In Chapter 2, I describe how a PEF can be used to interpolate stationary, aliased data in two steps of least squares, similar to Claerbout 1992b. The first step calculates a PEF from the recorded data. The PEF ``learns'' the dips in the data; its spectrum tends towards the inverse of the data spectrum (in the limit as the number of filter coefficients approaches the number of data samples). The data spectrum is aliased, and so the PEF spectrum is as well. But taking advantage of the scale invariant properties of the PEF allows us to easily ``unwrap'' its aliased spectrum. The PEF then has a dealiased representation of the data spectrum.
In the second step of the interpolation, a closely related problem is solved, except that the filter is known and the missing data is unknown. By ``missing'' we mean all the hypothetical data that was not included in the acquisition geometry. Typically this translates into every other trace, when the offset sampling interval or the source sampling interval is being halved in the interpolation. Whereas the PEF acquired the inverse of the aliased data spectrum in the first step, the data winds up with the inverse of the dealiased PEF spectrum in the second step. The result is a version of the original data, dealised by virtue of having double (or some other integer multiple) the spatial sampling rate of the original.
Of course, seismic data do not tend to be stationary, so the data is divided into small patches, and assumed to be stationary within a patch. Each patch is a separate, independent problem.