  This paper is concerned with the inverse elastic scattering problem for a
random potential in three dimensions. Interpreted as a distribution, the
potential is assumed to be a microlocally isotropic Gaussian random field whose
covariance operator is a classical pseudo-differential operator. Given the
potential, the direct scattering problem is shown to be well-posed in the
distribution sense by studying the equivalent Lippmann--Schwinger integral
equation. For the inverse scattering problem, we demonstrate that the
microlocal strength of the random potential can be uniquely determined with
probability one by a single realization of the high frequency limit of the
averaged compressional or shear backscattered far-field pattern of the
scattered wave. The analysis employs the integral operator theory, the Born
approximation in the high frequency regime, the microlocal analysis for the
Fourier integral operators, and the ergodicity of the wave field.
