In many practical applications of active target scattering in ocean acoustics, there is a significant amount of uncertainty in both the scattering properties of the target and the propagation characteristics of the medium. In order to consistently combine a reasonable description of this uncertainty with the established physics of scattering from submerged objects, a statistical model is developed that describes both the spatial and temporal characteristics of the scattered field. An essential aspect of this model is the multiplicative nature of the scattering process. The acoustic field propagates from the source through a random waveguide. It is then scattered from a random target that undergoes fluctuations that are uncorrelated with those of the incident field. This scattered field then propagates into azimuthal directions where the random waveguide to the receiver may be partially or fully uncorrelated with the incident waveguide. Standard assumptions for statistical fluctuations of the waveguide channels and target lead to a non-Gaussian field at the receiver. Possible randomness of the sound source leads to a further departure from Gaussianity in the received field statistics. Expressions for the spatio-temporal mutual intensity structure of the received field will be provided, with illustrative examples for a shallow water scenario.