The shape-slope relation in observed gamma raindrop size distributions: Statistical error or useful information?

The three-parameter gamma distribution n(D) = N_oD^μ exp(-AD) is often used to characterize a raindrop size distribution (DSD). The parameters μ and A correspond to the shape and slope of the DSD. If μ and A are related to one another, as recent disdrometer measurements suggest, the gamma DSD model is simplified, which facilitates retrieval of rain parameters from remote measurements. It is important to determine whether the μ-A relation arises from errors in estimated DSD moments, or from natural rain processes, or from a combination of both statistical error and rain physics. In this paper, the error propagation from moment estimators to rain DSD parameter estimators is studied. The standard errors and correlation coefficient are derived through systematic error analysis. Using numerical simulations, errors in estimated DSD parameters are quantified. The analysis shows that errors in moment estimators do cause correlations among the estimated DSD parameters and cause a linear relation between estimators ̂μ and ̂Λ. However, the slope and intercept of the error-induced relation depend on the expected values μ and Λ, and it differs from the μ-Λ relation derived from disdrometer measurements. Further, the mean values of the DSD parameter estimators are unbiased. Consequently, the derived μ-Λ relation is believed to contain useful information in that it describes the mean behavior of the DSD parameters and reflects a characteristic of actual raindrop size distributions. The μ-Λ relation improves retrievals of rain parameters from a pair of remote measurements such as reflectivity and differential reflectivity or attenuation, and it reduces the bias and standard error in retrieved rain parameters.