Abstract

Three-dimensional seismic wavefields may be extrapolated in depth, one frequency at a time, by two-dimensional convolution with a circularly symmetric, frequency- and velocity-dependent filter. This depth extrapolation, performed for each frequency independently, lies at the heart of 3-D finite-difference depth migration. The computational efficiency of 3-D depth migration depends directly on the efficiency of this depth extrapolation.McClellan transformations provide an efficient method for both designing and implementing two-dimensional digital filters that have a particular form of symmetry, such as the circularly symmetric depth extrapolation filters used in 3-D depth migration. Given the coefficients of one-dimensional, frequency-and velocity-dependent filters used to accomplish 2-D depth migration, McClellan transformations lead to a simple and efficient algorithm for 3-D depth migration.3-D depth migration via McClellan transformations is simple because the coefficients of two-dimensional depth extrapolation filters are never explicitly computed or stored; only the coefficients of the corresponding one-dimensional filter are required. The algorithm is computationally efficient because the cost of applying the two-dimensional extrapolation filter via McClellan transformations increases only linearly with the number of coefficients N in the corresponding one-dimensional filter. This efficiency is not intuitively obvious, because the cost of convolution with a two-dimensional filter is generally proportional to N 2 . Computational efficiency is particularly important for 3-D depth migration, for which long extrapolation filters (large N) may be required for accurate imaging of steep reflectors.

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