This study offers a unified formulation of single- and multimoment normalizations of the raindrop size distribution (DSD), which have been proposed in the framework of scaling analyses in the literature. The key point is to consider a well-defined "general distribution" g(x) as the probability density function (pdf) of the raindrop diameter scaled by a characteristic diameter D_c. The two-parameter gamma pdf is used to model the g(x) function. This theory is illustrated with a 3-yr DSD time series collected in the Cévennes region, France. It is shown that threeDSD moments (M₂,M₃, and M₄) make it possible to satisfactorily model the DSDs, both for individual spectra and for time series of spectra. The formulation is then extended to the one- and twomoment normalization by introducing single and dual power-law models. As compared with previous scaling formulations, this approach explicitly accounts for the prefactors of the power-law models to yield a unique and dimensionless g(x), whatever the scaling moment(s) considered. A parameter estimation procedure, based on the analysis of power-law regressions and the self-consistency relationships, is proposed for those normalizations. The implementation of this method with different scaling DSD moments (rain rate and/or radar reflectivity) yields g(x) functions similar to the one obtained with the three-moment normalization. For a particular rain event, highly consistent g(x) functions can be obtained during homogeneous rain phases, whatever the scaling moments used. However, the g(x) functions may present contrasting shapes from one phase to another. This supports the idea that the g(x) function is process dependent and not "unique" as hypothesized in the scaling theory.

This study offers an approach to estimate the rainfall kinetic energy (KE) by rain intensity (R) and radar reflectivity factor (Z) separately or jointly on the basis of a one- or two-moment scaled raindrop size distribution (DSD) formulation, which contains (1) R and/or Z observations and (2) the dimensionless probability density function (pdf) of a scaled raindrop diameter. The key point is to explain all variability of the DSD by the evolution of the explaining moments (R and Z); hence the pdf is considered as constant. A robust method is proposed to estimate the climatological values of the parameters with a 28 month DSD data set collected in the Cévennes-Vivarais region of France. Three relationships (KE-R, KE-Z, and KE-RZ), which link the observations (R and/or Z) to rainfall kinetic energy (KE), are established. As expected, the assessment using the disdrometer data indicates that (1) because of the proximity of the moment orders, the KE-Z relationship exhibits less variability than the KE-R relationship and (2) the combination of R and Z yields a significant improvement of the estimation of KE compared to the single-moment formulations. Subsequently, a first attempt to spatialize the kinetic energy using radar and rain gauge measurements is presented for a convective event, showing a promising potential for erosion process studies. Different from the application with the disdrometer data, the performance of the KE-Z relationship degrades compared to the KE-R relationship as a result of a bias and/or the sampling characteristics of the radar data.

In radar hydrology the relationship between the reflectivity factor (Z) and the rainfall intensity (R) is generally assumed to follow a power law of which the parameters change both in space and time and depend on the drop size distribution (DSD). Based on disdrometer data, this study tries to improve our understanding of the temporal variability of the power-law relationship between Z and R using a scaling-law formalism for the raindrop size distribution proposed in previous contributions. In particular, this study focuses on the inter-event variability of Z-R coefficients and associated DSD-parameters and their relationship to the type of precipitation. This is crucial for developing improved quantitative precipitation estimation algorithms for extreme, flash flood triggering rainfall.Within the DSD scaling-law framework a new normalized parameter estimation method is presented, which calculates significantly faster than the original method and leads to bulk event estimates of the DSD-parameters and associated Z-R coefficients. Based on a 2.5-year disdrometer dataset collected in the Cévennes-Vivarais region in the south of France, comprising a total of 70 events, it is shown that the quality of the resulting Z-R relationships obtained by the new method compares well to two standard least-squares fitting techniques. A major benefit of the new implementation, as compared to such purely statistical methods, is that it also provides information concerning the properties of the DSD.For each of the 70 events this study also estimates the convective activity based on a threshold technique. Results show that convective events generally tend to have smaller Z-R exponents, which is assumed to result from an increased amount of drop interaction. For stratiform events, a much larger range in exponents is obtained, which is thought to depend on differences in meteorological origin (snow vs. ice). For the types of precipitation events observed in the Cévennes region, for a given value of the exponent, the prefactor of the Z-R relation tends to be larger for the more convective type of events. This emphasizes the different meteorological origin of the heavy rainfall observed in the south of France as compared to other regions of the world.