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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 [1] 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 [1], the random incident angle variation Δϑ is ignored, and the incidence plane angle of rotation b is assumed to be uniformly distributed in an interval (−β1, β1). 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 β and Δϑ 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, α) can be also retrieved from the co-pol and cross-pol ratios.