Step 1: Equalization of irregular data

The first step in our method involves applying the inverse of the matrix ${\bf A}$ to correct for the interdependencies between the data traces. Each element of ${\bf A}$ is a 2D AMO transformation from one data trace with a given offset to another offset geometry. The offset-continuation operator is applied as an integral operator in the time-space domain. For small offset continuations, the operator is very compact and inexpensive to apply. The implementation defines a true amplitude transformation with amplitude weights following Fomel (1996). A phase factor is also taken into account and consists of applying a causal half differentiation for continuations to low offsets and anticausal half differentiation for continuations to large offsets. No antialiasing filter was implemented for the computations of the trace-to-trace AMO transforms.

A satisfactory solution based on a conjugate gradient scheme was obtained after 5 iterations. Given the affordable cost of these 2D transformations, we experimented with large numbers of iterations but without much improvement in the results. Judging the quality of the equalized data is not a straightforward procedure. The number of output traces is the same as in the input volume and the equalized data is still irregularly sampled. Only its amplitude and phase information were equalized for the varying fold coverage. We can only evaluate the equalization step through the quality of the final image after applying an imaging operator to the equalized data.