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., L1 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 L1 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 L1 norm. Such an evident improvement on error suppression is critical for numerical simulations adopting more flexible grid-spacing ratios.