Electromagnetic Scattering from Fractional Brownian Motion Surfaces via the Small Slope Approximation

Marine and terrestrial natural surfaces exhibit statistical scale invariance properties that are well modelled by fractional Brownian motion (fBm), two-dimensional random processes. Accordingly, for microwave remote sensing applications it is useful to evaluate the normalized radar cross section (NRCS) of fBm surfaces. This task has been accomplished in the past by using either the Kirchhoff approximation (KA) or the small perturbation method (SPM). However, KA and SPM have rather limited ranges of application in terms of surface roughness and incidence angle: a wider range of application is achieved by the small slope approximation (SSA), more recently developed, but the latter has not been applied yet to fBm surfaces. In this paper, the first-order SSA (SSA-1) is applied to the evaluation of scattering from fBm surfaces obtaining an analytical formulation of their NRCS. It is then shown that the obtained SSA-1 expression reduces to the KA and SPM ones at near-specular and far-from-specular scattering directions, respectively. Finally, the results of the proposed method are compared to experimental data available in the literature.