For radar rain estimation, the number of independent remote measurements is limited and they are prone to measurement errors. Ground-based polarimetric radar measures reflectivity (Z) and differential reflectivity (ZDR), while space-borne dual-wavelength radar provides measurements of reflectivity (Z) and dual-wavelength ratio (DWR). Retrieval of the raindrop size distribution (DSD) from a dual-polarization radar and from a dual-wavelength radar shares a same fundamental problem: to retrieve rain DSD parameters from two remote measurements (Z and ZDR or Z and DWR). It is widely accepted that the Gamma distribution can be used to represent the variety of rain DSDs. The three-parameters of a Gamma DSD, however, cannot be determined from the two remote measurements and another relation is needed. In the past, one of the Gamma DSD parameters, mu has been fixed in rain DSD retrieval. During the analysis of video-disdrometer measurements, we found that Gamma DSD parameters mu and lambda are highly correlated and constructed a constrained Gamma DSD model, and developed an algorithm to retrieve DSD parameters from dual-polarization radar measurements.

In this paper, we describe justification and usefulness of the constrained Gamma DSD model in radar-based retrieval. We will show that the mu-lambda relation comes from the rain physics rather than statistical error, by converting the mu-lambda relation into a relation between median volume diameter and the width of mass distribution. An error analysis indicates that the errors in moment estimates can enhance the mu-lambda correlation, but not necessarily causes the observed mu-lambda relation. Then, we apply the constrained Gamma DSD model in rain DSD retrieval and rain estimation from polarimetric radar measurements. The retrieved DSD parameters and rain physical parameters are shown as a time series comparison with disdrometer measurements. The instantaneous comparisons of the DSD parameters and rain parameters show that radar retrieved values based on the constrained Gamma DSD model agree with disdrometer measurements.