Scattering on Rough Surfaceswith Alpha-stable Non-gaussian Height Distributions

We study the electromagnetic scattering problem on a random rough surface when the height distribution of the profile belongs to the family of alpha-stable laws. This allows us to model peaks of very large amplitude that are not accounted for by the classical Gaussian scheme. For such probability distributions with infinite variance the usual roughness parameters such as the RMS height, the correlation length or the correlation function are irrelevant. We show, however, that these notions can be extended to the alpha-stable case and introduce a set of adapted roughness parameters that coincide with the classical quantities in the Gaussian case. Then we study the scattering problem on a stationary alpha-stable surface and compute the scattering coefficient under the first-order Kirchhoff and Small-slope approximation. An analytical formula is given in the high-frequency limit, which generalizes the well-known Geometrical Optics approximation. INTRODUCTION In the theory of scattering by random rough surfaces, a systematic yet crucial assumption is the Gaussianity of the height distribution. This hypothesis is adopted for technical reasons but is often far from being realistic. Gaussian distributions miss essentially two important features of natural surfaces. First, they are unskewed, that is unable to explain an eventual asymmetry between crests and troughs. Second, the tail of these distributions is exponentially decreasing, thereby not allowing the occurrence of “big” events; Gaussian distributions are therefore also irrelevant for rough surfaces that can exhibit peaks of very large amplitude with respect to the mean level. In this paper, we will focus on the second aspect only and discard for this first attempt the modelization of skewness. Surfaces with a non-negligible probability of very large peaks are well described by heavy-tailed height distributions, that is distributions whose tail is polynomially (instead of exponentially) decreasing. The archetype of such distributions are the stable distributions. This family of distributions share with the Gaussian distribution the essential property of being stable under linear combination of independent variables and also a Generalized Central Limit Theorem for normalized sums of independent identically distributed variables. This is why they are found in the descriptions of many physical and economical systems. To our best knowledge, very few works have been concerned with non-Gaussian surface scattering [1] [2][3] [4] [5]. We take another step in direction of non-Gaussian surfaces by studying the family of alpha-stable surfaces. Contrarily to the previous models, they are not second-order (that is they have infinite variance) and thus require a complete redefinition of the roughness parameters. We will show that, in spite of their apparent complexity, alpha-stable surfaces yield to simple expression for the scattering computations. We have chosen to work in the one-dimensional case, which means that the surface is assumed to be invariant in one direction and hence only described by its profile. This restriction has been adopted to make the presentation clearer in introducing new concepts and to avoid some technical difficulties. The extension to the two-dimensional case is, however, straightforward. The SCATTERING PROBLEM A -invariant infinite surface separates the vacuum (upper medium) from a perfectly conducting medium. A linearly polarized time-harmonic plane wave with wave vector and wave-number !#" $#% is impinging at incidence & on the top of the surface. The total field ' in the upper medium writes '( )'+* '-, , where ' * is the incident field, ' * . / 0 1 2 3 547608 9 :<; => ? A@CB# 2 D 54E6F8 9C: ; =G H I J=> 1 K and ' , is the scattered field. Here ' stands for the electric field L or magnetic field M according to whether the polarization is s or p. It satisfies the Helmholtz equation in the upper domain ( NO' 'P DQ ) and vanishes in the lower domain ( 'R DQ1 S UT(VW ) with Dirichlet (resp. Neumann) boundary conditions on the interface in the s (resp. p) polarization case. The harmonic time-dependence X 4 *ZY5[ will always be implicit. Above the highest excursion of the surface, the scattered field admits a Rayleigh expansion: