AbstractRainfall retrieval algorithms often assume a gamma-shaped raindrop size distribution (DSD) with three mathematical parameters N_w, D_m, and μ. 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 that μ is either a constant or a function of D_m. Previous studies have suggested μ–Λ constraints [where Λ = (4 + μ)/D_m], but controversies exist over whether μ–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 diameter D_m and mass spectrum standard deviation σ_m. To remove correlations between DSD attributes, a normalized mass spectrum standard deviation is constructed to be statistically independent of D_m, with representing the most likely value and std representing its dispersion. Joint PDFs of D_m and μ are created from D_m and . A simple algorithm shows that rain-rate estimates had smaller biases when assuming the DSD breadth of than when assuming a constant μ.