Backscattering from a Gaussian-distributed perfectly conducting rough surface

An analytical approach to the problem of scattering by composite random surfaces is presented. The surface is assumed to be Gaussian so that the surface height can be split (in the mean-square sense) into large ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\zeta_{l}</tex> ) and small ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\zeta_{s}</tex> ) scale components relative to the electromagnetic wavelength. A first-order perturbation approach developed by Burrows is used wherein the scattering solution for the large-scale structure is perturbed by the small-scale diffraction effects. The scattering from the large-scale structure (the zeroth-order perturbation solution) is treated via geometrical optics since <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4k_{0}^{2}\bar{\zeta_{l}^{2}} \gg 1</tex> . The first-order perturbation result comprises a convolution in wavenumber space of the height spectrum, the shadowing function, a polarization dependent factor, the joint density function for the large-scale slopes, and a truncation function which restricts the convolution to the domain corresponding to the small-scale height spectrum. The only "free" parameter is the surface wavenumber separating the large and small height contributions. For a given surface height spectrum, this wavenumber can be determined by a combination of mathematical and physical arguments.