Finite-difference migration by optimized one-way equations

Some considerations on the steep-dip finite-difference migration

“…The coefficients a. and b. may be obtained either by the continuedfractions method (Hildebrand, 1956, p. 406) or by the least-squares optimization method (Lee and Suh, 1985). A polynominal expression of equation (4) represents the unknown P in terms of the unknown P z xxz Therefore, equation (4) is an implicit equation.…”

“…The program uses the explicit finite-difference algorithm and controls the difference error quite accurately. Lee and Suh (1985) studied the dispersion relations of explicit and implicit one-way equations and showed that the former is less accurate than the latter for the same order of approximation. In figure 3A, the 45-degree segments are properly migrated, but the steeper segments are still under-migrated.…”

“…The current version of the program supports time migration by the optimized one-way wave equations from the second-order to the tenth-order equation (Lee and Suh, 1985), and by the optimized second-order difference operator. Since the migration is a part of a stream of other seismic processes, the program is designed to handle the output of other processes.…”

“…Figures 3B and 3C are computed by the implicit method using the optimized second-order equation and the optimized fourth-order equation (Lee and Suh, 1985), respectively. The figures show remarkable differences.…”

“…Ma (1981) developed a solution method to high-order implicit equations. Lee and Suh (1985) introduced optimized one-way equations of the implicit type which significantly reduce the dispersion error in conventional one-way equations.…”