In this paper we use Born approximations to derive the mean square amplitudes of the scattered field for P-P, P-S, S-P, and S-S scattering by an elastic random medium characterized by perturbations of elastic constants and density. We also obtain the total scattered power or the scattering coefficient for the case of an incident P wave. We find that, in both the spatial scattering pattern and the frequency dependence of the scattering coefficient, there are some significant differences between scalar wave scattering and elastic wave scattering. These differences are most striking when the wavelength is comparable to the size of inhomogeneities, which is often encountered in the study of short-period seismic body waves.

Under certain conditions, the perturbations of the medium parameters can be decomposed into an impedance term and a velocity term. In the forward direction, scattered waves are primarily controlled by the velocity perturbations. For backscattering, scattered waves are generated mainly by impedance perturbations. We derive low- and high-frequency asymptotic forms of the directional and total scattering coefficients. In the low-frequency range, Rayleigh scattering with fourth-power frequency dependence occurs. For the high-frequency range the scattered power for common-mode scattering has a second-power frequency dependence, which is attributed to velocity perturbations.

The scattered power of converted waves reaches a maximum, for the case of an exponential correlation function, in the high-frequency range. We find that the scalar wave theory can be only approximately used for the forward scattering problem in the high-frequency range, such as the phase and amplitude fluctuations in large seismic arrays. The case of coda wave excitation by local earthquakes, which is a backscattering or a large-angle-scattering problem, must be handled by the full elastic wave theory. A preliminary analysis of past observations using our theory suggests that the lithosphere may have multiple-scale inhomogeneities. Besides the 10--20 km scale velocity inhomogeneities revealed by the forward scattering observations at LASA and NORSAR, the lithosphere in tectonically active regions may be rich in small-scale (less than 1 km) inhomogeneities.