Determining the optimal coefficients of the explicit finite-difference scheme using the Remez exchange algorithm

Explicit finite-difference (FD) schemes are widely used in the seismic exploration field due to their simplicity in implementation and low computational cost. However, they suffer from strong artifacts caused by using coarse grids for high-frequency applications. The optimization of constant coefficients is popular in reducing spatial dispersions, but current methods could not guarantee that the bandwidth of the tolerable dispersion error is the widest. We have applied the Remez exchange algorithm to optimize the constant coefficients of the explicit FD schemes, for conventional and staggered grids. The resulting dispersion errors are distributed alternately between the maxima and minima in the passband of the filter, which is consistent with the most important equal-ripple property of the error magnitude for the optimal solution according to the Chebyshev criterion. The Remez exchange algorithm can determine the optimal coefficients of the FD method with only a few iterations, and the resulting operator has a wider bandwidth compared with previous solutions. It can handle arbitrary orders without the influence of local minima. Its computational cost for solving the objective function is comparable to that of the least-squares method, but its bandwidth is wider. Its accuracy is also higher than that of the maximum norm solved by the simulated annealing algorithm, but its computational cost is much lower. Theoretically, the equal-ripple error can offer the widest bandwidth for suppressing numerical dispersions among all solutions obtained by the constant-coefficient optimization. In other words, we can obtain a smaller error limitation than traditional methods under the same bandwidth. This superiority over traditional methods is essential for reducing the total error accumulation, which is helpful to avoid rapid error accumulations especially for large-scale models and long-term problems.