The time-space-domain finite-difference method (FDM) is widely used in forward modeling of wave equations. Conventional explicit FDMs (EFDMs) (with high order in space and the second order in time) would result in apparent temporal and spatial dispersion for high frequencies and large time steps. Moreover, the saturation effect of Taylor expansion seriously restricts the improvement of bandwidth coverage and efficiency of EFDMs. We have developed a variable-order optimization scheme of the implicit FDM (IFDM) to improve the computational efficiency and numerical accuracy of forward modeling. Then, we applied time-dispersion transforms (TDTs) to filter out the temporal dispersion generated by the second-order temporal approximations. Our method greatly alleviates the saturation effect of the high-order spatial finite-difference (FD) operators. Dispersion analysis indicates that the optimized coefficients of the IFDM based on the Remez algorithm can achieve the widest bandwidth (close to the Nyquist wavenumber), which corresponds to the shortest length of the spatial FD operator under a given error threshold. Our method has great potential to approach the highest spectral accuracy but with minimal increase in computational cost. Numerical experiments indicate that the combination of the variable-order optimization of IFDM and TDTs can significantly reduce numerical dispersion. Compared with traditional methods, our scheme is more advantageous for numerical simulation of large-scale geologic models because it has the least amount of calculation burden under the same accuracy requirements.

中文翻译：

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波动方程显式时间行进解的变阶最优隐式有限差分格式

时空域有限差分法（FDM）被广泛用于波动方程的正向建模。常规的显式FDM（EFDM）（在空间上具有高阶，在时间上具有第二阶）会导致明显的时间和空间色散，以用于高频和大时间步长。此外，泰勒扩展的饱和效应严重限制了EFDM带宽覆盖率和效率的提高。我们已经开发了隐式FDM（IFDM）的可变顺序优化方案，以提高正向建模的计算效率和数值精度。然后，我们应用时间色散变换（TDT）来滤除由二阶时间逼近产生的时间色散。我们的方法大大减轻了高阶空间有限差分（FD）算子的饱和效应。色散分析表明，基于Remez算法的IFDM优化系数可以实现最宽的带宽（接近奈奎斯特波数），对应于给定误差阈值下空间FD算子的最短长度。我们的方法具有接近最高光谱精度的巨大潜力，但计算成本的增加却很小。数值实验表明，IFDM和TDT的可变阶优化的组合可以显着减小数值离​​散。与传统方法相比，我们的方案在相同精度要求下具有最小的计算负担，因此在大规模地质模型的数值模拟中更具优势。