  Scattering from a non-smooth random field on the time domain is studied for
plane waves that propagate simultaneously through the potential in variable
angles. We first derive sufficient conditions for stochastic moments of the
field to be recovered from correlations between amplitude measurements of the
leading singularities, detected in the exterior of a region where the potential
is almost surely supported. The result is then applied to show that if two
sufficiently regular random fields yield the same data, they have identical
laws as function-valued random variables.
