Experiments with Higdon's absorbing boundary conditions for a number of wave equations

Abstract

Simulation of wave propagation for seismic purposes is usually restricted to a small portion of the earth. Artificial boundary conditions are required where the subsurface model is truncated. Absorbing boundaries should ensure that waves hitting the artificial boundaries are not reflected. The vast amount of literature on the subject suggests that “good” conditions have not been found, and only “reasonable” solutions exist. A cursory overview of existing and a few new ideas is presented that may guide the construction of suitable boundary conditions. Because the intended application of the boundary conditions was a high-order finite-difference code that runs on a parallel computer, we have restricted our attention to local boundary conditions. A fundamental problem in the design of accurate local boundary conditions is pointed out: accuracy is required to keep the amount of reflected energy small, but at the same time allows for growing low-frequency modes. We have settled for Higdon’s boundary conditions. Higdon proposes to include some damping to suppress the growing low-frequency modes. We show that third-order conditions provide acceptable results for the simple scalar wave equation and the acoustic equation. In the elastic case, an additional low-frequency growing mode may occur. This mode can be suppressed by using a dissipative boundary scheme and by increasing the amount of damping. The increase in damping results in an increase in the amount of reflected energy, which is larger than in the scalar case. Numerical experiments exhibit a reasonable performance, although some improvement would be useful, particularly in the anisotropic elastic case.