Decon and interpolation with nonstationary filters

Building on the notions of time-variable filtering and the helix coordinate system, we develop software for filters that are smoothly variable in multiple dimensions. Multiscale prediction-error filters (PEFs) can estimate dips from recorded data and use the dip information to fill in unrecorded shot or receiver gathers. The data are typically divided into patches with approximately constant dips. Instead, we estimate a set of smoothly varying filters, up to one PEF for every data sample. They are more memory-intensive to estimate, but the smoothly varying filters do give more accurate interpolation results than discrete patches. Finally, we offer an improved method of controlling the smoothness of the filters. We design filters like directional derivatives that we call “flag filters”. They destroy dips in easily adjusted directions. We use them in residual space to encourage dips in the specified directions. We develop the notion of “radial-flag filters”, i.e., flag filters oriented in the radial direction (lines of constant x/t in (t, x) space). Break a common-midpoint gather into pie shaped regions bounded by various values of x/t. Such a pie-shaped region tends to have constant dip spectrum throughout the region so it is a natural region for smoothing estimates of dip spectra or of gathering statistics (via 2-D PEFs). In this paper we use smoothly variable PEFs to interpolate missing traces, though obviously they have many other uses.