However, Z is approximated by the sixth moment of the RSD, and R is approximated by the 3.67th moment. For example, Marshall and Palmer (1948) utilized the measured raindrop size distribution (RSD), the simulated reflectivity Z, and a power-law relation Z = aR b ( ZāR) to estimate rainfall rate. Retrieval of rain rate R requires accurate radar measurements and a relation that relates radar observations to rain rate. Improving radar-based quantitative precipitation estimation (QPE) has been one of the major goals of the radar-meteorology community for more than 50 years. The variational quantitative precipitation estimation (QPE) retrieved more accurate rainfall estimation than did the power-law dual-polarization QPE in this particular event, despite the fact that both algorithms used the same dual-polarization radar measurements from the Next Generation Weather Radar (NEXRAD). Furthermore, the variational algorithm is less sensitive to the observational error of differential reflectivity Z DR than is the R( Z HH, Z DR) algorithm. On the contrary, the usage of ad hoc or improper observational error degrades the performance of the variational method. The results from OSSE show that the variational algorithm with event-based observational error consistently estimates more accurate rain rate than does the ā R( Z HH, Z DR)ā power-law algorithm. The rain event occurred between 11 and 12 September 2013. Rain-rate estimation using the variational algorithm that uses event-based observational error and background rain climatological values is evaluated using observing system simulation experiments (OSSE), and its performance is demonstrated in the case of an epic Colorado flood event. A variational technique that concurrently takes into account radar observational error and dynamically varying rain microphysics is proposed in this study. In the literature, a variety of power-law relations between polarimetric radar measurements and rain rate are described because of the dynamic or varying nature of rain microphysics. The accuracy of rain-rate estimation using polarimetric radar measurements has been improved as a result of better characterization of radar measurement quality and rain microphysics.
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