With the assumption of stationarity, a large hole in the data does not adversely affect PEF estimation as long as there are sufficient contiguous data present to constrain the data elsewhere. However, when non-stationary PEFs are used to interpolate data, there is a large gap in the PEF coefficients as well as in the data. In the stationary case those filter coefficients were assumed to be known, but in the non-stationary case that assumption is no longer valid.
For a simple non-stationary test case, a herringbone pattern has previously been used to test interpolation and simulation methods in geophysics with stationary PEFs Brown (1999); Claerbout (1999) as well as more recently in the geostatistical community as a test case for multiple-point geostatistics Journel and Zhang (2005).
Estimation of a non-stationary PEF is an under-determined problem, so a regularization term is added to the estimation which ensures spatial smoothness of filter coefficients. This regularization term looks a lot like an isotropic interpolation, but this paper shows that the isotropic interpolation of filter coefficients is not a successful approach.
A much simpler method is to replace the unknown filter coefficients with the regularized filter coefficients from the nearest known filter, which is tantamount to a nearest-neighbor type of interpolation of filters. This wreaks substantially less havoc than other attempts to interpolate filters, as it does not manipulate PEF coefficients.
The herringbone pattern used in this paper has an obvious preferential direction, so by only regularizing and searching for a nearest-neighbor vertically a much better result can be produced. For seismic data this direction would be along radial lines in the cmp domain.
Manipulating non-stationary filters during the estimation process with regularization terms to fill in missing filters appears to be ineffective. Instead, using a very simple method which uses the nearest known portion of the non-stationary PEF to interpolate shows promising results for a simple test case. By incorporating some prior information of which PEF to use, a much better interpolated result can be obtained.