# Describing the shape of raindrop size distributions using uncorrelated raindrop mass spectrum parameters

Christopher R. Williams, V. N. Bringi, Lawrence D. Carey, V. Chandrasekar, Patrick N. Gatlin, Ziad S. Haddad, Robert Meneghini, S. Joseph Munchak, Stephen Nesbitt, Walter A. Petersen, Simone Tanelli, Ali Tokay, Anna Wilson, David B. Wolff

January 2014### Abstract

Rainfall retrieval algorithms often assume a gamma-shaped raindrop size distribution (DSD) with three mathematical parametersNw,Dm, and μ. If only two independentmeasurements are available, as with the dualfrequency precipitation radar on the Global Precipitation Measurement (GPM) mission core satellite, then retrieval algorithms are underconstrained and require assumptions about DSD parameters. To reduce the number of free parameters, algorithms can assume that μ is either a constant or a function of Dm. Previous studies have suggested μ-∧ constraints [where ∧ = (4 + μ)/Dm], but controversies exist over whether μ-∧ constraints result from physical processes or mathematical artifacts due to high correlations between gamma DSDparameters. This study avoidsmathematical artifacts by developing joint probability distribution functions (joint PDFs) of statistically independent DSD attributes derived from the raindrop mass spectrum. These joint PDFs are thenmapped into gamma-shapedDSD parameter joint PDFs that can be used in probabilistic rainfall retrieval algorithms as proposed for the GPM satellite program. Surface disdrometer data show a high correlation coefficient between the mass spectrum mean diameterDm andmass spectrum standard deviation σm. To remove correlations betweenDSDattributes, a normalizedmass spectrumstandard deviation σ′m is constructed to be statistically independent of Dm, with σ′m̄ representing the most likely value and std(σ′m) representing its dispersion. Joint PDFs of Dm and μ are created from Dm and σ′m. A simple algorithm shows that rain-rate estimates had smaller biases when assuming the DSD breadth of σ′m̄ than when assuming a constant μ.

Publication

*Journal of Applied Meteorology and Climatology*