PML Absorbing Boundaries

It has been previously demonstrated that no reflection is generated when elastic (or electromagnetic) waves enter a region with Perfectly Matching Layer (PML) absorbing conditions in a continuous medium. The practical application of PMLs, however, is in numerical modeling, where the medium is discretized by either a finite-element or a finite-difference scheme thus introducing a reduced amount of reflection. In such a case what is the practical and quantitative efficiency of PML absorbing boundaries? Assuming a regular spatial mesh, we start by evaluating analytically the reflection of body waves introduced by the discrete transition toward PML properties, under variable angle of incidence and wavelength. We then extend our evaluation with numerical tests for both body and Rayleigh waves. Surprisingly enough, the absorption remains equally efficient at wavelengths far larger than the PML thickness itself. As a consequence, the PML thickness can be kept minimal even for studies involving relatively low frequencies, and no rescaling with model size is required. Another pleasant feature is that it is all the more efficient at shallow angles of incidence. Finally, we show through numerical examples that a major advantage of using PMLs is their efficiency in absorbing Rayleigh waves at the free surface, a point where more classical methods perform rather poorly. Although previous authors essentially limited the description of their discrete implementation to 2D, we develop to some level of detail a 3D finite-difference scheme for PMLs and provide numerical examples.