An empirical study of instability and improvement of absorbing boundary conditions for the elastic wave equation

One of the persistent problems with numerical solutions to the elastic wave equation is the finite size of the numerical grid. As with a physical body, the grid boundaries will reflect incident energy. If not eliminated or reduced substantially, these reflections will invade the grid interior and interfere with the desired solution. One method for eliminating reflections is creating a large and/or expanding grid. This method may be impractical since it can be quite costly in both computer time and memory. Another method is making the grid boundary “transparent” to outgoing energy. This method is ideally done by designing absorbing or nonreflecting boundaries which are mathematically equivalent to a one‐way, or outgoing, elastic wave equation only. In practice, an outgoing elastic wave equation is an approximation since the wave equation is not generally separable into outgoing and incoming parts. Two absorbing boundary condition approximations commonly used are those from Reynolds (Reynolds, 1978) and Clayton and Engquist, (Clayton and Engquist, 1977).

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Absorbing boundary condition for the elastic wave equation

Geophysics ◽

10.1190/1.1442496 ◽

1988 ◽

Vol 53 (5) ◽

pp. 611-624 ◽

Cited By ~ 47

Author(s):

C. J. Randall

Keyword(s):

Boundary Condition ◽

Wave Equation ◽

Finite Difference ◽

Elastic Wave ◽

Grazing Incidence ◽

Scalar Fields ◽

Absorbing Boundary Conditions ◽

Absorbing Boundary Condition ◽

Elastic Wave Equation ◽

Absorbing Boundary

Extant absorbing boundary conditions for the elastic wave equation are generally effective only for waves nearly normally incident upon the boundary. High reflectivity is exhibited for waves traveling obliquely to the boundary. In this paper, a new and efficient absorbing boundary condition for two‐dimensional and three‐dimensional finite‐difference calculations of elastic wave propagation is presented. Compressional and shear components of the incident vector displacement fields are separated by calculating intermediary scalar potentials, allowing the use of Lindman’s boundary condition for scalar fields, which is highly absorbing for waves incident at any angle. The elastic medium is assumed to be homogeneous in the region immediately adjacent to the boundary. The reflectivity matrix of the resulting absorbing boundary for elastic waves is calculated, including the effects of finite‐difference truncation error. For effectively all angles of incidence, reflectivities are much smaller than those of the commonly employed paraxial absorbing boundaries, and the boundary condition is stable for any physical Poisson’s ratio. The nearly complete absorption predicted by the reflectivity matrix calculations, even at near grazing incidence, is demonstrated in a finite‐difference application.

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Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

Journal of Computational Methods in Physics ◽

10.1155/2014/108713 ◽

2014 ◽

Vol 2014 ◽

pp. 1-14 ◽

Cited By ~ 5

Author(s):

Ke-Yang Chen

Keyword(s):

Wave Equation ◽

Finite Difference ◽

Elastic Wave ◽

Computing Time ◽

Staggered Grid ◽

Difference Operators ◽

Elastic Wave Equation ◽

Absorbing Boundary ◽

Wave Separation ◽

Time Requirements

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.

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Instabilities in applying absorbing boundary conditions to high‐order seismic modeling algorithms

Geophysics ◽

10.1190/1.1444379 ◽

1998 ◽

Vol 63 (3) ◽

pp. 1017-1023 ◽

Cited By ~ 11

Author(s):

Antonio Simone ◽

Stig Hestholm

Keyword(s):

Boundary Condition ◽

Acoustic Wave ◽

Elastic Wave ◽

Wave Equations ◽

Absorbing Boundary Conditions ◽

Absorbing Boundary Condition ◽

Seismic Modeling ◽

Absorbing Boundary ◽

Numerical Grid ◽

Numerical Discretization

The problem of artificial reflections from grid boundaries in the numerical discretization of elastic and acoustic wave equations has long plagued geophysicists. Even if modern computers have made it possible to extend the synthetics over more wavelengths (equivalent to larger propagation distances), efficient absorption methods are still needed to minimize interference from unwanted reflections from the numerical grid boundaries. In this study, we examine applicabilities and stabilities of the optimal absorbing boundary condition (OABC) of Peng and Toksöz (1994, 1995) for 2-D and 3-D acoustic and elastic wave modeling. As a basis for comparison, we use exponential damping (ED) (Cerjan et. al., 1985), in which velocities and stresses are multiplied by progressively decreasing terms when approaching the boundaries of the numerical grid.

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