Requirements for Model-based Polarimetric Decompositions

Several authors have proposed decompositions of measured polarimetric covariance matrices into the weighted sums of individual covariance matrices that represent known scattering models. In this paper we critically examine some minimum requirements that these model-based polarimetric scattering decompositions must satisfy. Starting with the definition of the covariance matrix, we show that a necessary requirement for these decompositions is that the eigenvalues of the individual covariance matrices must be non-negative. We then show with examples that this is not guaranteed for some of the proposed decompositions. Finally, we propose a method of decomposition that would ensure that the eigenvalues of the individual covariance matrices would be non-negative.