A Generalized Deconvolution Approach for Local Radon Transforms

A chief problem in seismic data processing is the filtering of unwanted events like ground roll and multiples. Methods to deal with this problem often exploit moveout or curvature differences between offending events and the events one would like to preserve (primaries). In particular, removal of multiples based on moveout discrimination can be attained via parabolic and hyperbolic Radon transforms. In the parabolic transform, seismic data (after normal-moveout correction) are assumed to be composed of a superposition of parabolas; in the second case, the data are assumed to be a superposition of hyperbolas. Methods exist to enhance the resolution of both hyperbolic and parabolic Radon transforms (Thorson and Claerbout, 1985; Sacchi and Ulrych, 1995). In both cases, the operator capable of inverting the Radon transform is constructed in such a way that the Radon panel exhibits minimum entropy or maximum sparseness (synonymous used to describe a distribution of isolated events in the Radon panel). The sparseness assumption might not be optimal when there is a mismatch between the integration path of the Radon operator and the spatial-temporal signature of the seismic event. Amplitude variation with offset can further complicate the problem, as described by Spagnolini (1994). It is clear that assumption of sparseness or simplicity can lead to erroneous results when there is a mismatch between the operator and the data waveforms. The latter can be overcome by constructing local operators. In other words, we propose to use operators that match the structure of the wavefield on small spatio-temporal apertures. Alternatively, one could attempt a much more ambitious path where the operators are extracted from the data (data driven process). This paper describes a method to construct local Radon operators. We show that these operators can be designed with any integration path. This new class of Radon operators is implemented through a strategy that is based on Generalized Convolution (GC) and Generalized Deconvolution (GD) (Sacchi et. al, 2004). We describe this idea in the following section.