Seismic data interpolation beyond aliasing using regularized nonstationary autoregression

Theory

A common constraint for interpolating missing seismic traces is to ensure that the interpolated data, after specified filtering, have minimum energy (Claerbout, 1992). Filtering is equivalent to spectral multiplication. Therefore, specified filtering is a way of prescribing a spectrum for the interpolated data. A sensible choice is a spectrum of the recorded data, which can be captured by finding the data's PEF (Crawley, 2000; Spitz, 1991). The PEF, also known as the autoregression filter, plays the role of the `inverse-covariance matrix' in statistical estimation theory. A signal is regressed on itself in the estimation of PEF. The PEF can be implemented in either $t$ -$x$ (time-space) or $f$ -$x$ (frequency-space) domain. Time-space PEFs are less likely to create spurious events in the presence of noise than $f$ -$x$ PEFs (Crawley, 2000; Abma, 1995). When data interpolation is cast as an inverse problem, a PEF can be used to find missing data. This involves a two-step approach. In the first step, a PEF is estimated by minimizing the output of convolution of known data with an unknown PEF. In the second step, the missing data is found by minimizing the convolution of the recently calculated PEF with the unknown model, which is constrained where the data are known (Curry, 2004).

Seismic data interpolation beyond aliasing using regularized nonstationary autoregression