  We study an inverse acoustic scattering problem in half-space with a
probabilistic impedance boundary value condition. The Robin coefficient
(surface impedance) is assumed to be a Gaussian random function $\lambda =
\lambda(x)$ with a pseudodifferential operator describing the covariance. We
measure the amplitude of the backscattered field averaged over the frequency
band and assume that the data is generated by a single realization of
$\lambda$. Our main result is to show that under certain conditions the
principal symbol of the covariance operator of $\lambda$ is uniquely
determined. Most importantly, no approximations are needed and we can solve the
full non-linear inverse problem. We concentrate on anisotropic models for the
principal symbol, which leads to the analysis of a novel anisotropic spherical
Radon transform and its invertibility.
