Ca. Guerin et M. Saillard, Electromagnetic scattering on fractional Brownian surfaces and estimation of the Hurst exponent, INVERSE PR, 17(3), 2001, pp. 365-386
Fractional Brownian motion is known to be a realistic model for many natura
l rough surfaces. It is defined by means of a single parameter, the Hurst e
xponent, which determines the fractal characteristics of the surface. We pr
opose a method to estimate the Hurst exponent of a fractional Brownian prof
ile from the electromagnetic scattering data. The method is developed in th
e framework of three usual approximations, with different domains of validi
ty: the Kirchhoff approximation, the small-slope approximation of Voronovit
ch and the small-perturbation method. A universal power-law dependence upon
the incident wavenumber is shown to hold for the scattered far-field inten
sity, irrespective of the considered approximation and the polarization, wi
th 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 perfo
rmance of this estimator and propose an improved version by allowing the sc
attering angle to vary. The theoretical performance of these estimators is
then checked by numerical simulations. Finally, we present a rigorous numer
ical computation of the scattered intensity in the resonance domain, where
none of the aforementioned approximations applies. The numerical results sh
ow the persistence of a power-law behaviour, but with a different and still
non-trivial exponent.