An early version of two-dimensional deconvolution was the FX-decon process developed by Canales1984. This two-dimensional deconvolution was done by using a one-dimensional filter within each frequency in the frequency-space domain. Burg Claerbout (1992b) established the form of a two-dimensional prediction-error filter that would whiten the two-dimensional spectrum of its output. Claerbout1992b specialized Burg's filter to remove plane waves from a seismic section with a process referred to as LOMOPLAN.
Here, we specialize Burg's filter to do a lateral prediction that will be equivalent to the FX-decon process by using a modification of Claerbout's LOMOPLAN program. We show that FX-decon is almost equivalent to lateral prediction with two-dimensional deconvolution, and that the added ability of two-dimensional deconvolution to control the length of this filter in time gives us a result that should be superior to the FX-decon result. A symmetry for the two-dimensional deconvolution operators calculated forward and backward in space that will save filter calculation time is also found. We also review some of the effects of different filter sizes and shapes and speculate about the uses of other filter configurations.
Burg's two-dimensional filter offered too much freedom to produce good lateral predictions, because the predictions within a trace overwhelmed the lateral predictions. By fixing the filter coefficients along the output column, we avoided the strong predictions within a trace and also found the form of the two-dimensional filter then became equivalent to that of the FX-decon operator. We show a method of producing better results than FX-decon by using lateral prediction with two-dimensional deconvolution and derive a two-dimensional filter from the collection of the FX-decon operators. Other two-dimensional deconvolution applications are also reviewed. These multi-dimensional filters appear to have a flexibility that other techniques do not. We are now beginning to understand what the possibilities and limitations are. The capability of choosing a filter shape to control the action of the filter may allow many new applications with seismic data.