Abstract We derive modified matrix operators that minimize the numerical error of solutions of the discretized elastic equation of motion. The criterion for obtaining the modified matrix operators is that the net error of the discretized equation of motion must be approximately equal to zero whenever the operand is an eigenfunction and the frequency is equal to the corresponding eigenfrequency. As it is not necessary to know the explicit values of the eigensolutions, our approach can be applied to arbitrarily heterogeneous media. In this paper we primarily consider frequency domain solutions calculated using the direct solution method (DSM) (Geller et al. 1990; Hara, Tsuboi & Geller 1991; Geller & Ohminato 1994). We present explicit formulations of the modified operators and numerical examples for P-SV and SH wave propagation in laterally homogeneous, isotropic media. The numerical solutions obtained using the modified operators are about 30 times more accurate than those obtained using the unmodified operators for the same CPU time. Our methods are readily applicable to problems in spherical coordinates or involving laterally heterogeneous media, as well as to time-domain solutions. It should also be possible to apply the methods of this paper to numerical methods other than the DSM.

Abstract We previously presented an optimally accurate time-domain finite difference method (FDM) scheme for computing synthetic seismograms for one-dimensional (1-D) problems [Geller, R.J., Takeuchi, N., 1998. Optimally accurate second-order time domain finite difference scheme for the elastic equation of motion: 1-D case. Geophys. J. Int. 135, 48–62]. This scheme was derived on the basis of a general criterion for optimally accurate numerical operators obtained by Geller and Takeuchi [Geller, R.J., Takeuchi, N., 1995. A new method for computing highly accurate DSM synthetic seismograms. Geophys. J. Int. 123, 449–470]. In this paper, we derive optimally accurate time-domain FDM operators for 2-D and 3-D problems following the same basic approach. A numerical example shows that synthetics for a 2-D P-SV problem computed using the modified scheme are 30 times more accurate than synthetics computed using a conventional FDM scheme, at a cost of only 3.5 times as much CPU time. This means that the CPU time required to compute synthetics of any specified accuracy using the modified scheme is only 1/47 that required to achieve the same accuracy using the conventional scheme; the memory required by the modified scheme is 1/18 that of the conventional scheme. We have not conducted computational experiments for the 3-D case, but we estimate that the CPU time advantage of the modified scheme will be a factor of over 100. The stability condition (maximum time step for a given spatial grid interval) for the various modified schemes is roughly equal to that for the corresponding conventional scheme.