With radar sounders, coherent backscattering simulations from global
planetary DEMs typically display a deficit in diffuse clutter, which is
mainly due to the implicit assumption that roughness at scales below the
resolution of the DEM is absent. Indeed, while polynomial approximations
of the phase evolution across the facet allow for fast and
mathematically rigorous simulators, the coarse resolution of these
planetary DEMs leads to a potentially significant portion of the
backscattering response being neglected. In this paper, we derive the
analytical phase response of a rough rectangular facet characterised by
Gaussian roughness and a Gaussian isotropic correlation function under
the linear phase approximation. Formulae for the coherent and incoherent
power scattered by such an object are obtained for arbitrary bistatic
scattering angles. Validation is done both in isolation and after
inclusion in different Stratton-Chu simulators. In order to illustrate
the different uses of such a formulation, we reproduce two lunar
radargrams acquired by the LRS instrument with a Stratton-Chu simulator
incorporating the proposed rough facet phase integral, and we show that
the original radargrams are significantly better-reproduced than with
state-of-the-art methods, at a similar computational cost. We also show
how the rough facet integral formulation can be used in isolation to
better characterise subglacial water bodies on Earth.