Since the phenomena measured by seismic techniques are multi-dimensional, the manipulation of the measurements of these phenomena will generally require multi-dimensional filters. While in the case of deconvolution in time, the reverberation effect is approximately contained in one dimension, this one-dimensional approximation breaks down for long-period reverberations. In the case of f-x prediction, two-dimensional filtering is done by decomposing the problem into a large set of one-dimensional complex filters. The emphasis in this thesis is on the more general case of both calculating and applying multi-dimensional filters directly with multi-dimensional convolutions.
One characteristic of multi-dimensional filters is that the number of samples contributing to an output point increases rapidly with the dimensionality of the data. This is an advantage since much more information is available to predict an output point, or, from a different point of view, given a constant number of samples from which to predict, more dimensions allow the predictions to be done from smaller distances. A disadvantage of using many dimensions is that the cost of computation also rises quickly with the dimensionality of the data. Fortunately, the complexity of the programs needed for applying and calculating multi-dimensional filters rises slowly with the number of dimensions, especially when the filters are calculated with Claerbout's conjugate-gradient methodsClaerbout (1992a, 1995). The relative simplicity of this approach is important, since calculating even two-dimensional filters using the traditional techniques used to calculate one-dimensional filters appears to be a formidable task, and the problem becomes more complex as the dimensionality increases. Almost all of the filters used in this thesis have been calculated with the conjugate-gradient methods.