Globally optimized finite-difference extrapolator for strongly VTI media

Abstract

Implicit finite-difference (FD) migration is unconditionally stable and is popular in handling strong velocity variations, but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult. Traditional local optimizations generate the optimized coefficients for each pair of Thomsen anisotropy parameters independently, which can degrade results substantially for large anisotropy variations and lead to a huge table. We developed an implicit FD method using the analytic Taylor-series expansion and used a global optimization scheme to improve its accuracy for wide phase angles. We first extended the number of the constant coefficients; then we relaxed the coefficient of the time-delay extrapolation term by tuning a small factor such that error is less than 0.1%. Finally, we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0.5% simultaneously. The extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range. Compared with traditional local optimization, our scheme does not need any table and table lookup. For each order of the FD method, only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy. More importantly, our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters, no matter how strong the anisotropy is. For the globally optimized second-order FD method, the accurate phase angle is up to 58°, and the increase is about 18°–22°. For the globally optimized fourth-order FD method, the accurate phase angle is up to 77°, and the increase is about 22°–27°.