Interpolation with smoothly nonstationary prediction-error filters

Building on the notions of time-variable filtering and the helix coordinate system, I develop software for filters that are smoothly variable in multiple dimensions, but that are quantized into large enough regions to be efficient. 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, with the requirement that the patches contain enough data samples to provide a sufficient number of fitting equations to determine all the coefficients of the filter. Each patch of data represents an independent estimation problem. Instead, I estimate a set of smoothly varying filters in much smaller patches, as small as one data sample. They are more work to estimate, but the smoothly varying filters do give more accurate interpolation results than PEFs in independent patches, particularly on complicated data. To control the smoothness of the filters. I use filters like directional derivatives that Clapp et al. (1998) call “steering filters”. They destroy dips in easily adjusted directions. I use them in residual space to encourage dips in the specified directions. I describe the notion of “radial-steering filters” (Clapp et al. , 1999), i.e., steering 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 I use smoothly variable PEFs to interpolate missing traces, though they may have many other uses. Finally, since noisy data can produce poor interpolation results, I deal with the separation of signal and noise along with missing data.