Grazing Behavior of Scatter and Propagation Above Any Rough Surface

At grazing, propagation and scatter become inextricably connected. For sufficiently low source/observer heights, free-space inverse-distance propagation no longer applies and plane-wave descriptions of scatter give way to surface-wave modes. Concepts like surface radar cross section must be reinterpreted; lack of awareness of these facts in attempts to correlate measurements with grazing-angle laws has led to contradictions. When plane-wave depictions hold, a regime is entered where backscatter follows a grazing angle-to-the-fourth power dependence for surfaces of any roughness scales for both polarizations and for perfectly conducting as well as impedance boundaries above penetrable media. Propagation is described in terms of a roughness-modified effective impedance/admittance that approaches a constant at grazing for all roughness profiles. These facts are first explored with numerical examples, after which we establish universal laws that confirm these suspicions. We derive expressions for the first Taylor-series expansion terms for scatter and impedance/admittance versus grazing angle. Statistics are neither required nor excluded—the laws hold for single arbitrary deterministic profiles as well as averages over ensembles of random surface samples. Proofs of these claims are based on two-dimensional (2-D) fields over one-dimensional (1-D) impedance/admittance boundaries.