An Unsplit CFS-PML Scheme for the Second-Order Wave Equation With Its Application in Fractional Viscoacoustic Simulation

Abstract

The unsplit complex frequency-shifted perfectly matched layer (CFS-PML) has been widely used in the first-order wave equation in velocity and stress while rarely formulated for the wave equation recast as a second-order system in displacement. Among different variants of PML, the unsplit CFS-PML for the second-order wave equation enjoys better absorbing performance and numerical stability, compared to the traditional PML, due to the presence of the general form of CFS stretching factors, as well as higher computational efficiency over the split PMLs since it avoids wave equation order reduction and splitting the state variables into multiple directional components. This study aims to develop an unsplit CFS-PML scheme for the second-order wave equation and devote specific attention to fractional viscoacoustic simulation where fractional time derivatives are involved and hard to be reformulated into a first-order form. In the complex space, PML is typically regarded as an analytical continuation of the real coordinates; thus, we define an explicit coordinate stretching operator acting on the Laplacian operator. This stretching operator consists of several convolution terms; each of them can be efficiently resolved by a recursive convolution updating strategy. Viscoacoustic simulations on homogeneous Pierre Shale, Marmousi model, and 3-D SEG/EAGE overthrust model verify the feasibility and absorbing the performance of our proposed scheme.