## 1. Introduction

[2] Soil erosion due to rain is a major issue in the fields of agriculture, environment, and water management. All studies on soil erosion have suggested that increased rainfall amounts and intensities lead to greater rates of erosion [e.g., *Intergovernmental Panel on Climate Change*, 2007]. In particular, rainfall kinetic energy has often been suggested as an indicator of rainfall erosivity [*Fornis et al.*, 2005]. Over the past decades, many erosion models have been developed, such as the Water Erosion Prediction Project model (WEPP) [*Laflen et al.*, 1997] and the European Soil Erosion Model (EUROSEM) [*Morgan et al.*, 1998]. These models require rainfall time series with moderate to high temporal and spatial resolution [*Van Dijk et al.*, 2005], which is a restriction due to the large rain variability and the limitation of rain gauge observations. The most common approach to estimate rainfall kinetic energy is by means of an empirical relationship between the kinetic energy flux density KE and rain intensity *R* [*Mihara*, 1951; *Kinnell*, 1973; *Sempere Torres et al.*, 1992]. Various mathematical expressions and parameterizations for the KE-*R* relationship have been presented in the literature. In order to study the physical interpretation behind different KE-*R* relationships, several mathematical distributions have been introduced to account for the variation in raindrop size distribution (DSD). *Uijlenhoet and Stricker* [1999] developed an approach to link KE to *R* on the basis of the exponential DSD. Later, *Salles et al.* [2002] proposed a KE-*R* relationship according to the one-moment scaling formulation of the DSD. Their work suggested that the varying character of the DSD, which depends on the type of rain (convective or stratiform) and the geographical location, are the main factors explaining the variability of KE-*R* relationships. *Fox* [2004] investigated theoretical KE-*R* relationships on the basis of the gamma distribution [*Ulbrich*, 1983] and pointed out that the KE-*R* relationship is poorly defined unless some assumptions about the parameters of the gamma distribution are made. He also found that the assumption of an exponential DSD leads to an overestimation of the kinetic energy flux density. Assuming that the DSD is determined mainly by the breakup process rather than the initial DSD [*Assouline and Mualem*, 1989], *Mualem and Assouline* [1986] proposed a Weibull distribution with two parameters to derive the KE-*R* relationship. This approach showed advantages in KE estimation for light rainfall. Additionally, to overcome the limitation of a rain gauge observation network, *Steiner and Smith* [2000] showed the potential advantage of radar reflectivity factor (*Z*) for estimating KE, which can provide detailed spatial and temporal information about rain storms.

[3] The first purpose of this paper is to investigate the KE-*R*, KE-*Z*, and the KE-*RZ* relationships using one-moment [*Sempere Torres et al.*, 1994] and two-moment normalizations of the DSD following a number of previous contributions regarding a revised mathematical formulation of the DSD [*Testud et al.*, 2001; *Illingworth and Blackman*, 2002; *Lee et al.*, 2004; N. Yu et al., Unified formulation of single and multi-moment normalizations of the raindrop size distribution based on the gamma probability density function, submitted to *Journal of Applied Meteorology and Climatology*, 2012, hereinafter referred to as Yu et al., submitted manuscript, 2012]. It is hoped that the radar reflectivity factor (*Z*) in combination with the rain rate (*R*) can improve the estimation of rainfall kinetic energy flux density. The study is organized as follows: section 2 gives a brief description of the observed data used in this study. Section 3 outlines the general DSD scaling formulations on the basis of the concept of the probability density function (pdf) of a scaled raindrop diameter. Section 4 explores a robust method to estimate the climatic values of the DSD parameters in the one- and two-moment formulations. Three relationships to estimate the rainfall kinetic energy flux density (KE-*R*, KE-*Z*, and KE-*RZ*) are derived using a 28 month DSD data set collected in the Cévennes-Vivarais region, France, which is a region prone to heavy precipitation events and subsequent flash floods [*Delrieu et al.*, 2005]. Finally, we select, in section 5, one particular rain event to start addressing the second objective of the article, which is to study how rain gauge and radar measurements could be used separately and/or in conjunction to spatialize the kinetic energy flux over a region of interest.