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

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 1017-1023 ◽  
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.

Absorbing boundary condition for the elastic wave equation

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 611-624 ◽  
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.

An optimal absorbing boundary condition for elastic wave modeling

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 296-301 ◽  
Author(s):  
Chengbin Peng ◽  
M. Nafi Toksöz
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Short Note ◽  
Absorbing Boundary Conditions ◽  
Parallel Computer ◽  
Absorbing Boundary Condition ◽  
Absorbing Boundary

Absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries (Lindman, 1975; Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan et al., 1985; Randall, 1988; Higdon, 1991). We are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage. This is practical for 3-D simulation of elastic wave propagation by a finite‐difference method. Peng and Toksöz (1994) developed a method to design a class of optimal absorbing boundary conditions for a given operator length. In this short note, we give a brief introduction to this technique, and we compare the optimal absorbing boundary conditions against those by Reynolds (1978) and Higdon (1991) using examples of 3-D elastic finite‐difference modeling on an nCUBE-2 parallel computer. In the Appendix, we also give explicit formulas for computing coefficients of the optimal absorbing boundary conditions.

An absorbing boundary condition for acoustic-wave propagation using a mesh-free method

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. T145-T154 ◽  
Author(s):  
Junichi Takekawa ◽  
Hitoshi Mikada
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Acoustic Wave ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Acoustic Wave Propagation ◽  
Absorbing Boundary ◽  
Mesh Free ◽  
Mesh Free Method

We have developed an absorbing boundary condition for acoustic-wave propagation using a mesh-free method without sacrificing the flexibility of the mesh-free framework. When we simulate acoustic-wave propagation using a numerical method, artificial reflections from model edges induced by a truncated computational domain should be avoided. Although many absorbing boundary conditions have been developed, most of them have been based on a regular latticed alignment of grids or nodes, and the efficiency of such absorbing boundary conditions for irregular arrangement of grids or nodes has not been examined yet. We have studied the artificial reflections generated at the boundaries of a model for a mesh-free method, and we have proposed a novel approach for suppressing the artifacts. The method uses a hybrid approach with a transition zone, in which the wavefield is estimated by a weighted average of solutions from the one- and two-way wave equations. Numerical experiments indicate that the proposed method can provide good performance in suppression of the artificial edge reflections even for irregular distributions of calculation points in the vicinity of model edges.

Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T301-T311 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xueyuan Huang ◽  
Yanjie Zhou
Keyword(s):  
Boundary Condition ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
Complex Frequency ◽  
Absorbing Boundary ◽  
Time Layer ◽  
The Time Domain

The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.

The implementation of an improved NPML absorbing boundary condition in elastic wave modeling

2009 ◽  
Vol 6 (2) ◽  
pp. 113-121 ◽  
Author(s):  
Zhen Qin ◽  
Minghui Lu ◽  
Xiaodong Zheng ◽  
Yao Yao ◽  
Cai Zhang ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Elastic Wave ◽  
Absorbing Boundary Condition ◽  
Absorbing Boundary ◽  
Wave Modeling
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