A Probabilistic View on Raindrop Size Distribution Modeling: A Physical Interpretation of Rain Microphysics

Francisco J. Tapiador Faculty of Environmental Sciences and Biochemistry, University of Castilla–La Mancha, Toledo, Spain

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Ziad S. Haddad Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Joe Turk Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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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 (NT), the sample mean [m = E(D)], and the sample variance [σ2 = E(mD)2] of the drop diameters (D) or, alternatively, in terms of NT, E(D), and E[log(D)]. Statistical analyses indicate that (NT, m) are independent, as are (NT, σ2). The ZR relationship that arises from this model is a linear R = T × Z expression (or Z = T−1R), with T a factor depending on m and σ2 only and thus independent of NT. The ZR so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the ZR arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters N0 or , which are often used in statistical approaches but lack physical meaning. The modeling presented here preserves a well-defined and consistent set of units across all the equations, also taking into account the effects of RDSD truncation. It is also shown that the rain microphysical processes such as coalescence, breakup, or evaporation can then be easily described in terms of two parameters—the sample mean and the sample variance—and that each of those processes have a straightforward translation in changes of the instantaneous ZR relationship.

Corresponding author address: Francisco J. Tapiador, University of Castilla–La Mancha (UCLM), Faculty of Environmental Sciences and Biochemistry, Avda. Carlos III s/n, 45071 Toledo, Spain. E-mail: francisco.tapiador@uclm.es

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 (NT), the sample mean [m = E(D)], and the sample variance [σ2 = E(mD)2] of the drop diameters (D) or, alternatively, in terms of NT, E(D), and E[log(D)]. Statistical analyses indicate that (NT, m) are independent, as are (NT, σ2). The ZR relationship that arises from this model is a linear R = T × Z expression (or Z = T−1R), with T a factor depending on m and σ2 only and thus independent of NT. The ZR so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the ZR arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters N0 or , which are often used in statistical approaches but lack physical meaning. The modeling presented here preserves a well-defined and consistent set of units across all the equations, also taking into account the effects of RDSD truncation. It is also shown that the rain microphysical processes such as coalescence, breakup, or evaporation can then be easily described in terms of two parameters—the sample mean and the sample variance—and that each of those processes have a straightforward translation in changes of the instantaneous ZR relationship.

Corresponding author address: Francisco J. Tapiador, University of Castilla–La Mancha (UCLM), Faculty of Environmental Sciences and Biochemistry, Avda. Carlos III s/n, 45071 Toledo, Spain. E-mail: francisco.tapiador@uclm.es
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