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S U M M A R YAs the first step towards a general analysis of the stability of optimally accurate predictorcorrector (P-C) time domain discretized schemes for solving the elastic equation of motion, we analyze the stability of two P-C schemes for a 1-D homogeneous case. Letting t be the time step, h be the spatial grid interval, β be the velocity of seismic wave propagation and C = β t/ h be the dimensionless Courant parameter, we find that each scheme has the following stability properties: stability for 0 < C ≤ C 1 , instability for C 1 < C < C 2 , stability for C 2 ≤ C ≤ C 3 and instability for C 3 < C, where 0 < C 1 < C 2 ≤ C 3 . We refer to the region C 2 ≤ C ≤ C 3 as the second island of stability. The values of C 1 , C 2 and C 3 are schemedependent. The existence of a second island of stability in a numerical scheme for solving the wave equation has not, to our knowledge, been previously reported.

The accuracy, efficiency and stability of numerical schemes for computing synthetic seismograms are important factors when the schemes are used for forward modeling and waveform inversion. We develop a method for evaluating the dispersion error directly for each volume element of the numerical grid using the criterion for optimal accuracy (eq. 2.20 of Geller & Takeuchi, 1995, GJI). This allows us to also conduct dispersion analyses of the OPT scheme for cylindrical coordinates. We conduct dispersion analysis and an evaluation of grid anisotropy for a stable and optimally accurate finite difference scheme (referred to below as "OPT") for a Cartesian grid (Mizutani et al., 2008). For 15 grid points per wavelength in a homogeneous medium, the maximum value of the dispersion error of the phase velocity for the OPT scheme implemented by a predictor-corrector (P-C) algorithm is about 0.02 %, which is 30 times less than that for a scheme which uses only the predictor step (e.g. conventional time marching). We find that the grid anisotropy for the P-C scheme is almost the same order as that of the predictor-only scheme, but the absolute value of the dispersion error is, as noted above, about 30 times smaller for the P-C scheme.