Efficient Implementation of CFS-CPML in FDTD Solutions of Second-Order Seismic Wave Equations

The complex-frequency-shifted convolutional perfectly matched layer (CFS-CPML) has been widely used in numerical simulations of the first-order seismic wave equations to avoid artificial reflections caused by truncated boundaries. However, numerically solving the second-order seismic wave equation is preferred for some large-scale geophysical algorithms, such as reverse-time migration (RTM). The extension of CFS-CPML to the second-order seismic wave equations has been realized by a strict derivation approach in the existing literature. However, the derived CFS-CPML formulations are complex, due to the introduction of a large number of intermediate variables, which decreases the overall computational efficiency, visibly. We present an efficient strategy to implement CFS-CPML in finite-difference time-domain (FDTD) modeling of the second-order seismic wave equations. We do not derive any continuous CFS-CPML formulations. Instead, we propose to split the high-order centered-grid finite-difference (CGFD) stencil that is applied to approximate the spatial second-order derivatives into two levels of CGFD stencils. The inner stencils are viewed as the second-order CGFD approximations of the first-order derivatives at symmetric grid points that are centered at the grid point where the second-order derivative is specified. With this viewpoint, the convolution term related to CFS-CPML can be introduced to each inner CGFD approximation, naturally. The outer CGFD stencil is also viewed as a CGFD approximation of a first-order derivative and it is augmented by a convolution term as well. In this way, CFS-CPML is incorporated into the high-order CGFD scheme, efficiently. We take three-dimensional (3-D) acoustic, viscoacoustic, and elastic wavefields FDTD modeling, for example, to verify the feasibility of the new implementation strategy of CFS-CPML.