In the context of remotely sensed data analysis, an important problem is the development of accurate models for the statistics of the pixel intensities. Focusing on Synthetic Aperture Radar (SAR) data, this modelling process turns out to be a crucial task, for instance, for classification or for denoising purposes. In the present report, an innovative parametric estimation methodology for SAR amplitude data is proposed, which takes into account the physical nature of the scattering phenomena generating a SAR image by adopting a generalized Gaussian (GG) model for the backscattering phenomena. A closed form expression for the corresponding amplitude probability density function (PDF) is derived and a specific parameter estimation algorithm is developed in order to deal with the proposed model. Specifically, the recently proposed «method-of-log-cumulants» (MoLC) is applied, which stems from the adoption of the Mellin transform (instead of the usual Fourier transform) in the computation of characteristic functions, and from the corresponding generalization of the concepts of moment and of cumulant. For the developed GG-based amplitude model, the resulting MoLC estimates turn out to be numerically feasible and are also proved to be consistent. The proposed parametric approach is validated using several real ERS-1, XSAR, ESAR and airborne SAR images and the experimental results prove that the method models the amplitude probability density function better than several previously proposed parametric models for the backscattering phenomena.