Estimating filters is routine in seismic processing. The simplest example might be deconvolution, but filter estimation is also valuable in many other aspects of seismic processing: interpolation Crawley (1998); Spitz (1991), noise attenuation Abma (1995); Canales (1984); Soubaras (1994), missing data Claerbout (1998a); Fomel et al. (1997), and coherency estimation Bednar (1997); Schwab (1998) to name just a few. All of these processes are based on the concept of finding a filter that minimize the energy when it is applied to a given set of data. The fundamental assumption is that that statistics of the data does not change spatially. This is often not the case. One solution to this problem is to separate the data into a number of overlapping patches Claerbout (1992d) where the stationary statistic assumption is more valid. Unfortunately, there is a limit to how small we can make our patches and still gather sufficient statistics.
A way around this limitation is to estimate a space varying prediction error filter (PEF) Crawley et al. (1998). In the extreme case you can think of estimating a filter at every data location, or more realistically, at a coarser grid spacing. With so many filters and, as result, so many filter coefficients, our estimation can quickly turn into an undetermined or at least poorly determined problem. Therefore we must impose some type of regularization to our estimation problem. Choosing an appropriate regularization then becomes an issue.
In this paper we argue that when estimating filters on seismic CMP data, you should smooth along radial lines. In a constant velocity medium the dip along a radial trace does not change, but in a more complex media it will vary slowly Ottolini (1982). By limiting filter variation in the radial direction we gather more data in our filter estimation thus enhancing stability. Here we show how to estimate the appropriate smoothing direction, and how to build and apply the appropriate regularization.