Extended Small Perturbation Method and retrieval of natural surface parameters

The extended Bragg (X-Bragg), or extended small perturbation method (X-SPM), has been recently introduced to widen the SPM range of validity in terms of surface roughness, and to account for cross-polarisation and depolarisation effects. This is obtained by modelling the scattering surface as composed by slightly rough randomly tilted facets. The facet random tilt causes a random variation of the local incidence angle and a random rotation of the local incidence plane around the line of sight, which in turn causes a random rotation of the facet scattering matrix. In, the random incident angle variation Deltathetav is ignored, and the incidence plane angle of rotation b is assumed to be uniformly distributed in an interval (-beta₁, beta₁). In this paper, we remove both these simplifying assumptions, in order to assess their validity and to improve them. In particular, we derive more realistic distributions of beta and Deltathetav by assuming that the facet slope (i.e., the slope of the large-scale surface roughness) is a Gaussian random variable, in agreement with both classical and fractal surface models. Finally, we show that, if the X-SPM is used, the same information about a bare soil surface that can be retrieved from the coherency matrix entropy and scattering angle (H, alpha) can be also retrieved from the co-pol and cross-pol ratios.