In this paper we determine the statistical distributions of the co- and cross-polarized Layered Rough Surface Index (LRSI) for three-dimensional layered structures with an arbitrary number of slightly rough interfaces illuminated by an electromagnetic plane wave. For infinite surface areas and Gaussian centered height distributions, we show within the framework of the first-order small perturbation method that the LRSI under a given observation direction is a random variable, whose statistical distribution is only function of two parameters. Contrary to the intensity ratio which follows a heavy-tailed distribution, the LRSI has finite mean and variance. For a structure air/clayey soil/rock, we analyze the influence of a snow layer upon the probability laws in the cases of Gaussian or exponential correlation functions.
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