Fully polarimetric model-based decompositions are developed by accounting for the physical scattering model and experimental polarimetric SAR data acquisition processes. These decompositions offer the promising straightforward interpretation and highly improved inversion models for visualizing images of scattering scenarios optimally. However, the attempts in existing decompositions to implement the split real and imaginary components of the T-13 element of the coherency matrix have been hampered by the absence of physical models to fit the coherency matrix. In this paper, two additional physical scattering submodels are derived. The real and imaginary parts of T-13 are accounted for by implementing two newly developed physical scattering models. (One is for oriented dipole scattering and the other is for oriented quarter-wave reflection.) Furthermore, this paper is extended by implementing these physical models into a six-component scattering power model-based decomposition. To this date, the developed novel decompositions account for the maximum elements of the coherency matrix in a physical manner compared to the existing model-based decompositions. The proposed novel decomposition is tested on L-band and X-band fully polarimetric SAR data sets of the Advanced Land Observing Satellite-2/Phased Array L-band Synthetic Aperture Radar-2 and the X-band TerraSAR-X, respectively. This new decomposition produces additional two scattering submatrix components. Such scattering components are prevalent in vegetation and urban areas and even dominant over highly oriented urban scenarios. The new method enhances the truly existing double-bounce scattering contributions and reduces the overrated volume scattering from double-bounce scatterers. By comparing the results, it is found that the proposed decomposition considerably enhances the SAR image quality and its more correct visualizing presentation compared to existing decompositions. It is also found to be more robust over the oriented urban areas than the existing decompositions, resulting from the utilization of both the real and imaginary components of T-13 polarimetric information in a physical scattering manner.