Optimizing finite-difference scheme in multidirections on rectangular grids based on the minimum norm

Abstract

The finite-difference (FD) method is widely used in numerical simulation; however, its accuracy suffers from numerical spatial dispersion and numerical anisotropy. The single-direction optimization methods, which optimize the FD coefficients along a single spatial direction, can suppress numerical spatial dispersion, but they are suboptimal for mitigating numerical anisotropy on rectangular grids. We have developed a multidirection optimization method that penalizes approximation errors among all propagation angles on rectangular grids with the minimum norm (i.e., [Formula: see text] norm) to mitigate numerical spatial dispersion and numerical anisotropy simultaneously. Given maximum absolute error tolerance and grid-spacing ratio, we first determine the optimal order of the FD operator in each spatial direction. Then, we penalize approximation errors within the wavenumber-azimuth domain to obtain the optimized FD coefficients. Theoretical analysis and numerical experiments find that our method is superior to single-direction optimization methods in suppressing numerical spatial dispersion and mitigating numerical anisotropy for square and rectangular grids. For homogeneous models with grid-spacing ratios of 1.0 (i.e., square grids), 1.2, and 1.4, the root-mean-square (rms) errors obtained by our method are 77%, 80%, and 72% that of the single-direction optimization method adopting the [Formula: see text] norm, respectively. For the Marmousi model with a grid-spacing ratio of 1.4, the rms error of our method is 36% that of the single-direction optimization method based on the [Formula: see text] norm. Such an evident improvement on error suppression is critical for numerical simulations adopting more flexible grid-spacing ratios.