Application of the multiaxial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves

Abstract

Perfectly matched layer (PML) absorbing boundaries are widely used to suppress spurious edge reflections in seismic modeling. When modeling Rayleigh waves with the existence of the free surface, the classical PML algorithm becomes unstable when the Poisson's ratio of the medium is high. Numerical errors can accumulate exponentially and terminate the simulation due to computational overflows. Numerical tests show that the divergence speed of the classical PML has a nonlinear relationship with the Poisson's ratio. Generally, the higher the Poisson's ratio, the faster the classical PML diverges. The multiaxial PML (M-PML) attenuates the waves in PMLs using different damping profiles that are proportional to each other in orthogonal directions. The proportion coefficients of the damping profiles usually vary with the specific model settings. If they are set appropriately, the M-PML algorithm is stable for high Poisson's ratio earth models. Through numerical tests of 40 models with Poisson's ratios that varied from 0.10 to 0.49, we found that a constant proportion coefficient of 1.0 for the x-and z-directional damping profiles is sufficient to stabilize the M-PML for all 2D isotropic elastic cases. Wavefield simulations indicate that the instability of the classical PML is strongly related to the wave phenomena near the free surface. When applying the multiaxial technique only in the corners of the PML near the free surface, the original M-PML technique can be simplified without losing its stability. The simplified M-PML works efficiently for homogeneous and heterogeneous earth models with high Poisson's ratios. The analysis in this paper is based on 2D finite difference modeling in the time domain that can easily be extended into the 3D domain with other numerical methods.

Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form

“…However, numerical experiments reported in the literature demonstrate that growth may occur. In particular, numerical experiments in [6], using a staggered grid approximation, suggest that growth can occur for the split-field PML of [2] when a free-surface boundary condition is imposed. Stable numerical approximations of the PML are also a challenge to finite/spectral element methods [8,15].…”

“…We repeated the experiments of [6] for a two-dimensional aluminum solid with the free surface boundary conditions on the top surface of the solid, and PMLs along the other boundaries. The equations are written in first order form and are discretized using a high order finite difference scheme [18,23], which is provably stable without the PML.…”

“…This situation leads to an initial boundary value problem (IBVP) for the PML. A typical set-up can be found in [6,10]; where for the elastic wave equation a traction free boundary condition is used on one boundary with PML truncation on all others. Further complicating the matter, the PML is derived in the continuous setting under the assumption that it is an unbounded layer surrounding the truncated computational domain.…”

“…For hyperbolic systems, the temporal stability of the PML Cauchy problem is well known; see for example [3,2]. When the domain is bounded or semi-bounded more care is required, since a PML which is stable in an unbounded domain, can support growth when boundaries are introduced [10,6]. Having said that, in [4] it was shown that the PML for second order elastic wave equations on a half plane with the free-surface boundary condition at a boundary tangential to the damping direction does not support temporally growing modes.…”

“…Efficient and reliable domain truncation becomes essential, since it enables more accurate numerical simulations. More than thirty years of extensive research in this area has resulted in two standard, competing approaches for artificial boundary closures: high order local non-reflecting boundary condition (NRBC) [19,20], and damping layers such as the perfectly matched layer (PML) [1][2][3][4][5][6][7][8][9][10][11][12], grid stretching techniques [14] and others [13,15]. An NRBC is a boundary condition defined on an artificial boundary such that little or no spurious reflections occur as a wave passes the boundary.…”