Kirchhoff Scattering From Fractal and Classical Rough Surfaces: Physical Interpretation

Scattering from both fractional Brownian motion (fBm) and classical rough surfaces under the Kirchhoff approximation is here considered. The focus is on the scattering integral analytical expression and on its physical interpretation. First, we show that, for an fBm surface, the Kirchhoff approach scattering integral is directly proportional to a symmetric alpha-stable (SαS) distribution. The interpretation of this intriguing result leads us to revisit the meaning of the Kirchhoff solution and of the geometrical optics (GO) even for a regular (classical, nonfractal) rough surface. Then, we conclude that, for both fractal and classical surfaces, the Kirchhoff scattering integral can be interpreted in terms of a sort of “intrinsic” two-scale model, and that, in the fractal case, the obtained SαS distribution can be interpreted as the probability density function (pdf) of the slopes of an equivalent rough surface whose GO scattered power density is equal to the scattered power density of the actual fBm surface.