Scale invariant relationship between rainfall kinetic energy and intensity in Paris region: An evaluation using universal multifractal framework

To calculate the effect of rainfall in detaching particles and initiating soil erosion, or in eroding wind turbine leading edge, it is important to measure recorded drop size distributions (DSD) and fall velocity over long period. Commonly used relationships between kinetic energy (KE) and rainfall rate (R) exhibit strong dependence on the temporal resolution at which the analysis is carried out. Here we aim at developing a new scale invariant relationship relying on the framework of Universal Multifractals (UM), which is widely used to analyze and characterize geophysical fields that exhibit extreme variability across wide range of scales. Rainfall data is collected using three optical disdrometers working on different underlying technologies (one Campbell Scientific PWS100 and two OTT Parsivel² instruments) and operated by the Hydrology Meteorology and Complexity laboratory of École des Ponts ParisTech in Paris area (France). They provide access to the size and velocity of drops falling through sampling areas of few tens of cm². Such data enables estimation of rainfall DSD, R and KE at various resolutions. The temporal variations of this geophysical data over wide range of scales are then characterized in the UM framework, which was never done for KE. A new power law relation is developed and tested against the theoretical framework assuming gamma DSD for describing the dependence between KE and R. The developed equation using scale invariant features of UM does not rely on gamma DSD assumption, performs as well as the existing tools, and is valid not only at a single scale, but also across scales.