Multi-parameter polarimetric radar has shown great utility in meteorology by improving measurement accuracy and microphysical understanding. This work concentrates on studies of the full polarimetric covariance matrix to retrieve the moment estimations over the shape distribution and orientation distribution of the precipitation medium.; The antenna-induced polarimetric errors that result from inter-channel contamination and sidelobe leakage are first examined. Antenna patterns are used to convolve the distributed medium with large gradients and a methodology to detect such polarimetric errors is developed. The methodology is illustrated with volume simulation of a synthetic storm and then is applied to real cases from the project of Thunderstorm Electrification and Precipitation Study (STEPS). The detection is evaluated by comparing the results between two coordinated research radars---CSU-CHILL and NCAR S-Pol. The evaluation of antenna performance is also given within this context.; In the second part, the full polarimetric covariance matrix is studied along with polarization basis transformation. The effect of backscattering canting, the effect of propagation, and the integrated antenna polarization errors on the covariance matrix are analyzed. Considering backscattering canting, a new method to estimate the orientation factors, i.e., the mean canting angle and its dispersion, is proposed based directly on the covariance matrix in the linear basis. The new method is compared with other approaches, such as that based on circular covariance matrix and that based on polarization optimization. The distortion due to propagation, especially due to non-diagonal propagation, is studied with simulation. Concerning the integrated antenna polarization errors, a novel approach is proposed to estimate the non-orthogonal error matrix.; In the last part, the proposed approaches are applied to a variety of real cases. For the purpose of orientation estimation, the full covariance matrix must be calibrated carefully on both power terms and phase terms. A modified phase filtering process is designed to obtain robust filtering over the "weak" cross-polar channels so that the phase offsets could be accurately estimated. The integrated antenna polarization errors are then estimated and corrected. Different types of precipitation are analyzed with the estimated orientation factors, the derived circular radar variables, and the conventional linear radar variables. It is shown that the linear depolarization ratio suffers ambiguity on presenting orientation information compared to the estimated orientation dispersion. The mean canting angle is very close to zero for most precipitation targets except the aligned crystals which can be usually observed at the top of electrified storms. The near zero mean canting angle also justifies the hybrid operation mode planned for the WSR-88D radars.
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