Sampling two‐dimensional seismic data and their Radon transforms

Abstract

2‐D seismic data are usually sampled and processed in a rectangular grid, for which sampling requirements are generally derived from the usual 1‐D viewpoint. For a 2‐D seismic data set, the band region (the region of the Fourier plane in which the amplitude spectrum exceeds some very small number) can be approximated by a domain bounded by two triangles. Considering the particular shape of this band region, I use 2‐D sampling theory to obtain results applicable to seismic data processing. The 2‐D viewpoint leads naturally to weaker sampling requirements than does the 1‐D viewpoint; i.e., fewer sample points are needed to represent data with the same degree of accuracy. The sampling of 2‐D seismic data and of their Radon transform in a parallelogram and then in a triangular grid is introduced. The triangular sampling grid is optimal in these cases, since it requires the minimum number of sample points—equal to half the number required by a parallelogram or rectangular grid. The sampling of 2‐D seismic data in a triangular grid is illustrated by examples of synthetic and field seismic sections. The properties of parallelogram grid sampling impose an additional sampling requirement on the 2‐D seismic data in order to evaluate their Radon transform numerically; i.e., the maximum value of the spatial sampling interval must be half of that required by the sampling theorem.