An optimal numerical solver for elastic wave simulation

Abstract

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.