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Bulletinofthe Seismological SocietyofAmerica.Vol.67, No. 6, pp. 1529-1540.December1977

Boundary conditions are derived for numerical wave simulation that minimize artificial reflections from the edges of the domain of computation. In this way acoustic and elastic wavepropagation in a limited area can be efficiently used to describe physical behavior in an unbounded domain. The boundary conditions are based on paraxial approximations of the scalar and elastic wave equations. They are computationally inexpensive and simple to apply, and they reduce reflections over a wide range of incident angles.

INTRODUCTION One of the persistent problems in the numerical simulationof wave phenomena is the artificial reflections that are introduced by the edge of the computational grid. These reflections, which eventually propagate inward, mask the true solution of the problem in an infinite medium. Hence, it is of interest to develop boundary conditions that make the perimeter of the grid "transparent" to outward-moving waves; otherwise a much greater number of mesh pointswould be required. One solution to the problem that has been proposed (Lysmer and Kuhlemeyer, ]969) is the viscous damping of normal and shear stress components along the boundary. This method approximately attenuates the reflected compressional waves over a wide range of incident angles to the boundary, but it does not diminish reflected shear waves as completely. Another method that can be madeto work perfectly for all incident angles has been proposed by Smith (1974). With this approach, the simulation is done twice for each absorbing boundary: once with Dirichlet boundary conditions, and once with Netunann boundary conditions. Since these two boundary conditions produce reflections that are opposite in sign, the sum of the two cases will cancel the reflections. The chief shortcomingof this method is that the entire set of computations has to be repeated many times. In this paper we present a set of absorbing boundary conditions that are based on paraxial approximations (PA) of the scalar and elastic wave equations. A discussion of these types of boundary conditions based on pseudo-differential operators, for a general class of differential equations, can be found in Engquistand Majda (1977). The chief feature of the PA that we will exploit is that the outward-moving wave field can be separated from the inward-moving one. Along the boundary, then, the PA can be used to model only the outward-moving energy and hence reduce the reflections. The boundary conditions that we present are stable and computationallv efficient in that they require about the same amount of workper mesh point for finite difference applications as does the full wave equation. In the first part of this paper, some paraxial approximations to the scalar and elastic wave equations are presented. In the second part, the PA are used as absorbing boundary conditions and expressions for the effective reflection coefficients along the boundary are given. Some numerical examples are presented inthe final section. 1529



PARAXIAL APPROXIMATIONSOF TI-IE WAVE EQUATION ParaxiM approximations of the scMar wave equation have been extensively developed by Claerbout (1970, 1976), and Claerbout and Johnson (1971), and in the first part of this section we present a brief review of that work. In the second part we develop paraxial approximations forthe elastic wave equation. The two-dimensionM scMar wave equation
P ~ + P= = v-~Ptt,


is usually considered for modeling purposes to b e initial valued in time. The stability of the equation for time extrapolation is ensured by the fact that in its dispersion relation :¢o = v(k~ 2 + k,2')~/2, (2) the frequency ¢0is a real function of the spatial wave numbers k~ and k,. If we now consider...
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