Introduction

Most geophysical problems are either under-determined or mixed-determined, requiring some type of regularization. The ideal regularizer is the inverse model covariance Tarantola (1986). In previous papers I have shown that a space-varying operator composed of small plane-wave annihilation filters, or steering filters, can be an effective regularization operator Clapp et al. (1997, 1998); Clapp and Biondi (1998, 2000). Steering filters are best suited to describing models with relatively simple covariance functions. For a certain class of velocity models, such as models with discontinuities, steering filters have difficulty accurately describing model covariance.

PEFs are able to describe a much wider class of models than steering filters. To robustly estimate a PEF we must have a model with stationary statistics, something that is rarely true with seismic problems. We can often satisfy the stationarity requirement by breaking up our problem into small patches Claerbout (1992b). Unfortunately, we can only make our patch size so small before we can't generate sufficient statistics to estimate our PEF Crawley et al. (1999).

An alternative approach, proven to be effective when dips change quickly, is to estimate PEFs in micro patches with a non-stationary PEF Clapp and Brown (1999, 2000); Crawley et al. (1999). When dealing with discontinuities, regularizing with a non-stationary PEF can be more effective in describing the model covariance than steering filters. Non-stationary filters do a better job honoring sharp boundaries and characterizing complex models.

I will begin by showing how steering filters perform poorly at discontinuities. I will then show how to build and estimate a non-stationary PEF. I will use the non-stationary PEF in the context of a missing data problem. I will conclude by using a non-stationary PEF for regularization to tomography.