Analytical Formulation of Scattering From Anisotropic Power-Law Spectrum Surfaces: Getting Rid of the Cutoff Wavenumber

Sea and soil surfaces exhibit power-law spectra over a wide range of spatial frequencies. An analytical formulation of the electromagnetic scattering from such surfaces can be obtained via the two-scale model (TSM). However, this approach requires the definition of a cutoff surface wavenumber, separating the low- and high-frequency parts of the surface spectrum. The final obtained normalized radar cross section (NRCS) value is dependent on the choice of this cutoff wavenumber, which is, to some extent, arbitrary. This problem can be avoided by describing power-law spectrum surfaces via the theory of fractional Brownian motion (fBm) two-dimensional (2-D) random processes. The bistatic NRCS of an fBm surface can be analytically evaluated by using the Kirchhoff approximation (KA) or the first-order small slope approximation (SSA-1): its expression is related to the probability density function (pdf) of an alpha-stable random process, and it can be efficiently evaluated by means of proper asymptotic series expansions. However, fBm surfaces are statistically isotropic, whereas natural surfaces are often anisotropic. Therefore, in this work, we first of all show that an anisotropic power-law spectrum surface can be considered as a generalized anisotropic fBm surface; then, we present an analytical formulation of its NRCS, based on SSA-1; and finally, we compare the obtained results with measured NRCSs of natural surfaces and with NRCS values obtained via more accurate but more computationally demanding methods that require the numerical evaluation of scattering integrals.