On the Co-Polarized Scattered Intensity Ratio of Rough Layered Surfaces: The Probability Law Derived From the Small Perturbation Method

Abstract

International audienceWe determine the probability law of the ratio between the co-polarized intensities scattered from a stack of two two-dimensional rough interfaces in the incidence plane. Calculations are carried out within the framework of the first-order small perturbation method. For slightly rough interfaces with infinite length and Gaussian height distributions, we show that the probability density function is only a function of two parameters and has an infinite average and an infinite variance. For a sand layer on a granite surface in backscattering configurations, we study the influence of the incidence angle and the cross-spectral density upon this probability law

Statistical Distribution of the Layered Rough Surface Index (Lrsi)

“…Let V (ba,b a ) = I ba /I b a the intensity ratio associated with the components (ba) and (b a ) of the wave scattered in the direction (θ, φ). For infinite surface areas (L → +∞) and centered Gaussian height distributions, we demonstrated in [8]…”

“…The quantity r designates the correlation coefficient of the two complex random variables A [1,8] and its value lies between 0 and 1. The quantity σ 2 R ba is the variance of the real part of the scattering amplitude A…”

“…For a structure with one, two or three slightly rough surfaces, the analytical expressions for both parameters p 0 and r can be derived from the co-and cross-spectrum formulas (Eqs. (5) to (8)) and from the closed-form formulas given in the appendix for the SPM-Kernels.…”

“…The interfaces were correlated or not, isotropic or not, but the spectra and cross-spectra were Gaussian. We have demonstrated that the PDF is fully defined by two parameters that depend on the second-order statistical moments of the real and imaginary parts of scattering amplitudes [8][9][10][11]. But, the co-and cross-polarized intensity ratios under an observation direction are non-negative real-valued random variables which follow heavy-tailed distributions and do not have a finite mean and a finite variance.…”

Scattering and Transmission of Waves in Multiple Random Rough Surfaces: Energy Conservation Studies With the Second Order Small Perturbation Method

“…The SPM is a low frequency approach valid for small roughness; most SPM studies consider only the first order terms [6,7]. High order corrections are included in [8][9][10][11][12][13][14], where fourth and higher order corrections are discussed in [11].…”