My ultimate goal includes all forms of parameter estimation. My immediate goal is data interpolation. Interpolation is a trivial consequence of a full inversion. I believe, however, that interpolation is an important stepping stone toward a full inversion.
In the code in this paper I have reformulated conventional practice to look like general inversion theory. To introduce the notion of covariance matrix into conventional practice, I replaced the notion of ``best crosscorrelation'' with the idea of ``2-D prediction-error filter''. The problem I have found with PEFs is they immediately suggest using a large number of undetermined coefficients, many more than required to parameterize the simple model we have in mind. These coefficients arise from the need to cover a large range of possible stepouts. Using them all allows for frequency dependent coherency between adjoining traces, and that is not the model we need here. Our first need is to reduce the number of coefficients. That is why I put a ``1'' on the trace on one side of the gap and a single adjustable coefficient on the other. Since many stepouts are possible, the question arises, ``which lag will have the free parameter?'' That is why I did a global search of all possible lags within some range to see which gives the best fit. You might recognize some similarity in what I am proposing here to the more conventional approach of finding the peak of the crosscorrelation. Although my real stratagy is to structure the method to match the conventional approach, you may notice that differences could arise when the two traces have differing autocorrelations. As with one-dimensional PEF theory (see FGDP) I want my 2-D PEF's to predict both ways. In other words, if you reverse both time and space axes, the stepout (or dip) remains unchanged and the same PEF applies.
In my beginning efforts I used a three-point filter, a ``1'' on one trace and two adjacent coefficients on the other. As I was finishing the coding but before I began debugging it, I judged the extra refinement to be unwarranted since this refined interpolation in data space will be overruled in the steps that I plan where I will invoke the monoplane principle in image space.