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## Abstract

The raindrop size distribution (RDSD) is defined as the relative frequency of raindrops per given diameter in a volume. This paper describes a mathematically consistent modeling of the RDSD drawing on probability theory. It is shown that this approach is simpler than the use of empirical fits and that it provides a more consistent procedure to estimate the rainfall rate (*R*) from reflectivity (*Z*) measurements without resorting to statistical regressions between both parameters. If the gamma distribution form is selected, the modeling expresses the integral parameters *Z* and *R* in terms of only the total number of drops per volume (*N*
_{T}), the sample mean [*m* = *E*(*D*)], and the sample variance [*σ*
^{2} = *E*(*m* − *D*)^{2}] of the drop diameters (*D*) or, alternatively, in terms of *N*
_{T}, *E*(*D*), and *E*[log(*D*)]. Statistical analyses indicate that (*N*
_{T}, *m*) are independent, as are (*N*
_{T}, *σ*
^{2}). The *Z*–*R* relationship that arises from this model is a linear *R* = *T* × *Z* expression (or *Z* = *T*
^{−1}
*R*), with *T* a factor depending on *m* and *σ*
^{2} only and thus independent of *N*
_{T}. The *Z*–*R* so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the *Z*–*R* arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters *N*
_{0} or *Z*–*R* relationship.

## Abstract

The raindrop size distribution (RDSD) is defined as the relative frequency of raindrops per given diameter in a volume. This paper describes a mathematically consistent modeling of the RDSD drawing on probability theory. It is shown that this approach is simpler than the use of empirical fits and that it provides a more consistent procedure to estimate the rainfall rate (*R*) from reflectivity (*Z*) measurements without resorting to statistical regressions between both parameters. If the gamma distribution form is selected, the modeling expresses the integral parameters *Z* and *R* in terms of only the total number of drops per volume (*N*
_{T}), the sample mean [*m* = *E*(*D*)], and the sample variance [*σ*
^{2} = *E*(*m* − *D*)^{2}] of the drop diameters (*D*) or, alternatively, in terms of *N*
_{T}, *E*(*D*), and *E*[log(*D*)]. Statistical analyses indicate that (*N*
_{T}, *m*) are independent, as are (*N*
_{T}, *σ*
^{2}). The *Z*–*R* relationship that arises from this model is a linear *R* = *T* × *Z* expression (or *Z* = *T*
^{−1}
*R*), with *T* a factor depending on *m* and *σ*
^{2} only and thus independent of *N*
_{T}. The *Z*–*R* so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the *Z*–*R* arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters *N*
_{0} or *Z*–*R* relationship.