A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion

The perfectly matched layer is an efficient tool to simulate nonreflecting boundary condition at boundaries of a grid in the finite-difference modeling of seismic wave propagation. We show relations between different formulations of the perfectly matched layer with respect to their three key aspects — split/unsplit, classical/convolutional, with the general/special form of the stretching factor. First we derive two variants of the split formulations for the general form of the stretching factor. Both variants naturally lead to the convolutional formulations in case of the general form of the stretching factor. One of them, L-split, reduces to the well-known classical split formulation in case of the special form of the stretching factor. The other, R-split, remains convolutional even for the special form of the stretching factor. The R-split formulation eventually leads to the equations identical with those obtained straightforwardly in the unsplit formulation. We also present an alternative time discretization of the unsplit formulation that is slightly algorithmically simpler than the discretization presented recently. We implement the discretization in the 3D velocity-stress staggered-grid finite-difference scheme — 4th-order in the interior grid, 2nd-order in the perfectly matched layer.