Interpolation with adaptive PEFs

  In this chapter, I introduce a new method for dealing with the nonstationarity of the input data. In the previous chapter, the data are divided into rectangular patches and assumed to be stationary within a patch. Each patch is an independent interpolation problem. This implies that seismic events have piecewise constant dips in prestack data. In general that is not true, though it can be nearly true in cases, particularly at long offsets, where seismic events tend to approach their asymptotes. In this chapter, instead of independent patches, we divide the data into smaller regions, as small as a single data sample, and assume that the dip of seismic events varies gradually with location in the data. We refer to the new, smaller regions as ``micropatches.'' They are no longer independent problems, but a series of related problems. Following the literature, we say we have moved from block estimation to adaptive or nonstationary estimation.

Instead of independent problems in patches, we smooth between micropatches. We apply a smoother between values at identical lags of different PEFs, in such a way that PEFs which are applied at similar data coordinates are averaged together. For statistical robustness, we want to smooth as much as possible, but we want to avoid making a roundabout assertion of stationarity where the data are not stationary. To maximize the area in data space over which PEFs are averaged, without averaging nonstationary regions together, we can choose the directionality of our smoother. CMP gathers tend to have nearly constant dips in regions along radial lines, so we choose the radial direction.

The filter calculation part of the problem becomes large and underdetermined. Happily, however, it converges in few iterations, and turns out to be cheaper and lead to better interpolation results than interpolation in independent patches.