Estimating a set of adaptive PEFs tends 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. Several prescriptions exist. In one dimension, Claerbout 1997 preconditions the filter estimation with a smoother, and limits the number of iterations to control the null space. Shoepp and Margrave 1998 use an estimate of attenuation of the input trace to characterize the time-varying behavior of the input data. In two and three dimensions, Brown 1999 regularizes the filters with a Laplacian. Clapp 1999 preconditions with a smoother and also damps the preconditioned (roughened) model variable.
In this section we compare some strategies for controlling the null space of the underdetermined filter calculation problem. Convergence of the filter calculation by itself is not very informative. Filter calculation is the first step in a sequence of two linear optimization steps, and solving either step by any of several methods will guarantee convergence. There is no guarantee of convergence for the real quantity of interest, which is the difference between the true data (unknown in principle, but known for test cases) and the output of the second step, the interpolated data. Here we track that difference, as well as the residuals of the two individual least squares steps, varying the filter calculation strategy.