3-D implicit finite-difference migration by multiway splitting

We show that 3-D implicit finite-difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome. The basic idea is as follows. We approximate the 3-D square root operator by a sequence of 2-D operators in three, four, or six directions to solve the azimuth symmetry problem. Each 2-D square root operator is then approximated by a sequence of implicit 2-D operators to improve steep dip accuracy. This sequence contains some unknown coefficients, which are calculated by a Taylor expansion technique or by an optimization technique. In the Taylor expansion method, the square root and its approximation are expanded into power series. By comparing the terms, the unknown coefficients are calculated. The more 2-D finite-difference operators for cascading are taken and the more directions for downward continuation are chosen, the more terms from power series can be compared to obtain a higher-degree migration operator with better circular symmetry. In the second method, optimized coefficients are calculated by an optimization procedure whereby a variation of all unknown coefficients is performed, in such a way that both the sum of all deviations between the correct square root and its approximation and the sum of all deviations from azimuth symmetry are minimized. A mathematical criterion for azimuth symmetry has been defined and incorporated into the optimization procedure.