Perfectly Matched Layers for the 2 D Elastic Wave Equation

The variable-split formulations of the perfectly matched layers absorbing boundary condition (PML) are derived for the 2D elastic wave-equation and tested for both homogeneous and layered velocity models by applying the 2-4 staggered-grid finitedifference scheme. Test results are compared with those by Cerjan’s sponge zone absorbing boundary condition. The comparison indicates that PML is significantly more effective and efficient in absorbing spurious reflections. INTRODUCTION Robust absorbing boundary conditions are highly desirable in eliminating spurious reflections from artificial at the boundaries of the computational domain. The absorbing boundary conditions proposed by Clayton and Engquist (1977), Reynolds (1978) and many other authors are still widely used. They are effective for small incidence angles, but they do not work very well for large ones. Another approach is the sponge absorber method (Cerjan et al, 1985) which adds a damping zone to the boundaries to attenuate the wavefield in the sponge zone from the inner region to the outer boundary. Although effective and commonly used, the sponge zone needs to be thick (30 to 50 nodes) and smooth (damp only by 5%). The extra computational efforts for this thick sponge zone are not trivial, especially for the 3-D simulation. Berenger (1994) proposed a highly effective absorbing boundary condition perfectly matched layer (PML) in 2D time-domain EM simulations. It has since been widely used for finite-difference and finite element methods in both EM and acoustic/elastodynamic wave propagation simulations (Hastings et al., 1996; Chew and Liu, 1996; Liu and Tao, 1997; Zeng and Liu, 2001; Zeng et al., 2001). In the continuum limit, PML is proved to be reflection free regardless of the incidence angle and frequency (Chew and Liu, 1996; Zeng et al., 2001; Festa and Nielsen, 2003). Although there are weak reflections associated with the discretization, PML provides excellent results with less computational cost than the sponge absorber method and does not have the instability problem (Mahrer, 1986; Stacey, 1988) associated with 1 the Clayton and Engquist absorbing boundary condition when the Vs Vp ratio is less than 0.5. In this report, I apply PML to the 2-D elastic wave-equation simulations with a 2-4 (second-order accurate in time and fourth-order accurate in space) staggered grid finite-difference method. The results are compared with those from the sponge method, a combination of Clayton absorbing boundary condition and Cerjan’s sponge absorber boundary condition. PML IMPLIMENTATION The PML absorbing layer is a non-physical region located outside the artificial numerical boundary as shown in Figure 1. Hereafter, the PML absorbing layer is referred to the PML region, and the internal model space is referred as the interior model region. In a Cartesian coordinate, the 2D elastic wave-equation can be expressed as a group of first-order differential equations in terms of the particle velocities and stresses: ∂ux ∂t = 1 ρ ( ∂τxx ∂x + ∂τxz ∂z ), ∂uz ∂t = 1 ρ ( ∂τxz ∂x + ∂τzz ∂z ), ∂τxx ∂t = (λ+ 2μ) ∂ux ∂x + λ ∂uz ∂z , (1) ∂τxz ∂t = μ( ∂ux ∂z + ∂uz ∂x ), ∂τzz ∂t = λ ∂ux ∂x + (λ+ 2μ) ∂uz ∂z , where u(x, t) is the particle velocity in term of position x and time t, τ(x, t) is the stress tensor, λ(x) and μ(x) are the Lamé elastic constants, and ρ(x) is density. Conventionally, the PML method can be formulated with a simple time-domain, variable-splitting procedure. In the 2-D elastic case, each wavefield variable is split into two components ux = u x x + u z x, uz = u x z + u z z, τxx = τ x xx + τ z xx, (2) τxz = τ x xz + τ z xz, τzz = τ x zz + τ z zz,