Describing the Shape of Raindrop Size Distributions Using Uncorrelated Raindrop Mass Spectrum Parameters

Rainfall retrieval algorithms often assume a gamma-shaped raindrop size distribution (DSD) with three mathematical parametersNw ,Dm , andμ. If only two independent measurements are available, as with the dual-frequency 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 thatμis either a constant or a function ofDm . Previous studies have suggestedμ–Λ constraints [where Λ 5 (4 +μ)/DDm ], but controversies exist over whetherμ–Λ constraints result from physical processes or mathematical artifacts due to high correlations between gamma DSD parameters. This study avoids mathematical artifacts by developing joint probability distribution functions (joint PDFs) of statistically independent DSD attributes derived from the raindrop mass spectrum. These joint PDFs are then mapped into gamma-shaped DSD 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 and mass spectrum standard deviationσm . To remove correlations between DSD attributes, a normalized mass spectrum standard deviation σ m ' is constructed to be statistically independent ofDm , with σ m ' ¯ representing the most likely value and std ( σ m ' ) representing its dispersion. Joint PDFs ofDm andμare created fromDm and σ m ' . A simple algorithm shows that rain-rate estimates had smaller biases when assuming the DSD breadth of σ m ' ¯ than when assuming a constantμ.