The precipitation particles’ vertical velocity retrieval with single Doppler weather radar. PartⅠ： Retrieval method’s analysis

The ill-conditioned coefficient matrix and error sensitivities of the Volume Velocity Processing (VVP) wind retrieval method with Doppler radar are analyzed in mathematics, and the equations for resolving vertical velocity are modified especially. Given the condition number varies largely when the different fitted parameters chosen, the partial parameters with small magnitude are always neglected in retrieval. The analysis of error's norm verified that model errors might be introduced, but simplified wind model can decrease the difficulty in solving and stabilize the retrieval results. In the VVP retrieval algorithm, the linear correlation among the coefficient matrix vectors caused the matrix singularity. It is demonstrated that the accuracy of the fitted parameters could be affected when combining or abandoning linear correlation items in the coefficient matrix, and the model bias varied along with the position of analyzed points, not with the magnitude of parameters only, but the partial fitted parameters could remain accuracy. Based on the error analysis of VVP retrieval method, the errors when abandoning some parameters are analyzed and examined. The demonstrations of solving equations' modification further provide a fundamental understanding for the accurate retrieval of the vertical velocity.