Recent studies have shown that the retrieval of raindrop size distribution (DSD) benefits from polarimetric radar measurements. Current operational WSR-88D (Weather Surveillance Radar 88-Doppler) radar network in the United States operates at the S-band and will be upgraded to dual polarizations over the next few years. A dense polarimetric radar network operating at shorter wavelength (e.g., X-band) would complement the S-band network in providing measurements of higher resolution and better low-level coverage. The experimental IP1 (Integrated Project One) X-band network of CASA (Center for Collaborative Adaptive Sensing of the Atmosphere) installed in southwest Oklahoma possesses such advantages. However, reflectivity attenuation is significant with X-band weather radars; the effects of attenuation have to be taken into account when the data are used or the data have to be corrected for attenuation before their use.

Conventional approaches for attenuation correction and rain estimation are generally not optimized in minimizing error effects. For example, conventional attenuation corrections are usually based on the power-law relations between attenuation and radar reflectivity, and/or the relations associated with differential phase measurements (e.g., differential phase ΦDP or specific differential phase KDP). They do not directly account for the error characteristics of data in the algorithms and are therefore sensitive to measurement errors; they tend to degrade quickly with increased errors. Based on optimal estimation theory, the variational approach can optimally account for the error effect while effectively utilizing useful information in the observations. With appropriate weighting related to their error, different polarimetric measurements are optimally used. Attenuation correction can be built into the retrieval system using all available observations. The variational framework also allows for solutions for over-determined problems.

In this study, we apply a variational method in two dimensions to retrieve the rain DSD from polarimetric X-band radar measurements that include radar reflectivity (ZH), differential reflectivity (ZDR) and specific differential phase (KDP). A constrained-gamma DSD model with two degrees of freedom is used. Two DSD parameters, instead of the integrated variables, are chosen as state variables. The retrieval of DSD parameters enables adaptive attenuation correction, which is incorporated into the forward model of observations and automatically performed during the minimization of cost function. Measurements from three CASA X-band IP1 radars are processed to retrieve the rain DSD and consequently the rainfall rate (R), intrinsic (corrected) ZH and ZDR. The retrievals in the overlapping regions of different IP1 radars show good consistence among the radars, suggesting the effectiveness of the variational approach for rain DSD retrieval in the presence of attenuation. Further, S-band polarimetric KOUN radar observations and video disdrometer measurements are used for validation.