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In this paper, a general scheme for complete model-based decomposition of the polarimetric synthetic aperture radar (POLSAR) coherency matrix data is presented. We show that the POLSAR coherency matrix can be completely decomposed into three components contributed by volume scattering and two single scatterers (characterized by rank-1 matrices). Under this scheme, solving for the volume scattering power amounts to a generalized eigendecomposition problem, and the nonnegative power constraint uniquely determines the minimum eigenvalue as the volume scattering power. Furthermore, in order to discriminate the remaining components, we propose two approaches. One is based on eigendecomposition, and the other is based on model fitting, both of which are shown to properly resolve the surface and double-bounce scattering ambiguity. As a result, this paper in particular contributes to two pending needs for model-based POLSAR decomposition. First, it overcomes negative power problems, i.e., all the decomposed powers are strictly guaranteed to be nonnegative; and second, the three-component decomposition exactly accounts for every element of the observed coherency matrix, leading to a complete utilization of the fully polarimetric information.