Empirical relations between size parameters of ice hydrometeor populations and radar reflectivity

Empirical power-law relations between the equivalent radar reflectivity factor Z(e) and the slope parameter of the gamma function Lambda (i.e., Lambda = cZ(e)(d); used to describe ice hydrometeor size distributions) are derived. The Lambda parameter can also be considered as a size parameter since it is proportional to the inverse of the hydrometeor characteristic size, which is an important geophysical parameter describing the entire distribution. Two datasets from two-dimensional microphysical probes, collected during aircraft flights in subtropical and midlatitude regions, were used to obtain Lambda by fitting measured size distributions. Reflectivity for different radar frequencies was calculated from microphysical probe data by using nonspherical-particle models. The derived relations have exponent d values that are approximately from -0.35 to -0.40, and the prefactors c are approximately between 30 and 55 (Lambda: cm(-1); Z(e): mm(6) m(-3)). There is a tendency for d and c to decrease when radar frequency increases from K-u band (similar to 14 GHz) toWband (similar to 94 GHz). Correlation coefficients between Z(e) and Lambda can be very high (similar to 0.8), especially for lower frequencies. Such correlations are similar to those for empirical relations between reflectivity and ice water content (IWC), which are used in many modeling and remote sensing applications. Close correspondences of reflectivity to both Lambda and IWC are due to a relatively high correlation between these two microphysical parameters. Expected uncertainties in estimating Lambda from reflectivity could be as high as a factor of 2, although estimates at lower radar frequencies are more robust. Stratifying retrievals by temperature could result in relatively modest improvement of Lambda estimates.