The optimum method for seismic modeling in random media must (1) be highly accurate to be sensitive to subtle effects of wave propagation, (2) allow coarse sampling to model media that are large compared to the scale lengths and wave propagation distances which are long compared to the wavelengths. This is necessary to obtain statistically meaningful overall attributes of wavefields. High order staggered grid finite‐difference algorithms and the pseudospectral method combine high accuracy in time and space with coarse sampling. Investigations for random media reveal that both methods lead to nearly identical wavefields. The small differences can be attributed mainly to differences in the numerical dispersion. This result is important because it shows that errors of the numerical differentiation which are caused by poor polynomial interpolation near discontinuities do not accumulate but cancel in a random medium where discontinuities are numerous. The differentiator can be longer than the medium scale length. High order staggered grid finite‐difference schemes are more efficient than pseudospectral methods in two‐dimensional (2-D) elastic random media.

The anisotropic behavior to be expected from various types of sediments is investigated by considering them as laminated media, with randomly varying velocity depth distributions. Two different stochastic processes are used to model transitional and cyclic layering. The kinematics of waves propagating through the laminated media is studied by evaluating overall elastic parameters of the transversely isotropic medium in the long wavelength limit using averaging techniques. Models with strong velocity fluctuations and high correlation between P‐ and S‐wave velocities exhibit significant anisotropy, comparable in magnitude to field and laboratory measurements. Elastic wavefields for the stochastic models were computed and the results were compared with analytical and numerical results for homogeneous anisotropic media computed with the derived overall parameters. The wavefield modeling shows that anisotropy and scattering are not simply effects influencing waves on the opposite ends of the wavelength scale but that there is an intermediate range where both effects profoundly influence wave propagation.

A down-hole experiment was carried out in the transversely isotropic Oxford Clay outcropping in the south of England. Different moveout curves for the two shear wave types and anomalous amplitude features for the SV-wave were found in the field data. Based on velocity measurements carried out formerly at the site a model study was performed to explain the results. Phase velocity and group velocity curves computed analytically with the method of characteristics, and synthetic seismograms computed with the AlekseevMikhailenko method, are presented. The field experiment and the model studies demonstrate that the occurrence of cuspidal triangles in the qSV-wavefront is an essential feature of wave propagation in transversely isotropic media. Even for weak transversely isotropic media there is a focusing effect into the direction of the cusp which leads to prominent shear wave amplitudes in this direction. Furthermore, we examined the effect of numerical anisotropy which can contaminate the synthetic seismograms. Velocity errors are one order of magnitude higher for shear waves than for compressional waves and increase with increasing Poisson's ratio. It was found that the error can be restricted to less than 1 percent only if using a spatial sampling of three times higher than a value that would generally be regarded as sufficient in finite difference computations.

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