarity is one of the most important details in interpolation. This thesis describes two strategies. The first strategy, well known in various filtering applications, is called patching. The input data are simply divided into patches (also known as analysis windows, design gates, etc.), and assumed to be stationary within the patch. Each patch gets a single PEF, and is interpolated as an independent problem. At the end, the patches are reassembled to form the interpolated data volume, with some overlap and normalization to hide the patch boundaries.

The second (and new) approach treats the data as a set of gradually varying dips rather than as independent dips. Seismic events are curvy in prestack data, and so have gradually changing dips. Instead of dividing the data into patches, we treat the data as a single nonstationary unit. We estimate many PEFs, as many as one per data sample, and the filter calculation problem becomes underdetermined. To control the null space we add a penalty function, which says that different PEFs which are calculated from adjacent portions of the data should be approximately equal. More specifically, we add directionality so that PEFs are approximately equal along radial lines extending from zero time and offset in CMP gathers.

On data with complicated dips, nonstationary filters work noticeably better than filters calculated in patches. Nonstationary filters also work noticeably better on data with noise and statics. Land data is an interesting interpolation problem because it so expensive to acquire. It can be much more difficult than marine data, because of the differences in acquisition geometry, and the relatively large amount of noise that tends to be in the data. Nonetheless, the interpolation results are promising.