3-D depth migration via McClellan transformations

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 [Formula: see text]. 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.