Failures of the perfectly-matched layer method in frequency-domain seismic wave modelling in elastic anisotropic media

The perfectly-matched layer (PML) technique is popular approach to remove the artificial edge effects in numerical wave modelling. However, there are stability conditions of the PML which all wavefronts must satisfy, such as the spherical wavefronts of P and S waves in elastic isotropic media. But the qSV wavefronts in anisotropic media may present severe instability of the PML because of triplications or cusps. In this paper, we give examples showing the failures of the PML in frequency-domain seismic wave modelling and how serious the problem can be. Our results demonstrate that the frequency-domain wave solutions may be destroyed by intersecting qSV wavefronts in singular directions. To overcome this issue, we introduce a generalized stiffness reduction method (GSRM) that aims at stabilizing the absorbing layer. We show that the GSRM successfully dissipates the wave energy to such an extent at the edge of the grid so that the spurious boundary reflections are minimized. Our numerical examples prove that the GSRM can replace the PML in frequency-domain seismic wave modelling in arbitrary elastic anisotropic media.