A nonreflecting boundary condition for discrete acoustic and elastic wave equations

TLDR

Discrete wave‑propagation methods suffer from boundary reflections that contaminate seismic signals, and while enlarging the mesh delays these effects, it is computationally costly; nonreflecting boundary conditions have been developed to mitigate this, though their application to globally coupled methods remains unclear. The nonreflecting boundary condition replaces the wave equation in the boundary region with one‑way equations that prevent energy from propagating from the boundaries into the numerical mesh. This approach is effective in most cases but its performance degrades for events that impinge on the boundaries at shallow angles.

Abstract

One of the nagging problems which arises in application of discrete solution methods for wave‐propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh. These undesired events eventually override the actual seismic signals which propagate in the modeled region. The solution to avoiding boundary effects used to be to enlarge the numerical mesh, thus delaying the side reflections and wraparound longer than the range of times involved in the modeling. Obviously this solution considerably increases the expense of computation. More recently, nonreflecting boundary conditions were introduced for the finite‐difference method (Clayton and Enquist, 1977; Reynolds, 1978). These boundary conditions are based on replacing the wave equation in the boundary region by one‐way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh. This approach has been relatively successful, except that its effectiveness degrades for events which impinge on the boundaries at shallow angles. It is also not clear how to apply this type of boundary condition to global discrete methods such as the Fourier method for which all grid points are coupled.