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Current polarimetric model-based decomposition techniques are limited to specific types of vegetation because of their assumptions about the volume scattering component. In this paper, we propose a generalized probability density function based on the nth power of a cosine-squared function. This distribution is completely characterized by two parameters; a mean orientation angle and the power of the cosine-squared function. We show that the underlying randomness of the distribution is only a function of the power of the cosine-squared function. We then derive the average covariance matrix for various different elementary scatterers showing that the result has a very simple analytical form suitable for use in model-based decomposition schemes.