Electromagnetic scattering on fractional Brownian surfaces and estimation of the Hurst exponent

Fractional Brownian motion is known to be a realistic model for many natural rough surfaces. It is defined by means of a single parameter, the Hurst exponent, which determines the fractal characteristics of the surface. We propose a method to estimate the Hurst exponent of a fractional Brownian profile from the electromagnetic scattering data. The method is developed in the framework of three usual approximations, with different domains of validity: the Kirchhoff approximation, the small-slope approximation of Voronovitch and the small-perturbation method. A universal power-law dependence upon the incident wavenumber is shown to hold for the scattered far-field intensity, irrespective of the considered approximation and the polarization, with a common scaling exponent trivially related to the Hurst exponent. This leads naturally to an estimator of the latter based on a log-log regression of the far-field intensity at fixed scattering angle. We discuss the performance of this estimator and propose an improved version by allowing the scattering angle to vary. The theoretical performance of these estimators is then checked by numerical simulations. Finally, we present a rigorous numerical computation of the scattered intensity in the resonance domain, where none of the aforementioned approximations applies. The numerical results show the persistence of a power-law behaviour, but with a different and still non-trivial exponent.