Estimating a bank of nonstationary prediction-error filters
is likely to create an underdetermined problem, at
least in those regions of the data (if any) where the filters are placed
close together.
The filter calculation then requires some damping
equations or some method of controlling the null space in order to get
satisfactory results.
In 1D, Claerbout 1997 just preconditions the
filter estimation with a smoother,
and limits the number of iterations to control the null space.
Clapp 1999 preconditions and also damps
the preconditioned model variable.
Shoepp and Margrave 1998 use an estimate of *Q* of the
input trace to characterize the time-varying behavior
of the input data.
Brown 1999 regularizes the filters with
a Laplacian.

Filling in missing seismic traces with PEFs is actually a series of two linear optimizations. The second step, the missing data calculation, has its own set of null space issues, depending on the number of dips that are present in the data Claerbout (1992), which I do not go into here. The second step also depends, naturally, on a good result from the first step, the filter coefficient calculation. Adding the true data that we are trying to replace (which is unknown in principle but is known in test cases where we want to make difference plots and such) gives us three things to track as a function of iteration and algorithm: the residual of the PEF calculation problem, the residual of the missing data calculation problem, and the misfit between the interpolated data and the true data. Solving the two linear optimization steps by any of several methods will guarantee that the two residuals decrease and converge. No such guarantee exists for the difference between the interpolated and real data, and in fact running either of the two optimizations for too many iterations (where ``too many" turns out to be far less than anything as intuitive as the number of model variables) increases that difference. Here I compare the behavior of those curves given different strategies for controlling the null space of the filter estimation step.

10/25/1999