# Seismic trace interpolation with nonstationary prediction-error filters. SEP-104 (2001)

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**Abstract**

Theory predicts that time and space domain prediction-error filters (PEFs) may be used to interpolate aliased signals. I explore the utility of the theory, applying PEF-based interpolation to aliased seismic field data, to dealias it without lowpass filtering by inserting new traces between those originally recorded. But before theoretical potential is realized on 3-D field data, some practical aspects must be addressed.

Most importantly, while PEF theory assumes stationarity, seismic data are not stationary. We can divide the data into assumed-stationary patches, as is often done in other interpolation algorithms. We interpolate with PEFs in patches, and get near-perfect results in those parts of the data where events are mostly local plane waves, lying along straight lines. However, we find that the results are unimpressive where the data are noticeably curved. As an alternative to assumed-stationary patches, I calculate PEFs everywhere in the data, and force filters which are calculated at adjacent coordinates in data space to be similar to each other. The result is a set of smoothly-varying PEFs, which we call adaptive or nonstationary. The coefficients of the adaptive PEFs constitute a large model space. Using SEP's helical coordinate, we precondition the filter calculation problem so that it converges in manageable time.

To address the difficult problem of curved events not fitting the plane wave model, we can control the degree of smoothness in the filters as a function of direction in data coordinates. To get statistically robust filter estimates, we want to maximize the area in data space over which we estimate a filter, while still approximately honoring stationarity. The local dip spectrum on a CMP gather is nearly constant in a region which is elongated in the radial direction, so I estimate PEFs that are smooth along radial lines but which may vary quickly with radial angle. In principle that addresses the curvature issue, and I find it performs well in practice on strongly curved data, noisy data, and data with somewhat irregular acquisition geometry or statics.