In seismic exploration, the dominant applications are still based on the acoustic wave assumption because of the simplicity and computational efficiency. Compared to the acoustic wave equation, the elastic wave simulation is more accurate, especially for land data, to characterise the elastic property of the real earth, but it is also far more expensive in terms of the computational cost. The traditional way to solve the first order velocity-stress elastic wave equation is based on a second order staggered grid time domain finite difference stencil, and this stencil is expensive because the time step is usually very small due to the limitation the stability condition and numerical dispersion relations. When the fourth order Lax-Wendroff stencil is applied, the time step could be larger but the computational cost triples that of the second order stencil for each time step. To improve the efficiency of the Lax-Wendroff stencil, we rewrite the first order velocity-stress equations into a combination of second order and zeroth order temporal derivative equations. When the second order stencil is applied to the new equation, there is no benefit compared to the original equation; when the fourth order Lax-Wendroff stencil is applied, the computational cost only doubles for each time step, which is 33% faster than the Lax-Wendroff stencil applied on the original equation. Both memory consumption and computational cost are compared among the four different numerical stencils, and a staggered grid Pseudo Spectral method is applied to prevent the numerical dispersion for computing the spatial derivatives. Numerical tests are performed on a 2D homogeneous media and a 2D line of the SEAM II Arid model. The results suggest that although the memory cost is increased, the Lax-Wendroff stencil applied to the second order temporal derivative equations is the optimal option in terms of the both accuracy and computational cost.

中文翻译：

---

弹性波模拟的最优数值求解器

在地震勘探中，由于简单且计算效率高，主要应用仍然基于声波假设。与声波方程相比，弹性波模拟更准确，尤其是对于陆地数据，可以表征真实地球的弹性特性，但在计算成本方面也昂贵得多。求解一阶速度-应力弹性波方程的传统方法是基于二阶交错网格时域有限差分模板，这种模板很昂贵，因为由于稳定性条件和数值的限制，时间步长通常很小色散关系。当应用四阶 Lax-Wendroff 模板时，时间步长可能更大，但每个时间步长的计算成本是二阶模板的三倍。为了提高 Lax-Wendroff 模板的效率，我们将一阶速度-应力方程重写为二阶和零阶时间导数方程的组合。当二阶模板应用于新方程时，与原始方程相比没有任何好处；当应用四阶 Lax-Wendroff 模板时，每个时间步的计算成本只会翻倍，这比应用于原始方程的 Lax-Wendroff 模板快 33%。在四种不同的数值模板之间比较内存消耗和计算成本，并应用交错网格伪谱方法来防止计算空间导数的数值分散。数值测试在二维均质介质和 SEAM II 干旱模型的二维线上进行。结果表明，尽管内存成本增加，但就准确性和计算成本而言，应用于二阶时间导数方程的 Lax-Wendroff 模板是最佳选择。