One- and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations

The approaches developed by Wagner [1] and Smith [2], [3] for computing the shadowing properties from a one-dimensional randomly stationary surface are investigated for an arbitrary surface uncorrelated height and slope probability density function (pdf) and extended to a two-dimensional surface in the monostatic and bistatic configurations. Bourlier et al [5] have expressed, from Brown's work [4], the Smith and Wagner average shadowing functions, for a one-dimensional surface, whatever the assumed uncorrelated slope and height pdf. They are then completely defined from both integrations over the surface slope pdf. The shadowing function is performed for Gaussian, Laplacian, and exponential slope probability density functions. With the method presented in [6], the one-dimensional monostatic shadowing function is also compared with the exact solution. It is obtained by generating the slope-height surfaces. The Gaussian and Laplacian slope pdfs are treated with a Gaussian surface height. The analytical results are extended to a one-dimensional bistatic configuration, and the case of a two-dimensional surface is investigated with a Gaussian and Laplacian surface slope pdfs. The last point is very relevant, because the classical shadowing functions of Smith and Wagner are assumed to be one-dimensional or isotropic.