Bulletin of the Seismological Society of America. Vol. 67, No. 6, pp. 1529-1540. December 1977 ABSORBING BOUNDARY CONDITIONS FOR ACOUSTIC AND ELASTIC WAVE EQUATIONS By ROBERT CLAYTON AND B~6R~ ENGQUIST ABSTRACT 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 wave propagation 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 re- flections over a wide range of incident angles. INTRODUCTION One of the persistent problems in the numerical simulation of 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; other- wise a much greater number of mesh points would 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 bound- ary. 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 made to work perfectly for all incident angles has been proposed by Smith (1974). With this approach, the simu- lation is done twice for each absorbing boundary: once with Dirichlet boundary con- ditions, 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 shortcoming of 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 Engquist and 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 reflec- tions. The boundary conditions that we present are stable and computationallv efficient in that they require about the same amount of work per 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 in the final section. 15291530 ROBERT_ CLAYTON AND BJORN ENGQUIST PARAXIAL APPROXIMATIONS OF TI-IE WAVE EQUATION ParaxiM approximations of the scMar wave equation have been extensively de- veloped 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 for the elastic wave equation. The two-dimensionM scMar wave equation P~ + P= = v-~Ptt, (1) is usually considered for modeling purposes to be initial valued in time. The stability of the equation for time extrapolation is ensured by the fact that in its dispersion re- lation :- ¢o = v(k~ 2 + k,2') ~/2, (2) the frequency ¢0 is a real function of the spatial wave numbers k~ and k,. If we now consider spatial extrapolation of (1) (say, in the z-direction), the ap- propriate form of the dispersion relation would be k~ = :l:(~/v)[1 - ('/22k:~2/502)]112, , (3) Clearly there are stability problems wl~en I vk~/~ I > 1 (evanescent waves), because k~ becomes imaginary. It is therefore necessary to modify the wave equation in such a way as to eliminate the evanescent components of the solution. A relatively simple way of accomplishing this is to restrict the range of solutions to those waves that are traveling within a cone of the z-axis (paraxial waves). Note that the :t: sign in equa- tion (3) is for wave fields moving in opposite directions in z, and we will model these fields separately. To form the paraxial approximation of (1), we expand the square-root operator of (3) as a rational approximation about small vk~/z. Three such approximations [for the -4- sign of equation (3)] are hl: vkJ~ = 1 + O(]vk~/~12), (4) 1 2 A2: vk~/~ = 1 - -~ (vkJ~) + o(IvkJ ,14), (5) A3: vk~ = 1 :~v,~/~/ ~-- I ~(vk~/.,)~ + nc~ ~ / ~ ~ . (6) A general order expansion, which leads to stable differencing schemes, can be found by the recursion relation for a Pad~ series approximation to a square root (Francis Muir, personal communication) at 1 (v =/~) +0(i 2j = vk~/,~ I ), al = 1, (7) 1 Jr" a~-i where the jth paraxial approximation is given by vk~/~ = aj. The dispersion relations A1, A2, A3, and the dispersion relation of the full wave equation are shown in Figure 1. The error term in the expansions indicates that the approximations are valid forABSORBING BOUNDARY CONDITIONS FOR ACOUSTIC AND ELASTIC WAVES 1531 waves traveling within a cone of the z-axis. A similar set of equations can be derived for the minus sign of equation (3). In this way, the incoming and outgoing wave fields are separated by the paraxial approximation. It is interesting to note that higher-order approximations based on Taylor series expansions of the square root in equation (3) lead to unstable differencing schemes (Engquist and Majda, 1977). In their differential form, equations (4 to 6) appear as AI: P~+ (1/v)Pt = O, (8) A2: P,t