Statistical description of rough-surface scattering using the quasi-small-slope approximation for random surfaces with a Gaussian multivariate probability distribution

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Abstract
In a previous paper (1993) we presented the quasi-slope expansion for the scattering amplitude for the problem of wave scattering by an arbitrary soft boundary. In this paper we consider the statistical description of this problem. Under the assumption that the probability density of elevations of N arbitrary points of a surface is a multivariate Gaussian distribution, we obtain an analytical expression for the scattering cross-section. This expression consists of different contributions that correspond to different terms of the quasi-slope expansion for the scattering amplitude. It is proved that under appropriate conditions the expression for the scattering cross-section corresponds either to the small-perturbation formula or to the Kirchhoff formula. The results of numerical calculation of the angular dependence of the scattering cross-section for several values of parameters are presented for a Gaussian correlation function for surface elevations. By continuously changing the wavelength, we show the continuous transition from the results of small-perturbation theory to results corresponding to the Kirchhoff case. To estimate the accuracy of the theory, we also calculate the contribution to the scattering cross-section caused by one of the second-order (in powers of slope) terms of the quasi-slope expansion. The comparison with the experimental results for the scattering of H-polarized light by a rough metal surface shows good quantitative agreement with our calculations including grazing angles of scattering.