In this paper we model time-domain plane-wave scattering from targets buried under a rough (random) air-ground interface. The properties of the interface are parametrized as a random process with known statistics. Since the fields incident upon a buried target first penetrate the rough interface, they are random processes as well, and so are the scattered fields. An optimal detector is built based on this model, which takes into account both 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., is non-random). The results presented here, as a function of angle and polarization, show that, when the target signature is properly treated as a random process, a gain in detector performance can be obtained. Also, we explain under which conditions this improvement is expected to be significant.