We consider pulsed plane-wave scattering from targets buried under a rough air-ground interface. The properties of the interface are parametrized as a random process with known statistics, and therefore the fields scattered from a particular surface constitute one realization of an ensemble, characterized by corresponding statistics. Moreover, since the fields incident upon a buried target must first penetrate the rough interface, they and the subsequent scattered fields are random processes as well. Based on this understanding, an optimal detector is formulated, accounting for the clutter and target-signature statistics (the former due to scattering at the rough surface, and the latter due to transmission); the statistics of these two processes are in general different. Detector performance is compared to that of a matched filter, which assumes the target signature is known exactly (i.e., nonrandom). The results presented here, as a function of angle and polarization, demonstrate that there is often a significant gain in detector performance if the target signature is properly treated as a random process.