Statistical parameters of the travel time for signals reflected or refracted at a rough surface

The travel time of signals reflected or refracted by a rough surface is investigated in the geometrical optics approximation. It is shown that surface roughness typically decreases the mean travel time in the case of large-scale roughness when only one specularly reflecting point moves randomly around its unperturbed position, resulting in a negative travel-time bias (toward early echoes). In the opposite limiting case of multipath propagation, when many specular points exist on a random surface, the travel-time bias is always positive.