  In this paper we consider an inverse problem for the $n$-dimensional random
Schr\"{o}dinger equation $(\Delta-q+k^2)u = 0$. We study the scattering of
plane waves in the presence of a potential $q$ which is assumed to be a
Gaussian random function such that its covariance is described by a
pseudodifferential operator. Our main result is as follows: given the
backscattered far field, obtained from a single realization of the random
potential $q$, we uniquely determine the principal symbol of the covariance
operator of $q$. Especially, for $n=3$ this result is obtained for the full
non-linear inverse backscattering problem. Finally, we present a physical
scaling regime where the method is of practical importance.

Analysis of PDEs