Use of the Telegraphy Equation to Improve Absorbing Boundary Efficiency for Fourth-order Acoustic Wave Finite Difference Schemes

Finite difference methods are becoming increasingly popular for calculating synthetic seismograms because of the ease with which they can be applied to model the low-frequency response of complex geometries for which no analytical solutions can be derived. In addition to the obvious advances in computing speed and storage capabilities which have made numerical solution of large, realistic geometries possible, Clayton and Engquists' (1977) development of absorbing boundaries for acoustic and elastic wave equations greatly reduced the physical storage and computational burden necessary to solve a given problem. In the past, most difference calculations used second-order temporal and spatial difference operators (Boore, 1970; and many others). Recently, more attention has been given to higher order spatial difference operators as a means of improving the bandlimiting criteria required to minimize grid dispersion (Alford et al., 1974; Frankel and Clayton, 1984). Unfortunately, no clear guidelines have emerged in the literature for developing absorbing boundary conditions for higher order operators. (In the following, I will refer to secondor fourth-order operators and equations, either wave or telegraphy, meaning secondor fourth-order accurate approximations to the Laplacian. Time derivatives are always approximated with second-order difference operators.) The usual practice is to solve the internal grid with a fourthorder operator, and link the absorbing boundary condition to the fourth-order operator with a single row or column in which a second-order operator is used (see Figure 1A). This results in an unexpectedly large, high-frequency reflection from the absorbing boundary. Clayton (personal communication) has identified at least part of this noise as being due to an artificial mismatch in impedance resulting from the difference in the dispersion relations of the secondand fourth-order schemes at high frequencies. Clayton has suggested that the impedance mismatch can be handled by increasing the material velocities in the region of the second-order operators. Another cause of the noise is due to undersampling of the field at the edge of the numerical grid. If the mesh takes advantage of the reduction in the number of gridpoints per wavelength offered by the fourth-order operator, then the field will be locally undersampled in the second-order region. The undersampling cannot be simply treated. My experiments with the mixed secondand fourth-order code show a significant reduction in backscattered energy using Clayton's rescaling of material velocities. The backscattered waves are, however, larger than those generated from the simple second-order internal grid wave equation used with the Clayton-Engquist A2 condition [Clayton and Engquist, 1977, equation (9)] and are generated by waves at any angle of incidence to the boundary. What is needed is an absorbing condition for the fourth-order approximation to the Laplacian or, lacking that, a means to attenuate the wave field scattered from the absorbing boundary. I have used secondand fourth-order approximations to the telegraphy operator (Coulson and Jeffrey, 1977, p. 15) in a zone adjacent to the boundaries to introduce